Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: 1-Are matrices and second rank tensors the same thing? 2-If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)?


title: Gauss' law - changes in the magnitude of E field inside the closed surface

text: Gauss's law says that the flux through a closed surface which contains neither a sink nor a source will be zero. It's quite clear that all field lines will have to exit somehow, but the strength of the E-field is also proportional to the inverse of the distance squared. So if, for example, we have a cube, and the E field is perpendicular to one of the sides, the electric flux through that one side will be $A * E$ = $A$ * $kq \over r^2$. But on the opposite side, the distance from the source of the E field will be larger, so the magnitude of the E field should be smaller. Where is my misconception? Thank you. EDIT: Okay, the point charge was just an example. I'll ask it differently:   All the proofs I've seen of this concept state that "all field lines that enter the closed surface must also leave the closed surface, hence the total flux will be zero".   But how does this account for the differences in the distances of the sides of the closed surface from the source of the charge? Can someone refer me to a proof or give an explanation of why the differences in distances always balance out with the differences in area in order to give you a zero result?


title: Jacobi's matric formulation for tensors

text: Hi I am trying to derive the E field equation and am stuck using the Jacobi formula, is this correct: $$\delta \det g_{\mu \nu} = Tr(ADJ(g_{\mu\nu}\delta g_{\mu\nu})=\det(g_{\mu\nu})Tr(g^{\mu\nu}\delta g_{\mu \nu}) $$ Then how can we remove the trace, or should it be: $$\delta \det g_{\mu \nu} =\det(g_{\mu\nu})Tr(g^{-1}\delta g)=\det(g_{\mu\nu})(g\delta g)^{\mu}_{\mu}=\det(g_{\mu\nu})(g\delta g)^{\mu}_{\mu}=\det(g_{\mu\nu})(g^{\mu\nu}\delta g_{\mu\nu}) $$


title: Mathematics and Physics prerequisites for mirror symmetry

text: I am a physics undergrad interested in Mathematical Physics. I am more interested in the mathematical side of things, and interested to solve problems in mathematics using Physics. My current knowledge is some QFT(beginnings of QED), no string theory, differential geometry confined to riemannian manifolds, and some knowledge on Riemann surfaces . One such area is Mirror Symmetry. What are the QFT and string theory prerequisites, and also how much algebraic geometry should I know (is this confined to complex manifolds)? Also books and references which build up this background, and also directly on mirror symmetry would be appreciated.


title: Orbits within a $-\vec{r}$ field

text: Let's say that we have a cold dark matter theory, so we imagine weakly interacting particles. Now, let's say that one of those dark-matter particles has a rare interaction while traveling through the Earth's mantle. Let's also imagine that this interaction causes the particle to lose most of its energy in Earth's reference frame (also rare, but nonetheless realistic), such that it's not energetically possible for it to rise above the surface. Since interactions are fairly improbable it also can't stop moving. Is it then correct to imagine that it will "orbit" within the center of the Earth for a while? If we put the Earth's center at the origin and assume constant density, as I understand, the field should be: $$ \vec{F} = - \frac{4}{3} G \rho \pi \vec{r}$$ I've read from several sources about the consequences of gravity being $1/r^p$ with $p \ne 2$, that is, not inverse-square. What would happen in this case where $p=-1$? Will the orbit be periodic? Or will its trajectory over time just be a big knot?


title: Why Do Sausages Always Split Lengthwise?

text: Sausages universally split parallel to the length of the sausage. Why is that? ![enter image description here](http://i.stack.imgur.com/9vLa7.jpg)


title: Applying conservation of energy to any system?

text: How would the conservation of energy be applied in any system? Can we always be certain that every system will always obey COE? How can we be certain without measuring the system? Input & Output? Shouldn't we analyze all the variables involved? And calculate all the Work done in the system?


title: String total cross sections at asymptotically high energy

text: I only have a vague understanding of string theory, but a solid understanding of particle physics. At asymptotically high energy (Regge limit), the string cross section is dominated by the exchange of the leading trajectory -- the graviton trajectory, corresponding to an intercept $\alpha(0)=2$. Regge phenomenology dictates that the string-string total cross should rise linearly with energy: $$\sigma_\text{tot} \sim s^{\alpha(0)-1} = s$$ which is startling to me. Given that gravitons are massless, they are not subject to the Froissart-Martin bound $\sigma_\text{tot}<\log(s)^2$. So, is it really true that string-string total cross sections exhibit this growth?


title: Two charges in a medium where different permittivity is adjacent

text: I was thinking about the following: Assuming that you have NON SPHERICALLY SYMMETRIC charges and they are both in medium 1. Does it make a difference(whereby I am referring to the total force) if there is a second medium 2 in the local area of both, such that both charges are NOT in the second medium, but very close to it to the case where only medium 1 is around them?


title: The Faint young Sun paradox: an alternative solution?

text: The faint young Sun paradox describes the contradiction between the fact that there must have been liquid water 3.5 billion years ago for life to have begun then and the fact that the Sun's power output would only have been 70% as intense as it is now. The issue was raised by astronomers Carl Sagan and George Mullen in 1972. A recent explanation is that there must have been a lot of carbon dioxide (20,000 ppm) and methane (1000 ppm) in the early atmosphere to produce enough greenhouse effect at that early time to raise the Earth's temperature above water freezing temperature. However, perhaps the solution lies in the fact that the Earth was nearer to the Sun 3.5 billion years ago and therefore received enough solar energy to warm its surface above water freezing temperature even though the Sun's power output was 70% of what it is today. There is evidence from radiometric measurements that the Earth-Sun distance is increasing by about 15 meters per year. This is in fact approximately the rate of increase one would expect if the solar system is expanding in the same manner as the Universe at large. A rough calculation of the present temperature of the Earth The solar flux density, $F$, going into the Earth's climate system is: $$F = \frac{S}{4} (1 - A),$$ where $S$ is the solar constant and $A$ is the albedo. The solar constant $S$ is the power per unit area at the Earth's current mean distance from the Sun. The albedo $A$ is the fraction of solar power that gets reflected back into space. The factor of 4 comes from the fact that the Earth receives solar energy over its cross-sectional area but that energy is delivered to the climate over its surface area. We take the value $S=1366\ $Watts/m$^2$ for the current solar constant at our present distance from the Sun. According to NASA the current albedo of the Earth is $A=0.306$. Therefore we find that the current flux density $F$ is given by: $$F = \frac{1366}{4}\left(1 - 0.306 \right) = 237\ \mbox{Watts/m}^2.$$ Now, by conservation of energy, the Earth must radiate out as much energy as it receives (in the long run) so that by inverting the Stefan-Boltzmann law we have: $$T_e = \left(\frac{F}{\sigma}\right)^{1/4},$$ where the Stefan-Boltzmann constant $\sigma=5.6704\times 10^{-8}$W m$^{-2}$ K$^{-4}$. Plugging in our value for $F$ we find the Earth's current temperature is predicted to be: $$T_e = 254\ K = -19\ C.$$ In fact the current surface temperature of the Earth is about $288$ K or $15$ C. The extra $34$ K is due to the greenhouse effect created largely by water vapor and carbon dioxide in the Earth's atmosphere. We have good evidence that the first life probably started in bodies of water about 3.5 billion years ago. The standard model of the Sun indicates that it was 30% less bright then as it is now. Let us calculate the temperature of the surface of the Earth at that time assuming no greenhouse effect. The solar constant $S$ would then be $S=0.7 \times 1366=956$ Watts/m$^2.$ Assuming the same albedo $A$ as we have at present we calculate the solar flux density $F$ going into the Earth's climate as $F = 166$ Watts/m$^2.$ Using the Stefan-Boltzmann law we find the surface temperature of the Earth, without greenhouse effect, would have been: $$T_e = 233\ K = -40\ C.$$ Thus under the assumption that the Earth-Sun distance does not change we would require a hefty greenhouse effect 3.5 billion years ago to ensure that there was an appreciable amount of liquid water on the surface of the Earth to allow life to develop. Finally, let us assume an alternate hypothesis that the Earth was nearer to the Sun 3.5 billion years ago. As mention above there seems to be some experimental evidence that the Earth is currently receding from the Sun at a rate of $15$ meters per year. At present the semi-major axis of the Earth's orbit is $a_{now}=1.49598\times 10^{11}$ meters. If we extrapolate back using the current rate of Earth recession we find that 3.5 billion years ago the semi-major axis would have been $a_{past}=9.7098\times 10^{10}$ meters. Therefore if the Earth was closer to the Sun the intensity of solar flux density, $S$, hitting the Earth would be increased over the present value by a factor $f$: $$f = \left(\frac{1.49598\times 10^{11}}{9.7098\times 10^{10}}\right)^2 = 2.37.$$ Therefore applying this factor as well as the 30% reduction in the early Sun's intensity we find that 3.5 billion years ago the solar flux density would be: $$S = 2.37 \times 0.7 \times 1366 = 2266\ \mbox{Watts/m}^2.$$ Assuming the same albedo $A$ as we have at present we calculate the solar flux density $F$ going into the Earth's climate as $F = 393$ Watts/m$^2.$ Using the Stefan-Boltzmann law we find the surface temperature of the Earth, without greenhouse effect, would have been: $$T_e = 288\ K = 15\ C.$$ By coincidence that is the same as the present surface temperature of the Earth but without greenhouse warming. Certainly warm enough to sustain early life in liquid water.


title: Finding the odd parity bound state wave function for a particle in one dimension

text: Let a particle of mass $m$ and energy $E$ be moving on the $x$-axis in a potential $V(x)$ given by $$V(x)=\left\\{\begin{matrix} -V_{0}, & -a<x<a\\\ 0, & otherwise \end{matrix}\right.$$ where $a>0, V_{0}>0$ are constants and energies $E$ satisfying $0<-E<V_{0}$. I'm then asked to show that the _odd parity bound state_ wave functions have energies $E$ which satisfy $$\eta^{2}+\xi^{2}=\frac{2mV_{0}a^{2}}{\hbar^{2}}, \eta=-\xi cot(\xi)$$ where $\eta=\sqrt{-2mEa^{2}/\hbar^{2}}$ and $\xi=\sqrt{2m(V_{0}+E)a^{2}/\hbar^{2}}$ So, I think I'm nearly there but I have a feeling I might be misunderstanding exactly how to get to the _odd parity bound state_. My understanding of "odd parity" is that $\psi(x)=-\psi(-x)$ as stated here Definite Parity of Solutions to a Schrodinger Equation with even Potential? and my understanding of "bound state" is just that the wave function is normalizable (i.e. $\int_{-\infty}^{\infty}|\psi(x)|^{2}dx<\infty$). My working so far gives: For $-a<x<a$: $$\frac{-\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}-V_{0}\psi=E\psi \\\ \Rightarrow \frac{d^{2}\psi}{dx^{2}}=\frac{-2m(E+V_{0})}{\hbar^{2}}\psi \\\ \Rightarrow \psi=Acos(\lambda x)+Bsin(\lambda x), \lambda=\sqrt{2m(E+V_{0})}/\hbar$$ Otherwise: $$\frac{-\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}=E\psi \\\ \Rightarrow \frac{d^{2}\psi}{dx^{2}}=\frac{-2mE}{\hbar^{2}}\psi \\\ \Rightarrow \psi=Cexp(\mu x)+Dexp(-\mu x), \mu=\sqrt{-2mE}/\hbar$$ I then used the fact that both the wave function and its derivative must be continuous to get: $$Acos(\lambda a)+Bsin(\lambda a)=Cexp(\mu a)+Dexp(-\mu a)\\\ Acos(\lambda a)-Bsin(\lambda a)=Cexp(-\mu a)+Dexp(\mu a) \\\ -A \lambda sin(\lambda a)+B \lambda cos(\lambda a)=C \mu exp(\mu a) - D \mu exp(-\mu a) \\\ -A \lambda sin(\lambda a)+B \lambda cos(\lambda a)=C\mu exp(-\mu a) - D \mu exp(\mu a)$$ Right, now this is where I run into difficulty. Normally, I'd try and use this system of equations to get rid of a some of the constants. But that doesn't seem to work. I feel like I'm supposed to impose the fact that I want $\psi$ to be normalizable and have odd parity. I thought maybe I could just assume that it was and then work from there? But that seemed quite an arbitrary assumption to make. But then I thought it might not be so arbitrary as the solutions to the Schrodinger equation are elements of the Hilbert space and could we just be looking for the specific one that has odd parity? Thanks in advance for your help!


