stackexchange_physics_binary_stars.jsonl

{"unit_id": "stackexchange:physics.stackexchange:binary-stars:31201:0000", "text": "Question: Might a planet perform figure-8 orbits around two stars?\n\nMight a planet perform figure-8 orbits around two stars?\r\n\r\nI'm thinking that if the two stars were equal mass (and not orbiting each other) then a planet that were to go right between them would continue in a straight line, with no preference for either star. But since the two stars would in fact be orbiting each other, the system would be rotating and thus there would be a Coriolis preference for one of the stars. Might that preference be made to alternate stars?\r\n\r\nAnother possibility would be if each star were in turn orbited closely by another planet, which would perform three orbits for each orbit of our planet of interest. Then things could be timed where on one pass star A's inner planet were aligned right to pull the planet of interest into an orbit, and on another pass star B's inner planet would be aligned right to pull the planet of interest into an orbit. So we have a system of five bodies, two massive (stars) each orbited by a minor, and one minor performing figure-8s.\r\n\r\nIs this at least plausible if contrived?\n\nAccepted Answer:\n\nIt would be possible, but very unlikely, since the orbits wouldn't be stable.\r\n\r\nTry to take a look at this visualization of the gravitational potential of a binary star system (from the Wikipedia [Roche Lobe](https://en.wikipedia.org/wiki/Roche_lobe) entry): \r\n\r\n![Roche potential][1]\r\n\r\nIf the planet orbits just one of the stars, its orbit will be inside one of the lobes of the thick-lined figure eight at the bottom part, analogous to a ball rolling around inside one of the \"bowls\" on the 3D-figure. Such an orbit will be stable, just like the Earth's around the sun (bar perturbations from other planets, but let's leave them out for now), and there will be many different orbital energies for which this is true.\r\n\r\nThe same goes for an orbit around both stars: the planet will have many different energy levels at which it would simply experience the two stars' gravity combined as the gravity of one single body (and in which case the figure wouldn't apply, since it would be practically unaffected by the two stars orbiting each other).\r\n\r\nIn order to orbit in a figure eight, you have to imagine that the ball has to roll across the ridge between the two indentations in the 3D part of the figure. It is clear that this is possible, but also intuitively clear that this would only be possible for a narrow range of orbital energies (a little less and it would go into one of the holes, a little more and it would simply just orbit them both), and that it would not be a stable orbit. The ball would have to roll in an orbit where it *exactly* passes the central saddle point at the ridge (L1) in order to stay stable, the tiniest little imperfection will get it perturbed even further away from its ideal trajectory. \r\n\r\nYour 5-body system could possibly be timed in such a way that it would work, but it would suffer the same fundamental flaw, and as far as I can see, it would also introduce even more sources of instability into the system. \r\n\r\nThis is, by the way, the gravitational potential in the **rotated coordinate system**, and you can see from the symmetry of the system that the coreolis preference you mention is not present. A simple **symmetry argument** should convince you of the same, though: Assume the system is rotating clockwise. This should allegedly give you a preference for one of the stars. But if you now let the system continue, while you rotate yourself 180 degrees up/down, it will now be rotating counter clockwise, which should give a coreolis preference for the other star, which of course cannot be the case, since there is **no preferred up/down direction** in a system like this.\r\n\r\n  [1]: https://i.sstatic.net/EGDew.jpg\n\nAnswer (score=7):\n\nConsidering that there are no other answers, I will answer.\r\n\r\nFrom user9886's links I have found several types of figure-eight orbits. Here is one nice one:\r\nhttp://www.ams.org/samplings/feature-column/fcarc-orbits1\r\n\r\nHowever, I don't know if the two orbits postulated in the OP are possible.\n\nAnswer (score=0):\n\n1\r\n\r\nGenerally speaking, binaries cannot have shared planets orbiting at close distances. Neither dynamics nor kinematics holds.\r\n\r\nIf a binary has a common planet, the planet must be far enough, that is, the radius of the circle is large enough to be far greater than the distance between the two stars. This approximates two stars as one star at the center of mass.\r\n\r\nThe acceleration of the winding provides centripetal force, so the system can be stable. It's impossible to circle two stars at close range.\r\n\r\nEach star of the binary can have its own planet, in this case, the planet need to close to its own star and far away to other star", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 0, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-mechanics", "newtonian-gravity", "orbital-motion", "three-body-problem"], "page_start": null, "page_end": null, "quality_flags": ["mixed_license"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 31201, "title": "Might a planet perform figure-8 orbits around two stars?", "url": "https://physics.stackexchange.com/questions/31201/might-a-planet-perform-figure-8-orbits-around-two-stars", "share_url": "https://physics.stackexchange.com/q/31201", "content_license": null, "owner": {"display_name": "dotancohen", "user_id": 4877, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/4877/dotancohen"}}, "answers": [{"answer_id": 67260, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/31201/might-a-planet-perform-figure-8-orbits-around-two-stars/67260#67260", "share_url": "https://physics.stackexchange.com/a/67260", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Thriveth", "user_id": 17556, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/17556/thriveth"}}, {"answer_id": 32406, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/31201/might-a-planet-perform-figure-8-orbits-around-two-stars/32406#32406", "share_url": "https://physics.stackexchange.com/a/32406", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "dotancohen", "user_id": 4877, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/4877/dotancohen"}}, {"answer_id": 490763, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/31201/might-a-planet-perform-figure-8-orbits-around-two-stars/490763#490763", "share_url": "https://physics.stackexchange.com/a/490763", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Cang Ye", "user_id": 227290, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/227290/cang-ye"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:534036:0000", "text": "Question: Why do gravitational waves circularize a binary?\n\nI understand that a binary orbiting around one another will circularize due to the emission of GWs due to Peters equations and that highly eccentric binaries evolve faster. \r\nBut GW emission also removes energy and angular momentum (wouldn’t the latter increase the eccentricity from the relation between the eccentricity and the angular momentum?).\r\nWhat is the physical picture behind this?\n\nAccepted Answer:\n\nThe shape of a Keplerian  orbit can be characterized geometrically by its semimajor axis $a$ and eccentricity $e$ or dynamically by its energy $E$ and angular momentum $L$.\r\n\r\nThe latter are expressible in terms of the former as\r\n\r\n$$E=-\\frac{Gm_1m_2}{2}\\frac{1}{a}\\tag{1}$$\r\n\r\nand\r\n\r\n$$L^2=\\frac{Gm_1^2m_2^2}{m_1+m_2}a(1-e^2)\\tag{2}.$$\r\n\r\nThe former are expressible in terms of the latter as\r\n\r\n$$a=-\\frac{Gm_1m_2}{2}\\frac{1}{E}\\tag{3}$$\r\n\r\nand\r\n\r\n$$e^2=1+2\\frac{m_1+m_2}{G^2m_1^3m_2^3}EL^2\\tag{4}.$$\r\n\r\nNote that $E$ is *negative* for a bound orbit.\r\n\r\nGravitational radiation carries energy and angular momentum (and also linear momentum) off to infinity. $E$ decreases and becomes more negative, so its absolute value increases; $L^2$ decreases and becomes less positive, so its absolute value decreases. Whether the absolute value of the negative number $EL^2$ increases or decreases, and thus what happens to the eccentricity, is not obvious.\r\n\r\nOne must do the calculation! Here is what [Peters](https://journals.aps.org/pr/abstract/10.1103/PhysRev.136.B1224) did.\r\n\r\nHe first derives/rederives the formulas\r\n\r\n$$\\frac{dE^\\text{rad}}{dt}=\\frac{G}{c^5}\\left(\\frac15\\dddot{Q}_{ij}\\dddot{Q}_{ij}\\right)\\tag{5}$$\r\n\r\nand\r\n\r\n$$\\frac{dL_i^\\text{rad}}{dt}=\\frac{G}{c^5}\\left(\\frac25\\epsilon_{ijk}\\ddot{Q}_{jl}\\dddot{Q}_{kl})\\right)\\tag{6}$$\r\n\r\nfor the rate at which energy and angular momentum are carried to infinity by gravitational waves, in the leading order of a multipole expansion. Here\r\n\r\n$$Q_{ij}=\\sum_n m^{(n)}\\left(x_i^{(n)}x_j^{(n)}-\\frac13\\delta_{ij}x_k^{(n)}x_k^{(n)}\\right)\\tag{7}$$\r\n\r\nis the system's traceless mass quadrupole moment tensor when the system is considered as $n$ point masses.\r\n\r\nHe then applies this to a Keplerian binary and averages over one elliptical orbit. Using the conservation of energy and angular momentum, he finds that the binary's energy and momentum decrease at the average rate\r\n\r\n$$\\left\\langle\\frac{dE}{dt}\\right\\rangle=-\\frac{32}{5}\\frac{G^4}{c^5}\\frac{m_1^2m_2^2(m_1+m_2)}{a^5}\\frac{1+\\frac{73}{24}e^2+\\frac{37}{96}e^4}{(1-e^2)^{7/2}}\\tag{8}$$\r\n\r\nand\r\n\r\n$$\\left\\langle\\frac{dL}{dt}\\right\\rangle=-\\frac{32}{5}\\frac{G^{7/2}}{c^5}\\frac{m_1^2m_2^2(m_1+m_2)^{1/2}}{a^{7/2}}\\frac{1+\\frac{7}{8}e^2}{(1-e^2)^2}\\tag{9}.$$\r\n\r\nDifferentiating (3) gives\r\n\r\n$$\\frac{da}{dt}=\\frac{Gm_1m_2}{2}\\frac{1}{E^2}\\frac{dE}{dt}\\tag{10}$$\r\n\r\nand differentiating (4) gives\r\n\r\n$$e\\frac{de}{dt}=\\frac{m_1+m_2}{G^2m_1^3m_2^3}\\left(L^2\\frac{dE}{dt}+2EL\\frac{dL}{dt}\\right)\\tag{11}.$$\r\n\r\nSubstituting (1), (2), (8), and (9) into (10) and (11) gives\r\n\r\n$$\\left\\langle\\frac{da}{dt}\\right\\rangle=-\\frac{64}{5}\\frac{G^3}{c^5}\\frac{m_1m_2(m_1+m_2)}{a^3}\\frac{1+\\frac{73}{24}e^2+\\frac{37}{96}e^4}{(1-e^2)^{7/2}}\\tag{12}$$\r\n\r\nand\r\n\r\n$$\\left\\langle\\frac{de}{dt}\\right\\rangle=-\\frac{304}{15}\\frac{G^3}{c^5}\\frac{m_1m_2(m_1+m_2)}{a^4}\\frac{e(1+\\frac{121}{304}e^2)}{(1-e^2)^{5/2}}\\tag{13}.$$\r\n\r\nYou can see that the rate of decrease of the eccentricity is very rapid for a highly eccentric orbit with $e$ near $1$, due to the $(1-e^2)^{5/2}$ in the denominator. In others words, the orbit rapidly circularizes.\r\n\r\nFrom these equations, Peters proceeds to find $a$ as a function of $e$ (with two unusual exponents, $12/19$ and $870/2299$) and a differential equation for $e(t)$ from which the lifetime of the binary can be found.\n\nAnswer (score=40):\n\nThe other answers have given good “rigorous mathematical” arguments for why this happens, but I'd like to add a simpler hand-waving one.\r\n\r\nGravitational waves are emitted when massive bodies _accelerate_. The acceleration is strongest at periapsis (i.e. when the bodies are closest). The GW emission removes energy. As a result, the orbiting body has less kinetic energy left to rise out of the periapsis, i.e. the next apoapsis will be lower.\r\n\r\nBy comparison, at the apoapsis there's not so much acceleration, thus the height of the periapsis doesn't change very much. But of course the next time you are at that periapsis, more energy is lost. Repeat, until eventually the apoapsis isn't really higher than the periapsis anymore: you have a circular orbit.\n\nAnswer (score=29):\n\nNote that the emission of gravitational waves does **not** necessarily makes an orbit more circular.\r\n\r\nThis happens to  be the case in the weak field (as detailed in G. Smith's answer). However, for binaries with sufficiently different masses, emission of gravitational waves can actually increase the eccentricity in the strong field regime. This effect was discovered by Glampedakis and Kennefick in [gr-qc/0203086][1], and has since been confirmed by many independent calculations.\r\n\r\nSince whether gravitational wave increase or decrease eccentricity, apparently depends  on the precise properties of the binary, we should not expect a simple qualitative argument explaining why its does so one way or the other.\r\n\r\n\r\n  [1]: http://arxiv.org/abs/gr-qc/0203086\n\nAnswer (score=14):\n\nA circular orbit is the one that minimises the energy (kinetic plus potential) for a given angular momentum.\r\n\r\nIf a process radiates energy away from the system without carrying angular momentum away, then the orbit will naturally relax to a circular configuration. This might be the approximate case for an object on an eccentric orbit interacting with a disc of gas and dust for example, where energy can be dissipated and radiated away.\r\n\r\nA similar things happens for gravitational waves, except the complication here is that the gravitational waves also take away some angular momentum. It turns out - and the the answer of @G.Smith shows why - that the rate of angular momentum loss is not high enough, compared to the energy losses, to prevent circularisation.\r\n\r\nThe effect is quite extreme in short period eccentric binary systems because the luminosity in gravitational waves is dimensionally proportional to $L \\sim M^2R^2 \\omega^6$, where $\\omega$ is the angular velocity and $R$ is the orbital separation. But if we set $v \\sim R\\omega$, then $L \\sim M^2 R^{-4} v^6$. This strong dependence on orbital speed (and inverse dependence on orbital radius) is what makes this more efficient in eccentric binaries, because they spend parts of their orbits with a smaller orbital radius and faster orbital speed than a circular binary.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 1, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "forces", "gravitational-waves", "post-newtonian"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 534036, "title": "Why do gravitational waves circularize a binary?", "url": "https://physics.stackexchange.com/questions/534036/why-do-gravitational-waves-circularize-a-binary", "share_url": "https://physics.stackexchange.com/q/534036", "content_license": null, "owner": {"display_name": "Warrenmovic ", "user_id": 255246, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/255246/warrenmovic"}}, "answers": [{"answer_id": 534043, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/534036/why-do-gravitational-waves-circularize-a-binary/534043#534043", "share_url": "https://physics.stackexchange.com/a/534043", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "G. Smith", "user_id": 199630, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/199630/g-smith"}}, {"answer_id": 534172, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/534036/why-do-gravitational-waves-circularize-a-binary/534172#534172", "share_url": "https://physics.stackexchange.com/a/534172", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "leftaroundabout", "user_id": 3540, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/3540/leftaroundabout"}}, {"answer_id": 534111, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/534036/why-do-gravitational-waves-circularize-a-binary/534111#534111", "share_url": "https://physics.stackexchange.com/a/534111", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "TimRias", "user_id": 101892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/101892/timrias"}}, {"answer_id": 534102, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/534036/why-do-gravitational-waves-circularize-a-binary/534102#534102", "share_url": "https://physics.stackexchange.com/a/534102", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:795809:0000", "text": "Question: Would there be one rainbow, a double rainbow or bisecting rainbow on a planet with two suns?\n\nI'm just curious if binary stars are low over the horizon and the conditions are just perfect for the formation of rainbow, would I see a single rainbow, double rainbow or two rainbows intersecting each other?\n\nAccepted Answer:\n\nA rainbow occurs when a viewer is looking towards refractive droplets (typically droplets of water) that are illuminated by a light source behind the viewer. One could imagine standing looking North towards a rain storm on a planet with one local star near the Southwest horizon and one near the Southeast horizon. In this case the Southwest star would form a rainbow towards the Northeast and the Southeast star would form a rainbow towards the Northwest.\r\n\r\n[Rainbows subtend 42 degrees of angle][1]. Depending how close the stars are to each other in the sky it is possible the rainbows from each star could overlap.\r\n\r\n[![][2]][2] \r\n\r\nI believe this is a shot of the two stars of the fictional Tatoo system as viewed from the planet Tatooine. These stars are very close in the sky. I imagine if there was a rainstorm behind the camera, and the camera was turned around, there would appear two overlapping rainbows.\r\n\r\nNote: as pointed out in the comments by Vorbis, XKCD has a [what if][3] with this exact question. I'll just copy some of the figures from that page since it does a better explanation than I do here:\r\n\r\nThe geometry of rainbows (note in this picture it is important that there is a volume filled with refractive droplets at the right hand side where the rainbow appears):\r\n\r\n[![enter image description here][4]][4]\r\n\r\nTypes of binaries and planets:\r\n[![enter image description here][5]][5]\r\nIt is thought that Tatooine is a circumbinary. This would mean that you would *usually* see overlapping rainbows when you see any rainbow on Tatooine. In \"the other kind\" you would only see overlapping double rainbows for certain configurations of the three bodies.\r\n\r\n\r\nWhat the rainbows might look like:\r\n\r\n[![enter image description here][6]][6]\r\n\r\n\r\n  [1]: https://en.wikipedia.org/wiki/Rainbow#:~:text=Substituting%20back%20into%20the%20earlier,radius%20angle%20is%2040.6%C2%B0.\r\n  [2]: https://i.sstatic.net/2Jq4x.jpg\r\n  [3]: https://what-if.xkcd.com/150/\r\n  [4]: https://i.sstatic.net/2IRu7.png\r\n  [5]: https://i.sstatic.net/A5pws.png\r\n  [6]: https://i.sstatic.net/19iS9.png", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 2, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "optics", "refraction", "soft-question"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 795809, "title": "Would there be one rainbow, a double rainbow or bisecting rainbow on a planet with two suns?", "url": "https://physics.stackexchange.com/questions/795809/would-there-be-one-rainbow-a-double-rainbow-or-bisecting-rainbow-on-a-planet-wi", "share_url": "https://physics.stackexchange.com/q/795809", "content_license": null, "owner": {"display_name": "user6760", "user_id": 75502, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/75502/user6760"}}, "answers": [{"answer_id": 795810, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/795809/would-there-be-one-rainbow-a-double-rainbow-or-bisecting-rainbow-on-a-planet-wi/795810#795810", "share_url": "https://physics.stackexchange.com/a/795810", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Jagerber48", "user_id": 128186, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/128186/jagerber48"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:692017:0000", "text": "Question: Time dilation effects at the center of a binary black hole system\n\nImagine two identical black holes in a circular orbit, and Alice is smack-dab in the middle of the system (at the barycenter). Bob is at infinity.\r\n\r\nLet's assume that Alice and Bob are stationary relative to each other. Alice does not experience any net acceleration (from being at the barycenter of the system).\r\n\r\n**Does Alice experience any time dilation relative to Bob?**\r\n\r\n---\r\n\r\nHere are some arguments for why she should not:\r\n\r\n1. Alice is stationary relative to Bob - so no special relativistic effects.\r\n\r\n2. Alice is not accelerating - so by the equivalence principle, she is not experiencing gravitational effects and therefore, no time dilation.\r\n\r\nHowever, this *feels* like the wrong conclusion because she's still stuck in the gravitational well of this binary system and requires a non-zero escape velocity to exit the system and meet with Bob.\r\n\r\nIn literature, lot of discussion about time dilation talks about escape velocity which is straightforward when talking about single spherically symmetric masses - but I am not sure how it applies to this system.\r\n\r\nOf course, the resolution here might just be they both don't experience any relative time dilation, but any signals they try to send each other will always be gravitationally redshifted. And there is no way for them to meet up and compare clocks that show the same passage of time.\n\nAccepted Answer:\n\nLets suppose that the orbit is very large, so we could apply the linearized theory at the center.  The spacetime metric is then\r\n$$\\tag{1}\r\nds^2 \\approx (1 + 2 \\phi) \\, dt^2 - (1 - 2 \\phi) (dx^2 + dy^2 + dz^2),\r\n$$\r\nwhere $\\phi$ is the newtonian potential.  For two black holes on the circular orbit:\r\n$$\\tag{2}\r\n\\phi = \\sum_k \\phi_k = -\\, \\frac{2 G M}{r}.\r\n$$\r\nThe time dilation is defined by the following formula (for stationary observers at the center and at infinity):\r\n$$\\tag{3}\r\nd\\tau = \\sqrt{g_{00}} \\, dt \\approx (1 + \\phi) \\, dt.\r\n$$\r\nSo the time retardation of the central observer would be\r\n$$\\tag{4}\r\n\\Delta \\tau \\approx \\frac{2 G M}{r} \\, \\Delta t.\r\n$$\r\nNotice that if the orbit is very large so that $r \\gg 2 G M$, then $\\Delta \\tau \\approx 0$.  The time dilation would be negligible.\n\nAnswer (score=8):\n\nFor your particular example, the question is: \"is spacetime curved more at Alice's location than at Bob's\", or equivalently \"is the gravitational potential higher for Alice than for Bob\"? And the answer is \"yes\": it would in fact take energy for Alice to go visit Bob. She is located at a local maximum of potential, but not a global maximum. So she experiences time dilation relative to Bob.\r\n\r\nTime dilation is more about gravitational potential than about acceleration (to be very precise it's about the length of the path through spacetime). Acceleration has no effect on time dilation, except in so far as it changes the instantaneous speed of the object; this is the [clock postulate][1] and has been experimentally verified in particle accelerators, where particles experience literally millions of g's of acceleration but their time dilation relative to the laboratory frame is entirely due to their speed.\r\n\r\nThe equivalence principle says that there's a kind of pseudo-potential created in an accelerating coordinate system (such as a coordinate system co-moving with an accelerating rocket). That is, it does indeed take energy for someone at the bottom of an accelerating rocket to move to the top. But from an outside observer's perspective, the different times counted by clocks at the top and bottom of the rocket are due to their different paths through spacetime, which is only indirectly due to acceleration.\r\n\r\n  [1]: https://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html\n\nAnswer (score=4):\n\nA partial answer / extended comment.  \r\n\r\nThink about how a time dilated observer must observe distant light sources to be blueshifted to the amount of her time dilation. \r\n\r\nFor example: Suppose Bob sees Alice get into a blue space ship and fly into position, while Alice sees Bob ready a red laser. Bob sees Alice's space ship gradually turn red as she gets there. When Alice gets to position, Bob shines his laser at her space ship. \r\n\r\nAlice sees a blue laser which reflects off of her ship back to Bob. Bob sees a red laser reflect off of a red space ship. But Alice knows that Bob's laser is red in Bob's frame and Bob knows that Alice's space ship is blue in Alice's frame. If either of them does the calculation, they will find that the red laser was shifted in frequency by $+\\Delta \\nu$ as it traveled from Bob to Alice, then by $- \\Delta \\nu$ as it traveled back from Alice to Bob. The reason for the change in frequency is Work: the gravitational field does work on the beam as it goes from Bob to Alice, and the beam does work on the gravitational field as it goes from Alice to Bob. \r\n\r\nChange in photon energy is proportional to change in frequency: $\\Delta T = h \\Delta \\nu$. Energy is conserved. Therefore, if we know the work per unit mass-energy required to move a parcel of mass-energy from one point to another, we know the time dilation factor between those points, regardless of the value of the gravitational (pseudo)force vector at either point.