title: Action Reaction when I pushing a trolley?

text: I tried to explain how those force work but I can hardly figure it out. I exerted a force on the trolley and there will be a force on trolley on me as well. This is the newton's third law. But why the trolley will move? Where's the force come from


title: Dissipation when the temperature is not constant

text: Consider a process where some chemical species diffuses from one part of a system (which I'll call $A$) to another ($B$) at a rate $r$ $\text{mol}\cdot \mathrm s^{-1}$. If the system's temperature is constant and homogeneous, we say that energy is dissipated at a rate $$ D = r(\mu_B - \mu_A), $$ where $\mu_A$ and $\mu_B$ are the chemical potentials of the particles in parts $A$ and $B$ of the system respectively. The dissipation is always positive, because $D=T d_i\\!S/dt$, where $d_i\\!S/dt$ is the rate at which entropy is produced within the system due to the transport process. However, if the temperatures of the two parts of the system are different then the second law says $$ \frac{d_i\\!S}{dt} = r\left( \frac{\mu_B}{T_B} - \frac{\mu_A}{T_A} \right) \ge 0, $$ which means that the above expression for $D$ can be negative if $T_A>T_B$. So my questions are   * Is the term "dissipation" generally thought to be meaningful in systems that don't have a constant, homogeneous temperature?   * If yes, what is the correct expression for it in the above scenario?   * Most importantly, does anyone know of a reference where the concept of dissipation in systems with a non-constant or non-homogeneous temperature is discussed?


Why are polymer representations called "polymer representations"? Polymer representations deal with non-continuous unitary representations of groups acting on nonseparable Hilbert spaces (see e.g. http://arxiv.org/abs/0704.0007). But what has this to do with polymers?


title: Turbulence forming between the coaches of rail

text: is there any specific reason why rail coaches are connected by a rubber sheet; not of any other material ? I watched a documentary on trains (Discovery Channel) which said it connecting coaches by rubber sheets reduces turbulence that are formed between the rail coaches. What exactly does it mean ?


title: Why are two eigen-state-kets with different eigenvalues orthogonal?

text: The operators $J_1^2$, $J_2^2$, $J_{1z}$, and $J_{2z}$ are mutually commuting operators. Likewise, $J_1^2$, $J_2^2$, $J^2$, and $J_z$ are mutually commuting operators. The two groups are incompatible, and the simultaneous eigenkets along with their eigenvalues are given by: ${J_1}^2 \left|j_1,j_2;m_1,m_2\right> = j_1 \left(j_1+1\right) \hbar^2 \left|j_1,j_2;m_1,m_2\right>$ ${J_2}^2 \left|j_1,j_2;m_1,m_2\right> = j_2 \left(j_2+1\right) \hbar^2 \left|j_1,j_2;m_1,m_2\right>$ $J_{1z} \left|j_1,j_2;m_1,m_2\right> = m_1 \hbar \left|j_1,j_2;m_1,m_2\right>$   $J_{2z} \left|j_1,j_2;m_1,m_2\right> = m_2 \hbar \left|j_1,j_2;m_1,m_2\right>$ and ${J_1}^2 \left|j_1,j_2;j,m\right> = j_1 \left(j_1+1\right) \hbar^2 \left|j_1,j_2;j,m\right>$ ${J_2}^2 \left|j_1,j_2;j,m\right> = j_2 \left(j_2+1\right) \hbar^2 \left|j_1,j_2;j,m\right>$   $J^2 \left|j_1,j_2;j,m\right> = j \left(j+1\right) \hbar^2 \left|j_1,j_2;j,m\right>$   $J_z \left|j_1,j_2;j,m\right> = m \hbar \left|j_1,j_2;j,m\right>$ I read that each set of eigenkets are mutually orthogonal [1] (for eigenkets corresponding to different sets of eigenvalues). This is what I don't understand. In principle it makes sense, but when I plug in numbers I don't get zero for the inner product. For example take the first eigenket: $\left|j_1,j_2;m_1,m_2\right>$. If I choose different eigenvalues for this eigenket (e.g. let $j_1 = 0$ and then let $j_1 = 1$) I get the following: for $j_1 = 0$ I can have:   $\left|0,j_2;0,m_2\right>$ for $j_1 = 1$ I can have any of the following, since $\left|m_1\right| \leq j_1$:   $\left|1,j_2;-1,m_2\right>$   $\left|1,j_2;0,m_2\right>$   $\left|1,j_2;1,m_2\right>$ If I take the inner-product of the $j_1 = 0$ eigenket with any of the $j_1 = 1$ eigenkets I do not get zero, e.g.: $\left<0,j_2;0,m_2 \mid 1,j_2;-1,m_2\right> = {j_2}^2+{m_2}^2$ which is non-zero unless $j_2 = 0$. What am I misunderstanding here? How do you show that eigenkets with different eigenvalues are orthogonal?


title: How do I calculate the induced metric in the Gibbons–Hawking–York boundary term?

text: This question concerns the expression for the induced metric in the explicit variation of the GHY boundary term. Just how is that expression derived in detail from the definition of the induced metric in this case? The expression is h = g - nn where indices have been left off.


title: What's the difference between “evidence of a new particle” and “discovery of a new particle”?

text: Today’s exciting press release from Tevatron on the Higgs boson keeps its head cool and say that physicists saw a “hint” of the Higgs boson because the signal is barely above the two-sigma level. In an effort to explain the reasons for this, the press release says: > Physicists claim evidence of a new particle only if the probability that the > data could be due to a statistical fluctuation is less than 1 in 740, or > three sigmas. A discovery is claimed only if that probability is less than 1 > in 3.5 million, or five sigmas. I wonder: what's the difference between “claim[ing] evidence of a new particle” and “a discovery” (of said new particle)? Is this the common vocabulary for these levels (three and five sigmas) of signal? The reason I ask is that I can understand the need for having different vocabulary to talk about discoveries with different levels of certitude, and I could see, say, the need for using “discovering a particle” and “confirming the discovery of a particle”, but “evidence” and “discovery” seem semantically very close to me. * * * **Edit —** to be clearer, the question is: how universal are the choices of terms (“evidence”, “discovery”) and associated incertitudes?


title: Has anyone studied a statistical scaling law for the universe?

text: How do named objects in the universe scale? Is there a predictable curve for an ordered list, say {atom, animal, planet, solar system, galaxy, etc}? Can you then use the analysis to predict when the next larger structure will be seen?


title: Ricci identity/Riemann curvature tensor and covectors

text: Can somebody please explain to me how the following statement is true? > The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity > $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv R^c_{dab}V^d$$ where > $\nabla_a$ denotes the covariant derivative. It is linear in $V^c$, hence > may be shown by the Quotient theorem to be a tensor. Now, I can see that the $R^c_{dab}$ is a tensor by construction -- based on the LHS of the Ricci identity. However, I don't understand how the linearity in $V^d$ comes to play. * * * Also, it is given that for covectors, the Ricci identity takes the form > $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V_c\equiv -R^d_{cab}V_d$$ How does this follow from the Ricci identity for (contravariant) vectors? If I write $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V_c=(\nabla_a\nabla_b-\nabla_b\nabla_a)(g_{cd}V^d)$$ and in GR, the Levi-Civita connection has that the metric is covariantly constant, we have $$(\nabla_a\nabla_b-\nabla_b\nabla_a)(g_{cd}V^d)=g_{cd}(\nabla_a\nabla_b-\nabla_b\nabla_a)V^d\\\=g_{cd}R^d_{eab}V^e=R_{ceab}V^e=R^d_{cab}V_d$$ Where has my minus sign gone? I have read that you can the Ricci identity for covectors by arguing using the fact that the Levi-Civita connection is symmetric, but I don't know how they mean. Thanks in advance for any help!


title: Geometrical interpretation of complex eigenvectors in a system of differential equations

text: Let's consider a system of differential equations in the form $$\dot{X} = M X$$ in two dimensions ($X = (x(t), y(t))$). In the case that $M$ has real values, it is easy to give a geometric interpretation of the eigenvectors in the $(x, y)$ plane: they are the directions along which the dynamical system is "sucked" or "expelled" from the stable point. If the eigenvalues are complex, then the eigenvectors are complex too. Let's say the eigenvalues are purely imaginary, so that the trajectory is an ellipse. Can I draw anything in the $(x, y)$ plane that is related to the eigenvectors? In particular, do the eigenvectors have any simple relation to the rotation and eccentricity of the ellipse?


title: can we have a parallel earth made of dark matter?

text: In the sense that the two planets move together and has intelligent beings with their own forces and particles, but they just do not interact with (are invisible to) us?


title: What are the determining factors for the velocity of orbiting bodies?