title: Parseval's Theorem on a Random Signal

text: NB - I'm re-posting this question in physics because I haven't had any luck getting a response from the maths StackExchange site - it's a rather applied problem so is probably better suited here anyway. ** I'm struggling with Parseval's Theorem. I'm trying to relate variation in the time domain to the average value in the frequency domain. To do this, I'm performing the Fourier Transform on an arbitary random signal that I've generated with 2048 points (although the graph below only shows 100 of them) and a standard distribution (in this particular case) of 0.58: ![Time domain trace of random signal - NB only showing 100 of 2048 points!](http://i.stack.imgur.com/O0zlF.png) The frequency domain, after an FFT, then looks like this: ![Frequency domain trace of random signal](http://i.stack.imgur.com/coWig.png) with an average value of 0.022 - although this depends on the number of samples used. Now when I try to apply Parseval's Theorem, where: $\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N}\sum_{n=0}^{N-1} |X[k]|^2 $ I run into a problem. When I do the summations, I get 678 for the time domain and 0.662 for the frequency domain. When I apply the factor of $1/N$ I end up with 0.0003 for the frequency side - clearly a long way off! Obviously I'm going wrong somewhere, but can't see where. This many orders of magnitudes off isn't very encouraging... Thanks for your help!


title: Dielectric with polar molecules

text: Suppose a dielectric slab contains polar molecules (which are not further polarisable). When placed in an electric field, (for simplicity, an uniform field), align themselves according to the field.   * Do they perfectly align so that all molecules have no net torque on them, or it is only a general arrangement which is less randomized and slightly prefers to have a net dipole moment according to the field?    * What effects in a dielectric will determine the extent of alignment?  There was a very poorly executed attempt to explain this with the thermal energy of the molecules i.e. $1/2k_b T$ at a temperature $T$. Can someone explain how the strength of the field and the thermal energy interact to determine the extent of alignment?


title: Numerical schemes, time integration algorithms and energy conservation

text: What does it mean when someone says a numerical scheme or a time integration algorithm is "energy conserving". How can a numerical scheme "gain" or "lose" or "conserve" energy apart from the numerical diffusion that is inherent.


title: (Exercise review) Find the moment of the electric dipole

text: I have solved an exercise, but the result I have obtined is wrong and I can't understand why. If you can help me, I'll be so grateful. Let's consider a charged cylinder, its radius is equal to R and its height=4R. Its center of is in the origin of the cartesian axes and its axis is parallel to z axis; we know the volumic density of charge: $\rho= 2az \epsilon$ where $a, \epsilon$ constant $>0$. There is an electric field inner to the cylinder, described by: $E_{ox}=0, E_{0y}=0, E_{0z}=az^2$ We want to find the electric potential to a great distance from the origin of the axes. I have thought as follow. Using Maxwell's first equation, I have found the volumic density of charge: $\rho=2\epsilon_0 a z$. At a very big distance to the origin, I can consider the cylinder like an electric dipole, with the positive charge concetrated in the center uf the upper part of the cylinder and the negative one in the center of the lower part: So $Q_+$ will be in (0,0,R) and $Q_-$ in (0,0,-R). Now, I need the moment of dipole, ${\bf p}=q{\bf a}$ where ${\bf a}$ is the vector from the negative charge to the positive charge. I have integrate $\rho$ and I have obtained $q=\int_0^{2R}\int_0^R\int_0^{2\pi}2az\epsilon r d\theta dr dz$ and I have obtained $q=4\pi a \epsilon_0 R^5$ Then I have posed a=2R, so I have obtained $p=8\pi a \epsilon_0 R^4$.The correct result is $p=32/3 \pi a \epsilon_0 R^5$ I can't understand where I was wrong and I wouldn't like to have made an error of proceeding.


title: What is the equation for Lin-Shu density wave theory?

text: Lin-Shu density wave theory is a (very good) theory to explain the spiral arm structure of spiral galaxies. Where can I find the equations that underlie this theory? * * * I've seen its computer simulation, thats the way I found it a very good theory...etc


title: Black bodies, energy per unit volume

text: While reading about black bodies, I found that the average energy per mode per unit volume is $KT$ classically, but I don't understand why this is per unit volume. The linear oscillators are considering to be in the walls of a cavity modeling a blackbody, so shouldn't it be modes per area?


title: Could a mirror array used for thermal solar power plant double as a telescope at night?

text: I was looking at the Ivanpah Solar facility and it occurred to me that the large array of mirrors could double as some sort of telescope array at night. The climate in the desert would be ideal for this, and there would be no power generated at night anyway.


title: Surface tension between water (liquid) methane (gas)

text: In my project I should calculate surface tension between water and methane with lammps. I write the program.I use SPC model for water and $l_j$ for methane and water - methane. But now I don't know how I calculate the surface tension. Is it right I find $P_{xx}$, $P_{yy}$ and $P_{zz}$ , then finally using the equation $(P_{yy}+P_{zz})/2 - P_{xx}$? Would appreciate comments on this.


title: Resource(s) for developing a good understanding of surface tension?

text: I have read through several junior undergraduate level explanations of surface tension. Here is a typical presentation at that level: > Molecules at the surface of a fluid experience approximately half of the > force that molecules in the bulk of the fluid experience, thus putting > surface molecules in an energetically unfavourable position. Thus the liquid > tries to minimize surface area (in order to minimize the molecules at the > surface) and macroscopically this is evidenced by the presence of a "surface > tension" that resists deformation of the surface from its energetically > favourable conformation. There are several finer points that are not really explained in such an explanation:   * What really is the difference between the force felt by a molecule in the bulk, versus a molecule at the surface? Is it that a molecule at the surface experiences a net force pulling it towards the bulk, while a molecule in the bulk feels zero net force?    * How can such an intuition be properly connected with an intuition of "energetically favourable"? How should one develop an understanding of "energetically favourable"?    * What is the nature of these intermolecular forces?  Simplistic explanations also leave the reader without enough of an understanding to definitively answer questions like:   * Do elastic solids return to their original shape after deformation due to similar reasons?   * What is the general mathematical framework that connects behaviours all the way from elastic solids, to capillarity in fluids?   * Does the Earth's atmosphere have a surface tension?  **Can you recommend resources that can help one develop a well-grounded understanding of surface tension?**


title: Camp Fire Crackle

text: I was sitting around the camp fire and there was a periodic sound of cracking and pieces of the burning wood flew off. My question is what is the mechanism for the cracking of the wood? I was thinking it was a pocket of air in the wood or something, but that still doesn't explain why the piece of wood flew well over a meter high. Could this be a chemical reaction causing this?


title: Newtonian gravity vs. general relativity: exactly how wrong is Newton?

text: Is there a simple function I can use to describe the difference between simple Newtonian dynamics and the actual observed motion? Or maybe some ratios for common examples of, say, the motion of stars and planets? I know for example that Newtonian physics is sufficient to fire a rocket at the moon, so the error there must be miniscule. What exactly is that ratio, and could a layperson calculate it generally for other bodies? The context is this: in a debate regarding how drastically wrong science can be, I want to make the case that even though Newton was wrong regarding gravity in the bigger picture, he was only ___ % wrong regarding what he'd observed. I know science can often be wrong, but I want to emphasize the fact that our system of observation isn't fundamentally dysfunctional.


title: Two impulses on body

text: Say I have a physics simulation with a single body with three degrees of freedom (two axis and rotation). I have two impulses of equal magnitude in opposite directions. When applying these impulses in the same location, the body should not move in any way, the same if I apply them in spots that aligned in the same direction as the impulses. If I apply them outside of this axis, the object would gain momentum of some sort. In some situations there would be a balance, in others a combination. What is the right way of calculating the actual gain in momentum for the object? Calculating one impulse after the other would not (by what logic I use) create the right solution, as both impulses might add momentum in the same direction even if their torques might cancel one-another out.


title: Faster than critical damping for harmonic oscillator?

text: The image below shows damping for spring oscillator with Hooke law F=-kx and damped with F=-cv where: k is spring constant x is oscillator position c is damping coefficient v is velocity of oscillator Critical damping happens when $\zeta = { c \over 2 \sqrt{m k} } = 1$ where m is mass of oscillator The question is, is there another method for damping function to get to rested position faster than critical damping of F=-cv?![enter image description here](http://i.stack.imgur.com/DJvOi.jpg)


title: Proving two forms of atom-field interaction perturbation Hamiltonian are equivalent

text: In the presence of an electromagnetic field in the dipole-approximation (${\boldsymbol A} = {\boldsymbol A}(0,t)$) we have the two forms $$H_{{\boldsymbol d}\cdot {\boldsymbol E}} = - q {\boldsymbol r}\cdot {\boldsymbol E}(t)\,\,\,\,\, ,\,\,\,\, H_{{\boldsymbol p}\cdot {\boldsymbol A}} = - \frac{q}{m} {\boldsymbol p} \cdot {\boldsymbol A}(t) $$ The question asks to show that the first-order time dependent transition probabilities from some $t \to \pm \infty$ are in fact equal for the two choices of the interaction. For a refresher, I'm working with Schiff's notation of the perturbation theory where we are given $$ a_k^{(1)} = \frac{1}{i \hbar} \int\limits_{-\infty}^{\infty} \langle k | H'(t) | n \rangle e^{i \omega_{kn} t'} dt'$$ (The first part of this question asks to prove $\langle k | {\boldsymbol p} | n \rangle = i m \omega_{kn} \langle k | {\boldsymbol r}| n \rangle $ by using Heisenberg's equation of motion and the atomic Hamiltonian $H = \frac{{\boldsymbol p}^2}{2 m} + V({\boldsymbol r})$ ) which is easily done using commutation relations. The only other information we are given is that ${\boldsymbol A}(t)$ is a smooth function on the boundaries of $t \to \infty$. I've tried a few things by using the relation from the first part in the form $\langle k | {\boldsymbol p}\cdot {\boldsymbol A} | n \rangle = i m \omega_{kn} \langle k | {\boldsymbol r}\cdot {\boldsymbol A} | n \rangle $ and the fact that ${\boldsymbol E} \approx -\frac{\partial {\boldsymbol A}}{\partial t}$ in our dipole-approximation but things are not reducing to what I want them to. Thanks


