title: How explain this perturbing equation about the 43 arcseconds?

text: The planetary orbits have been studied as ellipses but the solar system is in motion in relation to the distant stars. Their path is along the tip of an **helix** and the ecliptic plane is a convenient plane of projection. I think that the studies were never conducted under this viewpoint. The sunlight we see now was emitted more than 8 minutes ago when the Sun was ‘below’ the ecliptic but we see it centered in the plane. I wonder why we do not see any consequence of this. I’m following a line of reasoning that the motion can have consequences and I revisited the anomalous precession of the perihelion of Mercury, settled long time ago by Einstein, and I found this perturbing equation: $$\frac{43}{5557} = 2\pi\frac{369.2\ \text{km}\ \text{s}^{−1}}{299792.458\ \text{km}\ \text{s}^{−1}}$$ Where 5557 is the predicted theoretical value (in mathpages) for the advance and the 43 is the anomaly. The simplicity of the formula $$\text{error}/\text{theoric}=2\pi\ V/c$$ and because $V$ is  **0.054%** off the central measured value of the speed of solar system \- $369(\pm0.9$) - makes me wonder if this can be more than a coincidence. Any kind of reasoning on the why's will be helpful.


title: Identify the coefficients of Operator Product Expansion (OPE)

text: Sorry I have a stupid question in Polchinski's string theory book vol 1, p46. For a holomorphic function $T(z)$ with a general operator $\mathcal{A}$, there is a Laurent expansion $$T(z) A(0,0) \sim \sum_{n=0}^{\infty} \frac{1}{z^{n+1}} \mathcal{A}^{(n)}(0,0). \tag{2.4.11}$$ Under transformation $\delta \mathcal{A}=-\epsilon v^a \partial_a \mathcal{A}$, why the OPE is determined as $$T(z) A(0,0) \sim \cdots + \frac{h}{z^2}\mathcal{A}(0,0) + \frac{1}{z} \partial \mathcal{A}(0,0)+\cdots? \tag{2.4.14}$$ How to derive this equation?


title: Surface terms for field path integrals?

text: My question relates to something that I´ve seen in many books and appears in all its glory here: Ryder, pg 198 My question is about eq. 6.74. Which I repeat below: $$i \int {\cal D}\phi \frac{\delta \hat{Z} [\phi] }{\delta \phi} exp \left(i\int J(x) \phi(x) dx\right) = i\; exp \left(i\int J(x) \phi(x) dx\right) \hat{Z}[\phi] \Bigg|_{\phi\rightarrow\infty}+ \int {\cal D}\phi J(x) \hat{Z}[\phi] exp \left(i\int J(x) \phi(x) dx\right)$$ $\phi$ is a scalar field, J is a source, $x = x_{\mu}$ in 4D Minkowsky space and $\hat{Z}[\phi] = \frac{e^{iS}}{\int {\cal D}\phi\; e^{iS}}$ The author is clearly doing a integral by parts and the first term on the right hand side is a kind of surface term for the path integral. He then considers this term to be zero and the second one gives us: $$i \int {\cal D}\phi \frac{\delta \hat{Z} [\phi] }{\delta \phi} exp \left(i\int J(x) \phi(x) dx\right) = J(x) Z[J]$$ The trick thing here is that integral limits for $\int{\cal D}\phi$ are not very obvious (at least not to me). You are in fact summing up for all field configurations. So, there are actually two problems in my mind:   1. For what configuration of $\phi$ is the surface term calculated? (the author says it is $\phi \rightarrow \infty$)   2. Assuming the author is right about taking huge $\phi$: why is this term zero? This applies to path integrals in general: can we do the usual trick of throwing out surface terms safely?


title: Metric Perturbations in General Relativity and quasi-normal modes?

text: I am familiar with the tools that appear in (linear) perturbation theory for general relativity, that is namely that one writes: $$g_{\mu \nu} = g^{(0)}_{\mu \nu} + \epsilon g^{(1)}_{\mu \nu} + \mathcal{O}(\epsilon^2) \tag{*}$$ Where $g^{(0)}_{\mu \nu}$ is typically assumed, and then one assumes a perturbation of some kind and solves some equations. Let us now perturb the background space-time manifold, $(M,\mathbf{g}^{(0)})$, by a (massless) scalar field, we have that the equations of motion for the scalar field $\Phi$ are given by the Klein-Gordon equation: $$(\square + \xi R) \Phi := (\nabla_{\mu} \nabla^{\mu} + \xi R) \Phi = 0 \tag{**}$$ Where the covariant derivative is taken with respect to the background metric tensor $g^{(0)}_{\mu \nu}$. Suppose we can solve these. How does one translate the solution for $\Phi$ into terms of the metric components $g^{(1)}_{\mu \nu}$? In the sense that we have considered a physical perturbation on the space-time and now the metric tensor field must be modified via $(*)$. What if we generalise a bit and consider a spin $\sigma$ perturbation; electromagnetic, Dirac or gravitational (via the Teukolsky equation)? Following the excellent review article of Kokkotas and Schmidt (http://arxiv.org/abs/gr-qc/9909058) he says that (on page 10): "The variation of the Einstein equations: $$\delta G_{\mu \nu} = 4 \pi \delta T_{\mu \nu}$$ by assuming a decomposition into tensor spherical harmonics for each (in my notation) $g^{(1)}_{\mu \nu}$ of the form $\chi(t,r,\theta,\phi) = \sum_{\ell m} \chi_{\ell m}(r,t) Y_{\ell}^{m}(\theta,\phi)$ the perturbation is reduced to a single equation." I believe this immediatly in light of separation of variables and equation $(**)$ above. But they do not describe how the metric components in $g^{(1)}_{\mu \nu}$ are recovered. Is there a general procedure by which I can match a spin $\sigma$ perturbation solution (given that I can solve the EoM for the spin $\sigma$ field) with the metric? **Question:** **How does one match $\Phi$ in $(**)$ with $g^{(1)}_{\mu \nu}$? How do the quasi-normal modes in terms of $\omega$ explicitly couple to the metric?** Edit: I have re-written the question slightly in the hope that it is now more cogent. Thanks!


title: Sound difference between musical instruments

text: I know that the difference between two musical notes is given by the sound frequency, and the difference in volume is given by the amplitude. What I am wondering is why does the same note sound different on different musical instruments? What in the wave makes the difference between the sound of a harmonica and the sound of a violin singing the same note?


title: Is it possible to generate electric energy from the radiation of radioactive materials

text: I wonder if it is possible to generate electric energy from the radiation of radioactive materials like nuclear waste? If it is then wouldn't that also mean that it could be used as an energy source for a very long time since the radiation takes a long time to decay?


title: If components of differing resistance are wired in parallel, how does a common voltage drop arise?

text: I've been teaching myself DC electronics as a hobby and, although I have a feeling i'm "missing something obvious", I was wondering if someone could help me out. If two components of differing resistance are wired in series, they likewise drop different voltages; yet when they are wired in parallel they apparently share a single magnitude of voltage drop. I would like to better understand why a common drop is exhibited across components of differing internal resistance wired in parallel?


title: Are these sunglass lenses linearly polarized or what?

text: This is a difficult question to phrase, so please bear with me. I found some cheap sunglasses and pulled out the plastic lenses which are polarized. For clarity, I have labeled them as lens-1 and -2 with faces A and B. ![enter image description here](http://i.stack.imgur.com/iKSi9.jpg) There are three cases I need help with in explaining what is going on. I wrote a letter “A” on a sheet of white paper that I am looking at through these lenses. **Case (1): Sides 1A and 2A (or sides 1B and 2B)** ![enter image description here](http://i.stack.imgur.com/7C5SM.jpg) If put side 1A in front and 2A in back (or vice versa), there is a color change depending on the rotation. With $0^0$ or $180^0$ rotation, there is a brownish color while a $90^0$ or $270^0$ rotations produces a greyish color. If on the other hand, one switches the A-sides to B’s, you get the exact same colors as above (as expected). Here is where I was really surprised. If you have combinations of 1A and 2B, I found to my surprise that it matters which lens goes in front. **Case (2): Side 1A in front with 2B behind (or sides 1B and 2A)** ![enter image description here](http://i.stack.imgur.com/O9g1e.jpg)   With lens-1A in front, you can barely see the letter A with rotations of $0^0$ or $180^0$ whereas with rotations of $90^0$ or $270^0$ it appears to have perpendicular polarizations. **Case (3): Side 1A behind while 2B is in front (or sides 1B and 2A)** ![enter image description here](http://i.stack.imgur.com/4sSqu.jpg) Reversing the order of 1A and 2B allows me to clearly see the letter A with rotations of $0^0$ or $180^0$. On the other hand, rotations of $90^0$ or $270^0$ appear to have perpendicular polarizations. So apparently, the order in which lenses 1A and 2B appear matter. Here are my questions:   1. There are 4 different color changes when these lenses are overlapped – why?   2. Why does the order of the lenses (either in front or behind) in case (2) and (3) matter? I would have expected the same outcome, however, that is not correct.   3. What role does polarization play in these lenses? I think that case (3) is a situation of linear polarizers but the order dependence has thrown me through a loop. Thank you in advance for any help.


title: If two observers pass each other in opposite directions at $.5c$ what would effect on each others clocks be?

text: Assume an alternate universe with same physical laws as here. In this universe nothing exists except 3 observers, each in a transparent box with a clock. Observer A is travelling at .5c. Observer C is travelling in the opposite direction at .5c. Observer B is at rest. In the absence of outside landmarks each would feel they were at rest and their clocks were functioning normally. Suppose they pass each other. To A it would appear B had zoomed by at .5c and C at c. To B it would seem A and C had zoomed by at .5c in opposite directions. Suppose they noted the functioning of each others' clocks as compared to their own in passing, and they met later to compare notes on their observations. What observations would they have concerning clock function? If C shone a light beam ahead of himself, how would it appear to A?


title: Quantum Mechanics - Finding momentum probability density

text: everyone. I got a bit stuck on 2(iii), this is supposed to be a easy question, but i don't know how you get the square term? I thought you just do the Fourier transform, but then I got some exponential out of it and I don't know what to do? Can anyone suggest or shed some light on the problem? THANKS If you just plug in infinity into the integral, how can you avoid the problem of getting zero? ![enter image description here](http://i.stack.imgur.com/8A8e4.png) ![enter image description here](http://i.stack.imgur.com/6jd2U.png) You may use the integral provided.


title: Why is it said that without quantum mechanics we would not have modern computers?

text: I've heard this in many quantum mechanics talks and lectures, nevertheless I don't seem to grasp the idea behind it. What I mean is, at which point is that our modern understanding of quantum mechanics led to a technological development so fundamental for today's computers that we could not have got it working other way? Why is it not enough with Maxwell, Bohr, Lorentz, (Liénard)?


title: Would ferromagnetic space debris have been magnetized by their orbit in the Earth's magnetosphere?

