Are the limits to prediction by QM provable?

Quantum mechanics states that only two aspects of a quantum system can be predicted with certainty: 1) the average and 2) the standard deviation of many measurements of identically prepared quantum systems. (At least, that's the message I took from my Quantum Chemistry textbook). I believe Einstein did not like the fact that you couldn't predict more than those two things -- that there's a fundamental limit to what science can help us predict about events that do occur. Any other measured information from the system is essentially "random". What I'm wondering is whether QM has definitively proven that we are unable to obtain any other information about the system? For example, assume that with each quantum system there is associated a pseudo-random number generator that we don't know about. The "random" information is actually the result of the next iteration of the generator. All statistical tests would say that the information is random, but if we knew what function the PRNG was using, we could predict this data! I remember reading that experiments have ruled out hidden variable theories. Is a PRNG function an example of a hidden variable? (By the way, I'm not trying to challenge QM; I learn best by asking contradictory questions like this).


Is there current flowing though a power source?

Would there be any current flowing though an ideal voltage source? I am asking this more specifically when you are analyzing a circuit and you are doing nodal analysis. I know i=v/r, but since the source is ideal, resistance would be zero or infinity?


Does Dirac's idea of filled negative energy states make sense?

Please bear with me a bit on this. I know my title is controversial, but it's serious and detailed question about the _explanation_ Dirac attached to his amazing equations, not the equations themselves. Imagine for a moment that someone signed onto this group and proposed the following:   1. Empty space is a lot like a metal, or maybe a semiconductor, because like those materials its chock full of negatively charged electrons occupying different velocity states, only...   2. Unlike metals or semiconductors, the density of electrons in any one region of space is _infinite_ , because there is no limit to how fast the electrons can move. That is because these are negative energy states in which an electron can _always_ move faster simply by emitting a photon, so there's not "bottom" to how far they can drop and how dense they can become, and...   3. Unlike metals or semiconductors, there is no exactly balancing sea of positive atomic charges, well, unless maybe there are infinite numbers positively charged atoms too, and...   4. The resulting infinite negative charge density of real electrons not only doesn't matter but is in fact completely and totally invisible for some reason, and...   5. The resulting infinite mass density of electrons (recall that these are quite real electrons, only in odd negative-energy kinetic states) also doesn't matter, and...   6. Unlike the Fermi sea of a metal conduction band, removing an electron from this infinitely dense sea of electrons for some reason _doesn't_ cause other electrons to collapse into it and fill it, even though the negative kinetic velocity electrons are pushed by exactly the same Pauli exclusion forces as the ones in a Fermi band; in short, for reasons not clear, semiconductor-style hole stabilization applies while metal-style hole filling does not (is there a band gap going on here?), and...   7. Since the infinitely dense negative charges become invisible for no particular stated reason when the electrons fall into negative energy states, these unexpectedly stable open states in the negative energy sea have net positive charge, even though...   8. ... such missing states categorically should have _zero_ charge, since in sharp contrast to the positive ionic background of metallic and semiconductors, the vacuum has no background charge at all, which should leave holes in the mysteriously invisible negative kinetic energy just as uncharged and invisible as that sea for some reason is, and...   9. Even if you _do_ assume that the negative kinetic state electrons have visible charge, their infinite density would make the "comparatively" but infinitesimally smaller positive charge of such a hole invisible, and...   10. Repeat this process for every other kind of particle in existence, and...   11. If you have done all of this and done it correctly, congratulations: You now understand conceptually what anti-electrons (positrons) and other anti-particles are. First question: Have I misrepresented any of the implications of Dirac's explanation of positrons as holes in an infinite sea of negative-kinetic- energy electron states? What I have tried very hard to do is nothing more than make a list of the implications of a physics idea, just as people do all the time on this group. Who said it should not really be the issue, not if we are talking about an unelaborated _explanation_ rather than the math itself. Second question: If someone had proposed a theory in this forum like the one I just described, _and you had never heard of it before_ , what would you have thought of it? Please be honest. My point in all of this obviously is this: While Paul Dirac's amazing equations (they really are) managed to predict antimatter, his _explanation_ for why his equations require antimatter is... shall we say incompletely analyzed, to put a nice spin on it? A final thought: Has anyone ever seriously tried to make Dirac's _conceptual_ negative energy sea ideas, the ones that he espoused in his Nobel Lecture, into a real, working theory? And if so, how did they deal with the various issues I described above? (Me, I just think antiparticles are regular particles moving backward in time. Yeah, that's a pretty weird idea too, I know...)


Dimensional reduction from $3+1$ to $2+1$ for $\cal{N}=2$ vector superfield

Let the supersymmetry transformations for the chiral multiplet $(z_k,\psi_{kL},f_k)$ be, $\delta z_k = 2i \bar{\alpha} \psi_{kL}$ $\delta \psi_{kL} = D_\mu z_k \gamma ^\mu \alpha_R + f_k \alpha_L$ $\delta f_k = 2i\bar{\alpha} \gamma^\mu D_{\mu} \psi_{kL} - 2ie\bar{\alpha}\lambda _R z^k$ Similarly let the gauge multiplet $(A_\mu,\lambda,D)$ transform as, $\delta A_\mu = i \bar{\alpha}\gamma_\mu \lambda$ $\delta \lambda = -\frac{1}{2}F^{4}_{\mu \nu}\gamma_4 ^{\mu \nu} \alpha + D \gamma_5 \alpha $ $\delta D = i \bar{\alpha}\gamma_5 \gamma ^ \mu \partial _ \mu \lambda$ Here in $F^4$ the $\mu$ and $\nu$ go over indices $0,1,2,3$ and $\gamma_4 ^{\mu \nu} = \frac{1}{2} [\gamma_4^\mu, \gamma_4^\nu]$. (these gamma matrices are defined below) One does something called "dimensional reduction" of these to $2+1$ dimensional spacetime by assuming that the fields are independent of the third spatial coordinate. In $2+1$ dimensional spacetime the gamma matrices are defined as, $\gamma ^0_3 = -i\sigma^2$, $\gamma^1_3 = \sigma ^3$ and $\gamma^2_3 = \sigma ^1$ In the so called "Majorana representation" the $4-$dimensional Gamma matrices can be written such that $\gamma_4 ^{0\1\2} = \left [ \begin{array}{cc} & \gamma_3 ^{0\1\2} \\\ \gamma_3 ^{0\1\2} & \end{array} \right ] $ and $\gamma_4 ^3 = \left [ \begin{array}{cc} I & \\\ & -I \end{array} \right ]$ One renames the third component of the gauge field $A_3$ as $C$ and splits the $4-$component fermions into $2-$component spinors in $2+1$ dimensions as, $\lambda = \left [ \begin{array}{c} \lambda_1 \\\ \lambda_2 \end{array} \right ]$ $\alpha = \left [ \begin{array}{c} \alpha_1 \\\ \alpha_2 \end{array} \right ]$ One substitutes the above in the first set of supersymmetric transformations and sets to $0$ all derivatives with respect to the $3^{rd}$ spatial direction. Then one is supposed to get, $\delta A_\mu = i \bar{\alpha}_a \gamma_\mu \lambda _a$ $\delta \lambda _a = - \frac{\epsilon ^{\mu \nu \rho}F_{\nu \rho}}{2}\gamma _ \mu \alpha_a + \partial _\mu C \gamma ^\mu \alpha ^a + D \alpha ^a$ $\delta C = i\bar{\alpha}^a \lambda _a$ $\delta D = i \bar{\alpha}^a\gamma ^\mu \partial_\mu \lambda_a$ where $a$ runs over $1,2$ and $\alpha^1 = \alpha_2$ and $\alpha^2 = -\alpha_1$ Doing the above I can get the above transformations for all the fields. (Just that I need to use a fact that is true only for these gamma matrices in the $3$-dimensions that $[\gamma_3^\mu,\gamma_3^\nu]=2\epsilon ^{\mu \nu \rho}\gamma_{3\rho}$)   * In this "dimensional reduction" the particular choice of the gamma matrices seemed crucial. Is this true?    * I don't know how but this is supposed to match the transformations of the $\cal{N}=2$ vector multiplet in $2+1$ dimensions which has componets, $A_\mu, \lambda_a,C,D$ (why?)    * I don't know how one derives the field content (as above) of the $\cal{N}=2$ vector multiplet in $2+1$ dimensions.    * From the above it is claimed that the following forms a lagrangian respecting the above symmetry and is what gets called the ``supersymmetric Chern-Simons theory" $L = \frac{\kappa}{2} (\epsilon ^{\mu \nu \rho} A_\mu \partial _\nu A_\rho - i \bar{\lambda_a}\lambda_a + 2CD)$ But to establish the above claim I need to assume two results,   * First is a property which again seems typical of $3$ dimensional gamma matrices that, $\gamma_3^{\mu \dagger} \gamma_3 ^0 = -\gamma_3^0\gamma_3^\mu$    * Secondly I need to assume the following variation for the fermion self-coupling,  $\delta (\bar{\lambda} \lambda) = 2\delta (\bar{\lambda}) \lambda$ It is not clear to me as to why should the above hold.   * A crucial difference between the above and the usual Abelian supergauge theory is that the kinetic term for the fermion is missing. How does one understand this?


Idea of integrable systems

I do not quite understand the idea an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system is integrable?


What is the motivation for the definition of a manifold?

In Wald's _General Relativity,_ an $n$-dimensional $C^{\infty}$ manifold $\mathit{M}$ is defined as a set, with subsets $\lbrace{O}_{\alpha}\rbrace$, which satisfies 3 properties. In particular, the third one (abbreviated) is: > _If any two sets $\mathit{O}_{\alpha}$ and $\mathit{O}_{\beta}$ overlap, we > can consider the map $\psi_{\beta}\circ \psi_{\alpha}^{-1}$, which takes > points from subset $\psi_{\alpha}[\mathit{O}_{\alpha} \cap > \mathit{O}_{\beta}]$ to points in subset $\psi_{\beta}[\mathit{O}_{\alpha} > \cap \mathit{O}_{\beta}]$. We require these subsets of $\mathbb{R}^{n}$ to > be open and this map to be $\mathit{C}^{\infty}$._ My question is: what is the purpose of this clause in the definition of a manifold? It appears to allow the existence of a transformation from different coordinate systems (from $\psi_{\alpha}$ to $\psi_{\beta}$). It requires the existence of an inverse. Other than that, is there any other motivation for defining the manifold in this way?


The definition of entropy in quantum mechanics

I have seen entropy with several different definitions. Like Von Neumann entropy and Rényi entropy, etc. So I am curious why there are so many different definitions in quantum mechanics while only one in classical mechanics named after Boltzmann?


Forces in a balloon popping?