text: Please bear with me, as I'm not in the field of physics, this question may seem a bit simple. The scenario is the following; > A specific stable orbit radius of a small body, say a satellite, to large > body, say a planet, is known. My questions then are;   1. Is the velocity of said body a single value, or is there an interval of possible velocities?   2. What are the determining factors for the value/interval of velocit(y/ies)? (Assuming the orbit radius stays constant, and is stable)      * The masses of the two bodies?     * Orbit radius?


title: Why nuclear tests move under ground since 1964?

text: According to this database: http://www.ga.gov.au/oracle/nuclear-explosion.jsp Since 1962, 99% of USA Nuclear tests was underground. Since 1964, 99% of Soviet Nuclear tests was underground. Is it ecologies nuance, or what?


title: Force needed to rotate a load horizontally?

text: I need to rotate a 500lbs(226.8kg and balanced at center of load) telescope horizontally, at up to 90 degrees per sec. Ideally I would like to reach the max speed in a tenth of a sec. I need to buy a stepper motor capable of achieving the force needed to quickly accelerate and rotate the telescope. I found the following equation, and I just need someone to double-check my work and make sure I don't screw up a detail, and buy an unnecessarily over-powered motor, and definitely not an under-sized model (:P): \begin{align} kg &= 226.796185 \\\ radius \text{(meters)} &= 0.4572 \\\ MOI \text{(moment of inertia)} &= kg*radius*radius \\\ degrees &= 90.0/360.0 \\\ rotsPerSec &= degrees/60.0 \\\ radiansPerSecSQRD &= 2*π*rotsPerSec \\\ newtonMeters &= MOI * radiansPerSecSQRD \end{align} So basically, just moment of inertia multiplied by the radians per sec gives me the Newton-Meters of force that I need. Does this check out?


title: What if a normal digital clock used to measure time in twin paradox?

text: Last night I was watching a youtube video about twin paradox. It still continues to baffle me :). In that video, the speaker conceptualizes a clock that measures time by reflections of light. If we consider speed of light as reference to our time measurement, I guess it is natural to expect that our time measurements will be warped if we are traveling at speeds comparable to that of light. I was wondering, what if if we measure time with something else as reference? Say, a digital clock, assuming it works under such extreme speed/acceleration... So, my questions:   1. Would our time measurements then be still warped?   2. I think, even if the measured times indeed gets warped, is it not just the perceived time for the traveling twin? Logically, the twins have lived for the same amount of time (as measured from a digital clock on earth) and should have grown/aged similar. Why would the twin who stayed back on earth be aged more? I might be totally wrong, but I would appreciate if somebody helps me understand where I went wrong. Thanks for your time.


title: Electromagnetic field of unpolarized light

text: I need help in finding an expression for the instantaneous electric and magnetic field of unpolarized light in order to write down and evaluate the time-averaged norm of the Poynting vector (i.e. the intensity of unpolarized light). I expect this to be a superposition of some basis functions, but how many and how do they look like? Are these linear polarized waves with random orientation and phase shift? $$\bar{E}(t,\bar{x})=\sum_{i}\bar{E}_{0,i}e^{i(\bar{k}\cdot\bar{x}-\omega t+\delta_{i})}$$ with $\|\bar{E}_{0,i}\|=\|\bar{E}_{0,j}\|$ ?


title: Is this a valid understanding of Newtonian mechanics?

text: This is a conceptual understanding of Newtonian mechanics. What the laws mean, how we know they're true, etc. I'm looking for criticism. I know this is really border line on the "don't ask questions that can't be answered" rule, but here we go anyway. **The Laws**   **First Law:** _A body retains its velocity unless acted upon by an outside force._   This first law is actually a definition, not an empirical statement. _Body_ can be defined based on sense data, as can be _velocity_. But _force_ is as yet undefined. As it is the only undefined term in the statement, the statement must be a definition. A _force_ is defined as "that which is said to 'act upon' a body when that body's velocity changes, the immediate cause for a change of velocity". This contrasts with the Aristotelian definition, which is "that which is said to 'act upon' a body when that body has non-zero velocity, the immediate cause for a change in position". **Second Law:** _A body's acceleration is a function of its mass and the force acting upon it, according to the relation F = ma._   This second one is still just a definition. It's implied that mass is a function of the specific body. We can apply this to predict accelerations (see _Applications_ , below). **Third Law:** _When one body exerts a force on another, that other exerts an equal and opposite force on the first._ This is the only of the three laws that is empirical. It is not a definition as force was already defined by the first two laws, and it cannot be proven logically from the first two. It would have to be proven by some sort of experiment. **Empirical observations**   The most glaring omission from the Laws is what can cause a force. Newton probably just meant it to be implied that forces were caused by collisions between bodies. In any case, you can demonstrate experimentally that collisions cause forces. This is the first apparent strength of Newtonian physics over Aristotelian physics. The Aristotelian definition of a force is valid (no such thing as a false definition), but in Newtonian mechanics it's much easier to express the relationship between collisions and forces. **Applications**   We can apply these laws to calculate masses, forces, and accelerations.   1. _Define a unit mass._ Mass is an unchanging property of a body and therefore we may simply take an arbitrary body and define its mass as the unit.   2. _Notice that when the same object collides with the same object in the same way, it has the same acceleration._ For example, roll the same ball into it from the same height down the same slope, or hit it with a pendulum dropped from the same height. This provides inductive evidence that that sort of collision always produces the same force (F = ma, m is unchanging, a is unchanging, therefore F is unchanging).   3. _Now apply that collision to other objects to deduce their mass._ If an object accelerates x times as much as the object of unit mass, it has 1/x mass.   4. You can now measure mass.


title: Why if the torque equals zero measured from one point in space it equals zero measured from any other point?

text: I've heard it from two teachers and saw a task with a solution based on this assumption: > If the net torque is zero when measured from one frame of reference, it is > equal to zero in all other frames of reference This actually seems clear, especially if you put it to simple words like: > If a body doesn't rotate from one point of view, it won't start rotating if > you look at it from another point of view Still, I wanted to test it and made such an image: ![torques](http://i.stack.imgur.com/P8m9v.png) I calculated the net torque measured from point A (I'll skip the designations such as $m_1, r_1, m_2$ etc. and get to the value): $$1m\cdot1N + 5m\cdot1N - 4m\cdot1.5N = 0$$ Now, let's calculate the net torque from B: $$-2m\cdot1N + 2m\cdot1N - 1m\cdot1.5N = -1.5Nm$$ From this it seems, that the body doesn't rotate around an axis in A, but does around the axis in B.


title: Chemical potential of a Bose gas

text: In my course, there is this fact : > In a Bose gas, the chemical potential $\mu$ must always be lower than the > smaller level of energy $\epsilon_0$. I find this strange, because if we put a Bose gas in a big container of $\mu > \epsilon_0$, what happens ?


title: how does a sniper scope work?

text: How does a sniper rifle scope enables us to pinpoint the exact location even though the lens in situated 5-6 inches above the muzzle. The bullet leaves the muzzle and hits the target exactly where the cross-hairs in the scope pinpointed? How?


title: Thermal equilibrium in general relativity

text: The Newtonian condition for thermal equilibrium for a static system is $T = \mathrm{const}$. In this homework I'm asked to show that it's curved space generalization is $T(-g_{00})^{\frac{1}{2}} = \mathrm{const}$, where $g_{\mu \nu}$ is the metric (assumed stationary). But I have no idea how to start.


title: What fraction of peak horsepower do typical 4 door passenger vehicles use?

text: I was surprised when I looked at the power rating of the engine used on a Humvee. It's only ~190 horsepower, which is exceeded by many sedan engines. So an obvious question is why doesn't my Camry SE burn more gas than a Humvee and I think it's just because during regular use, it doesn't use anywhere near 250 HP. Does anyone know about how much mechanical power a typical sedan engine puts out during highway driving? Also, can someone speculate what the torque vs RPM graphs of a sedan engine and a Humvee engine look like and what the typical operating points are?


title: Tensor perturbation inflation

text: During inflation the metric is de-Sitter so $dt^2-d\underline{X}^2 $. I know that the eqn.motion governing GW's from inflation (tensor perturbations) is $$2H\dot{h}+\ddot{h}-\nabla^{2}_{i}h~=~0,$$ derived from varying $$S^{(2)}~=~\int \frac{a^{2}(t)}{2}~\partial^{\mu}h ~\partial_{\mu}h ~d^{4}x.$$ See paper 1 and paper 2. Surely this isn't valid during inflation because it isn't de-Sitter. I have read lots of papers/lectures where quantum fluctuations are found by deriving equations of motion from a metric with an $a(t)$ and then Fourier decomposed. Surely during inflation the metric can't contain an $a(t)$. So why do all these lectures/papers use it?