text: Given a nail, and a magnet it is possible to magnetize the nail by patiently rubbing the nail until it is magnetized; albeit the field strength may be less than that of the source. With several score kilograms of space debris in orbit, all ferromagnetic material up there may each well have been subjected to several thousand revolutions around their primary; some sooner than the other. Would such ferromagnetic debris be magnetized by their orbit within the magnetosphere? p.s. If yes, I would assume the material that magnetize quicker might attract those that are yet to be magnetized to form a larger mass, and so on ad infinitum. What prevents such an occurence? Is it precluded merely because of the volume of open space and the possible repulsion if the similar poles face each other?


title: Randomly sampling a "well-mixed" solution of Brownian particles

text: I place $N$ Brownian particles in $V$ liters of solution, shake until I assume that the particles are "well-mixed", and sample and randomly sample an $S$ liter volume. What is the probability distribution for the number of particles in my sample of $S$ liters?


title: band gaps in tight binding model

text: What happens at the zone boundaries of the brillouin zones in the tight binding model? How does the band gap originate in the TB model?


title: Simple explanation on how to roll up a graphene sheet into a nanotube?

text: I have read some articles such as http://sinnott.mse.ufl.edu/Backgrounds/theo01_CNT.html and https://www.rose- hulman.edu/math/seminar/seminarfiles/2006-07/abstract2006-11-01.pdf which talk about how to roll up a graphene sheet into a nanotube. Now I am confused about the chiral angle and $n,m$ lattice vectors they talk about and the complex math equations make it a little blurry to me. Can someone provide a simple an layman explanation of how this works and how it relates to things like this http://demonstrations.wolfram.com/RollingUpASheetOfGraphene/


title: Melting and Boiling Points of Odd Materials

text: In Chemistry, I was taught that there are three main states of matter: solid, liquid, and gas, and that heat and pressure determine that state. For some substances, the line is blurry between them. Some materials don't seem intuitively to do this--nor have I been able to find data on them. For example, what is a reasonable estimate of a melting point for brick? What is the boiling point of paper? When will a carpet sublimate? The common theme seems to be that these are all composite materials. Certainly all the elements have melting points (as applicable) and boiling points. Many compounds do too. However, something like cardboard is a mixture of fiber, glue, pigment, and possibly other things. Each of these might be made up of several compounds, with each compound having its own boiling point. My suspicion is that for composite materials, individual compounds would exhibit properties roughly individually--so to melt wood, the water would boil off first, and then maybe it would start melting into a glucose-protein slag. Is this truly the right idea for what happens?


title: Photon Mass Term in Schwinger's 2D QED Model

text: Why does the vacuum polarization in 2D massless Fermion QED, $$ i\Pi^{\mu\nu}(q) = i(\eta^{\mu\nu}-\frac{q^\mu q^\nu}{q^2})\frac{e^2}{\pi}, $$ have the structure of a photon mass term, as is claimed on Peskin chapter 19 page 653?


title: What is an observer in quantum mechanics?

text: My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts. **What is an observer?** What are the conditions required to be qualified of **observer** , both mathematically and experimentally?


title: What is an "observer" really?

text: > **Possible Duplicate:**   >  What is an observer in quantum mechanics? I'm sick of quantum physics explanations which term experiments where the outcome depends on "if you observe it or not". For example, the two-slit photon experiment. I figure that something physical happens, irrespective of if a "being" of some kind is there to consciously watch it. Is it really to do with absorption and emission? Like the only way to know for sure where a photon is, you need to have it absorbed and re-emitted by an atom/electrons so that you can detect it (on one level, seeing with your eyes is about absorbing photons on the atoms in your eye and having that create a potential on your nerves which can travel through your brain). So in a sense, if a tree fell in the forest and nobody heard, it still would have fallen because the atoms in the tree would have absorbed thermal energy, and the atoms of the tree would have caused a vibration in the tree and the ground, which vibrated the air, and so this energy would have dissipated into the environment, and that counts as observation without a human ever getting involved? Do I understand things or am I very wrong?


title: Quantum observation in the double slit experiment?

text: As far as quantum observation, when a human observes electrons going through the double whole experiment, have scientist tried having a blind person observe the experiment to see if the electrons actually can tell if a person is able to view it or not? This will tell us a lot as far as how intelligent the electrons are and if a persons eyes gives off a certain radio signal to warn the electrons of its observation.. This may seem crazy but i had to ask.


title: What does it mean 'the observer' in Quantum Physics?

text: Is it only necessary a human consciousness? A measurement device? Can it be said that any of them cause the wave function collapse?


title: Some very basic questions on the Higgs Boson

text: What exactly is a boson? Is the Higgs boson the cause of gravity or a result of it? Does the collision of particles at the LHC create a gravity field or waves or somehow interact with the gravity field of the earth? The Higgs Boson is supposed to be quite massive and equivalent to a large number of protons. Were many particles needed to create it or only a few travelling at high speeds? Was the high energy converted into the large mass? Why is the particle so short lived and what does it decay into?


title: What colour is nothing?

text: To me this is very confusing, but I hope we can discuss it and find a solid answer to the question. If you were somewhere where there was absolutely nothing, what colour would your eyes see?


title: diffusion by an external potential in quantum field theory

text: I'm studying quantum field theory and I encountered some problems of diffusion of particles by an external potential. Until now I have to do with diffusion of the type particle-particle obtaining the Feynman rules from the Lagrangian of the theory and I don't know how to implement the presence of an external potential. How can I implement it?


title: Variation of delayed choice quantum eraser

text: Let me base the discussion on the pictorial description of the delayed choice quantum eraser experiment on wikipedia. ![quantum eraser diagram](http://upload.wikimedia.org/wikipedia/commons/thumb/2/21/Kim_EtAl_Quantum_Eraser.svg/1000px- Kim_EtAl_Quantum_Eraser.svg.png) First suppose that we do precisely the same thing with the lower parts of the blue and red beams as we do with the upper parts, i.e. we put them through a lens which ends at a new detector $E_0$. > **Question**. Would there be the interference pattern at $D_0$ (or - I > suppose equivalently - at $E_0$)? If there woudn't be - why? I thought there would be, but then it occured to me we could use it to communicate faster than light - by shooting the lower parts of the red and blue beams to a distant partner. Then the partner would send the mesage either by preparing the setup above (and so causing the patterns at $D_0$ to interfere), or by preparing the setup of the original delayed quantum choice eraser (and so causing the interference pattern at $D_0$ to vanish.) Actually as I've described it, it would be even a message from the future. > **Question** If there would be interference pattern, what's wrong in the > above faster-than-light communication protocol?


title: Fermion field structure in non-abelian gauge theories

text: I am trying to understand the structure of the fermions in non-abelian gauge theories. Disclaimer: my question might be very trivial (I suspect the answer could simply be "a change of basis"), but I would be grateful is someone could shed some light if there's something deeper lurking around the corner. Let's consider the Yang-Mills Lagrangian $$ \mathcal{L} = -\frac{1}{4} (F_{\mu \nu}^a)^2 + \bar{\psi}(i \gamma^\mu D_\mu -m)\psi $$ where $ D_\mu = \mathbf{1} \partial_\mu - i g A_\mu^a t^a$, then $\psi$ needs to have both Dirac ($\mu$) and colour (a) degrees of freedom. I get confused when I change the regular approach to the problem and I am not sure if it is a real issue or if I'm just overcomplicating/overthinking this. For instance, let's consider $SU(N)$ YM where the generators $t^a$ are $N \times N$ matrices, and there are $N^2-1$ of them. Therefore, the covariant derivative $D_\mu$ is an $N \times N$ matrix, and the index $a$ above runs over $N^2-1$ value. When we start by contracting $D_\mu$ with the $\gamma$-matrices, $$ (\gamma^\mu D_\mu)_{ij} = \delta_{ij} \gamma^\mu \partial_\mu - i g (\gamma^\mu A_\mu^a) (t^a)_{ij} $$ one gets an $N \times N$ matrix of $4 \times 4$ matrices. The corresponding $4N$-component object this matrix acts upon is the $N$ Dirac spinors arranged in a column. However, notice that if we start in the following way instead: $$ (D_\mu \psi)_i = \partial_\mu \psi_i - i g A_\mu^a (t^a)_{ij} \psi_j $$ we get that the covariant derivative acts on $\Psi \equiv (\psi_1, \cdots, \psi_N)$. For all we know, $\Psi$ has no spinor structure since we haven't contracted with the gamma matrices yet. Contracting with the $\gamma^\mu$, one gets a $4 \times 4$ matrix of $N \times N$ matrices that encodes the same information as before. This time, it would seem that we only have one Dirac spinor, where each component is a $N$-valued singlet. However, it seems that the matrix $\gamma^\mu D_\mu$ and the $4N$-component spinor look different, even though all we did was to change the order in which we constructed things. In the first case, we get $N$ spinors corresponding to the $N$ colors of the adjoint representation. In $SU(3)$, this would be like saying we effectively have 3 spinors which correspond to the red/blue/green colors as $(\psi_R, \psi_B, \psi_G)$. In the second case, this identification fails since we only have one big complicated object. What went wrong? Is this difference simply a change of basis for $\psi$? Is there something relevant we can learn from looking at the YM Lagrangian in these two different ways? Also, in the first case, when we get $N$ spinors, I am confused about their signification. I always assumed that in QCD, $\psi$ would correspond to a quark, which is a fermion by itself. Does it mean that quarks are fermions that can be described by fermionic fields/degrees of freedom that we call color?


title: Gauss-Bonnet term in Physics

text: Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$ with $R^{ab}$ is the curvature 2-form. Perturb the connection 1-form (represent by $\delta \omega^{ab}$), $E_4$ should be unchanged. How can I proof that, and what will be change if the manifold is not torsion free anymore? Given that: Connection 1-form from torsion-free condition: $$T^a ~=~ De^a ~=~ de^a + \omega^a_b \wedge e^b ~=~ 0$$ and $$\omega^{ab} ~=~ \delta^{ac} \omega^b_c.$$ Curvature 2-form: $$R^{ab} ~=~ D\omega^{ab} ~=~ d\omega^{ab} + \omega^a_c \wedge \omega^{cb}.$$ Gauss-Bonnet term appears as a topological term in some theoretical gravitational actions. I don't see why it's unchanged, so I post this question of mine here.


title: Thermal exfoliation of graphite oxide

text: I have been reading up on the production methods of graphene, and one that I found interesting in particular was the thermal exfoliation of graphite oxide. From what I gather the basic idea is that the oxides are located in between the graphene layers of graphite, and that you then heat the material up, causing gasses such as CO and CO2 to form in between the layers. This sort of lifts the layers from each other, and then they become separate. However, what I don't get is why the layers don't simply re-aggregate on top of each other. The reason I wonder about this is because thermal exfoliation is done (at least it is possible to do so) without putting the graphite oxide in a liquid or anything; it can just be done 'dry'. Now liquid exfoliation, on the other hand, is based on using ultrasound to separate graphite layers, which are submerged in a fluid with surface energies such that it is unfavorable for the graphite to re-aggregate. So is it simply that this re-aggregation is not troublesome when your graphite is not in a liquid? I suppose if this is the case, then the only reason for liquid exfoliation is if it is more efficient/effective/cheap than doing the thermal version.