How much difference between internal and external pressure can a balloon withstand? how would the value be calculated? I found an equation: http://library.thinkquest.org/C003758/Function/laplacelaw.htm But, how does that relate to a material?


Does our current notion of velocity/acceleration (based on calculus) require absolute time?

When people explain special relativity, they say that a velocity in one frame leads to time dilation in that frame relative to a stationary frame. When we say something like $v=\mathrm{d}x/\mathrm{d}t$ or $a=\mathrm{d}v/\mathrm{d}t$. Isn't dt something that cannot change? A smallest infinitesimal unchanging amount. Newton described it in terms of absolute space and time. How does it make sense to define a velocity of one frame as $\mathrm{d}x/\mathrm{d}t$ yet say that the rate of time itself changes? Can $\mathrm{d}t$ change?


Position vector, length/modulus function and differentials

If $\vec r$ and is the position vector of a point in motion and $r$ is its length/modulus/magnitude/size, then: Can it be true that: $$\|\mbox{d} \vec r\| \neq \mbox{d}r? $$ I think that this is true in circular motion, but if this question is generic, i.e., for almost all cases, then do you think this is correct to say? **EDIT:** Thanks for the very prompt answers and comments, but please tell me, if this question is given in an exam, which option would you mark? (As a student, I am very interested in understanding physics, but at the end of the day, I also have to think of the exam point of view...)


Time Reversal Operator

I know that time reversal operator is an antiunitary operator. How does it work on wavefunctions? I believe in this way: $$T \psi (k,+)=e^{i\pi S_y/\hbar} K \psi (k,+) = \psi^*(-k,-),$$ but I am not sure. ("+" and "-" represent spin-up or spin-down states) Does anybody have a good explanation?


Significances of energy conditions

I know about these different energy conditions in GR, namely strong, weak and null, but never really understood the full physical significance of them or for example how to 'derive' them or how compelling is a certain condition (for that particular one to be applied). I know that they exist because you need some constraints for $T_{\mu\nu}$ to satisfy as otherwise its completely unconstrained. Can someone please elaborate?


Diffraction-Long versus short waves

Why is it that long waves diffract more than short waves? I know that refraction has a quantum mechanical explanation, does diffraction have a similarly complex explanation? My teacher in high school said that using ocean waves would help me understand-but it didn't. Is there something in the math, or some related physics concept that can help me better understand diffraction? Related Question: Bass and Treble-Car Steroes


Does a hydrogen atom today have same mass as a hydrogen atom in the future?

Does an atom of hydrogen today have the same rest mass energy as an atom of hydrogen a billion years in the future? Standard cosmology seems to tacitly make this assumption. But surely one can only make this assumption if the FRW metric has a time-like Killing vector? From V.Moretti's answer to my previous post the FRW metric does not have a time-like Killing vector so that we cannot assert that the energy of an atom of hydrogen is constant over cosmological time. Is this reasoning correct? I'm sorry to keep bringing this question up but I think it's an important point that is overlooked in current cosmology. **Answer to responses** I guess I am following the spirit of Sean Carroll's Energy is not conserved Blog post where he says: > When you thought energy was conserved, there was a reason why you thought > that, namely time-translation invariance. A fancy way of saying “the > background on which particles and forces evolve, as well as the dynamical > rules governing their motions, are fixed, not changing with time.” But in > general relativity that’s simply no longer true. Einstein tells us that > space and time are dynamical, and in particular that they can evolve with > time. When the space through which particles move is changing, the total > energy of those particles is not conserved. > > Energy and momentum evolve in a precisely specified way in response to the > behavior of spacetime around them. If that spacetime is standing completely > still, the total energy is constant; if it’s evolving, the energy changes in > a completely unambiguous way. According to the FRW metric spacetime is evolving so surely the energy of particles cannot be assumed to be constant? A hydrogen atom today does not necessarily have the same energy as it did a billion years ago or will have a billion years in the future. Taking the spatially flat FRW metric in Cartesian co-ordinates for simplicity we have: $$ds^2 = -dt^2 + a^2(t)(dx^2+dy^2+dz^2)$$ Now from this representation of the FRW metric one might conclude that it is only the spatial co-ordinates that are evolving. In fact we can transform to conformal co-ordinates: $$ds^2 = a^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ where $d\tau = dt/a(t)$. Now we can clearly see that spacetime is evolving so we cannot assume that the energy of massive particles is constant. However according to V.Moretti massless particles like photons that are described by conformally invariant fields do not have their energy changed by a conformally changing metric as described above. Thus we can rigorously argue that photon energy is constant if we are in fact moving forward in conformal time $\tau$. So an increasing length scale is associated with a constant energy. Thus a massive particle with a _fixed_ length scale is associated with an energy that increases with the scaling factor $a(\tau)$. In seems that the energies of all massive particles must scale with $a(\tau)$ including the Planck mass. In fact I suppose as all particle masses are ultimately linked to the Planck mass then the the scaling particle masses is ultimately caused by the Planck mass increasing.


volume of phase space of composite microcanonical ensemble

Let $N_1$ be number of particles in volume $V_1$ with momenta and coordinates $(p_1, q_1)$ and $N_2$ be particles with momenta and coordinates $(p_2, q_2)$ in $V_2$, if $E_1, E_2$ be the energies of subsystems satisfying $E< E_1 + E_2 < E + 2 \Delta$ with $\Delta << E$. My book says that > Obviously the volume of the region of $\Gamma$ space that corresponds to > conditions with total energy lying between $E_1 + E_2$ and $E_1 + E_2 + 2 > \Delta$ is $\Gamma(E_1) \times \Gamma(E_2)$ Why is the volume of phase space multiplied? Shouldn't it be added?


Equation for null geodesic around schwarzschild metric?

I'm trying to find the path of a photon around the Schwarzschild black hole, given its initial conditions. After much tribulation, I've basically given up on solving the equations by myself. I just need equations which I can integrate for: $$ r(\lambda) = \text{function(initial direction, initial position)} $$ $$ \theta(\lambda) = \text{function(initial direction, initial position)} $$ $$ \phi(\lambda) = \text{function(initial direction, initial position)} $$ $$ t(\lambda) = \text{function(initial direction, initial position)} $$ given the initial conditions of the light ray. Surely this has been solved before?


Origin of Charge Neutrality

What is the origin of the "charge neutrality" requirement in solids? Why do we require the bulk to be charge neutral, yet the surface can have a net charge?


Question about Majorana fermions

I have a few questions about Majorana fermions.   1. What is Majorana mass? Does it have a different value compared to the mass in the Dirac equation for an arbitrary fermion? How exactly do they differ?   2. Can the Majorana equation be rewritten in form two-component spinor equation? Or it is two-component already?


Single photon double slit experiment

In the question Double Slit experiment with just one photon or electron, one of the answers says > There have been experiments recently where one can detect which the slit the > particle went through and still the interference pattern appears. This had 6 votes. Is it correct? If you put a detector to find out which way the particle went through won't that destroy the interference?


Photons, no energy?

I am only 14, so I don't know much about physics, and I would appreciate help on this topic. A photon, is said to have 0 mass. However, they carry energy, as electromagnetic radiation. Albert Einstein proved that $E = mc^2$. But if you plug the data in, $E = 0\times (299791)^2$, you try to multiply the speed of light squared by $0$. Therefore, the energy of a photon would be $0$. How is that possible? Since all the photon does is carry energy?


Compute closed line integral of electric field in circuit

I have a circuit where resistor is parallel to capacitor, which is charged with voltage U. How to compute line integral around closed loop to get the result of Kirchhof second law - $U_{capacitor}+I\cdot R=0$? For beginning I splited it in two parts, starting from the capacitor positive plate through resistor:   $$\oint \vec{E} \cdot d \vec{l} = \int_{+R-}\vec{E_{R}} \cdot d \vec{l} + \int_{-C+} \vec{E_C} \cdot d\vec{l}$$ Since intensity of resistor is with the same direction as current density $\vec{j}$: $$\int_{+R-}\vec{E_{R}} \cdot d \vec{l} = \int_{+R-} \rho \cdot \vec{j} \cdot d \vec{l} = I \cdot R$$ When I do the same for capacitor I gain negative line integral, because the $E_{c}$ is opposite to $d \vec{l}$ when positive charge is moved from negative plate to the positive. So in which part I have made a mistake?


Unequally charged hanging pith balls

I have done a problem which asks us to find the charge on two equally 'massed and charged' pith balls which are left hanging on a string with a certain length that repels each other and attains an equilibrium point making an angle (the string) with the vertical. The problem was quite clear, what left me wondering was: Won't the angle be the same if the balls are unequally charged (but equally massed) ? Or does it deviate from the initial case?


Divergence of non conservative electric field

I'm looking for the proof that the 1st Maxwell equation is valid also on non conservative electric field. When we are talking about a electrostatic field, the equation is ok. We can apply the Gauss (or Flux) theorem and get Gauss' law: $$\mathbf{\nabla} \cdot \mathbf{E} ~=~ \frac{1}{\epsilon _0} \rho (x,y,z).$$ The question is, why when there is a time dependent magnetic field, and then a time dependent (non conservative) induced electric field, the 1st Maxwell equation is the same? How we can prove that?


What are the applications of Gauss's law in technology?

Freshmen physics textbooks use Gauss's law plus symmetry to calculate the electric field. I was wondering if this method of finding the electric field using a symmetry is used in real applications in life, science, research, or technology. For example do researchers use symmetry to find the electric field due to a charged virus that is spherically symmetric, or a charged DNA that has some kind of symmetry or charged symmetrical objects in nanotechnology. I am looking for such specific applications.


What is the meaning of uncertainty in Heisenberg's uncertainty principle?

The Heisenberg's uncertainty principle states the following: $$\Delta p \cdot \Delta x \ge \frac{h}{4\pi}.$$ While studying for my high school physics exams, I fooled myself into believing that I understood the uncertainty principle (at least the implications). But suddenly the question that's nagging me is the following. If the uncertainty $\Delta x$ of an electron is 1.2 nm, does it imply that the probability that the x-coordinate lies within a 1.2 nm range, equal to 100%? Or does it mean that the probability is 95%? Or does it mean something totally different? I wonder why no author made it clear in the high school/junior college level textbooks. I am uncertain about what uncertainty means.


Projectile Motion

I know the angle at which a projectile is launched, how far it needs to go, and also the maximum height. How can I find the initial velocity needed (disregarding air resistance)? Currently, I am using: $Vf^2=Vy^2-2g\Delta y$ From that, I can find the velocity in the y direction needed for the projectile to reach the known maximum height. Next, I am finding the time it takes to reach that maximum height by using the found y velocity.


How does a lightning striking stall a moving car?

