title: Size of the Observable Universe

text: I wanted to know what the observable universe is so I was thinking and I thought, it must be age of the universe times 2. Well I was wrong. I found on one website that it is 46B LY across in each direction. How does this make sense? I get how the universe has expanded since then, but we should only be able to see light that is 13.7 billion LY old. Does this mean that the Universe is expanding faster than the speed of light? Or light from other objects is travelling to us faster than speed of light?


title: Many times speed of light

text: http://www.huffingtonpost.com/2014/03/24/theory-of-everything-big-bang- discovery_n_5019126.html What does "many times speed of light" really mean in this context? For a layman it's easy to draw wrong conclusions here. Is this just bad journalism or is there a good scientific explanation for this?


title: How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago?

text: I was reading through the Wikipedia article on Quasars and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the article exactly says > _The highest redshift quasar known (as of June 2011[update]) is > ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a **proper > distance** of approximately 29 billion light-years from Earth._ Now I come to understand that the Big Bang singularity is believed to be around 13.8 Billion years ago. So how is this possible? Does the presence of such a quasar negate the Big Bang Theory? I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the "proper distance" mentioned in the fact is a concept that explains this? Edit: My Bad! Here's how.. A simple google search led me to this article (sciencedaily.com/releases/2011/06/110629132527.htm) which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. Lol, what a waste of time! Although I did find out a lot of new stuff from the answers, so thanks for all your responses!


title: How did the universe get so big so fast?

text: * The universe started at the big bang around 15 billion years ago.   * The universe is now at least 92 billion light-years in diameter. Together, don't these mean that the universe, at some time in the past or present, was expanding at a rate of faster than 1 ly/y in every direction? Why is that possible?


title: If nothing can travel faster than speed of light then how the Universe is only 13.7 billion years old?

text: The light would take 93 billion years to reach the edge of universe but nothing can travel faster than the speed of light not even the big bang?


title: Can space expand with unlimited speed?

text: At the beginning, right after the Big Bang, the universe was the size of a coin. One millionth of a second after the universe was the size of the Solar System (acording to http://www.esa.int/Our_Activities/Space_Science/So_how_did_everything_start), which is much much faster than speed of light. Can space expand with unlimited speed?


title: Is there an absolute center of mass exist in the universe?

text: By the "absolute center of mass" I mean the center of all the matter that exists in a given moment. My friend asserted that today and it intrigued us very much but since we only have a CS degree and not very well versed in modern physics we couldn't be sure about that claim. It is based on the observation that; the total momentum of a given system never changes. Am example; considering a hypothetical situation where an astronaut is floating in the space, to change his speed, he needs to push an object against the direction he wants to move. He gains as much momentum as the object gains in the opposite direction, net momentum is zero and center of mass doesn't changes. Applying this principle to the beginning of our universe, we arrive to the aforementioned conclusion. Is this true? If not, why?


title: Diameter of the universe

text: **Should the diameter of our universe always be more than its age in light years?** As if the distance between any two points in the universe is equal to 13.5 billion light years then the light from the big bang could also reach there and certainly more length than its age in light yrs would violate the very basic laws of our universe.


title: Can the coordinate of the big bang point be calculated via observed universe or it is impossible?

text: We know all galaxies spread out after Big Bang theory.The key idea is that the universe is expanding after that theory. Can we play back the scenes via observable universe (galaxies) and can we calculate the coordinate of big bang point as a fixed reference point in the universe? Is there a such calculation in literature or claim? EDIT: After answers I have some questions: If the distribution of matter in the universe is homogeneous,how should we think the geometry of universe. Is it a sphere ? Does not a line go to endless as we imagine? Is there any idea of the border of observed universe? If the universe is homogeneous and like sphere geometry, can we see the beginning of our baby galaxy while observing in the sky? I really will appreciate if someone explains what the universe geometry structure is if we have a homogeneous universe after big bang. Thanks


title: Was Einstein wrong when he said nothing can go faster than the speed of light?

text: If the universe is constantly expanding faster than the speed of light, how could Einstein be right?


title: If nothing can travel faster than the speed of light, how can there be parts of the universe we can't see?

text: Assuming we originated from a single infinitely dense point in space time in the big bang, how can there be parts of the universe that we can't see as the light has not reached us yet, if nothing can travel faster than the speed of light relative to our frame of reference? Is that because the universe can expand faster than the speed of light? Or am I missing something?


title: $P$ symmetry that is apparent with one definition of fields but not with another

text: Suppose that we have a Lagrangian density like $$\mathcal L = -\frac{1}{4} \operatorname{tr} F_{\mu\nu}F^{\mu\nu} + \frac{\theta}{32\pi^2} \operatorname{tr} \big( \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}\big) + \overline{\psi}\gamma^\mu D_\mu \psi$$ where $F_{\mu\nu}$ is the gauge field strength and $D_\mu$ the gauge covariant derivative, and $\psi$ is a fermion field. This Lagrangian is not $P$ conserving because of the $\theta$ term. However if we redefine the fields $\psi \mapsto \exp(i\alpha \gamma_5)\psi$ we can make $\theta$ go away, by choosing $\alpha = \theta/2$ as per the Fujikawa method (described in [Weinberg], Chapter 22 or [Fujikawa]); this is due the path integral measure also transforming under the redifinition. With this redefinition of fields $\mathcal L$ is manifestly $P$ conserving. But surely I can't get more or less symmetry by redefining fields, so **how should I understand that the $P$ symmetry is not manifest with the original definition of the fields?** I suspect that the $P$ transformation too transforms the path integral measure, in a way that sends $\theta \mapsto -\theta$, but I do not know how to show this.   * [Weinberg] Weinberg, S. The Quantum Theory of Fields. 2: Modern Applications (Cambridge, 2005).   * [Fujikawa] Fujikawa, K. Path-Integral Measure for Gauge-Invariant Fermion Theories. Phys. Rev. Lett. 42, 1195{1198 (18 Apr. 1979).


title: Applying theorem of residues to a correlation function where the Fermi function has no poles

text: Let $n_F(\omega) = \large \frac{1}{e^{\beta (\omega)} + 1}$ be the Fermi function. A fermionic reservoir correlation function is given by: $$C_{12}(t) = \int_{-\infty}^{+\infty} d\omega~ \tag{5}J_R(\omega) \, n_F(\omega) \, e^{-i\omega t}$$ The Fermi function here is given in terms of Chebyshev polynomials. The coefficients of the Chebyshev polynomials are given by: $$c_k = \frac{2}{\pi}\int_{0}^{\pi}f(\cos\theta)\cos(k\theta) \;d\theta$$And the Fermi function itself is (Chebyshev approximated, if you will): $$n'_F(x) = \sum_{k=0}^{n}\left[ \frac{2}{\pi}\int_{0}^{\pi}\frac{\cos k\theta \; d\theta}{e^{\beta(E_F-\cos\theta)}+1} \right] T_k(x) \tag{6}$$ Where $T_k(x)$ are the Chebyshev polynomials of the first kind. Now, since the Fermi function here has no poles(as it's given in terms of Chebyshev interpolation polynomials which do not have any poles), the poles in the correlation function are only those of the Spectral Density: $$J_R(\omega) = \sum_{k=1}^{m}\frac{p_k}{4\Omega_k(\omega-\Omega_k)^2+\Gamma_k^2}\tag{7}$$ There's only pole at: $\omega = \Omega_k - i\Gamma_k=\Omega_k^-$, and the residue is $\left.\frac{1}{(\omega - \Omega_k) - i\Gamma_k}\right|_{\omega=\Omega_k^-} = \frac{1}{-2i\Gamma_k}$. Or the residue of $J_R(\omega)$ at $\omega=\Omega_k^-$: $$\mathop{\text{Res}}\limits_{\omega=\Omega_k^-} J_R(\omega) = \frac{p_k}{4 \Omega_k(-2i\Gamma_k)}\tag{8}$$ **My question is: How can the integral of the correlation function now be solved using the theorem of residues/Jordan's lemma? Is it still possible or another scheme should be employed?** If the Fermi function was given in terms of Matsubara frequency sum, it would have had poles and then it's residues could be calculated. Now it does not, and I can't see how the integral can be solved now. If the Fermi function had poles, we could have said: Noting that Poles of $J_R(\omega)$: $\Omega_k^-$, and the poles of $n_F(\omega)$: $\nu_{k'}^*$, we could have gotten: $$C_{12}(t) = (-)(2i\pi) \left \lbrace \sum_{k=1}^m \mathop{\text{Res}}\limits_{\omega=\Omega_k^-}\left[ J_R(\omega) \right] n_F(\Omega_k^-)e^{-i\Omega_k^- t} \\\\+ \sum_{k'} \mathop{\text{Res}}\limits_{\omega=\nu_{k'}^*} \left[ n_F(\omega) \right] J_R(\nu_{k'}^*)e^{-i\nu_{k'}^* t} \right \rbrace\tag{9}$$ And then the residues could have been calculated. _**A bit different Version of the same problem(If you decide to answer, kindly answer this first):_** Here's a function: $$C_{12}(t) = \int_{-\infty}^{+\infty} d\omega~ \tag{1}J_R(\omega)n_F(\omega)e^{-i\omega t}$$ Here's $n'_F(x)\approx n_F(\omega)$: $$n'_F(x) = \sum_{k=0}^{n}\left[ \frac{2}{\pi}\int_{0}^{\pi}\frac{\cos k\theta \; d\theta}{e^{\beta(E_F-\cos\theta)}+1} \right] T_k(x) \tag{2}$$ Where $T_k(x)$ are the Chebyshev polynomials of the first kind. Also, $n'_F(x)$ has no poles. And, $$J_R(\omega) = \sum_{k=1}^{m}\frac{p_k}{4\Omega_k(\omega-\Omega_k)^2+\Gamma_k^2}\tag{3}$$ Where $\Omega$, P and $\Gamma$ are only some numbers. **Are the prerequisites of Jordan's Lemma fulfilled? That is, can equation (1) be written as equation (4) after inserting (2) and (3) in (1) and then applying Jordan's Lemma?** $$C_{12}(t) = (-)(2i\pi) \left \lbrace \sum_{k=1}^m \mathop{\text{Res}}\limits_{\omega=\Omega_k^-}\left[ J_R(\omega) \right] n'_F(\Omega_k^-)e^{-i\Omega_k^- t}\right \rbrace\tag{4}$$


title: Does Fermi-Dirac Statistics explain anti-particles?

text: I wondered whether the Fermi-Dirac Statistics describes the anti-fermion particles. Does it include the anti-particles?