title: Peskin & Schroeder Chapter 3.1 EoM Lorentz Invariant under Lorentz Invariant Lagrangian

text: From Peskin & Schroeder _QFT_ page 35: > _The Lagrangian formulation of field theory makes it especially easy to > discuss Lorentz invariance. And equation of motion is automatically Lorentz > invariant by the above definition if it follows from a Lagrangian that is a > Lorentz scalar. This is an immediate consequence of the principle of least > action: If boosts leave the Lagrangian unchanged, the boost of an extremum > in the action will be another extremum._   1. Could anyone please help me translate the statement of this paragraph into a rigorous mathematical proof with symbols (and, in addition, to generalize it to proper orthochronous Lorentz transformations and not just boosts)?    2. Maybe as warm up: for boosts, how does one show that the boost of an extremum in the action will be another extremum?


title: Attraction of a Bullet due to Gravity in a Perfect Vaccum

text: _I realise that this might be conventially very difficult to answer because there's no KG or Newtons in space, only particles._ As far as I understand, every object creates a 'pull' due to the forces of gravity, that'll be felt noticeably by smaller objects (as well as the larger object being attracted 'less' to it). If we fire a bullet in a straight line on earth it'll curve down towards the earth and hit it because of the force of gravity. If we say had a bullet fired in a straight line in space, and an object 1km away at say 30 degrees from the main path, is there any way we can work out how massive that object would have to be for the bullet to hit it. Say the bullet is fired at 10m/s and is on Earth 20g - it's not really the figures I'd just wonder if it's possible.


title: Flat poster on a wall gaining curvature over time

text: Assuming you have a flat poster with no curvature, why is it that when you pin it to the wall (with thumbtacks) it gains curvature as seen in the picture below. When I put the poster up it was entirely flat to the wall with no curvature, but over time it somehow curls and develops this sort of curvature (it looks like positive curvature at the corner). ![enter image description here](http://i.stack.imgur.com/oPDQ9.jpg) Is there a physics-based explanation for why this happens?


title: Charge and current density fields

text: The charge and current density fields in classical electromagnetism are scalar real number fields on space time manifold. But these fields diverge/become infinite in case of point charges, how is this justified and mathematically consistent ?


title: Space curvature based on net energy = 0

text: In Neil DeGrass Tyson's epic video, at 2:26:50 https://www.youtube.com/watch?v=AdHlVY8pEk0&list=PL1L9zQimONkUOxXJyM0xcJja9ItQu1P8l&src_vid=AdHlVY8pEk0&feature=iv&annotation_id=annotation_4179310099#t=12s He mentions how if net energy is negative, the spacetime curvature is spherical, and if it's net positive, saddle-shaped. He uses the terms "flat" and "saddle-shaped" which are 2d, but he's actually just using that as an analogy for 3d curvature, which we can't really visualize, correct? If so, what does it mean for 3d space to be "saddle- shaped" or "spherical"? 1) II'm guessing spherical implies that if we keep traveling in one direction in space, it would be possible to end up back where you started? (kind of like how you can end up in the same position traversing the surface of a planet if the 2d plane is curved? 2) If net energy is 0. But they also say there is far more dark energy and dark matter than energy and matter. So isn't this a contradiction? How does the accounting work out?


title: How is graphene a 2D substance?

text: How is graphene a 2D substance? It has length, width and some thickness to it, else it would be invisible. Why is it considered a 2D substance?


title: What is the difference between hierarchy picture and composite Fermion in explaining FQHE

text: Are they equivalent? I came up this question because they both explain states beyond Loughlin state.


title: Trampoline physics - what is the right time to crouch and to jump to maximize your height?

text: Please bear in mind that I'm neither a physics guy nor particularly a math guy. So I will probably need a bit more hand-holding than just a complex equation. For a game, I'm trying to calculate a _reasonable_ simulation of a human body on a trampoline. It seems to me that there are two phases to consider once the body has contacted the trampoline: the deceleration of the body once it has contacted the surface, which gets stored as potential energy in the trampoline, and then the combination of the release of that potential energy once the body has reached the nadir plus any extra force that body might apply (assuming he has bent his knees prior to that point). I'm interested in understanding the interaction between the trampoline's storage and release of energy and the body's own contribution (addition) to that energy with its own kinetic energy. CLARIFICATION: In short, I want to know when the best time to crouch is (pull your knees up) and when the best time to release (jump) is. Anyone who's been on a trampoline knows you can get better height by crouching and jumping at the right time - you can also bring yourself to a complete stop by crouching and jumping at different times. I can't seem to find anything documented though about what these times actually are. Since I couldn't find the information from the sports / athletics side, I thought I'd come over here to the physics side and see if anyone knew. I'm surely not being very clear, which only further reveals my own shortcomings in this matter. Any guidance would be appreciated. I understand that this forum typically operates at a much higher level than what I'm looking for, but I thought I'd ask here, since this is where the great minds seem to gather. EDIT I can very easily do a "good enough" simulation of a mass "bouncing"off the trampoline by taking its incoming velocity and simply inverting it and multiplying by an arbitrary restitution amount to simulate the springiness of the material. Nothing more complicated really need from the perspective of a game. The part that's killing me is getting the body to continue moving past the plane of the trampoline while decelerating to 0 and then accelerate again up to final "release velocity". The part that baffles me is the part that happens between the time the body touches the material to the time it leaves contact, taking into account any additional force added by the person through his legs (I've decided not to model any additional lift sure to arm motion etc, to keep both the simulation and the user controls thereof straightforward). SECOND EDIT I've read through the answers here and they're very good! And surprisingly, I seem to be following about 75% of it. The challenge I'm having is that I'm principally interested in Velocity, rather than Energy, even though it's obviously energy that is behind all of this. Every 30th of a second (every "step") I need to know the velocity of the body so that I can plot its new position. It's really easy to plot the position of the body once it has left "contact": starting at some arbitrary position and velocity, v = v + g, y = y + v. See, it's a very different way of representing the same facts when you're dealing with visual graphics than physics equations. So now y = the plane of the trampoline. v = v * c, where 'c' is a factor of how far the springs will compress, less than 1. y = y + v until v approaches 0 at which point we start the "bounce back". Here I suppose v = v * (1/c) * s, where 's' represents whatever factor of 'loss' we want to simulate so that we don't get 100% bounce back. Once y > the plane of the trampoline, we can then forget the more complicated stuff and go back to letting gravity take over. Obviously this whole bit ignores mass, which is unfortunate, but I have to keep it simple for now. The "mass" factor gets rolled into the numbers for 's' and 'c' that I fudge until it 'feels' right. Okay, now that you guys have stopped shuddering in utter horror at this wishy- washy approach, the bit I'm trying to figure out is the part about the human body bending his legs and then performing the jumping motion to add his force to the rebound of the trampoline. It seems to me, having spent a little time on a trampoline, that you can jump at the right time and at the wrong time. So, from a physics perspective, when is the optimal time to crouch? When is the optimal time to jump / release?


title: If it was possible to dig a hole that went from one side of the Earth to the other...

text: ...And you jumped in. What would happen when you got to the middle of the Earth? Would you gradually slow down, until you got to the middle and once you were in middle would every direction feel like it was up?


title: What happens to gravity in the middle of the earth?

text: So I was curious about, what if we make a tunnel from one side of the earth, to the other side of the earth? Gravity is ofcourse always negative, which makes you "fall" and not "float". If we take this image, let's imagine that the BLACK LINE in the midle of the earth, is a tunnel from one side of the earth, to the other side of the earth. The red DOT is the midpoint, where something in theory, should happen to gravity that is out of our forces. What would happen, if a human were to be in the red dot? Would the human just stay there for eternity? **Notes** : Imagine that the being in the center couldn't kill you, neither would it bring you in contact of any substance. Just a long tunnel with nothing but the tunnel itself. ![enter image description here](http://i.stack.imgur.com/FPC7p.png)


title: Earth's graviational pull at the centre

text: Suppose, I somehow happen to dig a hole diametrically through the Earth (neglecting all constraints like molten core etc ) ... and I throw an object from one side of the hole, will it reach the other end?


title: Does gravity act in the centre of the Earth?

text: If we we dig a hole from north pole to the south pole of the Earth and we throw a ball in this hole where would it stop? Will it come back on the surface or it will stop at the centre?


title: Expected behavior of the gravity under some experiment

text: I would like to know the **expected behavior** of the **Gravity** under the following mentioned **imaginary experiment** : What if we dig a **well** or a **boar** or a **straight hole** (say, its diameter is **100** meter) **throughout the earth** (i.e. Start digging from the one end of the Earth until you reach the exact opposite end of the Earth) and then one heavy and rigid **spherical stone** (say, its diameter is **1** meter) is to be **fallen down** into that well or boar or hole? What would **happen** to that **sphere**? Assume all the **best conditions** for this experiment: a perfect round-shaped well with no inner disturbances, sphere is made from the very hard material, a perfect vertical free-fall of the sphere, etc. How long would it travel through this well? Will sphere **stop its journey** by reaching the **center of the earth**? If yes, then could you **justify this situation** by the various **forces applying** on that sphere **at that time**? Please share your views and justifications on this experiment.


title: Dig a hole to the other side of the Earth

text: > **Possible Duplicate:**   >  If it was possible to dig a hole that went from one side of the earth to > the other… This question may not be suitable here and it could be closed in seconds but I wanna give it a try.   1. is it possible to dig a hole to the other side of the Earth?   2. If we did somehow manage to dig a hole to the other side of the Earth, would we fall through?


title: If you could ride an elevator through the earth

text: What if it was possible to ride an elevator straight through earth. If we take all the heat and pressure problems away, and assume it would be possible: What would happen with gravity? Would gravity suddenly turn around as i pass through the center and hit my head into the roof? Or would i gradually get weightless as i descend? **EDIT:** This is not a duplicate. The other questions does not involve an elevator car. The other ones are about jumping into it.


title: "Observed" angular velocity

text: I have a homework question based on the following diagram: ![velocity diagram](http://gyazo.com/f0de3827222c4c5651df70ae25b657e9.png) I need to find the **angular velocity of the object as seen by an observer at the origin** of the frame. The question says that the observed angular velocity is given by $\omega_0 = \Delta \varphi_0 / \Delta t$, where $\Delta t = t_2 - t_1$ and $\Delta \varphi_0 = \varphi_0(t_1^*) - \varphi_0(t_2^*)$ is the change of the angle $\varphi$, the polar angle which the object had at times $t_1^*$ and $t_2^*$. These are the times when the light detected by the observer (at $t_1$ and $t_2$) was emitted. The object is located at $(x_1, y_1)$ at time $t_1^*$ and it is located at $(x_2, y_2)$ at time $t_2^*$. I started to answer this question by just finding an expression for $\Delta \varphi_0$ in terms of the coordinates ($\Delta \varphi_0 = \arctan{(y_1/x_1)} - \arctan{(y_2/x_2)}$) and dividing this by an expression for $\Delta t$. However, I was told that this was not correct. Apparently, I have to get an expression for the angular momentum in terms of the angle $\theta$ in the diagram. My lecturer told me that this was could be done with some basic geometry. I can't quite see what to do though. Also, I was told that the angle $\Delta \varphi_0$ was assumed to be very small. I have a feeling that what the lecturer wants us to derive is the expression $\omega = \frac{|\vec{v}|\sin{(\theta)}}{|\vec{r}|}$, but I can't see how to get this. Can anyone help? Edit: This question is from a special relativity course, but I believe this can be answered without using any knowledge of special relativity. As I said, the lecturer told me that it was basically just geometry.