I am currently an undergraduate student leading a discussion section for an introductory electricity and magnetism course. I was asked this question that I was unable to answer and I am hoping for some help in understanding what is going on. I was told by this student that he and his girlfriend were driving along a highway when lightning struck the roof of their car. Although they were unharmed by the lightning (Faraday cage), my student claimed that his car stalled and was unable to turn it back on right away. However, after about 5 to 10 minutes, the car started up immediately. So my question is **how does lightning striking the roof of a car stall a moving car?**


Thermographic camera vs infrared camera

AFAIK, all objects above 0K emit infrared radiation. It is an electromagnetic radiation, like visible light. I guess thermographic and infrared cameras work on the same principles, but their images are very different, the former is false color and very different from the visible image, the latter is black and white and usually correlates to the actual visible image, at least that's what I've seen so far. So physically, how exactly are they different?


A single valued function from a multi-valued function

In Schrieffer's book "Theory of Superconductivity", there is said when he deals with multiple connected superconductors (and discuss London equations), that if one takes the line integral of the potential vector on the boundary of a hole where there is a magnetic flux inside we have : $$\int{\vec dl\cdot \vec A} = \Phi$$ Then, if we go inside the superconductor ($\vec B=0$ thus $\vec A = \nabla \chi$ where $\chi$ is a scalar function) and again take the line integral of $\vec A$, we deduce that $\chi$ must be multi-valued : $$\int{\vec dl\cdot \vec A} = \Delta \chi = \Phi$$ But it is stated that now, if the magnetic flux is known in each hole, $\vec A$ is determined and the physics is saved. But for me, it is not clear that if we know how much $\chi$ raises with every $2 \pi$ rotation around a hole, we don't know how it varies... Could someone help me on this ?


Sound Produced due to motion of air inside a tube

I'm trying to understand the sound produced by Air motion inside the respiratory system. If you have a long tube with a suction apparatus at one end , You turn on the apparatus and the air is sucked in. Now Sound waves are produced due to vibration of the molecules in the wall so that each molecule behaves like a harmonic oscillator and the whole is a system of coupled oscillators . I have some questions. If The air inside the tube is highly collaminated in its motion , That is it has zero component of velocity perpendicular to the axis , The there will not be any sound waves that are produced . If there is a very small inhomogeinity at the beginning then inside the tube will these inhomogenities be amplified ? How does The Frequency and intensity depends on the velocity of air , The diameter of the tube and the degree of disturbance in the motion of air ?(I expect that sound will be a linear combination of waves with different frequencies , the more the turbulence , the more terms will be in the series expansion , The less diameter the more velocity and subsequently the bigger values of frequencies that appear in the series) Will sound waves emitted have the same frequency along the tube, that is the velocity of and kinetic energy of air increases as they move along the axis and subsequently the energy of sound will increase or may be the higher velocity the more turbulence that occurs and subsequently sound waves will be a linear combinations of many waves with different frequencies ? Why is the sound produced from a cavity amplified and has low frequency?(I expect this is due to turbulence , but in this case why does turbulence amplify sound?, I think that the more turbulence , the more frequencies produced and when you add them together , the amplitudes will tend to vanish for some frequencies and increase for other frequencies , right? ) Lastly , What exactly is the pitch of the sound and how is it related to the frequency?


Why is the Earth so fat?

I made a naive calculation of the height of Earth's equatorial bulge and found that it should be about 10km. The true height is about 20km. My question is: why is there this discrepancy? The calculation I did was to imagine placing a ball down on the spinning Earth. Wherever I place it, it shouldn't move. The gravitational potential per unit mass of the ball is $hg$, with $h$ the height above the pole-to-center distance of the Earth (call that $R$) and $g$ gravitational acceleration. Gravity wants to pull the ball towards the poles, away from the bulge. It is balanced by the centrifugal force, which has a potential $-\omega^2R^2\sin^2\theta/2$ per unit mass, with $\omega$ Earth's angular velocity and $\theta$ the angle from the north pole. This comes taking what in an inertial frame would be the ball's kinetic energy and making it a potential in the accelerating frame. If the ball doesn't move, this potential must be constant, so $$U = hg - \frac{(\omega R \sin\theta)^2}{2} = \textrm{const}$$ we might as well let the constant be zero and write $$h = \frac{(\omega R \sin\theta)^2}{2g}$$ For the Earth, $$R = 6.4*10^6m$$ $$\omega = \frac{2\pi}{24\ \textrm{hours}}$$ $$g = 9.8\ m/s^2$$ This gives 10.8 km when $\theta = \pi/2$, so the equatorial bulge should be roughly that large. According to Wikipedia, the Earth is 42.72 km wider in diameter at the equator than pole-to-pole, meaning the bulge is about twice as large as I expected. (Wikipedia cites diameter; I estimated radius.) Where is the extra bulge coming from? My simple calculation uses $g$ and $R$ as constants, but neither varies more than a percent or so. It's true the Earth does not have uniform density, but it's not clear to me how this should affect the calculation, so long as the density distribution is still spherically-symmetric (or nearly so). (Wikipedia also includes an expression, without derivation, that agrees with mine.)


Why do the drift and diffusion components cancel for each type of carrier if EHP generation plays such big role in p-n-junctions?

I have always argued to myself that drift and diffusion components of the current though a p-n-junction cancel for each type of carrier because any electron diffusing from n into p will sooner or later require the same electron to drift back. (And analogously for holes). However, it now seems to me that the generation of EHPs (electron-hole-pairs) profoundly contributes to the drift current: > "The supply of minority carriers on each side of the junction required to > participate in the drift component of current is generated by thermal > excitation of electron-hole pairs." (In "Solid-State-Electronic Devices" , > Streetman and Banerjee) In thermal equilibrium, that implies an equal amount of recombination events. Anyway, one can imagine an EHP being generated on the p-side, with the electron drifting and the hole diffusing to the n-side, where thy recombine. In that scenario, there is no hole drift and no electron diffusion, but hole diffusion and electron drift. So, how in earth does one dare to argue that "the drift and diffusion components must cancel for each type of carrier"? I feel like I am missing the real argument.


Clarifying what metric counts as flat space

In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers greater or less than zero, would this still count as flat space even though it is no longer Euclidean? I tbink the answer is yes but just want to make sure. Also, when a physicist says that the local tangent space on a manifold is flat, do they imply that the metric is locally Euclidean, or can the diagonal elements be any non-zero real number in that local tangent space?


Field Strength Renormalisation in Peskin and Schroeder

In chapter 7 of Peskin and Schroeder they define the field strength renormalisation $Z$ for a quantum field to be the residue of the Fourier transform of the correlation function $$\langle \Omega | \phi(x)\phi(0) | \Omega\rangle$$ They go on to refer to $Z$ variously as $Z_1$, $Z_2$ and $Z_3$ at different points in the text. Is the subscript meant to denote the **order to which we've computed $Z$**? In other words do we have $$Z = Z_1 + O(e^2) = Z_2 + O(e^3) = Z_3 + O(e^4)$$ and so on? This would seem to make no sense because then naturally $Z_1 = Z_2$ due to the Feynman diagram stucture. Alternatively are they just meant to be $Z$s for different systems (Lagrangians)? Or are they just genuinely different quantities depending on context? I'm getting confused because P&S generally use subscripts to denote the order to which a quantity is valid, but it doesn't seem to be the case here! If anyone has the book and wants page references, see p215, 221 and 243.


Statistical Weighting Factor on thermal neutron importance

The problem is concerning the use of a thermal fluxed squared weighting factor in a thermal reactor. I have seen in sources the thermal flux in a reactor is squared as a statistical weighting factor, for example looking at temperature feedback. Say a number of thermocouples are measured in a reactor, the weighting of the importance of the thermocouples are to flux squared. Why is this when the reaction rate is equal to the flux? Is it simple statistics I'm not understanding.


A question about Lorentz invariance of the Polyakov action

I have a super basic and stupid question about the Lorentz invariance of the Polyakov action (cannot skip the disclaimer..) $$S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \gamma^{ab} \partial_a X^{\mu} \partial_b X_{\mu} $$ I may write the action as (using metric tensor $\gamma^{ab}$) $$S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \partial^a X^{\mu} \partial_a X_{\mu} $$ The Lagrangian is obviously invariant under proper Lorentz transformation. $\partial^a X^{\mu} \partial_a X_{\mu}$ is a Lorentz scalar. $d\tau d\sigma$ transforms with a determinant equals 1 for proper Lorentz transformation. $(-\gamma)^{1/2} $ also transforms with a determinant equals 1. But for the range of integration $[0,l]$, there is length contraction under a boost. Why the action is still invariant under Lorentz transformation? (or I completely mistaken something....)


Why did they omit the action-reaction pair force in this problem?

**Problem** ![](http://img577.imageshack.us/img577/2205/problemse.jpg)! **Solution** ![](http://img543.imageshack.us/img543/858/solutionq.jpg)! **My solution** ![](http://img853.imageshack.us/img853/4305/solution2.jpg)! Let me explain $n_{bg}$ means the force exerted by the block on the ground. This is because the normal force of the slab exerts (or touches) a force on the block and by Newton's Third Law, it repels with a force n. So $n_{ground}$ is actually a force exerted by the ground and the force exerted by the ground on the block Also I kinda intuitively knew to put a friction force on the slab on the right. Is this a consequence of Newton's third Law? As the block slides to the right, the friction force drags it from the left. On the slab, (imagining now) wouldn't the heat/friction be sliding on top from the left?


traditional transformer design improvement

The traditional tranformers suffer a lot of losses in various forms, some of them are as follows : 1\. Flux leakage 2\. Eddy currents 3\. Hysteresis Figure : ![Figure](http://lh3.ggpht.com/-8_dBB49n_3Q/UmPS6S2qMUI/AAAAAAAAAd4/FQy4MSOMjG8/s1600/2013-10-18%25252020.04.41.png) But as we are taught induction we come across the attached figure of two inductors placed coaxially various times. My point is that if we would actually place inductors in this way, we would have the following benefits : Note : taking inner coil as primary   1. NO NEED OF CORE : As the flux of inner coil passes from outer on its own, no core is needed for transferring flux from primary to secondary   2. NO FLUX LEAKAGE : As entire flux of inner inductor passes from outer, no flux is leaked.   3. NO EDDY CURRENTS: As no core is used no eddy currents are generated.    4. NO HYSTERESIS : Similarly due to absense of core, no loss due to hysteresis is encountered. So what really is the problem in creating transformers in such a way to reduce losses much more than the current status ? All relevant opinions and comments appreciated My original post : http://thinklo.blogspot.in/2013/10/improving- transformers.html


Why do physicists believe protons and electrons are present in equal numbers?

I tended to consider that negative and positive charges are present in equal numbers in the universe to be a known, obvious fact. But is it so? How can we rule out the possibility that there is some kind of asymmetry in the numbers of protons and electrons? Of course, matter is neutral and even tiny deviations would give rise to enormous forces... But not all protons and electrons are in atoms in the universe. More importantly, is there some conceptual reason why we believe that this equality is exactly perfect or do modern theories allow tiny deviations?


