Moment of inertia as a tensor

A professor at my university briefly stated that moment of inertia is a tensor and can be represented by a $3×3$ matrix. I don't have a good idea of what a tensor is, so I would be grateful if someone could explain how to **intuitively** think of moment of inertia as a tensor.


H-theorem and Boltzmann equation applied to Boltzmann distribution

Using the Boltzmann equation: $$ \frac{dH}{dt} = \int_0^{\infty} dr \int_0^{\infty} ds W(r,s)[p_r - p_s][\ln{p_r} - \ln{p_s}]$$ and assuming $p_r = e^{-\beta r}$, the equation looks like $$ \frac{dH}{dt} = \beta \int_0^{\infty} dr \int_0^{\infty} ds W(r,s)[e^{-\beta r} - e^{-\beta s}][s - r]$$ I would like to prove that as long as the transition rate satisfies detailed balance ($W(r,s) = W(s,r)$) the system will be in equilibrium (meaning that the whole expression is equal to zero) However, if I assume that $W(r,s)=1$, the integral doesn't seem to converge Just for giggles I attempted $W(r,s)=e^{-(r-s)^2}$, which makes the integral converge, but it is clearly non-zero Any idea what step is required here?


status of +4/3 scalar as explanation of $t\bar t$ asymmetry

One of the early proposals for the Tevatron asymmetry on $t \bar t$ was a "fundamental diquark" with a charge (and hypercharge) +4/3, either in a triplet or a sextet colour. I am interested on the current status of this proposal: Generically, has some particular model of been favoured in ulterior research? For starters, you can see the short report in JHEP 1109:097,2011, at the end of section 2, points 5 and 6. I only become aware of the survival of this hypothesis after seeing http://arxiv.org/abs/1111.0477 last week.


Imaginary masses

While watching this video, at around 5:00, the man mentions a certain type of particle having imaginary mass. He also says that these kind of particles can go faster then light. But how it is possible that a mass is imaginary? For me, this make no sense. And also what does this even mean? For example, how would you calculate a gravitational force involving particles having imaginary masses? The result of the force will also be imaginary. Or will it interact gravitationally with other objects?


How can one get the eccentricity of the orbit of the Sun around center of the Milky Way? Can it be measured?


Light-Particle Wave Duality

There is a lot of reading to do on this to fully understand it, but without doing that reading is there a short explanation as to why and how light behaves as a wave and a particle?


How long would it take to scan the visible universe for unique signals?

The article _Amazing rays as star succumbs to dark side_ talks about a very large black hole swallowing up a star. The report goes on to say that the only reason it was discovered was because it shot out a signal directly at our Solar System, by chance. A discovery that was 3.8 billion years in the making, as our Solar System spins around and happened to collide with that signal at the right moment in time. Given that the universe is around 14 billion years old and this was only 3.8 billion years ago (the Earth may not have existed when this event occurred). But I digress. This discovery implies to me that it's very hard to even detect significant events. That is, we have to be pointing our listening equipment at very minute areas of the sky to detect an astronomical event. So even with our available array of detection hardware here on Earth as well as orbital telescopes, how much of the visible sky have we already scanned? And how long will it be until we've "scanned" the universe? To help clarify the question, I know that there may be areas of the sky that may be completely void, so perhaps scanning "the relevant universe" is a more appropriate scope of this question, but if you can guestimate the whole visible sky, it would be interesting to know.


Using CO2 to air condition a room

I'm trying to determine how much dry ice or liquid nitrogen I would need to cool 3300 cubic feet, about 90,000 liters of air, from about 100F (37.78C or 310K) to about 90F (26.67C or 299.81K). I'm assuming that the air is comprised of about 21% oxygen. How would I go about doing this?


Wavefront RMS errors, mirror surface roughness and Gaussian beam scattering

What I'm interested is in the scattered power of Gaussian beams reflected from mirrors with a given surface RMS. Usually the surface RMS $\sigma_{s}$ of a mirror translates in an error for the wavefront $ \sigma_{w} \sin{\theta} = 2 \sigma_{s}$, where the beam has incidence angle $\theta$. Now, What I' concerned is specifically, what fraction of power is lost from a Gaussian beam that reflects in a mirror with a given $\sigma_{s}$ **(Ignoring the portion of beam that is absorbed or transmitted, considering only the light that is scattered)**. It sounds like _maybe_ , this is somewhat related to the Strehl ratio $1 - (\frac{2 \pi \sigma_{w}}{\lambda})^2$. What I know is that the Strehl ratio gives a qualitative broadening of the diffraction- limited ideal beam. But it is usually applied to incoherent sources, like point sources generating spherical wavefronts. The interesting part here is to ask: if a perfect Gaussian beam, with far- field divergence of $\frac{\lambda}{\pi W_0}$ is incident on a mirror with surface RMS $\sigma_s$, what portion of the reflected beam will have the same divergence in the far-field? The reflected beam divergence is usually characterized by $\frac{M^2 \lambda}{\pi W_0}$, where $M^2$ is a factor describing the beam quality: **What is the relationship between $M^2$ and the surface errors?** **Important insight** one of the reasons that I'm not quite convinced that Strehl ratio is the only figure of merit for analize the power loss is that a lower-quality beam that is not diffraction limited will have a beam divergence larger than the Gaussian beam. But if the mirror reflects the beam near its Rayleigh range, and tries to focus it at focal length $D$, the Strehl ratio only says that the beam waist will be thicker by an amount $W_0'= MW_0$. **But there is no inmediate reason to conclude that the beam will be any wider at the Rayleigh range at twice the focal length $2D$.** , since the Strehl ratio only concerns itself with the focal plane, which concides with the waist in the case of a Gaussian beam: a lower-quality beam will have a higher divergence, but a thicker waist also means a lower divergence, so both effects seem to cancel each other, leaving the beam width at the Rayleigh range unchanged. If this conclusion is wrong, please explain


Do Pulsar Stars produce sound?

Can I hear electromagnetic radiation coming from a pulsar star? Or can I hear it if I stand outside it?


Orbital motion with varying radius

The model of a orbital motion in many Physics textbooks and online resources (e.g., Wikipedia on circular motion) assumes that the orbit is a circle, i.e. the radius is constant and the speed is constant, when deriving the many relationships such as $v = R \omega$ and $a_{\text{centripetal}} = R\omega^2 = \frac{v^2}{R}$. Would the many relationships hold when the orbit's radius is no longer constant but varies with time, for example? I argue that the many relationships would not hold because the model from which the relationships are derived does not take into account the change in the orbit's radius. If my argument is wrong, what is the logical basis to say that the relationships still hold even when the model does not take into account the change in the orbit's radius? If my argument is right, any pointer to a better model that takes into account the change in the orbit's radius?


Is the curvature of spacetime invariant? Could it be characterized as the ether?

I'm writing a paper for a Philosophy of Science course about GR/SR and I'm wondering if I can (1) characterize the curvature of spacetime as invariant and (2) argue that this is what Einstein referred to in 1920 when he said "space without ether is unthinkable." I take (1) from Gauss' proof that the curvature of 2-surfaces has one invariant which seems to be an intrinsic quality of the space (i.e. non- reference-frame-dependent). Shown by: $K=\frac{(\nabla_{2}\nabla_{1}-\nabla_{1}\nabla_{2})e_{1},e_{2}}{det(g)}$ where $\nabla_{i}=\nabla_{e}$ is the covariant derivative and $g$ is the metric tensor. And (2) from a rather terse paper (which I don't fully understand) found here. I would just like to know if I'm completely off-base as even though I have the math background, my physics knowledge is spotty at best.


Thought experiment-shell theorem

![](http://i.stack.imgur.com/p2Al9.png) Consider a uniformly dense perfectly spherical planet. At any point inside the planet, the gravitational attraction is only due to the mass below the point. (red circle) according to shell theorem. * * * Now onto the thought experiment, as per Newtons law of gravitation, _every particle_ attracts _every other particle_ with $\frac{G m_1 m_2}{r^2}$ By symmetry, the particles in the blue region should nullify their(net) attraction at the point. Similarly, all particles in the green region would have a component of the force along the -ve X-axis. _Even the particles in the green region **but not inside the red circle** would have a small component of attraction along -ve X-axis at the point._( **contradicting shell theorem** ) Again, their Y-axis components would nullify each other by symmetry. The same would hold true by spinning this cross-section along the X-axis to a 3d planet. * * * Now what's wrong with this explanation?


Compatibility of economics and physics?

This is an open-ended and broad and multi-part question; but I would be willing to accept an answer that simply corrects my misconceptions. I am sorry if it is inappropriate or too soft for this site. Essentially I am wondering how can a financial system generally operate in the presence of the two actual pinnacles of physical theory, General Relativity and Quantum Mechanics? Please forgive or else enjoy the following science-fiction... In the former case, suppose in the future we make contact with an alien civilization on Alpha Centauri which is about four light-years away. Assume even that it is possible to travel at close to the speed of light, so ambassadors from each of our stars can travel back and forth within a lifetime. And suppose that subsequently many economic benefits of the interaction become apparent: perhaps they wish to buy our computers because they are faster than theirs, and we wish to buy their rocket ships because they are more efficient than ours. Soon an interstellar trade is established at a certain exchange rate. Alien corporations are established on both worlds. What are some general strategies to minimize risk in these situations? Given that a single currency is superior to a multi-good barter system here on Earth, is the same true in such an interstellar distributed system with high communication latency? Does the nature of stocks and bonds change in the presence of this latency? For example, does the Black-Scholes formula apply just as well as it does locally? This aspect is quite analogous to foreign trade prior to radio. How did radio influence the theoretical development and applications of economics? I recall reading somewhere that HFT benefits from the placement of data centers at intermediate points (e.g. the mid-Atlantic or mid-Pacific). Is this true, and is there a reference for the existence or effectiveness of any such practice? One tool that comes to my mind is the CAP theorem which seems to place limits on the effectiveness. In a context of distributed systems, one might imagine bartering persistent storage space for serial computation speed (e.g., I have a 386 with a 1TB hard-drive; you have a 3GHz processor with only a floppy disk; and we are connected with a certain latency). On the other hand, Quantum Mechanics suggests the possibility of win-win situations that would not be possible in a classical or purely relativistic universe. Can this somehow mitigate the confusion caused by considering General Relativity? If we place a computer in interstellar space midway between here and Alpha Centauri, that helps, but what if we place a quantum computer there? Would that help even more? Ultimately I am wondering how can financial risk be calculated in the future when our current physical knowledge is exploited in the form of technology?


Why does reparameterisation invariance lead to gauge-fixing?