title: Is my textbook teaching an incorrect concept of Work?

text: I fear my textbook is teaching an incorrect concept of Work. I am very frustrated right now since I was struggling to understand the concept in the way that was explained in the textbook and my instincts tell me it is wrong. Also note that this is a online highschool course, so the textbooks are not published per se and it is common for these courses to be incorrect. **background information:** Here is the definition of work given in the textbook: `We say that work is done on an object when a force that is exerted on it causes a displacement.` * * * I'll give two examples here as I consider both to be incorrect. **Textbook Example 1:** If a 10 N force causes an object to be displaced 5.0 m, what is the work done on the object? **Textbook Answer:** W = Fd W = 10 N x 5.0 m W = 50 Nm W = 50 J * * * **Textbook Example 2:** A 0.0782 kg hockey puck was shot with an acceleration of 2.8 m/s2 over extremely fast ice (we will assume that it is frictionless). How much work was done when the puck passed the 12.0 m mark and the 112.0 m mark? **Textbook Answer:** In order to determine the work done, we need to know the distance traveled (which we do) and the force (which we do not). However, because we know the mass of the puck and its acceleration, then we can determine the force and then, use this force to determine the work. Fnet = ma Fapplied = 0.0782 kg x 2.8 m/s2 F = 0.218 96 kg/m/s2 F = 0.218 96 N F = 0.22 N Now we know the force exerted on the puck, we can now determine the work done on the puck as it passes the 12.0 m mark. W = Fd W = 0.22 N x 12.0 m W = 2.64 Nm W = 2.6 J We can also determine the work done on the puck as it passes the 112.0 m mark. W = Fd W = 0.22 N x 112.0 m W = 24.64 Nm W = 25 J Notice that the distance the puck travels affects the amount of work done! * * * **Why I think these are incorrect:** I believe these answers are incorrect because of what the unit Joules means to me. Newton meters would appear to be a meaningless unit of the distance in the calculation was the distance traveled by the object since friction is not considered and any distance might be possible that being the case. As well, it means that, as shown in the hockey puck problem, the resulting joules may be found to be thousands of Joules because the object is sliding on a frictionless surface, but there is no more a transfer of energy--and work IS defined as a transfer of energy. In the puck problem it doesn't make sense for the resulting Joules to be more after a longer distance since there is no more energy transferred. Once the puck is accelerated, it will remain at a constant velocity and its kinetic energy will be constant (since it is a frictionless surface). Therefore, the Joules of energy transfer should remain the same after the acceleration. I think that Work should really be defined as: **the distance the force was exerted over** as opposed to **the distance the object traveled**. Please correct me if I'm wrong. If I'm right, I will send the link of this question to the head of the science department or school as it is a huge oversight to teach a completely wrong concept to students (including myself).


title: What did recombination look like?

text: I recently remembered that someone worked out what the big bang sounded like and that got me thinking... About 377,000 years after the Big Bang, electrons became bound to nuclei to form neutral atoms. Because of (?) this, the mean free path of photons became effectively infinite, i.e. the universe became transparent to radiation. **What would this have _looked_ like?** More precisely I could ask: did the sky suddenly become dark, or was the amount of radiation basically same after as before? What would the distribution and timescale have been like? In a perfectly homogeneous universe it would happen at the same rate everywhere. Did the universe have any structure at this point? Would you have been able to see blotches of lighter and darker patches of the sky (assuming you were within one of the more transparent patches), these being proto-shapes of galactic filaments perhaps, or the noisy grit of the CMB? Or would it just be a vague cloud at any scale, and in any part of the spectrum?


title: Nonzero space-like commutation relation in string theory?

text: How did Nima Arkani-Hamed come up with the following nonzero space-like commutation relation in string theory? $$\left\langle \left| \left[ \hat{\mathcal{O}}(\vec{x}), \hat{\mathcal{O}}(\vec{y}) \right] \right|^2 \right\rangle \sim \exp \left( -\frac{k\left| \vec{x} - \vec{y} \right|^{D-2}}{8\pi G} \right)$$ His explanation isn't too clear.


title: What is the linear momentum of an EM wave in a medium?

text: In free space, the linear momentum density of an EM wave is given by the Poynting vector $\vec S$ over the speed of light squared, $\vec g=\frac{\vec S}{c^2}$. In a medium, $S$ is generally not directed along the wave vector $\vec k$. If $v_g$ is the group velocity and $W$ is the density of EM wave energy, then $S = v_g W$. On the other hand, it is often assumed that the momentum of a photon is still $\hbar\vec k$. What is the macroscopic momentum density of EM wave in an anisotropic medium such as magnetized plasma ?


title: Comparing Energy Technologies in terms of: Cost per Power Capacity  and Research Spendings

text: ### Cost per Power Capacity For a number of energy technologies I'd like to know what the minimum costs are to install a given power capacity. Are there any such comparisons available? ### Research Spendings Technologies often own much to publicly funded research efforts so I'm also interested in estimations regarding development cost of the technologies themselves. Of course foundations to a given technology may have been readily available, so let's limit this to spendings on larger research projects _dedicated_ to developing a technology and motivated by the need of energy. I'm aware it's unlikely any exhaustive comparison in full detail exists but maybe something related? (Maybe someone should tag this question 'energy-economics' -- I'm lacking the reputation)


title: Classic home experiments for an 8-year-old child

text: My 8-year-old daughter's school report says that she's good at understanding the basic science she's doing, but she's having trouble seeing how experimental results lead to conclusions. Specifically, it says she struggles to appreciate how changing parameters in an experiment can be used to prove or rule out a hypothesis. I'm a biologist/chemist by degree, and I found it hard to think of any parameter-based experiments in my field that we could do at home that wouldn't bore a child. They like creepy-crawlies well enough, but biology experiments tend to need time and repeat observations to get a result. Chemistry tends to need more specialist equipment and reagents. I've got a couple of books of home science, and I scoured the net as well, but most of the living-room science experiments I found that were suitable for an 8-year-old were more demonstrations than actual experiments. There were few parameters: nothing you could really vary. So can any of your learned people suggest some simple, parameter-based physics experiments we could do at home or in the garden, without specialist equipment, which involve science at a level an eight-year-old could engage with?


title: Double Slit Experiment with Two Independent Sources

text: Imagine a variation on the double slit experiment. I'll describe it in 2D using the $x-y$ plane. The $x$-axis is impenetrable other than the two slits, which are positioned at $(-1,0)$ and $(+1,0)$. Detectors will be positioned in the upper half plane, say along $y=10$ or something. Here's the twist: the negative $y$-axis is impenetrable as well. Two sources will be needed, say one at $(+1,-\infty)$ and the other at $(-1,\infty)$. These sources create plane waves impinging on the two slits. The basic question: will one observe simple interference at the detectors? For photons, I think the answer is obviously yes. (Maxwell's equations in free space are linear and there's no problem "superposing" or simply summing the contributions of each of the slits.) But what if one uses electrons as the source? The wall separating the two sources clearly seems to preclude a single electron from taking both paths to the destination, as would be the case in the usual setup. That suggests you'll have to have more than one electron at a time involved to achieve interference. But electrons interact electromagnetically, and that interaction may wreck any interference pattern. In contrast, photons don't interact, so simple interference occurs there. Alternatively, one might think that interference will still occur with single electrons, since there is no way to determine which source the electron came from. Intuitively, I lean toward this answer being correct, but if it is, how would you understand the underlying Hilbert space and Hamiltonian of such a system? I don't think it can possibly be understood as a single electron system with a single wave function throughout space. Or can it?


title: Functional Derivative in the Linear Sigma Model

text: In the linear sigma model, the Lagrangian is given by $$ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) +\frac{1}{2}\mu^2\sum_{i=1}^{N}\left(\phi^i\right)^2-\frac{\lambda}{4}\left(\sum_{i=1}^{N}\left(\phi^i\right)^2\right)^2 \tag{11.65} $$ (for example see Peskin & Schroeder page 349). When perturbatively computing the effective action for this Lagrangian the derivative $ \frac{\delta^2\mathcal{L}}{\delta\phi^k(x)\delta\phi^l(x)} $ needs to be computed. (for instance, Eq. (11.67) in P&S): $$ \frac{\delta^2\mathcal{L}}{\delta\phi^k(x)\delta\phi^l(x)} ~=~ -\partial^2\delta^{kl} +\mu^2\delta^{kl}-\lambda\left[\phi^i\phi^i\delta^{kl}+2\phi^k\phi^l\right].\tag{11.67}$$ My question is, how is one supposed to handle the derivative term? This seems to be completely implicit in the presentation of P&S, but from what I could gather it should go like so: 1) Because we are computing the effective action, $\mathcal{L}$ is actually under an integral and we can replace $\left(\partial_\mu\phi^i\right)\left(\partial_\mu\phi^i\right)$ with $-\left(\partial^\mu\partial_\mu\phi^i\right)\phi^i=-\left(\partial^2\phi^i\right)\phi^i$ using Stokes' theorem. 2) Then when performing the first derivative I get $\frac{\delta}{\delta\phi^l}\left[-\left(\partial^2\phi^i\right)\phi^i\right]=-\partial^2\phi^l$. 3) It is the second derivative I get stuck at, for as far as I can see, $\frac{\delta}{\delta\phi^k}\left[-\partial^2\phi^l\right]=0$, for there is only dependence on the 2nd derivative of $\phi^l$ and not $\phi^l$ itself. If, as is usual in field theory, the field and its derivatives are treated as independent dynamical variables, then the second derivative should also be an independent dynamical variable. How is it explained then, that the result of this computation should be $\frac{\delta}{\delta\phi^k}\left[-\partial^2\phi^l\right]=-\delta^{kl}\partial^2$?


title: Q factor of parallel RLC circuit in series with a capacitor and resistor

text: I know that for parallel RLC circuits, the $Q$ factor is given by: $$ Q = R \sqrt {\frac{L}{C}} $$ But now suppose it is connected in series to a resistor $R_2$ and capacitor $C_2$. Would the $Q$ factor be changed? ![enter image description here](http://i.stack.imgur.com/dpLzs.png)


title: why is mass of air bubble in material medium considered to be negative?

text: The mass of air bubble in any medium is considered as negative. Is the air bubble is massless. I m in confusion. can we not neglect the mass of air bubble in material medium. But i have found in many books the mass of air bubble in material medium as negative. please discuss.....