title: Uncertainty and Thermodynamics

text: ## Dilemma The uncertainty principle of energy and the 2nd law of thermodynamics don't add up : the uncertainty principle of energy says that $\Delta \tau \cdot \Delta E \ge \frac{h}{4\pi} = \frac{\hbar}{2}$ where $\Delta$ is the uncertainty in measurement. Now lets consider a situation: lets say that an isolated system $A$ is in thermodynamic equilibrium. It has two particles $b$ and $c$ so that it is in the state of maximum possible entropy. To preserve the uncertainty principle, some net energy must flow from $b$ to $c$ or from $c$ to $b$, and that results in a non-equilibrium state, may be for a fraction of a fraction of a second. But the system must move from the state of maximum entropy (equilibrium) to the state of lesser entropy, which is a sure violation of the second law of thermodynamics. Does anyone have an explanation?


title: Where can I find a list of when I can see something in the sky?

text: My son is very keen on Astronomy but I don't know when there are meteor showers or something else that would be worth going out to the country to see. Last year we went to see a meteor shower, but I only knew about it because I'd heard it on the radio. Which websites list events that we can either see with the naked eye, or with a decent pair of binoculars? (He doesn't have a telescope because the advice I found on the web was that decent binoculars are better than cheap telescopes.)


title: How much gas is in elliptical galaxies?

text: it is widely known that elliptical galaxies have little or no gas, but how is this determined? What is the amount of gas? Is there a decent ratio of stellar mass to gas for ellipticals? How does this vary between normal ellipticals and those that were formed by mergers (or still look like merger remnants)? Thanks for your help!


title: Radiaton of black body

text: We have : $E=h/f$ I realised that the problem what quanta solved was that $h/0$ equals infinity but energy can't be infinity. But when frequency is zero we haven't any energy to calculate - there is no real solution. But Planck came with another solution: The light is packages with different energy - so we always have integer energy score. Why solved he solved problem? Thank you a lot.


title: If we were to travel through space (sci-fi style), how close to the false-color images would the galaxies we see be?

text: I understand that the black-and-white images you see looking through a household telescope are only like that due to the intensity of the light that reaches us, and that most of the astronomy images we find online have some color modification. But if we were to eventually travel between galaxies, how would those galaxies appear to the naked eye? How close to the false color images would they be, and how accurately do we know this?


title: Accuracy and assumptions in deriving the Tully-Fisher relation

text: I understand the mathematical derivation of the Tully Fisher relation from basic physics formulas, as shown on this site. However, after using the physics equations, it seems that several assumptions are made from this point on. First are statistical assumptions. There are statistical errors because the observable, luminous mass of the galaxy is less than the actual mass of the galaxy, and the mass of the galaxy is assumed to be only the observed mass, not the actual mass. Second, this relationship seems to assume that all galaxies are perfectly circular (with negligible thickness). So, with all these assumptions necessary for the Tully-Fisher relation, how is the relationship derived, how can corrections be made for the assumptions when attempting calculations using this relationship, and why is the Tully-Fisher relation generally accepted by the astrophysics community?


title: Proof of Thevenin and Norton theorem

text: How can I prove Thevenin's and Norton's theorem? Thévenin's theorem can be used to convert any circuit's sources and impedances to a Thévenin equivalent.


title: Copper mean free path dependence on temperature

text: I'm doing simulations of copper, where the temperature can reach up to ~1300 K. Some calculations depend on the mean free path (MFP) of copper. The only value I've found for it is 39nm and it's always accompanied with the clause of "at room temperature". Can anybody tell me (or provide some references), how does the MFP depend on temperature? Copper's resistivity dependence on temperature is known to me, can that be used to derive the mean free path dependence in the following way? So we got the conductivity $\sigma(T)$ and that can be related to the relaxation time as $\sigma = \frac{n e^{2}}{m} \tau$ And the relation between the mean free path and relaxation time is $\Lambda = v_F \tau$, where $v_F$ is the Fermi velocity (is this an approximation?). Thus we get $\Lambda(T) = \frac{v_F m}{n e^{2}} \sigma(T)$ Is this a valid way of finding the dependence of the mean free path of copper on temperature?


title: I am learning Quantum Mechanics and I have some questions about some basic concept

text: 1. What does a "STATE" exactly mean in quantum mechanics?    2. What is the equivalence of "STATE" in classical mechanics?   3. If we have a wave function $\Psi$ , its absolute square $|\Psi|^2$ is the probability density of finding the particle somewhere in the space, and I know it can be written as $\langle \Psi |\Psi \rangle$. But what is the physical meaning of $\langle \Phi | \Psi \rangle$, where $\Phi$ and $\Psi$ are two different wave functions. Why we need to take inner product of two different wave functions.


title: Dropping a ball in a fast moving train

text: If I drop a ball in a train moving at a constant speed, will it land on the spot I aimed it at or a little away as the train has moved while it was in air? If it lands away, will the observer not know that he is in a moving frame of reference? If it lands on the intended spot, how did the ball know it is inside a train?


title: What's a "noninertial frame"?

text: In some PSE questions or answers such as here (and comments below) there appears the notion of " _accelerating frame_ " or (more or less equivalently) " _noninertial frame_ ". What's the definition of this notion?,   _i.e._   How are given participants (or, if you prefer, " _point particles_ "$\,\\!^{(*)}$) who "keep sight of each other" supposed to determine whether they (pairwise) belonged to the **same** " _noninertial frame_ ", or not? $(*$: Cmp. the notion of "inertial frame", in distinction to "inertial coordinate system", of http://www.scholarpedia.org/article/Special_relativity:_kinematics   > We should, strictly speaking, differentiate between an inertial frame and an > inertial coordinate system, although in sloppy practice one usually calls > both IFs. An inertial frame is simply an infinite set of point particles > sitting still in space relative to each other. $)$. Follow-up:   The new (follow-up) question to be asked to fully address this question has been submitted as How should observers determine whether they can be described as being "defined on a Lorentzian manifold"?


title: pendulum is a inertial frame of reference or non inertial frame of reference?

text: Example of pendulum is inertial frame of reference or non inertial frame of reference? because if pendulum starts moving its continuously moves without changing there period of time but is changes its velocity with there motion...


title: Why is this not a realisable operation on a quantum system?

text: Let $\rho = \begin{bmatrix}\ 1&0 \\\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\\ 1&1 \end{bmatrix}$ (all density operators). Consider a physical operation $\phi$ such that $\phi(\rho) = \rho$, $\phi(\rho') = \rho'$, $\phi(\rho'') = \dfrac{1}{5}\begin{bmatrix}\ 4&2 \\\ 2&1 \end{bmatrix}$. Why is $\phi$ not a realisable physical operation? It certainly preserves trace and positivity...


title: Leap from photon gas energy distribution to black body radiation?

text: I remember considering in class in college, the case of a photon gas trapped in a d-dimensional box as a subject of interest, whose energy distribution, heat capacity, etc. should be calculated. This momentum/energy distribution was then related to that emitted by a black body - that up to a constant factor, they are one in the same. I could understand the argument that making a small hole in such a box would constitute a "black body", as the hole can't reflect back light impinging on it, but I can't see how to make the conclusion that this is the exact same distribution. Shouldn't the atomic structure of matter be brought into account as well, or is it considered "hollow" on sufficiently large scales?


title: Pauli matrices identities

text: There are some definitions and properties for Pauli matrices and their combinations: $$ \varepsilon^{\alpha \beta } = \varepsilon^{\dot {\alpha} \dot {\beta} } = \begin{pmatrix} 0 & 1 \\\ -1 & 0 \end{pmatrix}_{\alpha \beta }, \quad \varepsilon_{\dot {\alpha} \dot {\beta}} = \varepsilon_{\alpha \beta} = -\varepsilon^{\alpha \beta }, $$ $$ (\sigma^{\mu})_{\alpha \dot {\alpha} } = (\hat {\mathbf E} , \hat {\mathbf \sigma})^{\mu}_{\alpha \dot {\alpha}}, \quad (\tilde {\sigma}^{\mu})^{\dot {\beta } \beta } = \varepsilon^{\alpha \beta}\varepsilon^{\dot {\alpha }\dot {\beta }}(\sigma^{\mu})_{\alpha \dot {\alpha}} = (\hat {\mathbf E}, -\hat {\mathbf \sigma})^{\mu , \dot {\beta }\beta }, $$ $$ (\sigma^{\mu \nu})_{\alpha \beta} = -\frac{1}{4}\left( (\sigma^{\mu} \tilde {\sigma}^{\nu})_{\alpha \beta } - (\sigma^{\nu} \tilde {\sigma}^{\mu})_{\alpha \beta }\right), \quad (\tilde {\sigma}^{\mu \nu})_{\dot {\alpha }\dot {\beta }} = -\frac{1}{4}\left( (\tilde{\sigma}^{\mu} \sigma^{\nu})_{\dot {\alpha} \dot {\beta} } - (\tilde {\sigma}^{\nu} \sigma^{\mu})_{\dot {\alpha} \dot {\beta} }\right), $$ $$ (\tilde {\sigma}^{\mu})^{\dot {\alpha }\alpha}(\sigma_{\mu})_{\beta \dot {\beta }} = 2\delta^{\dot {\alpha}}_{\dot {\beta}}\delta^{\alpha}_{\beta}. $$ How to show, that $$ (\sigma^{\alpha \beta})_{a b }(\tilde {\sigma}^{\mu \nu})_{\dot {c} \dot {d}}g_{\alpha \mu} = 0? $$ (It helps to show, that spinor irreducible representation of the generators of Lorentz group expands on two spinor subgroups). **My attempt**. I tried to show this, but only got following. $$ (\sigma^{\alpha \beta})_{a b }(\tilde {\sigma}^{\mu \nu})_{\dot {c} \dot {d}}g_{\alpha \mu} = \frac{1}{16}\left( (\sigma^{\alpha})_{a \dot {n}}(\tilde {\sigma}^{\beta })^{ \dot {n}}_{\quad b}(\tilde {\sigma}^{\mu})_{\dot {c}}^{\quad m }(\sigma^{\nu })_{m \dot {d}} - (\sigma^{\alpha})_{a \dot {n}}(\tilde {\sigma}^{\beta })^{ \dot {n}}_{\quad b}(\tilde {\sigma}^{\nu})_{\dot {c}}^{ \quad m}(\sigma^{\mu })_{m \dot {d}}\right) $$ $$ \+ \frac{1}{16}\left(-(\sigma^{\beta })_{a \dot {n}}(\tilde {\sigma}^{\alpha })^{ \dot {n}}_{\quad b}(\tilde {\sigma}^{\mu})_{\dot {c}}^{\quad m}(\sigma^{\nu })_{m \dot {d}} + (\sigma^{\beta })_{a \dot {n}}(\tilde {\sigma}^{\alpha })^{ \dot {n}}_{\quad b}(\tilde {\sigma}^{\nu})_{\dot {c}}^{\quad m }(\sigma^{\mu })_{m \dot {d}} \right)g_{\alpha \mu} $$ After that I transformed each summand like $$ (\sigma^{\alpha})_{a \dot {n}}(\tilde {\sigma}^{\beta })^{ \dot {n}}_{\quad b}(\tilde {\sigma}^{\mu})_{\dot {c}}^{\quad m }(\sigma^{\nu })_{m \dot {d}}g_{\alpha \mu} = (\sigma^{\alpha})_{a \dot {n}}(\tilde {\sigma}_{\alpha})_{\dot {c}}^{\quad m }(\tilde {\sigma}^{\beta })^{ \dot {n}}_{\quad b}(\sigma^{\nu })_{m \dot {d}} = $$ $$ = \varepsilon_{\dot {c}\dot {\gamma}}(\sigma^{\alpha})_{a \dot {n}}(\tilde {\sigma}_{\alpha})^{\dot {\gamma} m }(\tilde {\sigma}^{\beta })^{ \dot {n}}_{\quad b}(\sigma^{\nu })_{m \dot {d}} = 2\varepsilon_{\dot {c}\dot {\gamma }}\delta^{\dot {\gamma}}_{\dot {m}}\delta^{n}_{a}(\tilde {\sigma}^{\beta })^{ \dot {n}}_{\quad b}(\sigma^{\nu })_{m \dot {d}} = $$ $$ 2\varepsilon_{\dot {c}\dot {m}}(\tilde {\sigma}^{\beta })^{ \dot {m}}_{\quad b}(\sigma^{\nu })_{a \dot {d}} = 2(\sigma^{\beta } )_{b \dot {c}}(\sigma^{\nu })_{a \dot {d}}. $$ Finally, I got $$ (\sigma^{\alpha \beta})_{a b }(\tilde {\sigma}^{\mu \nu})_{\dot {c} \dot {d}}g_{\alpha \mu} = \frac{1}{8}\left( (\sigma^{\beta})_{b\dot {c}}(\sigma^{\nu})_{a\dot {d}} + (\sigma^{\beta })_{b \dot {d}}(\sigma^{\nu})_{a \dot {c}} + (\sigma^{\beta })_{a \dot {c}}(\sigma^{\nu})_{b \dot {d}} + (\sigma^{\beta})_{a \dot {d}}(\sigma^{\nu})_{b \dot {c}}\right). $$ What to do next?