In Becker, Becker and Schwarz, the point particle action is given in terms of an auxiliary field $e(\tau)$ as: \begin{align} \tilde{S}_0 = \frac{1}{2}\int \,d\tau \left(e^{-1}\dot{X}^2 - m^2e\right) \end{align} It is then shown that under infinitesimal reparametrizations of $\tau$, the action is unchanged. This allows us to pick a gauge, in particular $e(\tau) = 1$. I'm not sure I'm understanding this right, but I have a few issues with this.   1. Doesn't this assume that $e$ takes the value $1$ somewhere?   2. Although $\tilde{S}_0$ (sorry, not $e(\tau)$) may be reparametrization invariant, I don't see how you could pick a reparametrization that can leave $e$ constant. Such a reparametrization $\tau'(\tau)$ would need to map all $\tau$ to a constant, but then $\frac{d\tau'}{d\tau} = 0$, which can't be the case.   3. Invariance is only under infinitesimal transformations. This is related to 2: how do we know that an infinitesimal reparametrisation could make $e$ constant?


Hamiltonian flow is volume preserving

I was reading about advantage of Hamiltonian over Lagrangian. One of the advantage is "Hamiltonian flow is volume preserving". Can anyone help me to understand this? Means what is advantage of preservation of volume vs volume is not preserve?


Caimir effect regularization for every divergent sum or series

can we use the tools of renormalization of casimir effect to get finite results for any divergent series in QFT ?? for example let be the divergent series $ \sum_{n=1}^{\infty}n^{l} $ for positive 'l' then instead of the sum we interpretate this as the difference $$ \sum_{n=1}^{\infty}n^{l}e^{-n\epsilon}-\int_{0}^{\infty}dtt^{l}e^{-t\epsilon}=finite$$ this is made to evaluate the infinite sums of the vaccuum energy in casimir effect but could it be done for any divergent series??


At what angle does a single atom "reflect" a single photon?

Does this question make sense in the quantum world? Imagining a single photon (wave packet?) interacting with a single atom (its electrons etc) how do we currently describe/define the emitted photon in terms of its direction in relation to the incoming photon? Now "scaling up" to a surface of atoms actually reflecting "light" according to the simple reflection rules like angle-in equals angle-out how do we manage to explain this effect in terms of the quantum world? How comes the probabilities work out for the out-going angle depending on the incoming- angle?


How can you prove generalised Jacobi identity?

This $$[A, [B,C]] + [C, [A,B]] + [B, [C,A]] = 0$$ This proof is not wanted, since there is an attachment of some variables. **How can you prove the equation?**


What is the experimental evidence for quantization of lattice vibrations?

Is there any direct experiment to show the quantization of lattice vibrations?


Can water exist as liquid in a vacuum chamber at room temperature?

I know boiling point at vacuum is far below the room temperature. But I still wondering if the humidity reaches the saturated humidity (100% relative humidity), will the vapor condense to liquid, although it is probably going to evaporate again soon?


Difference between parallel and series capacitors

Basically, I can't figure out that whether a capacitor is connected in series or parallel. The same problem occurs while solving for resistors. Can someone help me understand this difference conceptually?


What does one need to learn to become a experimental particles physicist?

I know one needs E&M, classical, quantum, statmech at the graduate level. But what more does one need? Are quantum field theory, relativistic QM, advanced math needed? or just an advanced course in elementary particles on top of basics suffice? Please bear in mind that the intended profession is an experimenter and not a theorist.


Radians to Rotate Earth to Match ECI Lat/Lon with ECEF Lat/Lon

I am attempting to model GPS Satellite positions on the globe for a set of ephemerides. I have a verified set of ECI ( http://en.wikipedia.org/wiki/Earth-centered_inertial ) XYZ Coordinates and a verified set of ECEF ( http://en.wikipedia.org/wiki/ECEF ) XYZ coordinates.               For example ( m ):     ECEF X 4076514.50     ECEF Y 14673598.00     ECEF Z -21793674.00          ECI X 8004604.50     ECI Y 12956032.00     ECI Z -21793674.00      Sometimes the signs of the ECI X and Y coordinates are opposite of the signs for the ECEF X and Y coordinates. When I plot these locations on a globe and, obviously they do not match unless you rotate the earth by some number of radians. I can see that if you rotate the earth, the ECI coordinates follow a groundtrack that is consistent with the ECEF groundtrack, but with my current code, it is offset by different varying amounts, depending on the time of day I plot it. Here is my code:               // The radians earth rotates in 1 second     double earthRotation = 0.000072921151467;          // Do something different depending on the switch position     if ( self.shouldAnimate == true ) {              // The seconds since we started rotating         double newRotationMultiplier = [self.eciDate timeIntervalSinceDate:self.lastEciDate];              // Seconds           // Rad/Sec        // Multiplier         self.timeSinceOpenGlStarted = newRotationMultiplier * earthRotation;         matrixToRotate = GLKMatrix4Rotate(matrixToRotate, self.timeSinceOpenGlStarted, 0.0, 0.0, 1.0);              // Latch it so that when we stop rotating, it doesn't revert         self.effect.transform.modelviewMatrix = matrixToRotate;          }      How do I determine the radians to rotate the earth to show an ECI positions for a time to be the same location on the rotated globe ( in terms of latitude and longitude )? That is, how do I calculate for any UTC time in a day, how much to rotate the earth about the Z ( "Up" ) axis so that the ECI coordinates agree with the ECEF coordinates and the satellites appear to be in the same position over my 3D globe? UPDATE: I got it to work - thanks to the answer below! Here is the code:                   // Get the NSDate components         NSCalendar *currentCalendar = [NSCalendar currentCalendar];         [currentCalendar setTimeZone:[NSTimeZone timeZoneWithAbbreviation:@"UTC"]];         NSDateComponents *components = [currentCalendar components:NSCalendarUnitDay | NSCalendarUnitMonth | NSCalendarUnitYear | NSHourCalendarUnit | NSMinuteCalendarUnit | NSSecondCalendarUnit fromDate:self.eciDate];         [components setTimeZone:[NSTimeZone timeZoneWithAbbreviation:@"UTC"]];         NSInteger hour = [components hour];         NSInteger minute = [components minute];         NSInteger second = [components second];              // Get UTC in terms of decimal hours         float utc = hour + (float)minute/60.0f + (float)second/3600.0f; // UTC is expressed in hours as a decimal number              //NSLog(@"UTC: %f H %f M %f S %f",utc,(float)hour,(float)minute/60.0f ,(float)second/3600.0f);              // Get the seconds since the last vernal equinox         double secondsSinceVernalEquinox = [self.eciDate timeIntervalSinceDate:appDelegate.lastVernalEquinox];              // Do the math         // http://physics.stackexchange.com/questions/98466/radians-to-rotate-earth-to-match-eci-lat-lon-with-ecef-lat-lon         double d = secondsSinceVernalEquinox  / ( 60.0 * 60.0 * 24.0 ); // Give decimal days         double p = 365.242187; // Length of tropical year         double result = ( M_PI * 2 / 24 ) * ( utc - 12.0 ) + ( 2 * M_PI * d  / p );              //NSLog(@"Radians = %f",result);              // Seconds           // Rad/Sec        // Multiplier         self.timeSinceOpenGlStarted = result;         matrixToRotate = GLKMatrix4Rotate(matrixToRotate, self.timeSinceOpenGlStarted, 0.0, 0.0, 1.0);              // Latch it so that when we stop rotating, it doesn't revert         self.effect.transform.modelviewMatrix = matrixToRotate;


What happens (in terms of pressure) when there is a leakage in a horizontal pipe?

Consider a horizontal pipe that has water being pumped continuously through it, with a leak at a given point. What will the leak's effect be on the pressure in the pipe, both above and below it in the flow?


induced emf when a wire or coil travel through a magnetic field

I've recently learned that If you move a conductor through a magnetic field, an emf is induced across the ends of the conductor: $E = BLv $ I've also been told that the same is true for a coil but that the equation is: $E = BLvN$ And that for a coil, there is only an emf induced when the current is entering or leaving the magnetic field as once the coil is inside, there is no change in magnetic flux linkage. I've been told that this is not the case with just a wire moving through a magnetic field, but I haven't been told why and I can't seem to figure it out. I've thought about it, and I think it perhaps has something to do with the area covered by the wire moving at speed $v$ per unit time compared to the area covered by a coil moving at the same speed $v$ in the same unit time. Unfortunately, I can't quite work it out. Thank you.


Can a particle with non-zero angular momentum pass through the center of a spherical potential?

Suppose you have a particle of mass $m$ moving in a potential $V(r) = -\frac{k}{r^2}$, with $r^2 = x^2+y^2+z^2$ and $k > 0$. Since the angular momentum $l$ is conserved, the particle will move in a plane: Let's use polar coordinates $(r, \theta)$. The expression for the energy is then: $$ E = \frac12m\dot{r}^2+\left(\frac{l^2}{2m}-k\right)\frac1{r^2} $$ Suppose $l < \sqrt{2mk}$. Then the effective potential is of the form $V_{eff} = -\frac{a}{r^2}$ with $a > 0$. This implies that no matter what the initial energy $E$ of the particle is, it can and will at some point pass through $r=0$. Indeed, if you find the trajectory $r = r(\theta)$, you get things that look like either $r = \cos \theta$ or like $r = \sinh \theta$, which are $0$ at some point. I don't understand how this can be. When the particle is very close to $r=0$, its momentum has to pointed towards $r=0$, but this would imply that $\vec{r}$ and $\vec{p}$ are parallel, and so $l$ would be $0$! How can this be? Doesn't conservation of angular momentum prevent the particle from reaching the center, no matter how strong the potential is?


Electric field represented by parallel curves

In a region of space, I have an electric field represented by parallel curves as shown in the diagram. ![enter image description here](http://i.stack.imgur.com/GnB8T.jpg) The curves are equally spaced. We know that a uniform electric field can be represented by lines that are parallel and evenly spaced. Can this definition of uniform electric field be extended to parallel and equally spaced curves?


How is it possible to calculate the energy liberated by a given fission process? For example, in the fission of a $^{235}$U induced by capturing a neutron?


Production of antiproton at the Tevatron

The Tevatron is a proton-antiproton collider; it collides a beam of protons against a beam of antiproton. I can understand how we obtain the protons, but for the antiprotons ? How are they produced ?