title: Newton's principle of determinacy

text: I am a mathematician. I have a (somewhat long term) goal of understanding some of the physical insights that have influenced my area of research. To this end I read Arnold's Mathematical methods in classical mechanics a while ago, but something I didn't understand has been bugging me ever since. In the first chapter Arnold defines a motion of $n$ particles in $\mathbb{R}^3$ as a map $\mathbf{x}:\mathbb{R} \rightarrow \mathbb{R}^N$ for $N=3n$. The first chapter is then about what types of motion are allowed. In section 2D Arnold makes the following observation: > According to Newton's principle of determinacy all motions of a system are > uniquely determined by their initial positions ($\mathbf{x}(t_0) \in > \mathbb{R}^N$) and initial velocities ($\mathbf{\dot{x}}(t_0) \in > \mathbb{R}^N$). This seems important and I expect that this would have an impact on what types of functions $\mathbf{x}$ could be. He goes on: > In particular, the initial positions and velocities determine the > acceleration. In other words, there is a function $\mathbf{F} : \mathbb{R}^N > \times \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}^N$ such that $$ > \mathbf{\ddot{x}} = \mathbf{F}(\mathbf{x},\mathbf{\dot{x}},t). $$ So the implication of Newton's determinacy principle is that the acceleration obeys a second order differential equation. This seems completely vacuous to me. Any function $\mathbf{x}$ obeys a second order differential equation (as long as it is twice differentiable). Could someone please explain to me what Arnold is saying here. I feel like I am missing something important.


I have been doing some research on nuclear changes and I have found that the energy released during a nuclear change comes from a minuscule amount of mass that is converted in to energy. After making some more research it seemed completely logical because of Einstein’s famous energy equation, but I was left with a couple of questions roaming my mind.   1. Why does mass change in to energy during a nuclear change?   2. What prevents matter from changing to energy and vice versa in a normal state?


title: Temperature in an isobaric process

text: I have a certain conceptual issue - I'm solving heat engine problems and I found something difficult to understand. Let's take an isobaric part of a cycle of an engine, let's say that the ideal gas is compressed from $2V$ to $V$. How does the temperature change? From the Clapeyron equation in the earlier point it is $T_2 = 2Vp/nR$ and in the latter point $T_1 = Vp/nR$ (it's labeled "1", because the cycle closes there). This would mean that compressed gas has lower temperature that the uncompressed one? This is false isn't it? Could someone please explain this to me?


title: Does entropy apply to Newton's First Law or does "acted upon" always require an external factor?

text: > First law: Every body remains in a state of rest or uniform motion (constant > velocity) unless it is acted upon by an external unbalanced force. This > means that in the absence of a non-zero net force, the center of mass of a > body either remains at rest, or moves at a constant speed in a straight > line.   * Wikipedia -- Newton's laws of motion Doesn't the law of increasing entropy affect all objects though, since they are all in the closed system of the universe at large, and therefore they are all subject to slowing down, regardless of the containing medium, given enough time? I guess what I'm curious is, can there ever be a body that will remain at uniform motion or uniform rest given that entropy must increase?


title: How does the Lennard Jones Potential changes for interaction between particles of different sizes?

text: I am interested in incorporating a Lennard-Jones potential in a simulation. When the interaction only involves the same type of particle, with same characteristics, we can use reduced units, scaling the units according to the potential well. But I assume the potential well is not the same for a case with particles of different sizes. Neither the sigma parameter should remain the same in my opinion. How can I propose the units or incorporate such feature to my code? The question can be answered theoretically, but it would be nice to have examples with real code (in Fortran, Matlab, C, or whatever language you have experience with).


title: Why does the induced EMF oppose the change in magnetic flux? Lenz's Law question

text: Can anyone explain to me _why_ the induced magnetic field will oppose the change in magnetic flux? Is it an energy thing? I know that the induced emf is $$ emf= - \frac{d\phi}{dt} $$ but my book doesn't give a satisfactory (or any) explanation as to why this is the case. I recognize that an emf source supplies energy and does work on charge carriers, but I dont see what that has to do with the magnetic flux.


title: Property of stress-tensor in flat spaces

text: Let $T_{ab}$ be a stress-tensor in a flat space satisfying conservation equations. Define $$ P^i=\int T^{oi}d^3x, \;\; D^i=\int T^{00}x^id^3x $$ Can anyone show me how to prove $$ \frac{dD^i}{dt}=P^i $$ and $$ \frac{dP^i}{dt}=0~? $$ * * * Update: Following is my try: $$ \frac{dD^i}{dt}=\int\frac{d(T^{00}x^i)}{dt}d^3x=\int\frac{\partial T^{00}}{\partial t}x^id^3x $$ By conservation law we have $$ \frac{\partial T^{00}}{\partial t}+\frac{\partial T^{01}}{\partial x}+\frac{\partial T^{02}}{\partial y}+\frac{\partial T^{03}}{\partial z}=0 $$ However, I don't know how to move on...


title: Looking for an Time Series Analysis Text

text: (Apologies, I could not think of a better title). Two days ago, I was in the Library at Köln (Those who know it, knows that the books are not ordered by subject - but by date of buying, and one has to do catalogue research). and I was casually browsing the racks for things that might interest me. Now I did find a book. All what I remember is, it was a book of Non-linear analysis of time series. The author, at the beginning of some chapter (either 3,4 or 5 - or may be I am wrong), argues that in contrast to the previous chapters where he presented interpolation (e.g. B-Splines) and nearest neighbor techniques, in the present chapter he will present techniques that also make assumptions on the _underlying_ dynamic process that generate the time series. The actual sentences were like, not that i remeber the actual words for which I could possibly search insides of amazon or google books, but it went along these lines: "it thus seems to be a good idea [...] to account for the underlying processes that generate the time series". However, afterwards, I found this interesting. So I wanted to read more, but I had to come back to Bonn - thus I wanted to check the book out, but I had some fines and all - thus I just kept the book there and thought I will be back later. Problem is, as I was rushing back to catch the train to Bonn, I forgot to write down the name and author of the book. Going through the search result returned by the catalogue does not seem to lead me to the book I was looking for - the book feels like as if vanished. Can anyone help be by telling me the name of a book where the author argues as above?


title: Expectation value of Hamiltonian in different pictures of quantum mechanics

text: We start with the familiar Schrodinger equation: $$ i\hbar \frac{\partial \left|\psi_S\right\rangle}{\partial t} = \hat{H}_S \left|\psi_S\right\rangle $$ As we switch to a different picture than Schrodinger picture with a unitary transformation $\hat U$: $$ \left|\psi_S\right\rangle = \hat{U}\left|\psi_P\right\rangle $$($S$ indicating Schrodinger picture and $P$ indicating the arbitrary picture) If we plug in $\left|\psi_S\right\rangle = \hat{U}\left|\psi_P\right\rangle$ into the Schrodinger equation, we obtain: $$ i\hbar \frac{\partial \left|\psi_P\right\rangle}{\partial t} = \hat{H}_P \left|\psi_P\right\rangle $$ where $$ \hat{H}_P = U^\dagger \hat{H_S} U - i\hbar U^\dagger \frac{\partial U}{\partial t} $$ is the Hamiltonian in this arbitrary picture. So, the question is - if Hamiltonian is an observable, shouldn't it have the same expectation values in both pictures - yet the second term in $\hat H_P$ makes them unequal. Because: $$ \left\langle\psi_P\right|\hat{H}_P\left|\psi_P\right\rangle = \left\langle\psi_P\right| U^\dagger \hat{H_S} U \left|\psi_P\right\rangle - i\hbar \left\langle\psi_P\right| U^\dagger \frac{\partial U}{\partial t}\left|\psi_P\right\rangle $$ this simplifies to: $$ \left\langle\psi_P\right|\hat{H}_P\left|\psi_P\right\rangle = \left\langle\psi_S\right| \hat{H_S} \left|\psi_S\right\rangle - i\hbar \left\langle\psi_P\right| U^\dagger \frac{\partial U}{\partial t}\left|\psi_P\right\rangle $$ telling us that the expectation values in two different pictures are not the same. I do not see a reason why that the last term should be zero. What is wrong here? Is Hamiltonian somewhat different from other observables?


title: Why does vaccum have a nonzero characteristic impedance towards electromagnetic radiation

text: OK, so the characteristic impedance is calculated "as square root of the ratio of the permeability of free space (µ o ) in henrys per meter ( H/m ) to the permittivity of free space ( o ) in farads per meter ( F/m )" according to the sources I have come across on the internet. This comes to 377 ohms (approximately). Now impedance is described as an impeding effect to flow of something, it makes more sense for electric current travelling in a wire where the characteristic impedance of the line (as the line consists of capacitance and inductance per unit length) prevents the flow of AC/DC flow. My question is, why in the world does free space have a characteristic impedance? That makes no sense to me. Wires makes a lot of sense, but free space having 377 ohm of impedance is too much and not clear why such a value exists.


title: Derivation of the Landauer formula for phonons using Nonequilibrium Green's functions

text: I am currently trying to understand this paper: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.96.255503 I really like their derivation of the Landauer formula for phonons using Nonequlibrium Green's functions (NEGF) because of its compactness. I am having trouble understanding it though since it has been abbreviated a lot. Can someone with a deeper knowledge of quantum physics and/or Green's functions help me understand? Especially I am wondering about how they get to equation number (3): $J_{th} = -\lim_{t'-t} \sum_{i\in L, j \in S, \alpha\beta=x,y,z} \frac{k_{i\alpha,j\beta}}{2} [\mathrm{i} \hbar \frac{d}{dt'} G_{i\alpha,j\beta}^<(t, t') +h.c.]$. How do they get the limit and the time derivation from $J_{th} = -\langle \dot H_L \rangle = -\langle [H_L, H_{sys}]\rangle$? I don't see that coming from the commutator. Any ideas? Thanks a lot in advance!


title: Can General Relativity Metric Tensor be independent of a particular co-ordinate index in a local area?

text: For example in a particular local area, can the metric tensor be totally independent of $z$ co-ordinate in $(t,x,y,z)$ co-ordinate system? This way the distance function will not contain $z$ co-ordinate. Probably this is same as asking if a diagonal element be zero in metric tensor? Or this leads to degeneracy of the metric?


title: Conservation in space-time curvature

text: Pardon this possibly naive question. I'm starting to poke around in the topic of General Relativity (as soon as I can pull myself back up out of the vortex of underlying mathematics that I've gotten sucked into) and started to wonder this: is there any sort of "conservation" law(s) associated with space-time curvature? Perhaps I'm stuck trying to visualize the effects of mass (or acceleration), so let me explain my question a bit more. If the observable universe is expanding from every observer's viewpoint, one model that supports this consists of the observable universe on the surface of an expanding "sphere." In trying to visualize the curvature of the surface of that "sphere," I began to wonder if the "inward" bulging of the "sphere" might not somehow need to be compensated for by a corresponding "outward" bulge elsewhere? Does this question make sense?