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 3, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "time-dilation"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 692017, "title": "Time dilation effects at the center of a binary black hole system", "url": "https://physics.stackexchange.com/questions/692017/time-dilation-effects-at-the-center-of-a-binary-black-hole-system", "share_url": "https://physics.stackexchange.com/q/692017", "content_license": null, "owner": {"display_name": "XYZT", "user_id": 33807, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/33807/xyzt"}}, "answers": [{"answer_id": 692022, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/692017/time-dilation-effects-at-the-center-of-a-binary-black-hole-system/692022#692022", "share_url": "https://physics.stackexchange.com/a/692022", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Cham", "user_id": 98263, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/98263/cham"}}, {"answer_id": 692086, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/692017/time-dilation-effects-at-the-center-of-a-binary-black-hole-system/692086#692086", "share_url": "https://physics.stackexchange.com/a/692086", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Eric Smith", "user_id": 287817, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/287817/eric-smith"}}, {"answer_id": 692032, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/692017/time-dilation-effects-at-the-center-of-a-binary-black-hole-system/692032#692032", "share_url": "https://physics.stackexchange.com/a/692032", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "g s", "user_id": 285671, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/285671/g-s"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:747546:0000", "text": "Question: Does a double star system have more mass than its constituents?\n\nAccording to Einstein, energy is equal to mass. Consider a planet that is in gravitational attraction to two stars. Normally I would say that the gravitational attraction is proportional to the masses of the two stars. But if they are orbiting each other, they possess energy.\r\n\r\nIs it correct to say that this star system therefore has a stronger gravitational pull that is greater than just the two added masses of the stars?\n\nAccepted Answer:\n\nA double star has less energy than if the two stars were separated.\r\n\r\nIt is fairly easy to see why this is. If you have two stars orbiting each other you would need to add energy to separate them. That is, assuming you had some form of Star Trek-esque tractor beam you'd have to use that to grab the stars and physically pull them away from each other. Then you'd be putting work into the system and that added work means the two separated stars have a greater combined energy than the original double star system.\r\n\r\nThat means the gravitational field of a double star system is slightly smaller than you'd expect from the masses of the two stars, though in practice the difference is far too small every to be measured.\n\nAnswer (score=14):\n\nThe sum of the kinetic and potential energy of a bound system is *negative*. This must be the case, because you would have to inject more energy into the system to separate the components to infinity.\r\n\r\nTherefore the \"gravitational mass\" of the binary - what you would measure with another orbiting test mass at greater distance - would be *less* than you would expect from the sum of all the masses of the components of the system measured when they are far apart and stationary.\r\n\r\nWhether this is an important effect (i.e. the relative size of the correction) can be judged from the ratio of the absolute value of binding energy to the rest mass energy:\r\n$$\\alpha \\simeq \\frac{GM_1M_2}{2Rc^2(M_1+M_2)}\\ ,$$\r\nwhere $R$ is the separation and the result is exact for a circular orbit.\r\n\r\nThe effect can be important in binaries featuring compact stars (e.g. neutron stars) that have stellar masses and where $R$ can be quite small - the ratio above is a few per cent for a pair of neutron stars separated by 30 km.\r\n\r\n\r\nAs an aside, this consideration also applies to single stars, where the sum of their internal kinetic energy and gravitational potential energy is also negative. Again, this is important in white dwarf and neutron star physics.\n\nAnswer (score=2):\n\nYour question is not really different from a proton and an electron joining together to form a hydrogen atom.  \r\nWhen such a combination takes place approximately $13.6\\,\\rm eV$ of energy is released and the mass of a hydrogen atom is less by the mass equivalent of $13.6\\,\\rm eV$ $(E=mc^2)$ than the combined mass of an isolate proton and an isolated electron.  \r\n\r\nSo if you found the mass of two stars separately and then arranged for them to be in a bound state orbiting each other, the system of two orbiting stars would have a mass which was less than the combined masses of the two individual stars.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 4, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravity", "mass-energy", "binding-energy"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 747546, "title": "Does a double star system have more mass than its constituents?", "url": "https://physics.stackexchange.com/questions/747546/does-a-double-star-system-have-more-mass-than-its-constituents", "share_url": "https://physics.stackexchange.com/q/747546", "content_license": null, "owner": {"display_name": "Anon", "user_id": 337868, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/337868/anon"}}, "answers": [{"answer_id": 747548, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/747546/does-a-double-star-system-have-more-mass-than-its-constituents/747548#747548", "share_url": "https://physics.stackexchange.com/a/747548", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "John Rennie", "user_id": 1325, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/1325/john-rennie"}}, {"answer_id": 747547, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/747546/does-a-double-star-system-have-more-mass-than-its-constituents/747547#747547", "share_url": "https://physics.stackexchange.com/a/747547", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}, {"answer_id": 747556, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/747546/does-a-double-star-system-have-more-mass-than-its-constituents/747556#747556", "share_url": "https://physics.stackexchange.com/a/747556", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Farcher", "user_id": 104696, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/104696/farcher"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:98011:0000", "text": "Question: Binary Star system with one star stationary?\n\nCan a Binary Star system be possible where in one star is stationary and the other star revolves around it? (Just like a planet revolving a star. i.e planets in the system and the star revolving around their own center of masses to balance the system).\n\nAccepted Answer:\n\nAs others explained: When two masses interact gravitationally, it's not like the smaller mass is orbiting the larger mass. Both bodies orbit the common barycenter. When one of the two masses is extremely large compared to the other, the barycenter of the system is almost in the center of the larger mass, so the effect on the larger mass is negligible (like a satellite orbiting the earth). But it still won't be completely stationary. The effect would just be too small to measure.\r\n\r\nBut I would like to get back to your original question, **\"Can a Binary Star system be possible where *the mass difference is so large that the effect on the larger star is negligible*\"**: There isn't that much difference in the masses of stars. Stars need to be massive enough to generate enough pressure to maintain nuclear fusion, but not so massive that they collapse into black holes. Also, the larger the star, the faster does it undergo fusion and the shorter its lifespan. For that reason the heaviest known stars have just about 100 times the mass of our sun, while the lightest know stars have just one tenth of a solar mass. A mass difference of factor 1000 isn't small enough that the effect on the larger mass wouldn't be notable.\r\n\r\n  [1]: http://en.wikipedia.org/wiki/Barycentric_coordinates_%28astronomy%29\r\n  [2]: https://i.sstatic.net/zqOkD.gif\r\n  [3]: https://i.sstatic.net/0iRbd.gif\r\n  [4]: http://en.wikipedia.org/wiki/Hypergiant\n\nAnswer (score=17):\n\nWell, motion is relative so you can choose a frame of reference where one is stationary.  If you do though, it makes the equations of motion quite complicated.\r\n\r\nEven in our solar system, the Sun isn't stationary.  It orbits the center of mass of the whole solar system ([barycenter][1]), just as each planet orbits the center of mass.\r\n\r\nThe center of mass of our solar system moves relative to the sun due to the motion of the Sun and planets.  Here is a graph:\r\n\r\n![solar system barycenter][2]\r\n\r\n([Source][3])\r\n\r\nSo you could potentially call a body in a system where the barycenter stays inside that body \"stationary\" but that's not technically correct, no matter how lopsided the masses of the bodies are.  For example:\r\n\r\n![binary system orbit][4]\r\n\r\nThe larger mass will still always move relative to the system barycenter.\r\n\r\n  [1]: http://en.wikipedia.org/wiki/Center_of_mass\r\n  [2]: https://i.sstatic.net/0LTKL.png\r\n  [3]: http://en.wikipedia.org/wiki/Primary_%28astronomy%29\r\n  [4]: https://i.sstatic.net/5t8x0.gif\n\nAnswer (score=8):\n\nIn any system, components revolve around center of mass of the system. In our solar system, Sun has 99.9% mass of the system. So, center of mass of system is inside the Sun (nearly coinciding with center of Sun).\r\n\r\nSeeing the massive stars in the existence, yes, it's possible that a Sun type star can truly revolve around R136a1 type stars. R136a1 stars are 265 times solar mass. In a binary system with Sun, it'd contain 99.6% mass of the system. So, center of mass of the system would exist near its own center (so, size of star doesn't matter here).\n\nAnswer (score=0):\n\nThis is an instance of the classical two body problem. We know from Classical Mechanics that the movement of two particles orbiting eachother can be reduced to the study of the movement of a single particle with a mass equal to the system's reduced mass, $\\mu=\\frac{mM}{m+M}$ moving in a conservative potencial (for more on this subject, you can refer to virtually any CM book; I specially recommend Golstein's \"Classical Mechanics\", although Thornton's book also gets the job done). As to your proposed problem, say you have two bodies orbiting each other, of masses $m$ and $M$, where $m<<M$. Then, the mechanical energy of the system is given by:\r\n\r\n$$E = \\frac{1}{2}\\mu \\dot r^2 + \\frac{L^2}{2\\mu r^2}+V$$\r\n\r\nwhere $r$ is the distance between both bodies. Now, since the second star is much heavier than the first one, we could approximate the reduced mass as:\r\n\r\n$$\\mu = \\frac{mM}{m+M}\\cdot \\frac{M}{M} = \\frac{m}{1+m/M}\\approx m$$\r\n\r\nTherefore, Classical Mechanics correctly predicts that the heavier star remains stationary whilst the second one, which is much lighter, orbits it. We can introduce this approximation in the conservation of energy equation:\r\n\r\n$$E \\approx \\frac{1}{2}m\\dot r^2 + \\frac{L^2}{2mr^2} + V$$\r\n\r\nWhere, in your problem, the potential will be keplerian:\r\n\r\n$$V = -G\\frac{M}{r}$$\r\n\r\nNotice how the mechanical energy is conserved (since there are no external forces acting on the system), so you can actually calculate the orbit of the lighter star (more generally, you can reduce your problem to quadratures, which you can then integrate numerically):\r\n\r\n$$dt = \\sqrt{\\frac{m}{2}}\\cdot \\frac{dr}{\\sqrt{E-\\frac{L^2}{2mr^2}-G\\frac{M}{r}}}$$\r\n\r\nFurhtermore, if you'd like to calculate the trajectory as a function of the angle (which I would recommend more than $r(t)$), you can use the conservation of angular momentum:\r\n\r\n$$L = mr^2\\dot \\theta$$\r\n\r\nThen, applying the chain rule to $\\dot r$, you get:\r\n\r\n$$\\frac{dr}{d\\theta}\\cdot \\frac{d\\theta}{dt} = \\frac{dr}{d\\theta}\\frac{L}{mr^2}$$\r\n\r\nTherefore:\r\n\r\n$$\\frac{dr}{d\\theta}\\frac{L}{mr^2} = \\sqrt{\\frac{2}{m}(E-\\frac{L^2}{2mr^2}-G\\frac{M}{r})}$$\r\n\r\n$$\\boxed{d\\theta = \\sqrt{\\frac{L^2}{2m}}\\cdot \\frac{dr}{r^2\\sqrt{E-\\frac{L^2}{2mr^2}-G\\frac{M}{r}}}}$$\r\n\r\nIn short, your system would only be stable if one star were much lighter than the other, forming a binary system with an approximately circular orbit.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 5, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "orbital-motion", "planets", "stars", "celestial-mechanics"], "page_start": null, "page_end": null, "quality_flags": ["mixed_license"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 98011, "title": "Binary Star system with one star stationary?", "url": "https://physics.stackexchange.com/questions/98011/binary-star-system-with-one-star-stationary", "share_url": "https://physics.stackexchange.com/q/98011", "content_license": null, "owner": {"display_name": "user6123723", "user_id": 38634, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/38634/user6123723"}}, "answers": [{"answer_id": 98021, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/98011/binary-star-system-with-one-star-stationary/98021#98021", "share_url": "https://physics.stackexchange.com/a/98021", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Philipp", "user_id": 38302, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/38302/philipp"}}, {"answer_id": 98020, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/98011/binary-star-system-with-one-star-stationary/98020#98020", "share_url": "https://physics.stackexchange.com/a/98020", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Brandon Enright", "user_id": 22494, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/22494/brandon-enright"}}, {"answer_id": 98018, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/98011/binary-star-system-with-one-star-stationary/98018#98018", "share_url": "https://physics.stackexchange.com/a/98018", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Earth is a Spoon", "user_id": 2170, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/2170/earth-is-a-spoon"}}, {"answer_id": 800593, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/98011/binary-star-system-with-one-star-stationary/800593#800593", "share_url": "https://physics.stackexchange.com/a/800593", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Lagrangiano", "user_id": 354667, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/354667/lagrangiano"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:458444:0000", "text": "Question: On what timescale does gravitational wave emission circularise an orbit?\n\nGravitational waves remove both energy and angular momentum from a binary orbit. Both rates are enhanced in non-circular (eccentric) orbits and I presume that (like tidal friction) the net effect will be to circularise the orbits over time.\r\n\r\nBut is there a handy formula for the circularisation timescale and is it always (or never?) shorter than the timescale for merger?\n\nAccepted Answer:\n\nI have not seen any *nice* formula in the literature yet. There are some with baroque rational powers, though. At least for point masses circularisation may not happen fast enough to matter, surprisingly enough.\r\n\r\nPeters, P. C., & Mathews, J. (1963). [Gravitational radiation from point masses in a Keplerian orbit][1]. *Physical Review*, 131(1), 435. gives a formula for the average power loss over one period of a Keplerian orbit as $$\\langle P \\rangle = \\frac{32}{5}\\frac{G^4}{c^5}\\frac{m_1^2m_2^2(m_1+m_2)}{a^5(1-e^2)^{7/2}}\\left(1+\\frac{73}{24}e^2+\\frac{37}{96}e^4\\right ),$$ noting that it is equivalent to the standard circular orbit formula multiplied by an enhancement factor of $$f(e)=\\frac{1+(73/24)e^2+(37/96)e^4}{(1-e^2)^{7/2}}.$$ where $f(0.6)\\sim 10, f(0.8)\\sim 100, f(0.9)\\sim 1000$. So we should expect circularisation to happen on a timescale less than $1/f(e)$ of the normal energy loss timescale.\r\n\r\nPeters then went on analysing the decay rate of eccentricity over time in Peters, P. C. (1964). [Gravitational radiation and the motion of two point masses][2]. *Physical Review*, 136(4B), B1224 as $$\\left\\langle \\frac{de}{dt}\\right\\rangle = -\\frac{304}{15}\\frac{G^2}{c^5}\\frac{m_1m_2(m_1+m_2)}{a^4(1-c^2)^{5/2}}e\\left(1+\\frac{121}{304}e^2\\right)$$ which can be combined with his expression for $\\langle da/dt\\rangle$ to get an equation for $\\langle da/de\\rangle$ and $$a(e)=\\frac{c_0 e^{12/19}}{1-e^2}\\left (1+\\frac{121}{304}e^2\\right )^{870/2299}$$ where $c_0$ is determined by the initial condition. \r\n\r\n[![enter image description here][3]][3]\r\n\r\nAn eccentric system loses a lot of angular momentum until the eccentricity is $<0.5$ and then things level out - but to get rid of the last few percent eccentricity the semi-major axis has to shrink a lot more. \r\n\r\nPeters then calculates the lifetime for a system starting at $a_0,e_0$ as $$T(a_0,e_0)=\\frac{12}{19}\\frac{c_0^4}{\\beta}\\int_0^{e_0}\\frac{e^{29/19}\\left(1+\\frac{121}{304}e^2\\right )^{1181/2299}}{(1-e^2)^{3/2}} de$$ where $\\beta=(64/5)(G^4/c^5)m_1m_2(m_1+m_2)$. \r\n\r\n[![enter image description here][4]][4]\r\n\r\nThe result, compared to a circular system, is that initially eccentric systems have lifetimes that actually are shorter roughly like the $f(e)$ factor: they radiate away so much energy by being in eccentric orbits that the reduction in eccentricity doesn't have the time to \"take\" before final infall. \r\n\r\nHowever, this is all for point masses. Tidal effects will also allow spin-up and dissipation in actual objects, like in [this paper on white dwarfs near black holes][5].  [This numerical paper][6] found that black hole pairs in-spiralling over just 9 orbits with $e\\leq 0.8$ circularised by merger time, so the process looks very fast in the more extreme cases. \r\n\r\n\r\n  [1]: https://journals.aps.org/pr/abstract/10.1103/PhysRev.131.435\r\n  [2]: https://journals.aps.org/pr/abstract/10.1103/PhysRev.136.B1224\r\n  [3]: https://i.sstatic.net/2XcKw.png\r\n  [4]: https://i.sstatic.net/YdRvf.png\r\n  [5]: https://arxiv.org/abs/0709.0480\r\n  [6]: https://arxiv.org/abs/0710.5167", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 6, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "orbital-motion", "gravitational-waves"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 458444, "title": "On what timescale does gravitational wave emission circularise an orbit?", "url": "https://physics.stackexchange.com/questions/458444/on-what-timescale-does-gravitational-wave-emission-circularise-an-orbit", "share_url": "https://physics.stackexchange.com/q/458444", "content_license": null, "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}, "answers": [{"answer_id": 458467, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/458444/on-what-timescale-does-gravitational-wave-emission-circularise-an-orbit/458467#458467", "share_url": "https://physics.stackexchange.com/a/458467", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Anders Sandberg", "user_id": 165299, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/165299/anders-sandberg"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:27149:0000", "text": "Question: Two charged black holes in equilibrium\n\nConsider a pair of (possibly rotating) charged black holes with masses $m_1$ and $m_2$, and like charges $q_1$ and $q_2$. It seems that under certain conditions gravitational attraction should exactly cancel electrostatic repulsion and a stationary spacetime will result. \r\n\r\n> What are these conditions?\r\n\r\nThe point charges analogy suggests the equation\r\n$$k q_1 q_2 = G m_1 m_2.$$\r\nHowever, it is by no means obvious this equation is the correct condition except in the large distance limit. Also,\r\n> Is it possible to write down this solution of Einstein-Maxwell theory in closed form?\n\nAccepted Answer:\n\nThere is a quite instructive paper G. A. Alekseev and V. A. Belinski, *Equilibrium configurations of two charged masses in General Relativity*, [Phys.Rev. D76 (2007) 021501][1]; [arXiv:0706.1981 \\[gr-qc\\]][2], e.g. they mentioned a work about non-existence of static equilibrium configurations of two charged black holes by P. Chrusciel and P.Tod, [Commun.Math.Phys., 271 577 (2007)][3]; [arXiv:gr-qc/0512043][4] and found condition for equilibrium of two charged masses:\r\n$m_1 m_2 = (e_1-\\gamma)(e_2+\\gamma)$ with $\\gamma = (m_2 e_1-m_1e_2)/(l+m_1+m_2)$.\r\n\r\n\r\n  [1]: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.76.021501\r\n  [2]: http://arxiv.org/abs/0706.1981\r\n  [3]: https://doi.org/10.1007/s00220-007-0191-9\r\n  [4]: https://arxiv.org/abs/gr-qc/0512043\n\nAnswer (score=-1):\n\nA naive vision of black holes (which is mine) is that their total energy is zero.\r\n\r\nLet the total energy of an object be the sum of its mass energy, its electric energy, its rotating energy, and its gravitational energy.\r\nThe first three energies are positive, while the gravitational energy is negative.\r\n\r\nBut a total energy cannot be negative, so there exist a  limit where the total energy of the object is equal to zero, and this is object is a black hole.\r\n\r\nOf course, by equating the total energy to zero, you will find special values for the  radius of the black holes, and you will find qualitatively that this radius is decreasing when a black hole is charged or rotating.\r\nThe meaning of this radius is that you cannot put any value of energy in a sphere of given radius, there is a limit. \r\n\r\nSo maybe you could apply the same (naive) logic with two black holes, by adding not only the individual energies of each black hole, but also the interacting energies between the two black holes.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 7, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "electromagnetism", "general-relativity", "black-holes", "charge"], "page_start": null, "page_end": null, "quality_flags": ["mixed_license"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 27149, "title": "Two charged black holes in equilibrium", "url": "https://physics.stackexchange.com/questions/27149/two-charged-black-holes-in-equilibrium", "share_url": "https://physics.stackexchange.com/q/27149", "content_license": null, "owner": {"display_name": "Squark", "user_id": null, "user_type": "does_not_exist", "profile_url": null}}, "answers": [{"answer_id": 27151, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/27149/two-charged-black-holes-in-equilibrium/27151#27151", "share_url": "https://physics.stackexchange.com/a/27151", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Alex 'qubeat'", "user_id": 3351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/3351/alex-qubeat"}}, {"answer_id": 27150, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/27149/two-charged-black-holes-in-equilibrium/27150#27150", "share_url": "https://physics.stackexchange.com/a/27150", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Trimok", "user_id": 6316, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/6316/trimok"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:700379:0000", "text": "Question: Are orbital eccentricities in a binary system always the same?