Physics and what it means

I got into a debate with a friend about the meaning of physics and its purpose, he is the sort who will test you and if you get it wrong it somehow gratifies his own self-reflection and self-worth. That aside, I have loved physics since I got into it when I was at school, I had a great teacher who fostered that love and helped grow my appreciation for physics and ever since I have said to myself that I would one day go do my degree in physics. At the moment I have just finished my degree in computer security and forensics, I plan to study physics part-time hopefully in the next 5 years after I settle into permanent work in the software engineering field. Anyway the conversation started with a question from him asking what temperature water boils at and what temperature does water freeze at. For the life of me I couldn’t remember, even as a general question most would know the answer, well that’s what we thought. For him this was a big disgrace and he continued to laugh and "then" call a friend to verify that I am being moronic. Of course he phoned our friend and pitched the same question, of which he neither knew the boiling point nor the freezing point of water. Regardless, for him he classed this as "true physics", but to me it could have been any random number for all I cared. What he was trying to prelude to and was expecting as an answer was "it depends" and the dependency was altitude (from a documentary he watched). Of course I never got past that stage and all it became was a ridicule session. For me it didn’t matter I thought the true physics was understanding why water boiled in the first place. How exciting molecules or increasing the energy within water gradually increases its temperature and when the molecules become extremely excited water gets to its boiling temperature. Understanding this allows us to understand all the states of water (phases), from a solid (ice) to a liquid or to a gas (vapour, altho vapour is essentially two phases) in one, the phase change occurs via nucleation and growth - small liquid drops spontaneously form and evaporate, but if a drop is larger than some critical radius, it will continue to grow ('nucleates'). This is the most common phenomenon of a gas-liquid (or liquid-solid) phase transition. Of course I discussed this with him but he argued blind that "knowing" the boiling point of water was "physics" but to me that was essentially just a factoid. That the boiling point of water did not matter but understanding the process behind it did. So I ask those of you in the community who was wrong or what physics trully is. If we take the standard definition of Physics it is a natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves. For me knowing the boiling temperature of water does not equate to understanding how water as a liquid, boils. Or the transfer of energy to change a liquids state occurs. It is simply just a consequence of the action or a "factoid" of the means in which we can reliably get the same testable results.


Power, Force Velocity

In any case, if the force and power is constant , does velocity also need to be constant? If no, can you tell me a case. Further can this be extended for non-rigid bodies.


Allowed interactions for the neutral weak vector boson

Just wanted to double check a couple of things for my own sanity! I am looking at scattering amplitudes for 2 partons going to two partons with the emission of a Z boson (eventually decaying to e+ e-). I have used madgraph to plot the possible tree level Feynman diagrams for a couple of processes and I'm a bit surprised by the output...See link below for the output diagrams. Initially I thought that diagram 2 cause of the virtual anti-down quark (displayed as d~). It looks like a d quark has radiated a Z boson which changed the d quark to an anti-down quark which in turn radiated a gluon which changed it back to a down quark. I am 99% sure this is not allowed as the Z can't change q's to qbar's. An alternative way of thinking about it is that the u quark radiates a gluon which then _pair produces_ a d-dbar pair, one of which is final state particle but the other one goes and annihilates the incoming d quark to give a Z...But I am not certain that this is allowed either! ![Feynmann diagrams](http://i.stack.imgur.com/x5aYE.gif)


Shining object in the sky resembling a star

From last 2 days I am observing a shiny star like shining object in the sky in almost same place at same time. Is there any possibility that it could be a satellite? Will satellite shine like a star in the sky due to any kind of reflection from the sun. I see it in the evening and in white light. I don't know how to verify but I am very curious to know what it is.


What is the origin of flavor?

> **Possible Duplicate:**   >  Origin of lepton/quark generations? In the standard model (and in nature), Fermions appear in different generations, or flavors. Besides up and down quarks and electrons, there are strange, charm, muon, tau, etc.. I am wondering if there is any theoretical explanation for the existence of heavier generations. What I'd like to know is especially: _Why_ are there different, heavier copies of $e$, $u$, $d$? What makes these particles different from their lighter relatives _besides_ their mass? What is flavour, where does it come from, and why is it approximately conserved (at least for charged leptons)? * * * Note: I'm posting this question as a follow up on my previous one _"Origin of lepton/quark generations?"_, which didn't reallty get the type of answers I was expecting. Sorry if the old question caused some confusion. To be clear, I'm _not_ asking:   * "Why must there be _three_ generations (and not 2 or 4)?" -> e.g. CP Violation   * " _How do we know_ that there are 3 generations?" -> Experiment   * "Why do we _organize_ particles in this manner?" -> Quantum numbers But rather: What possible theoretical explanation is there that our universe exibits flavor quantum numbers, and maybe how are they are related to particle masses? Conceivable (although probably wrong) answers would be along the line of:   * "Because electrons and up quarks have substructure, and muons and charm quarks are the first excited states."   * "In XY theory, $e$, $\mu$ and $\tau$ are actually the same at first order, but the thingamajig symmetry is broken at low energies."   * "It is explained by string theory, if your vacuum has such-and-such properties it comes out naturally."   * "It's accidential in the SM and just experimentally observed, but in SU(...) GUTs it is a prediction."


What's the difference between background field and dynamical gauge field?

Dynamical gauge fields are assumed to be able to respond to sources. What's the difference in the _Lagrangians_ between a background field and a dynamical field?


Why does $\langle a_-\alpha|\alpha\rangle = \alpha $for harmonic oscillator

Why does $\langle a_-\alpha|\alpha\rangle = \alpha$ doesn't the ladder operator lower the $\alpha$ so that it became $\sqrt{\alpha}*\delta_{\alpha, \alpha_{-1}}$ Maybe its because that \alpha can be imaginary? In that case how would you prove the above relation?


Degenerate vibrating modes of a metal plate

My question is based around this video: https://www.youtube.com/watch?v=1yaqUI4b974 The video shows sand settling in vibrational nodes of a plate at certain resonance frequencies. It appears that all the patterns are radially symmetric however at 5:15 I see a pattern that is bilaterally symmetric. My question is why this pattern occurs horizontally instead of vertically? For a rectangular plate I understand that the two situations would occur at different frequencies, but what about a square plate? Is this a random occurrence?


What is the error on measuring the phase of a sine wave?

Let's say I have a wave, with frequency $\omega$ and phase $\phi$, of the form: $$y(t) = 1 + A \sin(\omega t+\phi)$$ where $A < 1$. I have measured this wave N times, and we can assume these measurements are evenly spaced over one period. Each of those measurements has an uncertainty in the y-direction of $\sigma_y$, and no uncertainty in the time of measurement (for practical purposes). How well can I measure the phase of this sine wave? I've tried constructing a Fisher matrix for the problem, but Mathematica thinks it's singular, and I can't see how to invert it either.


Confused on newton's second law being invariant under relaitivity

I am a math student with some interests in physics. I picked up a book called "A First Course in General Relativity", and I am confused on the second page. I am assuming by notation or convention. The chapter is on special relativity, and at this point they are just talking about how measurements of velocity are invariant by a constant. That is $v'(t) = v(t) - V$, where $v(t)$ is a measurement by one observer and $v'(t)$ is a measurement by another observer whose relative velocity to the original is $V$. Then it says Newton's second law is unaffected by this replacement. It offers as an explanation, $$a' \- dv'/dt = d(v - V)/dt = dv/dt = a.$$ I am confused in how this explains anything. Also as I read the notation, shouldn't $a' = dv'/dt$? So $a' \- dv'/dt = 0$. Also the first equality is confusing, since $v' = v - V$, I thought, so using that replacement I get $$a' \- dv'/dt = a' \- d(v - V)/dt.$$ And I don't see where they go from there. I assume I am just misunderstanding what is meant by a particular variable. Can anyone help shed some light on my confusion on this point? Thanks. **EDIT** I think I am being confused by a typo, replacing the $=$ with $-$ seems to make things make sense. I probably should have seen that. $$a' = dv'/dt = d(v - V)/dt = dv/dt = a.$$


How to simulate a crashing wave?

I'd like to create a very rough animation of a wave crashing on a beach. I'm guessing it would have to be a particle simulator, where you code in the forces between the particles and then integrate forward in time. I've done similar things, like simulations of charged particles, but there the forces are pretty straightforward, whereas here I guess I'd have to account for 1) tides 2) gravity 3) water surface tension. These seem like widely different forces acting on different scales. I don't even know where to begin. Any hints or links to papers related to this topic?


How do charged particles interact?

You'll have to forgive me if this question is too wrapped up in "classical" thinking. I've read that electrons and protons interact by trading photons, but this only raises more questions. What determines the energy and direction of the emitted photon? How often can a particle emit a photon? How often can a particle absorb a photon? Can one particle emit and absorb multiple photons at once? Where does the energy to emit a photon come from? Is the destination of the photon somehow pre-determined or is the photon simply emitted in the hopes of being absorbed?


Quantum Thrusters and Warp Drives

Any reservations about the potential of this (given in the link below/title above)? Does it seem like a helpless attempt or something which might have the potential of developing into something real? http://news.discovery.com/space/quantum-thruster-warp-drive-physics-130823.htm


Would an object hit the ground faster when it gets launched vertically upwards or downwards?

A while ago I had a discussion with someone about the speed of an object when it hits the ground. When does an object hit the ground faster:   * When you launch the object to the sky so it can fall down;   * Or when you launch the object to the ground so it doesn't have to deal with much air resistance. The attraction “The Booster” has an arm that has a constant speed (about 120 km/h) is about 80M high and undergoes full revolutions. ![enter image description here](http://i.stack.imgur.com/iAbtQ.png) In this situation the cart gets disattached at 90 degrees. ![enter image description here](http://i.stack.imgur.com/gLB1j.png) I think that the cart hits the ground with a higher speed when it's launched to the ground because of:   * The air resistance, when you launch it to the sky it has a longer path to travel, so it needs to travel through much more air.    * The cart has the same initial speed, when you travel a shorter path there will be less to slow the cart down;   * Nothing provides the cart with additional energy;   * I guess that the "final gravity" is the same.


Integrals of Motion

Landau writes in chapter 2 of his Mechanics book "The number of independent integrals of motion for a closed mechanical system with s degrees of freedom is 2s-1". Then he goes on to show how -- "Since the equations of motion for a closed system do not involve time explicitly, the choice of the origin of time is entirely arbitrary, and one of the arbitrary constants in the solution of the equations can always be taken as an additive constant t0 in time. Eliminating t + t0 from the 2s functions, we can express the 2s-1 arbitrary constants as functions of q and q’ (generalized co-ordinates and velocities) and these functions will be the integrals of the motions." Could someone elaborate?


tracking a satellite

The following article says that because the GOCE satellite has a polar orbit, it is impossible to track where it will land. > "The satellite is one of the few satellites in a Polar Orbit. Consequently, > it could land almost anywhere," Mark Hopkins, chair of the National Space > Society's executive committee told FoxNews.com. I was looking for an explanation based on the physics of the orbit, as I am not familiar with orbital mechanics. http://www.foxnews.com/science/2013/11/08/track-europe-falling-2000-pound- satellite-in-real-time/


Light bending by black holes

In the center of our milky way, it is assumed that a black hole exists with a mass of $\approx 4\times 10^6$ times our sun's mass. How much light bending (in degrees) would arise for stars that are in perspective near the position of this black hole in the sky? Could the mass of a black holes become so great that it actually distorts the area in the sky we see around it (through telescopes)?


Am I making the right assumption about a jump discontinuity in the acceleration?