title: How would a change in ambient temp affect a radiator?

text: I'm curious if you have a radiator or say a block of metal (lets say it's copper since it has the highest thermal conductivity) and on one side is a processor producing heat. At idle the processor heats the block to 50°C, and the block is 120mm x 120mm. The ambient room temp is 22.2°C. You put a fan on the block (120mm x 120mm @ 1000RPM) and it cools the block to 40°C. Now drop the ambient temp 2 degrees C to 20.2°C. Would the full change in ambient temp take effect upon the aluminum block? Would it have less effect due to the blocks efficiency? The processor's heat is constant and the fan blowing on the block is constant.


title: Kinetic energy of two charged balls at infinite distance between them

text: If I have two balls with masses and charges $m_1, q_1^{+}$, $m_2, q_2^{+}$, initially held at distance $d$, and then released, how can I know the kinetic energies of each of the balls at infinite distance between them? I'm quite stuck on that, because they both have the same potential energy at the beginning, and it decreases not in the same pattern, as if one of the balls was stationary. So it not only falls like $1/R$, because at the same time, the other ball that is causing this potential energy, is also being repelled. So how can I really find out the energies? I tried to apply the conservation of energy law, because I know that at infinite distance from each other they'll have zero potential energy, thus all the initial was transformed into kinetic form, however I'm stuck with the initial potential energy (they both have it, so should I put $2U_p$?), and even so, I can't find their kinetic energies separately, without having another equation.


title: antimatter moving back in time

text: This just blew my mind away! I was watching a video about imagining the fourth dimension and the narrator said that little line. Can some people elaborate on this. Also please keep it simple not too complicated. Thank you!


title: Do particles travel backward and forward in time?

text: > All these classical ideas are pointless and obsolete today, because in > quantum mechanics, the particles are completely different objects, defined > by quantum motion of fields, not by the location of classical points (at > least not in a causal field picture). The notion of a point particle was > replaced by the more subtle notion of a quantum point particle, which has a > probability amplitude to be at various places. This quantum point particle > can reproduce the quantum field if it is allowed to go backward and forward > in time. _Ron Maimon_ That was quoted by Ron Maimon. From this - Can I gather that in reality, particles travel backward and forward in time or, is this just a mathematical expression or model?


title: Missing terms in Hamiltonian after Legendre transformation of Lagrangian

text: **Short question** Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms of the original fields, will it contain all of the terms originally in the Lagrangian but with the signs of the potentials 'flipped'? Or are there cases when terms will be dropped in the transformation? Or is my statement altogether wrong. This question is inspired by the long version of my question below, relating to a problem I am currently working on. **Long, specific question** Given the Lagrangian density ![enter image description here](http://i.stack.imgur.com/bIVKZ.png) I have derived (several times to check for errors) using a Legendre transformation, the Hamiltonian density: ![enter image description here](http://i.stack.imgur.com/pZJVT.png) which is basically the Lagrangian density with a few sign flips _apart_ from that it is missing the two terms outside the brackets in the Lagrangian density $\frac{1}{4\pi c}\frac{\partial\vec{A}}{\partial t}\cdot \nabla\phi + \frac{1}{c}\vec{A}\cdot\frac{\partial \vec{P}}{\partial t}$. This worries me, as naively I expected to obtain a Hamiltonian very similar looking to the Lagrangian with all the same terms but some of the signs flipped. However I noticed that the 'missing' terms are the only ones which contain non-squared time derivatives so thought that might have something to do with it.


title: How can I compute the angular velocity of a triangle formed by three particles knowing their instantaneous positions and velocities?

text: I have a set of trajectories of three particles and their instantaneous velocities. I would like to compute the 3 components of the angular velocity pseudovector of the fictive triangle formed by these three particles. How can I proceed for that?


title: How can interstellar space have a temperature of 2-3K?

text: Several different sources online state that the average temperature of interstellar space (or the universe in general) is around 2-3K. I learned that temperature is basically the wiggling of matter, and I find it somewhat counterintuitive that the wiggling of so few particles can cause a temperature of 2-3K. Is there a (order-of-magnitude) calculation which can show that this average temperature estimation is correct, using an estimation of the average density of interstellar space (or the universe in general)?


title: Partition function for a microcanonical ensemble

text: Is it possible to write down a partition function for a microcanonical ensemble?


title: Easy formula for ultrarelativistic bremsstrahlung?

text: I am very curious if an easy calculable formula for the bremstrahlung radiation of deeply relativistic, charged particles exists, if they are moving on circular orbit: $P(E,m_0,Q,r)=?$ ...where   * $P$ is the power of the Bremstrahlung radiation;   * $E$ is the total kinetic energy of the particles (we are in deeply relativistic case, thus $E\gg{m_0}c^2$);   * $m_0$ is the total rest mass of the particles;   * $Q$ is the total charge of the particles;   * and $r$ is the radius of the orbit. If a such clean, trivial formula doesn't exist, a link were also okay, where it can be found.


title: Unstable states and imaginary (complex) energy?

text: I came across the notion of complex energy while studying instanton method to study the unstable state. Unstable states are those which have energy with an imaginary part. But as we know Hamiltonian is a Hermitian operator. So how can the energy be imaginary?


Where do the terms microcanonical, canonical and grand canonical (ensemble) come from? When were they coined and by whom? Is there any reason for the names or are they historical accidents?


title: Is Bremermann's limit redundant with Landauer's limit for all practical purposes?

text: Trying to understand the physical limits to computation, I notice that among these we have two types of limits that constrain the minimum allowable energy for a computation.   * Limits that constrain **the product of energy and time taken** , which include Bremermann's limit and Margolus-Levitin. These two laws pretty much state the same thing as far as I can tell, with the only difference being a constant which doesn't concern me since only order of magnitude matters for my own purposes.   * Limits that constrain **only energy** required for computation, notably Landauer's principle. So let's just write these. I'll utilize the Bremermann's limit since it seems to be more commonly referenced. $$ E \ge \frac{ 2 \pi \hbar }{ \Delta t } $$ $$ E \ge k T \ln{(2)} $$ If you believe what these two equations are telling you literally, then obviously there is a certain time range where the energy-time limit will be more restrictive and a range where the energy-alone limit will be more restrictive. With trivial algebra, we set the RHS equal to each other and find the time for computation where the crossover happens. I used the temperature of space here. For room temperature, it is much smaller/faster. $$ \Delta t = \frac{ 2 \pi \hbar }{ k T \ln{(2)} } \approx 2 \times 10^{-13} s $$ If we're considering serial computations, this would leave us in the $\text{THz}$ range. It's not anywhere near Planck time or anything like that, but it also doesn't seem like much of a practical limitation. Is the energy-time limitation only discussed for academic purposes, with the understanding that it will be swamped by the more restrictive limit? Or is there some deeper reason why it should matter?


title: Partition function of an interacting gas

text: By reading an article, I found a partition function that, according to the author, describes an interacting with random variables as coupling constant. $$Z =\int \mathrm{d} \lambda_i e^{i(K^{ij}\lambda_i\lambda_j + V^{ijk}\lambda_i\lambda_j\lambda_k)}\mathrm{exp}(e^{iS_{eff}(\lambda)})$$ This expression is totally unfamiliar to me. Could someone show me how to derive that, providing a reference (online course, textbook, etc.) if necessary?


title: Refractive Index of Super Critical Fluids

text: I'm studying optics at the moment and I am wondering how the refractive index changes when a liquid is in super critical state? Thank you.


title: Shouldn't Quantum Mechanics change in a black hole?

text: I recently learnt that the conservation laws are a consequence of the symmetries of space and time (the Lagrangian in Newton mechanics). Since space-time change in a black hole wouldn't quantum mechanics also change? Let me give an example by what I mean:   In classical space:   $$x' = x + e \\\ L(x') = L(x)$$ But the above transformation is not true in relativistic space. Hence, the conservation of momentum changes in special relativity (i.e momentum is not $mv$). So similarly shouldn't all the conservation laws change for a black hole and therefore it's action? Side note: the reason I was thinking of a black-hole was that the incoming particle of mass will not drastically affect the space-time (due to the heavy centre)


title: Gauge invariance of gg->gg scattering amplitude?

text: I'm trying to calculate the spin and color averaged gg->gg cross section, and I am stumbling upon gauge invariance: Must the amplitude not be invariant under replacements $\epsilon_i \to \epsilon_i + \kappa_i p_i$ where $p_i\cdot\epsilon_i=0$ and $\kappa_i$ is arbitrary? For me it is not :/ I have summed the 4-gluon diagram and s,t,u-channel diagrams with two 3-gluon vertices each. When I square the amplitude and carry out the spin averaged polarization sum in axial gauge, the extra "n"-vector does not vanish. Am I missing something trivial? Can someone point me to some resource? I do not want to calculate the amplitude with spinor helicity formalism. I am thankful for any hints because I've been sitting on this problem with a collegue for quite a while. Thanks, Tobias


title: squeezed radiation astronomy

text: Squeezed electromagnetic vacuum does have a renormalized energy density smaller than the vacuum. So it makes it in my opinion a inconspicuous candidate for a dark energy carrier. Are there observatories at the moment attempting to detect squeezed radiation from astrophysical and cosmic background sources? If not, What sort of equipment do you need to measure squeezed vacuum radiation?