title: How much can a thin layer of hi-speed material within a low-speed volume block a wave due to total internal reflection?

text: Consider a block of isotropic material with compression wave velocity associated with it, $v_1$. Consider a thin flat layer of high compression wave velocity $v_2, v_2>v_1$ that is buried within the block. It is known that if the wave hits the layer at an angle of incidence above a `critical angle`, the phenomenon of `total internal reflection` occurs. What is the estimate of the reflection coefficient for that phenomenon? How near it approaches $1$? How it depends on the hi-speed layers thickness in relation to compression wave length? Is it reasonable to say that the transmission of wave energy through the hi-speed layer occurs only within the cone with apex at the wave source and the apex angle equivalent to critical angle? In the picture below `f` is the thickness, in pale red is the low-speed volume, in purple is the hi-speed layer, in blue is the total internal reflection raypath, V1 and V2 are velocities. ![enter image description here](http://i.stack.imgur.com/J0g3S.png)


title: Sign Conventions for Dirac equation

text: Is it possible to have the Dirac sign convention, (-,+,+,+) and at the same time use the metric $$dt^2-dx^2-dy^2-dz^2$$ i.e have opposing Dirac and metric tensor conventions?


title: Is Hamilton-Jacobi equation valid for only conserved systems?

text: From derivation of Hamilton-Jacobi (HJ) equation one can see that it is only applicable for conserved systems, but from some books and Wikipedia one reads the HJ equation as $$\frac{\partial{S}}{\partial{t}}+H(q,\frac{\partial{S}}{\partial{q}},t)~=~0$$ there is an extra variable $t$ in Hamilton's function. When Hamilton's function explicitly depends on time , this means that energy is not conserved. So where is truth?


title: What is a good model for computing water dropping on a surface?

text: _Before introducing my question I would like to outline the fact that I'm a coder, so I can be wrong when using some kind of terminology._ What is a good model for computing the flow, the shape and all the geometric properties of the water? ( and other similar liquids in general, like mixtures based on water or some more oily and viscous substances ) I assume that this kind of simulation will be based on particles, which means entities that are part of the domain that defines the "drop" of water, but what I need is the math, the equations and the mathematical model/s; I have no idea what is the math behind a liquid that meets a surface and if there is any algorithm or model that can be translated into machine code to compute that. To keep things simple I'm not considering the air and what kind of properties the air can possibly modify, or any other extra variable outside the domain of this drop of liquid, like adding a inclined plane where the drops, let's say that for now I'm considering a vector and an X quantity of liquid, this quantity is than sprayed or just dropped on the surface and I should be able to reproduce a realistic dispersion of the water on the surface, even better I should be able to compute different liquids with different properties at the same time, like water and oil.


title: Curvature based derivation of Schwarzchild Metric

text: I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzchild metric, if there exists such a proof?


title: What are the correct initial conditions for the moon (in a simulation)?

text: So I've modeled the interactions between the sun and all the planets (and the interactions between the planets) using Verlet integration. I've used data from Wikipedia for masses, distance from the sun etc. I initialized the initial velocities of the planets via the critical velocity equation. This produces nice stable velocities. I'm unsure of how to calculate the initial velocity of the moon so that it stays in orbit around the earth.


title: Using a Plasma Globe to generate real random numbers

text: I have been thinking how to easily generate real random numbers when I cam across the small USB-powered Plasma globe. My idea would be to place it in front of an light sensor and read the sensor's values with an Arduino microcontroller. Does this approach make sense? Would I need to convert the sensor data in some way?


title: Can atmospheric pressure literally push electromagnetic waves?

text: I work for an IT company and some time ago we had an issue with our wireless internet. We are 5 miles away from the ISP's antenna. Our Sys Admin expressed the view that the electromagnetic waves are being literally pushed to the ground because of high atmospheric pressure and the signal gets disrupted by EM waves being scattered by objects on the ground. Is this a plausible explanation? I would be interested in the limiting conditions (what must be happening for this to occur?).


title: Need to find the Vab of this circuit!

text: Find Vab in the circuit http://imgur.com/IqZ7Yis I uploaded a picture of the circuit. I tried adding 10 and 5 (top) because they appear to be in series. I also added 10 and 15 (bottom two) because they also look in series. Then I did Req = 1/(1/15+1/25) = 9.375 ohms. Then I used V = IR = 9.375(4A) = 37.5 V. That is not even one of my answer choices. The answers to choose from are: 10V , -5V, 15V, and -10V. What did I do wrong? http://imgur.com/IqZ7Yis


title: A question about a vibrating membrane

text: The usual way to model a vibrating membrane is by using the wave equation. Is it possible to do that from "within"? Probably the answer is yes, but where can I see it done explicitly. What I mean is can we model the membrane as a two dimensional Riemannian manifold without any specific embedding in $\mathbb R^3$ plus equations involving the metric (or form, tensors etc.) such that at any time it is isometric to the surface given by the graph of the function satisfying the wave equation at that time. Higher dimensional generalizations are also interesting.


title: Rotation motion like the number 8

text: I'm a college student majoring in culinology and I'm trying to find out the reason or method of the number 8 motion. Responses doesn't have to be in culinology examples, but that would be a great help if it is. From my observations of the circular motions v.s. the figure 8 motions, if I was to mix cake batter with big circular motions in a bowl; the center would not be affected. Mixing the cake batter with small circular motions would have the opposite effect as making big circles. Now mixing the cake batter in a figure 8 motion would hit all the spots, (center & outer area) but you'll have to turn the bowl to make this true. I think the amount of velocity and the force used to mix in these ways would change things.


title: Electromagnetic fields vs electromagnetic radiation

text: As I understand, light is what is more generally called "electromagnetic radiation", right?   The energy radiated by a star, by an antenna, by a light bulb, by your cell phone, etc.. are all the same king of energy: electromagnetic energy, i.e. photons traveling through space. So far, so good? (if not please clarify) On the other hand, there are these things called "electromagnetic fields", for example earth's magnetic field, or the magnetic field around magnets, the electric field around a charge or those fields that coils and capacitors produce. Now here is my question:   * Are these two things (electromagnetic fields and electromagnetic radiation) two forms of the same thing? or they are two completely different phenomena?    * If they are different things, What does light (radiation) have to do with electromagtetism? I'm not asking a complex theoretical answer, just an intuitive explanation. Thanks.


title: Einstein's Field equations and impulse-energy tensor

text: I premise that I haven't yet studied General Relativity, but in Relativistic Electrodymaics I have knowed impulse-energy tensor of Electromagnetic Field. I know in Einstein's equations there is impulse-energy tensor $T_{\mu\nu}$ too: $$ R_{\mu \nu} - {1 \over 2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} $$ I suppose that it's connected with matter in the space, being that equation about gravitation theory. My question is: has got any sense put in the Einstein's field equation the impulse-energy tensor of EM field? If yes, what is the physical meaning? What are the consequences?


title: Normal force at the inclined plane

text: A particle mass $m$ moves down a frictionless inclined plane. Does a normal force cancels a normal component of $m$ **g**?


title: Box normalisation and Particle in a box - Quantum Mechanics

text: I have been long itched by this issue of subtle difference between box- normalised free particle and infinite-dimensional potential well. Choosing a one dimensional case, the Hamiltonian in two cases respectively are : $$ \hat H_f = \frac{\hat p^2}{2m} $$ $$ \hat H_b = \frac{\hat p^2}{2m} + V(x) $$ where $V(x)$ is zero inside the box and infinity everywhere else. Now the eigen-functions of both the hamiltonian satisfy similar boundary conditions(BC), that the wavefunction has to vanish at the end points of the box. So, $$ \psi_f(x) = L^{-1/2}e^{ikx} $$ $$ \psi_b(x) = \sqrt{\frac{2}{L}}sin(\kappa x) $$ where $k = \frac{2n\pi}{L} $ and $ \kappa = \frac{n\pi}{L} $ Now the difference in the functional form is what is reflected in the boundary conditions. And I have tried to convince myself that the difference in the functional form is due to the fact that $\psi_f$ is eigenfunction of $\hat p$operator as well (since $ [\hat H_f,\hat p] = 0 $). But that it is not so for $\psi_b$ since its hamiltonian($H_b$) doesn't commute with momentum. So my question is, is this all the difference it makes or is there more to it ?