> A train heads from Station A to Station B, 4 km away. If the train begins at > rest and ends at rest, and its maximum acceleration is 1.5 m/s^2 and maximum > deceleration is -6 m/s^2, what's the least time required to complete the > journey? **My attempt at a solution** : I assumed the acceleration function, $a(t)$, has a jump discontinuity at some time, say $t_1$, where it jumps from 1.5 to -6. Under this assumption, it's easy to use the velocity boundary conditions to get the total time $t_f$ is $\frac{5}{4}t_1$, because the velocity $v(t)$ for $t>t_1$ is just $1.5t_1-6(t-t_1)=7.5t_1-6t$. Setting this equal to 0 at $t_f$ gives this result. The position $x(t)$ for $t\leq t_1$ is just $\frac{1}{2}(1.5)t^2$ and for $t>t_1$ it's just $\frac{1}{2}(1.5)t_1^2+(1.5t_1)(t-t_1)-\frac{1}{2}(6)(t-t_1)^2$. Using the fact that $x(t_f)=4000$, and substituting $t_f=\frac{5}{4}t_1$, I find $t_f\approx 86.06$ seconds. But my teacher's solution says the minimum time is $81.65$ seconds. Who's wrong? Is my assumption about the jump discontinuity of $a(t)$ what's wrong?


Why removing one slab in Michelson-Morley experiment causes an elliptical fringe pattern?

I performed Morley experiment using He-Ne laser. Two glass slabs were there and circular fringes were formed but when I removed one slab elliptical fringes were there (and less intense fringes). Why?   In general is there any benefit of using two slabs?


Would the north poles of two magnets repel each other if a weaker south pole was inserted between them?

My son asked me this question and I was stumped - my intuition says that is the south pole was strong enough the attraction between the north and south poles would outweigh the repulsion between the two north poles - but how strong would the south pole need to be in this case?


Ideas for high school physics paper, including simulations

I have to write, with two friends, a simulation based physics paper at school. The problem is, that will I am quite familiar with the concepts of calculus and algebra, my friends aren't. So, we could find a good topic. The task is, to write a paper about a physical topic, with the use of a selfmade program. The development of the program shouldn't be a problem at all. we have a good programmer. Any suggestions, for a topic that involves only pre-calculus math or only just some calculus?


How to calculate the electrostatic force applied on an object that is along the axis of a charged tube's surface?

Let's say there is a charged tube(a cylinder with no top and bottom) with radius r and length l, charge q1 which also made out of insulating material. And also if there is an object with charge q2 along the axis of the surface of the tube, how can I calculate the force applied on this object?(the distance between tube and object is d) Here is my calculations: From the Gauss' law: $$ \phi_E= \frac Q{\epsilon_0} $$ I took cylinder as Gaussian surface. $$ E \times 2\pi dl= \frac {\lambda \bullet l}{\epsilon_0} $$ $$ E = \frac {\lambda}{\epsilon_0\times 2\pi d} $$ or $$ E = \frac {q_1}{\epsilon_0\times 2\pi dl} $$ Thus we found the electric field applied to a point d away from the tube.To calculate the force we use $F=qE$. And finally the electrostatic force is: $$ \frac {q_1q_2}{\epsilon_0\times 2\pi dl} $$ However what I found is the force applied to an object d away from the tube but perpendicular to the surface of the tube. Would this calculation be valid for an object which is along the axis of the surface of the tube? P.S: I know I asked that question but I couldn't get any answers so I hope this question won't be closed. Thanks in advance.


How to obtain Dirac equation from Schrodinger equation and special relativity?

I'm reading the Wikipedia page for the Dirac equation: > _The Dirac equation is superficially similar to the Schrödinger equation for > a free massive particle:_ > > A) $-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial > t}\phi.$ > > _The left side represents the square of the momentum operator divided by > twice the mass, which is the non-relativistic kinetic energy. Because > relativity treats space and time as a whole, a relativistic generalization > of this equation requires that space and time derivatives must enter > symmetrically, as they do in the Maxwell equations that govern the behavior > of light — the equations must be differentially of the same order in space > and time. In relativity, the momentum and the energy are the space and time > parts of a space-time vector, the 4-momentum, and they are related by the > relativistically invariant relation_ > > B) $\frac{E^2}{c^2} - p^2 = m^2c^2$ > > _which says that the length of this vector is proportional to the rest mass > m. Substituting the operator equivalents of the energy and momentum from the > Schrödinger theory, we get an equation describing the propagation of waves, > constructed from relativistically invariant objects,_ > > C) $\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi > = \frac{m^2c^2}{\hbar^2}\phi$ I am not sure how the equation A and B lead to equation C. It seems that it is related to substituting special relativity value into quantum mechanics operators, but I just keep failing to get a result...


Physical meaning of Impedance

So I have been thinking about the way impedance is defined for electrical systems, and the way it is derived. Even after looking through some websites, I cannot seem to grasp something, which every website I visited seemed to skip on. In an ideal resistor, if we take the ratio of the voltage to the current at ANY time, the ratio is a constant. This ratio is then the resistance (also the impedance) of the resistor. I have been interpreting this as the opposition the resistor bears to the current flow through it upon application of a voltage. I thought of it like a constant that relates the otherwise-would-be- infinite current to the applied voltage. Coming to a capacitor, there seems to be some sort of change in the definition of impedance. The impedance for a capacitor is no longer the ratio of voltage to the current, but the ratio of the complex voltage to the complex current. In other words, mathematically it is equivalent to $$\dfrac{1}{i\omega C},$$ and not $$\dfrac{\tan(\omega t)}{\omega C},$$ for a capacitor with capacitance C at a frequency w. I am finding this difficult to interpret because the ratio of the voltage to current is definitely varying in time. How can the analogy of a resistor's impedance be applied to the impedance of a capacitor when the ratio of the voltage across it to the current through it is changing (it's even 0 or undefined sometimes). Why is the magnitude of the quantity $$\dfrac{1}{i\omega C}$$ now the resistance of the capacitor circuit? What happened to the plain old voltage to current ratio? Have books been skipping this subtlety, or I am just not understanding something very simple? Clarification: $$ if \hspace{.5 pc} v(t) = V_m \sin (\omega t), \hspace{0.5 pc} then \dfrac{d V(t)}{dt} = \omega V_m \cos (\omega t) $$ $$ so \hspace{0.5 pc}the\hspace{0.5 pc}ratio\hspace{0.5 pc} \dfrac{v(t)}{i(t)} = \dfrac{V_m \sin (\omega t)}{C\omega V_m \cos (\omega t)} = \dfrac{\tan (\omega t)}{C\omega} $$ $$ In \hspace{0.5 pc}phasor \hspace{0.5 pc}notation, \hspace{0.5 pc}this \hspace{0.5 pc} ratio\hspace{0.5 pc} seems\hspace{0.5 pc} to\hspace{0.5 pc} be\hspace{0.5 pc}different \hspace{0.5 pc}in \hspace{0.5 pc}that: \\\ \dfrac{\mathbf{V_c}}{\mathbf{I_c}} = \dfrac{1}{Cj\omega} = -\dfrac{1}{C \omega }j $$ $$ and \hspace{0.5 pc} \Re\\{-\dfrac{1}{C \omega }j \\} = 0$$ $$but \\\ |\dfrac{\mathbf{V_c}}{\mathbf{I_c}}| = \dfrac{1}{C \omega } $$ So my question is why does the magnitude of the ratio of the complex voltage to the complex current now suddenly carries a physical meaning (if my understanding is correct, it is the resistance which can be measured in ohms, just like in the resistor). Also, why is going from the complex domain to real domain not possible for this ratio, because clearly the real part of the complex impedance is 0 (@Alfred Centauri mentions that the impedance is not itself a phasor). I understand that the math works, but the thing that is not making sense to me is this ratio of two complex quantities and the apparent emergence in its physical meaning.


How can we represent the motion of a particle in 2D space using Lagrange's equations?

Can we represent the motion of a particle in 2D space using Lagrange's equations? This is what I tried. Please tell me what is wrong? Consider a particle on a plane have the co-ordinates $(x,y)$ with a velocity $v$ and mass $m$. Now $v=\dot{x}+\dot{y}$ where $\dot{x}$ and $\dot{y}$ are the derivatives with respect to time. According to Lagrange's equation, $L=\frac{1}{2}m(\dot{x}+\dot{y})^2-mgy$ $\frac{{\partial}{L}}{\partial{y}}=-mg$ ---(1) $\frac{{\partial}{L}}{\partial{\dot{y}}}=m(\dot{x}+\dot{y})$ Therefore $\frac {\mathrm d t}{\mathrm d t}(\frac{{\partial}{L}}{\partial{\dot{y}}})=\frac{{\partial}{L}}{\partial{\dot{y}}}=m(\ddot{x}+\ddot{y})$ Now $\ddot{x}$ and $\ddot{y}$ are the vertical & horizontal components of the acceleration $a$ of the particle. So let $\ddot{x}=a_x$ and $\ddot{y}=a_y$ Hence $\frac {\mathrm d t}{\mathrm d t}(\frac{{\partial}{L}}{\partial{\dot{y}}})=m(a_x+a_y)$ ---(2) From Lagrange's equation, Eq (1)= Eq(2) => $-mg=m(a_x+a_y)$ $-g=a_x+a_y$ Does the above equation make any sense? I tried to apply it to a real problem and it gave me a wrong result. What is wrong with my working? Thanks in advance. ## **Update:** Thanks a lots guys. Yes, the expression for $v$ was wrong. When I calculated with the correct expression I got the result $a_y=-g$ What does it mean? Does it mean that the Y component of the acceleration is always equal to acceleration due to gravity? And can someone explain how Lagrangian mechanics can be used to solve simple mechanical problems? When I tried applying it to problems all I got were few known results & other nonsensical ones (like the one in this question. Probably due to my erroneous application of the Lagrange.) PS: Will someone explain why this question was downvoted?


Vertex for quartic interaction of complex scalar multiplet

Since I'm new to QFT and I tend to do a lot of errors during calculations, I would like you to tell me if I got the four-point vertex of the quartic interaction with a multiplet of complex scalar fields right: $$ L = (\partial_\mu \Phi)^\dagger (\partial^\mu\Phi) - m^2\Phi^\dagger\Phi-\lambda(\Phi^\dagger\Phi)(\Phi^\dagger\Phi),$$ where $\Phi= (\phi_1,\ldots,\phi_N)$ and $\phi_i$ are complex scalar fields. I have $$-2i\lambda ( \delta^{ik}\delta^{jh} + \delta^{ih}\delta^{jk}),$$ where $i,j$ are the indices of the two $\Phi^\dagger$, while $k,h$ are the indices of the two $\Phi$. Is it correct?