Energy loss of Earth

How does the Earth lose the energy that it gains from the sun's radiation if it is surrounded by - and in contact with - nothing?


Maxwell's equations and wave equations

As can be proven, Maxwell's equations don't work for spherical or cylindrical waves, but these waves satisfy the wave equation. Are there any other examples of this type that satisfy the wave equation but not Maxwell's equations?


Principle of Least Action

Is the principle of least action actually a principle of least action or just one of stationary action? I think I read in Landau/Lifschitz that there are some examples where the action of an actual physical example is not minimal but only stationary. Does anybody of know such a physical example?


Is proper time renormalization gauge invariant?

Proper Time Renormalization is achieved by putting: $$ \int_0^\infty e^{iat} dt = {1\over ia} $$ Is it true that this is the only kind of normalization that is gauge invariant? If so, why do famous books ignore it even though it has such an important property (gauge invariance)? Does the reason behind the name have anything to do with this?


Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field

I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows: $$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 + q\phi $$ and also $$H=\frac{1}{2m}(\vec{p}-\frac{q}{c}\vec{A})^2 + q\phi $$ with $\vec{p}$ the momentum, $q$ the electric charge, $\vec{A}$ the vector potential, $\phi$ the scalar potential and $c$ the speed of light. So basically the difference is in the term $1/c$ multiplying $\vec{A}$, present in the second form (which I use in my lectures) but not in the first one (used by Griffiths in _Introduction to quantum dynamics_ to treat the Aharonov-Bohm effect). Why does this difference exist and what does it mean? And how does the term $1/c$ affect the dimensional analysis (the units) of the problem?


Book against a wall and forces

If you take a book with mass of 1kg and push it against the wall. With how much force do you have to push the book so it does not fall? The problem is I know how to calculate this problem, you say $F_{friction}$=$F_{gravitational}$ and $F_{wall / normal}$ = $F_{human / push}$. The problem is solvable if you say that $F_{gravitational}$ = $F_{normal / wall}$, but why is this true? How would you calculate this problem if you didn't know that $F_{normal}$ = $F_{gravitational}$, how would you prove this statement? For me the problem is that $F_{normal}$ = $F_{gravitational}$ * $\cos(\alpha)$, but cos(angle) is 0, I don't understand the relationship between $F_{gravitational}$ and $F_{normal}$ in this scenario.


Why is there no massless spin 1/2 particles with charge? Is there any underlying theory?

Whether any massless spin 1/2 particles cannot have charge? If so, why?


Are there (or could there be) electrically charged particles that move at the speed $c$?

A photon, a neutrino (if it has zero rest mass) move at $c$ but what about charged particles? If the answer is no, is there a fundamental reason or just because of the radiation it emits?


How might a resonant antenna and black body radiation interact?

How does an antenna behave when it is cooled so that its black-body radiation is emitting energy at its resonant frequency? Edit: To clarify, its not how they're related in general, but how might thermal radiation and resonance interact with each other when their spectra are aligned well? Edit: Also, I'm sure that the thermal radiation spectra that have a significant peaks are associated with incredibly high temperatures, and peak at incredibly small wavelengths, rendering such an antenna completely impractical to build. Still, I'm still interested in the theoretical concept.


Air pressure relative to a force on a bag?

Assume an airtight bag occupied by air such that the pressure inside the bag is equal to the atmospheric pressure. Assume the surface tension of the bag is negligible. What is the change in air pressure inside the bag relative to a weight placed on it? Is it possible to approximate the weight of the object on top of the bag by the air pressure inside the bag, without knowing the volume of the bag or the area of the object?


How would natural (resonant) frequencies affect amplitudes?

I read $y=A\sin(2\pi ft)$, where $A$=Amplitude, $f$=Frequency, $t$=Time and $y$=$Y$ position of the wave. Since natural frequencies only take the most effect when they are close to the frequency. How would one natural frequency and several natural frequencies affect the equation? Would I be correct in thinking it's something to effect of: `y=Y_Position*NaturalFrequency` where Y_Postion is the first equation and NaturalFrequency is similar to the first equation but with a low amplitude?


Hamiltonian of the charged current in SM (related to the lorentz invariance)

recently when I was studying the scatterings which involves a vector boson (like W boson) as an intermediate particle, I saw that the propagator is not Lorentz invariant, I read that there is another source of Lorentz non- invariance in the Hamiltonian of the charged current. I mean the Hamiltonian is not simply the minus of the Lagrangian in this case h=-l. the extra term is not Lorentz invariant and, these two source of the Lorentz non-invariance will cancel each other eventually in the S-Matrix. (you can see :The standard Model, A primer written by Burgess chapters 4,5 and problem 5.4). I don't understand this completely, I'll be grateful if anyone can help me and explain it or introducing a reference for it. I apologize for my bad English writing. Thanks


Physics for taffy pulling?

I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea. Is there a general equation for an objects threshold and pulling it apart?


Could entropy explain dark energy?

This was 3rd beer idea, so please bear with me. What if the universe was not actually expanding but the speed of light was slowing? Wouldn't that be indistinguishable to our observations? Either way we should see an increase in the redshift the further a light source is from us? Suppose we're in a giant extremely advanced compute cluster, and the speed of light has to do with how long it takes data to travel between compute resources on the network/bus. As more data travels across this bus (entropy increases in the universe) it takes longer to reach its destination (speed of light slows.) This seems vaguely reasonable considering fundamental limitations past and present on computers of varying types, there's always a trade off between size of data and speed of access that has it's roots in physics. So I know it's a bit wacky, but it sounds easily falsifiable to me, simply plot entropy over time and dark energy over time and see how well they correlate (or don't most likely.) Can anyone help me test my armchair theory? I think that would be a fun exercise. I know dark energy has been plotted over time, but what about entropy, is that possible?


Basic question on experimental plots

On the following Higgs $\rightarrow$ Tau Tau plot, since we are plotting the ratio of $\frac{\sigma}{\sigma_{SM}}$ on the y axis, shouldn't the expected for this be 1? i.e., shouldn't the expected 68% and 95% be centered at a dotted line at 1? Anything else seems to imply we are expecting something other than the Standard Model... ![Higgs ->Tau Tau](http://i.stack.imgur.com/dffNF.png)


Does tea stay hotter with the milk in it?

A little thought experiment, similar to this one: Imagine you are making a cup of tea when the door bell rings. You've poured the boiling water into a cup with a teabag in it. As you're just about to pour milk in to the mix, the door bell rings. My question is: what will keep the resulting drink hottest after my chat with the person at the door: quickly pouring the milk in, or leaving it in the carton until after the interruption ends? Being a bit of a mathematician I'm happy to have the questions answered with a formula in terms of $t$ the length of the interruption, $m$, the temperature of the cold milk, $w$ the temperature of the hot water, $a$ the ambient temperature etc. I guess you'll have to use a model of how an insulated body loses/gains heat... feel free to use simplifying assumptions like the environment stays the same temperature etc. My starting point from a physics point of view, is that heat is lost (or gained) proportional to the temperature difference between the object and the environment. So I imagine that would create some sort of exponential curve?


How is the speed of light an absolute maximum speed in the universe?

Under the heading _which came first, the chicken or the egg,_ which came first: the maximum velocity of electrons orbiting the atom, or the speed of light as the maximum velocity limit? Is it possible that when an object reaches the velocity at which its forward velocity equals its electron(s) orbital speed around its nucleus, then its advancing electrons would cease to spin around their orbit, lose their centrifugal force, and collapse into its nucleus, and the retreating electrons would lose their forward speed and the atom would run away from them, and shed them as radiation? Could this limit the maximum velocity at which a mass can travel, and this define the speed of light? If this is true, then at the speed of light its mass would not become infinite, in according to Einstein, but the mass would transmute into pure energy, in other words, become light or some other radiation frequency.


Ratio of distance between mirror and person

In perspective of a given example, if a man was to stand $2\ m$ away from a mirror which was $0.9\ m$ in height and was able to see his full reflection, what would the height of the mirror have to be if the man was now $6\ m$ away from the mirror and was to maintain a full reflection? Would the mirror be equal to or less than the original height and why? This scenario seems to have caught me out multiple times. So, some reasoning would be much appreciated.


General Thermodynamic equation of state

I heard my professor saying that the equation $$ PV = \frac{2}{3}U $$ is valid for any **non-relativistic** gas, be it Ideal or Real gas(includes quantum ideal gases). Is this true, If it is how can we prove with all generality ?


Why (infra-)red light is not used in diagnostics?

I noticed that human body conducts red visible light quite well. A hand placed over a powerful red lamp seems to be semi-transparent while over any other color it does not. Possibly the light conductivity in near infra-red range is even better. So my question is why (infra-)red light is not used for diagnostics instead of harmful X-rays?


Dynamic equation

I'm asked to give the dynamic equations of a charged particle moving on a potential, under relativistic considerations, basically $dp/dt$, with $p$ the 4-momentum vector. The question for the three space coordinates is simply $d\vec p =-q\nabla\Phi$, being $\Phi$ the potential, that is: $$\frac{d(\gamma m\vec v)}{dt}=-q\nabla\Phi$$ For the first term of the four vector, the answer given is this, and I have no idea where this comes from: $$\frac{d(\gamma mc^2)}{dt}=q(\vec E\cdot\vec v)$$, being $\vec E$ the electric field. Where does this come from? IS it even correct, isn't the first term of the four momentum vector just $\gamma mc$ instead of $\gamma mc^2 $


Electromagnetic Inductance: Different voltmeter readings from different positions

In the setup below, the voltmeter on the right would read differently than the voltmeter on the left even though they are both connected to point D and point A. ![enter image description here](http://i.stack.imgur.com/3yrq7.png) This picture is taken from Lecture 16 of MIT course 8.02. It shows a circuit with a solenoid in the middle. The magnetic field of the solenoid is pointing out of the blackboard and at the moment it is turned on, there is an increase in magnetic field inducing a clockwise current in the wire. The EMF is given to be 1V. $ Given: V = 1 $   $ V=I(R_{1} + R_{2}) = I(100+900) = 1 $   $ \Rightarrow I = 0.001 $ By Ohm's Law,   $ Clockwise: V_{D} - V_{A} = IR_{2} = 0.9 $   $ Counter clockwise: V_{D} - V_{A} = -IR_{1} = -0.1 $ So the Voltmeter on the right will read 0.9 while the left will read -0.1. This is shown in an experiment where two voltmeters connected to the same points A and D read different values. How is this possible? How does the voltmeter "know" whether it is on the left or right? It seems entirely symmetrical. How would the experiment need to be setup? If I were to just take a voltmeter and connect it to point A and D, how would I know what it would read?