title: confusion in discrete transform to solve kronig penney matrix equation in fourier space

text: I have a periodic potential $$V(x) =\sum_{K}e^{iKx}V_{K} =\sum_{n}e^{\iota2\pi nx/a}V_{n} $$ where $K =\frac{2\pi n}a$ is the reciprocal lattice vector and $a$ is the lattice constant and $n =\pm 0,\pm 1,\pm 2,\pm 3... $so on. I would like to find the fourier coefficients $V_{K}=V_{n}$ corresponding to a particular $K$ or $n$. Suppose I have a vector for $V(x)$ having 10000 points for $$x = 0,0.01a,0.02a,...a,1.01a,....2a....99.99a$$ such that the size of my lattice is $100a$. Now $n$ will also go from $-50$ to $+49$. Thus I have defined the potential for 10000 points on a 1D lattice of 100 atoms. FFT on this vector gives 10000 Fourier coefficients. By theory of discrete fourier transform (http://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l7.pdf), the $K$ values corresponding to these fourier coefficients are $\frac{2\pi n}{(NX)}$ where $N$ is the no. of readings = 10000 each separated by spacing $X=0.01a$ with $n=0,1,2,3,...9999$. But I started with $K$ that had the form $K =\frac{2\pi n}a$. What am I missing ? How do I correctly find the fourier coefficients $V_{K}$ numerically using DFT (Fast Fourier transform method in Matlab can be used) ? For reference on the Kronig Penney fourier space matrix equation, look here http://www.physics.buffalo.edu/phy410-505/topic5/index.html


title: Bulk flow of air in a long tube between Antarctica and Australia?

text: I have a 5km diameter clear plastic tube which is open at each end and runs from the center of Antarctica to Lake Eyre in Australia. The tube is on the ground where it can be and at sea level on the ocean. Will there be bulk flow of the air in the tube? If so, which way will the air flow?


title: Unitary representations of the diffeomorphism group in curved spacetime

text: In (special) relativistic quantum mechanics there is a standard argument that says that the (rigged) Hilbert space of states $H$ should be equipped with a projective unitary representation $U$ of the Poincaré group $P$ that goes something like this: Suppose we have two observers $O_1$ and $O_2$ and $x\in P$ is the Poincaré transformation that maps $O_1$s coordinate system into $O_2$s coordinate system. Then, if $O_1$ and $O_2$ attempt to measure the same thing, they will actual find different states, $\left| \psi _1\right>$ and $\left| \psi _2\right>$ respectively, in $H$ (for example, if you first find the vector $(1,0,0)$, rotate your axes by $\pi/2$ and measure the _same_ vector, the numbers you now record are going to be $(0,1,0)$). In fact, this gives us a map of states: $\left| \psi _1\right> \mapsto \left| \psi _2\right>$ (it's only going to be well-defined up to phase). Let us call this map $U(x)$, so that $\left| \psi _2\right> =U(x)\left| \psi _1\right>$. However, if we accept the principle of relativity that physics should be the same in two reference frames related by a Poincaré transformation, then we better have that $$ \left| \left< \phi _1|\psi _1\right> \right| =\left| \left< \phi _2|\psi _2\right> \right| =\left| \left< U(x)\phi _1|U(x)\psi _1\right> \right| $$ for all (normalized) $\left| \psi _1\right>$ and $\left| \phi _1\right>$ because this represents a probability (of course, $\left| \psi _2\right> :=U(x)\left| \psi _1\right>$ and $\left| \phi _2\right> =U(x)\left| \phi _1\right>$). Wigner's Theorem then tells us that this gives a projective unitary representation of $P$ on $H$. (Note: I am allowing some elements of $P$ to be represented by antiunitaries.) Now, if you try to do the same argument in curved spacetime, you run into the obvious problem that you will not in general have an analogue of the Poincaré group (I believe there exist spacetimes which possess no Killing fields); however, it naively seems as if the principle of general covariance suggests that we should 'upgrade' the isometry group of spacetime to the entire diffeomorphism group of the spacetime in the above argument. (In particular, I don't see how this argument makes crucial use of the fact that both observes coordinate bases are orthonormal.) That then, would imply, that the Hilbert space of states in a quantum theory of curved spacetime should possess a projective unitary representation of the diffeomorphism group of that spacetime. This, however, at first glance, seems like it would be false. So, where does the argument break down if you replace the Poincaré group in special relativity with the diffeomorphism group in general relativity? Or, is it in fact the case that we should obtain a unitary representation of the entire diffeomorphism group.


title: Effective Field Theories of QCD

text: Recently, I am studying the online course Effective Field Theory provided by MIT OCW. Prof. Stewart gives a nice picture to summarize the effective theories: ![Example Effective Field Theories derived from Quantum Field Theory](http://i.stack.imgur.com/lXZ23.jpg) As a newbie in this field (I only have a knowledge of QFT on the level of Peskin & Schroeder's book), I am quite confused with so many terminologies. What do they stand for and Why do we need these different theories? Could someone explain this picture in depth (especially on their advantages and limitations)? Any help will be greatly appreciated.


title: Meaning of $k$ in Sachs-Wolfe formula for angular power spectrum

text: I've seen the formula for the angular power spectrum of the CMB written as $$C_\ell = \frac2\pi \int\left|\Theta_\ell(k) \right|^2 k^2dk, $$ where $\Theta_\ell(k)$ is the temperature contrast at a given wavevector $k$ (the specific reference I'm using is the book "Primordial Cosmology" by Peter and Uzan (2009), pg. 313). The expression for $\Theta_\ell(k)$ is derived from linear perturbation theory in a cosmological background, using _conformal time_ $\eta$, as opposed to cosmic time $t$. So it seems to me that $k$ in this formula is _not_ the physical wavenumber, but rather the conformal wavenumber: $$ k = ak_\text{phys},$$ where $a$ is the scale factor. My issue is that if I'm using the conformal wavevector $k$ in the above formula for $C_\ell$, it seems to be non-gauge invariant. I can make a transformation where I send $a\rightarrow a/\lambda$, $k\rightarrow k/\lambda$, $\eta\rightarrow \lambda \eta$, and $x^i\rightarrow \lambda x^i$. This transformation doesn't seem to affect $\Theta_\ell(k)$, but does rescale $k$ so that the transformation of $C_\ell$ is $$C_\ell\rightarrow C_\ell/\lambda^3 $$ This does not seem like a very good feature for a physical observable. My question is how do I get a scaling invariant observable, and more generally what is the correct interpretation of $k$ in that formula?


title: Calculating Dynamics of a Thick Sheet of Material

text: So I'm working on a computational neuroscience problem, and I'm attempting to model skin dynamics for use in some perception models. The problem is as follows: Imagine you had a two dimensional sheet with some thickness. Imagine stimulating it with a probe with negligible spatial extent (i.e. a point source of stimulation) which can deliver arbitrary time varying stimuli (let's say for our purposes it's a sinusoid). The question is basically how can you determine $z(x,y,t)$, the lateral position of the sheet as a function of time for each point on the sheet, given knowledge of $z(0,0,t)$ (which is just how the probe is stimulating the sheet). The idea is basically that the probe will induce travelling waves that will propagate to every other point on the sheet. Now, imagine instead that your probe has spatial extent, in other words, now you know $z(x,y,t)$ for $(x,y)\in A\subseteq \mathbf{R}^2$ where $A$ is essentially the shape of the probe (which we can assume to be simple for now, like a circle or square). Then how can we determine $z(x,y,t)$ for $(x,y)\in\mathbf{R}^2$? My suspicion is that you could use the solution to the problem above and essentially integrate over $A$ (i.e. compute the response of the sheet elsewhere as the summation of all the contributions of the point sources of $A$). Finally, imagine instead of a sheet, you had some non-convex shape (like the hand for instance). How would you compute the sheet (skin) dynamics at another point given knowledge of how the probe is moving laterally. I think we've solved the first problem. Essentially, we experimentally derive propagation speeds and how sinusoids decay as a function of distance from the point at which the probe contacts the skin (for instance, we may decide that if you're stimulating the skin with a wave $z(0,t) = A\cos(2\pi f t)$, then $$z(r,t) = \frac{1}{r^{\gamma(f)}}z(0,t)$$ where the decay parameter as a function of frequency is experimentally derived. Then we extend this to arbitrary stimuli by decaying the wave in fourier space and then inverting it back into a time series. I'm wondering $(a)$ Is this solution correct even in the simple case? $(b)$ How does the solution change when you incorporate the spatial extent of your probe? $(c)$ How do you compute the skin dynamics at another point, where there isn't a straight line connecting contact point of the probe to the point in question? Any help would be greatly appreciated, or at least reference to some literature that might help me begin to think about how to solve this problem?


title: What is the axial transformation of a group, i.e. $SU(3)$?