Air pressure in balloon

I have to calculate the air pressure inside of an hot air balloon. After some searching I found out that I can use the ideal gas law: PV = nRT (from Wikipedia) So to get the pressure in the balloon I would have to know n, which is the amount of gas in moles. For that amount I'm currently using the pressure (kg/m3) * the volume. So for the pressure I need the pressure which of course doesn't work. Is there a way to calculate this? Right now I do know the normal air pressure (outside of the balloon) volume of the balloon etc. etc., but I can't know the weight of the amount of air in the balloon, because I use the pressure to calculate it. One thing to note is that it isn't an actual hot air balloon. It's just one made up from physics formulas in a program that calculates them every 0.1 seconds (for example) for me. Is there some thing I'm missing?


Viscous force in a conductor

I was told that in a conductor there is a dissipative force and if Ohm's Law is valid for a component, the component has to be dissipative. I'm trying to prove it, but I have some problems. I know that for a viscous force: $$\tag{1}\mathbf {F}=m\mathbf {a}=-\gamma \mathbf {v}$$ so $$\tag{2}\mathbf {a}=\frac{-\gamma \mathbf {v}}{m}$$ and in a conductor (considering the motion of electrons): $$\tag{3}-e\mathbf {E}=m\mathbf {a}$$ so $$\tag{4}\mathbf {a}=\frac{-e \mathbf {E}}{m} $$ Considering the drift velociy given by $$\tag{5}\mathbf {v_d}=\frac {-e \tau}{m}\mathbf {E}$$ I could find $\mathbf {E}$, put it in (4) and obtain an acceleration in function of $\mathbf {v}$. So I could say that I'm treating a viscous force. But I'm not sure that this is a correct way... How could I verify that if Ohm's Law is valid for a component, the component has to be dissipative?


$\Delta T$, temperature units. How to use fahrenheit in this formula

To calculate the power in a fluid with a flow, the following formula is applied: $$P = \Delta T \rho c_p \times flow$$ $\Delta T$ can be entered as both Kelvin and Celsius. But what if I want to use Fahrenheit? Is there any formula for this? Reason is I'm making a web application that calculate some stuff. I have full unit support. I live in Europe so we always use Celsius or Kelvin. But, are americans using fahrenheit for calculations in general? If not, then maybe I can skip it all together.


Definition and difference between the R-symmetry and the $U(1)_R$ internal symmetry

* For a general ${\cal N}$ the R-symmetry group is $U({\cal N})$ but for the ${\cal N}=2$ case why is it $SU(2)$ ? I guess it is again different for ${\cal N}=4$. How does one understand this? One denotes the generators of the R-symmetry group by $B^i$ and let $\alpha$ be the spinor index and let $a,b$ be the R-charge indices in a ${\cal N}=2$ theory. Then one writes the defining equations as, $[Q_{\alpha a}, B^i] = -\frac{1}{2} (\tau ^i)_a ^b Q_{\alpha b}$ and $[\bar{Q}_{\dot{\alpha}}^a, B^i] = \frac{1}{2} \bar{Q}_{\dot{\alpha}}^b (\tau ^i)_b ^a$   * But it is not clear to me as to why over and above the aforementioned equations there should be yet another charge $R$ (called the $U(1)_R$ charge) with the defining equations, $[Q_{\alpha a},R] = Q_{\alpha a}$ and $[\bar{Q}_{\dot{\alpha} a}, R] = - \bar{Q}_{\dot{\alpha} a}$ I would like to know why the above charge should exist in ${\cal N}=2$ supersymetry separate from the R-symmetry. Is there an analogue for this $R$ for other values of ${\cal N}$ ?   * Typically this R-charge as defined above is anomalous. How does one see that? Also what is the analogous statement for R-symmetry?


How efficient is the human body?

This question sort of comes to mind when hearing how efficient an internal combustion engine is turning chemical energy in mechanical energy (something like 20-40%) with lots of excess heat. As an analog, how efficient is (or potentially) the human body at turning food into energy? Please bare with me, I realise there LOTS of different variables (how much the person weighs vs mass, metabolism, diet, etc). But I would imagine that there shouldn't be much margin of error given that most people maintain the same constant temperature (98 F +/- 1 degree).


Can an Electromagnetic Gauge Transformation be Imaginary?

The Hamiltonian of a non-relativistic charged particle in a magnetic field is $$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2$$. Under a gauge transformation of the magnetic potential: $$\vec A ~\rightarrow~ \vec A + \vec\nabla \chi,$$ the wavefunction of the particle transforms as $$\Psi~\rightarrow~ \Psi\exp(\frac{iq\chi}{\hbar c}).$$ When $\chi$ is real, the wavefunction simply gains an extra phase factor. However, when $\chi$ is imaginary, there is a measurable change to the wavefunction. This seems to contradict the fact that the magnetic field is invariant under the gauge transformation. How do I resolve this?


How to be sure that a He-Ne laser light is monochromatic

How can I be sure that the emission of a He-Ne laser contains only one single mode of laser cavity? The only thing that I know is that if I use a diffraction grating and the light isn't monochromatic, I'll see maximums of the same order $m$ at different angles, but I also know that if wavelengths are very close I may not see them. I have to mind the resolutive power of the grating ($R=mN$). If N1=1000 lines/mm and N2=500 lines/mm and the grating paces are D1=10^-6 m and D2=2*10^-6 m, will I see different maximums if the light isn'tmonochromatic? Do you know other ways to know if the light of a He-Ne is monochromatic?


I reached a result concerning displacement with quantized time intervals. Am I on to something?

A few days ago, I realized a similarity between distance with constant acceleration, $d = v_i t + 1/2 a t^2$, and the sum of integers up to n, $(n^2 + n)/2$. This came up again today when I decided to work out some formulas for distance and velocity with constant acceleration updated at discrete intervals, as happens in physics simulations I've programmed. > Discretely incremented $v$ and $x$ with constant acceleration, $v_i = 0$ > >   * Add $a\mathrm{d}t$ to $v$ every tick >   * Add $v\mathrm{d}t$ to $x$ every tick > > > $$\begin{align} v_f &= a\mathrm{d}t + a\mathrm{d}t + a\mathrm{d}t + \cdots > \\\ x_f &= v_1\mathrm{d}t + v_2\mathrm{d}t + v_3\mathrm{d}t + \cdots \\\ v_2 > &= (a\mathrm{d}t) + (a\mathrm{d}t) = 2(a\mathrm{d}t) \\\ v_n &= > n(a\mathrm{d}t) \\\ x_f &= (a\mathrm{d}t)\mathrm{d}t + > 2(a\mathrm{d}t)\mathrm{d}t + 3(a\mathrm{d}t)\mathrm{d}t + \cdots \\\ x_n &= > \sum_{i=0}^{n}i(a\mathrm{d}t)\mathrm{d}t = \frac{n^2 + > n}{2}(a\mathrm{d}t)\mathrm{d}t \\\ n &= > \left\lfloor\frac{t_\text{total}}{\mathrm{d}t}\right\rfloor \\\ x_f &= > \frac{1}{2}\Biggl[\biggl(\frac{t_\text{total}}{\mathrm{d}t}\biggr)^2 + > \frac{t_\text{total}}{\mathrm{d}t}\Biggr](a\mathrm{d}t^2) = > \biggl(\frac{t_\text{total}^2}{2\mathrm{d}t^2} + > \frac{t_\text{total}}{2\mathrm{d}t}\biggr)(a\mathrm{d}t^2) \\\ x_f &= > a\mathrm{d}t^2\times\frac{1}{2}\times\frac{t_\text{total}^2}{\mathrm{d}t^2} > + \frac{1}{2}a\mathrm{d}t\frac{t_\text{total}}{\mathrm{d}t} \\\ x_f &= > \frac{1}{2}at_\text{total}^2 + a\mathrm{d}t\frac{t_\text{total}}{2} \\\ > \lim_{\mathrm{d}t\to 0} &= \frac{1}{2}at^2 = \text{normal $v_f$ equation > where $v_i = 0$ and $\Delta a = 0$} \end{align}$$ I'd like to know what relations to theory this or an idea like it has if any, and if it touches on anything else.


How do I know the position of some mass in space (starting at rest) over time when applying earth's gravitational force to it?

I know how to use Newton's law of universal gravitation. But I'm not sure where to go from there. This is for my understanding while working on a game. I'm not trying to do anything fancy here.


The Role of Gravity among the Fundamental Forces of Nature

If we look at the standard model, we have 4 fundamental forces which include   1. Gravity,   2. Electromagnetism,   3. Nuclear weak force,   4. Nuclear strong force. I would like to look at Gravity for a minute. Scientists are still searching for what causes gravity. But is it possible that gravity is only a property of the space-time fabric? For example take a blanket that is stretched tight and put different weight balls on the blanket. As one would see, the balls sink into the blanket differently. If you look at the space-time fabric as the blanket and the different balls as objects in the space-time fabric when it comes to planets and more on the macro scale, thinking gravity as a property of space-time fabric seems to work. Since the standard model deals more at the quantum level though and we are looking at it here on earth where gravity is $9.8\,m/s^2$, it seems to me that the standard model would also need to stand up to say if we had gravity that was $11.4\, m/s^2$ or something more absurd like $20.6\,m/s^2$ . Also could we not say that gravity is really just a property of space-time fabric?


Why does a superconductor obey particle-hole symmetry?

We normally solve the Bogoliubov-de Gennes (BdG) equations in order to compute the energy spectrum of a superconductor. The Nambu spinor is a common object that is used in formulating these equations. In other words, the problem is formulated with the a priori assumption that the superconductors obeys particle-hole symmetry. But how do we know that it will obey particle-hole symmetry? I'm sure that by assuming particle-hole symmetry, computing the spectrum by solving the BdG equations, and comparing with experiments we get a good agreement. So we know it's true. I just don't understand why that is the case. Is there any physical reason behind it?


Why is my approach to the equation of time off by a constant?