title: electromagnetic interference

text: If the atmosphere is filled with electromagnetic waves all oscillating at different wavelengths and speeds how is it that they don’t all interfere with each other? For example turning on your light seems to have no effect on the sound coming from your radio.


title: What's the difference between space and time?

text: I'm having a hard time understanding how changing space means changing time. In books I've read people are saying "space and time" or "space-time" but never explain what the difference is between the two concepts or how they are related. How are the concepts of space, time, and space-time related?


title: Space-Time Continuum

text: In special relativity it is said that " Time and space cannot be defined separately from one another. Rather space and time are interwoven into a single continuum known as spacetime. " What is the exact meaning of this statement ? In SR, even though time is relative too but still in classical physics, we still needed time to define an event and what is the meaning of the term continuum ?


title: Space as a function of time?

text: So - a little bit of background. Obviously from Einstein's equations it's shown that energy can be converted to matter and vice versa; in essence, energy and matter are different manifestations of the same "stuff". Now, I've done a bit of reading about the interplay between space and time. Some of what I've read would seem to indicate that just like energy and matter, space and time are different manifestations of the same "stuff", though the majority of what I've read seems to indicate that this isn't the case. So I got to thinking...what if space and time aren't the same thing, but there's interplay between them? What if one is a function of the other? And so I come to my question: Could space be a function of time? Or vice versa? From what I've gathered, our universe is expanding at a faster and faster rate. If space is a (seemingly exponential) function of time, then it makes complete sense that as more and more time passes, the amount of space in existence would increase. In addition (also from what I gather), nothing can go faster than the speed of light. We know that the closer to the speed of light that something is going, the more time slows down for that something, until the speed of light is reached and time stops for whatever it is. Since we're outside observers, a light year is the distance _we perceive_ light to travel in one year. But since (for the light) time stops completely, "how far it travels in a year" becomes meaningless to the light itself. Because no time is passing for it, wherever light goes, it effectively goes that distance in a relative time frame of zero seconds. This phenomenon would also make sense if space is a function of time; if there is no time affecting that light, there would be no space affecting it either. If from the light's perspective time has stopped completely, then from its perspective there would be no distance between it and any destination in the universe; any journey the light made would be instantaneous. Again, that's from the perspective of the light itself, not the perspective of someone who's observing the light. So does that make any sense? Could space be a function of time? Note: I am not a physicist. I am a lay-scientist, in the very loosest definition of that term. I ask this question assuming there are glaring errors and inaccuracies, which is why I'm asking it here - so you can all point me in the right direction.


title: Inductor rise time and inductor wire length

text: So here is an interesting question about inductors or coils in general. Suppose you had a inductor which was 12 inch's in diameter and say 12 inch's in length and has 100 turns of wire. The total length of that wire would be 12(diam) x 3.14(pie) x 100(turns) = 3,768 inches/314 ft in length. So if you apply a voltage across the coil, I assume the time needed for the current to start moving in the whole coil would about the time needed for the signal to traverse the length of the wire which would take about 314 nanoseconds at 1 ft per nanosecond. Otherwise if the electric field from the first wire just permeates out from wire to wire, then it would take a max of 1 nanosecond before the current in every wire was moving. So which is it? Does the electric field need to move through all of the 314 ft of wire before the current is moving at the end of the coil or does electric field move through the length of the wire and ultimately cause the current in the end of the coil to start moving after 1 nanosecond?


title: A projector equal to its own conjugate by a unitary

text: For projector $p$, in finite dimension say, some unitaries $u, v$ does $upu^\dagger = vpv^\dagger$ implies $u = v$ ? Intuitively, can we not say that a unitary is matrix permuting the basis and since $p$ is diagonal then obviously $u$ is $v$ ? But for an exact proof ? what if further, $p = upu^\dagger = vpv^\dagger$ ?


title: Can you read and write a quantum state to a particle without collapsing its probability field?

text: I know the main problem with a quantum computer is that you can't read and write data from it until after it has run though all of its calculations. But would reading and writing quantum data to particles _without collapsing their probability fields_ make this possible?


title: Interacting fermionic SPT phases in 2d with time-reversal symmetry

text: Interacting fermionic SPT phases in 1d and 3d with $\mathbb{Z}_2^T$ symmetry are classified by $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ respectively, as shown in the paper by Fidkowski and Kitaev http://arxiv.org/abs/1008.4138, and Wang and Senthil http://arxiv.org/abs/1401.1142. I'd like to know the classification for the 2d case. Can anyone suggest some reference on that?


title: Optimal door opening

text: This is a problem that has been periodically bugging me, so I finally decided to work on it. I haven't done any physics since high school, so I'm a bit out of practice: Consider a doorway with two hinged doors, one being the mirror of the other. Someone opens one of the doors (which open outward) and lets go, allowing the door to begin shutting. You want to walk though the doorway, but you need one of the doors to be fully open. When is it more optimal to open the closed door completely, as opposed to pulling the closing door back open? By optimal, I refer to the torque over time I would have to exert to open the door fully in time $t_{open}$ **Specifications**   * The doors use 2 manual door closers (a spring that is stretched when the door is open and pulls the door back) which are connected the top and bottom of the door (for simplicity) in such a way that they always pull perpendicular to the door frame   * The doors have two sets of hinges equidistant from the center of the door (again, for simplicity)   * The doors are fully open when they reach an angle $\theta_{open} < {\frac{\pi}{2}}$ from the frame. **Assumptions:**   * Ignoring friction/air resistance/gravity entirely   * ignoring dampening mechanisms of door closer (I have no idea how they work)   * Treating the door as though it had no width **Attempt** So, we have a door of mass $m$ with length across $l_d$, and a handle offset some distance $l_h$ from the hinges of the door. There is are two springs connected to the inner side of the door some distance $l_{s}$ from the hinge edge. Let the two springs each have a spring constant $\frac{k}{2}$. When offset some distance $x$, they pull on the door with force of magnitude $kx$, giving a torque on the door of $l_{s}kx\cos{\theta}$. The offset $x$ is related to $\theta$ by $l_{d}\sin{\theta} = x$, so torque is equal to $l_{d}l_{s}k\sin{\theta}\cos{\theta}$, and torque over time is equivalent. Since we can treat the door as a thin rod, moment of inertia $I$ of the door as $\frac{ml_{d}^{2}}{3}$, we know the angular acceleration $\alpha$ of the door: $$\alpha(t) = \frac{3l_{s}k\sin{\theta}\cos{\theta}}{ml_{d}}$$ Then, the angular velocity $\omega$ is : $$\omega(t) = \frac{3l_{s}k\sin{\theta}\cos{\theta}}{ml_{d}}t $$ and $\theta$ is: $$\theta(t) = \frac{3}{2}\frac{l_{s}k\sin{\theta}\cos{\theta}}{ml_{d}}t^2 $$ At this point, I pretty sure I've taken a wrong turn somewhere, probably involving conversion of angular acceleration to velocity, since theta is a function of time as well... I feel like this is a problem I should be able to do relatively simply. Anyone want to help me along? Also, it seems like I'm probably over-complicating the situation...


title: Do any good theories exist on why the weak interaction is so profoundly chiral?

text: I find the profound asymmetry in the sensitivity of left and right chiral particles to be one of the most remarkable analytical observations captured in the Standard Model. Yet for some, I've not found much in the way of discussions that worry about _why_ of such as truly remarkable fact is true. I can't help but be reminded a wee bit of views on the motions of planets before Newton... you know, "it be Angels that do push them around, ask ye not why!" Seriously, I know the Standard Model has a lot of givens in it... but surely _someone_ has mulled over why the universe might exhibit such a non-intuitive and thus _interesting_ asymmetry? And perhaps even developed some solid speculations or full theories on why such in-your-face chiral asymmetries exist in nature? Do such theories exist, or is this asymmetry truly just a "given" and nothing more?


title: Does the Higgs mechanism address the spin statistics problem?

text: Since the Higgs mechanism is so intimately tied to binding together massless chiral fermions, does it happen to have anything to say about the spin statistics issue? I'm actually assuming the answer is no, but perhaps there is a deeper tie there that I'm missing completely. So, just curious...


title: Relating generalized momentum, generalized velocity, and kinetic energy: $2T~=~\sum_i p_{i}\dot{q}^{i}$

text: According to equation (6) on the first page of some lecture notes online, the above equation is used to prove the virial theorem. For rectangular coordinates, the relation $$ 2T~=~\sum_i p_{i}\dot{q}^{i} $$ is obvious. How would I show it holds for generalized coordinates $q^{i}$?


title: where does the photon go after scattering?

text: My question is about `photo electric` but it could be applied to other daily routine phenomenon. As we know rest mass of photon is zero. When a photon strikes the metal surface it transfers its energy to the electrons. Whether electron will be emitted or not, it depends upon work function. But my question is about that photon which was hit on metal surface. Does that photon vanished? Does that photon turned into nothing? where does it go after scattering? When I study this I only find the story about the emitted electrons but not about photons after collision. Am i missing some basic concept?


title: On a trick to derive Noether current

text: Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the variation local: the standard argument, which does't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the _only_ term appearing in the variation will be proportional to _derivatives_ of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on shell: $$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon $$ This is stated, e.g., in _Superstring Theory: Volume 1_ by Green Schwarz Witten on page 69 and _The Quantum Theory of Fields, Volume 1_ by Weinberg on page 307. In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden? * * * Taking from the answer below, I believe two nice references are   1. theorem 4.1   2. example 2.2.5


title: Breaking two ropes. What can be said about the tensions?

text: This is a theoretical problem. Suppose we have two ropes of the same material and diameter, but differing in length. We excert force on both ropes until they break. Let rope A be the longer one. What can be said about the stress in rope A compared to the stress in rope B at the moment of breaking, taking in account the deformation required to keep a constant volume? You would expect them to be the same as the stress-strain diagram is derived for a specific material. But at the same strain of both ropes, the stress in rope A would be higher as it crosssection decreases more. ![!](http://i.stack.imgur.com/aj7XJ.jpg) Stress-strain diagram from http://cdn.transtutors.com/ 1


title: Is a large system just a set of smaller systems?

text: All particles exhibit wave-particle duality. And I have a strange question. Why does a larger system, liken an atom that is just a set of smaller systems, itself exhibit wave-particle duality? In principle all large systems can be defined as a set of smaller systems. An atom is a set of nucleus (a smaller set of up and down quarks held together by the strong interaction) and some electron(s) bound to the set of quarks by the electromagnetic force. How is this set of smaller systems being able exhibit wave-particle duality as a whole as if it is a particle itself? Does this imply that electron and all other elementary particles are indeed just another set of smaller sets and any sets can exhibit wave-particle duality? And here comes the question: how do you define a set? We naturally define composition such as atom, molecule or "elementary particle" like electron as a set. Can the composition of electrons and up quarks be defined as a set (a system with wavefunction) and the down quarks that present in the atom be defined as a separated set (another system)?