title: Max. speed that a space probe can communite with the Earth?

text: **Question** : _Which is the average or maximum speed of a telematically sent message that a space probe can send_? **Is as simple as radio waves travel at the speed of light in a vacuum?** **Context** : I have been asked to make a first estimation about how much time is needed to send a space probe and then receive the signal. I am supposing that the space probe is sent at $15 \ km/s$ but I have not idea of the telematic signal speed. Thank you in advance!


title: Definition of entropy

text: In my textbook there is written > $$\Delta S = \int_R \frac{\delta Q}{T}$$ > > where the $R$ means calculated along a reversible transformation. > > The variation of entropy only depends upon the initial and the final state, > and **thus has the same value regardless if the transformation was > reversible or not.** > > [...] > > We can then found the relation > > $$\Delta S = \int_R \frac{\delta Q}{T}$$ $$\Delta S > \int_I \frac{\delta > Q}{T}$$ I understood that the RHS in the second equation is just an integral and does not represent the variation of entropy whatsoever. It is then added that > in a close system $\delta Q = 0$ implies > > $$\Delta S \ge 0$$ The equal sign holds for a reversible process, the $>$ > for an irreversible one. Now, I don't understand this. $\Delta S = \int_R \frac{\delta Q}{T} = 0$. Why the difference between reversible and irreversible ones, while above there is clearly stated that the the value should be the same? Thanks in advance


title: Force acting on center of mass

text: ![enter image description here](http://i.stack.imgur.com/NLGyf.png) Pictured above is a biker (traveling in a curve) whose center of mass has two forces acting on it. These two forces - a horizontal centrifugal force F(centripetal)* and a vertical gravitational force mg - yield a resultant force which pulls the biker's center of mass diagonally towards the ground. Here is my question: why does the biker [who I've pictured as a black stripe] stay in place _if_ (and only if) he positions himself according to the resultant force as shown in the image? My instructor told me that the biker would begin to tilt left or right for any other configuration, and that _kind of_ makes intuitive sense, but I'd rather not base my knowledge on iffy intuition alone. *I have mistakenly named the centrifugal force as centripetal in the image EDIT: Could it be that the position of the biker as shown in the image is the _only_ one in which a "conversion" _from_ tilting to the left _to_ tilting to the right occurs (so that the system is balanced, in a sense)?


title: Quadratic versus other types of degrees of freedom

text: In my text book I have the following question and after thinking about this one for hours, I can't figure out what they're getting at: In Section 2.5* I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of such a system is proportional to $U^{Nf/2}$, where $Nf$ is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula $\Omega$ cannot be valid when the total energy is very small? * * * (*) = In section 2.5 of the text book, the formula for multiplicity $\Omega$ is given to be $$\Omega\left(U,V,N\right)=f\left(N\right)V^NU^{3N/2}.$$ with $f\left(N\right)$ some function of $N$. (This formula is for three degrees of freedom.) * * * What I got so far is: The initial expression (for $f$ degrees of freedom) is $$\Omega_N\approx\frac{1}{N!}\frac{V^N}{h^{3N}}\frac{\pi^{Nf/2}}{\frac{Nf}{2}!}\left(\sqrt{2mU}\right)^{Nf}.$$ Using $S=k\ln\Omega$ and solving $T=\left(\partial S/\partial U\right)^{-1}$ for $U$. the final answer for $U$ becomes $$U=\frac{1}{2}NfkT.$$ (Which is really just the equipartition theorem all over again???? Or did I make a mistake?) Now, I don't get why this formula can somehow tell me that my initial formula for multiplicity is not sufficient when the energy is small:   * Both $N$ and $k$ are constants.   * Although $f$ sort of "jumps" per the rule of Dulong and Petit, assuming that more degrees of freedom freeze out would seem baseless to me. Do I have to treat this as a constant as well?   * From this I conclude that $U$ is proportional to $T$, which really doesn't seem so problematic to me. How do I have to interpret the role of the quadratic vs. the non-quadratic degree of freedom here? What is the consequence?


title: If $S$ is a closed achronal set in a spacetime, any timelike curve starting at a point in $I^+[S]$ and ending at a point in $I^-[S]$ interset $S$?

text: Suppose $S$ is an achronal set in a spacetime $M$. And $S$ is closed. At the same time, any null geodesic of $M$ intersects $S$. Then, why does any timelike curve from $I^+[S]$ to $I^-[S]$ intersect $S$, too? I understand that, any point $r\in M$ belongs to either $S$, or $I^+[S]$ or $I^-[S]$. Because if $r\notin S$, and there is a past null geodesic $\gamma$ starting at $r$, then $\gamma$ must intersect $S$ at $q$. Pick any point $p\in\gamma$ and $p$ is in the causal future of $q$, then there is a second null geodesic $\eta$ from $p$ and intersecting $S$ at $s$. Therefore, we can find a timelike curve connecting $r$ to $s$ which implies that $r\in I^+[S]$. But unfortunately, I cannot figure out how to show any timelike curve from $I^+[S]$ to $I^-[S]$ intersects $S$. Would you give me some hint, please?


title: Accelerating an object vertically with a constant power source

text: Assume we have an object (e.g. a ball) of mass $m$. We want to accelerate this object vertically over a distance $L$ (e.g. a canon pointed vertically) up to a velocity $v$. We have a power source of constant output $P$, and we have some ideal assembly (e.g. a rope / pulley) connected to the object though which the power is applied. Neglect any friction / dissipation, etc. After a distance $L$, and assuming zero initial velocity, the object's energy is $E=\frac{1}{2}mv^2+mgL$. This means it will take $t=E/P$ seconds for the object to reach this energy level. For example, if $m=1\:kg$, $L=1\:m$, $g=10\:m/s^2$, $P=1\:W$, $v_{final}=1\:m/s$, then $E=0.5+10 = 10.5\:J$. This means it will take $t=10.5/1=10.5\:s$ for the object to reach that terminal velocity. Is this correct?


title: Why cannot we apply perturbation theory in Born-Oppenheimer approximation

text: In Weinberg's _Lectures on Quantum Mechanics_ , he mentions > Unfortunately, we cannot simply use first-order perturbation theory, with > $T_{nuc}$ taken as the perturbation and the state vectors $\Phi_{a,X}$ taken > as unperturbed energy eigenstates. This is because we are looking for > discrete eigenvalues of the full Hamiltonian, for which the eigenvectors > $\Psi$ would be normalizable, in the sense that $(\Psi,\Psi)$ is finite, > while $(\Phi_{a,X},\Phi_{a,X})$ is infinite. We cannot expand in powers of a > perturbation that converts a state vector with continuum normalization into > one that is normalizable as a discrete state. I cannot catch here. Why perturbation theory fails here? About the notations: follow from http://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation, $T_{nuc}$ is the kinetic energy of nucleus, i.e. $T_n$ in wiki. $\Psi$ in Weinberg is the same as $\Psi$ in wiki. $\Phi_{a,X}$ is $\chi_k (\mathbf{r}; \mathbf{R})$ in wiki, where $k\leftrightarrow a$ is the label of energy eigenstates.


title: Can a D-brane be closed and contractible?

text: Let's consider for simplicity D-branes in bosonic string theory. I have a very basic question whose answer I couldn't find clearly stated in the few textbooks where I looked for it. Take for instance a D1-brane. On the 1+1 dimensional world-volume live a gauge field (a vector with 2 real components) and 24 scalars, which presumably parametrize the shape of the brane in space-time. For instance, I quote from Johnson's textbook that _they are exactly analogous to the embedding coordinate map $X^{\mu}(\sigma,\tau)$" used to describe fundamental strings._ But there is in my opinion a crucial difference, because we have only 24 scalars and not 26. This means that the world-volume of the brane should be viewed as the **graph of a map** $\mathbb{R}^2 \longrightarrow \mathbb{R}^{24}$. This is a severe restriction on the allowed shapes for a brane. It implies that the D-string can not be closed (hence the question in the title - I added a condition of contractibility to rule out a trivial solution like space compactification), but is of course far more restrictive. What worries me is that I have never seen this restriction much emphasized, so maybe I missed something and I am completely wrong... I thought the explanation could come from the RR charge carried by any D-brane when you go to the superstring, but that is just a guess.


title: Changes in planetary orbit?

text: Forgive a novice and her naive curiosity, but I was wondering about planetary orbit and the aging of planets. As a **planet ages** , does its transit around its sun/star **decrease** or **increase** in **speed**? ## Does the speed and proximity of a planet's orbit around the star, have more to do with the age and type of the star, and less to do with the age of the planet and its distance from the star? Also, is it possible for a planet to be **pushed** or **pulled** closer to its star, into a closer orbit, for example by increased gravitational pull of the star, or the planet being knocked off orbit by a comet or asteroid, or even a shockwave from an implosion -- one far enough away so that no debris comes near, but near enough that the shockwave is powerful enough to knock the planet. I realise such an occurance would probably cause catastrophic upset on the planet, but if it remained relatively intact, could it find itself on a new path around its star??   Or could the comet/asteroid/shockwave hit the planet from the opposite direction of its transit, perhaps glancing it at such an angle as to cause it to slow in pace around the star, or to turn faster or slower on its own axis? And would its poles be reversed? Living in a world with sci-fi like **Doctor Who** , **Eureka** , **Stargate** , **Star Trek** , **X-files** , **Fringe**.. it makes you wonder what is **theoretically possible**. Often the explanations they give sound **far- fetched, yet plausible**. Just as every myth evolves from a truth, you tend to wonder how much truth exists within these stories. Hope one of you can shine a light on my wonderings and find a kernel of truth among them.


title: What software do publishing companies use to create drawings for text books?

text: What software is used to create free body diagrams, inclines, springs and so on in the introductory physics textbooks ? I know Adobe, Luxology, and many others could, but I am more interested in software with ready made objects, that could still create good looking diagrams with less effort. Most of the textbooks have 100s of such diagrams that are kind of good looking.


title: Are "phase-specific" masks ever used in the context of double exposure lithography? Can these sorts of things exist?

text: I was recently reading an 08' paper on double exposure lithography: and I was wondering if there existed _some_ material that could be used to create a mask for e.g. a 193 nm light source that was _only opaque to light with a specific phase_ $\theta$. The idea, in the context of double exposure lithography, would be to overlay two masks, one opaque for light with some phase $\theta_a$ and another opaque for light with some phase $\theta_b$, where (in radians) $$\theta_a = \theta_b + \pi\cdot n$$ where $n$ is any non-zero integer value. Thus, one could just phase-shift the source of coherent light used for lithography by $\pi$ radians to pattern with one mask and then the other. This would save an alignment step for the substrate being patterned. Is there any precedent for this sort of thing? Or is this question nonsense?


title: How might eddy currents and terminal velocity be broken down?

text: This is a branch from this question. With regards eddy currents and terminal velocity I have located a homework question but would appreciate more detail (and translation of the symbols described therein) about the forces at play. Eddy currents are generated though the resistance of electrons interacting with magnetic fields as they cut across coils and force them downwards. Since there is resistance it would stand to reason that this would reduce the speed at which a magnet falls - but that there would be an equilibrium whereby further slowing a magnet would reduce the flux generation required to sustain that reduced descent. So the forces at play include:   * gravity   * mass of the magnet   * coils of the coil (solenoid?)   * air resistance (may be discounted?) May these be further broken down? Are other variables at play?