I'm trying to better understand the causes for the equation of time by deriving an approximation from first principles. My naive approach, $EOT_{NAIVE}$, is to take the difference between the right ascension of the mean sun, $\alpha_M(t)$, and the right ascension of the "real" sun $$EOT_{NAIVE}(t)=\langle\dot{\alpha }\rangle\cdot(t-t_0)-\alpha(t)$$ where $\alpha(t)$ is simply the "actual" position of the sun at time $t$ from ephemeris data (e.g. either of the RA values reported by JPL HORIZONS, or a similar source) and where $\langle\dot{\alpha }\rangle = 24/365.242$ and $t_0=t:\alpha_M(t_0)=0$. To confirm that this is about right, I compare my result with what I get using the USNO definition of GMST $$EOT_{GMST}(t)=GMST(t)-(t-t_{noon})-\alpha(t)$$ and with $EOT_{POLY}$, a standard polynomial expression from Dershowitz & Reingold as described in The Clock of the Long Now documentation. But the values I get for $EOT_{NAIVE}$ (blue dashed line) are consistently about $7.4$ minutes greater than these reference methods (exactly $7.4537$ for $EOT_{GMST}$, red line; and within a couple seconds of that for $EOT_{POLY}$, gray outline) give: ![enter image description here](http://i.stack.imgur.com/cqWAE.png) Surprisingly, I can fix this by simply changing $t_0$ from the date, $t_E$, on which the most recent vernal equinox occurred (2012-03-20T05:14:33Z) — which I had assumed would give a "starting" RA of 0 — to 2012-03-22T02:20:32.41Z, which brings my approximation (blue line) exactly in register with the GMST approach (gray band), and within a couple seconds of the polynomial approach (red line), as can be seen by taking the difference between the GMST approach (gray band) and each of these approaches (note the change of scale, this figure essentially takes the difference between $EOT_{GMST}$ and each of the plots in the first figure above, and "zooms in" on the gray band): ![enter image description here](http://i.stack.imgur.com/njj15.png) Why should changing the date in this way "fix" my approximation? Why doesn't my approximation work with $t_0=t_E$? Why does it work perfectly with a different date? Are the approaches I'm using as references ($EOT_{GMST}$ and $EOT_{POLY}$) the right ones for this kind of exploration? * * * Note that the reported RA of the sun at 2012-03-22T02:20:32.41Z is $0.1146^h$, and that the fit between $EOT_{GMST}$ and $EOT_{NAIVE}$ using that date is quite good (this figure zooms in further on the gray band in the second figure above): ![enter image description here](http://i.stack.imgur.com/1Mtl5.png) Note also that despite the documentation for the ephemeris I use claiming that "light time delay is not included", the values I get match those from JPL's "airless apparent right ascension and declination of the target center with respect to the Earth's true-equator and the meridian containing the Earth's true equinox-of-date. Corrected for light-time, the gravitational deflection of light, stellar aberration, precession and nutation."


Does a particle in a spherically symmetric infinite square well potential exert a force on the inner and outer shell barrier?

For a particle in the potential: $$V(r) = \begin{cases} 0 & \text{a < r < b}\\\ \infty & \text{otherwise.} \end{cases}$$ Does this guy in the ground-state exert a force on the shells a and b? I don't think so because at ground-state both the centrifugal term of the radial equation and the angular momentum of the particle is zero. What say you brethren?


Why isn't temperature measured in units of energy?

Temperature is the average of the kinetic energies of all molecules of a body. Then, why do we consider it a different fundamental physical quantity altogether [K], and not an alternate form of energy, with a dimensional formula derived from three initial fundamental quantities, length [L], mass [M] and time [T]?


Simple friction formula for a car

I am making a 2D driving video game, and I would like to know the "simple" formula for calculating the friction force between the car and the road. I have read lots of friction diagrams involving balls rolling down inclined planes, but I'm having trouble applying it to this problem. Consider a car driving on a horizontal road, with some coefficient of friction, $\mu$. Its velocity is $v$, mass is $m$, and a constant driving force is accelerating the car at $a\ \mathrm{m/sec}^2$ in the forwards direction. What is the friction force, $F$, that acts on the car in the backwards direction? I _guess_ that the friction force is proportional to the car's speed, but I can't explain this guess. The reason I think that is because of a handful of case studies. Say that the driver's foot is on the pedal such that the car will get up to $16\ \mathrm{m/sec}$:   * If the car's velocity is $0\ \mathrm{m/sec}$, there is no force accelerating it backwards (it is stationary).   * If the car's velocity is $8\ \mathrm{m/sec}$, the friction force accelerating it backwards must be less (in magnitude) than the driving force, and hence the car keeps accelerating.   * If the car's velocity is $16\ \mathrm{m/sec}$, the friction force accelerating it backwards is equal (in magnitude) and opposite to the driving force, and hence the car maintains a constant velocity.   * When the driver releases the pedal, the same friction force accelerates the car backwards, but now there is no driving force, so the car drifts slowly to a halt. It would therefore make sense that $F \propto -\mu v$, such that in the first case, $F = 0$; in the second case $F = -\frac{1}{2}ma$; in the third and fourth cases, $F = -ma$, where $a$ is whatever acceleration is required to maintain a constant velocity of $16\ \mathrm{m/sec}$. But I can't figure out the exact relationship between $F$ and $v$, or explain why they are proportional.


Change of variables in calculating the integral of multivariable differential entropy

I have already asked this question in math.SX but here might be more proper. So I decided to put a copy here and delete the one which is not the one that got an answer: I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int p(x)\log(p(x))dx=\int\limits_{0}^{1}\frac{d}{dp}[\log F^{-1}(p)]dp$$. Where $F$ is the cumulative distribution function of $p$. To extend the idea for multivariate density functions my idea was to try to integrate on level sets of $p(x)$ to simulate somehow what happens in one dimension. As a results I took a look at this question and topics relevant to pullback method in differential geometry and co-area formulation but couldn't come up with anything by myself. Does anybody know of any method to get a similar formula for differential entropy when $p$ is a multivariable density function? **EDIT** : As it has been clarified by "NowIGetToLearnWhatAHeadIs" CDF does not have a unique generalisation. What I thought to be the natural one is: $$F(x_1,...,x_n)=\int\limits_{-\infty}^{x_1}\int\limits_{-\infty}^{x_2}...\int\limits_{-\infty}^{x_n}p(x_1,...,x_n)dx_ndx_{n-1}...d_2d_1$$


Functions of time

Following are the functions of time:   $s(t)$ = distance a particle travels from time $0$ to $t$   $v(t)$ = velocity of a particle at time $t$   $a(t)$ = acceleration of a particle at time $t$. If we want to see how the position of a particle changes with respect to time only, then its velocity must remain constant with time. Likewise, if we want to see how velocity varies with time, then the distance between the former position of the particle and the current position should remain constant with time. Similarly, if we want to see how acceleration varies with time, then the difference between the initial velocity $U$ and final velocity $V$ should remain constant with time. Is this what the above functions of time tell us?


Momentum And Energy Problem

The information to the question is, "A $59.0~kg$ boy and his $38.0~kg$ sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with a velocity $3.00~m/s$ toward the west. Ignore friction." The specific question I am working on is, "How much potential energy in the girl's body is converted into mechanical energy of the boy–girl system?" Well, I said that $E_{mech,~i}=0$, because no one has kinetic energy (no one is moving), and neither of them have the potential to move, that is, until the girl commences with the push; but, $E_{mech,~f}=PE=KE$, when she uses her potential energy to apply a force over a distance, on her brother, and by doing so she changes the energy of the system, by changing the speed of both of them, meaning the potential energy will convert to kinetic energy. I've tried to plug every piece of data given to me, but I still couldn't find how much $PE $ was converted to mechanical energy. How do I find it? Is my analysis correct; is there anyway to improve it? Edit: I have another question: why can't internal forces of a system cause momentum to not be conserved?


Solution of a partial differential heat equation with derivative and boundary conditions

I want to solve the following partial different equation. Find $u(x, t)$, satisfying $u_t = u_{xx}$ , $u(x, 0) = x − x^2$ , $u(0, t) = T_0$ , $u_x (1, t) = 0$ and $|u|$ is bounded. Using separation of variables, and eliminating solutions that diverge with time, one gets $$u(x,t)~=~e^{ -\lambda^2t}(A \cos(\lambda x)+B\sin(\lambda x)).$$ How do I proceed after this? The condition $u_x (1, t) = 0$ gives $ A=-B \tan(\lambda )$ which is giving me a contradiction for the condition $u(0, t) = T_0$ , as you get $$-e^{ \lambda^2}B\tan(\lambda)~=~T_0.$$


Number of Grassmann generators for Dirac field?

How many Grassmann generators are sufficient for the description of a Dirac spinor in 4 dimensions? _i.e._ The Dirac field is a map to $\Lambda_N$, the space of supernumbers with $N$ real Grassmann generators. What is $N$? This is a follow up question to my previous question Grassmann Paradox Weirdness. I am following Prakash's book Mathemaical Perspectives on Theoretical Physics, where they say a supernumber $z\in\Lambda_N$ can be thought of the extension of complex numbers by the addition of $N$ Grassmann generators $\zeta^1,\, \zeta^2,\,\ldots \zeta^N$. The most general supernumber is written $$z = z_0+z_i\zeta^i+\textstyle\frac{1}{2!}z_{ij}\zeta^i\zeta^j+\ldots,$$ where $z_i$, $z_{ij}$, $\ldots$, are complex-valued and antisymmetric. The odd part of this is anticommuting and is used to describe Fermion fields. The book says that for finite $N$, it takes $2^{N-1}$ complex numbers to specify an anticommuting number. How do I figure out how many Grassmann generators $\zeta^i$ I need to specify a Dirac spinor in 4 dimensions.


Finding an efficient strategy for walking

Let's say you are already walking at a maximally efficient combination of pace and stride (or $\omega$ and $X_0$ I guess) but you need to reach your destination faster. Should you increase/decrease your pace and/or stride to achieve this goal (assuming that any adjustment will result in a net increase in speed)? How can I interpret this question as a question about kinetic energy of oscillators? I have tried solving the equation $KE'=0$ to find a minimum $KE$ but this seems like the wrong approach, for all that seems to tell me is the if you do not walk (ie $w$ or $X_0$ is $0$) then yes, of course you have kept your kinetic energy at a minimum. Where does this idea about "efficiency" come into play? Edit: Pace is the frequency of steps, stride is the length of steps, and average speed is the product of pace and stride.


How to find that a molecule has zero spin?

I read that Ne has $S=0$. How can this be found, knowing the electron configuration? Electrons, protons and neutrons all have 1/2 spin. The Pauli exclusion principle implies that the even number of electrons are half up-spin and half down-spin, so that they sum to 0. Is that argument valid, and if so, can the same be applied to show that the total spin of neutrons and protons, which occupy the same space, is 0? I suppose that would imply that $S=0$ for all molecules with even numbers of electrons and protons+ńeutrons.


precise definition of "moduli space"

I'm curious what the precise definition of the moduli space of a QFT is. One often talks about the classical moduli space, which then can get quantum corrections. Does this mean the quantum moduli space is something like the set of minima of the effective potential in the 1PI action (or does the 1PI action already presume a choice of vacuum?)? Is there a non-perturbative definition, without referring to field configurations or effective actions (eg, how do we define it in string theory, where there are no fields)? Maybe it can be identified with some submanifold of the (projective) Hilbert space? Can the metric or other structure be defined in such an invariant way?


How to calculate the total angular momentum (J) values

How can I calculate the values of $J$ (total angular momentum) for a particular term, for instance, ${}^3P$?