title: Photosensitivity of Photodiodes, lower wavelength -> lower sensitivity?

text: I saw a diagram of the photosensitivity (Current per Power) of a photodiode. ![Photodiode sensitivity](http://i.stack.imgur.com/44MQw.png) So there is this diagonal stating the 100% quantum efficiency. I wondered why the sensitivity for bluer light lower than that of redder light. Is it because a blue photon has a higher energy? Thus fewer blue photons are needed to achieve the same power as with red photons? The photocurrent is proportional to the number of converted photons. So a 1 watt beam in red has more photons, which generate more photo-electrons, thus more current per watt. Is this reasoning correct and explains the curve in the diagram?


title: Ways to create controlled small bubbles in water within plastic tubing

text: What is a practical way in an engineering or physics laboratory to create bubbles of a specified size in water within plastic tubing? The tubing is a few mm inner diameter. We'd like to make bubbles smaller in size than the diameter, for example 1/2 the diameter. We'd like to have control over the size, especially volume, and rate of bubble generation. We have tried a metal hypodermic needle connected to a low-power air pump, but air accumulates over time to make one big bubble, which eventually breaks away. Maybe there is a way to knock the bubble off when it's the desired small size, instead of waiting for it to do as it pleases? The water is at roughly 1 to 1.3 atm pressure, at room temperature. We imagine possibilities of electrolysis, or acoustics, but haven't found any written experience through google searches.


title: Electron degeneracy pressure

text: Why is it that in stars undergoing gravitational collapse electron degeneracy kicks in? Why couldn't the electrons form energy bands like in semiconductors?


title: Which values of the Riemann zeta funtion at negative arguments come up in physics?

text: For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. Currently, I am studying the divergent series that arise when considering the Riemann zeta function at negative arguments. The Riemann zeta function can be analytically continued. By doing this, finite constants can be assigned to the divergent series. For $n \geq 1$, we have the formula: $$ \zeta(-n) = - \frac{B_{n+1}}{n+1} . $$ This formula can be used to find:   * $\zeta(-1) = \sum_{n=1}^{\infty} n = - \frac{1}{12} . $ This formula is used in Bosonic String Theory to find the so-called "critical dimension" $d = 26$. For more info, one can consult the relevant wikipedia page.    * $\zeta(-3) = \sum_{n=1}^{\infty} n^3 = - \frac{1}{120} $ . This identity is used in the calculation of the energy per area between metallic plates that arises in the Casimir Effect.  My **first question** is: do more of these values of the Riemann zeta function at negative arguments arise in physics? If so: which ones, and in what context? Furthermore, I consider summing powers of the Riemann zeta function at negative arguments. I try to do this by means of Faulhaber's formula. Let's say, for example, we want to compute the sum of $$p = \Big( \sum_{k=1}^{\infty} k \Big)^3 . $$ If we set $a = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} $, then from Faulhaber's formula we find that $$\frac{4a^3 - a^2}{3} = 1^5 + 2^5 + 3^5 + \dots + n^5 , $$ from which we can deduce that $$ p = a^3 = \frac{ 3 \cdot \sum_{k=1}^{\infty} k^5 + a^2 }{4} .$$ Since we can also sum the divergent series arising from the Riemann zeta function at negative arguments by means of Ramanujan Summation (which produces that same results as analytic continuation) and the Ramanujan Summation method is linear, we find that the Ramanujan ($R$) or regularised sum of $p$ amounts to $$R(p) = R(a^3) = \frac{3}{4} R\Big(\sum_{k=1}^{\infty} k^5\Big) + \frac{1}{4} R(a^2) . $$ Again, we know from Faulhaber's Formula that $a^2 = \sum_{k=1}^{\infty} k^3 $ , so $R(a^2) = R(\zeta(-3)) = - \frac{1}{120} $, so $$R(p) = \frac{3}{4} \Big(- \frac{1}{252} \Big) + \frac{ ( \frac{1}{120} )} {4} = - \frac{1}{1120} . $$ My **second** (bunch of) **question(s)** is: Do powers of these zeta values at negative arguments arise in physics? If so, how? Are they summed in a manner similar to process I just described, or in a different manner? Of the latter is the case, which other summation method is used? Do powers of divergent series arise in physics in general? If so: which ones, and in what context? My **third and last** (bunch of) **question(s)** is: which other divergent series arise in physics (not just considering (powers of) the Riemann zeta function at negative arguments) ? I know there are whole books on renormalisation and/or regularisation in physics. However, for the sake of my bachelor's thesis I would like to know some concrete examples of divergent series that arise in physics which I can study. It would also be nice if you could mention some divergent series which have defied summation by any summation method that physicists (or mathematicians) currently employ. Please also indicate as to how these divergent series arise in physics. I also posted a somewhat more general and improved version of this question on MO.


title: Determine pipe outlet temperature without length?

text: Let's say I am flowing a fluid through a pipe that has a constant surface temperature. Is it possible to determine the heat flux on the inner surface without knowing the pipe's length when the only two temperatures I know are the pipe surface temperature and the inlet flow temperature? I am able to assume the flow is thermally and hydrodynamically developed. A fully developed pipe with constant surface temperature has a Nusselt number of 3.6568 which allows me to then determine the convection coefficient. I was going to calculate the heat flux using $$q''=\bar h\Delta T_{lm}$$ The log mean temperature difference would need to use the outlet temperature found by $$\frac{T_{s}-T(L)}{T_{s}-T_{in}}=exp(\frac{-\pi DL}{\dot m c_{p}}\bar h)$$ The problem lies in that I don't have the pipe's length so I can't calculate the outlet temperature or the LMTD. The only other method I could think of is using $$q''=h(T_{s}-T_{in})$$ but that will greatly underestimate the actual heat flux.


title: Yang Mills theory and SU(N) groups

text: Trying to get a better understanding of the relation between a SU(N) Yang Mill theory and its number of "color" space. Most of the description I've found so far are either way to complex/specific. Yiannis answer on this post is almost what I'm looking for, except I was hoping someone could provide addition sources and readings corresponding precisely to what he is describing.


title: How to determine predicted CP violation for a given SUSY point?

text: I'm currently studying at the spectra of some supersymmetric models, and would like to know whether the parameter points I'm looking at are ruled out due to excessive CP violation. I am using SPheno, which allows me to test my spectra against various other experimental bounds. It provides a `Block SPhenoLowEnergy` in its SLHA output that has, for example, the predicted branching ratio $B_s \rightarrow \mu\mu$, or the anomalous muon $g-2$. What's the variable to look for when one colloquially says "ruled out because of CP violation"? How to get it in SPheno (or with any other spectrum generator / event generator, I can switch)? I feel like the answer is obvious but I'm missing the forest for the trees.


title: Are these assumptions correct?

text: Is it true:From equation $C=Q/V$ 1)If Potential difference is kept **constant** $C \propto Q$ 2)If charge is kept **constant** $C \propto 1/V$ Then if 2nd is true I do have a question: Don't we say that capacitance is the ability of a body to store charge,so my question goes this way,if the charge is kept constant and potential difference is decrease,then by the above assumption the capacitance should increase(and which is true).But we have said that charge is kept constant,then how would the capacitance increase? I need an answer


title: Graduate Physics Problems Books

text: Need to brush up on my late-undergrad and early-grad physics and was wondering if anyone can recommend books or lecture notes (hard copy, or on-line) that also have solutions. Two that I have come across are: **Princeton Problems in Physics with Solutions** \- Nathan Newbury **University of Chicago Graduate Problems in Physics with Solutions** \- Jeremiah A. Cronin **Spacetime Physics** \- Taylor & Wheeler (favorite book on special relativity; has a lot of problems with solutions at the back; a lot of the problems really enforce the material and discuss paradoxes) If possible, please also provide a reason why you like the books as opposed to just listing them.


title: Guides for solving E&M problems

text: I am an undergraduate physics student in a university and I am looking for a good guide or a handbook with lots of questions and full answers in subjects such as Gauss' law and Maxwell equations to study for my test.


title: A set of problems for "The Theoretical Minimum" Stanford course

text: I'm really excited by an opportunity to watch Leonard Susskind's course "The Theoretical Minimum". However, I couldn't find any physics problems on the website, and I'm a little bit wary of just watching videos. I want to think on some problems during that. Maybe there's a good collection of Physics problems on related themes, so I could try them between the videos?


title: Please recommend a physics problems book similar to Demidovich

text: > **Possible Duplicate:**   >  Graduate Physics Problems Books Does anyone have recommendation for a physics problems collection book (series) that is similar to Demidovich's _A Collection of Problems and Exercises in Mathematical Analysis_? I am looking for the encyclopediac coverage and Russian editorial style, if you like. # Update: I'd like problems for 1) senior undergraduates major in physics and/or 2) Ph.D. candidates in physics


title: Where can some worked problems in classical mechanics (and more specifically the Lagrangian and Hamiltonian formalisms) be found?

text: I've been looking for a textbook in classical mechanics that's readily available (like can be found in the library of James Cook University of Townsville, Australia) and full of fully-answered questions in the Lagrangian/Hamiltonian formalisms yet I can't find any. My first port of call was the Schaum's Outlines and Demystified series but the only member of these series I could find that was relevant was Lagrangian Dynamics which is difficult for me to track-down in real-life. It would be particularly helpful if one could point me to a free eBook with this material.


title: Problem book in Quantum mechanics with emphasis on physical(ly relevant) problems

text: I am a second year undergraduate and studying quantum mechanics from sakurai's 'Modern Quantum Mechanics'. Is it a good idea to solve problems from sakurai, which are mostly mathematical in nature? I need a textbook that has physically relevant problems, maybe going even into condensed matter, or field theory in its exercises. This would probably help me to appreciate and understand qm better. Sorry if this question is too localised but I just had to post it.


title: Problem books like I.E. Irodov for advanced physics

text: I really enjoyed doing problems from Irodov while learning introductory physics. But I am not able to find a book like that for Graduate level physics. Can you suggest me a book which has good (and hard) problems from advanced topics like Quantum Theory, QFT, Relativity, etc?


title: Requesting some research study problems on Classical Mechanics

text: Someone please tell (advice) me some research study problems on Classical Mechanics that can be tackled with Undergraduate level knowledge.


title: Goldstein's Classical Mechanics exercises solutions

text: Does anyone know where I can find some (good) solution of Goldstein's book _Classical Mechanics_?