\n\nSome excercises on Kepler laws and binary system use this relation $$\\frac{r_1}{r_2} = \\frac{a_1}{a_2},$$\r\nwhere $r$ is the distance from the center of mass to each object and $a$ is the semi-major axis of its orbit. Given this equation \r\n$$ r = \\frac{a(1 - e^2)}{1 + e\\cos\\theta} $$\r\nand the fact that both bodies are always opposite to each other, the only way the first relation holds is if $e_1 = e_2$, but I don't know how to prove that.\n\nAccepted Answer:\n\nPlace the center of your coordinates at the center of mass; this says that $M_{1}\\vec{r}_{1}+M_{2}\\vec{r}_{2}=0$.  Since there are no external forces, the center of mass does not move, and this equation holds for all time.  Therefore, we can express the position of one body in terms of the other and the mass ratio:\r\n$$\\vec{r}_{2}(t)=-\\frac{M_{1}}{M_{2}}\\vec{r}_{1}(t).$$\r\nTherefore, the second mass traces out an orbit that is exactly the same shape as the orbit of the first mass.  The sizes of the orbits differ by the factor $M_{1}/M_{2}$, but their eccentricities are identical.\r\n\r\nNote that the statement that the two trajectories have the same shape (just rescaled) is true in any two-body problem, not just the Kepler problem.  However, only in the Kepler problem are the orbits described by closed ellipses (and thus describable with an eccentricity and major axis).", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 8, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-gravity", "reference-frames", "orbital-motion", "celestial-mechanics"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 700379, "title": "Are orbital eccentricities in a binary system always the same?", "url": "https://physics.stackexchange.com/questions/700379/are-orbital-eccentricities-in-a-binary-system-always-the-same", "share_url": "https://physics.stackexchange.com/q/700379", "content_license": null, "owner": {"display_name": "bajotupie", "user_id": 195892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/195892/bajotupie"}}, "answers": [{"answer_id": 700380, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/700379/are-orbital-eccentricities-in-a-binary-system-always-the-same/700380#700380", "share_url": "https://physics.stackexchange.com/a/700380", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Buzz", "user_id": 70245, "user_type": "moderator", "profile_url": "https://physics.stackexchange.com/users/70245/buzz"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:484483:0000", "text": "Question: Theoretical solution to binary black hole merger based on Hawking and Ellis\n\nFollowing Hawking and Ellis, Chapter 9, Fig. 60, Pg. 322, the following figure is meant to illustrate the contrast between apparent horizons and event horizons in the case of a binary black hole merger.\r\n\r\n[![Taken from Hawking and Ellis book on the same][1]][1]\r\n\r\n  [1]: https://i.sstatic.net/jgO2S.jpg\r\n\r\nI understand the merger of two black holes in the long run settles into a Kerr black hole, and I more or less understand why this is correct. But, I wonder if there exists any simple solution of the binary black hole merger that can actually illustrate this phenomenon? Does there exist any simple metric solution to the Einstein's Field equation that models the black hole merger to illustrate the above concept in explicit form?\n\nAccepted Answer:\n\nThe first explicit example describing black hole coalescence is the Kastor–Traschen class of dynamical multi–black hole solutions:\r\n\r\n- Kastor, D., & Traschen, J. (1993). *Cosmological multi-black-hole solutions*. Physical Review D, 47(12), 5370, [doi:10.1103/PhysRevD.47.5370](https://doi.org/10.1103/PhysRevD.47.5370), [arXiv:hep-th/9212035](https://arxiv.org/abs/hep-th/9212035).\r\n\r\nThis is a family of exact solutions to the Einstein–Maxwell equations with a cosmological constant representing an arbitrary number of charged black holes, with $Q=M$ for each, in an otherwise closed universe. Charge to mass ratio equal to $1$ means that the net force that black holes exert on each other is precisely zero and so it is possible to analytically describe the merger. Some aspects of these coalescing solutions have been investigated here:\r\n\r\n- Brill, D. R., Horowitz, G. T., Kastor, D., & Traschen, J. (1994). *Testing cosmic censorship with black hole collisions*. Physical Review D, 49(2), 840. [arXiv:gr-qc/9307014](https://arxiv.org/abs/gr-qc/9307014).\r\n\r\n- Nakao, K. I., Shiromizu, T., & Hayward, S. A. (1995). *Horizons of the Kastor–Traschen multi-black-hole cosmos*, Physical Review D, 52(2), 796, [doi:10.1103/PhysRevD.52.796](https://doi.org/10.1103/PhysRevD.52.796).\r\n\r\n- McNutt, D., & Coley, A. (2018). *Geometric horizons in the Kastor-Traschen multi-black-hole solutions*. Physical Review D, 98(6), 064043, [doi:10.1103/PhysRevD.98.064043](https://doi.org/10.1103/PhysRevD.98.064043), [arXiv:1811.02931](http://arxiv.org/abs/arXiv:1811.02931). \r\n\r\nAnother situation amenable to analytic description is *extreme mass ratio* mergers, when one of the merging black holes is much larger than the other. While most of the literature focuses on obtaining gravitational waves from such [extreme mass ratio inspiral](https://en.wikipedia.org/wiki/Extreme_mass_ratio_inspiral), there are some works dedicated to the dynamics of event horizon during such process:\r\n\r\n- Emparan, R., & Martínez, M. (2016). *Exact event horizon of a black hole merger*. Classical and Quantum Gravity, 33(15), 155003, [doi:10.1088/0264-9381/33/15/155003](https://doi.org/10.1088/0264-9381/33/15/155003), [arXiv:1603.00712](http://arxiv.org/abs/1603.00712).\r\n\r\n- Emparan, R., Martínez, M., & Zilhão, M. (2018). *Black hole fusion in the extreme mass ratio limit*. Physical Review D, 97(4), 044004, [doi:10.1103/PhysRevD.97.044004](https://doi.org/10.1103/PhysRevD.97.044004), [arXiv:1708.08868](http://arxiv.org/abs/1708.08868).", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 9, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravity", "black-holes", "gravitational-waves"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 484483, "title": "Theoretical solution to binary black hole merger based on Hawking and Ellis", "url": "https://physics.stackexchange.com/questions/484483/theoretical-solution-to-binary-black-hole-merger-based-on-hawking-and-ellis", "share_url": "https://physics.stackexchange.com/q/484483", "content_license": null, "owner": {"display_name": "Sandesh Jr", "user_id": 83357, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/83357/sandesh-jr"}}, "answers": [{"answer_id": 484554, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/484483/theoretical-solution-to-binary-black-hole-merger-based-on-hawking-and-ellis/484554#484554", "share_url": "https://physics.stackexchange.com/a/484554", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "A.V.S.", "user_id": 180269, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/180269/a-v-s"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:819514:0000", "text": "Question: Extreme Mass Ratio Inspirals and GWs cycles\n\nI was reading through the following paper [GRMHD study of accreting massive black hole binaries in astrophysical environment: A review][1]. Therein, we have the following image\r\n\r\n[![enter image description here][2]][2]\r\n\r\nIt is not quite clear how long Post-Newtonian (PN) theory remains adequate and what distinguishes between an inspiral and a late inspiral. My questions are as follows:\r\n\r\n- How long does the inspiral phase last before it transitions into a late inspiral phase, rendering PN theory inadequate? In other words, how long do the early to mid-inspiral phases last, and how long does the late inspiral phase run?\r\n\r\n- Specifically, how long does a late inspiral phase last? In the book \"Gravitational-Wave Astronomy: Exploring the Dark Side of the Universe,\" it is stated that \r\n\r\n> As an example, a $10M\\odot$/$10^{6}M\\odot$ system will spend the last few\r\nyears of inspiral in a very tight orbit emitting $10^5$ −$10^6$ gravitational-wave cycles in the\r\nLISA band.\r\n- How do they determine that the late inspiral lasts this long, and how many gravitational wave cycles are emitted during this period?\r\n\r\n  [1]: https://https://ui.adsabs.harvard.edu/abs/2024APh...15402892C/abstract\r\n  [2]: https://i.sstatic.net/zO03DqJ5.png\n\nAccepted Answer:\n\nThere is no well established taxonomy of dividing the inspiral into \"late\" and \"early\" parts. When people do talk about the \"late inspiral\" they mean \"that part of the inspiral in which the PN approximation is inadequate.\" Where that is depends on a combination of things including the parameters of the binary (mass-ratio, spins, etc.), the order of the PN approximation, and how accurate you need the final answer.\r\n\r\nFor equal mass, non-spinning binaries, and if you don't need a particularly accurate answer a 3PN approximation can be accurate enough for the entire inspiral phase (and you need full Numerical Relativity only for the final plunge and merger.)\r\n\r\nThis changes a lot when the mass-ratio becomes smaller. One of the effects of making the mass-ratio smaller is that the (specific) energy lost to gravitational waves also becomes smaller, and binary evolves slower, completing more gravitational wave cycles during the inspiral. A consequence of this is that you need to know the energy lost per GW cycle much more accurately. Specifically, the accumulated error in the gravitational wave phase scales roughly with the reciprocal of the (symmetric) mass-ratio $\\nu$. Consequently, if we want the inspiral to stay roughly in phase all through the inspiral, this means that we can tolerate an relative error in the energy flux that is rough $O(\\nu)$.\r\n\r\nFor $\\nu=10^{-5}$, the relative error in the 5.5PN flux (chosen simply because I had the plot at hand) for quasicircular non-spinning binaries becomes larger than $10^{-5}$ at a separation of roughly $25 GM/c^2$. So, in this case the part of the inspiral between $25 GM/c^2$ and the last stable orbit ($6 GM/c^2$) can be considered \"late inspiral\". Since the shrinkage of the orbit is $O(\\nu)$ per orbital cycles, this means that the binary $\\nu=10^{-5}$ completes $O(10/\\nu) \\approx 10^6$ orbital cycles in the \"late inspiral\". These are very rough \"back of the envelope\" estimates. They can be made more precise by looking at numerical simulations of such inspirals, e.g. using the publicly available [fast EMRI waveforms][1] package, which uses interpolate numerical data obtained in the small mass-ratio limit.\r\n\r\nEdit:\r\nBelow is a plot showing the relative difference between the linear in $\\nu$ part of the the energy flux to its $n$PN approximation as a function of the \"compactness\" ($GM/(c^2r)$) of the binary, using PN approximations available from the [Black Hole Perturbation toolkit][2], and numerical data I generated myself. \r\n[![PN Flux residuals][3]][3]\r\n\r\n\r\n  [1]: https://github.com/BlackHolePerturbationToolkit/FastEMRIWaveforms\r\n  [2]: https://bhptoolkit.org/PostNewtonianSelfForce/\r\n  [3]: https://i.sstatic.net/VV5Lwjth.png\r\n  [4]: https://i.sstatic.net/Mi0he7pB.png", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 10, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "gravitational-waves", "post-newtonian"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 819514, "title": "Extreme Mass Ratio Inspirals and GWs cycles", "url": "https://physics.stackexchange.com/questions/819514/extreme-mass-ratio-inspirals-and-gws-cycles", "share_url": "https://physics.stackexchange.com/q/819514", "content_license": null, "owner": {"display_name": "RKerr", "user_id": 250454, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/250454/rkerr"}}, "answers": [{"answer_id": 819557, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/819514/extreme-mass-ratio-inspirals-and-gws-cycles/819557#819557", "share_url": "https://physics.stackexchange.com/a/819557", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "TimRias", "user_id": 101892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/101892/timrias"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:382847:0000", "text": "Question: Kepler's 3rd law applied to binary systems: How can the two orbits have different semi-major axes?\n\nI suddenly came to the realization that I don't understand something about Kepler's law when applied to binary systems, because I encountered an apparent paradox. There must be an error somewhere in my reasoning, but I can't figure out what.\r\n\r\nConsider a binary system of stars with masses $m_1$ and $m_2$. Both stars will be in an elliptic orbit around their common center of mass. If, say, star 1 moves on an ellipse with semi-major axis $a$, then  the period of the orbit of star 1 should be given by Kepler's third law,\r\n\\begin{align}\r\nT_1^2 = \\frac{4\\pi a_1^3}{G(m_1 + m_2)}.\r\n\\end{align}\r\nThe same holds for star 2:\r\n\\begin{align}\r\nT_2^2 = \\frac{4\\pi a_2^3}{G(m_1 + m_2)}.\r\n\\end{align}\r\nBut both stars must have the same orbital period (because otherwise the center of mass cannot be at rest), which seems to imply that the semi-major axes of the two stars should also be equal, looking at the above formulas. However, if we consider the Sun-Earth system, for instance (neglecting the other planets, etc) we see that this is clearly not the case.\r\n\r\nWhere is the error in the above reasoning?\n\nAccepted Answer:\n\nYou simply have your definitions wrong in your application of Kepler's third law. In the case where the masses of both bodies are considered, then $a$ is not the semi-major axis of each orbit, it is the sum of the semi-major axes of the two bodies (and is therefore the same in both equations).\r\n$$ a = a_1 + a_2$$\r\n$$ m_1 a_1 = m_2 a_2$$\r\n$$T^2 =\\frac{4\\pi^2}{G(m_1+m_2)} a^3$$\n\nAnswer (score=3):\n\n> If, say, star 1 moves on an ellipse with semi-major axis $a$, then  the period of the orbit of star 1 should be given by Kepler's third law,\r\n\\begin{align}\r\n{T_1}^2 = \\frac{4\\pi {a_1}^3}{G(m_1 + m_2)}.\r\n\\end{align}\r\nThe same holds for star 2:\r\n\\begin{align}\r\n{T_2}^2 = \\frac{4\\pi {a_2}^3}{G(m_1 + m_2)}.\r\n\\end{align}\r\n\r\nMinor issue: You are missing a factor of $\\pi$: You should have $T^2 = (2\\pi)^2 \\frac {a^3}{G(m_1+m_2)}$.\r\n\r\nThat's a minor issue. The major issue: The semi-major axis in that expression is **not** the semi-major axis of the orbit about the center of mass. It is instead the semi-major axis of one object about another, with the other object fixed, and it doesn't matter which object you choose to be fixed. Suppose one of the objects is a pea and the other a ten solar mass giant. You can view the pea as orbiting the star, the star as orbiting the pea, or view each as orbiting their common center of mass.\r\n\r\nAny way you do it, you'll get an ellipse. The math comes out much easier if you view the star as orbiting the pea (or equivalently, if you view the pea as orbiting the star). The math gets considerably messier (but still quite doable) if you view each as orbiting about the center of mass. You implicitly chose the route that requires the messier mathematics. You aren't doing that messier mathematics right, and hence your apparent dilemma.\n\nAnswer (score=3):\n\nKepler's Third law takes a slightly different form when you consider motion around the center of mass. The equations of motion are\r\n$$\r\n\\begin{align}\r\nm_1\\ddot{\\boldsymbol{r}}_1 &= - \\frac{Gm_1m_2}{|\\boldsymbol{r}_1 - \\boldsymbol{r}_2|^3}\\left(\\boldsymbol{r}_1 - \\boldsymbol{r}_2\\right),\\\\\r\nm_2\\ddot{\\boldsymbol{r}}_2 &= \\frac{Gm_1m_2}{|\\boldsymbol{r}_1 - \\boldsymbol{r}_2|^3}\\left(\\boldsymbol{r}_1 - \\boldsymbol{r}_2\\right).\\\\\r\n\\end{align}\r\n$$\r\nIf you want to describe the relative motion of one celestial body with respect to the other one, you can combine these equations to obtain\r\n$$\r\n\\ddot{\\boldsymbol{r}}_1 - \\ddot{\\boldsymbol{r}}_2 =\r\n- \\frac{G(m_1+m_2)}{|\\boldsymbol{r}_1 - \\boldsymbol{r}_2|^3}\\left(\\boldsymbol{r}_1 - \\boldsymbol{r}_2\\right),\r\n$$\r\nor in short\r\n$$\r\n\\ddot{\\boldsymbol{r}} = - \\frac{\\mu}{r^3}\\boldsymbol{r},\r\n$$\r\nwhere $\\boldsymbol{r} = \\boldsymbol{r}_1 - \\boldsymbol{r}_2$ and $\\mu =G(m_1+m_2)$. This is the familiar Kepler problem, with the corresponding 3rd Kepler law\r\n$$\r\nT^2 = \\frac{4\\pi^2}{\\mu}a^3.\r\n$$\r\nOn the other hand, if you wish to describe the motion of both celestial bodies with respect to the center of mass, you need to separate $\\boldsymbol{r}_1$ and $\\boldsymbol{r}_2$ in the equations of motion. You can do this by using the fact that the position of the center of mass remains constant\r\n$$\r\nm_1\\boldsymbol{r}_1 + m_2\\boldsymbol{r}_2 = \\boldsymbol{0},\\tag{1}\r\n$$\r\nso that\r\n$$\r\n\\boldsymbol{r}_1 - \\boldsymbol{r}_2 = \\frac{m_1+m_2}{m_2}\\boldsymbol{r}_1 =\r\n-\\frac{m_1+m_2}{m_1}\\boldsymbol{r}_2.\r\n$$\r\nTherefore,\r\n$$\\begin{align}\r\nm_1\\ddot{\\boldsymbol{r}}_1 &= -Gm_1m_2\\left(\\frac{m_2^3}{(m_1+m_2)^3r^3_1}\\right)\\left(\\frac{m_1+m_2}{m_2}\\boldsymbol{r}_1\\right),\\\\\r\nm_2\\ddot{\\boldsymbol{r}}_2 &= Gm_1m_2\\left(\\frac{m_1^3}{(m_1+m_2)^3r^3_2}\\right)\\left(-\\frac{m_1+m_2}{m_1}\\boldsymbol{r}_2\\right),\r\n\\end{align}\r\n$$\r\nor\r\n$$\\begin{align}\r\n\\ddot{\\boldsymbol{r}}_1 = -\\frac{\\mu_1}{r^3_1}\\boldsymbol{r}_1,\\qquad\\text{and}\\qquad\r\n\\ddot{\\boldsymbol{r}}_2 = -\\frac{\\mu_2}{r^3_2}\\boldsymbol{r}_2,\r\n\\end{align}\r\n$$\r\nwith \r\n$$\\mu_1 = \\frac{Gm_2^3}{(m_1+m_2)^2},\\qquad\\text{and}\\qquad\r\n\\mu_2 = \\frac{Gm_1^3}{(m_1+m_2)^2}.\r\n$$\r\nSo once again we have two Kepler problems, but this time the 3rd laws take the form\r\n$$\r\nT^2 = \\frac{4\\pi^2}{\\mu_1}a_1^3,\\qquad\\text{and}\\qquad\r\nT^2 = \\frac{4\\pi^2}{\\mu_2}a_2^3.\r\n$$\r\nNote that this implies $\\mu_2a_1^3 = \\mu_1a_2^3$, which simplifies to\r\n$m_1a_1 = m_2a_2$, consistent with Eq. (1). Also, $\\mu_1 a = \\mu a_1$ and\r\n$\\mu_2 a = \\mu a_2$ lead to $m_2 a = (m_1+m_2)a_1$ and $m_1 a = (m_1+m_2)a_2$, so that indeed $a = a_1+a_2$.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 11, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-mechanics", "newtonian-gravity", "mass", "orbital-motion"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 382847, "title": "Kepler's 3rd law applied to binary systems: How can the two orbits have different semi-major axes?", "url": "https://physics.stackexchange.com/questions/382847/keplers-3rd-law-applied-to-binary-systems-how-can-the-two-orbits-have-differen", "share_url": "https://physics.stackexchange.com/q/382847", "content_license": null, "owner": {"display_name": "Inzinity", "user_id": 110516, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/110516/inzinity"}}, "answers": [{"answer_id": 382867, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/382847/keplers-3rd-law-applied-to-binary-systems-how-can-the-two-orbits-have-differen/382867#382867", "share_url": "https://physics.stackexchange.com/a/382867", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}, {"answer_id": 382865, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/382847/keplers-3rd-law-applied-to-binary-systems-how-can-the-two-orbits-have-differen/382865#382865", "share_url": "https://physics.stackexchange.com/a/382865", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "David Hammen", "user_id": 52112, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/52112/david-hammen"}}, {"answer_id": 399056, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/382847/keplers-3rd-law-applied-to-binary-systems-how-can-the-two-orbits-have-differen/399056#399056", "share_url": "https://physics.stackexchange.com/a/399056", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Pulsar", "user_id": 24142, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/24142/pulsar"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:684864:0000", "text": "Question: Can two relativistic black holes' event horizons overlap and separate again?\n\nI have read this question:\r\n\r\n>What I have not seen is a purely classical argument for the non-separation of a black hole merger. One can obviously take the time reversed spacetime manifold of a merger and get a valid manifold that splits - but this is a black hole disturbed by a complex ingoing set of strong gravitational waves that splits, a bit like how a loud crash sound converging on some shards can in principle make them reassemble into a vase jumping back up on a shelf. Possible, but thermodynamically impossible.\r\n\r\n>If you mean overlap, and if you are asking if it's possible that merging can be avoided even if the event horizons of both black holes were to overlap, then the answer is still no. They will merge (regardless of how fast the black holes are moving). The escape velocity at an event horizon is equal to the speed of light, and nothing (with mass2) can have this speed, so even if two black holes approach with a great speed, they will merge if their event horizons overlap.\r\n\r\nhttps://physics.stackexchange.com/q/676293/ \r\n\r\n[![enter image description here][1]][1]\r\n\r\nImagine a setup where two black holes of similar size and energy moving both at relativistic speeds, but in opposite directions so, that when they pass by, their event horizons slightly overlap. Now as far as I understand, gravitational fields here are extremely strong, but not infinitely strong.  In my understanding, the kinetic energy of the black holes could overcome the gravitational attraction, and even if the event horizons (which are not physical objects, but just boundaries) slightly overlap, the black holes themselves could continue their way in the opposite directions, and separate the event horizons again (maybe by exchanging part of their energies, so that they change mass). \r\n\r\nThe first answer says so (it is possible), and the second answer only applies the speed of light limit to say no. The second one is arguing about the escape velocity, but that is not what I am asking about. I am not denying that the horizons will inevitably overlap. I think it is a given that the effects of gravity travel at the speed of light (faster then the black holes), and the horizons will overlap. But just because two boundaries overlap where the escape velocity is exceeding the speed of light, why would that affect the whole objects' (black holes) interior completely? Just because the two boundaries overlap (meaning a partial unification), that does not necessarily mean that the interiors of the objects will completely unify. The rest of the objects could have kinetic energies that overcome gravity. I believe that it is not correct to think that black holes are some kind of rigid solid objects (where if the boundaries touch it will grag the whole object inevitably), whereas in reality they are objects like stars but with a boundaries that signal the extreme gravitational field. Extreme does not mean infinitely strong, that is very important. What I am asking about is the possible domination of the kinetic energies over the attraction of gravity. \r\n\r\nSo there are two thought coming to mind:\r\n\r\n1. black holes are not solid rigid objects (we are not talking about two billiard balls touching). Why would the overlap of boundaries drag the whole object inevitably? Are we realistically saying that two black holes would slow down in a matter of seconds from 0.9c to close to 0 (or possibly go into a spiral)?\r\n\r\n2. gravity is extreme, but extreme does not mean infinite. Kinetic energies could dominate gravity.\r\n\r\nJust to clarify, I am asking whether the kinetic energies of the black holes could be enough to overcome the attraction of the gravitational fields, and yes, especially in the case when the horizons slightly overlap. The gravitational fields of the black holes is not infinitely strong. Theoretically, the kinetic energies of the black holes could overcome this. So no question, the horizons will overlap, and maybe the two black holes will exchange energies, and change mass, but could depart and continue on their opposite ways.\r\n\r\nQuestion:\r\n\r\n1.  Can two relativistic black holes' event horizons overlap and separate again?\r\n\r\n  [1]: https://i.sstatic.net/SdPbD.jpg\n\nAccepted Answer:\n\n### Short Answer\r\n\r\nNo, they can't. One two black holes touch, they won't split again.