In non-relativistic quantum theory $\hat{K}=\hat{p}^2/2m$, What is the Kinetic energy operator in Dirac's relativistic quantum theory?


title: Jackson equation 2.9; How to derive it?

text: I wanted to understand the derivation of Jackson's equation (2.9) that you can find here on page 31 (50/661) from the potential given earlier. This equation is written as: $$ \textbf{F}=\frac{q}{y^2}\left[Q-\frac{qa^3(2y^2-a^2)}{y(y^2-a^2)^2}\right]\frac{\textbf{y}}{y} $$ Does anybody know how to do this as there are apparently many steps left out in between?


title: The eigenvalue of Schrodinger Equation

text: I'm a student majoring in Mathematics.But now I'm studying the KDV equation which uses Schrodinger Equation. My question is that in time-independent Schrodinger Equation$$\psi_{xx}-(u-\lambda)\psi=0$$,and when $x\to|\infty|,u\to0,u_x\to0$,there are two questions that I have:   * Why are all the eigenvalues real?   * Why are there discrete eigenvalues for $\lambda<0$ and continuous eigenvalues for $\lambda>0$?


title: When does a pot not spill?

text: What shape for the edges of a pot and what surface properties are best for a tea pot not to spill? Apart from the cross-section from the side, how does the radius of curvature as seen from above play a role? Ideally it should also stop drops coming down. And how does that depend on the properties of the liquid inside?


title: QM state probability

text: What does $|<\psi|\psi>|^2=1$ physically mean ? Can we use born's rule for superpositioned states ? Does it carry any physical meaning ? Why don't wquwe ever observe a superosotioned state then?


title: D'Alembertian for a scalar field

text: I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$   1. Exactly when is this correct? Only for $\Box \phi$ where $\phi$ is a scalar-field?    2. How exactly is it shown?    3. Is it true only on-shell, from the Euler-Lagrange equations of a scalar-field?


title: Doing a Gaussian Integral

text: When you integrate over p you get: ![enter image description here](http://i.stack.imgur.com/yEUbh.png) by using ![enter image description here](http://i.stack.imgur.com/9gXXP.png) What are the steps to this? Do you integrate by parts?


title: Momentum and Energy in Oscillations

text: During oscillatory motion, whether it's a simple pendulum, a block on a spring, or liquid sloshing back and forth in a well...   1. Is energy or momentum or both conserved? Why?   2. Is energy or momentum (or both) the driving factor to get the mass through the equilibrium point? Why?


title: Why 'max' in $hf=\phi+{1\over{2}}mv_\text{max}^2$?

text: The equation for the photoelectric effect is $$hf=\phi+{1\over{2}}mv_{\text{max}}^2$$ How does this make sense given that $hf$ describes a _single_ photon and ${1\over{2}}mv_{\text{max}}^2$ describes the maximum kinetic energy over _many_ electrons? Why can't the photon release many electrons which _only just_ exceed $KE=\phi$? Or, if $hf$ is strictly related to the maximum $KE$, doesn't this mean _only one_ photon is released, in which case the $\text{max}$ subscript doesn't make sense?


title: Why aren't units with powers, like cm³, surrounded by parentheses?

text: Since $\renewcommand{\unit}[1]{\,\mathrm{#1}} 1\unit{dm} = 10^{-1}\unit{m}$, it follows that $1\unit{dm^3} = 10^{-1} \times 10^{-1} \times 10^{-1} \unit{m^3} = 10^{-3} \unit{m^3}$. However, in regular mathematics the following equation holds true: $$a\,b^{3} = a\,b\,b\,b$$ By the above, the cube unit should expand as follows: $$\mathrm{dm^3} = \mathrm{dmmm}$$ While in actual usage (as seen in the second equation) the expansion is $\mathrm{dddmmm}$, which would arise from using $\mathrm{(dm)^3}$ instead. $$\mathrm{(dm)^3} = \mathrm{dddmmm}$$ So shortly: why aren't parentheses (commonly?) used in units?


title: Is the 74.2km/sec per megaparsec accidental?

text: The rate of expansion of the universe is 74.2km/sec per megaparsec The age of the universe is 13.798 bln years That places the furthest theoretically observable edge of universe at 13.798 bln light years away - this is how long light would take to reach us from there, and there were no light sources before that date. Now, how fast does the observable part of the universe expand? A bit of unit conversion and multiplication: 13798000000 light years in megaparsecs = 4 230.39567 megaParsecs [1] 74.2km/sec in light years/year = 0.000247504559 light years / year [2] Multiply the two, and I get the expansion rate: 1.04704221 light years per year per distance to the observable edge of the universe. Speed of 1 light year per year is the speed of light. The 0.047 seems awfully small. Probably within error margin of the universe expansion speed and age of the universe estimates. Is this a coincidence, or is this somehow bound - and if so, what are the consequences?


title: Couple Masses - Change in Basis

text: I'm having trouble with the linear algebra used to solved a coupled mass problem. $\ddot{x}_1 = -(2k/m)x_1 + (k/m)x_2$ and $\ddot{x}_2 = (k/m)x_1 - (2k/m)x_2$ Shankar then sets the equation up in matrix form. $$\left \lbrack \matrix{ \ddot{x}_1 \cr \ddot{x}_2 }\right \rbrack = \left \lbrack \matrix{ \Omega _{1,1} & \Omega _{1,2} \cr \Omega _{2,1} & \Omega _{2,2}} \right \rbrack \left \lbrack \matrix{x_1 \cr x_2} \right \rbrack$$ Where $\Omega _{i,j}$ equals -$(2k/m)$ and $(k/m)$ as in the original equations. This is the part I don't understand, he goes on to say that this equation (the matrix above) is set in a particular basis. The general equation here is $\left| \ddot{x}(t) \right \rangle = \Omega \left| x(t) \right \rangle$.. His explanation is "The equation is obtained by projecting (the ket equation above) on the basis vectors $\left| 1 \right \rangle$ and $\left| 2 \right \rangle$ which have the following significance: $\left| 1 \right \rangle$ first mass displace by unity, second mass undisplaced, $\left| 2 \right \rangle$ first mass undisplaced, second mass displaced by unity. Hew goes on to talk about how we need to find a matrix tin a basis that is diagonalized. How does the change of basis work?


title: Integrating the divergence of a velocity field (Fluid Mechanics)

text: I'm looking more for verification of my answer as I'm not 100% sure how valid my integration is here: If I have a 2-D incompressible fluid velocity field given by $\vec{U}(x,y,t)=u(x,t)\vec{x}+v(y,t)\vec{y}$, I know that $\nabla \cdot \vec{U} = 0$, call this eqn (1), by the definition of incompressibility. In order to find out more about the velocity components $u(x,t)$ and $v(y,t)$, i have been told to integrate eqn (1). Is the following integration w.r.t. space valid?: \begin{align} &\int \nabla \cdot \vec{U} \mathbb{d} r = \int 0 \\\ &\int \left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) \mathbb{d}r = C = \text{const.} \\\ &\int \frac{\partial u}{\partial x}dx+\int \frac{\partial v}{\partial y}\mathbb{d}y = C \\\ &u(x,t) + v(y,t) = C(t) \end{align} In other words, the magnitude of the sum of the fluid velocity in the $x$ and $y$ directions are constant everywhere in the fluid at any instant in time? Is there a better way of obtaining more information?


title: Spinor formalism in QFT

text: We can describe fields by two formalisms: vector and spinor. This is the result of possibility of representation of the Lorentz's group irreducible rep as straight cross product of two $SU(2)$ or two $SO(3)$ irreducible representation. Vector formalism is more porular, because working with it is more convenient. But there are some theories (models of interaction), where an introduction of spinor formalism has some advantages; sometimes we can't avoid introduction of spinors (in the case of half-integer spin of the field) and sometimes we can create an interaction by introducing $SU(2)$ fields (like in the case of Yang- Mills theory). So, my question is follow: how often spinor formalism is suitable for using in quantum field theory (in addition to the above results)?


title: Pressure and height of water in a pipe

text: In a building, at what height does water rise from the ground-floor in a pipe, if the pressure there measures P=3atm? The correct answer is 31m. What formula should I apply?


title: Breakdown voltage

text: > Capacitor (Geiger counter) is made out of wire with radius $R_1 = 5 mm$ and > coaxial tube with radius $R_2 = 5 cm$. What is the maximum voltage on the > capacitor, given the breakdown voltage of air is 30 kV/cm? I have figured out the capacity $C = \frac{2\pi \epsilon_0}{\ln {R_2\over R_1}} = 2.42*10^{-11}$ and voltage $V_c = \frac{Q \ln {R_2 \over R_1}}{2\pi \epsilon_0}$. I think that the key here is to find the relationship between overall voltage on the capacitor and voltage between the electrodes, which is smaller, proportionally to the amount of space between them. But how much exactly?


title: Tuned Mass Damper

text: I was learning about the different applications of resonance and one of them is the tuned mass damper used in buildings. One thing I am confused about is as to why the mass attached to the building oscillates $180^\circ $ out of phase when in resonance. Is there any mathematical reason behind this or an intuitive way of understanding the concept?


title: Quantum Efficiency Estimation

text: Might there be a way to do a rough estimate of the quantum efficiency of a photo-detector like a CCD or CMOS sensor based only on a picture taken with it? I've read papers and guides (like this one: http://www.clarkvision.com/articles/digital.photons.and.qe/) that are along the lines of what I'm thinking, but I'm wondering if there might be a way to do it without the need for a well-calibrated light source. It wouldn't be an extremely _accurate_ estimate without the calibration, of course, but is it even possible?


title: Making a superadjustable barbell

text: My working knowledge of physics is the level of a 7th grader (if I'm lucky) so please excuse this probably totally stupid question, but my curiosity is greater than my shame: Would it be possible to create a barbell that works without weight plates?   What I have in mind is something the shape of a barbell with some motor or whatever at the two ends that would be adjustable in weight without loading additional steelplates onto it. (If the answer is "no": why?) It would be awesome if anyone had thoughts on this.