Finding current in a sphere while given a changing current density

im trying to calculate the current running through a sphere while given a current density $$J=Jcos(2\theta)e^{-t/r} \hat r $$ J is given in spherical coordinates. I know that: $$I=\int J\,da $$ But I can't seem to understand how to "build" my integral. I was thinking that i need to do something like $$ \int r\,dr\,sin\theta\,d\theta*J $$ but this gets very messy because of the exponent containing expression with "r". Any tips/help please? Thanks


Equilibrium of of water level in punctured bucket?

Would a bucket with a hole in the bottom being filled at different rates with water come to to different equilibrium levels with in the bucket? Or would at any fill rate above initial flow rate of the hole cause the bucket to overflow? An answer with more math is preferred. **My thoughts:** If flow rates of water into bucket less than initial flow rate of hole then water level stays at or below height of hole in bucket. If water flow rates begin to increase, so does the depth of the water above the hole then since the product of the hole's size and pressure is positively proportional to flow rate, the water leaving and coming reach a equilibrium point with in bucket x height above hole. If water entering bucket becomes to fast then a equilibrium point in the size of the current bucket can not be reached, so the bucket over flows.


What happens if light/particles exceeded the speed of light for a particular medium

While the speed of light in vacuum is a universal constant (c), the speed at which light propagates in other materials/mediums may be less than c. This is obviously suggested by the fact that different materials(especially in the case of transparent ones) have a particular Refractive Index. But surely, matter or even photons can be accelerated beyond this speed in a medium? If so, what are the effects?


A dictionary of string - standard physics correspondences

Motivated by the (for me very useful) remark ''Standard model generations in string theory are the Euler number of the Calabi Yau, and it is actually reasonably doable to get 4,6,8, or 3 generations'' in http://physics.stackexchange.com/a/22759/7924 , I'd like to ask: Is there a source giving this sort of dictionary how to relate as much as possible from general relativity and the standard model to string theory, without being bogged down by formalism, speculation, or diluted explanations for laymen? Or, if there is no such dictionary, I'd like to get as answers contributions to such a concise dictionary. [Edit April 30, 2012:] I found some more pieces of the wanted dictionary in http://www.physicsforums.com/showthread.php?p=3263502#post3263502 : ''branes and extra dimensions have proved to be implicit in standard quantum field theory, where they emerge from the existence of a continuous degeneracy of ground states. That multidimensional moduli space of ground states is where the extra dimensions come from, in this case! Branes are domain walls separating regions in different ground states, strings are lines of flux connecting these domain walls. Furthermore, in gauge theories with a small number of colors, it looks like the extra dimensions will be a noncommutative geometry, it's only in the "large N" limit of many colors that you get ordinary space.''


About star groupings and there apparent positions in the sky

Imagine a group of 3 stars that appear to be equally luminous ,star A, B, and C in the shape of an isosolese triangle. Imagine the 3 stars are in different $time-zones$. A,B and C look like they form an isosolese triangle right now on the Earth but when the light from star B reaches the Earth the light from star C still has say, 10 years of travelling to go. So when B is observed in its present position, the apparent position of C is from 10 years in the past. Where C is now relative to the present position of B might be different in such a way so as to not to appear to form an isosolese triangle. The 3 stars 'right now ' could form a zig-zag pattern as they appear in the sky but we might not 'see' that from the Earth for a 'while'. So should differences in many stars positions relative to the Earth be taken into account when the simultaneity or cause- and- effect of some stellar events are being analysed?


Linearizing Quantum Operators

I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$ The right hand side is a sum of the "mean" value and the fluctuations about the mean. While I understand that the physical picture is reasonable, is this mathematically correct? If so what are the constraints this imposes? In literature this is designated as a "linearization" process. My understanding of a linear operator is that it is simply a homomorphism. I have never seen anything done like this and I'm having a hard time finding references which justify this process. I would be grateful if somebody can point me in the right direction!


Lagrangian Coordinates in Fluid Flow

I apologize if this is not the right place to ask this question: I am currently reading a paper by Y. Brenier, where for the fluid flow he introduces a Lagrangian label $a$ instead of the vertical coordinate $z$, and defines a "Lagrangian foliation to be a family of sheets $z = Z(t,x,a)$, labelled by $a \in [0,1]$, where $Z$ is a smooth function such that: $$ 0\leq Z(t,x,a)\leq 1, \,\,\,Z(t,x,0) = 0, \,\,\,Z(t,x,1) = 1\\\ \partial_a Z(t,x,a) > 0\\\ \partial_t Z(t,x,a) + u(x,t,Z(t,x,a))\cdot \nabla_x Z(t,x,a) = w(t,x,Z(t,x,a))" $$ where $x \in \mathbb{R}^2$, $u = (u_1,u_2)$ is the horizontal flow velocity and $w$ is the vertical velocity and $\nabla_x = (\partial_{x_1},\partial_{x_2})$. A possible initial choice is $Z(0,x,a) = a$. My question is: What is the physical interpretation for this? It seems like instead of labeling each individual fluid parcel in the classical Lagrangian coordinates, we are only labeling horizontal planes instead, and then seeing how these sheets move and deform with the flow?


Calculation of error in focal length?

![enter image description here](http://i.stack.imgur.com/P8Wsp.png) $u=-10cm$ $v=10cm$ ![enter image description here](http://i.stack.imgur.com/wab5k.gif) Using the formula the focal length is 5 cm. But how do I get the fractional error in focal length when neither $\Delta u$ nor $\Delta v$ are specified? The options given are   A. (0.05±0.05) cm   B. (0.05±0.10) cm   C. (5.00±0.05) cm   D. (5.00±0.10) cm


Emitting gravitational radiation

Is the following true:   * Two massive bodies with variable distance between them do not emit GR in any direction   * Two bodies that revolve around common center will not emit in the plane of their orbits   * But they will emit in all other directions with the maximum along the axis of rotation perpendicular to the orbit's plane


How is formula for converting pressure from mmHg to Pa derived?

Today my younger brother asked me from where does the 1 Pa = 0.00750061683 mmHg formula for mercury barometer come. He needs a way to derive it, or an academic source which can be cited. After doing some calculations we got the formula for a standard U-tube manometer: $P=\frac{h_2}{h_1}P_0$ where $P_0$ is atmospheric pressure, $P$ is pressure being measured, $h_1$ is height is mercury column exposed to atmospheric pressure and $h_2$ is height of the column exposed to pressure being measured. The problem is that in the case of a barometer, the $h_2$ is exposed to vacuum and I don't know how to use that. I've searched the Internet, and got countless sites which explain how a mercury column barometer works, but I was unable to find a site which explains which forces are acting there and how the number was derived. To make things even worse, none of the physics books I have access to have a detailed explanation.


A question on IR cancellation caculation in peskin schroeder

In Peskin and Schroeder _Introduction to Quantum Field Theory_ book, above equation 6.64 on pg. 200, it was said that "to gain better understanding, of the divergence, let us evaluate the coefficient of the log in the limit of $-q^2\rightarrow\infty$." Why does he want to take $-q^2$ to infinity rather than $0$ to extract the divergence? I thought the form factor $F_1(q^2)$ contains an infrared divergence at $q^2=0$.


Determine viscosity using falling sphere (Stokes Law, Ladenburg correction)

## Introduction I am trying to determine the viscosity of a fluid. Therefore, I let a sphere of known mass `m` and radius `r` fall (sink) into the fluid. I then measure the time it takes to sink/travel a given height. Then there are the following forces having a effect on the sphere: $F_g = mg$ and $F_a = \frac{4 \rho \pi r^3 g}{3} $ (buoyant force) with $g = 9.81 \ ms^{-2}$ and $\rho$ being the density of the fluid. The result is a motion downwards (sinking), hence I get friction $F_r$. Assuming this friction gets that strong to have no resulting force on the sphere (i.e. its velocity remains constant), I can use the following equation: $F_g - F_a - F_r = 0$ According to Stokes Law, the friction of a moving sphere in a fluid with viscosity `n` is: $F_r = 6 \pi r n v$ where `v` is the constant velocity. Using these equations, I can determine the viscosity: $n = \frac{g}{6 \pi r} (m - \rho \frac{4}{3} \pi r^3)\frac{1}{v}$ with $v = \frac{s}{t}$. ## Question about Ladenburg correction The fluid is inside of a cylinder, so I cannot use Stokes Law "as it is" because the walls of the container add more friction. Hence I have to apply the so called Ladenburg correction with a correction factor $f = (1 + \frac{(2.1 r)}{R}) (1 + \frac{(3.3 r)}{H})$ When / where do I have to apply this factor? ## Question about extrapolation As I am doing this with spheres of different radiuses, I get multiple values for the viscosity:               radius/mm | viscosity/(kg/(m*s))     --------------------------------     1.0       | 0.594     2.0       | 0.608     2.5       | 0.631      Extrapolation to $r = 0 \ mm$ gives a value $n_0 = 0.575$. I do not understand: Why do I need this extrapolation?


What Physical Quantity has SI Unit $kg \cdot m$,?

What is the physical quantity that has SI (System International) unit of $kg \cdot m$ (kilogram meters)?


Will a hole cut into a metal disk expand or shrink when the disc is heated?

Suppose you take a metal disc and cut a small, circular hole in the center. When you heat the whole thing, will the hole's diameter increase or decrease? and why?


Detecting dim objects

What are the minimum conditions (switching times, sensitivity ...) for a camera to detect dim objects, e.g. Oort cloud or Kuiper belt objects, passing by in front of a star in the background? I've read the Hubble did this. Is it possible also from ground? How do these conditions change, if the camera would be in space? Or do I need Hubble?


Is it possible to see satellites with the naked eye?

Every now and then I notice some very bright "stars" in the sky. They tend to be very few (one or two, usually), and are quite much brighter than any other star out there. Often they're perfectly visible and bright even in dusk, when no other star is visible yet or will be for some time to come. Now, I'm writing the word "star" in quotes because I really don't think that they're stars. But what are they? Planets? Satellites? Latest Chuck Norris encounters leaving the atmosphere? My best guess would be that they are satellites, because planets should be about the same brightness as stars, but that sounds pretty implausible as well... And yes, I'm very certain that they are NOT airplanes.


How do we know the masses of single stars?

I have recently read that we can only know the masses of stars in binary systems, because we use Kepler's third law to indirectly measure the mass. However, it is not hard to find measurements for the mass of stars not in binary systems. So how is the mass of these stars determined?


Why does a photon colliding with an atomic nucleus cause pair production?

I understand that the photon needs to have enough energy to produce a lepton and it's antimatter partner, and that all of the properties are conserved, but why does the photon do this in the first place? What's going on "behind the scenes" to transform a single, neutral particle (the photon) into two charged particles (i.e. electrons)? Why is the nucleus necessary?