\r\n\r\nI'll sketch a little formal argument which can provide some intuition, but should not be taken too seriously. Nevertheless, I believe it provides an intuitive contradiction in the situation you presented. I invite other people to comment on possible issues and I will raise a few of them myself. At the end of this answer, I'll provide a more solid answer by referring to a rigorous theorem on black hole physics.\r\n\r\n### Intuitive Approach \r\n\r\nLet us suppose one can somehow realize the situation you mentioned, but let me add a minor thing: suppose there is also some point particle of negligible mass on the spacetime. Suppose further this particle is in the intersection of the two black holes when they are superposed. Now, once the black holes separate (assuming the situation you presented is possible), in which of them should the particle stay? It should not be able to leave either black hole, but since they are separating and the particle is in the intersection it will need to somehow choose which black hole it will stay in.\r\n\r\nLet me provide an alternative, but very similar, argument. for a Schwarzschild black hole, once a particle is inside the black hole, it must necessarily fall into the singularity within finite proper time. More specifically, one can show (see Problem 6 of Chap. 6 of Wald's [*General Relativity*](https://press.uchicago.edu/ucp/books/book/chicago/G/bo5952261.html)) that the maximum lifetime of of any observer within the event horizon is $\\tau = \\pi M$ ($c = G = 1$), where $M$ is the black hole's mass. However, this holds for both black holes (if we assume them to be Schwarzschild), so the particle should crash into singularity $r_i = 0$ within proper time $\\tau_i = \\pi M_i$. Since this must happen for both black holes and they are (by assuming the situation you proposed) getting further apart, we've reached a contradiction. If the black holes do not merge, then a particle would need to be able to escape the black holes (to be fair, I think the ). \r\n\r\nNow for some issues with these lines of thought.\r\n1. As mentioned in the comments, General Relativity is wildly non-linear. One can't simply superpose solutions. In particular, the metric of the system is far different than just \"summing\" two black holes and there are gonna be effects of \"gravitational energy\" making the gravitational field even stronger (I'm writing \"gravitational energy\" to refer to the non-linear effects of the Einstein Equations, even though one can't properly define gravitational energy in a sensible and local manner). This also means that we can't treat both black holes as independent and just think of a collision with each other: as they get closer, the gravitational field changes in a non-trivial way due to their interaction;\r\n2. The result I mentioned concerning the maximum lifetime within a black hole uses the Schwarzschild metric and I'm being sloppy when applying it here, since this spacetime is not even stationary. If I recall correctly, more general black holes, such as the Reissner–Nordstrom and Kerr black holes, do not necessarily have these sorts of results, as one can tell from their conformal diagrams;\r\n3. Once the black holes are together, the pair is the black hole region and, at least in principle, the rule is nothing escapes the black hole region. Since the event horizon is a global construction, we also can't really define where one of them starts and the other one ends, and hence it doesn't really make that much sense to say the particle is within both black holes at the same time (we can't even properly define which black hole is which). This also gives a different hint on why the black holes can't split. I'd say this point is a double-edged sword for the argument haha.\r\n\r\n### Rigorous Approach\r\n\r\nI can't really provide much more intuition, so in this section I'll just provide a statement and a reference for the detailed discussions. Proposition 9.2.5 of Hawing & Ellis' [*The Large Scale Structure of Space-time*](https://doi.org/10.1017/CBO9780511524646) reads (up to notation)\r\n> Let $\\mathcal{B}_1(\\tau_1)$ be a black hole on $\\mathcal{S}(\\tau_1)$. Let $\\mathcal{B}_2(\\tau_2)$ and $\\mathcal{B}_3(\\tau_2)$ be black holes on a later surface $\\mathcal{S}(\\tau_2)$. If $\\mathcal{B}_2(\\tau_2)$ and $\\mathcal{B}_3(\\tau_2)$ both intersect $J^+(\\mathcal{B}_1(\\tau_1))$, then $\\mathcal{B}_2(\\tau_2) = \\mathcal{B}_3(\\tau_2)$.\r\n\r\nIn the above, $\\mathcal{S}(\\tau)$ is a partial Cauchy surface for the space-time at time $\\tau$, properly defined on Proposition 9.2.3. Intuitively, it is a \"photograph\" of spacetime at time $\\tau$. A black hole at time $\\tau$ is defined at p. 317 and is the usual definition one would expect: a connected component of the region from $\\mathcal{S}(\\tau)$ from which nothing can escape to the null infinity. $J^+$ is the causal future. \r\n\r\nNotice that, from these definitions, once the black holes touch, they are then one black hole only, and from the proposition I mentioned it follows that it can no longer split. \r\n\r\nHawking & Ellis is, of course, not the only reference discussing this result. Wald also proves it as Theorem 12.2.1 on the book I mentioned above. It is also shown on p. 73 of the [Lecture Notes on Black Holes by H. S. Reall](http://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2016.pdf).\n\nAnswer (score=1):\n\n---\r\n\r\n**The Question**: Arpad's initial query revolved around a cosmic conundrum. Given the significant momentum of two speeding black holes and their distant centers of gravity, how could a mere touch bring them together to form one entity?\r\n\r\n**The Counter**: Nikolas, drawing upon Einstein's theory, proposed a simple yet profound idea: once black holes overlap, they effectively become one. \r\n\r\n**Resolving the Paradox**: But let's bring this scenario down to Earth for a moment. Imagine China and India, two vast nations, each claiming a region they both believe is theirs. Instead of their territories overlapping and creating confusion, they negotiate and draw a line somewhere in the middle. Similarly, rather than assuming black holes maintain their initial shape and borders, the immense gravitational forces at play would dynamically reshape their event horizons. This adjustment ensures that they border each other without overlapping, maintaining their distinct identities yet coming incredibly close.\r\n\r\n**Conclusion**: The initial puzzle concerning the counter-intuive necessary merge of two massive, fast-moving black holes is reconciled when we take into consideration that their boundaries will be reshaped by the gravitational waves, and they won't actually overlap, but border each other.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 12, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravity", "black-holes", "event-horizon"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 684864, "title": "Can two relativistic black holes' event horizons overlap and separate again?", "url": "https://physics.stackexchange.com/questions/684864/can-two-relativistic-black-holes-event-horizons-overlap-and-separate-again", "share_url": "https://physics.stackexchange.com/q/684864", "content_license": null, "owner": {"display_name": "Árpád Szendrei", "user_id": 132371, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/132371/%c3%81rp%c3%a1d-szendrei"}}, "answers": [{"answer_id": 685464, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/684864/can-two-relativistic-black-holes-event-horizons-overlap-and-separate-again/685464#685464", "share_url": "https://physics.stackexchange.com/a/685464", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Níck", "user_id": 168783, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/168783/n%c3%adck"}}, {"answer_id": 779807, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/684864/can-two-relativistic-black-holes-event-horizons-overlap-and-separate-again/779807#779807", "share_url": "https://physics.stackexchange.com/a/779807", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "raphael feliz", "user_id": 377373, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/377373/raphael-feliz"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:663733:0000", "text": "Question: As of 2021 in how many binary systems has the period decrease due to gravitational waves been measured?\n\nI am searching for data for the period decrease of binary systems due to gravitational waves. I am aware of three systems in which it was possible to measure this period decrease:\r\n\r\n - The Hulse-Taylor binary PSR 1913+16  \r\n - the double pulsar PSR J0737-3039\r\n - the pulsar-neutron-star-binary PSR B1534+12 \r\n \r\nHowever I know of these three from the book von gravitational waves from Maggiore which is from 2006. Have there been any new binary sytems entered the list, from which the period decrease has been measured up to this date? I'm thankful for any help. \r\n\r\nI am especially interested in the question whether there have been any measurements comparing results not only to the famous period decrease formula that can be derived from the quadrupole-approximaion of gravitational radiation\r\n\r\n\\begin{equation}\r\n\\dot{P}_b=-\\frac{192\\pi G^\\frac{5}{3}}{5c^5}\\Biggl(\\frac{P_b}{2\\pi}\\Biggr)^{-\\frac{5}{3}}\\frac{m_1m_2}{(m_1+m_2)^\\frac{1}{3}}\\frac{1+\\frac{73}{24}e^2+\\frac{37}{96}e^4}{(1-e^2)^\\frac{7}{2}}\r\n\\end{equation}\r\n\r\nbut to higher Post-Newtonian formulas that may exist...\n\nAccepted Answer:\n\nAccording to the [Australia National Telescope Facility Pulsar Catalogue][1], there are currently 31 pulsar binaries with a measured negative value for the derivative of the period.\r\n\r\nFor an overview of what tests of gravity have been performed using these binaries see this [Living Review][2] by Cliff Will, or [this 2020 review][3] by Wex and Kramer.\r\n\r\n\r\n  [1]: https://www.atnf.csiro.au/research/pulsar/psrcat/proc_form.php?version=1.65&table_top.x=0&table_top.y=0&Name=Name&Binary=Binary&PBDOT=PBDOT&startUserDefined=true&c1_val=&c2_val=&c3_val=&c4_val=&sort_attr=jname&sort_order=asc&condition=PBDOT%20%3C%200&pulsar_names=&ephemeris=short&coords_unit=raj%2Fdecj&radius=&coords_1=&coords_2=&style=Long%20with%20last%20digit%20error&no_value=*&fsize=3&x_axis=&x_scale=linear&y_axis=&y_scale=linear&state=query\r\n  [2]: https://link.springer.com/article/10.12942/lrr-2014-4\r\n  [3]: https://www.mdpi.com/2218-1997/6/9/156\n\nAnswer (score=2):\n\nThree things can affect the observed binary period of a pulsar system: GW emission, relative acceleration, and the Shklovskii effect.  All three can be fit as part of a binary model using the same $\\dot{P}_B$ parameter (or `PBDOT` in the ANTF pulsar catalogue naming).  Here's search for all pulsars with positive or negative $\\dot{P}_B$ [from ANTF][1].  (the parameter C1 is PBDOT, which I can't seem to get to display with the right tag)\r\n\r\nGW emission causes the period to decay resulting in __negative__ $\\dot{P}_B$.\r\n\r\nRelative acceleration doesn't affect the actual orbit but causes the observed orbital period to change.  The pulsar timing model is based in the solar system barycenter (SSB) frame, so it appears as $\\dot{P}_B$.  If a pulsar system has a significantly different galactic gravitation potential relative to the solar system, that would show up as a relative acceleration.  A host globular cluster could also cause this.  For instance globular cluster [47 Tucanae][2] hosts many pulsars including several binaries with observed $\\dot{P}_B$ (e.g. J0024-7204E, H, I, ...).  A relative acceleration can result a __positive or negative__ $\\dot{P}_B$.\r\n\r\nFinally, the Shklovskii effect is the result of transverse motion of the binary system relative to the line of sight.  It's sometimes called _secular acceleration_.  Like relative accelerations it doesn't affect the true binary period just the observed period in the SSB frame.  The Shklovskii effect causes __positive__ $\\dot{P}_B$.\r\n\r\nIn order to measure the GW part of the orbital decay one must first account for the others.  Since transverse motion shows up in other parts of the timing model, I believe the Shklovskii effect can be pretty straightforwardly corrected.\r\n\r\nIt turns out there are 31 pulsars in the ATNF catalogue with __negative__ $\\dot{P}_B$.  Unfortunately I am unsure which have been confirmed to be the result of GW driven decay and not some other effect.\r\n\r\n\r\n  [1]: https://www.atnf.csiro.au/research/pulsar/psrcat/proc_form.php?version=1.65&JName=JName&PB=PB&startUserDefined=true&c1=c1&c1_val=pbdot&c2_val=&c3_val=&c4_val=&sort_attr=jname&sort_order=asc&condition=pbdot%3E0+%7C%7C+pbdot%3C0&pulsar_names=&ephemeris=short&submit_ephemeris=Get+Ephemeris&coords_unit=raj%2Fdecj&radius=&coords_1=&coords_2=&style=Long+with+last+digit+error&no_value=*&fsize=3&x_axis=&x_scale=linear&y_axis=&y_scale=linear&state=query\r\n  [2]: https://en.wikipedia.org/wiki/47_Tucanae", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 13, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravitational-waves", "post-newtonian"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 663733, "title": "As of 2021 in how many binary systems has the period decrease due to gravitational waves been measured?", "url": "https://physics.stackexchange.com/questions/663733/as-of-2021-in-how-many-binary-systems-has-the-period-decrease-due-to-gravitation", "share_url": "https://physics.stackexchange.com/q/663733", "content_license": null, "owner": {"display_name": "Benito McLanbeck", "user_id": 112942, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/112942/benito-mclanbeck"}}, "answers": [{"answer_id": 663900, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/663733/as-of-2021-in-how-many-binary-systems-has-the-period-decrease-due-to-gravitation/663900#663900", "share_url": "https://physics.stackexchange.com/a/663900", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "TimRias", "user_id": 101892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/101892/timrias"}}, {"answer_id": 664021, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/663733/as-of-2021-in-how-many-binary-systems-has-the-period-decrease-due-to-gravitation/664021#664021", "share_url": "https://physics.stackexchange.com/a/664021", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Paul T.", "user_id": 47594, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/47594/paul-t"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:665290:0000", "text": "Question: Is there an approximate expression for the force between two black holes?\n\nJust curious: is there an approximate expression for the gravitational attraction between two Schwarzschild black holes of masses $M$ and $m$, held without relative speed at some center to center distance $d> 2G(M+m)/c^2$? Or is there even an exact expression?\r\n\r\nIn a book on relativity, a force expression is given as\r\n$$F= \\frac{GMm}{d^2} \\frac{1}{\\sqrt{1-\\frac{2GM}{dc^2}}}.$$ But if I recall correctly, this is the force between a small test mass $m$ and a black hole $M$. Here, the question is about an expression for *two* black holes.  (A wild guess: Does the final expression just have two square roots, one with $M$ and one with $m$?)\r\n\r\nIn any case, the approximation should be *more* precise than the Newtonian expression $F= GMm/d^2$. The answer should take into account the two horizon radii. Newtonian gravity is not a good approximation at all when the distance $d$ is of the order of any of the two Schwarzschild radii. (Ideally, the force is given for two black holes held at constant distance, assuming that were possible.)\r\n\r\nThe question is *not* about dynamics, *nor* about the calculation of orbits. The question is only about the value of the *force* of attraction between black holes, e.g., while kept fixed. That way, no dynamics is involved. (If it helps to simplify the question, the black holes can also be of equal mass. They can also move in any desired way, such as towards each other. The black holes have no spin. The only quantity of interest in this question is the force.)\n\nAccepted Answer:\n\nThe thing that one should keep in mind, is that forces gravitate. So if we eliminate dynamics by considering a static system of black holes that are “held fixed” then the thing that prevents those black holes from falling toward each other would necessarily contribute to the curvature of spacetime and would distort the geometry of black holes.\r\n\r\nHaving said that, there is a class of static vacuum solutions of Einstein equations with axial symmetry, the *Weyl metrics*, that allows one to consider superposition of two static black holes held  at a fixed distance apart. This is because the metrics are written in terms of a single potential, $\\psi$, that satisfies a linear flat-space Laplace equation written in cylindrical coordinates. For a single Schwarzschild black hole the source of the potential is a rod of length $2M$ placed along the symmetry axis. By using as a source two rods along the same axis but at some distance apart we would obtain the potential for the “superposition” of *two* Schwarzschild black holes.\r\n\r\nHowever, if we integrate all the components of the metric from the potential we would find that it must have cosmic string-like conical singularity(-ies) along the symmetry axis, either between black holes or stretching to infinity from one of (or from both) black holes.  The physical meaning of such conical singularities is simple: this is a string (or a strut) or a system of strings/struts enabling  the black holes to remain static without falling toward each other, conical singularity corresponding to $\\delta$-like distributional stress-energy tensor. But this enables us  to find the force that the string/strut exerts on the black holes from the  conical singularity's angle deficit/excess. The result for the force between two black holes with masses $M$ and $μ$ is:\r\n$$\r\nF_z = \\frac{M μ }{R^2 - (μ+M)^2},\r\n$$\r\nwhere  $R$ is the coordinate separation between centers of the rods in *Weyl coordinates* and a unit system with $G=c=1$ is used. This force diverges when horizons almost touch each other, and in the limit of large separation it approaches the expected Newtonian expression. In the limit of one of the masses being much smaller than the other, the expression would approach the one provided by OP after changing from Weyl to Schwarzschild coordinates (see e.g. [this paper](https://arxiv.org/abs/2001.00430) for an example of such limit that also includes the $m^2$ corrections).\r\n\r\n----\r\nAs a sanity check let us consider if two expressions agree with each other when applicable (when $μ\\ll M$). The equations needed are in Section III of LaHaye & Poisson paper cited above.  First, we note  that the mass $μ$ in the above expression does not have the same meaning as $m$ in OP, because $μ$ is the contribution from the second black hole to the  ADM mass of the entire system (which is thus $M+μ$), so for a light second black hole ($μ\\ll M$) it is its Killing energy, whereas $m$ in the formula from the question is the norm of particle's 4-momentum: \r\n$$\r\n  μ=E_\\text{K}=m u_\\alpha t^\\alpha,\r\n$$\r\nwhere $u^\\alpha$ is the 4-velocity and $t^\\alpha$ is timelike Killing vector (in static coordinates $t^\\alpha=\\partial_t$). In Schwarzschild coordinates small mass located  at $r=d$ would have:\r\n$$\r\n u^\\alpha=\\frac{t^\\alpha}{\\sqrt{f}},\\quad E_\\text{K}=m\\sqrt{f},\\quad a= \\frac{M}{d^2\\sqrt{f}},\r\n$$\r\nwhere $f=1-\\frac{2M}{d}$, and $a$ is the norm of 4-acceleration, which has only radial component. To obtain the force needed to keep the particle still we just take $F=ma$, recovering expression from OP. That same particle when viewed in Weyl coordinates of background black hole (particle is assumed to be on the symmetry axis) would have $\\rho=0$, $z=d-M=R$ ($R$ is the separation as defined above). \r\n$$\r\n\\mu = m \\sqrt{\\frac{R-M}{R+M}},\\quad F=\\frac{Mm}{(R-M)^{1/2}(R+M)^{3/2}}=\\frac{M\\mu}{R^2-M^2}, \r\n$$ \r\nwhich coincides with linear in $\\mu$ part of the above expression for the force $F_z$ as expected.\r\n\r\n**References**\r\n\r\nDiscussion of Weyl metrics could be found in a book:\r\n\r\n- J. B. Griffiths and J. Podolský, *Exact space-times in Einstein’s general relativity* (Cambridge University Press, Cambridge, 2009), Ch. 10.\r\n\r\nNote, there are *a lot* of sources discussing Weyl metrics going back to original works of H. Weyl circa 1919 so here is a couple of papers that discuss the forces and has open access versions:\r\n\r\n- P.S. Letelier  and S.R. Oliveira. *Superposition of Weyl solutions: the equilibrium forces*. Classical and Quantum Gravity 15, no. 2 (1998): 421, [arXiv:gr-qc/9710122](https://arxiv.org/abs/gr-qc/9710122).\r\n\r\n- P. Krtouš and A. Zelnikov, *Thermodynamics of two black holes*, JHEP 02, 164 (2020) [arXiv:1909.13467](https://arxiv.org/abs/1909.13467).", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 14, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "forces", "gravity", "black-holes"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 665290, "title": "Is there an approximate expression for the force between two black holes?", "url": "https://physics.stackexchange.com/questions/665290/is-there-an-approximate-expression-for-the-force-between-two-black-holes", "share_url": "https://physics.stackexchange.com/q/665290", "content_license": null, "owner": {"display_name": "user85598", "user_id": null, "user_type": "does_not_exist", "profile_url": null}}, "answers": [{"answer_id": 665592, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/665290/is-there-an-approximate-expression-for-the-force-between-two-black-holes/665592#665592", "share_url": "https://physics.stackexchange.com/a/665592", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "A.V.S.", "user_id": 180269, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/180269/a-v-s"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:795219:0000", "text": "Question: Merge of black holes with very different masses\n\nSince surface area of the remnant black hole must be more that the sum of the binary surface then the maximum energy released via gravitational waves is\r\n\r\n$$\r\nΔE = [M_1 + M_2 – \\sqrt{M_1^2 + M_2^2}]c^2\r\n$$\r\n\r\nThat means that when ratio of the initial masses is very large then almost all mass of smaller black hole may turn into gravitational waves.\r\n\r\nI went through \"[SXS Gravitational Waveform Database][1]\" - it has 8 simulations of mergers with ratio above 9. For all of them less then 1% of total mass is converted into gravitational waves while typical LIGO merge releases several percents. This implies that in typical super massive BH and stellar mass BH merge there is almost no emission.\r\n\r\nHow does typical (most likely in the wild) and extreme (with maximum loss) case of high mass ratio merge looks like?\r\n\r\n\r\n  [1]: https://data.black-holes.org/waveforms/catalog.html\n\nAccepted Answer:\n\nTo lowest order approximation the dynamics of a binary with a large mass-ratio can be approximated as the smaller object following a geodesic in the metric generated by the larger object, and the system slowly (adiabatically) evolving due to the emission of energy (and angular momentum) in the form of gravitational waves. \r\n\r\nThis adiabatic \"inspiral\" process continues until the system reaches the last stable orbit (LSO) after which it transitions to a plunging geodesic.\r\n\r\nThe total energy converted to gravitational waves during the inspiral phase is simply the difference between the initial energy of the system and the energy of the LSO. The lowest possible energy LSO is a circular orbit, the innermost stable circular orbit or ISCO. For a binary without spin the energy of the ISCO is\r\n\r\n$$E_{\\rm isco} = \\frac{2\\sqrt{2}}{3} m_2 c^2 \\approx 0.943 m_2 c^2,$$\r\n\r\nwhere $m_2$ is the mass or the lighter (secondary) object. During the inspiral phase a non-spinning binary can convert at most 5.7% of the rest mass of the lighter object to energy. For spinning binaries (which have ISCOs with much lower energy) this can increase substantially to 42.3% (for a maximally spinning primary).