title: Relativistic momentum

text: I have been trying to derive **why** relativistic momentum is defined as $p=\gamma mv$. I set up a collision between 2 same balls ($m_1 = m_2 = m$). Before the collision these two balls travel one towards another in $x$ direction with velocities ${v_1}_x = (-{v_2}_x) = v$. After the collision these two balls travel away from each other with velocity ${v_1}_y = (-{v_2}_y) = v$. Coordinate system travells from left to right with velocity $u=v$ at all times (after and before collision). Please see the pictures below where picture (a) shows situation before collision and picture (b) after collision. ![momentum before and after collision](http://i.stack.imgur.com/6uETV.png) Below is a proof that Newtonian momentum $mv$ is not preserved in coordinate system $x'y'$. I used $[\, | \,]$ to split $x$ and $y$ components. $p_z'$ is momentum before collision where $p_k'$ is momentum after collision. $$ \scriptsize \begin{split} p_z' &= \left[ m_1 {v_1}_x' + m_2 {v_2}_x'\, \biggl| \, 0 \right] = \left[ m_1 0 + m_2 \left( \frac{{v_2}_x - u}{1-{v_2}_x\frac{u}{c^2}} \right)\, \biggl| \, 0 \right]= \left[ m \left( \frac{-v - v}{1+ v \frac{v}{c^2}} \right) \, \biggl| \, 0 \right] \\\ p_z' &= \left[ - 2mv \left( \frac{1}{1+ \frac{v^2}{c^2}}\right) \, \biggl| \, 0 \right] \end{split} $$ $$ \scriptsize \begin{split} p_k' &= \left[-2mv \, \biggl| \,m_1 {v_1}_y' + m_2 {v_2}_y'\right]=\left[ -2mv \, \biggl| \, m_1 \left( \frac{{v_1}_y}{\gamma \left(1 - {v_1}_y \frac{u}{c^2}\right)} \right) + m_2 \left( \frac{{v_2}_y}{\gamma \left(1 - {v_2}_y \frac{u}{c^2}\right)} \right) \right]\\\ p_k' &= \left[ -2mv \, \biggl| \, m \left( \frac{v}{\gamma \left(1 - v \frac{v}{c^2}\right)} \right) - m \left( \frac{v}{\gamma \left(1 - v \frac{v}{c^2}\right)} \right)\right]\\\ p_k' &= \left[ -2mv \, \biggl| \, 0 \right] \end{split} $$ It is clear that $x$ components differ by factor $1/\left(1+\frac{v^2}{c^2}\right)$. **QUESTION:** I want to know why do we multiply Newtonian momentum $p=mv$ by factor $\gamma = 1/ \sqrt{1 - \frac{v^2}{c^2}}$ and where is the connection between $\gamma$ and factor $1/\left(1+\frac{v^2}{c^2}\right)$ which i got?


title: What velocity must an aircraft achieve for its shock wave to transform to plasma?

text: A follow-up to After what speed air friction starts to heat up an object? I understand there may be technological limitations at present ... but, is it theoretically possible for a body to travel through the atmosphere so fast that the shock wave transforms to plasma? What velocity would a body need to achieve for this to happen?


title: Is there any reproducible tested evidence for Ni-H cold fusion?

text: The main replications of cold fusion, the ones that are beyond reproach, used Pd/d as the system. But commercial developers have often claim to use Ni-H to achieve similar effects. The claims include those of blacklight power some time ago, and now Rossi (which is for sure a scam, see here: Is the E-cat for real? ), and also others. Ni-H power would almost certainly be inconsistent with the mechanism I proposed in the answer to Why is cold fusion considered bogus? . Only the heavy isotope can fuse, and it is present at far to low a concentration to make a chain reaction. The Nickel would have to segregate the deuterium from the protons, and it's too far fetched to believe. So, to test the theory, I would like to know if there is any credible evidence for the existence of Ni-H cold fusion.   * Are there any laboratory reproductions?   * Are there audited non-Rossi tests?


title: Normal Forces and Ferris Wheels

text: At the moment, I am reading an example problem regarding what was alluded to in the title. In this example problem, they say, "Based on experiences you may have had on a ferris wheel or driving over small hills on a roadway, you would expect to feel lighter at the top of the path. Similarly, you would expect to feel heavier at the bottom of the path." Quite honestly, this is not what my intuition speaks to me: I would expect to have a greater normal force at the top of a hill, due to my inertia wanting to keep me at the bottom of the hill, thus the seat pushing up into me; for driving down a hill, I would expect my normal force to be less, because my inertia wants to keep me on top of the hill, therefore, the seat travels from beneath me. I trust that the book is correct, but I would really like to know why I am wrong. Thank you! Edit: Also, in the example problem, they analyze the forces acting on a person in a ferris wheel while at the bottom and top of it. For the bottom, $\Sigma F = N_{bottom} - mg = m \frac{v^2}{r}$; for the top, $\Sigma F = mg - N_{top} = m\frac{v^2}{r}$. Why does the normal force and weight change signs in each case?


title: Does every measurement correspond to an eigenstate of an observable?

text: In the postulates of quantum mechanics, physical observables are described by Hermitian matrices on the state space of a system. In another of my questions, the measurements of Rydberg-Ritz spectral emission lines do not label the energy eigenstates of an emitting atom, but instead these label differences of frequencies associated to the energy eigenstates. If the algebra of transitions "completely describes" the system, then measurements of the emission lines are not eigenvalues of a hermitian operator in this algebra (these can be derived from the frequencies of energy eigenstates, though). Does the postulate of QM above still assert that the above observed frequencies label eigenstates of some quantum system? Is there some other "matrix algebra" corresponding to transitions between these eigenstates? Even if the above two paragraphs are fundamentally misguided in some way, I'd still like to know: > In what sense is every physical measurement an eigenvalue of some > observable? I know these questions have been pretty naive, but I'm intentionally trying to investigate things from an extraordinarily minimalist perspective... Perhaps the above is ridiculous, in the sense that the starting point for a _computation_ to yield a physical quantity always is the eigenvalue of an observable. For example, if I am using a quantity that is the result of some horrid formula, the thing I plug into this formula must be a measured observable. This would save the previous question, since knowing the transition frequencies is as good as knowing the frequencies of the energy eigenstates...provided we can ascertain one of the energy eigenstates. It is the ability to get this "starting" energy eigenstate that bothers me...it should be something we can directly measure.


title: Question about finding $k$ in Hooke's Law

text: My textbook ( _Advanced Engineering Mathematics_ by Dennis Zill) offers the following explanation of Hooke's Law: > _By Hooke's Law, the spring itself exerts a restoring force $F$ opposite to > the direction of elongation and proportional to the amount of elongation > $s$. Simply stated $F=ks$ where $k$ is a constant of proportionality called > the spring constant. The spring is essentially characterized by the number > $k$. For example, if a mass weighing $10lb$ stretches a spring > $\frac{1}{2}ft$, the $10 = k \frac{1}{2}$ implies $k=20 \frac{lb}{ft}$. > Neccesarily then, a mass weighing, say, $8lb$ stretches the same spring only > $\frac{1}{2}ft$._ I have two issues with this paragraph.   1. The explanation says that to find $k$, you take the weight of the load as your $F$, and solve. For example in this problem, $F = ks$ becomes $10 = k \frac{1}{2}$. My question is, why is $F$ equal to $10$? $10$ is just the weight - I was under the impression that $F$ was a mass times its acceleration ?   2. At the end of the paragraph, it says that a weight of $8lb$ would stretch the same spring only $\frac{1}{2}ft$, which is the same as a $10lb$ weight. Is that a typo? What would an 8 pound weight really stretch to?


title: Is the tide on Earth caused by curvature of spacetime

text: The tide on Earth appears absolutely whenever the moon is overhead. Is that tide caused by spacetime, re-curvature in space or attraction gravity?


title: How to measure trajectory regularity?

text: I have two animal running trajectories. A regular one with repeated back and forth running between point A and B, like the one on top in the figure. The other one is very irregular, animal paused and turned around a lot in the middle. Is there any algorithm to measure the regularity of a trajectory, like repeated activity on the top? And compare the extend of regularity between the two trajectories? By 'more regular trajectory', I mean animal run with higher mean speed, less turn back and less pauses in the middle, more laps within the same amount of time. The figure above is for demonstration purpose, since animal can not run straight with constant speed anyway. ![enter image description here](http://i.stack.imgur.com/E7ANA.jpg)


title: Determine the acceleration in a double Atwood machine

text: Trying to solve the following double Atwood machine: > _Suppose there is a mass of 12kg hanging on an ideal rope that wraps around > an ideal pulley, and that the other end of the rope is attached to another > ideal pulley's axis. This second pulley has a rope around it which on one > side holds a mass of 4kg and on the other a mass of 8kg. What is the > acceleration of the first mass?_ So here's what I've done so far: First I've determined that the tension in the first rope is twice the tension in the rope for the "subsystem", and that the acceleration of the first mass is the average of the accelerations of the two other masses but in the opposite direction. That is to say $$T' = T/2$$ $$a_{1}=-\frac{a_{2}+a_{3}}{2}$$ From free-body diagrams we can determine that $$-12a = T-12g$$ $$4a_{2} = T/2 - 4g$$ $$-8a_{3} = T/2 - 8g$$ However, when I solve this system I obtain $T=32a$ which seems impossible that the tension would be greater than the greatest force of gravity of any mass. Part of me wonders if this has anything to do with it: How do I know which direction the third mass is traveling? The first mass is causing the subsystem to accelerate upwards but the third mass is accelerating downwards relative to the subsystem. So how do I know the direction of acceleration relative to the ground?


title: Free body diagram of rod in sphere

text: I was finding the free body diagram part of dynamics quite easy until I found this question , Here's how it goes : > A rod AB is placed inside a spherical shell, whose inside surface is rough. > Draw the free body diagram (FBD) of the rod. > > ![Rod in Spherical shell](http://i.stack.imgur.com/D1XXj.gif) Thanks for any help :)


title: Asymmetry in muon energies of public CMS dimuon event data

text: CMS published for educational purposes (caveat) 100.000 dimuon events: https://cms-docdb.cern.ch/cgi-bin/PublicDocDB//ShowDocument?docid=11583 As one easily sees, column 4 shows the first muon's energy, column 12 the second's. Although the first energy is not higher in general (the data are not ordered),the distribution of energies is asymmetric. The first muons energy is higher on average, as one can easily see with the following Mathematica code: (download the first file dimuon.csv to your working directory)               SetDirectory["c:\\yourworkingdirectory\"];     qwe = Import["dimuon.csv", "CSV"];           wer = Drop[qwe, 1];    (* select the events labelled "GG"*)               rty = Select[wer, ((#[[1]] == "GG")) &]; ert = Transpose[rty];          energy1 = ert[[4]]; energy2 = ert[[12]];     relationOfEnergies = energy1/(energy1 + energy2);          bcEnergies = BinCounts[relationOfEnergies, {0, 1, .01}];          BarChart[bcEnergies, Ticks -> {Table[i, {i, 0, 100, 10}], Automatic}];      Here is the result ("GG" events oly), the first energy as a fraction of both muon's energies: ![relation of 1st muon's energy to both muon's energy](http://i.stack.imgur.com/ATMax.jpg) Could have a trivial reason, just curiosity...


title: Why does NASA use gold foil on equipment and gold-coated visors?

text: I've read several websites about equipment covered with gold foil and astronaut helmet visors are coated with gold. However, their explanations are devoid of almost all physics content. Can someone explain the basic concept of why gold foil is so popular with NASA as a coating on visors?


title: Electromagnetic radiation and quanta

text: since electromagnetic radiation possess the property of both wave and particle(photon). and both theory are applicable but how we have to find out that which theory is suitable or applicable in particular explanation. for example in laser traping of atom we use photon concept rather than wave why?