Five-component field

Recently I was reading about 5-component field $(\varphi , \psi_{\mu})$, for which $$ \hat {p}^{\mu} \varphi = mc\psi^{\mu}, \quad \hat {p}_{\mu}\psi^{\mu} = mc\varphi . $$ This field refers to the spin-zero representation of the Lorentz group and obeys the Klein-Gordon massive equation (so it reproduces Poincare group irreducible representations). What physical system is described by it?


Is every quantum measurement reducible to measurements of position and time?

I am currently studying Path Integrals and was unable to resolve the following problem. In the famous book _Quantum Mechanics and Path Integrals,_ written by Feynman and Hibbs, it says (at the beginning of Chapter 5 _Measurements an Operators,_ on page 96): > _So far we have described quantum-mechanical systems as if we intended to > measure only the coordinates of position and time. Indeed, all measurements > of quantum mechanical systems could be made to reduce eventually to position > and time measurements (e.g., the position of a needle on a meter or time of > flight of a particle). Because of this possibility a theory formulated in > terms of position measurements is complete enough to describe all > phenomena._ To me this seems to be a highly non trivial statement (is it even true?) and I was unable to find any satisfying elaboration on this in the literature. I would be thankful for any answer to resolve this question and any reference to the literature!


If light is linearly polarized, does it have some spatial extent?

If light (a photon) is linearly polarized, say vertically, does it have some vertical _spatial_ extent (perhaps in amplitude)?


How is celestial navigation done on a low-level?

When we send a probe off to Jupiter or Saturn, or even Earth orbit, how are the rocket firings timed and coordinated? For instance, when I want to drive to another city I pull onto the highway and use the lines on the road to adjust my steering. I also use intuition to adjust my velocity, even though that won't make a difference _where_ I wind up (only _when_ ). However, spacecraft don't have lines on the road to line up with. Nor do they have a direct measurement to take for speed, which is critical in orbits and in planetary transfers. I suppose that some bright stars near opposition or off the ecliptic (Capella?) and GPS could be used for interplanetary navigation or LEO, but how was it done before GPS? Inertial navigation? How did the Soviets get the Vostok to enter a clean orbit (I understand that the Sputnik orbit was very elliptical)? How did early interplanetary probes orient themselves?


How to distinguish between the spectrum of an atom in motion and the one of a scaled atom?

**Galaxies are moving** dragged by the space expansion. When atoms are in motion the doppler effect will shift the spectra of the emitted photons. The proton-to-electron mass ratio, $\frac{m_e}{m_p}$ has been measured constant along the history of the universe, but nothing was said about the constancy of the electron or proton's masses. The photon's energy obey the Sommerfeld relation, $E_{jn}=-m_e*f(j,n,\alpha,c)$, as seen here, and **it is evident that a shifted (1) spectrum is obtained with a larger $m_e$.** The spectra lines are not only due to the Hydrogen atom; there are other spectral lines due to molecular interactions, due to electric/magnetic dipoles, etc, and so the electromagnetic interaction,the Coulomb's law, $F_{}=\frac{1}{4\cdot \pi\cdot \varepsilon}\cdot \frac{q1\cdot q2}{d^2}$ must be analyzed. If we scale all masses by the relation $\alpha(t)$ (not related with the above fine structure constant), where $t$ is time (past), and also scale the charges and the distances and time by the same factor, gives exactly the same value $F_{}=\frac{1}{4\cdot \pi\cdot \varepsilon}\cdot \frac{q_1\cdot q_2\cdot \alpha^ 2(t)}{d^2\cdot \alpha^2(t)}$. Thus the system with and without the transformation behaves in the same manner. The same procedure shows that the universal gravitational law is also insensitive to the **scaling of the atom** (2). This should not be a complete surprise because the scaling of masses, charges, time units and distances is routinely used on computer simulations that mimic the universe in a consistent way. **The conclusion is that there is no easy way to distinguish between the spectrum of an atom in motion and the one of a scaled atom.** The photons that were emitted by a larger atom in the past are received now without any change on its wavelength (here I'm assuming a static space). The mainstream viewpoint, not being aware that scaling the atom gave the same observational results, adopted the receding interpretation long time ago. As a consequence the models derived from that interpretation (BB, Inflation, DE, DM, ) do not obey the general laws of the universe, namely the energy conservation principle. My viewpoint offers **a cause for the space expansion**. Most physicists are comfortable with: 'space expands', period, without a known cause. Physics is about causes and whys, backed by proper references. **I used the most basic laws of mainstream physics to show that another viewpoint is inscribed in the laws of nature.** The question is strict, formulated in the title: How can someone make a distinction between those spectra? I've already opted for an answer, presented below to provide context, but that is not for discussion. When I graduated as electronic engineer, long time ago, I accepted naively that the fields (electrostatic and gravitational) are sourced by the particles, and expand at $c$ speed, without being drained. But now, older but not senile, I assume without exception, that in the universe there are no 'free lunches' and thus the energy must be transferred from the particles (shrinking) to the fields (growing). This new viewpoint is formalized and compared to the $\Lambda CDM$ model in a rigourous document, with the derivation of the scale relation $\alpha(t)$ that corresponds to the universe's evolution, at:   A self-similar model of the Universe unveils the nature of dark energy   preceded by older documents at arxiv:   Cosmological Principle and Relativity - Part I A relativistic time variation of matter/space fits both local and cosmic data Again, my question is:   **Can someone provide a way to distinguish between the spectrum of an atom in motion and the one of a scaled atom ?** maybe by probing the atom's nucleus and find the isotope ratio's abundance, D/H evolution and other isotopes as Mr Webb did with Mg (1998 paper) when in search of the $\alpha$ variability. **PS** : To simplify the argument this question skips some details and to avoid misrepresentations a short resumé of my position is annexed. (1) - after the details are done it is a redshifted spectrum at the reception (2) - LMTQ units scale equally by $\alpha(t)$, $c,G,\varepsilon$ and the fine structure constant $\alpha$ are invariant. * * * **A short introduction to the self-similar dilation model** The self-similar model arises from an analysis of a fundamental question: is it the space that expands or standard length unit that decreases? That analysis is not an alternative cosmological model derived from some new hypothesis; on the contrary, it does not depend on hypotheses, it has no parameters besides Hubble parameter; it is simply the identification of the phenomenon behind the data, obtained by deduction from consensual observational results. The phenomenon identified is the following: in invariant space, matter is transforming in field in a self-similar way, feeding field expansion. As a consequence of this phenomenon, matter and field evanesce while field expands since the moment when matter appeared. As we use units in which matter is invariant, i.e., units intrinsic to matter, we cannot detect locally the evanescence of matter; but, as a consequence of our decreasing units, we detect an expanding space. So, like the explanation for the apparent rotation of the cosmic bodies, also the explanation for another global cosmic phenomenon (the apparent receding of the cosmic bodies) lays in us. In units where space is invariant, named Space or S units, matter and field evanesce: bodies decrease in size, the velocity of atomic phenomena increases because light speed is invariant but distances within bounded systems of particles decrease. In standard units, intrinsic to matter, here called Atomic or A units, matter and all its phenomena have invariant properties; however, the distance between non-bounded bodies increases, and the wavelength of distant radiations is red-shifted (they were emitted when atoms were greater). The ratios between Atomic and Space units, represented by M for mass, Q for charge, L for length and T for Time, are the following: $$M=L=T=Q=\alpha(t)$$ The scaling function $\alpha(t)$ is exponential in S units, as is typical in self-similar phenomena: $$\alpha(t_S)=e^{-H_0 \cdot t_S}$$ Mass and charge decrease exponentially in S units, and the size of atoms decrease at the same ratio, implying that the phenomena runs faster in the inverse ratio; as A units are such that hold invariant the measures of mass, charge, length of bodies (Einstein concept of reference body) and light speed, they vary all with the same ratio in relation to S units. In A units, space appears to expand at the inverse ratio of the decrease of A length unit; the space scale factor in A units, a, is: $$a=1+H_0 \cdot t_A$$ Therefore, space expands linearly in A units. In what concerns physical laws, those that do not depend on time or space (local laws), like Planck law, are not affected by the evolution of matter/field and hold the same in both systems of units. The laws for static field relate field and its source in the same moment; as both vary at the same ratio, their relation holds invariant and so the laws – the classic laws are valid both in A and S units. The electromagnetic induction laws can be treated as if they were local, therefore holding valid in both systems, and then consider that, in S, the energy of waves decreases with the square of field evanescence (the energy of electromagnetic waves is proportional to the square of the field). In A, due to the relationship between units, the energy decreases at the inverse ratio of space expansion while the wavelength increases proportionally. This decrease of the energy of the waves (or of the photons) is a mystery for an A observer because in A it is supposed to exist energy conservation, which is violated by electromagnetic waves. The phenomenon is observed in the cosmic microwave background (CMB) because the temperature shift of a Planck radiation implies a decrease of the density of the energy of the radiation with the fourth power of the wavelength increase and space expansion only accounts for the third power (this is perhaps the biggest problem of Big Bang models, so big that only seldom is mentioned). Note that induction laws can be treated as time-dependent laws and the evanescence of the radiation be directly obtained from the laws; that introduces an unnecessary formal complication. Finally, there are the conservation laws of mechanics, which require a little more attention.   Although it is not usually mentioned, the independence of physical laws in relation to the inertial motion implies the conservation of the weighted mass summation of velocities and square velocities of the particles of a system of a particles. As the A measure of mass is proportional to the weighted mass, these two properties are understood as the conservation of linear momentum and of kinetic energy. Therefore, the correct physical formula of these conservation laws depends on the weighted mass and is valid in both systems of units; for simplicity, the A measure of mass can be used instead of the weighted mass; for instance, for the conservation of square velocity: $$\sum_i{\frac{m_i}{m_{total}}\cdot v_i^2=\text{const}}\Leftrightarrow \frac{1}{2}\sum_i{(m_A)\cdot v_i^2}$$ In the first expression the system of units is not indicated because the equation is valid in both systems; in the second equation, velocity can be measured in A or S (has the same value in both) but the measure of mass has to be the A measure. The second expression is the conservation of kinetic energy in A.   The conservation of the angular momentum is the only law modified in A units because the relevant measure of curvature radius is the S measure; the angular momentum L can be written as $$\textrm{L}=\mathbf{r}_s\times m_\text{A}\mathbf{v}$$ This is the quantity that holds invariant in an isolated system; the usual A angular momentum, function of $r_A$ $$\textrm{L}_\text{A}=\mathbf{r}_\text{A}\times m_\text{A}{v}$$, of an isolated system increases with time: $$\left ( \frac{d\textrm{L}_\text{A}}{dt_\text{A}} \right )_0=H_0\textrm{L}_0$$ This means that the rotation of an isolated rotating body increases with time. For an S observer, this is consequence of the decrease of the size of the body; an A observer can explain this considering that it is consequence of the local expansion of space, that tends to drag the matter.   Note that there is no conflict with the data that support standard physics because this effect is not directly measurable by now; yet, this insignificant alteration has an important consequence: the expansion of planetary orbits. A note on the value of constants in both systems of units. To change from one system to the other is like any change between two systems of units, but there is a difference: units are time changing one another. As a consequence, a constant in A may not be constant in S. For instance, the Planck constant change in S with the square of the scaling law (a simple image of the physical reason is the following: orbits radii are decreasing in S and also their associated energy; so, the wavelength of emitted radiation is decreasing with orbit radii and also the energy, which implies a Planck constant decreasing with the square of the scaling). Field constants are the exception: they hold constant in both systems of units. The Hubble constant is different: it is constant in S but not in A (in A, the Hubble constant is just the present value of the Hubble parameter). In short, the Hubble constant is the only time constant and is a S constant, field constants are constant is A and S and the other constants, including Planck constant, are relative to atomic phenomena and are constant in A; naturally, the dimensionless relations, like the fine structure constant, are independent of units.   Only the classical fundamental physical laws have been considered because that is what is required, the ground on which the analysis of all the rest of physics can be made, special and general relativity included.