\r\n\r\nThe plunge phase is relatively short for large marge ratio binaries, leaving little opportunity to convert energy into gravitational waves. The energy lost in this phase is proportional to $ \\frac{m_2^2}{m_1}c^2$. If $m_1 \\gg m_2$ this is always much smaller than the energy convert in the inspiral phase.\r\n\r\nIn fact, the next largest contribution to the energy convert to gravitational waves comes from the \"transition\" phase as the system transitions from the inspiral to the plunging phase in which an amount of energy portional to $m_2 \\left(\\frac{m_2}{m_1}\\right)^{4/5}$ is converted to gravitational waves see [this paper][1] by Ori&Thorne, which is still much smaller than the energy lost during the inspiral.\r\n\r\n\r\n  [1]: https://arxiv.org/abs/gr-qc/0003032\n\nAnswer (score=4):\n\nIf I understand \"[Gravitational wave snapshots of generic extreme mass ratio inspirals][1]\" (thanks to @andrew for the tip) correctly then typically almost all smaller mass body is converted in waves.\r\n\r\n[![EMRIs simulations table][2]][2]\r\n\r\n, where μ is the mass of the smaller body and E is wave energy.\r\n\r\nBut for head-on collision (plunging orbit) conversion ratio will be very small (see \"[Gravitational radiation from plunging orbits][3]\").\r\n\r\n  [1]: https://arxiv.org/pdf/gr-qc/0509101.pdf\r\n  [2]: https://i.sstatic.net/LWxHkl.png\r\n  [3]: https://arxiv.org/pdf/0809.2814.pdf", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 15, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "black-holes", "gravitational-waves"], "page_start": null, "page_end": null, "quality_flags": [], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 795219, "title": "Merge of black holes with very different masses", "url": "https://physics.stackexchange.com/questions/795219/merge-of-black-holes-with-very-different-masses", "share_url": "https://physics.stackexchange.com/q/795219", "content_license": null, "owner": {"display_name": "Vashu", "user_id": 111670, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/111670/vashu"}}, "answers": [{"answer_id": 796064, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/795219/merge-of-black-holes-with-very-different-masses/796064#796064", "share_url": "https://physics.stackexchange.com/a/796064", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "TimRias", "user_id": 101892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/101892/timrias"}}, {"answer_id": 795224, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/795219/merge-of-black-holes-with-very-different-masses/795224#795224", "share_url": "https://physics.stackexchange.com/a/795224", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Vashu", "user_id": 111670, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/111670/vashu"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:817703:0000", "text": "Question: Do we use transit photometry to look for a black hole star binary systems?\n\nWhat would a light curve look like for a black hole transiting a star? Initially I thought it would bend all light essentially blacking out a star but we would probably still detect some however the curve should look substantially different from a planet transiting.\n\nAccepted Answer:\n\nA binary black hole that passes in front of its companion star when observed from Earth will produce a self-lensing flare once every orbit.  Such flares are being searched for by TESS (The [Transiting Exoplanet Survey Satellite](https://tess.mit.edu/)), but none have been found so far (as of May 2024).\r\n\r\nAs discussed in [\"Prospects of Finding Detached Black Hole-Star Binaries with TESS\"]( \t\r\nhttps://doi.org/10.3847/1538-4357/ab3a4f) (Kento Masuda and Kenta Hotokezaka, Astrophysical Journal 883 (2019) 169), the light from the star in a black hole binary system varies because of:\r\n\r\n- ellipsoidal variations caused by the black hole's gravity physically distorting the star,\r\n- Doppler beaming (which includes relativistic aberration, time dilation, and Doppler shift) due to the star's motion relative to Earth, and\r\n- self-lensing when the Black hole passes in front of the star and gravitionally focuses the light travelling towards the Earth.\r\n\r\nFor example (from Figure 2 of Masuda & Hotokezaka), here is a typical (idealized) light curve for a binary whose orbit is exactly edge on to Earth.\r\n\r\n[![Light curve for black hole binary for a 10 solar mass black hole and a Sun-like star with an orbital period of 3 days][1]][1]\r\n\r\nThe narrow spike in the middle is the gravitational micro-lensing \"flare\" when the black hole is between the Earth and the star (\"inferior conjunction\"). Doppler beaming is modelled sinusoidally with a maximum when the star is in the part of its orbit when it is moving towards the Earth and an minimum when it is moving away. The ellipsoidal variations have minima when the star and black hole are aligned with the Earth, which happens twice each orbit at inferior and superior conjunction.  (Note: The ellipsoidal variations in the figures in the paper – including the one reproduced above – seem a factor of 2 larger than I calculate from Equation 1 in the paper.)\r\n\r\nYou can search for black hole binary flares in TESS data yourself by joining the [Black Hole Hunters](https://www.zooniverse.org/projects/cobalt-lensing/black-hole-hunters) citizen science project. TESS is looking at millions of stars and the expected black-hole-stellar-binary fraction is expected to be $10^{-5}$ to $10^{-6}$, so tens to hundreds are expected. \r\n\r\nAlthough no black hole binary flares have yet been seen, the Kepler Space Telescope observed five self-lensing binary systems where a white dwarf gravitationally lenses a Sun-like star. Some of these also have since been [observed by TESS](https://doi.org/10.3847/2041-8213/ad19dc \"The First TESS Self-lensing Pulses: Revisiting KIC 12254688, Nicholas M. Sorabella et al 2024 ApJL 961 L45\").\r\n\r\n\r\n  [1]: https://i.sstatic.net/8BpKyaTK.png", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 16, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "black-holes", "photometry", "transit"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 817703, "title": "Do we use transit photometry to look for a black hole star binary systems?", "url": "https://physics.stackexchange.com/questions/817703/do-we-use-transit-photometry-to-look-for-a-black-hole-star-binary-systems", "share_url": "https://physics.stackexchange.com/q/817703", "content_license": null, "owner": {"display_name": "Joe", "user_id": 64376, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/64376/joe"}}, "answers": [{"answer_id": 818278, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/817703/do-we-use-transit-photometry-to-look-for-a-black-hole-star-binary-systems/818278#818278", "share_url": "https://physics.stackexchange.com/a/818278", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "David Bailey", "user_id": 145491, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/145491/david-bailey"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:362181:0000", "text": "Question: How to identify binary stars in $N$-body simulation?\n\nBinary stars constitute a significant portion of the stars of a globular cluster.\r\n\r\nI would like to verify that this is true in my $N$-body simulation, but I don't know how to decide whether a star in the system is a binary.\r\n\r\nVisually this is easy to do, as binaries are identified as two stars at very close distance orbiting about their center of mass, but I need a mathematical condition which I can then translate to code.\n\nAccepted Answer:\n\nYou'd need to calculate the binding energy of pairs of particles in your simulation. If for a pair this energy is negative then the pair is bound forming a binary system.\r\n\r\nI assume you already have an effective way of calculating the potential, so this should not add much more execution time, since you just need to check for points that are close enough", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 17, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-gravity", "computational-physics", "simulations", "star-clusters"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 362181, "title": "How to identify binary stars in $N$-body simulation?", "url": "https://physics.stackexchange.com/questions/362181/how-to-identify-binary-stars-in-n-body-simulation", "share_url": "https://physics.stackexchange.com/q/362181", "content_license": null, "owner": {"display_name": "math_lover", "user_id": 63691, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/63691/math-lover"}}, "answers": [{"answer_id": 362189, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/362181/how-to-identify-binary-stars-in-n-body-simulation/362189#362189", "share_url": "https://physics.stackexchange.com/a/362189", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "caverac", "user_id": 135145, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/135145/caverac"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:519115:0000", "text": "Question: Free fall and gravitational waves\n\nIf you take the Earth-Moon system, from what I understand, the Moon for instance is in free fall towards the Earth-Moon center of mass. However Einstein's equivalence principle says that a body in free fall is in inertial motion. Then why does the Moon (and any binary system in general) emit gravitational waves?\n\nAccepted Answer:\n\n> However Einstein's equivalence principle says that a body in free fall is in inertial motion.\r\n\r\nThis holds in the limit of a small body (a \"test mass.\"). It is not true except in that limit, and for exactly the reason you've expressed.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 18, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravitational-waves", "free-fall", "equivalence-principle"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 519115, "title": "Free fall and gravitational waves", "url": "https://physics.stackexchange.com/questions/519115/free-fall-and-gravitational-waves", "share_url": "https://physics.stackexchange.com/q/519115", "content_license": null, "owner": {"display_name": "Opt", "user_id": 4665, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/4665/opt"}}, "answers": [{"answer_id": 519117, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/519115/free-fall-and-gravitational-waves/519117#519117", "share_url": "https://physics.stackexchange.com/a/519117", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "user4552", "user_id": null, "user_type": "does_not_exist", "profile_url": null}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:405295:0000", "text": "Question: How does the delay of pulsar signals prove that gravity travels at the speed of light?\n\nIn the [Hulse-Taylor][1] binary pulsar system, the orbit of the two neutron stars results in the warping of space time causing the pulses to arrive earlier and later because of the longer distance travelled when the pulse crosses the gravitational field. How does this prove that gravity moves at the speed of light?[![enter image description here][2]][2]\r\n\r\n\r\n  [1]: https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_binary\r\n  [2]: https://i.sstatic.net/phNDx.jpg\n\nAccepted Answer:\n\nYou'll have to forgive me if I misinterpret anything, I just learned about this binary system now and have only done some preliminary research. However, from what I can tell, the delay in signal from gravitational warping is *not* the evidence for the speed of gravity that this system provides.\r\n\r\nInstead, this binary pulsar system demonstrates a distinct orbital decay. That is, the periapsis (or to be pedantic, the periastron) distance decreases noticeably over time. This is predicted by general relativity, which states that two bodies like this should lose energy in the form of gravitational radiation (all information your link provided). In fact, the amount of orbital decay predicted and that observed matches to within half a percent (part of the reason why they got the Nobel prize for this).\r\n\r\nThis makes it extremely likely that gravitational waves are a thing. Of course, you couldn't have gravity waves if the speed of gravity were infinite. So, at first glance, it certainly seems to confirm that the speed of gravity is finite. Furthermore, the amount of energy radiated away with gravity waves [is dependant on the speed of the wave](https://journals.aps.org/pr/pdf/10.1103/PhysRev.131.435). The theories relate the radiated power as proportional to $1\\over v^5$, where $v$ is the speed of the wave. It turns out that if you make $v=c$, you get the exact amount of energy radiated away as what corresponds to the orbital decay of the system to within no more than $1\\%$. So it would stand to reason that the speed of gravity is likely the speed of light.\r\n\r\nNo messing around with the delay of pulses due to spacetime warping is necessary.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 19, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravity", "speed-of-light", "pulsars"], "page_start": null, "page_end": null, "quality_flags": ["single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 405295, "title": "How does the delay of pulsar signals prove that gravity travels at the speed of light?", "url": "https://physics.stackexchange.com/questions/405295/how-does-the-delay-of-pulsar-signals-prove-that-gravity-travels-at-the-speed-of", "share_url": "https://physics.stackexchange.com/q/405295", "content_license": null, "owner": {"display_name": "user5389726598465", "user_id": 133814, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/133814/user5389726598465"}}, "answers": [{"answer_id": 405302, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/405295/how-does-the-delay-of-pulsar-signals-prove-that-gravity-travels-at-the-speed-of/405302#405302", "share_url": "https://physics.stackexchange.com/a/405302", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Jim", "user_id": 23473, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/23473/jim"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:380870:0000", "text": "Question: Spacetime geometry with a system of $n$ interacting black holes\n\nThe Schwarzschild metric is given by $ds^2=-\\left(1-\\frac{2GM}{rc^2}c^2\\right)dt^2+\\left(1-\\frac{2GM}{rc^2}c^2\\right)^{-1}dr^2+r^2d\\Omega^2,$ for $d\\Omega^2$ the round metric on the $2$-sphere, in local coordinates $(x^0,x^1,x^2,x^3)=(ct,r,\\theta,\\phi),$ with corresponding metric tensor $$g_{\\mu\\nu}=\\begin{bmatrix} \\left(1-\\frac{2GM}{rc^2}\\right) & 0 & 0& 0\\\\0&\\left(1-\\frac{2GM}{rc^2}\\right)^{-1}&0&0\\\\ 0&0&r^2&0\\\\0&0&0&r^2sin^2\\theta\\end{bmatrix}.$$\r\n\r\n>Given this what, then, is the metric (tensor) of a spacetime $(M,\\mathcal{O},\\mathcal{A},g)$ with a system of $n$ interacting black hole bodies?\n\nAccepted Answer:\n\nEven Newtonian gravity is extremely complicated if the number of moving bodies exceeds 2. If you add nonlinearity of general relativity and existence of gravitational radiation, then generally, a system of several black holes could be investigated only by numerically solving full system of Einstein equations: a very computationally intensive system of nonlinear PDEs. However, in some cases there are solutions which admit a much simpler analysis. Because of their high degree of symmetry, even in cases where no explicit solutions are known, one can analyze such solutions with much simpler methods. Let us mention some of them. \r\n\r\n**Majumdar-Papapetrou**  There is a family of Majumdar-Papapetrou (MP) solutions of Einstein-Maxwell equations (general relativity with electromagnetic field). Metric $g$ and  potential $A$ can be\r\nwritten in the  form\r\n$$\\begin{array}{c}\r\ng = -u^{-2}dt^2 + u^2(dx^2+dy^2+dz^2)\\,, \r\n\\\\\r\n A = u^{-1}dt\\,, \r\n\\end{array}$$\r\nwith some nowhere vanishing, say positive, function\r\n$u$ independent of $t$. Einstein-Maxwell equations read then as\r\n$\\Delta u=0$, with a 3D flat space Laplacian $\\Delta$. A family of solutions for function $u$ has the form\r\n$$\r\nu=1+\\sum_{i=1}^I \\frac{m_i}{|\\vec x - \\vec a_i|} \\,,\r\n$$\r\nfor some positive constants $m_i$. It could be shown that this metric is describing a system of charged black holes with degenerate horizons. Gravitational attraction is exactly compensated by electrostatic repulsion because all charges here are equal to corresponding masses (in appropriate system of units). The original works are:\r\n\r\n> S.D. Majumdar,\r\n*A class of exact solutions of Einstein’s field equations*,\r\nPhys. Rev. 72 (1947), 390–398.\r\n\r\n> A. Papapetrou, *A static solution of the equations of the gravitational field for an arbitrary charge–distribution*, Proc. Roy. Irish Acad.\r\nA51 (1945), 191–204.\r\n\r\nInterpretation as a system of black holes is in following work:\r\n\r\n>  J.B. Hartle, S.W. Hawking,\r\n*Solutions of the Einstein–Maxwell equations\r\nwith many black holes*, Commun. Math. Phys.\r\n26 (1972), 87–101.\r\n\r\nBut, of course,  there are a lot of more modern treatments, in particular, because such solutions admit Killing spinor and thus could be interpreted as solutions possessing supersymmetry.\r\n\r\n **Black holes with cosmic strings**. There are also static solutions with multiple black holes in which the gravitational attraction is compensated by forces exerted on such black holes by [cosmic strings](https://en.wikipedia.org/wiki/Cosmic_string). Cosmic strings exibit $\\delta$-like singularity of Ricci tensor and so such solutions could be seen as containing exotic matter. A simplest would be a pair of black holes hanging on from a pair of semi-infinite strings at some distance from each other. Gravitational attraction between black holes is compensated by finetuned string tension.\r\n\r\n**Black hole universes**. Another class of metrics susceptible to simple analysis is \"black hole universes\" where multiple black holes embedded into cosmological model. For example, one could place black holes in highly symmetric positions on the 3-sphere. Although black holes would be attracting each other, because of the symmetry of their configuration they would be evolving in a cosmological manner with sphere expanding or contracting as a whole. And if we add a finetuned cosmological constant we could even obtain static Einstein universe with multiple black holes! Pioneering work for this would be \r\n\r\n> Lindquist, Richard W., and John A. Wheeler. *Dynamics of a lattice universe by the Schwarzschild-cell method*. Reviews of Modern Physics 29.3 (1957): 432.\r\n\r\nAnd a piece of modern development:\r\n\r\n> Yoo, Chul-Moon, et al. *Black hole universe: construction and analysis of initial data.* Physical Review D 86.4 (2012): 044027. [arXiv:1204.2411](https://arxiv.org/abs/1204.2411)\r\n\r\n**Higher dimensions**. If spacetime dimension $D\\ge 5$ then, the no-hair theorem no longer applies. And so among the exact solutions there is such beauty: black saturn. A central spherical black hole is surrounded by a toroidal black ring. Angular momentum of the ring balances its attraction toward the center.\r\n\r\n> Elvang, H., & Figueras, P. (2007). *Black saturn*. Journal of High Energy Physics, 2007(05), 050. [arXiv:hep-th/0701035](https://arxiv.org/abs/hep-th/0701035).\n\nAnswer (score=6):\n\nThere is no analytic solution of the Einstein equations for a system of more than one black hole. Indeed, the Schwarzschild metric only describes a black hole if:\r\n\r\n1. it is the only object in the universe\r\n\r\n2. it has already existed for an infinite time and will continue to exist for an infinite time into the future\r\n\r\nSo given the universe has only existed for 13.8 billion years, and that Hawking radiation means all black holes will eventually evaporate, [black holes don't exist][1].\r\n\r\nBut leaving this aside, the Schwarzschild metric is a solution to the Einstein equations for a single isolated spherically symmetric mass but it exploits the high symmetry of the situation. If you consider even just two masses no analytic solution can be found. You can't just superpose two Schwarzschild metrics because the Einstein equations are not linear so the sum of their solutions is not a solution.\r\n\r\n\r\n  [1]: https://physics.stackexchange.com/questions/95366/why-does-stephen-hawking-say-black-holes-dont-exist", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 20, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "metric-tensor", "many-body"], "page_start": null, "page_end": null, "quality_flags": ["low_score"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 380870, "title": "Spacetime geometry with a system of $n$ interacting black holes", "url": "https://physics.stackexchange.com/questions/380870/spacetime-geometry-with-a-system-of-n-interacting-black-holes", "share_url": "https://physics.stackexchange.com/q/380870", "content_license": null, "owner": {"display_name": "Sergio Charles", "user_id": 131313, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/131313/sergio-charles"}}, "answers": [{"answer_id": 380888, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/380870/spacetime-geometry-with-a-system-of-n-interacting-black-holes/380888#380888", "share_url": "https://physics.stackexchange.com/a/380888", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "A.V.S.", "user_id": 180269, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/180269/a-v-s"}}, {"answer_id": 380872, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/380870/spacetime-geometry-with-a-system-of-n-interacting-black-holes/380872#380872", "share_url": "https://physics.stackexchange.com/a/380872", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "John Rennie", "user_id": 1325, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/1325/john-rennie"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:703008:0000", "text": "Question: Gravitational wave radiation power from dimensional analysis\n\nLet us try to find a formula for the power emitted through gravitational waves (GW) from a binary system in quasi circular orbit. The relevant quantities are the Newton's constant $G_N$, speed of light $c$, a mass scale $M$, orbital frequency $\\omega$. So I write $P=G_N^a c^b M^c \\omega^d$. Demanding both sides have the same dimension gives \r\n\\begin{align}\r\nP_{GW}=\\frac{c^5}{G_N}\\left(\\frac{G_NM\\omega}{c^3}\\right)^d\r\n\\end{align}\r\nwith arbitrary exponent $d$. It turns out that $d=10/3$ from the quadrupole approximation. My question is the following: Is there a way to get $d=10/3$ from some sort of argument without having to get into the details of the computation? One possible answer is that we know from post-Newtonian theory that radiation effects start at $1/c^5$, and this implies $d=10/3$. But I thought  maybe there is a better argument.\n\nAccepted Answer:\n\nThe short answer is no.  The problem is that a simple dimensional analysis argument tells you that you need the power to be energy per unit time, given the mass $M$, frequency $\\omega$, and $G$ and $c$. Well, the energy part is \"easy\" in the sense that $Mc^2$ has the right dimensions. However, the problem is that there are two quantities with dimensions of time\r\n\\begin{equation}\r\n\\frac{G M}{c^3}, \\ \\ \\frac{1}{\\omega}\r\n\\end{equation}\r\nSo dimensional analysis is only powerful enough to tell you that the answer must look like\r\n\\begin{equation}\r\nP_{\\rm GW} = \\left(M c^2\\right) \\times \\omega^d \\times \\left(\\frac{c^3}{GM}\\right)^{1-d} = \\frac{c^5}{G} \\left(\\frac{GM\\omega}{c^3}\\right)^d\r\n\\end{equation}\r\nfor some $d$, as you said. Another way to express the same point is that dimensional analysis only fixes the dependence on dimensionful quantities, but there is a dimensionless ratio $GM \\omega/c^3$ which dimensional analysis cannot help you with. \r\n\r\nIn fact, in general the situation is even worse than this. If you allowed for the two objects in the binary to have different masses, there would even be another dimensionless quantity, which we can take to be the mass ratio $q=m_1/m_2$. The dependence on $q$ is also not fixed by dimensional analysis; doing a more careful calculation ends up telling you that the power depends on the *chirp mass*. And if you allow the black holes to have spins, you need even more associated dimensionless ratios to describe the system. \r\n\r\nSo, the bottom line is that you do need information about the leading order scaling of the post-Newtonian corrections to fix $d$ (as well as mass ratio dependence, and spin dependence).