Migration of an ion in a non-conductive medium

Someone asked me why, when performing DNA electrophoresis we need to put the gel inside a conducting buffer. Couldn't we just run it in distilled water? My answer was that if we had distilled water we would not be able to close the circuit between the anode and the cathode of the electrophoresis apparatus, no current (or very little) would flow and no movement would be possible. The next question then was: why do we need a current to pass through the system in order to have movement of the DNA (or, in general, any charged particle) towards the anode? What physical forces enter into play? In other words: if we had a bath of distilled water, we put two electrodes connected to a generator at each end and put a charged molecule in the middle of the bath, would it be able to migrate? Why? Which forces would be involved? I found this document which essentially confirms my first answer: > **Electrophoresis Buffer:** Several different buffers have been recommended > for electrophoresis of DNA. The most commonly used for duplex DNA are TAE > (Tris-acetate-EDTA) and TBE (Tris-borate-EDTA). DNA fragments will migrate > at somewhat different rates in these two buffers due to differences in ionic > strength. Buffers not only establish a pH, but provide ions to support > conductivity. If you mistakenly use water instead of buffer, there will be > essentially no migration of DNA in the gel! Conversely, if you use > concentrated buffer (e.g. a 10X stock solution), enough heat may be > generated in the gel to melt it. However, I have not been able to find a proper explanation of the physical forces involved in the process. PS: of course in a real DNA electrophoresis the charged molecule is trapped in a gel, which itself has been made using a conducting buffer, so the question becomes probably a tad more tricky...


what is the 'resolution' of the reality in pixel terms

1.What resolution should a TV screen have so that its image were so faithful as reality as if the TV were a window ? 2.Also what would happen if Physics could reproduce a 'pixel' of the size $ l_{p}^{2} $ the square of the Planck length ? Would be then this the resolution of our reality like watching through a real window ?


Rigorous definition of frame of reference

I'm looking for a mathematical definition of frame of reference. Most of the textbooks I have seen take it for granted and they just refer to some set of spacetime coordinates. A more mathematical definition I found defines spacetime $E$ as an affine space over $\mathbb{R}^4$ and then defines a frame of reference as a 5-tuple $(P,v_1,v_2,v_3,v_4)$ where $P\in E$ and $(v_i)$ is a basis of $\mathbb{R}^4$. But with this definition I cannot make sense of the phrase "a frame moving with respect to another".


What is the cause of Special Relativity

Why does special relativity exist? Via studying the outcome(s) of such relativity, one can end up with all the equations involved in special relativity. Thus one runs into things such as contraction, dilation, transformations, velocity addition, etc. But these are the outcomes. My question is what is the cause of special relativity, and is the knowledge of this cause currently being shared in the world of physics? Or, to put it another way, is it currently accepted despite there being no cause that is known of, thus an acceptance of an effect without a cause? If you give this question a negative vote, please present your reasoning and or evidence to support such a vote.


Thermodynamics of a bar, specific heat

I have a problem and I don't understand how the book solves it. It says I have a bar such that $$F=aT^2(L-L_0)$$ where $F$ is the force or tension of the bar, $T$ is the temperature, and $L$ the longitude. $L_0$ is the longitude when $F=0$. It says that $C_{L_0}=bT$ and it asks me about the general function $C_L(L,T)$, being $C_L$ the specific heat at constant $L$. The book starts solving it with this expression: $$\left(\frac{\partial C_L}{\partial L} \right)_T=-T\left(\frac{\partial^2 F}{\partial T^2}\right)_L$$ From there, I know how to go on, but I have no idea where that comes from. I have tried to get that out of the generalised Mayer relation but if that's the way, I can't do it.


Where are exactly the charges of charged capacitor?

If a capacitor is connected to a battery and is charged, are there charges inside the wires or do they just accumulate on the surfaces of capacitor?


The units of gain and number of atoms in population inversion in a laser

I am following my university course notes on amplification in laser media, and have come across expressions for the **gain** of a medium, but the notes are not exactly rigorous... The expression given for the gain $G$ in units of dB m$^{-1}$ is: $$G = 10 \log_{10}(e^\gamma) \mathrm{dB} \mathrm{m}^{-1}$$ where $\gamma$ is _also_ the gain coefficient for the media with units m$^{-1}$. However later in the notes it gives the expression $$\gamma = \Delta N \sigma$$ where $\Delta N$ is the "number of atoms in population inversion" and $\sigma = [\mathrm{m}]^{-2}$ is the "cross-section for stimulated emission". This clearly shows that $\Delta N = [\mathrm{m}]$, which doesn't make much sense to me. Also the first expression implies that $\gamma$ is unitless, so I am mighty confused.


Derivation of the centrifugal and coriolis force

I was wondering how easily these two pseudo-forces can be derived mathematically in order to exhibit a clear physical meaning. How would you proceed?


Displacement of a body in free fall

Suppose a ball is thrown up from the ground with some velocity. Let vertically up be the positive direction and vertically down be the negative direction and let the thrower be at the origin. What will be the sign of displacement after it has reached it's maximum height and is still descending? I think it will be negative since it's displacement will be in the negative direction, although it's position will always be positive. Am I correct?


Where does the idea gravity=curvature of spacetime really come from?

I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I know that the starting point for Einstein's considerations was the equivalence principle. Nevertheless I can't make the connection from there, to why gravity is the curvature of spacetime. Any help or reference to where this is explained would be much appreciated.


Definition of force, kinetic energy and momentum

**I've edited the post. Q1 and Q4 are the important ones but I didn't delete Q2 and Q3 since some older answers would not make sense anymore.** To begin with, the formula of the kinetic energy $T$ is $\frac{mv^2}{2}$. Furthermore momentum is conserved $\Sigma m_{i}\vec{v_{i}}=const.$ Then you have the definition that force is the change of momentum with respect to time $\vec{F}=\dfrac{d(m\vec{v})}{dt}$. I've read the chapters concerning mechanics of Physics for scientists and engineers by Giancoli and the Feynman Lectures. Giancoli introduces the arbitrarily work as $W=\int\vec{F} \cdot d \vec{s}$. From this definition of work he derives the kinetic energy to be $\frac{mv^2}{2}$. In contrast to that, in the Feynman lectures you never get a derivation of $\frac{mv^2}{2}$, but it is shown that $\dfrac{dT}{dt}=\vec{F}\cdot\vec{v} = \vec{F}\cdot\dfrac{d\vec{s}}{dt}$. Then it is shown that $dT=\vec{F}\cdot d \vec{s}$ and as a consequence $\Delta T = \vec{F} \cdot \vec{s}$ which is called work. http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S1 **Now I've got some questions:** Q1 _Is $F=\dfrac{d(mv)}{dt}$ **just an arbitrary definition** or is there something "more" behind the formula for force?_ Q2 _Is $W=\int\vec{F} \cdot d \vec{s}$ **just a definition** or is there something more behind? I mean, can you derive the formula for work not by taking the formula for the kinetic energy as given._ Q3 **How to derive the formula for the kinetic energy and work only form the conservation of momentum $\Sigma m_{i}\vec{v_{i}}=const.$?** Q4 **How are work and kinetic energy defined?** I have found both: A) Kinetic energy is doable work and work is $F=\dfrac{d(mv)}{dt}$. B) The formula for kinetic energy is: $\frac{mv^2}{2}$. Then, after some maths it follows that $\Delta T = \int \vec{F} \cdot d\vec{s}$. **It doesn't make sense to define 2 things in that way. This would be circular logic.** P.S.: I am not a native English Speaker so feel free to edit.


The possibility of free electrical energy?

Please excuse my lack of knowledge/understanding. **Question:** Why Nikola Tesla's Free Energy concept was never worked upon? Even today. **Context:** Now that we know Nikola Tesla was a genius and did exceptionally well with his works and the **Free Energy** concept. Why his idea was never adopted or was worked upon, even now? I know, back then, Edison didn't let him do much. But now, in our times, why GE or any other company or well the physics `PhD`s aren't following up? Why CERN is not working over it? By the articles that I have read over the internet I am under an impression that everyone (now) feel that we have "missed" the free-energy _chance_ that we had as a planet, that Nikola Tesla was trying to bring in. I see a lot of _brag_ (to an extent, as well) which leaves no doubt that his Free-Energy concept would have been **must-have**. But if I look at the actual followups, no one _it seems_ got **actually** interested to pursue his theories; or is it that no one is capable enough to do it? Which would be strange, given that we are at the peaks of technological advancements. **_How hard is it to build a prototype? Really?_** Or people might have worked over it, but Tesla's blue-prints were not correct, and that is why no one came up with the actual implementation of it? Or is it that he was _really_ a crackpot as politically-publicized by Edison? I would really appreciate an answer, because this is beyond me, to understand why no one followed up on this concept? Thank you, in advance.


A small object on the opposite side of the earth's orbit with 0 velocity,"in line" with the core - would it collide with the earth?

If a small object (say mass 1 kg), at the opposite side of the earth's orbit (six months away), was placed in the same orbital position as the earth's core, at zero velocity in all directions, would the object collide with the earth or would it have been drawn by the sun's gravity away from a collision with earth? A naive answer would simply calculate how far "inward" the object would travel in six months, when accelerated by the sun's gravity, and if the distance covered is greater than the earth's radius, a collision would be averted. But I started thinking about the earth's own gravitational influence as it orbited closer to the object. How would we get an answer to this question?


How did Jackson get the exact solution to potential in polar coordinates

On page 45 of Jackson's Classical Electrodynamics (3ed), He solved a polar potential problem with known source distribution. $$ \frac{1}{\rho}\frac{\partial}{\partial \rho}\left( \rho\frac{\partial \psi}{\partial \rho} \right)=-g(\rho)\\\ \psi(\rho)\bigg|_{\rho=1}=0\\\ $$ with the given source $$ g(\rho)=-5(1-\rho)+10^4\rho^5(1-\rho)^5 $$ To me, this second order Sturm-Liouville ODE can only be solved if we have two boundary conditions, but in this case, we don't know the other BC.


Must every isometry have an associated Killing vector?

I understand that the flows of Killing vector fields are isometries, and that one-parameter groups of isometries have an associated Killing vector which generates them, but are your Killing vectors guaranteed to give you all possible isometries of the manifold? I guess what I'm trying to ask is, do isometries necessarily come in families, or can you have "isolated" isometries which can't have an associated Killing vector field?


Twin Paradox, calculating spacetime intervals from both perspectives

I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me. The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path. * * * **According to Alice:** $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$ Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$. * * * The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations. * * * **According to Bob:** $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$ Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated!


Why is carbon dating limit only 40,000 years?

For an example, when they tried to get the carbon dating for presence of Aboriginal people in Australia they get to the number 40,000. But it could be much earlier. Why is that 40,000 years limit for carbon dating methods?