title: Questions about spacetime curvature and gravity

text: These are some further important queries regarding the question here Why would spacetime curvature cause gravity? Q1. Explain the statement “Everything in spacetime moves at the speed of light”. Is this statement even true or false? My naïve understanding is as follows: If an object is stationary in space (relative to an observer?) then it will be moving in time, so, it will, say, move from (x=0,t=0) to (x=0,t=10). The velocity is the space interval / time interval, which from above coordinates is still zero. So it is moving through spacetime, but at zero velocity. So velocity will just determine the angle of its path to the time axis. If this angle is zero, ie it is moving parallel to the time axis, then velocity = zero, and it can really move in time from one time point to the other at zero velocity. Where am I wrong, and what is the real explanation? Q2. Suppose there are two objects (see fig), an apple A above the earth at x=0, and the earth E at x=10 as shown in the fig. If there is no earth and just the apple in open space, then the spacetime is not curved due to the gravity of the earth, and the apple stays at x=0 but moves in time from A to B (fig a). Now the earth is at x=10 and presumably the spacetime curves and the axes tilt as in fig b. Then the apple moves from A to C, just following the geodesic. But if we assume this, then the apple has not moved at all, because, due to the tilt of the axis, point C is also at x=0. So fig b cannot be the correct situation, otherwise the apple will arrive at point C which is still on the x=0 axis. So I assume that in fact the apple has not moved from A to C in fig b, but has moved from A to D in fig. a, where D is really at x=10. But if I assume this, then it is not spacetime itself that has curved. The spacetime is still straight, but the apple has moved from (x=0,t=0) to (x=10,t=10). Again, where am I wrong, and what is the correct explanation? How exactly has the spacetime curved or tilted for the apple at A due to the presence of the Earth at E, and how does the apple move to the earth by following the geodesic in free fall? (assume only one spatial dimention) ![spacetime curved by gravity?](http://i.stack.imgur.com/9QQXe.png) Q3: We say that gravity is not just a “force of attraction” between two pieces of mass, and it does not “pull” the two pieces of mass towards each other. Instead, acceleration is manifested because the two pieces of mass are simply following their now curved geodesics. But it can be shown that there really is an attractive force due to gravity. E.g. Take the apple in the earth’s gravity and suspend it on a spring balance. The spring will extend. Yes, you can say that since it is suspended motionless, so it is no longer in free fall, and since the spring balance exerts an external force upwards on the apple, so it no longer follows its free fall geodesic. But it is not only the spring balance which is exerting an upward force on the apple. The apple is also exerting a downward force on the spring, which causes the spring to extend. Surely it is too naïve to say that the downward force which the earth exerts on the apple and in turn the apple exerts on the spring suddenly vanishes if the apple is released into free fall. If two masses were remotely attracting each other with a force, why would they cease to attract each other with a force when in free fall? It is logically a much more satisfying explanation that the gravitational attraction force which was there at rest remains in free fall too and it is this gravitational attraction force which causes the freefall acceleration by F=ma, just like any force on any object would cause an acceleration. Also, it is the net force on the object which causes the acceleration. When the object was suspended, the net force was zero, so acceleration was zero. When in free fall, the net force IS THE GRAVITATIONAL ATTRACTION FORCE, so acceleration is accordingly. This seems totally opposite to the statement that there is no gravitational force in free fall. The reason why the spring of the spring balance does not extend when in free fall is that there IS a gravitational force downwards, but there is no force exerted by the spring upwards, so the net force on the object is downwards due to gravity, which causes the acceleration F=ma. Where am I wrong? What is the correct explanation?


title: Pauli matrices and the Levi-Civita symbol

text: This is just a quick question. I would figure this out myself if I wouldn't have an exam about this tomorrow. I am working on the non-relativistic approximation of the Dirac equation for an electron in an EM field. On one point, I need the following relation: $$ \epsilon^{klm} \sigma^{m} = \sigma^m \epsilon^{mkl} $$ where $\sigma^m$ denotes the $m$th Pauli matrix and $\epsilon^{klm}$ denotes the Levi-Civita symbol and the Einstein summation convention is used. The question is: does this relation hold in general for the Levi-Civita symbol or is this specific for the Pauli matrices? TIA, Eric.


title: Feynman Diagram in $\phi^3$ theory

text: I'm slightly befuddled by is what it means when I'm asked to > Draw the Feynman diagram in momentum space for the two point function of > $\frac{\lambda}{3!}\phi^3$ theory for order $O(\lambda^2).$ I can draw Feynman diagrams, and I thought two-point function meant $$\langle0\|\phi(x)\phi(y)\|0\rangle$$ and what I know about $ O(\lambda^2)$ is that it will have more diagrams than $ O(\lambda).$ Other than that, I'm a bit lost. I mean, I'm not even sure if this is a really simple calculation or quite a long one. Apologies to myself if anything I've written above is embarrassing.


title: Sum of energy in a free rotation with Algodoo sofware

text: I try to understand why Algodoo software (freeware) give an efficiency greater than unity. I tested several scenes and at each time I can look the software give energy, I don't say it's all the time the case, a lot of scene have energy constant or decrease but with particular forces Algodoo give energy. The energy is not very high compare to sum of energy, I only have 50 J each second (the system give 50 W) with a total energy of 10000 J, it's small only 0.5 % of the total energy. But I can test until 12000 Hz for some scenes and the energy don't decrease. The system can be a 2 objects or better 3 objects in a free rotation, there is no axis. I compute with a large scale of frequencies from 100 Hz to 12000 Hz. I can't increase more frequency because forces become unstable: forces are unstable. It seems 1200 Hz give good results in a lot of scene and for a lot of scenes results don't change from 800 Hz to 1200 Hz. I compute datas with a spreadsheet for test the result of energy, when Algodoo give energy (efficiency >1) datas give energy. I looked at center of gravity and with good frequency it don't move, sometimes like 0.1 mm for an object of 4 m * 4 m for the main object and 0.3 m for the second object (for example). No friction and no gravity in this study. Friction and restitution is at 0.5 for all material. I changed only density for have best result. **Example of a scene: (not the best error but easy)** ![enter image description here](http://i.stack.imgur.com/6MjbY.png) **Forces:** ![enter image description here](http://i.stack.imgur.com/zOnz2.png) **Energy and informations:** ![enter image description here](http://i.stack.imgur.com/KeCbC.png) **Position of center of gravity:** ![enter image description here](http://i.stack.imgur.com/pMt5N.png) I'm not sure, but with some tests (not all), I find a phase angle between 2 objects (or 3 with 3 free objects in rotation). Like sum of force from one object is the same (value) than other, if there is a phase angle between two center of gravity, there is a torque from these center of gravity. Or at least one center works in positive and other in negative. Like centers of gravity don't move in the same distance, the work is not the same. **Angle: 6°** ![enter image description here](http://i.stack.imgur.com/9y1BB.png) **A link for download an example:** https://drive.google.com/file/d/0B63Jbse1IMAkUUo2Sm0tcnI5ak0/edit?usp=sharing **I have several questions:** 1/ For you, the error come from this phase angle ? If not, where the error come from ? 2/ If there is a phase angle, there a torque ? 3/ Is it possible in reality that centers of gravity are in phase angle ? I think it's not possible but I asked the question to be sure. 4/ Why Algodoo find a phase angle ? Is there a physical explanation ? Maybe the error don't come from the phase angle but when I compute sum of energy, center of gravity of red object give energy but in the same time green object don't consume this energy. I made a lot of tests and at each time it's the energy come from the small object. If you have an idea for find the error ? Or a test I can do for find the error ? If necessary I can give another scenes with more power with 3 objects. **For example, this 3 objects:** ![enter image description here](http://i.stack.imgur.com/t7v7X.png) **The energy increase like:** ![enter image description here](http://i.stack.imgur.com/u68hf.png) **I thought 4800 Hz is a good frequency, but some seconds after the sum is:** ![enter image description here](http://i.stack.imgur.com/GPfWF.png) **I study the system and find something: the energy increase only when forces are like :** ![enter image description here](http://i.stack.imgur.com/GYihS.png) **Details of forces:** ![enter image description here](http://i.stack.imgur.com/UCtRp.png) **The energy increase more if red and orange objects have forces like before (but in opposite direction):** ![enter image description here](http://i.stack.imgur.com/cw6ZH.png) **When I compute energy I have more than Algodoo:** ![enter image description here](http://i.stack.imgur.com/bb0g6.png) **A link for download the scene:** https://drive.google.com/file/d/0B63Jbse1IMAkTUUzTzNlc1VORXM/edit?usp=sharing In this scene, I don't find always a phase angle or small. But each time forces are like image shows, the energy increase and if I compute datas with a spreadsheet, it's red and orange objects (small objects) that give energy. In the contrary, green object don't consume energy, the sum is near zero. * * * It's not possible it's the moment of inertia the problem ? If one part of object give a torque from friction it receive a torque in the same time, but the work from a torque is torque by angle of rotation. If inertia is not the same, angle of rotation is not the same too even the torque is equal in value, no ? I don't find the error from Algodoo and it's the only idea I have found. At least, if someone can say if it's a possibility or not.


title: Translations and Noether's Theorem

text: I'm fine with $U(1)$ symmetry and Noether's Theorem, but struggling with the translations of the field; namely $$\phi'(x^{\mu})=\phi(x^{\mu}-a^{\mu}),$$ where $a^{\mu}$ constant four-vector $$x^{\mu}=x^{\mu}+a^{\mu},$$ and the Lagrangian density $${\cal L}=\frac{1}{2}\partial_{\mu}\phi^*\partial^{\mu}\phi-V(\phi^*\phi).$$ So a few questions:   1. I can't show the Lagrangian is invariant under this transformation. Is it just a case that as $a^{\mu}$ is constant then the first term in the Lagrangian will obviously stay the same? But what about $V$? How I can show that's invariant?   2. Infinitesimally, is the transformation $\phi'(x^{\mu})=\phi(x^{\mu})-a^{\mu}\partial_{\mu}\phi(x^{\mu})?$    3. If I'm right in point 2., how can I apply Noether's Theorem to this?


title: What are the specific electronic properties that make an atom ferromagnetic versus simply paramagnetic?

text: As I understand it, paramagnetism is similar in its short-term effect to ferromagnetism (spins of the electrons line up with the magnetic field, etc.), though apparently the effect is weaker. What is it exactly that determines whether an atom or molecule is ferromagnetic versus paramagnetic, and why is the paramagnetic effect weaker?


title: Is it correct to say that electromagnetic waves does not require a medium?

text: I can conceive of a particle existing in empty spacetime, but not a wave. A wave appears to me at least, to insist upon a medium for its very definition. I understand that the 19C physicists postulated the lumineferous aether for this very reason, but abandoned it in favour of the electromagnetic field. In the textbooks I've looked at, its always said that the physicists eventually understood that a wave does not necessarily require a medium. But it seems to me tht they abandoned the idea of a medium mechanically conceived (I assume because of the high prestige of newtonian mechanics, its status as a defining, fundamental paradigm which privileges our naive & intuitive understanding of mechanical phenonema) for a medium conceived in a a more general manner. Afterall, this field permeates spacetime, and its state evolves through time. This is how I would naively conceive of a medium in a general sense. I should add this is maybe more of a history of physics question, and I've added a tag as such to stop people getting upset.


title: How to simulate a full-suspension bike?