title: Does the spinning of a nucleus alter its charge distribution?

text: I put a nucleus in a magnetic field. It spins. Does the electric charge distribution remain homogeneous or does the charge redistribute? Can this be measured? Will accept reference as answer.


title: Never ending inflation

text: There are a lot of different models of inflation in cosmology, with plenty of different features and issues. I read that the problem with some models is that they introduce a never-ending inflation, but I can't find an example of it. Anyway, I thought that indefinite inflation could only happen in a true de- Sitter space, while if we have a quasi-de-Sitter space, it would always be an end at some point. Eventually after a very long stage of inflation, but still - there is an end. So, am I correct? Is the problem of never ending inflation a tough one? Is it a problem that often arise?


title: Are Newton's gravity waves detectable by a laser interferometer?

text: Newton's theory of gravity supports "gravity waves" in that moving objects cause changing gravitational fields. For example, two bodies rotating around their center of mass will have a stronger gravitational field when they are longitudinally oriented than when they are transverse oriented. Given two masses of mass $M$ orbiting on a circle of radius $r$, at a distance $d$ from an observer, the strength of the attraction is: $$\begin{matrix} F_{min} &=& \frac{2GM}{d^2},\\\ F_{max} &=& \frac{2GM}{d^2}\frac{1+r^2/d^2}{(1-r^2/d^2)^2} \end{matrix}$$ ![enter image description here](http://i.stack.imgur.com/dIjHy.jpg) This should be detectable at long distance. My question is this: With that sort of gravity, would the laser interferometer based gravity wave detectors be able to detect the gravity wave? An example is the LIGO, Laster Interferometer Gravitational-wave Observatory.


title: How can a point-particle have properties?

text: I have trouble imagining how two point-particles can have different properties. And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume? Not only that, but it can also detect all fields without having any structure. Maybe it can check curvature of spacetime to account for gravity, but how can a point contain the information of what the other fields-vectors are? This seems to mean that also the information/volume in space is infinite. Mathematically, a point cant have any intrinsic structure, so how does physics which is a mathematical theory explain this?


title: Absorption Spectrum

text: I just learnt what absorption and emission spectrum are. And our teacher introduced us to what Quantum Dots are. And showed us their absorption and emission spectra. And they look something like this ![enter image description here](http://i.stack.imgur.com/CuGuK.gif) I understand that the peaks of the spectra is supposed to match the transition energy in the QD. But why does the absorbance increase as the wavelength decreases? I am not aware of any QD transition energy at that wavelength. And this seems to be true for all the QD spectra I googled. Is it dependent on other substances in the solution or the film? Please correct if I am mistaken.


title: What predictions can a quantum gravity theory make?

text: Some of the major challenges that heralded the need for quantum mechanics we're explaining the photo-electric effect, the double-slit experiment, and electrons behavior in semi conductors.   1. What are some of the predictions we can expect to see from a theory of quantum gravity?   2. What types of experiments have shown the necessity for a quantum gravity theory?


title: Bomb trajectory from an airplane

text: If a plane is flying at a constant speed at some altitude like 5-6 km and it releases a bomb:   * does the bomb move forward at the same horizontal speed as the airplane?   * or does its horizontal speed decrease due to drag and no thrust to push it forward?   * or does it outrun the airplane because it's more aerodynamic than the airplane and the gravity is adding to its horizontal speed?   * or does it depend on the bomb? If so, what would be the typical scenario? I looked at a demonstration on Wolfram Alpha (http://demonstrations.wolfram.com/TrajectoryOfABomb/ \- requires a plugin) and it looks like the bomb moves forward at the exact same speed as the plane, and I don't know if that's accurate.


title: If I have created a polarization-entangled photon source, would I be able to create a qubit?

text: In my laboratory, I have already obtained a polarization-entangled photon source. It was created via the Spontaneous Parametric Downconversion process of 2 BBO crystals. The next question is that would I be able to use this source to create qubits? If so, how? Thank you in advance.


title: The 4 R's of radiotherapy - redistribution

text: I'm having some trouble of getting a grasp of this term, so I hoped someone could enlighten me. As far as I understand, the idea behind it is, that cells have different cycles in their "life", and in each cycle there is different phases where the radiosensitivity is different compared to the others. Then, by using radiation you can sync the cycles of tumour cells for example, so they start in the same phase, then you wait until you reach a good radiosensitive phase, and then irradiate again. But unfortunately because cells are so different, the speed of which they go through different phases are not the same, so therefore it can be difficult to sync the phases as wished. That is how I currently understand it. Is that totally wrong, or...?


title: Physics talk with an emphasis on Mathematics

text: I have to give a 10 minute physics talk that have to involve a fair bit of mathematics -- i.e. not just qualitative/handwaving material to some undergrads. I have wasted the last 3 hours looking for appropriate topics and have pretty much lost the will to live. Any suggestions would be greatly appreciated.


title: Is there a published upper limit on the electron's electric quadrupole moment?

text: I understand an electric quadrupole moment is forbidden in the standard electron theory. In this paper considering general relativistic corrections (Kerr-Newman metric around the electron), however, there is a claim that it could be on the order of $Q=-124 \, \mathrm{eb}$. That seems crazy large to me, but I can't find any published upper limits to refute it. Surely someone has tested this? Maybe it's hidden in some dipole moment data? If not, is anyone planning to measure it soon?


title: Can quark-gluon plasma ever be close to an ideal gas of asymptotically free quarks?

text: The question inspired by an upcoming colloquim at UCB. A naive interpretation of quark asymptotic freedom seems to imply that at high enough energies they should be weakly interacting. On the other hand, the quark-gluon plasma accessible experimentally via heavy ion collisions is claimed to be close to an ideal liquid. There is also a lot of excitement about this because of AdS/CFT correspondence (see this and related questions at Physics.SE). Shouldn't the quark-gluon matter phase diagram be close to an ideal gas at least in some low-density & high-temperature region? In other words, is quasi- independent quark gas picture of QCD ever valid? And is there a simple reason why?


What does it mean to say that mass "approaches infinity"? I have read that mass of a body increases with the speed and when the body reaches the speed of light, the mass becomes infinity. What exactly does it mean to say that the mass "approaches infinity" or "becomes infinity"? I am not able to get a picture of "infinite mass" in my mind.


title: Electron Wave Interference

text: In the double slit interference pattern for the wave of an electron, what will happen if I make the slits to be smaller than the size of an electron ? Will I still observe an interference pattern on the opposite side of the screen or no electron will be able to cross the slit? If no, then how can a quantum prisoner escape ?


title: Would the microwave background radiation have an "observer effect" on all systems in the universe due to its permeation?

text: Or more specifically, does the CMB radiation have an observer effect on us in our local system due to gravitational lensing? Acknowledging this effect, if any would be nearly negligible I have an also: would this same effect be "felt" in the void spaces between galaxies?


title: River model of spacetime for arbitrary situations

text: This paper describes black holes as space flowing inward (the rotating hole also twists in a weird way): http://arxiv.org/abs/gr-qc/0411060 The proper time given by the objects is the same as special relativity except for that fact that they are on a treadmill: $$ds^2 = dt^2-|v - w|^2$$ where t is "time" in the sense that the (static) geometry is translationally invariant along t, v is the velocity of you, w is the velocity of space. w is a function of the coordinates. The rotating black hole adds a "twist" term that I don't understand well; space itself doesn't rotate! Is it possible to describe arbitrary spacetimes (i.e. two unequal mass, unequal spin neutron stars colliding to become a black hole) in this fashion? Of course, there may be more than one right answer as you no longer have time translational invariance. What kind of coordinates singularities/caustics would such a description entail?


title: Uncertainty principle in atomic clocks?

text: How does the uncertainty principle limit the accuracy of atomic clocks. I know line width and measurement time are important but not exactly why?


title: Why is everything colder nowadays?

text: I know that when the universe began it was incredibly hot. Ever since, it is been cooling, and nowadays the average temperature of the universe is quite close to 0 K. Is this a consequence of having the same energy spread in a wider space, or is there something else going on?


title: About Stellar Evolution

text: During its formation, the Sun contracted under the force of its own gravity, until hydrostatic equilibrium was established. In this contraction, its temperature increased from $3\times10^4\text{ K}$ to $6\times10^6\text{ K}$. How to find the energy radiated during this contraction?


title: Gauge pressure clarification

text: Is gauge pressure always zero-referenced against ambient air pressure? Or is it referenced against the sum of all pressures acting on a fluid, which usually just happens to be ambient air pressure? To clarify, here is a potential question: Imagine a container with two immiscible fluids of different densities. 1atm of atmospheric pressure is acting at the surface of the first fluid. An object at the deepest point of the first fluid has a gauge pressure of 3atm, so the absolute pressure at this depth is 4atm. If the absolute pressure of the deepest point of the second fluid is 8atm, what would the gauge pressure of the object be if it was submerged to the deepest point of the second fluid? If gauge pressure is always zero-referenced to air pressure, the gauge pressure would be 8atm - 1atm = 7atm. However, if it accounts for the sum of the pressure from all the fluids above it, then the gauge pressure would be 8atm - 1atm - 3atm = 4atm. Which is the correct gauge pressure?


title: Why does a short circuit generate fire?

text: When a short circuit occurs it's obvious that there is fire. How come electric energy turns out to be heat energy? What causes the conductors to get hot when short circuit is present.


title: Lie derivative of Riemann tensor along killing vector ( = 0 )

text: I'm currently learning the mathematical framework for General Relativity, and I'm trying to prove that the Lie derivative of the Riemann curvature tensor is zero along a killing vector. With the following notation for covariant differentiation, $A_{a||b} $ (instead of $\nabla_b A_a$ ), I have the following: $\it\unicode{xA3}_\xi R_{amsq} = R_{amsq||x} \xi ^x + R_{xmsq} \xi^x{}_{||a} + R_{axsq} \xi^x{}_{||m} + R_{amxq} \xi^x{}_{||s} + R_{amsx} \xi^x{}_{||q}$. I suspect that I need to invoke the second Bianchi identity. However, before I can do this, I need to somehow get this into a different form. There has to be some property of either killing vectors or maybe covariant derivatives that I'm forgetting/failed to learn. Any help would be appreciated.


title: Diffraction Grating Spectrometry Question

text: Yes, this is a homework question, but I've already failed to solve it enough times that the online system hosting it isn't going to give me any marks, so I figure it's a good time to stop hitting the wall and actually ask. The question is as below: > A spectroscopist uses a spectrometer that has a grating with 600 grooves/mm. > This grating can illuminate a CCD with a range of 1228 nm, in the 5th order > (m = 5). One day, she buys a larger format CCD -- one larger than her old > one by a factor of 3. What spectral range of wavelengths can be measured > when she now obscures the 4th order lines (m = 4) with her larger CCD? Came out with answers like $1.228 \times 10^{-6}$ and $4.17 \times 10^{-7}$ so far because I have no idea what the process is. Hints would suffice. Basically, everything I've tried so far revolves around a formula: $m\lambda=d\sin\theta$. What did I do so far:   * $5(1.228\times10^{-9})=\frac{0.001}{600}\times\sin\theta$ to get $\sin\theta$ and then put it back in to replace $\sin\theta$ as $\frac{4}{5}\sin\theta$. Then I went and searched up a diffraction grating to find out that the orders are not equidistant.   * $5(1.228\times10^{-9})=\frac{0.001}{600}\times\sin 90^\circ$ where I assumed the angle is perpendicular.   * Other stuff I can't remember I feel like I understood the wavelength portion of this incorrectly, but I have no idea why.