Boltzmann distribution: derivation from canonical distribution

I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not sure what's wrong with it :). Any guidance would be much appreciated! I believe that the probability density function for the energy, $E$, of a random particle in a system at equilibrium is meant to look like this: $$f_E(x) = Ae^{-\frac{x}{kT}}$$ To simplify things I'll just consider the particular case: $$f_E(x) = e^{-x}$$ The cumulative distribution function is then: $$Pr(E\leq x) = F_E(x) = 1-e^{-x}$$ Now suppose all the particles in our system are of mass $1$, so, ignoring potential energy, the random variable governing the speed of a given particle is $V = \sqrt{2E}$. Now we want to find the distribution of the speeds. We can do this as so: $$F_V(v)=Pr(V\leq v) = Pr(\sqrt{2E}\leq v) = Pr\left(E\leq \frac{v^2}{2}\right) = F_E\left(\frac{v^2}{2}\right)$$ and so $$F_V(v)=1-e^{-\frac{v^2}{2}}$$ and finally $$f_V(v)= ve^{-\frac{v^2}{2}}$$. This is _almost_ right, however, the Maxwell-Boltzman distribution would predict a pdf with $v^2$ rather than $v$, something like: $$f_V(v)= \sqrt{\frac{2}{\pi}}v^2e^{-\frac{v^2}{2}}.$$ Can anyone point out the mistake in my logic? Thanks!


Commutator problem

I am trying to calculate the following commutator $$[\mathcal{H}_0(r',t'),\psi(r,t)]_-$$ where $\mathcal{H}_0 = (\frac{1}{2m}\nabla^2 + e\mathbf{A}(r',t'))^2 + e\phi(r',t') - \mu$, and $\mu$ is the chemical potential. It is supposed to be $0$, but I don't understand why it is not a a delta- function $\delta(r-r')[\mathcal{H}_0(r',t'),\psi(r,t)]_-$ which is not necessarily $0$. Could someone please help me understand why the anticommutator is 0?


Why is there a Majorana zero mode in the $\pi$ flux core of the p+ip superconductor?

In this review paper (http://arxiv.org/pdf/1202.1293.pdf), the author shows that threading a $\pi$ flux through a 2D $p_x+ip_y$ superconductor will trap a Majorana zero mode at the flux center. The deduction involves solving differential equations, and is hard to generalize to arbitrary dimensions or arbitrary gauge defects. Since Majorana zero mode is topologically robust, I would expect a simpler and less detail-dependent way to show that there is indeed a Majorana zero mode in the $\pi$ flux of a $p_x+ip_y$ superconductor. Does anyone know such a simpler approach? PS: The answer I expect should be general enough that can be applied to $\pi$-flux tubes in 3D topological insulator or topological superconductors as well.


Why does the index of refraction change the direction of light

I've been studying in optics the macroscopic maxwell's equations, and how electromagnetic fields propagate through different mediums. Over there, the index of refraction appears, as a complex function that generally depends on $\omega$, with both refraction and absorbtion terms: $n_c=n+i\kappa$. I understand how this affects the speed at which light propagates, both macro and microscopically, and how this affects to how light is absorbed when it transmits through the medium, but I don't get yet how this index changes the direction of light, producing dispersion. I mean, all time I'm seeing those effects as something that separates dispersive from non dispersive mediums and all that stuff, and I don't know how light is actually dispersed. As it's something that happens when changing medium, I guess it's an interface effect, and we haven't seen those effects yet, but I would like to know an explanation.


Why is the energy spectrum continuous for a plane wave when it has energy less than the potential barrier?

Please explain it in the context of this task: we have a potential barrier that looks like $\prod$, with $E<U$. There are 3 regions: 1) no field   2) barrier   3) no field Solution could be found as * * * 1) $\psi_{1} = A\cdot e^{-ikx}+B\cdot e^{ikx} $   2) $\psi_{2} = C\cdot e^{-ik_{0}x}+D\cdot e^{ik_{0}x}$   3) $\psi_{3} = 0 +F\cdot e^{ikx}$ (cause we haven't got a reflected wave here) where $k_{0}^{2}=2m_{0}(U_{0}-E)/h^{2}$ and $k^{2}=2m_{0}E/h^{2}$ * * * So why in this case spectrum is continuous?


Why are von Neumann Algebras important in quantum physics?

At the moment I am studying operator algebras from a mathematical point of view. Up to now I have read and heard of many remarks and side notes that von Neumann algebras ($W^*$ algebras) are important in quantum physics. However I didn't see where they actually occur and why they are important. So my question is, where they occur and what's exactly the point why they are important.


Question about the parallel axis theorem

I've a question about the parallel axis theorem. So I'm perfectly OK with the derivation of the proof in 2D. However in the typical derivation they just say this in my textbook "Because the Zi coordinates are not taken into account into this calculation, we can generalise the summations to include all particles for all slices" How is this possible? I don't get it how you can generalise for all slices! Please help me, and thank you very much.


Is there a mistake in this example in my mechanics textbook?

This is a worked example in my mechanics book: ![enter image description here](http://i.stack.imgur.com/1FKOF.png) ![enter image description here](http://i.stack.imgur.com/rxymK.png) But I think this worked example is wrong. In the 6th line. shouldn't $X=4R$ as the expression for distance above the surface is $(X-R)$ and when the particle P is a distance of 3R ABOVE the surface, $v=0.5(gR)^{0.5}$ could someone please tell me if I am wrong or the book is wrong... I mean books a re rarely wrong and so I am not sure what to follow


How to count quarks using Deep Inelastic Scattering?

The Wikipedia article on deep inelastic scattering suggests that the experiment shows baryons have three point of deflections (corresponding to three quarks) and mesons have two points of deflection. How are the electrons fired in this experiment being detected, and how exactly do the two or three points of deflection appear in the data? Are they fired at a target consisting entirely of baryons, or are collisions with non-baryons somehow filtered from the data?


Electric arc due to static discharge in a T-Shirt possible?

Yesterday, when I came home, I went to the bathroom (lights off) and i took off my T-Shirt (100% cotton) which I wore under a Shirt (50% cotton, 50% polyester). I _believe_ to have seen a small but well visible light arc. **Question:** Is the amount of static discharge in T-Shirt big enough to produce a visible light arc?


Can it be considered as time travel?

India is about 8.5 hours ahead of US. If some person starts from India at 9 pm(according to his wrist watch) and reaches US in 2 hours(just assume),at that time his wrist watch will b showing 11 pm but as he his in US so he will have to match his watch according to US time so he will adjust his watch to 4:30 pm.This means that he is goin to live those 6.5 hours again in his life,but this time in US.So can this be considered as time travel or i am wrong.Please explain.


Why is air invisible?

I think that something is invisible if it's isolated particles are smaller than the wavelength of visible light. Is this correct? Why is air invisible? What about other gases and fumes which are visible?


Why glass is transparent?

Once I asked this question from my teacher and he replied "because it passes light", "and why it passes light" I asked and he said "because it is transparent". Same question again, Why glass is transparent. Why it passes light while opaque objects do not?


Find vector position in time give this graph?

Graph is position x position. There are 3 points, $A$, $B$ and $C$.   * $A(0,2)$    * $B(4,2)$    * $C(6,0)$  Particle travels from $A$ to $B$ and from $B$ to $C$ at constant $v = 2 ~m/s$. Find vector position in time for the $BC$ part of the motion? For the AB part I did this: $v$ is constant, so position in time is given by $2\hat{i}\cdot \mathrm{t} + 2\hat{j}$. Velocity in $i$ direction x time, plus the constant $2\hat{j}$ vector component. For BC the "angled" displacement vector is making things a bit harder. Displacement vector has two components, $2\hat{i} + (-2\hat{j})$. Absolute value of it is $2\sqrt{2}$. Tried to decompose the velocity vector in two: $\hat{i}\sqrt{2}$ $-\hat{j}\sqrt{2}$. Then integrate each component and add up the two. But the result is not quite right... Integrated each component independently: $\hat{i}\mathrm{t}\cdot\sqrt{2} + c$ and $-\hat{j}\mathrm{t}\cdot\sqrt{2} + c$


Why do Engineers manipulate coordinate systems?

When tackling a physics problem, An Engineer will manipulate the axes/coordinate system where a Mathematicians and/or Physicists will use the original coordinate system and math. Why do Engineers think differently? I know its likely because that is how they are taught, but why are they taught that way?


How does someone move space?

I've heard that we've known components of the universe appear to be flying apart faster than $c$, and that this can happen because the space containing these materials are also moving apart. How does someone move space? How efficiently can it be done? I would like to get on a rocket and then move the space very close to Sedna and see what it looks like. What are the physical barriers to doing that?


Orthochronous Lorentz transformations are time-preserving and $SL(2,\mathbb{R})$

Let's consider the psuedosphere/hyperboloid in $\mathbb{R}^{1,2}$ given by $$x^2+y^2-z^2=-R^2.$$ We know that the Lorentz group $$O(1,2)=\\{ A \in Mat(3,\mathbb{R}): A^tGA=G \\},$$ where $G=diag(-1,-1,1)$ leaves the pseudosphere invariant. Now we are interested in the following facts:   1. How can we show that the orthochronous Lorentz group $O_+(1,2)=\\{ A: a_{33}>0 \\}$ is subgroup and, more important, maps upper cone to upper cone?   2. What is the relation between groups $O_+(1,2)$ and $SL(2,\mathbb{R})$?


Bohm, David (1980), Wholeness and the Implicate Order, London: Routledge, ISBN 0-7100-0971-2 As we can see from the book above, David created the hypothesis of the holopraphic universe in 1980. And we can see from the words below, the holographic universe of string theory is created in 1995. Using this approach he has argued that near a black hole, quantum fields could be described by a theory in a lower dimension.[Stephens, C. R.; 't Hooft, G.; Whiting, B. F. (1994). "Black hole evaporation without information loss". Classical and Quantum Gravity 11 (3): 621] This led to introduction of the holographic principle by him and Leonard Susskind.[Susskind, L. (1995). "The world as a hologram". Journal of Mathematical Physics 36 (11): 6377–6371.] So Whether the holographic universe of string theory prove the hypothesis of the holographic universe created by David Bohm or not? Or what is the link or differences between them? I have read the book of "The Elegant Universe" and the book of "The Fabric of the Cosmos:Spaca,Time, and the Texture of Reality", but Brian Greene doesn't mention about David Bohm. Thank you in advance!


Latent heat of vaporization

The molecular weight of water is 18.015 gram. The number of moles of water in one liter (1000 gram) will be: $3.34\times 10^{25}$ molecules (in 1kg). We know that latent heat of vaporization of water is $L_v= 2.26\times 10^6$ then the amount of heat required to rise one molecule of water will be $1.47\times 10^{-19}$ joule: Does this calculation is correct according to macroscopic and microscopic physics?


Understanding Tensors

I don't seem to be able to visualize tensors. I am reading The Morgan Kauffman Game Physics Engine Development and he uses tensors to represent aerodynamics but he doesn't explain them so I am not really able to visualize them. Please explain in very simple ways. I just want to understand the basics.


Size of universe after inflation?

Wikipedia states the period of inflation was from $10^{-36}$sec to around $10^{-33}$sec or $10^{-32}$sec after Big Bang, but it doesn't say what the size of the universe was when inflation ended. Just saw a Brian Greene show on the Multiverse and I thought I heard him say size was galactic scales when inflation ended. However I've also read size was about a basketball. Are there multiple theories with different resulting sizes? Does 'size' even mean anything in this context?