title: CFT Entanglement Entropy - relation between translations and the stress-energy tensor

text: In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \times \mathbb{H}^{d-1}_q$ with metric: $$ ds^2_{H_q^{d-1}} = d\tau^2 + du^2 + \sinh^2 u \; d\Omega_{d-2}^2 $$ Here $d\tau$ is probably a Wick-rotated time, $u$ is a radial variable and $d\Omega$ is the spherical area measure. Then they define a partition function $ Z_q = \mathrm{tr}(e^{-2\pi q H_\tau}) $ where $H_\tau$ "generates translations along the $S^1$". What does that mean? Could it mean this? $$ H_\tau = \frac{d}{d\tau}$$ This is the generator for translations along the $\tau$-axis. However, they also say this is related to the stress-energy tensor: $H_\tau = \int_{\mathbb{H}^{d-1}} dx^{d-1} \sqrt{g} T_{\tau\tau}$ This seems like a very complicated way of describing translations. Could there be another meaning for the phrase "generates translations along $S^1$? * * *   1. Jeongseog Lee, Aitor Lewkowycz, Eric Perlmutter, Benjamin R. Safdi   Renyi entropy, stationarity, and entanglement of the conformal scalar


title: Can the Kramers–Kronig relation be used to correct transfer function measurements?

text: In experimental physics, we often make measurements of linear transfer functions; these are complex-valued functions of frequency. If the underlying system is causal, then the transfer function must be analytic, satisfying the Kramers-Kronig relations. Our measurements, however, are corrupted by random (and perhaps systematic) errors. **Is it possible to improve a measurement of a linear transfer function of a causal system in the presence of noise by applying some kind of constraints derived from the Kramers-Kronig relations?**


title: Which of these two textbook equations of geodesic deviation is correct?

text: My previous question Geodesic deviation on a 2-sphere - is this the right track? got shot down as “off topic”, so I'm having a second stab at it. Misner _et al_ 's _Gravitation_ (p34) gives the geodesic deviation equation as$$\frac{D^{2}\xi^{\alpha}}{D\tau^{2}}+R_{\phantom{\mu}\beta\gamma\delta}^{\alpha}\frac{dx^{\beta}}{d\tau}\xi^{\gamma}\frac{dx^{\delta}}{d\tau}=0,$$ with the right-hand side $\xi$ index $\gamma$ equal to the _second_ lower index on the Riemann tensor. Lambourne's _Relativity, Gravitation and Cosmology_ (p185), on the other hand, gives $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\alpha\beta\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0,$$ with the right-hand side $\xi$ index $\alpha$ equal to the _first_ lower index on the Riemann tensor. My question is, which of these two equations is correct? I tried to answer this question myself by using the two equations to calculate the geodesic deviation on the surface of a unit 2-sphere. With Misner's equation (substituting $\lambda$ for $\tau$) I got $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\theta}\right)\xi^{\phi}-\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\phi}\right)\xi^{\theta}$$ and $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=\xi^{\theta}\left(u^{\theta}u^{\phi}\right)-\xi^{\phi}\left(u^{\theta}u^{\theta}\right).$$ You can see my calculation on my previous question Geodesic deviation on a 2-sphere - is this the right track? With Lambourne's equation I got $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=0$$ and $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=0.$$ This didn't seem right to me so I concluded that Lambourne's equation is incorrect.


title: Is This The Answer To Artificial Gravity?

text: Einstein teaches that as an object gets faster, its relativistic mass increases... Newton teaches that as an object's mass increases, so does it's gravitational pull... So... if you a tethered some objects to a wheel, and then accelerated them to a very great speed... Each object would eventually be going fast enough to create sufficient gravity to pull us towards them, correct? Then just stick this contraption in the centre of a spherical space ship and you've got semi artificial gravity?... I don't mind that it might take an unpheasable amount of energy to get the objects to the speed needed... I'd just like to know if the basic theory is correct? Or is there something major I'm overlooking...


title: Will physic object in a perfect environment last/exist forever?

text: We know,i.e. wood/steel tables in earth will completely broken or disappear in a very future day. If we put the table in a perfect/ideal environment (maybe in vacuum), will the disappear/broken still happen again? "Ideal" means every outside factors, temperature, pressure and gravity etc, is just perfect for the table and the table can endure. Table wouldn't move, only the particles of the table move. I always think that the movement of the particles eventually makes the table or other physics object die/decay. **Every objects decay for the internal cause**. I don't know if this is true, so I ask this question.


title: Polarization vectors of massive and massless particles

text: I read from Mandl & Shaw that when quantizing massless vector particles such as photons in Lorentz gauge, there are 4 linearly independent polarization vectors (2 of them being able to "gauged away"), while when quantizing massive vector particles such as W bosons, there are only 3 linearly independent space-like polarization vectors. It also says that when W and Z gauge bosons "eat" the Goldstone bosons from Higgs field, they become massive and acquire longitudinal polarization. So apparently there is some rule that says massive vector particles have longitudinal polarization, while massless particles don't. Why is that? From what principle is it derived? Their Euler-Lagrangian equations are essentially the same (both look like Klein-Gordon equation), just one of them has $m=0$. Is it the case that when $m=0$, it has four linearly-independent solutions while only three when $m\neq0$ ?


title: Initial pressure of explosion

text: I would like to know, if given a TNT equivalent of explosive (or its Energy) is it possible to determine what would be the initial pressure that will be the result of that explosion ? To be more precise let's say I have a vessel and given the vessel volume can I determine pressure inside that vessel (assuming vessel won't break).


title: Distance between 2 copper rods

text: A balance was used to investigate the relationship between current and force. The balance was set up with one copper rod fixed to it and a second rod fixed above it, as shown in the diagram. Each rod was connected to a source of current. The diagram is not to scale. ![enter image description here](http://i.stack.imgur.com/wsHSZ.png) The copper rods were rigid, each was 2.6 m long, and they were parallel. The current in the upper rod was kept constant at 50 A. Different currents were passed through the lower rod and the balance reading recorded for each current. The readings are given in the table below. ![enter image description here](http://i.stack.imgur.com/XcNjO.png) My question is: how would you calculate the distance between the 2 copper rods?


title: CPT symmetries for a free Klein-Gordon equation and in minimal coupling

text: I'm studying for an exam on relativistic quantum mechanics and one of the issues to prepare is about symmetries of Klein-Gordon equation concerning $C$, $P$, $T$ transformations for a free field, and with an electromagnetic interaction. So could you help me out a bit? As I understand the general free solution can be written as: $$ \Psi(\mathbf{x},t) = \int\limits_{ \mathbb{R}^3 }\frac{d^3p}{(2 \pi \hbar)^3} \frac{1}{2 E_p} \left( A_\mathbf{p}^{(+)}e^{-\frac{i}{\hbar}E_p t} e^{\frac{i}{\hbar}\mathbf{p} \cdot \mathbf{x}} \+ A_\mathbf{p}^{(-)}e^{\frac{i}{\hbar}E_p t} e^{\frac{i}{\hbar}\mathbf{p} \cdot \mathbf{x}} \right) $$ where: $E_p := \sqrt{\mathbf{p}^2c^2-m^2c^4}$ For a special solution (in a box): $$ \psi^{\pm}_{\mathbf{p}}(\mathbf{x},t) = \frac{1}{\sqrt{V}} e^{\mp\frac{i}{\hbar}E_p t} e^{\frac{i}{\hbar} \mathbf{p} \cdot \mathbf{x}} $$   * $P$ transformation gives: $P\psi_{\mathbf{p}}^{\pm}(\mathbf{x},t) = \psi_{-\mathbf{p}}^{\pm}(\mathbf{x},t)$,   * $T$ transformations gives: $T\psi_{\mathbf{p}}^{\pm}(\mathbf{x},t) = \psi_{\mathbf{p}}^{\mp}(\mathbf{x},t)$ and   * $C$ transformations gives: $C\psi_{\mathbf{p}}^{\pm}(\mathbf{x},t) = \psi_{-\mathbf{p}}^{\mp}(\mathbf{x},t)$ //Corrected a mistake thanks to a comment by TwoBs So what conclusion (if any) I can draw from this about the general solution? And do the conclusions stay valid for a KG equation with minimal coupling ($\partial_{\mu} \rightarrow D_{\mu} := \partial_{\mu} + \frac{i e}{\hbar c}A_{\mu}$)?


title: Capture reflecting image

text: Suppose you have following installation:                   +-------------------+         | O               M |         +-------+   +-------+                 |   |                 | C |                 +---+      View from top. In a corridor object $O$ gets reflected by a mirror wall $M$. Is it theoretically or practically possible to capture any information about the object using camera $C$?


title: Can a $CFT_2$ which can't be factorized into chiral and antichiral parts and/or have a central charge not a multiple of 24 have AdS duals?

text: In the article Three dimensional gravity reconsidered by Ed Witten, he remarked that the CFT dual to three dimensional quantum gravity has to admit a holomorphic factorization and have a central charge of $c=24k$ for some integer $k$. He argues this by some handwaving from a Chern-Simons crutch even though he admits the nonperturbative theory can't be Chern-Simons. Anyway, does this mean if we have a two dimensional conformal field theory which does not factorize into chiral and antichiral parts and/or has a central charge which is not an integer multiple of 24, it does not have an AdS quantum gravity dual?


title: Low frequency electromagnetic waves

text: Some frequencies of electromagnetic waves are used for transmission of information, like radiowaves, microwaves, light, but some not.. What about Low Frequencies?: Perhaps low frequencies aren't used for transmision simply because the waves haven't got enough energy to reach destiny, but is well known that if you go near a high voltage transmission tower, you can turn on a fluorescent lamp http://www.electric-fields.bris.ac.uk/ Perhaps the intensity of the high voltage gives more photons to interact there, than in a home installation, but the fact is waves are traveling anyway at 50/60 Hz without need any modulation. There is a limit on how low a frequency can be to put photons into travel?


title: Is the frequency of light restricted?

text: What are the factors that limit the frequency of light? Can it have wavelengths ranging between zero and infinity?


title: Why are lasers inefficient?

text: Why are lasers inefficient? Is it because of the heat lost during lasing? Why couldn't there be thermocouples or turbines in parts of the cooling circuits to extract something out of that heat?


title: Topological insulators: why K-theory classification rather than homotopy classification?

text: I am reading a 2009 paper by Kitaev on K-theory classification of topological insulators. In the 4th page, 1st paragraph in the section "Classification principles", he says, > Continuous deformation, or homotopy is part of the equivalence deﬁnition, > but it is not sufﬁcient for a nice classiﬁcation. Why is not homotopy sufficient? The only shortcoming of homotopy classification I can think of is that, a general homotopic deformation of the Hamiltonian may close some band gaps, so the satisfactory classification in terms of homotopy must be "equivalence class (of maps from Brillouin zone to Hamiltonians)classified up to homotopies that respect band gap", clearly this ought to give a finer classification than just "classified up to homotopy". Is this the reason for introducing the K-theory classification? I know a bit algebraic topology but K-theory is quite over my head, and Kitaev's paper is too terse for me to figure out if I got it right. Could someone more familiar with the paper explain it to me? Or is there some more expository paper on the subject? **EDIT** : Although it is already clear from Heidar's answer, let me stress here in the main post that the homotopy classification scheme does take into account the band gap nonclosing condition, I misunderstood this fact before seeing Heidar's answer and some other materials. I hope this edit makes the point clearer and eliminate the possiblity of misleading new learners of the subject who read this post.