title: Non-constant angular velocity

text: In this paper about Backstepping controll of a quadrotor helicopter an algorithm for control is described, but I have hit a dead end. In equation 15 it is described the part of state space for the angular and translation motion of a rigid body. ![Equation 15](http://i.stack.imgur.com/MSmYq.png) The author states $R_r$ is the rotation velocity matrix between Earth fixed reference frame and Body fixed reference frame. I assume the "between" means a rotation from body coordinates to earth coordinates. $R_t$ is the translation velocity matrix between Earth fixed reference frame and Body fixed reference frame. I also assume the "between" means a rotation from body coordinates to earth coordinates. One question is how do I calculate the value of the angular acceleration described by the partial derivative of $\dot{\phi}$ and $\dot{\theta}$. The author does not state and I would like to know if it is numerically or if it is analytically possible. The other question is in which referential is $\dot{\zeta}$ and why did the author make a rotation and "derotation" on $Kt$? From the paper $G$ is a vector with the $z$ element set to $g=9.81$. Last question, more like a curiosity, would anybody give me a pointer to state-space formulation? I do not follow how the author composed the state space system.


title: If a spaceship was pulled toward a sun, would it spin?

text: I was watching a movie. A spaceship was forced into "warp speed". The co- ordinates could not be set. The spaceships trajectory was that of a nearby sun. Forcing the spaceship to power down was the solution. Now out of "warp speed" and with no computer aid (steering etc) the spaceship was seen to be spinning toward the sun trapped in its gravitation field. My question is, would the spaceship (typically aerodynamically shaped) spin toward the surface? My opinion is no. The spaceship would just fall flat due to the surface area provided at the bottom of the fuselage


title: Exciting Surface Plasmon-Polaritons with Grating Coupling

text: I'm very new the topic of SPPs and have been trying to understand this particular method of exciting surface plasmons using a 1D periodic grating of grooves, with distance $a$ between each groove. If the light incident on the grating is at an angle $\theta$ from the normal and has wavevector ${\bf k}$, then apparently if this condition is met: $\beta = k \sin\theta \pm\nu g$ where $\beta$ is corresponding SPP wavevector, $g$ is the lattice constant $2 \pi/a$ and $\nu={1,2,3,...}$ then SPP excitation is possible. I haven't ever really had a formal course in optics, so my question is where this condition comes from. It seems like Fraunhofer diffraction, but only for the light being diffracted at a $90^\circ$ angle to the normal. Most books don't state how they get this result, they just say it's because of the grating "roughness" which really confuses me. Any help would be greatly appreciated.


title: Jumping sewer lid - WHY?

text: **Intro:** Few hours ago, there was a storm. We heard some constant banging which couldn't be explained by thundering. Then we found out, it was a sewer lid jumping. Maybe it's normal in other parts of the world, but for me it was like the first time in my life. I've captured the video. **The question** is what was causing this to happen. It's kind of clear that it was air pressure so strong that it was capable of lifting this metal lid. But where did this air pressure appear?   * Wind blowing into the sewer? If so, would it be so strong to lift the lid? And shouldn't be the sewers protected against wind somehow?   * Water filling the sewer so quickly, that it made the air pressure this strong? Would this really be the easiest way for the air to leave the sewer?   * Something else (such as bored sewer worked :) )?


title: What determines the pitch of a resonant object

text: What determines what pitch an object such as a bell or tuning fork produces when struck? I have heard that the box in the "king's chamber" of the great pyramid at Giza is tuned to 438 Hz. I know that in hand-bell choirs, the bigger the bell, the lower the tone, but I have noticed that size does not seem to be the determining factor in a bell's tone.


title: Minimum size of an asteroid to actually impact earth

text: From what I understand, an object entering the atmosphere will start to burn up from the tremendous resistance of the atmosphere. Presumably, for asteroids under a certain size, they will burn up completely and never impact the surface of the earth. Do we have a way of determining the minimum size needed for actual impact? If so, roughly what is the size and how does it compare to the average size of asteroids that pass by us regularly?


title: Yang-Mills Coulomb Gauge

text: My Question is how to explicitly move into the "Coulomb gauge" in Yang-Mills theory. Using the answer provided by QMechanic, one can move into the "temporal gauge" for Yang-Mills fields: Gauge fixing choice for the gauge field $A_0$ . Once in this gauge, to move into the "Coulomb gauge," does one take an explicitly _time independent_ $g$, and demand that it satisfy: $$ i\partial^\mu (g^{-1}(\partial_\mu - i A_{\mu})g) = 0 $$ Thus giving $\partial ^{\mu} A'_{\mu} = 0$? Are we guaranteed that such a $g$ exists? Thanks.


title: Any link between decoherence and renormalization?

text: I have been studying decoherence in quantum mechanics (not in qft, and don't know how it is described there) and renormalization in QFT and statistical field theory, I found at first a similarity between the two procedures: on one side decoherence tells us to trace over the degrees of freedom we don't monitor, in some way intrinsically unknown to recover a classical picture picture from quantum mechanics, on the other side by renormalizing we also integrate over "our ignorance" but this time, the U.V. physics or high energy modes to get the infra-red physics we observe. Beyond the technical similarity (taking a trace, for discrete Kadanoff-Wilson transformations) it feels that in both cases we are forced to do these procedures because we start from a wrong picture where we separate the free object (purely quantum in the first case, with bare parameters in the second one) and then calculate the effects of the interactions, that are responsible from what we, observers, see, classical and infra-red physics. This where it comes to me that some interested links between the two concepts may exist or be pointed out (and also wonder what decoherence becomes in QFT). I still see one huge asymmetry between the two, decoherence is dynamical, it has a typical time of decay, where renormalization is static. I hope I could explain my interrogation clearly, and some interesting comments will come.


title: Why does it take so long to get to the ISS?

text: I don't understand why when first launched Space X's Dragon capsule had to orbit the Earth many times in order to match up with the ISS? Was this purely to match it's speed, or to get closer (as in altitude) to the ISS? In the stages when it gets to about 200m, it seemed like it was able to go directly up to the ISS, how come it couldn't do that the entire way. (Additionally, in Scifi movies you see smaller shuttles able to go directly to space stations in orbit, is that type if travel not possible?)


title: How large can you make a tokamak?

text: I've seen questions on how small you can make a tokamak. But I haven't yet seen any "physical" upper limit on the tokamak design. If you take a wind turbine for example, doubling the linear dimensions will increase the sweep area by a factor 4 but the structural mass with a factor 8, which clearly explains why you don't want to make (convensional) wind turbines above certain dimensions. With a tokamak, I imagine that if you double the linear dimensions, the plasma volume (and hence the power production) will increase eightfold, whereas area that you have to protect against fast neutrons will only quadruple. So once you master the tokamak technology, you would only need to scale it up appropriately to bring down capital costs. What do I miss? What cannot be scaled up easily in a tokamak?


title: How to formally write down the Boltzmann equation?

text: Can someone write down the Boltzmann equation, not neglecting any of the variables of the involved functions and integrals? Specifically, how to concisely capture the "primed" variables in a sensible manner?


title: Solving the Tolman–Oppenheimer–Volkoff (TOV) equation

text: The Pressure of a static spherical object (say star), which has the Schwarzchild metric outside it, satisfies the following differential equation called the TOV equation. $$\frac{\mbox{d}P}{\mbox{d}r}=-\left(P+\rho\right)\frac{m(r)+4\pi r^3P }{r\left(r-2m(r)\right)}$$ Here $\rho$ the mass density is a function of $r$, and $m(r)=4 \pi\int_0^r\rho(r^{\prime}){r^{\prime}}^2 d{r^{\prime}} $. How do I solve this equation? Wald states the solution as follows; but Please could someone suggest **how I solve this analytically?** ($\rho_0$ is the density at the radius of the star $R$, and $M$ is the total mass $M=4 \pi\int_0^R\rho(r^{\prime}){r^{\prime}}^2 d{r^{\prime}} $) $$P(r)=\rho_0\left(\frac{\left(1-\frac {2M}{R}\right)^\frac12 - \left(1-\frac {2Mr^2 }{R^3 }\right)^\frac12 }{\left(1-\frac {2Mr^2 }{R^3 }\right)^\frac12 -3\left(1-\frac{2M}{R}\right)^\frac12 }\right)$$


title: Hamiltonian matrix off diagonal elements?

text: I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I don't understand is: Why are the $\mu B$ parts not diagonal? If the Hamiltonian is $\vec{\mu} \cdot \vec{B}$, why aren't all the components just diagonal? How is this matrix built systematically? Can someone please explain? > We now consider the effect of a uniform magnetic field $\mathbf{B} = > B\hat{z}$ on the hyperfine levels of the ${}^2 S_{1/2}$ ground state of > hydrogen. Initially, we will neglect the effect of the nuclear (proton) > magnetic moment. The energy eigenstates for the Hamiltonian describing the > hyperfine interaction are also eigenstates of the operators $\\{F^2, F_z, > I^2, S^2\\}$. Therefor if we write out a matrix for the hyperfine > Hamiltonian $H_\text{hf}$ in the coupled basis $\lvert Fm_F\rangle$, it is > diagonal. However, the Hamiltonian $H_B$ for the interaction of the magnetic > moment of the electron with the external magnetic field, > > $$H_B = -\mathbf{\mu}_e\cdot\mathbf{B} = 2\mu_B B S_z/\hbar,\tag{4.20}$$ > > is diagonal in the uncoupled basis $\lvert(SI)m_S, m_I\rangle$, made up of > eigenstates of the operators $\\{I^2, I_z, S^2, S_z\\}$. We can write the > matrix elements of the Hamiltonian in the coupled basis by relating the > uncoupled to the coupled basis. (We could also carry out the analysis in the > uncoupled basis, if we so chose.) > > The relationship between the coupled $\lvert Fm_F\rangle$ and uncoupled > $\lvert(SI)m_Sm_I\rangle$ bases (see the discussion of the Clebsch-Gordan > expansions in Chapter 3) is > > $$\begin{align} \lvert 1,1\rangle &= > \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2}\tfrac{1}{2}\rangle,\tag{4.21a} > \\\ \lvert 1,0\rangle &= > \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle > + > \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21b} > \\\ \lvert 1,-1\rangle &= > \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2},-\tfrac{1}{2}\rangle,\tag{4.21c} > \\\ \lvert 0,0\rangle &= > \frac{1}{\sqrt{2}}\biggl(\lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr)\tfrac{1}{2},-\tfrac{1}{2}\rangle > - > \lvert\bigl(\tfrac{1}{2}\tfrac{1}{2}\bigr),-\tfrac{1}{2}\tfrac{1}{2}\rangle\biggr),\tag{4.21d} > \end{align}$$ > > Employing the hyperfine energy shift formula (2.28) and Eq. (4.20), one > finds for the matrix of the overall Hamiltonian $H_\text{hf} + H_B$ in the > coupled basis > > $$H = \begin{pmatrix} \frac{A}{4} + \mu_B B & 0 & 0 & 0 \\\ 0 & \frac{A}{4} > - \mu_B B & 0 & 0 \\\ 0 & 0 & \frac{A}{4} & \mu_B B \\\ 0 & 0 & \mu_B B & > -\frac{3A}{4} \end{pmatrix},\tag{4.22}$$ > > where we order the states $(\lvert 1,1\rangle, \lvert 1,-1\rangle, \lvert > 1,0\rangle, \lvert 0,0\rangle)$. And for Eq. (2.28) the other part is > $$\Delta E_F = \frac{1}{2}AK + B\frac{\frac{3}{2}K(K + 1) - 2I(I + 1)J(J + > 1)}{2I(2I - 1)2J(2J - 1)},\tag{2.28}$$ > > where $K = F(F + 1) - I(I + 1) - J(J + 1)$. Here the constants $A$ and $B$ > characterize the strengths of the magnetic-dipole and the electric- > quadrupole interaction, respectively. $B$ is zero unless $I$ and $J$ are > both greater than $1/2$.