text: The Gell-Mann matrices $\lambda^\alpha$ are the generators of $SU(3)$. Applying an SU(3) - transformation on the triple $q = ( u , d, s )$ of 4-spinors looks like this: $$ q \rightarrow q' = e^{i \Phi_\alpha \lambda^\alpha / 2} q.$$ So far I can follow and I also understand why the expression $\bar{q}q$ is invariant under this transformation. * * * Now my book defines axial transformations as $ q \rightarrow q' = e^{i \Phi_\alpha \lambda^\alpha / 2 \gamma_5} q$ and states that the expression $\bar{q}q$ is not invariant any longer under this transformation. What confuses me is the fact that the $\lambda$ generators of $SU(3)$ and $\gamma$ matrices are being multiplied in the exponent, even though the $\lambda$ have 3 and the $\gamma$ have 4 dimensions. * * * Maybe this is not a matrix product but some sort of tensor product? In that case, how should the exponential expression be understood? I suspect $\lambda$ and $\gamma$ commute as they act on different vector spaces. Or maybe it is a typo? Or maybe the $\gamma_5$ is not 4-dimensional in this context? * * *


title: Inclined plane problem with friction

text: A block of mass 6.0 kg slides from rest at a height of 2.0 m down to a horizontal surface where it passes over a 1.5 m rough patch. After crossing this patch it climbs up another incline which is at an angle of 30° to the ground. The rough patch has a coefficient of kinetic friction μ k = 0.25. What height does the block reach on the incline before it comes to rest? (I got the answer 1.625m. Is it correct ?)


title: For the two identical particles scattering, How can i identify two particles are bosons or fermions?

text: If two particles are scattered. How can i know those two particles are bosons or fermions?


title: Maxwell's equations of Electromagnetism in 2+1 spacetime dimensions

text: What would be different in the theory of electromagnetism if instead of considering the equations of Maxwell in 3+1 spacetime dimensions, one would consider 2+1 spacetime dimensions?


title: Pascal's law and hydrostatic forces

text: We know that Pascal's law states that the pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all direction through out the fluid such that the initial pressure variations remains the same. The force exerted is transmitted to every atoms as they are so close to each other. These atoms are sliding pass each other to spread the force to all atoms. If my understanding is true then why the same hydrostatic pressure is not getting transmitted same way as external force?


title: How to calculate the $x$-component of the angular momentum $L_x$

text: I am considering the hydrogen atom. Given $L_z$ and each of the $n,\ell,m$ values, is there a way to calculate $L_x$? In the same way that $L_z=\hbar m$ is there a similar expression for $L_x$?


title: Capacitors in parallel and dieletric

text: I'm trying to solve this problem: > _Two capacitors of capacitance $C_1=200pF$ and $C_2=1000pF$ are connected in > parallel and loaded to a potential difference of $400V$. Subsequently the > space between the plates of $C_1$ is completely filled with distilled water > (with $ε = 80$). Calculate the variation of the difference $ΔV$ voltage > across $C_2$, the polarization charge $q_p$ on the faces of dielectric, the > variation of electrostatic energy of the system $ΔU$._ Now in order to obtain a lot of information I calculated $Q_1=8.0 \times 10^{-8}C$, $Q_2=4.0\times 10^{-7}C$, $C_{total}=1.2\times 10^{-9}F$ and $Q_{total}=4.8\times 10^{-7}C$, then I calculated the new capacitance of $C_1$ because of the dieletric so $C_{1ε}=1.6\times 10^{-8}$ Here I'm stuck, how can I calculate the variation of the difference of potential $ΔV$ voltage across $C_2$ since the capacitors are in parallel? Then how can I calculate the polarization charge $q_p$ on the faces of dielectric and the variation of electrostatic energy of the system $ΔU$. I also want to understand how the system change after the introduction of the dieletric. Can someone guide me to the solution in a fine way?


title: Which is the "roundest man-made object": Gravity probe B rotor or the Avogadro project sphere?

text: The Avogadro project is part of the efforts to replace the artefact-based standard kilogram with a definition based on naturally reproducible objects. The main challenge is to make a very precise sphere of isotopicaly pure silicon. There are often-repeated popular claims that the silicon sphere from the Avogadro project is the "roundest object ever made by man". However, the same title is ascribed to rotors of Gravity Probe B's gyroscopes, even earning then a certificate from Guinness book of World Records. So, which one is rounder (in appropriately defined way) -- the silicon sphere or the Gravity Probe B rotor?


title: Lagrangian for FRW metric

text: For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~? $$ I guess we get 1 because $g_{tt}=-1$, but what about the other terms?


title: What is the probability of ice in boiling water?

text: Ice crystals are spatially ordered, and in every randomness there is a low possibility of temporarily order. If given enough boiling water, and sufficient time, could local clusters water molecules happen to be in a crystalized state? This may seem absurd, but I believe it must be possible, imagine dropping an ice cube in boiling water(and water vapor) in a perfect closed system, the ice cube melts while the water keeps boiling (because of the water vapor), then because of Poincare recurrence theorem, there will be an similar ice cube after a sufficiently long but finite time. EDIT: The temperature is 100 Celsius and 1 atm (not at triple point)


title: Can we see a spaceship falling into a black hole and entering the event horizon?

text: Or it pauses in time because the spaceship reaches the speed of light, _c_?


title: Light in the absence of colloids?

text: If Earth had no colloids (Don't take LIFE into account) or any kind of scattering medium (like atmosphere) that promote to Tyndall effect, how would we able to receive the sun's rays? In the other way, How would the sun rays or light from any interstellar object reach or appear to us? (just like a distant star?) Is there anything that I missed here? (Please comment)


title: Why is the measured distance to the Triangulum Galaxy much more uncertain than to the Andromeda Galaxy?

text: Why is the measured distance uncertainty for M33 (835 ± 105 kiloparsec) six times bigger than the measured distance uncertainty for M31 (778 ± 17 kiloparsec)? They are approximately at the same distance, are both spiral galaxies and in the same region of the sky (within 15 degrees).


title: Calculation of Distance from measured Acceleration vs Time

text: I have an Accelerometer connected to a device that feeds the instant values of the acceleration in the 3 directions. I've tried to calculate the distance for a vertical movement using these values with excel (applying two times an integration), but somehow it doesn't seem to work properly. How would be a good way to calculate the traveled distance from the measured acceleration using excel tables?


title: What new breakthrough energy technologies are close to being realized?

text: Energy production is one of the burning issues for humankind. There has been some talk about future energy technologies including Fusion, Anti-matter annihilation and Zero-point-energy (from most to least plausible). I'm interested in hearing what people know about developments in the field what they think will be the next real breakthrough in energy technologies.


title: How can we make an order-of-magnitude estimate of the strength of Earth's magnetic field?

text: The source of Earth's magnetic field is a dynamo driven by convection current in the molten core. Using some basic physics principles (Maxwell's equations, fluid mechanics equations), properties of Earth (mass, radius, composition, temperature gradient, angular velocity), and properties of materials (conductivity and viscosity of molten iron) or other relevant facts, is it possible to estimate the strength of the field to order of magnitude (about one gauss)? Descriptions I've seen of the geodynamo all refer to extensive numerical computation on a computer, but can we get a rough idea with simple estimation?


title: What are good mechanics experiments for 10 year olds?

text: I'm trying to explain elementary mechanics - without the benefits of calculus or even algebra - and struggling. I'd like to find reasonable ways to demonstrate Newton's laws, minimally, and possibly continuing though basic projectile motion, although that may be too much to hope for.


title: Chirality and helicity operators for the massless bispinor rep and their generalisation on arbitrary (tensor, 4-vector etc) cases

text: Let's have chirality projection operator $$ \hat {C}_{\pm} = \frac{1 \pm \gamma^{5}}{2}. $$ We introduce it and called it chirality, because $$ \hat {C}_{+}\psi = \begin{pmatrix} \psi_{\alpha} \\\ 0 \end{pmatrix}, \quad \hat {C}_{-}\psi = \begin{pmatrix} 0 \\\ \eta^{\alpha} \end{pmatrix}, $$ which accords to the definition of left- and right-handed spinor (by the classification of irreducible representations of the Lorentz group). Then let's have helicity projection operator: $$ \hat {h} = \frac{ 1 \pm \left(\hat {\mathbf S} \cdot \hat {\mathbf p}_{\mathbf 1}\right)}{2}, \quad \hat {\mathbf p}_{1} = \frac{\hat {\mathbf p}}{|\mathbf p|}. $$ It can be showed, that for the case $v = c$, where $v$ is the speed of the particle, these operators acts on bispinor the same way. So for bispinor formalism in a massless case helicity and spirality are the same. But I don't know how to show analogical thing for the particles of different spines (and, according to this, different irreducible Lorentz reps). Maybe, the answer is in the words below? We can construct different reps of Lorentz group by using spinor formalism. We can construct rep $\left(\frac{1}{2}, \frac{1}{2}\right)$ by $\left(\frac{1}{2}, 0\right)\times \left(0, \frac{1}{2}\right)$, rep $\left(0, 0\right) +\left(1, 0\right)$ by $\left(\frac{1}{2}, 0\right) \times \left(\frac{1}{2}, 0\right)$. So, maybe, we can generalize the calculations with chirality and helicity for the bispinor case on arbitrary cases? If yes, how exactly we can do it? Related questions: Chirality, helicity and their relationship for the massless case , Why helicity is proportional to the spin of particle and has two values? .