Gravity mitigated by velocity regardless of mass?

We know that a small object moving fast enough can pass by a planet and escape its gravity. Would this be (theoretically) true in reverse? Meaning a planet moving fast enough past a stationary smaller object would not assert its gravitational pull on the small object? Would this mean that gravity is mitigated by velocity regardless of whether the more massive object is moving as opposed to the small object?


Does physics recognize the particle of light separately from the wave of light?

To frame this question, we need to assume that time freezes when traveling at the speed of light. This is theoretically congruent with Einstein's theory of relativity and the theory of time dilation, as discussed here: Would time freeze if you can travel at the speed of light? If this is true, then the photon, which may only exist at the speed of light due to it's "mass-less" property, would not be observable or measurable by any experiment or method that required the passage of time. For the photon's entire existence, the rest of time and existence would not experience change. If the photon traveled 1,000 light years from it's origin in a star to where it was absorbed into some matter, the rest of time and existence would remain changeless for the duration of the photon's 1,000 light year journey. There is no opportunity for anything that experiences the passage of time to observe or measure a photon which never experiences the passage of time. In the fabric of space time, instead of traveling, the photon stretches. While any other particle not traveling at the speed of light would leave it's point of origin and travel, one point at a time along a path to its destination, it is different for a photon. Instead, a photon could be thought of as being created on all points along a path, including the origin and the destination, simultaneously, and for absolutely no amount of time. Can this be represented with a physics equation? So if the photon is the light particle and there is also an aspect of light that is known to be an electromagnetic wave form, then the relationship between the two is not that they are the same thing, it is that the photon particle exists before, and causes, the electromagnetic wave form, the way that a boat moving through water causes a wake or lightning through the atmosphere causes thunder. Is this currently recognized in physics?


Why doesn't a resistor dissipate reactive power?

Is there a (preferably microscopic) explanation for why a resistor does not dissipate reactive power?


Is everything in the physical world composite?

In philosophy there is a principle that anything composite cannot have existed eternally, since it is preceded by its parts and whatever forces assembled it. Is everything in the physical world composite in one sense or another or is there something not composite?


Why do physicists believe that particles are pointlike?

String theory gives physicists reason to believe that particles are 1-dimensional strings because the theory has a purpose - unifying gravity with the gauge theories. So why is it that it's popular belief that particles are 0-dimensional points? Was there ever a proposed theory of them being like this? And why? What reason do physicists have to believe that particles are 0-dimensional points as opposed to 1-dimensional strings?


Rotational Potential Energy of a Hamster Wheel

# Background (unimportant back story)A colleague of mine showed me what i considered flawed statistics, that Internet Explorer had faster Index and array functions than Chrome(we are software developers...FWIW I'm not proud that this is what we talk about either) the graphs showed was in operations per seconds. I replied by trolling him with an example of a Hamster in a Hamster Wheel, saying _Sometimes more work, isn't always faster_ . on the flawed premise that $S=d/t$ for $0$ $d$, and $(PE_{initial} + KE_{initial}) + W = (PE_{final}+ KE_{final})$ where $KE = \frac{1}{2}I\omega^2$ are correlated. The basics of the equation make sense to me. However, apparently I've trolled myself, because i believe that the Potential Energy of the hamster wheel remains constant across the equation, but i don't believe it has 0 $PE$, I'm just not sure how to calculate the constant amount of $PE$ latent in a Hamster Wheel? # Research I like to show that i've done some work, I've done some searching on Rotational Potential Energy, i've found a few results such as $PE = T_{av} \times θ$ but this seems specific to rotational $PE$ of a spring, which should not factor into a Hamster Wheel, I've found loads of simple examples of rotational $KE$ and Inertia like this site https://www.boundless.com/physics/rotational-kinematics-angular-momentum-and- energy/conservation-of-energy/conservation-of-energy-in-rotational-motion/ # Summary and Question If a Hamster Wheel has inherent and constant Potent Energy as I believe, how would i calculate it, for a hamster wheel of a given size?


Reason for establishing a magnetic hysteresis curve in Zeeman effect experiment

I am new to StackExchange, and so apologize in advance for any issues with the formatting of the question/appropriateness of the question to this forum. My lab partner and I were conducting a standard undergraduate laboratory experiment investigating the Zeeman splitting of the red spectral line from a cadmium lamp. The procedure is simple: one takes an electromagnet, driven by a DC power source, a cadmium-mercury lamp, which is placed in the region of a strong magnetic field generated by the electromagnet, and looks at the spectral lines through an interferometer of some kind (we used a Lummer- Gehrcke plate). In the process of looking up laboratory manuals from different universities to consider different techniques and methods of going about the measurements, we stumbled upon this one. In the very last paragraph of the document, the authors suggest establishing a magnetic hysteresis loop before conducting the measurements of the splitting. The apparatus in this manual differs from the one we use, but not significantly: both use a DC driven electromagnet with a cadmium lamp. The question is, what is the purpose of establishing this hysteresis loop? We have followed the instuctions in the last paragraph of the manual and concluded that doing so results in a degree of permanent magnetization in the electromagnet at zero current (as one would expect with a hysteresis loop) - but then why ask the students to first degauss the electromagnet, as in the second to last paragraph? Any comments/ideas would be highly appreciated.


How much water is destroyed in photosynthesis, relative to the world's supply?

Water is involved in the photosynthesis. How much water are we talking about compared with the total amount on water on Earth? Is it enough to have an effect on the average age of water molecules?


If you suddenly move a piece of metal, will that disturb the free electron density?

If we have a hollow pipe sitting at rest filled with gas and we moved the pipe suddenly along its length to the right, then the gas density will be momentarily higher near the rear of the pipe and lower near its front because gas molecules have inertia. If we start rotating the pipe then gas molecules near the pipe wall will have maximum speed and the speed will go down as we approach the axis of the pipe. What if this pipe was a long metallic cylinder. Metals have free electrons. Will the free electron density change in a similar way if we translate the cylinder suddenly? What if we rotate the cylinder? how the free electrons will respond to that motion? What if the metallic cylinder was charged (so now there is charge resides on its outer surface). How the charge on the surface of the cylinder respond to the sudden translation or rotation of the cylinder? (And what theoretical model(s) one should consider to answer the previous questions?)


Thermal energy generated due to loss in kinetic energy when observed from two different frames of reference

A body is moving with a velocity $v$ with respect to a frame of reference $S_1$. It bumps into a very heavy object and comes to rest instantaneously, its kinetic energy $$\frac{1}{2}mv^2$$ as seen from the frame $S_1$ is completely converted to thermal energy. Now, a man moving with a uniform velocity $V$ (in the direction of the body) with respect to $S_1$ observes the body. He notes that its initial kinetic energy of the body is $$\frac{1}{2}m(v+V)^2,$$ and that after it rams into the heavy body as $$\frac{1}{2}mV^2,$$ and concludes that the thermal energy produced is the difference $$\frac{1}{2}m(v+V)^2-\frac{1}{2}mV^2~=~\frac{1}{2} mv^2+mvV.$$ Which of the two answers is correct and why?


Using Polyakov-Alvarez Anomaly Formula

Take $\Sigma=\mathbb{D}$ to be the unit disk with metric $g=\frac{4}{(1+|z|^2)^2}\,|dz|^2$. If $\phi$ is a nice enough function on $\mathbb{D}$, then I want to compute $$\int_{\partial \Sigma} k_g \phi\,ds_g$$ and $$\int_{\partial\Sigma} \partial_n \phi\,ds_g,$$ where $\partial_n$ is the outer normal derivative, $k$ is the curvature of the boundary and $ds$ is an element of arclength. How exactly does one go about computing these quantities?


The the title is self explanatory, I guess. Why can two (or more) electric field lines never cross?


Differentiating the gravitational redshift and the cosmological redshift?

If general relativity accounts for a redshift, independent of inflation, how can we still know that inflation is viable? Moreover, how do we differentiate the the gravitational redshift and the cosmological redshift observationally?