title: Photons-Wave/particle duality

text: I know that photons and electrons and such are said to have a wave particle duality, but what does that mean for a photon? When light strikes an object, are many photons emitted, enough to draw infinitely many rays, is only one emitted, or something in between? In particular, I'm having trouble with thin film interference: ![enter image description here](http://i.stack.imgur.com/E53nS.gif) The two resulting rays are said to constructively interfere, which is confusing to me. The two rays are clearly parallel, but not coinciding, so how do the two interfere at all? I think my problem is that I imagine light to be a single ray, with a linear oscillating magnetic field- what is the proper way to address these rays? Are they photons? Or are they small instances of a wave front? I've heard Huygens' Principle, but in this case we present single rays at the end, so I'm led to believe they really ARE rays, in which case they would be photons, and the interference problem would be a result of the wave/particle duality. The only other thought I've had with regards to the interference is that, as opposed to looking at the rays as one dimensional rays, they could be some kind of representation of a wave 'centered' around that vector, but that doesn't make sense either. I know it's a heavy question, but it's really confusing me.


title: Bloch's theorem and Bloch's state

text: The question is not so much about the theorem, but more about what it means in this context: see this link. So yes, because of Bloch's theorem the Hamiltonian eigenstates in a crystalline system can be written as \begin{align} \psi_{n,\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}u_{n,\vec{k}}(\vec{r}), \end{align} and so the Berry connection can be defined: \begin{align} A_{n}(\vec{k})=i\langle n(\vec{k})|\nabla_{\vec{k}}|n(\vec{k})\rangle, \end{align} but what in the world is $|n(\vec{k})\rangle$? I've read a few articles on topological insulators and they always seem to start off with the Bloch wavefunction $e^{i\vec{k}\cdot\vec{r}} u_k(\vec{r})$, and then somehow they magically get the ket $|u(\vec{k})\rangle$ from which the Berry connection is defined... is $|u(\vec{k})\rangle$ the column vector comprised of the Fourier coefficients of $u_\vec{k}(\vec{r})$ w.r.t. $e^{i\vec{G}\cdot\vec{r}}$ or what?


title: Sauna thermodynamics

text: Why does it get hotter (feel hotter) in a sauna when one pours water over the hot stones? Wikipedia says that the water condenses onto the skin, but the actual air humidity is so low that I doubt anything is condensing there. The water (sweat) evaporates from skin instead.


title: pressure exerted by fluid

text: If I had a flexible tube sealed at both ends and I submerged it in water (held vertical) Would the bottom half of the tube compress and the top half expand? What would the pressure in the tube be? Say its a 2" length of tube with the bottom being in 12" of water


title: Propagation of light in transparent media: absorption and reemission or scattering?

text: In the two Phys.SE questions What is the mechanism behind the slowdown of light/photons in a transparent medium? and Why glass is transparent? transparent media were discussed. But I'd like to clarify one detail: is a photon absorbed (and delayed) by the medium and then reemitted, or scattered instantly? Is e.g. a laser beam still coherent after passing glass? As medium molecules are disordered, this should distort the phase of photons taking different paths.


title: Why did my liquid soda freeze once I pulled it out of the fridge?

text: Can someone explain in both layman's terms and also technically why when I pulled my glass filled with liquid soda from the freezer, the liquid soda quickly froze? Doesn't this violate the 2nd law of thermodynamics since heat moved away from the glass with soda and to the ambient? Thanks


title: Confusion in biasing a NPN transistor in CE config

text: In CE configuration of NPN transistor, collector emitter junction is reverse biased. But how is a N-N junction reverse biased?


title: Superluminal interactions

text: > The fact that photons emitted from an electric-dipole active atom cannot be > spatially localized better than to the near-field zone of the atom is seen > as the origin of genuine superluminality. http://www.ncbi.nlm.nih.gov/pubmed/11309086 Question: What are superluminal interactions? No detailed information available on the web. I know that electromagnetic waves cannot be spatially localized better that to the near-field zone of the location where they are created.


title: $\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

text: I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1} \end{equation} where the $\mathbf{1}$ represents the trace. However, I do not understand why we cannot further decompose the traceless part into a symmetric and an antisymmetric part. In order to understand my logic: A general tensor $\varphi^i$ transforms as: \begin{equation} \varphi^i \rightarrow U^i{}_j \varphi^j \end{equation} whereas $\varphi_i$ transforms as: \begin{equation} \varphi_i \rightarrow (U^*)_i{}^j \varphi_j \end{equation} where $U \in \mathrm{SU(3)}$ is a $3 \times 3$ matrix. Now, I will let $S^i{}_j$ denote the traceless part of $T^i{}_j$ (i.e. $S^i{}_j$ has dimensions $\mathbf{8}$) and we can decompose this in the "symmetric" and "antisymmetric" part as usual: \begin{equation} S^i{}_j = \frac{1}{2}(S^i{}_j + S_j{}^i) + \frac{1}{2}(S^i{}_j - S_j{}^i) \end{equation} Then under an $\mathrm{SU(3)}$ transformation: \begin{equation} S^i{}_j + S_j{}^i \rightarrow U^i{}_k (U^*)_j{}^l S^k{}_l + U^i{}_k (U^*)_j{}^l S^k{}_l = U^i{}_k (U^*)_j{}^l (S^i{}_j + S_j{}^i) \end{equation} and: \begin{equation} S^i{}_j - S_j{}^i \rightarrow U^i{}_k (U^*)_j{}^l S^k{}_l - U^i{}_k (U^*)_j{}^l S^k{}_l = U^i{}_k (U^*)_j{}^l (S^i{}_j - S_j{}^i) \end{equation} Therefore, the symmetric part keeps its symmetry and the antisymmetric part keeps its antisymmetry. Thus two invariant subspaces are created and the representation is reducible? To sum up, I would think we decompose $T^i{}_j$ as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{3} \oplus \mathbf{5} \oplus \mathbf{1} \end{equation} where $\mathbf{3}$ denotes the dimensions of the antisymmetric part and $\mathbf{5}$ denotes the dimensions of the symmetric part. Where am I going wrong? Edit: I got my convention from "Invariances in Physics and Group Theory" by Jean-Bernard Zuber: ![enter image description here](http://i.stack.imgur.com/JduAD.jpg)


title: Electrons in CRT

text: In a CRT, where do the ejected electrons go after they cause fluorescence on the screen, have they lost most of their energy, or do they actually go through the glass?


title: Why is BTZ black hole asymptotically $AdS_3$?

text: The metric for the BTZ black hole is $ds^2=-N^2dt^2+N^{-2}dr^2+r^2(N^\phi dt +d\phi)^2$ where $N^2=-M+\frac{r^2}{l^2}+\frac{J^2}{4r^2}$ and $N^\phi=-\frac{J}{2r^2}$. It is often said that BTZ black hole is asymptotically AdS$_3$, but if I take $r\rightarrow \infty$ limit, then the BTZ metric, to the leading order of each component, becomes $ds^2\rightarrow-\frac{r^2}{l^2}dt^2+\frac{l^2}{r^2}dr^2+r^2d\phi^2-Jdtd\phi,$ while the AdS$_3$ metric reads, asymptotically, $ds^2_{AdS_3}\rightarrow-\frac{r^2}{l^2}dt^2+\frac{l^2}{r^2}dr^2+r^2d\phi^2.$ My question is that since the off-diagonal term $-Jdtd\phi$ doesn't approach to zero at infinity, how can one claim that BTZ is asymptotically AdS$_3$?


title: Degenerated Anderson Model Simulation

text: I'm trying to simulate the degenerative Anderson model. So depending on an energy difference first orbital and afterwards spin magnetism occurs. First i try to solve an easier ansatz with a limitation of only two orbitals to $l_z = \pm 1$. In this case i end up with a set of 4 equations: $\Delta \tan^{-1} (1/\pi * n_{m \sigma}) = E_0-E_f+(U-J)n_{\overline{m}\sigma}+U(n_{1\overline{\sigma}}+n_{2\overline{\sigma}})$ Where here $\sigma$ is the direction of the spin of certain electron in the orbital $m$, here only $1,2$ and the overline always is the opposite. U is a Hubbard potential and $J$ an intraatomic exchange. So we gathered a self consistent set of non-linear equation and i want to find solution in terms of $n_{m \sigma}$ satisfying all 4 equations. I already did some simulation and were able to plot it on dependancy of the energy difference with respect to the fermi energy, which is my parameter. So i change it in a certain range and one finds magnetic solution for $E_0 < E_f$. Is there a good method to solve such a problem with not too much calculation power? And in the end i want to play around with both potentials and see how the number of electron is changing for each situation. Thanks for any advice, /no PS: i already asked a question concerning to my code i wrote in SO, but i was rethinking the method of mine for solving this problem. I did an iterative attempt and reformulated the equation for a root searching algorithm and varied the energy difference $E_0 - E_f$. The programming stuff is on ipython plus a result of one simualtion run.