title: Can we build spinorial eigenstates of Time reversal symmetry?

text: In the SM, and general theories with spinors, we can build the Parity left/right eigenspinors. Indeed, there are also ELKO fields, eigenstates of Charge operator (non-standard Wigner classes). Can we build Time reversal operator self-conjugated spinors with non-standard features like we did with ELKO spinors?


title: Orbit reversal in a gravitational well

text: A friend recently asked me this, and without wishing to spend a huge time thinking about it, I wondered if anyone knew the answer (or at least had an informed guess), or was familiar enough with these sorts of calculations to figure it out efficiently (rather than painfully, which is how I would end up doing it). > What is the most (energy) efficient way to reverse your orbital motion? More specifically, assume you are in a non-relativistic regime in a spherical gravitational well, in some 'directed orbit' $O$. What is the most efficient way to end up in $-O$, the 'reversed orbit'? Some fairly obvious suggestions include attempting to make the orbit highly eccentric and then maneuvering at apoapsis, or performing a gradual 180 degree plane-change. I have no immediate idea which is better, and **whether there are other better techniques**. Any insight appreciated! _Edit:_ Added emphasis above. I'm interested in whether there could be any other approach; I have no idea what it might look like, and suspect the above are the right possibilities to look at, but would like to be convinced about it.


title: Distinction between Larmor frequency and Rabi frequency

text: Wikipedia said "In the context of a nuclear magnetic resonance experiment, the Rabi frequency is the nutation frequency of a sample's net nuclear magnetization vector about a radiofrequency field. (Note that this is distinct from the Larmor frequency, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)" However the two definitions seem equivalent to me... Any clarifications?


title: Why don't we use the concept of force in quantum mechanics?

text: I'm a quarter of the way towards finishing a basic quantum mechanics course, and I see no mention of force, after having done the 1-D Schrodinger equation for a free particle, particle in an infinitely deep potential well, and the linear harmonic oscillator. There was one small mention of the fact that _expectation values_ obey classical laws. I was wondering why we don't make more use of this fact. For example, in the linear harmonic oscillator problem, one could obtain the temporal evolution of $\langle x \rangle$ using the classical expression $\left(-\frac{dV(x)}{dx}=m\frac{d^2\langle x\rangle}{dt^2}\right)$, and if we could get the time-evolution of $\sigma$ and tack this on, we could re-create the Gaussian and get back $|\Psi(x,t)|^2$. Of course, that last part may not be very easy. I was just wondering if anybody has tried doing something like this, or if there an obvious flaw in thinking about it this way.


title: Galilean transformations and Frenet Frame

text: How I can prove that the curvature and torsion of a curve are invariant under the Galilean transformations? In my physics book a hint is the isometries of Galilean transformations, but it's still difficult. I don't have any idea.


title: Momentum vector transformation

text: I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx \delta_{ij} + \epsilon \frac{\partial f_i(q)} {\partial q_j}$. Then it (Nakahara's " _Geometry, Topology, and Physics_ " Chapter 1 Theorem 1.1) says that momentum transforms under this coordinate change as $$p_i \to p_i' = \sum_j p_j \Lambda_{ji}^{-1} \approx p_i - \epsilon \sum_j p_j \frac{\partial f_j}{\partial q_i}.$$ Why?


title: Why isn't there a centre of charge?

text: When determining the gravitational attraction between 2 solid bodies, we can simplify computations by taking their masses to be concentrated at their respective centres of mass. However, had they been electrically charged bodies and we needed to compute electrostatic attraction, there is no "centre of charge" notion available to us. ( We would evaluate a 2-dimensional integral, or equivalently, apply Gauss's law. ) Is there any simple reason for the absence of such a centre of charge? Frequently, I've got the answer: "charge just doesn't work that way".


title: Calculating work done on an ideal gas

text: I am trying to calculate the work done on an ideal gas in a piston set up where temperature is kept constant. I am given the volume, pressure and temperature. I know from Boyle's law that volume is inversely proportional to pressure that is, $$V \propto \frac{1}{p}$$ using this I can calculate the two volumes I need for this equation to calculate work done: $$\Delta W = - \int^{V_2}_{V_1} p(V)dV$$ but what I do not understand is how to use this equation to help me calculate the work done, I think I am confused by the fact that I need to have $p(V)$ but I am not sure what this is. If you could help me to understand this, that would be great


title: Infinitesimal Lorentz transformation is antisymmetric

text: The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that under a infinitesimal transformation $\Lambda^\mu_{\ \ \ \nu}=\delta^\mu_{\ \ \ \nu} + \omega^\mu_{{\ \ \ \nu}}$, that $\omega_{\mu\nu} = -\omega_{\nu\mu}$. I tried expanding myself: \begin{align*} \eta_{\rho\sigma} &= \eta_{\mu\nu}\left(\delta^\mu_{\ \ \ \rho} + \omega^\mu_{{\ \ \ \rho}}\right)\left(\delta^\nu_{\ \ \ \sigma} + \omega^\nu_{{\ \ \ \sigma}}\right) \\\ &= (\delta_{\nu\rho}+\omega_{\nu\rho})\left(\delta^\nu_{\ \ \ \sigma} + \omega^\nu_{{\ \ \ \sigma}}\right) \\\ &= \delta_{\rho\sigma}+\omega^\rho_{\ \ \ \sigma}+\omega_{\sigma\rho}+\omega_{\nu\rho} \omega^\nu_{{\ \ \ \sigma}} \end{align*} Been a long time since I've dealt with tensors so I don't know how to proceed.


title: Is special relativity a special case of general relativity, qualitatively?

text: Since Einstein name his theory Special Relativity and General Relativity, there should be some expected relationship between them, particularly "Special Relativity" being a special case of the more general "General Relativity". However, I can't seem to relate them in anyway; Special relativity concerns the fact that the speed of light is universal and General relativity is mainly about the curvature of spacetime. BTW, there is already another similar question on Stackexchange: Reducing General Relativity to Special Relativity in limiting case , but I am in for a more qualitative explanation.


title: Are electrons on a conductor, in electrostatic equilibrium, stationary?

text: Electrostatics basically means dealing with time independent electric fields (which was produced by stationary charges) Now consider a neutral conductor. We know that putting a net negative charge on the conductor, the charge will very quickly spread over the surface till electrostatic equilibrium is reached. What does this exactly mean? does it mean that electrons on the surface are not "moving" anymore and they become stationary? So in this situation we have 2 kinds of electrons: 1-electrons inside the meat of the conductor whose dynamics is described by quantum mechanics 2-electron on the surface which are not moving in the classical sense?


title: Form of Dilaton term in polarization tensor

text: The vertex operator associated with massless state is $$V(k,\epsilon) = -\frac{2}{\alpha}\epsilon_{\mu\nu}(k)\bar{\partial}X^\mu(\bar{z})\partial X^\nu(z)e^{ik\cdot X(z,\bar{z})}$$ The polarization tensor can be decomposed into symmetric (Graviton), antisymmetric and Trace bit (Dilaton) Lust,theisen 16.9 $$\epsilon^{(h)}_{\mu\nu} = \epsilon^{(h)}_{\nu\mu},\qquad\epsilon^{(h)}_{\mu\nu} \eta^{\mu\nu} = k^{\mu}\epsilon^{(h)}_{\mu\nu} = 0, $$ $$\epsilon^{(B)}_{\mu\nu} = -\epsilon^{(B)}_{\nu\mu},\qquad k^{\mu}\epsilon^{(B)}_{\mu\nu} = 0, $$ $$\epsilon^{(D)}_{\mu\nu} = \frac{1}{\sqrt{d-2}}(\eta_{\mu\nu}-k_{\mu}\bar{k_{\nu}}-k_{\nu}\bar{k_{\mu}})$$ $\bar{k}$ is an arbitrary light like vector orthogonal to $k$. Can you please tell me why we took that particular form for the Dilaton?


title: What is the origin of the factor of $-1/4$ in the Maxwell Lagrangian?