Naive quantum gravity

My question involves an analogy I have to point out. Consider the Lagrangian density for the a complex scalar field: \begin{equation} \mathcal{L}=\frac{1}{2}\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{\dagger}\phi \end{equation} which is invariant under the global gauge transformation $$\phi \rightarrow e^{i\alpha}\phi.$$ The divergence-less 4-current is $j=(\rho,\underline{j})$ and the associated conserved quantity is the electric charge. This illustrates a global $U(1)$ symmetry. To see the analogy we now look at spacetime translations $x \rightarrow x + a$ which give $$\phi(x) \rightarrow \phi(x+a) = e^{a^{\mu}\partial_{\mu}}\phi.$$ The Lagrangian is no longer invariant but the action: \begin{equation} S = \int \mathcal{L} d^{4}x \end{equation} is invariant due to the translation invariance of the Lebesgue measure. The Noether current is the stress-energy tensor $T^{\mu\nu}$ and the conserved quantity is the 4-momentum. Due to the role played by $T^{\mu\nu}$ in the Einstein field equations we can say that the 4-momentum plays the role of gravitational charge. Promoting the global U(1) symmetry to a local one couples $\phi$ to the electromagnetic field. This involves modifying the Lagrangian by introducing a covariant derivative and adding the Lagrangian for the EM field. My question is: does promoting the global translation invariance of S to a local one give the correct coupling of $\phi$ to the graviton field? An action invariant under general coordinate transformations is: \begin{equation} S = \int \mathcal{L} \sqrt{- g}d^{4}x \end{equation} where $\mathcal{L}$ should also be modified appropriately and $g_{\mu\nu}$ is the metric. I rephrase and extent my question: (1) Do general coordinate transformations include local translations $x \rightarrow x + a(x)$ or are they only $SO(1,3)$ gauge-like transformations? (2) In view of the above analogy the gauge group for gravity is the translation group in Minkowski spacetime. The group is not compact. Shouldn't the group be compact to guarantee a positive-definite kinetic term for gravitons? (3) The translation group is also abelian. Isn't this inconsistent with the fact that gravitons self-interact? (4) In essence what I am saying is that the classical theory of gravity is likely incomplete. Are there any approaches to gravity that modify the underlying spacetime such that the isometry group of the new space admits a compact version of the translation group of Minkowski spacetime? Maybe the universe after formation was compact but due to expansion the compact nature is now hidden. (5) Related to (4). Shouldn't elementary particles correspond to irreducible representations of the isometry group of the universe? If the universe is Minkowski then the isometry group is the Poincare group. Have other possibilities in this regard been explored? (6) In c = h = 1 units Newton's constant G has mass dimension -2. The fine structure constant is given by: $$\alpha = \frac{e^{2}}{4 \pi \epsilon_{0}}$$ In gravitoelectromagnetism we have the correspondence $G \leftrightarrow \cfrac{1}{4\pi \epsilon_{0}}$. As the momentum $p$ plays the role of gravitational charge, shouldn't the coupling constant to gravity be given by $$g \sim p^{2} G = m^{2} G$$ where $m$ is mass of the particle in question? $g$ is then dimensionless. Does this imply that the covariant derivative to couple matter to gravity is given by something like $D_{\mu} = \partial_{\mu} - p^{\nu} A_{\mu\nu}$? where $A_{\mu\nu}$ is some tensor potential related to $g_{\mu\nu}$. (7) The above ideas might imply something about spin 0 bosons and their mass renormalization. Any thoughts? Could the $\Lambda^{2}$ dependence of $m^{2}$ be just an indication that gravity was ignored?


Why does a street light cast a wavey shimmer across the water (at night)

See image You will have to zoom in. Why is the reflection a **wavey** one? Could someone explain how this is occurring, to the layman, (with the style of feynman if possible -heh) :)


Are there any objective wavefunction collapse theories which are local and forbid superluminal signalling? GRW is nonrelativistic and nonlocal.


Why are rebar grooved?

Just what the title states, please. Iron reinforcement bars used to provide the framework over which concrete solidifies always appear to bear grooves. Do the grooves behave as the thread of a screw and provide better grip? Are the grooves merely a manufacturing artifact? Are the grooves a historical artifact - the first bars had to be manufactured grooved for some reason, and hence they continue to be so produced without rhyme or reason. Why are rebar grooved?


Principle Axes of inertia and moments of inertia

1. Principal axes of inertia is defined as the eigenvectors of the inertia tensor matrices. I understand that a diagonalised tensor can yield these axes, but why are they necessarily form the axes of a rotating frame? Can anyone explain why?   2. Lastly, why is it true that any symmetric axis is always a principal axis?    3. If $I_{xx} = I_{yy}$, why is any linear combination of two orthogonal directions on the place spanned by the eigenvector $e_1$ and $e_2$ can always be chosen as principal axes? I'm really confused... Hoping that anyone could shed a light for me.


Looking for a simple proof of symmetry of linear susceptibility tensor

**Getting a diagonal susceptibility tensor:** Define the linear susceptibility tensor as $\chi_{ij}$: $P_i = \epsilon_0\chi_{ij}E_j$, using standard notation for the electric field and the polarization. If one assumes a toy model with atoms having a harmonic potential: $$\ddot{x} + 2\gamma\dot{x} + \omega_0^2x = -eE_x(t)/m$$ Now if we put $E_x(t)$ in terms of exponentials/sinusoid with a forcing frequency $\omega$, we can find the steady-state solution in the same manner as a forced harmonic oscillator. Then the $x$ can be used to write $P = -Nex$, with $N$ denoting number density. Finally, we can end up with an expression where $P \propto E$ and obtain the _scalar_ susceptibility. In principle, we can change the value of $\omega_0$ for $y$ and $z$ and write three different equations for the Cartesian axes to generate a diagonal susceptibility tensor, where not all non-zero terms are equal. Obviously, this tensor is symmetric and if we do a similarity transformation, it will stay symmetric. **Proving symmetry?** I'm not sure how to extend this basic idea to actually prove that the susceptibility tensor ought to be symmetric in general, which is pointed out here on page 2. > The matrix $\chi$ is known as the susceptibility tensor, and is a tensor of > rank 2. This is the most general representation of the susceptibility of a > linear and uniform dielectric medium. It may be shown that, for a lossless > and non-optically active material, $\chi_{ij}=\chi_{ji}$. While the optically-inactive part does seem intuitive, I'm not sure what is meant by lossless medium in this case. Does it mean $\gamma = 0$ in the toy model? As far as I understand, the $\gamma$ is introduced to account for retarding motions due to presence of other atoms in surroundings. Another question: how do you describe a material to be optically inactive, i.e., what equations can one write using it? Also, I did manage to find a proof for the symmetry of dielectric tensor. There are actually two proofs, one using Onsager's theorem and one using Fluctuation-Dissipation theorem. However, I'm looking for a much simpler proof, hopefully in the terms of energy and momentum conservation only, and not having any Thermodynamic machinery.


What is optical confinement?

What is optical confinement? What is it intuitively? How does this 'optical confinement' affect the performance of a laser?


Why is the relationship between atomic number and density not linear?

What are the factors that affect the density of an atom?


Most textbooks on quantum field theory never mention dispersion relations at all. Where can one learn about dispersion relations for S-matrices?


Minus Sign in Feynman Diagram

I've been reading these notes and I can't figure out the why on P.120, it is said that > The fermionic statistics mean that the ﬁrst diagram has an extra minus sign > relative to the ψψ scattering of Figure 25. Would someone mind explaining it to me? I see that the calculations below, illustrating this, but is there an intuitive way of seeing that the signs of the respective first diagrams in Figures 25, 27 should be opposite? Thank you.


Moduli Stabilization in 6D Einstein-Maxwell theory - Fluxes and O3 planes

I'd like to do the maths for the moduli stabilization of 6D Einstein-Maxwell Gravity $$ S= \int d^6X \sqrt{-G_6}(M_6^4R_6[G_6]-M_6^2|F_2|^2), $$ where the 6D metric is specified by $$ ds^2 = g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+R^2(x)\hat{g}_{mn}(y)dy^mdy^n, $$ and $\hat{g}_{mn}$ is the metric of a compact 2D manifold with unit volume. The setup of this model can be found in Denef, Douglas, Kachru, starting on page 10. Now I want to perform the dimensional reduction of the theory and thereby obtain the correct effective potential $V(R) = \frac{\chi}{R^4}-\frac{n^2}{R^6}$. I could manage to derive the first part of the potential by rewriting $$ ds^2= R^2(x)\\{\frac{g_{\mu\nu}(x)}{R^2(x)}dx^{\mu}dx^{\nu}+\hat{g}_{mn}(y)dy^mdy^n\\} $$ which allows me to rescale the 6D action by $G_{MN}=R^2\tilde{G}_{MN}$. Then we are left with a product space, i.e. we can write $R_6[\tilde{G}_6]=R_4[\tilde{g}]+R_2[\hat{g}]$, where $\tilde{g}_{\mu\nu}=\frac{1}{R^2}g_{\mu\nu}$. Further rescaling brings a factor $R^{-4}$ for the term $\chi$, while a factor $R^2$ has been absorbed in order to write $M_4^2=M_6^4R^2$. However, I'm not sure how to obtain the term $\frac{n^2}{R^6}$. I'd appreciate any help. My idea is as follows: $$ \int d^6X \sqrt{-G_6}M_6^2 |F_2|^2 = \int d^6X R^6 \sqrt{-\tilde{G}_6}M_6^2 |F_2|^2 = \\\ = \int d^4x \sqrt{-\tilde{g}}\int d^2y\sqrt{\hat{g}} R^6M_6^2 |F_2|^2 = \int d^4x \sqrt{-g}\int d^2y\sqrt{\hat{g}} R^2M_6^2 |F_2|^2 \sim \\\ \sim M_6^2 \int d^4x \sqrt{-g}R^2 (\frac{1}{R^2})^2 \cdot (\frac{n}{R^2})^2 = M_6^2 \int d^4x \sqrt{-g} \frac{n^2}{R^6} $$ In the last line, we obtain a factor $R^{-4}$ because of $\gamma^{m\hat{m}}\gamma^{n\hat{n}}F_{mn}F_{\hat{m}\hat{n}}$ and the second factor comes from $F_2 \sim \frac{n}{R^2}$, since $$ \int_{M_2} F_2 =n. $$ However, I'm puzzled because the factor of $M_6^2$ remains. If I want to rewrite the action s.t. we have a factor $M_4^2$ in front, then the term I obtain reads $\frac{1}{M_4R} \frac{n^2}{R^6}$ (reminder: $M_4^2 =M_6^4R^2$). So, where is my mistake? Furthermore, it is claimed that O3 planes give rise to an additional term $\frac{m}{R^4}$ (after Weyl rescaling), but I've no idea how it arises. I would start with a CS term $\int C_4$ for the action... It would be great if anyone could help me with these questions. Thanks in advance!


What would be the minimal size of an aerial vehicle capable of sustained suspersonic flight?

I wonder, what's the smallest possible size for a UAV capable of sustained supersonic flight at the current technology level? Let's say 10 minutes of flight at 1.1 M.


A change of sign in the electron-hole second quantization form

It is common to see people do a change of sign in the so called electron-hole representation, namely, $$ b^{\dagger}_{-k}=a_{v,k} $$ similar argument also seen in 1992 mattuck's book "guide to feynman's diagrams" pp.138, eq.7.78b. Can somebody explain what is the physical meaning of changing the sign of momentum, since it make a difference in specific calculations, clearly it is not just a convention, meanwhile it seems to change momentum conservation law in a way that is unambiguous in the electron-electron representation..


Is it easier to move or rotate an object?

This is a problem I'm facing on designing a moving bookcase for my home, I just don't have enough physics background to tackle the problem. Which of the following requires more force/effort?   * A bookcase with a pivot point in one of the back corners that is rotated 180° to its new position (therefore we now face the back of the bookcase).   * A bookcase that is pushed/pulled to the same relative position but with a diagonal push (guided by one rail at the top). In both cases the bookcase will be supported by equivalent fixed caster wheels aligned with the direction of movement or rotation, with the pivot and the rail acting only as guide. I don't think it should matter, but the bookcase will be about 1 ft deep and 3 ft wide, I don't have any good source, but I think it would be around 200 kg if fully loaded.