title: Angular momentum of particle rolling around inside of sphere

text: I have a hemispherical bowl in which I roll a small particle around the edge, starting from the top at point _A_ with a velocity $v_o$. It travels halfway around the sphere and reaches point _B_ , which is a vertical distance _h_ below _A_ , with a velocity $v_f$. Point _A_ is a radial distance of $r_o$ from the vertical centerline and point _B_ is a radial distance of $r$ from the vertical centerline. There is no friction. The goal is to solve for the angle, $\theta$, between the horizontal and the velocity $v_f$. Here is a diagram of the problem scenario: ![diagram of sphere](http://i.imgur.com/57qgEHI.png) My solution relies on the assumption that angular momentum only relies on the velocities in the plane perpendicular to the vertical centerline. Is that a safe assumption? Also, when dealing with energies, is rotational KE and linear KE the same? Should I be taking RKE into account? * * * $$ L_o=L_f $$ $$ mr_ov_o=mrv_f\cos \theta $$ $$ \theta = \arccos(\dfrac {mr_ov_o}{mrv_f}) = \arccos(\dfrac {r_ov_o}{rv_f}) $$ * * * $$ KE_o + PE_o = KE_f $$ $$ \frac 12 mv_o^2 + mgh = \frac 12 mv_f^2 $$ $$ v_o^2 + 2gh = v_f^2 $$ $$ \sqrt {v_o^2 + 2gh} = v_f $$ * * * $$ \theta = \arccos(\dfrac {r_ov_o}{rv_f}) = \arccos(\dfrac {r_ov_o}{r\sqrt {v_o^2 + 2gh}}) $$


title: Question regarding FTL information transfer between two inertial frames

text: Why can’t information travel faster than the speed of light, if the two endpoints to and from which the information is being sent are moving relatively to each other, as long as the information travels slower than the inverse of the velocity which the planets are moving apart at. According to the velocity addition formula: w=(u−v)/(1−uv/c2), as long as u < 1/v, then w is positive and the laws of causality are not violated. Is there a reason then that information cannot travel faster than the speed of light between two points moving relatively to each other. Also, couldn’t information travel an infinite number times the speed of light, if two points were not moving relatively to each other.


title: The Hole Argument

text: I have read explanations of this but haven't really understood. Given a spacetime $(M,g)$ I have read that if I represent the metric in some coordinates $(x,y,z,t)$ as $g(x,y,z,t)$ and then in another coordinate system as $g'(x',y',z',t'),$ that $g'(x,y,z,t)$ (now using the old coordinates) will also solve the Einstein equations. Now $g$ and $g'$ are two different metrics on the manifold and so _should_ predict different physics, but somehow they are the same? If I measure the distance between two points on the manifold I should get different answers if the metrics are different, shouldn't I? This is the part I don't really understand. Is it something like $(M,g')$ isn't a solution, but only $(M',g')$ where $M'$ is diffeomorphic to $M$? To give an explicit example, let's look at the Schwarzschild metric: \begin{equation} ds^2=-\left(1-\frac{2GM}{c^2 r}\right)c^2dt^2+\left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2+r^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \;. \end{equation} Apparently (according to http://faculty.luther.edu/~macdonal/HoleArgument.pdf) \begin{equation} ds^2=-\left(1-\frac{2GM}{c^2 f(r)}\right)c^2dt^2+\left(1-\frac{2GM}{c^2 f(r)}\right)^{-1}f'(r)^2dr^2+f(r)^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \; \end{equation} is also a solution for any diffeomorphism $f$ though distances aren't the same. Of course when deriving the Schwarzschild metric it seems (according to Carroll's book) $(r,\theta,\phi,t)$ are just symbols that you only interpret after you find the metric, which is confusing. Note I didn't do a change of coordinates here, I changed the actual metric. Are they really both solutions on the same manifold?


title: Can non-free forces change the rest mass?

text: While reading Hobsen et al.'s "General Relativity: An Introduction for Physicists", I came across a bit confusing derivation. Multiplying the 4-force and 4-velocity, the following derivation can be made $ \boldsymbol{u} \cdot \boldsymbol{f} = \boldsymbol{u} \cdot {d\boldsymbol{p} \over d\tau} = \boldsymbol{u} \cdot ({dm_0 \over d\tau}\boldsymbol{u} + m_0{d\boldsymbol{u} \over d\tau}) = c^2 {dm_0 \over d\tau} + m_0 \boldsymbol{u} \cdot {d\boldsymbol{u} \over d\tau} = c^2 {dm_0 \over d\tau} $ After this derivation, the authors make the following conclusion: > where we have (twice) used the fact that $\boldsymbol{u} \cdot > \boldsymbol{u} = c^2$. Thus, we see that in special relativity the action of > a force can alter the rest mass fo a particle! A force that preserves the > rest mass is called a pure force and must satisfy $\boldsymbol{u} \cdot > \boldsymbol{f} = 0$ But I have the following objections and questions about this derivation:   1. The rest mass is by definition a constant, so it should have been considered a constant while differentiating.   2. If we go back to Newton's second law, which is still valid under the special theory of relativity (though with some correction), the mass is the resistance of a body to changes in velocity, i.e. the larger the mass is, the stronger the force we need to change its velocity. But a non-free force seems to contradict this basic concept when $dm_0 \over d\tau$ is negative, because this means that the force is reducing the resistance of the body towards the force. As a funny comparison, imagine that the harder you push a heavy box, the lighter it becomes (which is obviously not the case even in Newtonian mechanics, not to mention that special relativity predicts the opposite, i.e. the faster the body is, the harder it becomes to increase its velocity)!!    3. Unless the mass is being converted to energy or transferred somewhere else (which is not inferred from the derivation, as the derivation comes straightforward from the force equation without depending on any other equation), where is the mass going?! Isn't this contradictory to the conservation of mass an energy law?   4. If we assumed in this derivation that the rest mass is variable, why didn't we do so in many other derivations in the special theory of relativity?   5. Do we have examples of such forces anyway? :-)


title: What force is acting on the charge in the dielectric?

text: For example I have a dielectric solid with a small charged ball in it. And I have external electric field $E$. So what force is acting on this ball? The field in dielectric is $\frac{E}{\epsilon}$, so the force should be $\frac{Eq}{\epsilon}$. On the other hand. If I remove a small piece of dielectric then the field in the hole will be $E$. Now I put charge in this hole and the force is $Eq$. The hole is just for charged ball and there is no free space. So what is the force?


title: How does water carve rock?

text: _How does water carve rock?_ _And more generally, how does a soft material carve a hard material?_ Obviously it happens, but is it a continual process (every drop of soft water carries away a minute amount of hard material) or is it a stochastic thing (every once in awhile, the force of the water just happens to exceed what's necessary to snap off a grain of sand)? Or is it something else (the erosion is actually caused by hard pieces of grit carried in the water)? UPDATE: _Grit certainly plays a major role in actual weathering, but is it **necessary**? Would pure water ever erode rock?_


title: What is the current physical theory of what causes the 'Morning Glory' roll cloud over Northern Australia?

text: The Morning Glory roll cloud (pictured below): ![enter image description here](http://i.stack.imgur.com/I5c0J.jpg) from this Wikipedia page, which briefly describes the landform and meteorological causes, but states > The Morning Glory cloud is not clearly understood because their rarity means > they have little significance in terms of rainfall or climate is there any definitive physical theory as to how these meteorological phenomena form?


title: Length of orbit on Bloch sphere

text: I'm not sure whether this question is well defined, but I am interested in the volume of the (obviously not linear) subspace of qubits $$ \left|\Psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta\, e^{i\varphi} \left|\downarrow\right\rangle$$ where $\varphi \in [0, 2\pi)$ and $\alpha$ and $\beta$ are fixed. Looking at the Bloch sphere representation $$ \left|\Psi\right\rangle = \cos\left(\tfrac{\theta}{2}\right) \left|\uparrow\right\rangle + \sin\left(\tfrac{\theta}{2}\right) e^{i\varphi} \left|\downarrow\right\rangle$$ shows that this is a circle with $\theta=\mathrm{const}$ and radius $\sin\theta$, so the size of the space is just the circumference of this circle, $2\pi \sin\theta$. On the other hand, the state only differs by an overall phase from, and is therefore equivalent to $$ \left|\Psi\right\rangle = \beta\, \left|\downarrow\right\rangle + e^{-\varphi}\alpha \left|\uparrow\right\rangle$$ so this leads to another Bloch sphere representation $$ \left|\Psi\right\rangle = \cos\left(\tfrac{\theta'}{2}\right) \left|\downarrow\right\rangle + \sin\left(\tfrac{\theta'}{2}\right) e^{-i\varphi} \left|\uparrow\right\rangle$$ where $\cos\left(\tfrac{\theta'}{2}\right) = \sin\left(\tfrac{\theta}{2}\right) $ which now leads me to think that this family of states form a circle of circumference $2\pi\cos\theta$. Is this whole line of reasoning flawed from the beginning?


title: Proper time along path in Minkowski Space

text: Consider the path $x^\mu(u)$ in Minkowski space; such that: $$t = \frac{a}{c} \sinh(u) , \quad x = a \cosh(u) ,\quad y = 0 ,\quad z = 0 $$ where $a$ is a positive constant and $u$ is a parameter Use equation: $$ c \nabla \tau = ds = \sqrt{\eta_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} du $$ to find the proper time elapsed along the path starting from $u = 0$, as a function of $u$. I'm having quite a bit of trouble with using the notation if someone could help me please? So far I have established: $$ d_u x^{\mu} = \left(\frac {a}{c}\cosh(u),\, a\sinh(u),\,0,\,0\right)) $$ Then: $$\eta_{\mu\nu} d_u x^{\mu} d_u x^{\nu} = \sum_{\mu =0}^{3} \sum_{\nu =0}^3 \eta_{\mu\nu} d_u x^{\mu} d_u x^{\nu} $$ Then I get a bit lost...


title: What is the symmetry difference between simple-cube and body-centered-cube structures

text: If the lattice types are categorized according to the point group symmetries, then what is the difference, for example, between sc and bcc structures?


title: Does sound propagate further in freezing weather?

text: A few days ago I went for a walk in the evening. We're having winter with a little snow and freezing temperatures. We're in a quiet, shallow valley with a train station about 1km from us. I heard a train coming so I wanted to wait for it to watch it arriving to the station. To my surprise, although I was hearing the sound coming from behind a hill, the train wasn't appearing. After several minutes, I gave up, and went back, and finally I saw the train arriving after another a few minutes. The train must have been several km away when I first heard it. I watched this phenomenon later and I realized that also I could hear and understand people talking on much larger distances than usual. This has not happened before, and my only idea is that it's because of the cold weather. I have two ideas how to explain it:   1. Cold air propagates sound better for some reason.   2. We have a few cm of snow covered by ice crust, as we had freezing rain a few days ago. I guess this can mean sounds aren't absorbed by earth and are reflected instead, which makes them propagate further. (I'd say this is more probable than 1.) Is any of this reasonable, or is there another explanation? (I'm not a native speaker so please feel free to correct any language errors.)


title: What does the term 'equation of motion' refer to?

text: What does the term equation of motion refer to? If I am asked a question of the form 'What is the equation of motion of this object?', what should I write?


title: Why does a salt solution conduct electrical current?

text: How does e.g. sodium chloride (aq) conduct electricity? By accepting electrons (unlikely since they already have a full outer shell)? But they can't be hopping around themselves, can they? I mean, if I have two poles made of metals inserted into a beaker with this solution, and I try to let a current go through from the pole through the solution to the other pole, the ion itself moving can't be conducting electricity, right? The metal pole won't accept such an ion instead of an electron...