text: I have seen numerous 'derivations' of the Maxwell Lagrangian, $$\mathcal{L} ~=~ -\frac{1}{4}F_{\mu \nu}F^{\mu \nu},$$ but every one has sneakily inserted a factor of $-1/4$ without explaining why. The Euler-Lagrange equations are the same no matter what constant we put in front of the contraction of the field strength tensors, so why the factor of $-1/4$?


title: Can the term «shadow» pertain to anything else than light?

text: Can the term shadow pertain to anything else than light? Feel free to interpret this question in the widest sense possible.


title: In a glass of water, will an ice cube ever touch the bottom?

text: No matter the mass of the ice cube or the level of the water, will the ice cube ever touch the bottom of the glass or will there always be a layer of water in between?


title: Hurricanes and Atmospheric Temperature

text: EDIT -- It appears that everyone is confused by the question. That's probably my fault. Let me say it another way. Do hurricanes cool the atmosphere (by taking heat out of the ocean/air and using that energy to essentially power the winds)? Does this counter the effect of global warming in any significant way? Original question wording: Does the energy expended by a hurricane/typhoon have a cooling effect/counter the effects of average temperature increase caused by climate change? If so, do any climate predication models account for this? Or is the effect so negligible that there is no need?


title: Filling ising model with basis

text: This questions involves how to fill a lattice with ordered basis: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For example, a triangular lattice can be filled with ![basis](http://i.stack.imgur.com/2OvLI.png) to get![filling](http://i.stack.imgur.com/WM6h1.png) While ![enter image description here](http://i.stack.imgur.com/ceIFE.png) alone cannot give any filled colored graph. Is there any general systematic mathematical approach in solving this problem?


title: Why is it written 'High voltage' in danger boards if current is the one which actually causes the shock?

text: After reading the answers given to this question I could understand that the amount of shock is dependent on the current and not majorly on the voltage even-though in some cases it depends. But, in that case if current is the actual villain why do I see danger boards on which it is written '10,000 V-Danger' etc rather than showing the amount of current that can pass on two somebody's body.


title: How to derive the form of the parity operator acting on Lorentz spinors?

text: I'm reading Berestetskii (Volume 4 of Landau & Lifshitz) section 19 on inversion of spinors. Berestetskii says parity $P$ maps undotted spinors into dotted spinors and vice-versa as $\xi^{A}\rightarrow i\eta_{\dot{A}}$ and $\eta_{\dot{A}}\rightarrow i\xi^{A}$. I think he means there are parity tensors $P_{\dot{A}B}$ and $P^{A\dot{B}}$ which act as, $$ \eta_{\dot{A}}=P_{\dot{A}B}\xi^{B} \\\ \xi^{A}=P^{A\dot{B}}\eta_{\dot{B}} $$ and which satisy $P^{2}=-1$, $$ P^{A\dot{C}}P_{\dot{C}B}=-\delta^{A}_{B} \ . $$ Berestetskii's map then says, $P_{\dot{A}B}=P^{A\dot{B}}=i\delta_{AB}$ which is only true in (presumably) the rest frame of the fermions. I've tried to derive the components of the parity tensors by imposing various mathematical and physically motivated restrictions, but I can only get $P^{A\dot{B}}$ diagonal with entries $a,1/a^{*}$ and $P_{\dot{A}B}$ diagonal with entries $-1/a,-a^{*}$. Berestetskii's parity is recovered by taking $a=i$ and so my question is, "What are the extra restrictions needed to be able to set $a=i$?"


title: What happens if an already excited electron gets hit by another photon (lasers)?

text: Slightly vague title but it is the best I can do. My question stems from some interest in how 3 stage lasers function. A photon from the pumping source comes in and strikes a atom in the active medium. The photon is absorbed, an electron is excited, and the energy instantly falls to a metastable state. My question is what happens when I try to excite that electron again with another pump photon. Does the electron jump to an even higher state if one is available (the pump photon provides the right dE again to excite the electron to a valid higher energy state)? Or do we choose our active medium properly such that no "double-pumped" state exists and the excited electrons become transparent to the pumping photons?


title: Why the generators of boosts transform like a vector under rotation?

text: $$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$ $$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$ $$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ is the generator of boost of Lorentz group In many textbook of QFT, they say that the second one implies that the generators of the boosts transform like a vector under rotations. But I can't see it explicitly. Can anyone give me the explanation.


title: How can we interpret polarization and frequency when we are dealing with one single photon?

text: If polarization is interpreted as a pattern/direction of the electric-field in an electromagnetic wave and the frequency as the frequency of oscillation, how can we interpret polarization and frequency when we are dealing with one single photon?


title: Aren't all collisions elastic at some level?

text: The more I think about it, inelastic collisions produce heat and sound which imply motion at some scale, right? Are inelastic collisions macroscopic events that ignore motion at microscopic levels? Maybe I'm just misunderstanding the definition of an inelastic collision.


title: Anomalously broken conformal symmetry

text: I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential contains $\mu^2\phi^2$. Bardeen argues that   1. Setting $\mu^2=0$ would increase the symmetry of the Standard Model by a conformal symmetry. > _But the things we can measure - which are basically the Green's functions - > are no more symmetric with $\mu^2=0$ than with $\mu^2\neq0$! Because the > symmetry is anomalously broken._   2. The conformal symmetry is broken anomalously by non-zero beta-functions.   3. Quadratic divergences do not contribute to the beta-functions.  > _Whilst its true that they don't contribute to the running of the > renormalised parameters, they contribute to the running of the bare mass._ >   4. Quadratic divergences must therefore be a separate (explicit) source of > symmetry breaking. > _Not true if you consider that quadratic divergences contribute to the > running of the bare mass._   5. Quadratic divergences must therefore be an non-physical artefact of our re-normalisation procedure, and we must remove them with, for example, counter-terms, and this is not a fine-tuning. I have added my comments. My main question is, > If conformal symmetry is broken anomalously, why should our Lagrangian > respect conformal symmetry? The Green's function are not more symmetric with > $\mu^2=0$. I am interested in this for any symmetry, but especially > conformal symmetry. Also, I don't understand 3. and 4. Quadratic divergences would contribute to the running of the bare mass. Wouldn't that break conformal symmetry anomalously? or is it only the renormalised parameters that must have vanishing beta-functions? The distinction seems artificial. I can't understand why quadratic divergences are an explicit source of symmetry breaking, whereas logarithms etc are anomalous. This is key to solving the naturalness problem, and I can't follow the argument. My feeling is that these arguments are faulty (which makes me think I must be making mistakes because Bardeen is a real expert who has surely thought a lot about it!). I'm certainly unconvinced. Have they been confirmed/refuted at length in the literature?


title: Why must the particles of an ideal gas be point-like?

text: > Why is a gas of elastically colliding hard balls of finite size not ideal? Respectively: > Why is it essential that the particles of an ideal gas are point-like? Especially: > **Which differing thermodynamic properties of a gas of _not_ point-like > particles are most striking in comparison with an ideal gas (of point-like > particles)?**


title: Amplitudes involving Goldstone bosons

text: Does anyone know some theorem or statement about amplitudes involving only Goldstone bosons in theories with spontaneous symmetry breaking in the limit of low energies?


title: Imagining to speed up the Earth's revolution around the Sun until we feel its motion

text: If the Earth were revolving around the Sun much faster than it does, let's say 100 or 1000 times faster or whatever big-enough multiple of its actual speed, would we feel it? What sensation would we feel? How could I estimate how big the revolution speed would be for us to feel it? Would it be possible to speed up the Earth motion around the Sun? How? What basic physics concepts can be used to explain what would happen?


title: Are Negative Eigen Values of a Hessian Matrix physically acceptable?

text: Suppose I have a Hessian Matrix of a System with 3N degrees of freedom, What are the physical significance of eigen values of the Hessian, Are negative Eigen Values physically acceptable?


title: The problem of a relativistic path integral

text: Many books have described the path integral for non-relativistic quantum. For example, how to get the Schrödinger equation from the path integral. But no one told us the relativistic version. In fact, the relativistic version is impossible to be perfect, it must be replaced by quantum field theory. But why? The answer I want is not that the quantum mechanics will give us a negative energy or negative probability. We need a answer to explain why non- relativistic Lagrangian $$L=\frac{p^2}{2m}$$ can lead a correct Schrödinger equation? Why if we replace it by relativistic Lagrangian $$L=-mc^2\sqrt{1-v^2/c^2},$$ we can not obtain any useful information? So how can we think of the correctness of the non-relativistic path integral? And how a quantum field can give a relativistic quantum theory? Here give us a simple answer Special relativity version of Feynman's "Space- Time Approach to Non-Relativistic Quantum Mechanics" but it is useless. It doesn't tell us how to develop it step by step in mathematical frame and physical frame...