\r\n\r\nThe longer answer, though, is that it isn't too hard to get the leading order PN scaling to fix the frequency dependence, if you are willing to accept that gravity is a spin-2 field and so must couple to the quadrupole moment $Q$. (As @ProfRob pointed out in the comments, there are other ways you can justify the coupling to $Q$. For example: $Q$ is the lowest order multipole that gravitational waves *can* couple to, given that the monopole and dipole moments correspond to the mass and linear momentum of the system, which are conserved and so cannot contribute to radiation). In order to estimate the frequency dependence, because of Kepler's laws, we also need to know the dependence on the size of the system, $R$. Given that $h \\sim \\ddot{Q}$ where $Q \\sim R^2$, we have $h \\sim \\omega^2 R^2$. Using Kepler's laws, we use that $R \\sim \\omega^{-2/3}$, so $h \\sim \\omega^{2/3}$. Therefore, $E \\sim \\int dt \\dot{h}^2 \\sim \\omega^2 h^2/\\omega \\sim \\omega^{7/3}$ and $P \\sim \\dot{E} \\sim \\omega E \\sim \\omega^{10/3}$.\n\nAnswer (score=2):\n\nYou can't without further physical arguments, as described by Andrew.\r\n\r\nAn approach is to assume that the power radiated depends on $G$, $c$ and some time derivative of a mass multipole. It is straightforward to show that the first time-derivative of the monopole and dipole moments is zero because of conservation of mass, linear momentum and angular momentum. One is left then with using the quadrupole moment as the lowest viable order.\r\n\r\nSince power is a scalar and must be $\\geq 0$, then the time derivative of the quadrupole moment must be squared and thus we can say\r\n$$ P \\sim  G^a c^b \\left(\\frac{\\partial^c Q}{\\partial t^c}\\right)^2\\ .$$\r\n\r\nLetting $Q$ have dimension of $ML^2$, then dimensional analysis yields $a=1$, $b=-5$ and $c=3$.\r\n\r\nIn a binary, we have $Q \\sim Ma^2 \\sin 2\\omega t$ (twice the frequency of the binary because the quadrupole moment flips twice per orbital period). Differentiate this three times, square it:\r\n$$ \\dddot{Q}^2 \\sim M^2 a^4 \\omega^3$$\r\nand then note that $a^4 \\propto (GM)^{4/3}\\omega^{-8/3}$ from Kepler's third law and you get\r\n$$ P \\sim \\frac{G}{c^5}G^{4/3} M^{10/3}\\omega^{10/3} = \\frac{c^5}{G}\\left(\\frac{GM\\omega}{c^3}\\right)^{10/3}\\ .$$", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 21, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "dimensional-analysis", "gravitational-waves", "post-newtonian"], "page_start": null, "page_end": null, "quality_flags": ["low_score"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 703008, "title": "Gravitational wave radiation power from dimensional analysis", "url": "https://physics.stackexchange.com/questions/703008/gravitational-wave-radiation-power-from-dimensional-analysis", "share_url": "https://physics.stackexchange.com/q/703008", "content_license": null, "owner": {"display_name": "Ali Seraj", "user_id": 162488, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/162488/ali-seraj"}}, "answers": [{"answer_id": 703020, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/703008/gravitational-wave-radiation-power-from-dimensional-analysis/703020#703020", "share_url": "https://physics.stackexchange.com/a/703020", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Andrew", "user_id": 27732, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/27732/andrew"}}, {"answer_id": 703134, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/703008/gravitational-wave-radiation-power-from-dimensional-analysis/703134#703134", "share_url": "https://physics.stackexchange.com/a/703134", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:810965:0000", "text": "Question: How to relate a gravitational plane wave to the GW from a binary system?\n\nI have two different forms of gravitational waves that I am trying to reconcile.\r\n\r\n\r\n1. A monochromatic GW with angular frequency $\\Omega$ propagating in the $\\textbf{n}$-direction can be expressed as\r\n\r\n$$ h_{ij}(t,\\textbf{x}) = \\textbf{A}_{ij}(\\textbf{n}) \\cos \\left (\\Omega (t -\\textbf{n}\\cdot \\textbf{x}) \\right) \\tag{1}$$\r\n\r\nwhere $\\textbf{n}$ is a unit vector and $i,j$ label the spatial indices. The amplitude term can be separated into the usual $+, \\times$ polarisations:\r\n\r\n$$\\textbf{A}_{ij} = A^{+} \\textbf{e}_{ij}^{+} +  A^{\\times} \\textbf{e}_{ij}^{\\times}$$\r\n\r\nfor polarization tensors $\\textbf{e}_{ij}^{+, \\times} $ . For example see [this textbook][1]\r\n\r\n\r\n2. The GW from a circular binary BH system is,\r\n\r\n$$ h_{ij}(t, \\textbf{n}) = h^{+}(t) \\textbf{e}_{ij}^{+} + h^{\\times}(t) \\textbf{e}_{ij}^{\\times} \\tag{2}$$\r\n\r\nwith $h^{+}(t) = A_{\\rm gw} a(\\iota) \\cos \\Phi(t)$ and $h^{\\times}(t) = A_{\\rm gw} b(\\iota) \\sin \\Phi(t)$ and $\\iota$ is the source inclination, $\\Phi(t)$ the GW phase and $a(\\iota)$, $b(\\iota)$ some functions of the inclination. For example, see Equation 1 and Equation 25 in [this paper][2]\r\n\r\n---\r\n\r\nI cannot see how to get from Equation (1) to Equation (2). The literature for Equation (2) points me towards [this paper][3], but that is formulated in terms of changing mass quadrupoles of the binary system, rather than the plane wave form.\r\n\r\nAny help much appreciated. \r\n\r\n\r\n  [1]: https://academic.oup.com/book/41655\r\n  [2]: https://arxiv.org/abs/1003.0677\r\n  [3]: https://link.springer.com/article/10.1007/BF00759146\n\nAccepted Answer:\n\nThe first equation is the equivalent of a \"linearly polarised\" gravitational wave (GW). It consists of an arbitrary mixture of two polarisation states with no phase difference between them.\r\n\r\nThe GWs from a binary system are also a mixture of the two polarisation states, the relative amplitudes of which depend on the binary inclination. However, in this case, the polarisation components are 90 degrees out of phase and the GW could be said to be \"elliptically polarised\". $\\Phi(t)$ is the equivalent of $\\Omega t$ in the first equation and is roughly equal to $2\\omega t$, where $\\omega$ is the angular frequency of the binary system.\r\n\r\nTo go from the first equation to the second you just realise that the GWs from a binary consist of a superposition of two plane waves with the same frequency, but with orthogonal polarisations and 90 degrees out of phase.\r\n\r\nThe other trivial difference is that the second equation is not of a travelling wave. It only includes the $t$ dependence for the wave at some fixed spatial coordinate along the propagation direction.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 22, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravity", "orbital-motion", "gravitational-waves"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 810965, "title": "How to relate a gravitational plane wave to the GW from a binary system?", "url": "https://physics.stackexchange.com/questions/810965/how-to-relate-a-gravitational-plane-wave-to-the-gw-from-a-binary-system", "share_url": "https://physics.stackexchange.com/q/810965", "content_license": null, "owner": {"display_name": "user1887919", "user_id": 34470, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/34470/user1887919"}}, "answers": [{"answer_id": 810970, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/810965/how-to-relate-a-gravitational-plane-wave-to-the-gw-from-a-binary-system/810970#810970", "share_url": "https://physics.stackexchange.com/a/810970", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:637663:0000", "text": "Question: Could two planets trapped between two stars orbit each other?\n\nSay there are two stars, each exerting and equal force on a point in the midde of them. For simplicity, we'll say the two stars have the same mass and are equidistant from the point. Now suppose that there are two, equally massive planets, moving at a great speed around that point. Would the two planets be able to create a sustainable orbit between the two stars? Could such a situation ever naturally occur?\n\nAccepted Answer:\n\nWhat you are asking is essentially whether a double planet can have its barycenter at a Lagrange point of a binary star system. This is definitely true. As long as the pair can be well approximated as a point mass much smaller than the stellar distance and much lighter, it is a standard [restricted 3-body problem][1]. (There is a slight complication due to the minor perturbations from the star potentials maybe setting up some resonant wobble in the planets orbits, but this is a very weak and long-term issue.)\r\n\r\nThe more major issue is whether the planets are stable. The L1, L2 and L3 Lagrange points are unstable, so any deviation from them will make the planets eventually drift away and end up in some other orbit. L4 and L5 are neutral, and there is no strong tendency to drift away: here they could in principle stay forever. Even better, there are periodic [halo orbits][2] around the points that also allow the planets to remain. \r\n\r\nCould this ever happen? The universe is very big, so many rare things surely occurs somewhere. Small objects do get temporarily or permanently trapped around Lagrange points (trojan asteroids, for example) and planets may end up in a similar situation in a binary system - it is just that planets tend to be much rarer than asteroids, so there will be far fewer \"attempts\" for this. Similarly binary planets do not seem to be uncommon (we kind of live on one), so maybe a binary planet does end up in a L4 Halo orbit around a heavy binary here and there. It will be a rare coincidence, though.\r\n\r\n\r\n  [1]: https://jan.ucc.nau.edu/~ns46/student/2010/Frnka_2010.pdf\r\n  [2]: https://en.wikipedia.org/wiki/Halo_orbit\n\nAnswer (score=0):\n\nWould you count this? Imagine two copies of the [James Webb telescope in orbit][1]. As you can see, the Webb telescope travels in a small circle as it orbits the Sun. Two could orbit in opposite sides of the circle.\r\n\r\nIt isn't quite what you asked, but almost. They orbit outside the Earth's orbit.\r\n\r\nIt also isn't really right to say they orbit each other. They are held by the gravity of the Sun and Earth. Their mutual attraction is very weak and isn't what makes them travel around each other.  \r\n\r\nBut it fits the idea of what you asked. So yes, this is possible. \r\n\r\n---\r\nEdit\r\n\r\nThere is another possibility. It too isn't exactly what is asked for. But it requires equal size Suns, as Carl asks. \r\n\r\nYou start with two equal size Suns and equal size planets between them. But you make the planetary orbits so big that the Suns and planets form a square. This forms a [Klemperer rosette][2]. All 4 rotate at the same speed. It still isn't stable, but it is more stable than having planets in small orbits rotating with a different period from the Suns. \r\n\r\n  [1]: https://www.jwst.nasa.gov/content/about/orbit.html\r\n  [2]: https://en.wikipedia.org/wiki/Klemperer_rosette\n\nAnswer (score=0):\n\n[This wiki article][1] says , yes\r\n\r\n>In astronomy, a double planet (also binary planet) is a binary system where both objects are planets, or planetary-mass objects, that share an orbital axis external to both planetary bodies. \r\n\r\n....\r\n\r\n>There is debate as to what criteria should be used to distinguish \"double planet\" from a \"planet–moon system\". The following are considerations.\r\n\r\n>At its 2006 General Assembly, the International Astronomical Union considered a proposal that Pluto and Charon be reclassified as a double planet,[2] but the proposal was abandoned in favor of the current IAU definition of planet.\r\n\r\nAddition after comment\r\n\r\nSo there is no problem for two planets to revolve about each other and also their center of mass about a star.\r\n\r\nSo the doubt in your question is not about a binary system, but if there can be a stable orbit for a planet at the point of zero gravity, because the planet could be a binary planet as above. I guess there would be no orbit, but maybe a meta-stable point, where the planet ( or system) would hover. Meta-stable because it would be easy to be attracted to one or the other star from a small perturbation due to their extended masses.\r\n\r\n\r\n  [1]: https://en.wikipedia.org/wiki/Double_planet", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 23, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-mechanics", "newtonian-gravity", "orbital-motion", "three-body-problem"], "page_start": null, "page_end": null, "quality_flags": ["low_score"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 637663, "title": "Could two planets trapped between two stars orbit each other?", "url": "https://physics.stackexchange.com/questions/637663/could-two-planets-trapped-between-two-stars-orbit-each-other", "share_url": "https://physics.stackexchange.com/q/637663", "content_license": null, "owner": {"display_name": "TheDragonOfFlame", "user_id": 299923, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/299923/thedragonofflame"}}, "answers": [{"answer_id": 810141, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/637663/could-two-planets-trapped-between-two-stars-orbit-each-other/810141#810141", "share_url": "https://physics.stackexchange.com/a/810141", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Anders Sandberg", "user_id": 165299, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/165299/anders-sandberg"}}, {"answer_id": 637674, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/637663/could-two-planets-trapped-between-two-stars-orbit-each-other/637674#637674", "share_url": "https://physics.stackexchange.com/a/637674", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "mmesser314", "user_id": 37364, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/37364/mmesser314"}}, {"answer_id": 637698, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/637663/could-two-planets-trapped-between-two-stars-orbit-each-other/637698#637698", "share_url": "https://physics.stackexchange.com/a/637698", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "anna v", "user_id": 1492, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/1492/anna-v"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:345819:0000", "text": "Question: Do the stars of the Hulse-Taylor binary follow geodesics?\n\nThe Wikipedia definition of a geodesic is,\r\n\r\n> the worldline of a particle free from all external, non-gravitational forces\r\n\r\nNow for a binary system like Hulse-Taylor I have heard it described as a free-falling oscillating mass quadrupole and so the worldlines are perturbed from the geodesic. **How accurate is this?** It seems to me that if we have two stars that are just under the influence of gravity then their motion must follow geodesics?\n\nAccepted Answer:\n\nThe orbits are following geodesics, which are pretty much ellipses, just the geodesics can have minor perturbations that are still consistent with General Relativity. That is, the full orbits, incorporating all effects, would have some perturbations from elliptical, which makes it a geodesic of the total spacetime metric.\r\n\r\nStill, the slight perturbations to the orbits have not been measured. What has been measured is the effect of the rotation on the X Ray frequencies observed.  \r\n\r\nThe oscillations you are referring to are probably the quasi period oscillations (QPOs) of the X Ray frequencies that have been observed in many compact stars such as neutron stars and black holes. In the last few years it's been possible to determine the effects well enough to check that it is consistent with General Relativity. For tight binaries the effect is larger, but it's is generally due to the so called Lens Thirring effect. \r\n\r\nSee the article at http://sci.esa.int/xmm-newton/58072-gravitational-vortex-provides-new-way-to-study-matter-close-to-a-black-hole/\r\n\r\nThe QPOs are quasi periodic small changes in the frequency of the X rays. In the case reported in the article it was a black hole, and the frequency flickered, with variations in the frequency. The article that determined the effect was published in May 2016. See it at  https://academic.oup.com/mnras/article-abstract/461/2/1967/2608396/A-quasi-periodic-modulation-of-the-iron-line?redirectedFrom=fulltext\r\n\r\nThe Lens Thirring effect is due to frame dragging in a rotating spacetime. Near the neutron star or black hole the frames of reference rotate. That is part of what is known as one effect of rotation, and it is also try for instance in the Ker metric for a rotating spherical symmetric body. See the description of the year general effect at https://en.m.wikipedia.org/wiki/Lense–Thirring_precession", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 24, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "gravity", "geodesics"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 345819, "title": "Do the stars of the Hulse-Taylor binary follow geodesics?", "url": "https://physics.stackexchange.com/questions/345819/do-the-stars-of-the-hulse-taylor-binary-follow-geodesics", "share_url": "https://physics.stackexchange.com/q/345819", "content_license": null, "owner": {"display_name": "user1887919", "user_id": 34470, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/34470/user1887919"}}, "answers": [{"answer_id": 345881, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/345819/do-the-stars-of-the-hulse-taylor-binary-follow-geodesics/345881#345881", "share_url": "https://physics.stackexchange.com/a/345881", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Bob Bee", "user_id": 108333, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/108333/bob-bee"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:402051:0000", "text": "Question: What happens to the singularities of two black holes in the moment they merger?\n\nLet's assume the merger of a binary black hole and consider especially the moment of the transition from the last stable orbit to the merger, i.e. the transition where two black holes form one black hole and thus two singularities one. Here I‘m not sure if it makes any difference if we discuss mathematical black holes (i) or physical black holes (ii) (physical in the sense that the mass is not contained in a point and hence the break down of GR is avoided). While in (i) the singularities which contain the masses are a point in time instead in (ii) the masses are part of the manifolds (this is just my guess I can be wrong). Are in the latter case the two masses at two different „locations“ before and instantaneously(?) at one location after the merger? In general how would you describe and eventually distinguish these two cases?\n\nAccepted Answer:\n\n> in (ii) [physical black holes] the masses are part of the manifolds (this is just my guess I can be wrong)\r\n\r\nYep, you were wrong :-). In any black hole, we can't really localize the mass. The difference between a Schwarzschild black hole and an astrophysical one is that a Schwarzschild black hole has always existed. It didn't form by gravitational collapse.\r\n\r\n> While in (i) [a Schwarzschild black hole] the singularities which contain the masses are a point in time\r\n\r\nThe singularity is spacelike in both cases (which makes it similar to a spacelike surface, which is similar to a point in time).\r\n\r\n> In general how would you describe and eventually distinguish these two cases?\r\n\r\nBasically there is no interesting distinction between the two cases with respect to a black hole merger. An astrophysical black hole differs from a Schwarzschild black hole in the past, when they formed. The merger happens after they have already formed.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 25, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "event-horizon", "singularities"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 402051, "title": "What happens to the singularities of two black holes in the moment they merger?", "url": "https://physics.stackexchange.com/questions/402051/what-happens-to-the-singularities-of-two-black-holes-in-the-moment-they-merger", "share_url": "https://physics.stackexchange.com/q/402051", "content_license": null, "owner": {"display_name": "timm", "user_id": 192212, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/192212/timm"}}, "answers": [{"answer_id": 402116, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/402051/what-happens-to-the-singularities-of-two-black-holes-in-the-moment-they-merger/402116#402116", "share_url": "https://physics.stackexchange.com/a/402116", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "user4552", "user_id": null, "user_type": "does_not_exist", "profile_url": null}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:746100:0000", "text": "Question: Closest possible orbital radius for equal masses\n\nIf you have two objects of equal mass, then what’s the closest distance that they can orbit at in terms of their schwarzschild radii? How fast would they be orbiting?\r\nWhat About stable orbits?\n\nAccepted Answer:\n\nIf you have two objects with non-negligible mass there are no stable orbits due to loss of energy to gravitational radiation.\r\n\r\nWhen one of the two mass is much smaller than the other, there may still be an approximate sense in which you talk about orbits and stable orbits. However, in the equal mass limit, the effect of dissipation is so strong in the strong field regime, that there is no sensible way to disentangle the conservative dynamics from the dissipative dynamics. All you have is inspirals.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 26, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "orbital-motion", "gravitational-waves"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 746100, "title": "Closest possible orbital radius for equal masses", "url": "https://physics.stackexchange.com/questions/746100/closest-possible-orbital-radius-for-equal-masses", "share_url": "https://physics.stackexchange.com/q/746100", "content_license": null, "owner": {"display_name": "blademan9999", "user_id": 263465, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/263465/blademan9999"}}, "answers": [{"answer_id": 746127, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/746100/closest-possible-orbital-radius-for-equal-masses/746127#746127", "share_url": "https://physics.stackexchange.com/a/746127", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "TimRias", "user_id": 101892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/101892/timrias"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:827211:0000", "text": "Question: Physical plausibility of space station between binary black holes\n\nIn science fiction you'll sometimes see a space station (or ancient alien relic, or something like that) dramatically suspended between two black holes (probably stellar-mass, but of course the masses aren't specifically addressed), which are generally separated by only a few times their radius. Obviously with Newtonian gravity there'd be a metastable point there, where something could remain with some amount of active station-keeping (assuming it can survive the tidal forces, radiation, etc.), but Newtonian gravity surely isn't a great approximation under the circumstances. Taking into account GR, would there still exist a metastable point under such circumstances where something could be kept for a long time (in general, or at least in a significant variety of cases)?\r\n\r\nI can think of two differences between Newtonian gravity and GR that seem like they might be problematic, but I don't know enough to quantify their effect:\r\n\r\n 1. Eventually \"between the black holes\" will cease to be a place since the black holes will spiral in and merge due to gravitational waves. I have been unable to find out any quantification of how long it takes black holes to go from several-radii separation to merging, so I'm not sure whether it's too short for it to make sense to build a space station there.