text: As a fun project, I would like to roughly simulate the suspension operation of a full-suspension mountain bike. This is _not_ another one of those "How does a bicycle stay upright?" questions. Follow some metrics of interest.   * Pedal bob - the amount of movement in the rear triangle upon pedaling up an incline. An estimate of the lost energy due to this.   * Sharp obstacle transient response - for example how does the bike move, when hitting a curb at high speed.   * Large obstacle transient response - for example, how does the bike move when it falls (along with the rider) 1 meter. I am somewhat familiar with python, Mathematica, ScyLab, Matlab, Ansys. I guess an analytical solution would be too difficult. _Which approach will give quickly a parameterizable model and analyze it for the abovementioned information?_ Here several common frame variations are mentioned. EDIT: More concretely:   * What model to use? Is there a widespread "bicycle" model for dynamic analysis?   * What software to utilize, in order to help me solve the problem?


title: Determining Ether Drag - Aberration of Starlight

text: In order to determine the relative motion between the Earth and the ether (the medium through which light supposedly propagated. It has zero density and complete transparency), scientists used the concept of the aberration of starlight. If the Earth frame of reference was not dragging the ether frame of reference, then the direction of starlight as seen from both frames of reference separately would be different. Whereas if the Earth frame of reference was indeed dragging the ether frame of reference, then the starlight observed from both frames of reference separately would appear to come from the same direction. This is all intuitive. But how do we prove whether either is the case, when we are unable to identify the ether frame of reference itself, and conduct experiments within this frame of reference to determine the observed direction of starlight from within this frame of reference?


title: choosing the right color

text: Context: My room is being painted, and i sit and study in a corner of the room, surrounded by walls on 2 sides, such that i am facing the wall. A tube light is at my 4-5 o'clock, and a door(transparent) at my 7 o'clock. These are the only two sources of light. There is a TV mounted just in front of my desk. Problem: I want to paint only one wall with a dark color and the rest with a light yellowish green color. Now, which wall should be painted with the dark color so as to maximize light in the corner?? IE, the one facing me, on which the TV is mounted, or any other?


title: Does the method of an experiment always have to be numbered?

text: When writing a method for an experiment, does it always have to be set out in orderly numbered steps? Can it not also be a paragraph of text that outlines the method? **A mundane example** :   1. Place a tripod on a table, and the wire gause mesh ontop    2. Following that, place a busen burner underneath it, and hook it up to the gas   3. Fill a beaker with 100ml of water   4. Place the beaker on the wire gause, turn on the gas and lit the Busen burner   5. Heat the water until it boils **...** **VERSUS** Firstly, place a tripod, ensuring that you place a wire gause on on top. Following that step, ... Thanks.


title: Gravitational distortion of an object's diameter, at a distance,

text: Does the curvature of space-time cause objects to look smaller than they really are? What is the relationship between the optical distortion and the mass of the objects?


title: what causes virtual particle pair production to not occur in the space occupied by matter?

text: Are virtual particles only popping in and out of existence where the local energy density is below a certain point? What I wonder is, does any kind of matter prevent the pairs from appearing? Is there a shell surrounding an atom or maybe I should call it a boundary beyond which particle pair production occurs, and within the boundary it does not, I have wondered if the different orbitals around an atom are affected (set)by the influence of the virtual particles.


title: Close electric field lines in wave guides

text: In a wave guide, graphics of propagation of Transversal Magnetic modes show closed field lines for the electric field. For example, for a rectangular guide: $E_x (x,y,z) = \frac {-j\beta m \pi}{a k^2_c} B_{mn}\cos\frac{m\pi x}{a}\sin\frac{n\pi y}{b}e^{-j(\beta z + \omega t)}$ $E_y (x,y,z) = \frac {-j\beta n \pi}{b k^2_c} B_{mn}\sin\frac{m\pi x}{a}\cos\frac{n\pi y}{b}e^{-j(\beta z + \omega t)}$ $E_z (x,y,z) = B_{mn}\sin \frac{m\pi x}{a}\sin\frac{n\pi y}{b}e^{-j(\beta z + \omega t)}$ Is it possible to have closed lines for the electric field?


title: Relation between the determinants of metric tensors

text: Recently I have started to study the classical theory of gravity. In Landau, Classical Theory of Field, paragraph 84 ("Distances and time intervals") , it is written We also state that the determinanats $g$ and $\gamma$, formed respectively from the quantities $g_{ik}$ and $\gamma_{\alpha\beta}$ are related to one another by $$-g=g_{00}\gamma $$ I don't understand how this relation between the determinants of the metric tensors can be obtained. Could someone explain, or make some hint, or give a direction? In this formulas $g_{ik}$ is the metric tensor of the four-dimensional space- time and $\gamma_{\alpha\beta}$ is the corresponding three-dimensional metric tensor of the space. These tensors are related to one another by the following formulas $$\gamma_{\alpha\beta}=(-g_{\alpha\beta}+\frac{g_{0\alpha}g_{0\beta}}{g_{00}})$$ $$\gamma^{\alpha\beta}=-g^{\alpha\beta}$$ Thanks a lot.


title: The measurement of $g$ and air resistance

text: Suppose I am trying to determine the value of the acceleration due to gravity by simply dropping objects from a ledge. If I did not account for air resistance, would the acceleration due to gravity measured be less than if I did account for air resistance? This is what a thought: Suppose I had an object of mass $m$ that was being dropped and let $F_{drag}$ be the force due to air resistance. Then $mg - F_{drag} = ma$. Then with some algebra I get $a = g - \frac{F_{drag}}{m}$. If I do not account for air resistance, I get $a = g$. But if I do account for air resistance $a < g$. So I get that the value of $g$ is greater if I do not account for air resistance.


title: Finding Speed with the Energy Principle - Nuclear Fission

text: The question goes along the lines: > Uranium-$236$ fissions when it absorbs a slow-moving neutron. The two > fission fragments can be almost any two nuclei whose charges $Q_1$ and $Q_2$ > add up to $92e$ ($e$ is the charge of a proton, $1.6 \times 10^{-19}C$), and > whose nucleons add up to $236$ protons and neutrons. One of the possible > fission modes involves nearly equal fragments of the two palladium nuclei > with $Q_1 = Q_2 = 46e$. The rest masses of the two palladium nuclei add up > to less than the rest of the mass of the original nucleus. Make the > assumption that there are no free neutrons, jus the palladium nuclei. The > rest mass of the U-$236$ is 235.996 u (unified atomic mass units), and the > rest mass of each $Pd-118$ nucleus is 117.894 u, where $1$ $u = 1.7 \times > 10^{-27} kg$. > > (a) Calculate the final speed $v$, when the Pd nuclei have moved far apart > (due to the mutual electric repulsion). Is this speed small enough that > $\frac{p^2}{2m}$ is an adequate ($p$ is momentum) approximation for the > kinetic energy of one of the palladium nuclei? (make the non relativistic > assumption first, then compare $v$ is indeed small enough to $c$) > > (b) Using energy considerations, calculate the distance between centers of > the palladium nuclei just after fission, when they are starting from rest. So the beginning step is to chose the system, for which I include both the Pd nuclei. So the energy principle is now: $$\Delta E_{sys} = W_{ext}$$ We know that $W_{ext} = 0$ since my system is defined not including any external work done on the system. So now instead the system is $\Delta E_{sys} = 0 = \Delta K_{m_1} + \Delta K_{m_2} + \Delta U_{elec_1} + \Delta U_{elec_2}$ We can ignore the earth and the gravitational potential energy sine the masses are very small. If we expand the equation, we get: $$K_{f_1} - K_{i_1} + K_{f_2} + K_{i_2} + U_{f_1} - U_{i_1} + U_{f_2} - U_{i_2} = 0$$ getting rid of terms that are $0$, we get: $$K_{f_1} + K_{f_2} - U_{i_1} - U_{i_2} = 0$$ furthermore: $$K_{f_1} + K_{f_2} = U_{i_1} + U_{i_2}$$ We know that $K = \frac{1}{2}mv^2$ and that $K_{f_1} = K_{f_2}$ are equal since the masses are the same. So we can simplify that to $2K_f = mv^2$. Same for $U = \frac{1}{2 \pi \epsilon_0} \frac{Q_1 Q_2}{r}$. So $U_{i_1} = U_{i_2} = 2U_i = \frac{2}{2 \pi \epsilon_0} \frac{Q_1 Q_2}{r}$. So we can essentially write: $$mv^2 = 2 \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{r}$$ $$v = \sqrt{\frac{2}{m} \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{r}}$$ I then get lost n finding $r$ since it is not given, furthermore, (b) asks us to find the distance before they move apart. So I am approaching the problem incorrectly. I would need to figure out the error here and how to commence part b.


title: What does the Coulomb barrier really mean?

text: I realize I'm trying to get a simple answer on a complicated subject, but here goes anyway. I've done some research and I understand (well, mostly) about how Vc is the amount of energy required to get two nucleons past the electric repulsive force so the strong force can take over. I know about the Maxwell- Boltzmann distribution, quantum tunneling, and the Gamow peak. But all that is about how the Coulomb barrier is overcome by ambient temperatures in stars. The texts always go from how Vc is calculated, to how hot a star would have to be to overcome it, and how they aren't actually that hot, and why quantum tunneling comes to the rescue, yada yada, yada. I want a more concrete understanding of what Vc really represents. If I had two protons and was using my patented Nucleon Energizer 3000, does Vc represent the amount of energy I'd have to set the MeV dial to so that I could force the two protons together past the Coulomb barrier and get them to fuse? And how much would my electric bill be afterwards?


title: Does the curvature of spacetime theory assume gravity?

text: Whenever I read about the curvature of spacetime as an explanation for gravity, I see pictures of a sheet (spacetime) with various masses indenting the sheet to form "gravity wells." Objects which are gravitationally attracted are said to roll down the curved sheet of spacetime into the gravity well. This is troubling to me, because, in-order for objects on the locally slanted spacetime sheet to accelerate, gravity must be assumed. Therefore I ask; does the explanation of gravity as the curvature of spacetime assume gravity? If yes, what is the point of the theory? If No, what am I missing?


title: Is the space-time deformation only a way to say how gravity works?

text: Here we have the classical picture of the deformation of the space-time: https://blogs.stsci.edu/livio/files/2012/06/spacetime.jpg And I would to know if this representation is only a way to say how the gravity field of the earth is shaped, and how the gravity works in the area near it, or not.


title: Does Spacetime have a "This Side Up" arrow?