title: Orthogonality between $\vec{E}$ and $\vec{H}$ waves with space-dependent amplitudes

text: I am able to prove in a few lines that the electrodynamic field vectors $\vec{E}$, $\vec{H}$ and $\vec{S}$ are all orthogonal to each other considering that $\vec{E}$ and $\vec{H}$ are coherent plane waves with _constant complex amplitudes_ propagating in the $\vec{k}$ direction. See for example **JACKSON J.D.** _Classical Electrodynamics_ (§7.1, Plane Waves in a Nonconducting Medium, 1st ed. (1962), pp. 204-205, eqs. 7.9-7.15). _**My question is:_** is there any way to prove the same but considering waves of amplitudes varying with space $(x,y,z)$? I haven't seen this even mentioned in any of the books I've consulted, they all consider plane waves with constant amplitudes. I tried to prove it on my own using the different methods I've seen for constant amplitudes but they all fail.


title: Book suggestions for foundation of Newtonian Mechanics

text: I'm not looking for books which deal with the mathematical foundations of Newtonian mechanics. What I'm looking for are modern books which deal with conceptual foundations of Newtonian mechanics - by that I mean exact definition of force, inertia, frames of reference etc. It seems like much of the books that deal topics were written before the 80's or even older (like the one by Ernst Mach himself). And the people who seem to be bothered about these things today are mostly science educators, who again publish very little on these topics. I'm not looking for books written from a philosophical standpoint either. Something that'd be comprehensible by a undergrad would be just fine. Note: This question probably doesn't meet the guidelines as defined in the faq. That is why I added the 'soft-question' tag. Also, any moderator who wishes to make this question community-wiki is most welcome to do so!


title: Is the charge of an ion evenly distributed?

text: This question relates to: Gauss' law and ions? Is the charge distribution in an ion spherically symmetric due to quantum mechanical effects or do we assume it when using Gauss's law, as in the linked question, to make the calculation easier? I think it is the former but I am not sure.


title: How many molecules rub off when I press a key?

text: I have a lot of questions below but my overarching questions are: _Do surfaces rubbing lightly together always strip molecules off of each other?_ and _How can we model that?_ Clearly the answer to the first question is yes in general. We've all seen worn out keyboards and the like. But it is not clear (at least to my non- expertise) that two substances might exist such that rubbing them together does no damage on the molecular level. If you lightly brushed the face of a diamond with a feather, how long would it be before a visible dent would form? Is it more likely that the feather would be "used up" first? Does hardness play a role? When I hit a key on a computer keyboard, how many of my skin cells rub off? How much plastic rubs off the key (on average)? Where do the plastic molecules end up? Just in the air? Embedded in skin cells? What about very small mechanical systems: Does friction and loss of matter between elements make micro or nanoscopic mechanical devices unreliable? Can they be designed to avoid such issues?


title: Can we multiply an electromagnetic field?

text: ![photomultiplier](http://i.stack.imgur.com/vNvVw.jpg)   * can we multiply light by a sequence of reflections in a closed vessel?   * also, is light a part of EM wave?   * can we multiply the radio wave or EM wave from a transformer?


title: Are Stars getting 'Less Massive?

text: A star produces energy through Nuclear fusion with the 2H and 3H isotopes to create 4He for example. I read that there is a mass loss of 0.018884 (in atomic mass units). Of course this mass loss is turned into energy (E=MC^2). So, the amount of mass when the star was young and its mass when it is about to die is different. So, when this star does eventually die and disperse all left matter there is less matter than when the star started with, and in other words this means there is less 'material' or building blocks to form other stars. Wouldn't this point towards the idea that stars are getting smaller with the slowly decreasing matter count in the universe to make them. I would also predict that stars would be getting more and more 'metal rich' and it can be seen with the patterns of population star groups: http://hyperphysics.phy- astr.gsu.edu/hbase/starlog/pop12.html Lastly, if stars are getting more and more 'metal rich' does this mean they die younger and are ever increasingly dying younger until they are too unstable at birth to 'live' any significant life?


title: What knowns conflict with this spacetime curvature concept?

text: A 1-d line can be curved and made infinite by being on a 2d plane, and having the 2d plane wrap. We can do this for a 2d sheet in 3d as well. The hypothesis is that somehow 3d (4d spacetime) may be wrapped and infinite. The consequences of this would be that if you kept traveling through space (accounting for curvatures where mass distorts the spacetime fabric), it would be possible to end up right back where you started (e.g. earth). The idea is similar to initiating a wave on the oceans of a planet at the north pole, and having the wave ripple across the planet on the opposite side of the curved 2d space. So even though from a 2d perspective, it's infinite, defining an "opposite point in the 2d universe becomes possible. (On a side note, the strange three-90 degree triangles we get on a sphere would be somewhat analagous to the strange relativistic effects we get in 3d, due to curvature of spacetime) So assuming the analagous construct for 3d (4d spacetime), it would mean the big bang would presumably have occurred at one point. As the universe expands, all matter would actually be all converging in a big crunch, but not in the way we normally think of it. It keeps expanding, and converges on the "opposite" side of the 3d universe. Given our limited viewing horizon, perhaps we're in a phase of the universe's traversal toward the opposite side that give the present conditions today. This of course requires that spacetime is curved in such a way that ends meet, and do become continuous (analagous to the surface of a sphere) Neil Tyson mentions that perhaps much farther in time, when we won't be able to see any stars in the sky, due to expansion, we will be missing chapters or evidence to be able to piece together that the night sky even used to have stars. He mentions perhaps we're already in such a phase, only for some other permanently unobservable past event. If we assume this hypothesis of an opposite point of 3d space, then there would be a point in time of a maximum expansion point, but afterwards, things would come together, and stars would appear again. Eventually, the night sky would just get brighter and brighter (assuming stars haven't burned all their nuclear fuel, and still have light to shine), since they will be converging toward the opposite point of the big bang. This is all just an amateur's hypothesis inspired by the COSMOS series. I don't have lots of formal training in astrophysics, so I can generate some off-the-wall ideas, but I understand that there is most likely conflicting evidence for this hypothesis. I wanted to get some thoughts on what those might be. I decided to just post my idea, and have the relevant info get flung at it, rather than going about it in the opposite manner. Thanks for reading.


title: Derivation of the Fourier Transform, what is the Fourier series for non periodic signal

text: I had learned about the Fourier series for periodic signal and when it come to non-periodic signal, I got problem I don't understand. Here is the content that I learned. http://cnx.org/content/m0046/latest/ To put it simply, according to fourier series $$s(t) = \sum_{k=-\infty}^{\infty}c_ke^{j\frac{2\pi ft}{T}}dt$$ while for a signal that have infinite period it become like $$\lim \limits_{T \to \infty}S_T(f) \equiv s(t) = \int_{-\infty}^{\infty}S(f)e^{-j2\pi ft}df$$ So, the frequency is various but in the explaination it say that let $f$ be fixed and also, the figure 1 makes me confuse. At the beginning, k was integer index so each spectral line $c_k$ must correspond to the index k but when the spectral lines come closer that mean there must be more $k$ point in an interval $[a,b]$ For example: if k is integer only then in the interval $[0,10]$ there are only 11 index k that will yield 11 spectral lines at each index but when there are more spectral lines to make it continuum the number of k in the interval has to increase so k now is not integer anymore. Can anyone explain it to me By the way, just an extra question, does Fourier transform and Fourier coefficient are the same.


title: For semi-insulating GaAs is the bandgap bigger or smaller than the pure GaAs?

text: The carrier concentrations are lower in semi insulating semiconductors, therefore, can we say that their bang gap is wider in terms of energy (eV)? If they have wider band gap how much bigger is it approximately and what parameters affect it? i.e. is temperature one of them?


title: Ways of finding average accelleration

text: So I had a little debate with a professor on a simple problem of motion (See section). It basically comes down to how to find average acceleration. I did some research and couldn't find a definite answer. He claims that the equations of motion: $$x = x_i + v_it+0.5at^2$$ $$v_f^2 = v_i^2+2a\Delta x$$ only work for finding average acceleration, if the acceleration is constant (thus redundant). However, the equation $$v_f = at + v_i$$ works for finding average acceleration, regardless of $a$ being constant. Is he correct? **Problem** : You and your little brother are rolling toy cars back and forth to each other across the floor. He is sitting at $x = 0$, and you are at $x = 4.0 $ $m$. You roll a car toward him, giving it an initial speed of $2.1$ $m/s$ . It stops just as it reaches him in $3.0$ $s$. To find the accelleration the obvious answer is ${0-2.1 \over 3} = -0.7$ $m/s^2$ using $v_f = at + v_i$, and that's what the book says. However, (not having done physics in a full year), I choose the less simple way and used the equation: $v_f^2 = v_i^2+2a\Delta x$. So: $0 = 2.1^2+2a(4)$, then $a = -2.1^2/8 \approx -0.55$ $m/s^2$. Two completely different answers. So now I argue that this situation can never be, because when you use $x = x_i + v_it+0.5at^2$, for this problem: $0 \not= 4 + 2.1*3-0.5*0.7*3^2$. Moreso, is this problem realistic (assuming that acceleration was actually $0.7$ $m/s^2$)?


title: Sound Intensity Measurements convert to Sound Power

text: If you need to have Sound Power results and the location that the microphone needs to be to make the measurement(per the standard) is not accessible. There is a machine part in between the measurements surface and mic location. In addition the noise source is a tire rotating on a drum. I was told to make the multiple measurement using SI and then convert to Pw.   1. How much will the reflective surface influence the SI measurement?   2. To calculate the point required for the mic position is ~ 1/2 the distance specified by the standard. Can you using the inverse square law to correct the value?   3. What is the best way to convert SI to Pw?   4. Does anyone have a better way to make this measurement without a lot of R&D studies to understnad the results?   5. How can I validate my results if this approach is used?   6. Can anyone share a public reference paper to help with this issue? Thanks Tony


title: Reason for strange "canyons" of dust on the cooling platform

text: I took apart my cooling platform Something like this :![Cooling Platform](http://i.stack.imgur.com/kIPFr.jpg) And I found something like this inside : ![Canyon of dust 1 ](http://i.stack.imgur.com/IyZes.jpg) ![Canyon of dust 2 ](http://i.stack.imgur.com/4yX7K.jpg) ![Canyon of dust 3 ](http://i.stack.imgur.com/udFJ5.jpg) I wonder why around the hollows are the "canyon like" structures. PS: I also drop this platform on the floor, maybe this could be a reason ? EDIT : Everybody seems (except Brandon Enright answer) to ignore the word "dust" in this question. Those structures are not in plastic itself. It is dust on the ON the plastic. Proof : ![Dust canyon](http://i.stack.imgur.com/SCmXW.jpg) Completely wiped out with my finder : ![Dust canyon wiped](http://i.stack.imgur.com/my18U.jpg)


title: Basic buoyancy question

text: If I have a cup of water filled with air at the bottom of a pool, then when the cup is "upside down" the air doesn't leave because the water pressure is pushing it up against the top of the container. But then when I flip the cup, such that it is no longer upside down, the air rushes upwards. But why on earth does this happen? I mean the water's pressure is pushing this air downwards against the bottom of the container. If I were to draw an FBD which force is causing this air bubble to rush upwards? Thanks