\r\n\r\n 1. Realistically the black holes will be spinning, so there will be frame-dragging effects. I'm not sure if this will shift where the metastable point is or eliminate them entirely, and I'm not sure whether the answer to that depends on how close the black hole's rotational axes are to being aligned with each other and with the axis of the orbit.\n\nAccepted Answer:\n\nYou correctly identified that one of the problems with this scenario is that a configuration of two black holes separated by just a few Schwarzschild radii is not going to be very long lived.\r\n\r\nFor example, [GW151226][1] (the second observed black hole merger) was separated by about 17 Schwarzschild radii when it entered the LIGO sensitivity band. It merged 1 second latter after just 55 orbits. \r\n\r\nThis would be a bit longer for heavier black holes, however even for the heaviest supermassive black holes with billions of solar masses we are talking only hundreds to thousands of years for a similar configuration.\r\n\r\n\r\n\r\n  [1]: https://dcc.ligo.org/public/0126/G1601226/004/GW151226-FactSheet.pdf", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 27, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 827211, "title": "Physical plausibility of space station between binary black holes", "url": "https://physics.stackexchange.com/questions/827211/physical-plausibility-of-space-station-between-binary-black-holes", "share_url": "https://physics.stackexchange.com/q/827211", "content_license": null, "owner": {"display_name": "Alex", "user_id": 427861, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/427861/alex"}}, "answers": [{"answer_id": 827295, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/827211/physical-plausibility-of-space-station-between-binary-black-holes/827295#827295", "share_url": "https://physics.stackexchange.com/a/827295", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "TimRias", "user_id": 101892, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/101892/timrias"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:537580:0000", "text": "Question: Derive hamiltonian from equations of motion\n\nIs there a method for deriving the hamiltonian given that you know the equations of motion?\r\nFor example given the equation (equation 5 in paper linked) they simply the derive the Hamiltonian in equations 6-8. What is the method behind this?\r\nPaper link: https://arxiv.org/abs/1902.01344\n\nAccepted Answer:\n\nif you have this kind of differential equations:\r\n\r\n$$\\vec{\\ddot{r}}=-\\vec{F}(\\vec{r})\\tag 1$$\r\n\r\nyou can get the Hamiltonian.\r\n\r\nmultiply equation (1) from the left with $\\vec{\\dot{r}}$\r\n\r\n $$\\vec{\\dot{r}}\\cdot \\vec{\\ddot{r}}=-\\vec{\\dot{r}}\\cdot\\vec{F}(\\vec{r})$$\r\n\r\nthus:\r\n$$\\frac{1}{2}\\frac{d}{dt}(\\vec{\\dot{r}}\\cdot \\vec{\\dot{r}})=\r\n-\\vec{\\dot{r}}\\cdot\\vec{F}(\\vec{r})$$\r\nor\r\n$$\\frac{1}{2}\\int d(\\vec{\\dot{r}}\\cdot \\vec{\\dot{r}})=\r\n-\\int\\vec{F}(\\vec{r})\\cdot d\\vec{r}$$\r\n$\\Rightarrow$\r\n$$\\underbrace{\\frac{1}{2}\\vec{\\dot{r}}\\cdot \\vec{\\dot{r}}}_{T}=\\underbrace{-\\int\\vec{F}(\\vec{r})\\cdot d\\vec{r}}_{U}$$\r\n\r\nwith the Lagrangian $L=T-U$ you can obtain the Hamiltonian  \r\n\r\n**Example:**\r\n\r\n$$\\ddot{r}=\\underbrace{-\\frac{M}{r^2}}_{F(r)}$$\r\n\r\n$\\Rightarrow$\r\n\r\n$$T=\\frac{1}{2}\\dot{r}^2\\quad,U=-\\frac{M}{r}$$\r\n\r\nwith $L=T-U$ you get the Hamiltonian\r\n\r\n$$H=\\frac{1}{2}\\,p^2-\\frac{M}{r}=T+U$$\r\n\r\nwhere $p=\\frac{dL}{d(\\dot{r})}=\\dot{r}$", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 28, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "lagrangian-formalism", "orbital-motion", "hamiltonian-formalism", "hamiltonian"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 537580, "title": "Derive hamiltonian from equations of motion", "url": "https://physics.stackexchange.com/questions/537580/derive-hamiltonian-from-equations-of-motion", "share_url": "https://physics.stackexchange.com/q/537580", "content_license": null, "owner": {"display_name": "Warrenmovic ", "user_id": 255246, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/255246/warrenmovic"}}, "answers": [{"answer_id": 537659, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/537580/derive-hamiltonian-from-equations-of-motion/537659#537659", "share_url": "https://physics.stackexchange.com/a/537659", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Eli", "user_id": 196140, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/196140/eli"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:452988:0000", "text": "Question: On Planets orbiting binary stars\n\nSeveral years ago a discovery was made of planet orbiting a star of a binary system (two stars orbiting each other).  Since binary star systems are plentiful in our galaxy, I presume we will be discovering even more such planets.\r\n\r\nHowever, as far as I know, no planet has been discovered that orbits both stars of a binary star system.  Question, is this feasible and likely that a planet would orbit both stars of a binary system?\n\nAccepted Answer:\n\nA planet in such an orbit is called a [**circumbinary planet**](https://en.wikipedia.org/wiki/Circumbinary_planet). Since planetary systems originate from a rotating disk of matter, and since binary stars may also originate that way, the possibility of ending up with two stars and one or more planets all orbiting in a common plane seems intuitively plausible, and some candidates have been reported. The paper [1] says:\r\n\r\n> Following the first detection of a circumbinary planet with the Kepler space telescope, namely Kepler-16b, eight more binary star systems with a planet on a P-type orbit have been discovered. All these systems show striking similarities. They are all very flat, meaning that the binary and the planet orbit are in the same plane, suggesting that these planets formed in a circumbinary disc aligned with the orbital plane of the central binary. Furthermore, in all systems, the innermost planet (so far only Kepler-47 is known to have more than one planet) is close to the calculated stability limit...\r\n\r\nAnother theoretical analysis of instabilities in the orbits of circumbinary planets is presented in [2], whcih says:\r\n\r\n> Consider a planar three-body system of gravitating bodies: a central massive binary and a much less massive particle orbiting around the binary. Thus,\r\nthe particle’s orbit is circumbinary. ... the circumbinary orbits cannot be permanently circular. This phenomenon provides a natural universal mechanism of internal tidal friction and heating in circumbinary planets (CBP)... In this article, we describe and consider the planetary escape process that takes place as a result of this shrinkage. Indeed, a particle in the slowly shrinking circumbinary orbit enters eventually the chaotic zone around the central binary, and therefore escapes. Once in the zone, the particle escapes inevitably... We show that the effect of tidal decay may explain, at least partially, the observed lack of CBP of close-enough (with periods <$ 5$ days) stellar binaries.\r\n\r\nA recent search for circumbinary planets is reported in [3], which says:\r\n\r\n> We present the full survey results of the Search for Planets Orbiting Two Stars (SPOTS) survey, which is the first direct imaging survey targeting CBPs. The SPOTS observational program comprises 62 tight binaries that are young and nearby, and thus suitable for direct imaging studies... Results from SPOTS include the resolved circumbinary disk around AK Sco, the discovery of a low-mass stellar companion in a triple packed system, the relative astrometry of up to 9 resolved binaries, and possible indications of non-background planetary-mass candidates around HIP 77911. We did not find any CBP within 300 AU...\r\n\r\nAn older report [4] says:\r\n\r\n> Ranked near the top of the long list of exciting discoveries made with NASA's Kepler photometer is the detection of transiting circumbinary planets. In just over a year the number of such planets went from zero to seven, including a multi-planet system with one of the planets in the habitable zone (Kepler-47).\r\n\r\nSome other recent references are cited in the the introduction of [5], which says this:\r\n\r\n> One of the exotic type of planetary systems detected by the Kepler mission are transiting circumbinary planets (CBPs), i.e., planets in nearly-coplanar orbits around a stellar binary (and nearly coplanar with the plane of the sky), temporarily blocking the binary’s light and giving a generally complex light curve [references]. Currently, 10 confirmed Kepler CBPs are known in 9 binary systems (see Table 1 for an overview [with references]). \r\n\r\n---\r\n\r\nReferences:\r\n\r\n[1] \"Migration of planets in circumbinary discs,\" https://arxiv.org/abs/1806.00314\r\n\r\n[2] \"Tidal decay of circumbinary planetary systems,\" https://arxiv.org/abs/1808.02090\r\n\r\n[3] \"SPOTS: The Search for Planets Orbiting Two Stars. III. Complete Sample and Statistical Analysis,\" https://arxiv.org/abs/1807.08687\r\n\r\n[4] \"Recent Kepler Results On Circumbinary Planets,\" https://arxiv.org/abs/1308.6328\r\n\r\n[5] \"Stability of exomoons around the Kepler transiting circumbinary planets,\" https://arxiv.org/abs/1806.06075\n\nAnswer (score=0):\n\n1\r\n\r\nGenerally speaking, binaries cannot have shared planets orbiting at close distances. Neither dynamics nor kinematics holds.\r\n\r\nIf a binary has a common planet, the planet must be far enough, that is, the radius of the circle is large enough to be far greater than the distance between the two stars. This approximates two stars as one star at the center of mass.\r\n\r\nThe acceleration of the winding provides centripetal force, so the system can be stable. It's impossible to circle two stars at close range.\r\n\r\nEach star of the binary can have its own planet, in this case, the planet need to close to its own star and far away to other star.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 29, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "orbital-motion", "celestial-mechanics", "stability", "exoplanets"], "page_start": null, "page_end": null, "quality_flags": ["low_score"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 452988, "title": "On Planets orbiting binary stars", "url": "https://physics.stackexchange.com/questions/452988/on-planets-orbiting-binary-stars", "share_url": "https://physics.stackexchange.com/q/452988", "content_license": null, "owner": {"display_name": "K7PEH", "user_id": 59161, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/59161/k7peh"}}, "answers": [{"answer_id": 453005, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/452988/on-planets-orbiting-binary-stars/453005#453005", "share_url": "https://physics.stackexchange.com/a/453005", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Chiral Anomaly", "user_id": 206691, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/206691/chiral-anomaly"}}, {"answer_id": 490764, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/452988/on-planets-orbiting-binary-stars/490764#490764", "share_url": "https://physics.stackexchange.com/a/490764", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Cang Ye", "user_id": 227290, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/227290/cang-ye"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:448522:0000", "text": "Question: How do compute an energy momentum tensor, given some equations of motion\n\nThis problem can be found in a paper called \"_Gravitational Radiation From Point Masses In A Keplerian Orbit_\", but I do not have access to this, so cannot see how to do it.\n\n_____\n\n>I have been given a problem where we have two (point-like, non-relativistic) masses $m_1,m_2$, which are in some kind of binary orbit. They each trace out an ellipse (of different sizes), and we are given two formulae / equations of motion:\n>\n> - One for the distance between the two masses, $r(t)$, for all time (given in terms of the eccentricity)\n>\n> - One for the angular velocity $\\dot\\phi(t)$ in terms of $r(t)$\n>\n>Then I am asked to compute the energy momentum tensor for this system.\n\n___\nI don't know exactly where to start. I had some ideas - it would be helpful if I knew a particular action to apply a variation to to obtain the energy-momentum tensor, but I don't know which action to use. I don't know what form of metric to use either (if I did, I could perhaps compute the Ricci tensor and Ricci scalar, then use Einstein's equations to find $T_{\\mu\\nu}$). But again, I don't have this starting point. \n\nAny ideas or hints?\n\nAccepted Answer:\n\nLet us consider the energy-momentum tensor $T^{\\mu \\nu}$ of a perfect fluid. Then we specialize to the configuration in question.  \r\n$T^{\\mu \\nu} = (\\rho + p) U^\\mu U^\\nu + p g^{\\mu \\nu}$  \r\nwhere:  \r\n$c = G = 1$ natural units  \r\n$\\rho$ energy density in the rest frame  \r\n$p$ pressure in the rest frame  \r\n$U^\\mu$ four-velocity  \r\n$g^{\\mu \\nu}$ inverse metric tensor  \r\n\r\nA nonrelativistic system means:  \r\n1) Energy density close to mass density in the rest frame  \r\n2) Pressure negligible. In fact $p = (1/3) v^2 \\rho$ where the Newtonian velocity $v$ is negligible compared to $1$ (speed of light in natural units).  \r\n3) $U^\\mu$ close to $(1, \\vec v)$. Again the spatial part is negligible compared to the time component.  \r\n\r\nHence the energy-momentum tensor of a binary system with point-like nonrelativistic masses $m_1$ and $m_2$ can be approximated in a polar coordinates system $(t, r, \\theta, z)$ centered in one of the focal points as  \r\n$T^{t t} = m_1 (1/r) \\delta(r - r_1) \\delta (\\theta - \\theta_1) \\delta (z) + m_2 (1/r) \\delta(r - r_2) \\delta (\\theta - \\theta_2) \\delta (z)$  \r\nOther components negligible  \r\nwhere:  \r\n$\\delta$ Dirac delta function  \r\nThe factor $(1/r)$ allows for the unity when integrating the $\\delta$ function in polar coordinates  \r\n$r_1, \\theta_1, r_2, \\theta_2$ as functions of time $t$ are the laws of motion of the masses and are given  \r\n\r\nNote: The delta functions allow for a straightforward integration to get the quadrupole moment which in turn allows for the gravitational perturbation.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 30, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "orbital-motion", "stress-energy-momentum-tensor"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 448522, "title": "How do compute an energy momentum tensor, given some equations of motion", "url": "https://physics.stackexchange.com/questions/448522/how-do-compute-an-energy-momentum-tensor-given-some-equations-of-motion", "share_url": "https://physics.stackexchange.com/q/448522", "content_license": null, "owner": {"display_name": "John Doe", "user_id": 154004, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/154004/john-doe"}}, "answers": [{"answer_id": 448709, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/448522/how-do-compute-an-energy-momentum-tensor-given-some-equations-of-motion/448709#448709", "share_url": "https://physics.stackexchange.com/a/448709", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Michele Grosso", "user_id": 179105, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/179105/michele-grosso"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:675694:0000", "text": "Question: Dissipation caused by Gravitational Wave Emission\n\nTwo massive bodies orbiting each other can lose energy through gravitational wave emission until colliding.\r\n\r\nCan a single massive body, moving with constant velocity with respect to an observer, lose it's kinetic energy to gravitational waves? I get that changing the position like this without oscillating doesn't result in a wave exactly, but mustn't the new information about the gravitational field still be transmitted, resulting in energy loss?\n\nAccepted Answer:\n\nThe answer and reasoning for this are exactly the same as in electromagnetism. (Assuming the observer is far enough away and has light enough mass that we can ignore the effect the observer has on spacetime for the massive body).\r\n\r\nSince the massive body is traveling at a constant velocity (below $c$), we can go into its rest frame. Since it obviously isn't emitting waves in its rest frame, and since the laws of physics are the same in any inertial frame, it is also not emitting waves in the observer's rest frame.\r\n\r\nAt this point I am going to transition to talking about electromagnetism, because the correct words are probably more familiar. But, the logic here can also be transferred to gravity (to leading order in perturbation theory). The electric field due to the \"static\" part of the field (not the wave) falls off as $1/r^2$, where $r$ is the distance to the \"charged body\" (assuming, since we've moved to electromagnetism, that the body has a net charge). The electric field due to an electromagnetic wave sourced by an accelerating charged body falls off as $1/r$. If the charged body is moving at a constant velocity $v$, the information about the changing $1/r^2$ part of the field, does indeed travel at $v$. You can imagine that in some sense, the entire $1/r^2$ field configuration is moving at a constant velocity. If the charge accelerates, *this* information needs to propagate to the observer somehow, which it does in the form of a wave.\r\n\r\n[![Figure from Purcell's textbook showing the electromagnetic wave communicating the change of a particle's velocity to ][1]][1] \r\n\r\nThe above image, from the [textbook by Purcell and Morin](https://www.amazon.com/Electricity-Magnetism-Edward-M-Purcell/dp/1107014026) (which I grabbed from [this website](https://physics.weber.edu/schroeder/mrr/mrrtalk.html))\r\nshows how an electromagnetic wave communicating the change of a particle's velocity to a distant observer. Far away (far enough that light has not had time to propagate since the particle accelerated), the field lines move at a constant velocity, and point to where the particle *would be* if it hadn't accelerated.  Nearby, the field lines point to the current location of the particle. In between the near and far regions, there is a visible shell, which is the wave (remember that the electric field points perpendicular to the direction of propagation of a wave). The wave communicates to the distant observer, that the point charge has accelerated.\r\n\r\nIn gravity, at asymptotically large distances away from the massive body, the spacetime curvature falls off as $1/r$ for gravitational waves emitted from the body (if its quadrupole moment changes with time), and $1/r^2$ for a body that is static or moving with a constant velocity. We can make the same conceptual split; gravitational waves carry information that tell a distant observer when the quadrupole moment of the mass distribution of the source changes, much like electromagnetic waves carry information that the dipole moment of the charge distribution changes. A massive body moving at a constant velocity does not lead to gravitational waves. This description and connection to electromagnetism relies on perturbation theory; there are also more rigorous descriptions in terms of quantities describing the asymptotic behavior of the spacetime like the Bondi news, but this does not change the answer to your question.\r\n\r\n\r\n  [1]: https://i.sstatic.net/qKfsy.png", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 31, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "energy-conservation", "gravitational-waves", "dissipation"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 675694, "title": "Dissipation caused by Gravitational Wave Emission", "url": "https://physics.stackexchange.com/questions/675694/dissipation-caused-by-gravitational-wave-emission", "share_url": "https://physics.stackexchange.com/q/675694", "content_license": null, "owner": {"display_name": "rel-s", "user_id": 22024, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/22024/rel-s"}}, "answers": [{"answer_id": 675699, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/675694/dissipation-caused-by-gravitational-wave-emission/675699#675699", "share_url": "https://physics.stackexchange.com/a/675699", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Andrew", "user_id": 27732, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/27732/andrew"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:369397:0000", "text": "Question: Kerr BH effects on inspiral binaries\n\nIn the gravitational wave calculation for binary systems: what is the effect of rotation of two BH (or neutron stars, BH-NS,...) on the usual calculations? Is there any EXACT result known?\n\nAccepted Answer:\n\nThere are two questions; I will first answer the simpler one `is there an exact solution to two-body problem in GR`:\r\n\r\n* Currently, [there is no exact solution to the two-body problem in GR][1], as pointed out by Qmechanic.\r\n\r\nThe second question is `what is the effect of rotation of two BH (or neutron stars, BH-NS,...) on the usual calculations?`: \r\n\r\nThis is a bit more complicated. \r\n\r\nFirst, let me point out that there are approximate solutions to binary systems in the limit of roughly equal mass binaries up to the innermost stable circular orbit using [Post-Newtonian (PN) Expansion][2]. The [approximation breaks down after the innermost stable circular orbit][3] because the system becomes highly non-linear and the binaries move at relativistic velocities (PN expansion is in v/c, where v is the effective velocity of the binary). The merger part itself is computed using numerical relativity or semi-analytical models based on numerical relativity.\r\n\r\nThe spin-orbit coupling has contributions at higher post-Newtonian terms. What do I mean by higher post-Newtonian terms? Each post-Newtonian term is computed by solving the Einstein Field Equations (EFE) recursively to next order (0th Order is Newtonian, plugging this into EFE will yield 0.5 PN correction terms, which you can recursively insert back to EFE -- solving the equations up to 4 PN order takes a few years and so we only have PN expansion up to 4 PN for binaries of equal masses). The effect itself is somewhat known from Newtonian physics already, which is that the spin wobbles around the orbital angular momentum vector (see e.g. [this simulation for demonstration][4]).\r\n\r\nThis shows up as small wiggliness in the gravitational wave strain. See the following figure of the absolute value of plus-polarized gravitational wave strain from (5, 5) solar mass binary without spin (red) and with near extremal spin perpendicular to the orbital angular momentum (black) as a function of the gravitational wave frequency: [![enter image description here][5]][5]\r\n\r\nIn addition to showing up in the gravitational wave strain amplitude, it shows up in the phase of the gravitational wave. I'll leave solving the specifics of that as homework, since I don't think explaining it will be very informative. I'll just say that the phase of the gravitational wave is directly related to the phase of the orbit.\r\n\r\nIf you want to learn more about the math, I would suggest taking a look at [this reference][2]. It takes a bit of effort to read.\r\n\r\nHope that helps!\r\n\r\n  [1]: https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity\r\n  [2]: https://link.springer.com/article/10.12942/lrr-2014-2\r\n  [3]: https://arxiv.org/abs/1102.5192\r\n  [4]: https://www.youtube.com/watch?v=9s1VoTy_U3s\r\n  [5]: https://i.sstatic.net/HB8Sg.png\n\nAnswer (score=0):\n\nNo, there is no exact solution for the [2-body problem in GR](https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity); only approximative & numerical solutions.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 32, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-3.