text: > **Possible Duplicate:**   >  Does the curvature of spacetime theory assume gravity? Forgive my naivete as I am not schooled in Physics or Mathematics. I was watching NOVA's "The Fabric of the Cosmos" last night. The subject was basically spacetime and how our conception of it moved from the Newtonian description to the Einsteinian. i.e. That spacetime is like a fabric stretched across the universe and that large objects like planets and stars create a depression in the spacetime fabric and the orbiting of satellites is really just the satellite getting caught in the "swirl" of the depression. The narrator used a pool table surface as an example of the fabric of spacetime and then with some CGI he dropped a bowling ball on it which caused a depression and then shot a pool ball near it which caused the pool ball to get caught in the swirl. I found myself pondering the following though:   1. I find it really hard to believe that there is a "This Side Up" arrow for spacetime. But if there is, what causes it? You can see a diagram of what I'm trying to describe in this Gravity Probe B article. I understand that spacetime can be twisted so I'm guessing reality is more like the depressions are at all orientations through the universe (sorry, I lack the language to describe this). But I still wonder: Why is the depression one orientation and not the other?   2. What is the effect of the other side of the depression? i.e. If the swirl on the other side pulls objects in, does the other side of the depression repulse objects? Warning: any answers that include mathematics will be lost on me. Any suggestions for further layperson reading on this fascinating subject welcome. Thanks!


title: Misused physics analogies

text: Have you noticed that many questions and misconceptions in physics arise due to misuse of analogies, which were invented to "explain in simple words" some physical phenomena? For example there is a "stretching rubber band" analogy for Hubble expansion of the Universe, which make people ask about the "tension" of this rubber band or speculating about "cutting a hole" in it. I'm pretty sure that there are a lot of such misuses. Do you have any remarkable examples of these (together with a proper treatment n of course)?


title: How does relativity explain gravity, without assuming gravity

text: I have seen the "objects pull down on space-time" explanations, but they assume a "pull down" force themselves. Could anyone explain the space-time explanation without assuming gravity in the first place? ![](http://i.stack.imgur.com/RVFrW.png)


title: Is the concept of space-time curvature a recursive one?

text: A way some people explain (or try to explain) how gravity works is using space-time curvature: an object with high mass distorts the surrounding space- time plane like a bowling ball distorts a sheet of plastic and the surrounding objects are always trying to roll down to that other object. However, this explanation to me seems flawed, because in order to work, it needs to be influenced by the very thing it tries to explain. The reason a bowling ball distorts a sheet is because Earth's gravity is affecting it. However, the space-time curvature which this distortion is an analogy to is trying to explain gravity. The implication of this space-time curvature is that something is pulling down on the space-time plane, but this pulling itself requires gravity from a perpendicular source, it seems to me. I'm having difficulty wrapping my head around this. The method we use to explain gravity itself uses gravity to work, sort of like some kind of recursive loop. it's comparable to "to get fuel, we need to drive, and to drive, we need fuel", and I don't know enough about gravity to figure this out myself. Am I understanding this analogy correctly, or are there deeper things at play?


title: Bowling ball on a rubber sheet analogy - what pulls the ball down

text: > **Possible Duplicate:**   >  Does the curvature of spacetime theory assume gravity? Since I read Cosmos long ago, I see the same analogy about the balls rolling on a rubber sheet used to explain how gravity works. But a ball rolls on a surface because gravity is pulling it down. In space it will follow a straight line and go over any hole on the surface. So, in the analogy we all know, where the curvature of the rubber sheet is gravity, what is pulling the ball down? If space is curved, what keeps an object attached to that "surface" of space?


title: If the universe is 3D, how is space-time like a "fabric"?

text: I have been taught that space-time should be viewed as a fabric and that objects with a large gravitational influence indent that fabric. My question is, if the singularity of a black-hole punctures space-time, how is this accomplished if the universe is 3D? Can an object move completely around the black-hole in all directions? Would you be able to travel "below" a black- hole?


title: Why is getting a tan in the evening less likely than getting a tan in the morning?

text: When I was young I read one book in which is written that you get more tan on the morning than on evening even light angle from Sun to Earth is the same. Don't remember exact reason, I think because ultraviolet is more absorbed on the evening because air is more humid or something like this. Is it true?


title: Can PEPS explain the holographic principle in quantum gravity?

text: Condensed matter physicists have shown using quantum information that in many condensed matter systems, entanglement entropy only scales as the area of the boundary, and not the volume. This is the basis for the density matrix renormalization group and Projected Entangled Pair States (PEPS). Does this also explain the holographic principle in quantum gravity?


title: is the sun itself moving

text: I just thought of an interesting question.Is the sun actually moving? I have learnt that the way the lunar landers and the space ships of the Mercury and Apallo missions moved and controlled their positions through the use of thrusters.NoW apply that same concept to the sun are the constant explosions on the surface of the sun enough to move it?


title: Explanation for speed of an electrical impulse

text: Our calculus book, Stewart, has a problem where they claim that for a metal cable (inner radius $r$) encased in insulation (outer radius $R$), the speed of an electrical impulse is given by $$v = - k \left(\frac{r}{R}\right)^2 \ln \left(\frac{r}{R}\right)$$ where $k$ is a positive constant. **My question** What I would like to know is the physical justification for their claim. **My thoughts** There claim is somewhat surprising, since for sufficiently high insulation R, with r fixed, the speed of the impulse _decreases_ (by L'Hospital) with _more_ insulation. * * * _EDIT_ : I received this email after contacting Brooks/Cole, the publisher of the textbook. The response didn't really help unfortunately. > Hi Professor ..., > > I just heard back from the author regarding your query: “I can understand > why Professor ... thinks this equation is counterintuitive, but it is in > fact correct. I have been >trying to track down the source that I used in > devising this problem, but unfortunately I >can’t seem to find it right > now.” I will certainly let you know if he is able to track >down the source > information. I’m sorry I can’t give you a more concrete answer at this > >time. Best, ... > > [JIRA] (KYTS-1199) Content Feedback from Instructor for ISBN: 0495014281 > Essential >Calculus: Early Transcendentals 1st edition.


title: Calculate force of electric charges "suspended" by strings

text: In a question: > ![enter image description here](http://i.stack.imgur.com/C9LIo.png) > > Two small plastic balls hang from threads of negligible mass. Each ball has > a mass of 0.110g and a charge of magnitude q. The balls are attracted to > each other, and the threads attached to the balls make an angle of 20.0 deg > with the vertical, as shown in the figure How do I compute the magnitude of electric force acting on each ball? I tried using coulomb's law: $F = k \frac{Q_1\cdot Q_2}{r^2} = 2.248 \times 10^{13} q^2$ but seems like the answer is wrong. The hint was that the answer did not depend on $q$ I guess I have to use other info? But how?


title: What is the formula for the relativistic ellipse?

text: If an astronomer moves at relativistic speed, the stars and constellations are distorted. He sees the stars towards which he is moving blue shifted, while the ones he's moving away from are red shifted. In addition, the apparent direction of distant stars is modified. I think that the coolest way of representing this is the "relativistic ellipse". Think of a spherical shell with the inner surface reflective. Imagine a flashbulb going off at the center. The light is reflected and returns to the source. Now think of the same situation observed from a moving frame of reference. The light is still emitted and returns to the center at the same time, but that point has now moved. Since the speed of light is the same in all frames of reference, the distance traveled by each light ray has to be the same. So the paths of the light rays (and the boundary of the spherical shell) describes an ellipse, the "relativistic ellipse": ![RelativisticEllipse](http://i.stack.imgur.com/9PoOJ.png) The above drawing is from the book "Relativity in Curved Spacetime" which I highly recommend and can be purchased here or here. Perhaps when my copy arrives it will have the equation. Should be possible to work it out using Lorentz invariance or contraction or something similar. * * * Per the answer by Helder Velez, Chapter 4 of Hans de Vries's book has the following useful diagram:   ![HansEllipse](http://i.stack.imgur.com/gC4Gu.jpg)


title: What is the exponential form gamma matrix for a general rotation and boost?

text: It would be nice to have a cute method that uses Lorentz transformations of basis vectors by exponential transformation using gamma matrices. To avoid confusion, let's assume -+++ signature. Given $\gamma_\mu$ as gamma matrices that satisfy ${\rm tr}(\gamma_\mu\gamma_\nu) = 4\eta_{\mu\nu}$, then we have ${\rm tr}(\gamma'_\mu\gamma'_\nu) = 4\eta_{\mu\nu}$ if we put:   $\gamma'_\mu = \exp(-A)\gamma_\mu\exp(+A)$   where $A$ is any matrix. Boosts and rotations use $A$ as a bivector. For example, with $\alpha$ a real number, $A=\alpha\gamma_0\gamma_3$ boosts in the z direction while $A=\alpha\gamma_1\gamma_2$ gives a rotation around the z axis. Solving for the value of $\alpha$ that gives a boost with velocity $\beta = v/c$ appears to be straightforward. But how to do the rotations? And by the way, what happens when you generalize $A$ to be something other than bivectors?


title: What is the probability of quantum tunneling occurring in this CPU?

text: You may have noticed over the last few years that Moore's law is no longer applying to the real world. This observation states that over the history of computing hardware, the number of transistors on integrated circuits doubles approximately every two years. However, as microprocessors in computers have continued to become smaller, the architecture size has become incredibly small:- the new Skylake architecture to be released by Intel will use 14nm semiconductors. We are beginning to reach the limit in size where the information being passed is still completely reliable. Would it be reasonable to ask how much quantum tunneling would occur in a semiconductor 5nm in length? We can assume that in this CPU Voltage = 1.2V Amperage = 63A I enjoy physics, but I'm not that great at it; I'd love to hear the results! Even if massive approximations!


title: What is the extent of Earth's gravitational pull?

text: 1. Where the gravitational pull of Earth exist up to?    2. What distance from Earth it will be zero?   3. How do the skydivers fly at a same altitude?    4. Won't they feel gravitational pull?    5. What is the Earth's gravitational pull?


title: Dimensonal analysis of damping constant?

text: What are the units of the damping constant from the following equation by dimensional analysis? $$\zeta = \frac{c}{2\sqrt{mk}}$$ I'm assuming the units have to be s^-1, as the damping constant is present in the exponential equation which plots damping of y=Ae^kt (which plots amplitude vs time). Is that a correct assumption? If somebody could do a quick dimensional analysis to confirm it would be great.


title: Do we feel the full force of the earths gravity even though the earth is spinning

text: As far as I know, since the earth is spinning, centrifugal force should be throwing you off the earth. However, gravity counteracts that by being a stronger force. So if you stood an object with the exact mass and size of earth, but it wasn't spinning, would you perceive gravity to be stronger? If this is true, wouldn't an object outside of the atmosphere not in orbit of the earth have greater acceleration than an object in orbit?


title: Why doesn"t centrifugal force throw us off the earth?

text: I was drawing a diagram and trying to resolve all the forces. It seems to me that because of gravity, centrifugal force will produce a sort of "dragging" motion along the surface of the earth. The centrifugal force is quite small (I got around 2.5 N at the equator). This force is easily neutralized by friction. Is this right? EDIT: I did not consider any complexities that may arise due to revolution, I just considered rotation.


title: Net acceleration felt by a body on Earth

text: For any body on earth will the only acceleration it has be gravity or should we take into consideration even centrifugal force due to rotation? Is there any precise formula to determine the net acceleration?