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "metric-tensor", "gravitational-waves", "kerr-metric"], "page_start": null, "page_end": null, "quality_flags": ["low_score"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 369397, "title": "Kerr BH effects on inspiral binaries", "url": "https://physics.stackexchange.com/questions/369397/kerr-bh-effects-on-inspiral-binaries", "share_url": "https://physics.stackexchange.com/q/369397", "content_license": null, "owner": {"display_name": "riemannium", "user_id": 22916, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/22916/riemannium"}}, "answers": [{"answer_id": 369483, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/369397/kerr-bh-effects-on-inspiral-binaries/369483#369483", "share_url": "https://physics.stackexchange.com/a/369483", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "OTH", "user_id": 83359, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/83359/oth"}}, {"answer_id": 369407, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/369397/kerr-bh-effects-on-inspiral-binaries/369407#369407", "share_url": "https://physics.stackexchange.com/a/369407", "content_license": "CC BY-SA 3.0", "owner": {"display_name": "Qmechanic", "user_id": 2451, "user_type": "moderator", "profile_url": "https://physics.stackexchange.com/users/2451/qmechanic"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:600893:0000", "text": "Question: Solving for the mass of the larger star in a binary star system. (Larger star is much larger, so orbit is essentially circular)\n\nI'm a teaching assistant for a class on Newtonian mechanics, and was confronted with a way of solving for the larger mass in this two-star system that gets the right symbolic solution, but seems to forget the factor of $\\frac{1}{2}$ for kinetic energy. \r\n\r\nThe problem gives a distance between the two stars **r**, a velocity **v**, and states that the larger has such a large mass that the orbit of the smaller star is nearly circular. \r\n\r\nHere's their solution:\r\n\r\n$$U_g = E_k$$\r\n\r\nGravitational and kinetic energy must be equal if the star is in orbit. \r\n\r\n$$U_g = -G \\frac{m_1m_2}{r}$$\r\n\r\n$$E_k = mv^2$$\r\n\r\nThis is **missing its factor of** $\\frac{1}{2}$ but we ignore this and set the two energies equal to each other and solve. \r\n\r\n$$m_2v^2 = -G \\frac{m_1m_2}{r}$$\r\n\r\nWe will ignore the minus sign on gravitational potential energy, and  $m_2$ cancels out.\r\n\r\n$$v^2 = G \\frac{m_1}{r}$$\r\n\r\nWe solve for $m_1$ and get:\r\n\r\n$$\\frac{v^2r}{G} = m_1$$\r\n\r\nWhich happens to be the same symbolic solution that is in the answer key. \r\n\r\n**How is this correct when kinetic energy is equal to** $\\frac{1}{2} mv^2$ ?\n\nAccepted Answer:\n\n> Gravitational and kinetic energy must be equal if the star is in orbit.\r\n\r\nNo it need not be.  Forces must balance, and total energy must *constant* but not zero.\r\n\r\nSo why do you get the result you do ?  This comes down to the [viral theorem][1].  As the Wikipedia page shows, the kinetic energy (in a time averaged gravitational system) is equal to half the potential energy.  As a circular orbit means we do not need to worry about time averaging we can just use :\r\n\r\n$$E_{KE}=-\\frac 1 2 U$$\r\n\r\nI think you can probably already see where that factor of $2$ you are worried about is coming from.\r\n\r\nFor a circular orbit we balance forces :\r\n\r\n$$m\\frac {v^2} r = \\frac {GMm} {r^2}$$\r\n\r\nAnd we get :\r\n\r\n$$v^2=\\frac {GM} r$$\r\n\r\nSo KE is given by :\r\n\r\n$$E_{KE}=\\frac 1 2 m v^2 = \\frac 1 2 m \\frac {GM}r=-\\frac 1 2 U$$\r\n\r\nSo you can see that the viral theorem does indeed work here.\r\n\r\nThe other misconception (in the original question) is :\r\n\r\n> the larger has such a large mass that the orbit of the smaller star is nearly circular\r\n\r\nA gravitational closed orbit of any two objects (in Newtonian mechanics) can be any ellipse (which of course includes circles).  We know of thousands of examples of asteroids and comets in our own solar system that have extremely elliptical orbits and clearly their masses are much, much smaller than the Sun's.\r\n\r\n\r\n\r\n\r\n  [1]: https://en.wikipedia.org/wiki/Virial_theorem\n\nAnswer (score=1):\n\nThe first equation means that the system is at the dissociation limit as the total energy is zero. For the circular orbit that you probably want your students to study instead the virial theorem holds.\n\nAnswer (score=1):\n\nYour first equation is inaccurate. In a circular orbit, the kinetic energy is *half* the magnitude of the gravitational potential energy.\r\n\r\nAssuming the secondary mass is small, Newton's second law can be written\r\n$$ m \\frac{v^2}{r} = \\frac{GMm}{r^2}$$\r\nThus\r\n$$\\frac{1}{2} mv^2 = \\frac{1}{2}\\left(\\frac{GMm}{r}\\right)$$", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 33, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-gravity", "orbital-motion"], "page_start": null, "page_end": null, "quality_flags": ["low_score"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 600893, "title": "Solving for the mass of the larger star in a binary star system. (Larger star is much larger, so orbit is essentially circular)", "url": "https://physics.stackexchange.com/questions/600893/solving-for-the-mass-of-the-larger-star-in-a-binary-star-system-larger-star-is", "share_url": "https://physics.stackexchange.com/q/600893", "content_license": null, "owner": {"display_name": "Aaron Redd", "user_id": 282736, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/282736/aaron-redd"}}, "answers": [{"answer_id": 600910, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/600893/solving-for-the-mass-of-the-larger-star-in-a-binary-star-system-larger-star-is/600910#600910", "share_url": "https://physics.stackexchange.com/a/600910", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "StephenG - Help Ukraine", "user_id": 104964, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/104964/stepheng-help-ukraine"}}, {"answer_id": 600897, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/600893/solving-for-the-mass-of-the-larger-star-in-a-binary-star-system-larger-star-is/600897#600897", "share_url": "https://physics.stackexchange.com/a/600897", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "my2cts", "user_id": 186017, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/186017/my2cts"}}, {"answer_id": 600951, "is_accepted": false, "url": "https://physics.stackexchange.com/questions/600893/solving-for-the-mass-of-the-larger-star-in-a-binary-star-system-larger-star-is/600951#600951", "share_url": "https://physics.stackexchange.com/a/600951", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "ProfRob", "user_id": 43351, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/43351/profrob"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:852768:0000", "text": "Question: Do curvatures of gravitational fields add up?\n\nAs far as I’m aware, in Newton’s gravity only gravitational potential energy can add up. Forces can add up, but they can also cancel out each other.\r\n\r\nIn the framework of GR, do curvatures of spacetime of two (or more) bodies add up to more curvature?\r\nWould light curve around the center of two bodies orbiting around each other, producing Einstein ring? Or around two body system as a whole?\n\nAccepted Answer:\n\nIn general relativity, gravity is not a force but a result of the worldlines of objects being curved towards one another due to the mass-energy of the objects curving spacetime itself. This is described by Einstein's equation, which is given by (in natural units where $G=c=1$)\r\n$$\r\nR_{\\mu\\nu}-\\frac{R}{2}g_{\\mu\\nu} +\\Lambda g_{\\mu\\nu} = 8\\pi T_{\\mu\\nu}\r\n$$\r\nWhere on the left hand side, $R_{\\mu\\nu}$ is the Ricci tensor obtained from contracting the first and third indices of the full Riemann curvature tensor, which is itself given by a complicated nonlinear expression involving second derivatives of the metric $g_{\\mu\\nu}$. $R$ is the trace of the Ricci tensor, and $\\Lambda$ is a cosmological constant that is only included for completeness. This side describes the curvature of spacetime. \r\n\r\nOn the right hand side, we have the energy-momentum tensor which describes distribution of mass-energy in the spacetime. Since the expressions on the left are nonlinear, one cannot just add two energy-momentum tensors to get the metric for the combined system (Indeed, the full two-body problem in general relativity has not been solved analytically!). But one can write the energy-momentum tensor for a system of $N$ bodies, and at least solve numerically for the metric. In this sense, the fields do \"add up\". \r\n\r\nHowever, they can never add up in a way that creates a stronger curvature _away_ from the sources, i.e. concentrations of mass-energy.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 34, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "mass", "event-horizon"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 852768, "title": "Do curvatures of gravitational fields add up?", "url": "https://physics.stackexchange.com/questions/852768/do-curvatures-of-gravitational-fields-add-up", "share_url": "https://physics.stackexchange.com/q/852768", "content_license": null, "owner": {"display_name": "Kyrylo Lyskov", "user_id": 392601, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/392601/kyrylo-lyskov"}}, "answers": [{"answer_id": 852855, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/852768/do-curvatures-of-gravitational-fields-add-up/852855#852855", "share_url": "https://physics.stackexchange.com/a/852855", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "paulina", "user_id": 398159, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/398159/paulina"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:731185:0000", "text": "Question: Does the Centre of Mass of a binary star system accelerate?\n\nAt the COM of a binary star system (where both stars has a different mass), the net force at the centre of mass is non-zero.\r\n\r\nDoes this mean that the COM will be accelerating? And if so, wouldn't the velocity of the binary star system keep on increasing? \r\n\r\nHowever, it doesn't make sense if the velocity keeps increasing. So what is the error here?\n\nAccepted Answer:\n\nIt is not entirely clear what you mean by \"the net force at the centre of mass,\" since forces act on objects, and there need not be an object at the centre of mass. Presumably, you mean to say that the gravitational field is non-zero at the centre of mass; i.e. if there were on object at the centre of mass, the net force on it would be non-zero. This does not mean that the centre of mass will accelerate, but rather that an object at the centre of mass will accelerate.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 35, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "newtonian-mechanics", "newtonian-gravity", "orbital-motion", "inertial-frames"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 731185, "title": "Does the Centre of Mass of a binary star system accelerate?", "url": "https://physics.stackexchange.com/questions/731185/does-the-centre-of-mass-of-a-binary-star-system-accelerate", "share_url": "https://physics.stackexchange.com/q/731185", "content_license": null, "owner": {"display_name": "john", "user_id": 300818, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/300818/john"}}, "answers": [{"answer_id": 731188, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/731185/does-the-centre-of-mass-of-a-binary-star-system-accelerate/731188#731188", "share_url": "https://physics.stackexchange.com/a/731188", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Sandejo", "user_id": 195139, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/195139/sandejo"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:728908:0000", "text": "Question: Can orbiting point masses precess?\n\nIn a recent Science News, it mentioned two orbiting black holes (which later combined) were precessing; i.e. their motion did not remain in a stable plane, but rather the plane itself was changing.  Which seemed odd to me. \r\n\r\nAdmittedly I need to read up on precession. And maybe in the case of black holes there were other causes (gravity waves? electro-magnetic fields?)  But my first thought is: maybe the *size* of the objects matter. So could precession happen with point masses?\n\nAccepted Answer:\n\nBlack holes are usually rotating very quickly. If their two rotational angular momentum vectors are not parallel and perpendicular to the orbital angular momentum vector, then the system should be able to exchange orbital and rotational angular momentum in such a way that the orbital plane itself tilts, even though the total angular momentum has to be conserved (except for the radiation of gravitational waves, as you mentioned). This isn't just restricted to black hole motion, of course. It is a perfectly \"classical\" effect.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 36, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "orbital-motion", "precession"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 728908, "title": "Can orbiting point masses precess?", "url": "https://physics.stackexchange.com/questions/728908/can-orbiting-point-masses-precess", "share_url": "https://physics.stackexchange.com/q/728908", "content_license": null, "owner": {"display_name": "Daniel", "user_id": 274860, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/274860/daniel"}}, "answers": [{"answer_id": 728931, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/728908/can-orbiting-point-masses-precess/728931#728931", "share_url": "https://physics.stackexchange.com/a/728931", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "FlatterMann", "user_id": 346466, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/346466/flattermann"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:618158:0000", "text": "Question: Binary stars orbiting around each other are equidistant\n\nThis is a problem that was given to me in my Classical Mechanics course:\r\n\r\n> J.S.Plaskett's star is one of the most massive stars known at present. It is a binary star, that is, it consists of two stars bound together by gravity. From spectroscopic studies it is known that the period of revolution of each component is $14.4$ days and that the velocity of each component is about $220$ km/s. The orbit is nearly circular.\r\n\r\n> (a) Argue that the masses of two stars are nearly equal and that they are nearly equidistant from the centre of mass of the system, and (b) compute the reduced mass and and the separation of the two components.\r\n\r\nSo my path forward was pretty clear: I simply assumed that the stars were orbiting around their center of mass and then used the simple equation $v = \\frac{2\\pi r}{T}$ to obtain the separation, and then used the fact that the COM satisfies $m_1r_1 = m_2 r_2$ to conclude. \r\n\r\nI am not sure how to compute the reduced mass though - I know that in this case $\\mu = \\frac{m}{2}$ where $m$ is the mass of one of the stars, but I can't think of a way to get $m$ in the first place. Do we have to use energy considerations (or simplify the system into a one-body problem)?\n\nAccepted Answer:\n\nYour argument for the first part of the problem is good.  \r\n\r\nYou know that the two orbital velocities and masses are the same.  Think of the two stars as masses on the ends of a lever.  Where do you need to put the fulcrum (the center of mass) to ensure that when the lever moves, the masses necessarily move at the same speed in opposite directions?", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 37, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "homework-and-exercises", "newtonian-mechanics", "orbital-motion", "celestial-mechanics"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 618158, "title": "Binary stars orbiting around each other are equidistant", "url": "https://physics.stackexchange.com/questions/618158/binary-stars-orbiting-around-each-other-are-equidistant", "share_url": "https://physics.stackexchange.com/q/618158", "content_license": null, "owner": {"display_name": "Lt. Commander. Data", "user_id": 288380, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/288380/lt-commander-data"}}, "answers": [{"answer_id": 618189, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/618158/binary-stars-orbiting-around-each-other-are-equidistant/618189#618189", "share_url": "https://physics.stackexchange.com/a/618189", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "S. McGrew", "user_id": 183212, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/183212/s-mcgrew"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:493913:0000", "text": "Question: Why is relativistic beaming/ Doppler beaming occur at non-relativistic speeds\n\nThe reflexive motion of a binary star system causes the host star to occasionally wobble towards and away from an observer on Earth, which gives rise to an effect called relativistic beaming. This is when light becomes more concentrated in the direction of motion of the host star when viewed from an observer on Earth. In the rest frame of the star, light will be radiating isotropically (uniformly in all directions). \r\n\r\nIt is surprising to me why this effect is even considered relativistic when the host star wont be moving more than $10^3 \\frac{m}{s}$ but appariantly it's true.\r\n\r\nCan anyone explain why this effects occurs? \r\n\r\nThanks.\n\nAccepted Answer:\n\nThe effect is upon the brightness of relativistic jets (including those emitted by the binaries), not upon the brightness of the accreting matter (binaries themselves). \r\n\r\nHere's an explaination https://en.wikipedia.org/wiki/Relativistic_beaming", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 38, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "special-relativity", "doppler-effect", "exoplanets"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 493913, "title": "Why is relativistic beaming/ Doppler beaming occur at non-relativistic speeds", "url": "https://physics.stackexchange.com/questions/493913/why-is-relativistic-beaming-doppler-beaming-occur-at-non-relativistic-speeds", "share_url": "https://physics.stackexchange.com/q/493913", "content_license": null, "owner": {"display_name": "hwhorf", "user_id": 199137, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/199137/hwhorf"}}, "answers": [{"answer_id": 493917, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/493913/why-is-relativistic-beaming-doppler-beaming-occur-at-non-relativistic-speeds/493917#493917", "share_url": "https://physics.stackexchange.com/a/493917", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "JMLCarter", "user_id": 140002, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/140002/jmlcarter"}}]}}}
{"unit_id": "stackexchange:physics.stackexchange:binary-stars:650016:0000", "text": "Question: An apparent paradox in General relativity using a binary black hole\n\n**Note**: Quantum gravity effects are ignored in this question.\r\n\r\nImagine 2 black holes each with mass $m$ approaching each other and after some time the Event Horizons (EHs) of both black holes touch each other. But the singularities of both black holes are outside the event horizon of the other black hole. Now since **classically** EH itself is not made up of matter it can come out of another EH (in quantum gravity this may not be possible since the surface of EH has many properties and it also stores information on the surface [when observed from outside][1] and information cannot leave another EH) and we can do something and separate the black holes and their EHs. I think this is possible in principle. I am neglecting Gravitational waves radiated by this binary. **Can something happen and make this separation impossible even in principle?** \r\n\r\nOne way I think we can separate these black holes is if those 2 are Reissner–Nordström black holes with same charge q  (they don't repel because their mass dominates over charge) then in principle we can get some very highly charged but with opposite sign objects (not necessarily black holes and they can be moved easily) with mass $M$ and charge $Q$ ( $|Q|>>|q|$ and $|M|>>|m|$) and put one to the left of left black hole and the other to the right of the right black hole. These large objects will be almost at rest and the 2 black holes will be attracted to these large objects and their EHs will be separated from contact.\r\n\r\nNow in another similar situation a test particle is exactly in the middle of 2 black holes. After sometime the black holes attract and their EHs meet and the particle is in both EHs and since this test particle is at the center it will be always at rest even inside the EHs (since if it moves towards either singularity it has to move away from another singularity which is not possible since the radial coordinate behaves like a time like coordinate and moving radially away is impossible inside an EH).\r\n\r\n[![Test particle is in blue. Singularities are in black.][2]][2]\r\n\r\nTest particle is in blue. Singularities are in black.\r\n\r\nNow we can do something like earlier and separate the EHs. Now from the symmetry at the end the test particle should still be at the center, so it has to come out of both EHs. **But in classical GR no particle can come out of an EH.** What is wrong in this description and how to resolve this paradox?\r\n\r\n\r\n  [1]: https://en.wikipedia.org/wiki/Black_hole_complementarity\r\n  [2]: https://i.sstatic.net/3AYZv.png\n\nAccepted Answer:\n\nIt is impossible to use Reissner-Nordström black holes in you scenario, because these black holes assume spherical symmetry of spacetime. This is a good approximation if other sources are far away and produce only small distortions around the black hole (BH) you analyze, but in your case the symmetry is highly broken. Just the fact that the two event horizons meet means the distortions from BH1 around BH2 is not negligible. \r\n\r\nThe event horizon is also not a property of singularity but of spacetime itself. You cannot imagine two spheres around singularities and if the singularity 1 is not inside a sphere of singularity 2 it can go away. The event horizons will merge in nontrivial way and create one event horizon, and everything underneath it will not be able to get away. The nonlinearity of Einstein field equations also means that you cannot imagine binary BH as simple combination of two isolated BHs.\r\n\r\nAnother thing is, that it is know that the scenario in your picture is unstable and it radiates gravitational waves  until one symmetrical black hole is created. From what I know, the process is very quick, but someone else more knowledgeable in this area will be better suited to explain this.", "source": "stackexchange", "source_doc_id": "physics.stackexchange", "source_title": "Physics Stack Exchange", "domain": "physics", "subdomain": "qa_community", "level": "mixed", "order_index": 39, "metadata": {"module_id": "binary-stars", "source_format": "stackexchange_api_v2.3", "extraction_method": "stackexchange_api_v2_3_extractor_v1", "license": "CC-BY-SA-4.0", "language": "en", "hierarchy_path": ["Physics Stack Exchange", "binary-stars", "general-relativity", "black-holes", "spacetime", "event-horizon"], "page_start": null, "page_end": null, "quality_flags": ["low_score", "single_answer"], "ocr_confidence": null, "layout_confidence": null, "math_confidence": null, "stackexchange_attribution": {"network": "Stack Exchange", "site_name": "Physics Stack Exchange", "site_url": "https://physics.stackexchange.com", "api_site": "physics", "api_terms_url": "https://stackoverflow.com/legal/api-terms-of-use", "license_help_url": "https://physics.stackexchange.com/help/licensing", "question": {"question_id": 650016, "title": "An apparent paradox in General relativity using a binary black hole", "url": "https://physics.stackexchange.com/questions/650016/an-apparent-paradox-in-general-relativity-using-a-binary-black-hole", "share_url": "https://physics.stackexchange.com/q/650016", "content_license": null, "owner": {"display_name": "K. Sreeman Reddy", "user_id": 264772, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/264772/k-sreeman-reddy"}}, "answers": [{"answer_id": 650020, "is_accepted": true, "url": "https://physics.stackexchange.com/questions/650016/an-apparent-paradox-in-general-relativity-using-a-binary-black-hole/650020#650020", "share_url": "https://physics.stackexchange.com/a/650020", "content_license": "CC BY-SA 4.0", "owner": {"display_name": "Umaxo", "user_id": 204534, "user_type": "registered", "profile_url": "https://physics.stackexchange.com/users/204534/umaxo"}}]}}}