title: Magnification, and it's application

text: I'm having a problem with a very trivial thing- magnification makes things bigger. But if it doubles or triples the size AND distance away, why does it even help? I think I'm just ignoring something about how we see, but I don't understand why making a small object twice as tall but twice as far away would help at all.


title: The phrase "Trace Anomaly" seems to be used in two different ways. What's the relation between the two?

text: I've seen the phrase "Trace Anomaly" refer to two seemingly different concepts, though I assume they must be related in some way I'm not seeing. The first way I've seen it used is in the manner, for example, that is relevant for the a-theorem and c-theorem. That is, given a CFT on a curved background, the trace of the energy-momentum tensor is non-zero due to trace anomalies which relate $T^\mu_\mu$ to different curvatures, that is (in 4D) $T^\mu_\mu\sim b\square R+aE_4+cW^2$, where $E_4$ is the Euler density and $W^2$ is the square of the Weyl tensor. The second manner in which I've seen it used is in the context of relating $T^\mu_\mu$ to beta functions as the presence of a non-zero beta function indicates scale dependence, and hence breaks conformal invariance. For example, Yang-Mills is classically conformally invariant but it is quoted as having a trace anomaly which seems to be of different character than that of the previous paragraph. As in the Peskin and Schroeder chapter on scale anomalies, it's quoted that since the gauge coupling, $g$, depends on scale due to RG the theory is not quantum mechanically scale invariant (or more generally, I guess, Weyl invariant) and hence $T^\mu_\mu$ is non-zero. Slightly more precisely, given the YM lagrangian $\mathcal{L}\sim \frac{1}{g^2}{\rm Tr}F^2$, one finds $T^\mu_\mu\sim \beta(g){\rm Tr}F^2$, or something to that effect. It's my understanding that this second kind of trace anomaly is important for explaining the mass of nuclei, as most of their mass comes from gluonic energy. That might be wrong, though, and it's not so relevant anyway. What's the relation between these two types of anomalies? Are they the same thing in disguise?


title: What do we mean when we say the QM wave function is a section of the $U(1)$ bundle?

text: I have a couple questions here. To keep the discussion simple lets stick to the following case: what is the quantum mechanics of a single particle in the presence of a background EM field, such as that of a monopole, where multiple coordinate patches are required to define the vector field everywhere without singularity. I have learned about the geometric formulation of EM - where the field $B$ of a monopole is described as a real-valued two-form that is closed but not exact. I also know about using the vector potential as a connection for the complex wavefunctions, using the usual covariant derivative. The questions I have in mind are >   1. What (complex line) bundle is the wave function a section of? >   2. What are the bundle data (i.e. transition functions) that specify this? >   3. In what way can we see the monopole vector potential as a connection on > this bundle? >   4. People seem to talk about the monopole bundle only with the monopole > connection in mind - what would it look like to have a different connection > instead? > On a similar note, I've seen physicists refer to the classification of bundles, which there is one for every integer (based on the homology of $\mathbf{R}^3-\\{0\\}$), and call that integer the monopole number. But I've also seen people call the flux integral of the magnetic field two form over some closed surface, which relates instead to the curvature of some _connection_ on the bundle - which should be quantized by Gauss-Bonnet theorem - the monopole number. Is there some reason that these are the same? I was reading a great set of lecture notes around 10 pages which explained the need to use a section of a bundle for the wave function and the relation to the Dirac quantization condition, but I can no longer find them. Links to related resources would be very welcome.


title: Question on circular motion — car turning on road

text: Lets say a car is turning on a road that has friction. I made this small diagram ![enter image description here](http://i.stack.imgur.com/3fhVd.png) where the two parallel lines indicate the tires of the car. In order for the car to keep turning in a circle, there must be an force directed to the center of curvature — the centripetal force. I have read that this force comes as a result of friction. However, in order for there to be a frictional force towards the center, doesn't there have to be a force directly opposite pulling away from the center, as frictional forces cannot exist by themselves — they have to act in order to counter an opposite force. Thus, how can there be a frictional force towards the center without a corresponding force pulling away from the center? Does this have something to do with the centrifugal pseudo-force? Further, where exactly does this frictional force come from? Also, why is the frictional force directed towards the center? Shouldn't it be opposite the direction the car is traveling at the moment -- thus in a direction opposite the tangential?


title: Are Everettian branchings global or local?

text: Everett's theory of quantum mechanics is about the wavefunction of the whole universe holistically. If a branching occurs very far away at the Andromeda galaxy, do I also branch? Are branchings global or local? If the branching occurs outside the cosmological horizon, do I branch?


title: Can someone explain the color Pink to me?

text: I just finished watching this interesting video: http://youtu.be/S9dqJRyk0YM It does a very quick explanation of how pink light doesn't exist, and that the concept of pink is our brain's attempt at filling in a gap in the spectrum. My question is, how does our brain perceive this gap? What frequency is this pink color, if it truly doesn't exist? Is it a frequency greater than violet? Or is it greater than red? Sorry if this question seems trivial.


title: Kinetic energy puzzle

text: **System S1 moves at constant speed V with respect to S0 in one dimension** :                                |-------------------------------------->                    S1                                      x     |--------------------------------->     S0                               x      A particle with mass m, moves at speed v with respect to S1 then at a speed of $v+V$ respect to S0 (if we can neglect relativity effects) **Kinetic energy** $Es1=\frac{1}{2} m v^2$ (respect to S1) $Es0=\frac{1}{2} m (v+V)^2$ (respect to S0) **Then an external influence change its speed** (from S1) = $v+dv$ (from S0) = $(v+dv)+V $ **New Kinetic energy** $Es1=\frac{1}{2} m (v+dv)^2$ $Es0=\frac{1}{2} m (v+dv+V)^2$ Gain of energy from S1 $\frac{1}{2} m (v+dv)^2 - (\frac{1}{2} m v^2)= \frac{1}{2} m (dv^2+ 2 v dv)$ Gain of energy from S0 $\frac{1}{2} m (v+V+dv)^2 - (\frac{1}{2} m (v+V)^2) = \frac{1}{2} m (dv^2+2(v+V)dv )$ **Difference between energy gains** = $dv*m*V$ Where did that energy came from?


title: How to guess the content of a christmas present?

text: Let us assume that the present does not make any recognizable sounds when shaken (`meow splat` \- the present now contains a dead kitten). Let us furthermore assume that the internal state of the present does not change, so that any measurements can be repeated. I will consider it cheating to unwrap the present, or shine any form of light, rays or particles through the present. Not many people has x-ray or neutron scattering devises available anyway. - and obviously it is cheating to squeeze the present. * * * What can be measured?   * Size and shape of wrapping(hey mom ... my wish elephant will not fit in here)   * The mass    * Position of center of mass   * Moment of inertia tensor   * Vibrational resonance    * Magnetic resonance    * Thermal radiation * * * If the seven scalars in mass, position of center of mass and moment of inertia tensor were measured, what could be said about the shape and mass distribution of the present? If vibrational or magnetic resonance were measured, what could be said about the shape and materials of the present? Would the thermal radiation not just depend on the temperature of the wrapping? what if the room temperature were dropped, then the present would heat the wrapping depending on the heat capacity, and thermal conductance. -but can it be used for anything? * * * This question has been heavily edited to correct my previously mistakes, and reflect my progress. THANK YOU for the answers already given.


title: What would be pressure of 1 kg of photon gas at room temperature put in a volume of 1 liter?

text: Suppose a number of photons with spectrum corresponding to black body spectrum at 293 K with total energy corresponding to 1 kg put in a box with ideal mirror walls with volume of 1/1000 of a cubic meter (1 liter). What pressure this photonic gas will manifest on the walls of the box?


title: Which physically acceptable quantum interpretations do not require the existence of any observer at all?

text: Some interpretations of quantum mechanics -- like the Copenhagen interpretation in particular -- require the existence of an observer. The role of the observer is a bit mysterious. After all, observers are not needed in classical mechanics. Which physically acceptable interpretations of quantum mechanics do not require the existence of any observer at all? By that, I mean they would continue to make sense even in the absence of any observer. The many worlds interpretations appears to require an observer to subjectively determine which branch is selected.


title: Why does lightning emit light?

text: What exactly is causing the electric discharge coming from the clouds to emit light while traveling through the air. I've read and thought about it a little but with my current knowledge I cant really figure what is happening at a subatomic level. In other words, what causes the emission of photons during lightning. What I want to know is, is there some sort of collisions taking place that produces the photons we see, or is it some other interaction that is causing it?. I don't know too much about particle physics, I've only read a couple of pages in the Introduction to Elementary Particles by Griffihts. So if you could keep the answer simple then that would be great.


title: Do indirect optical transitions "cool" the material a little?

text: So I'm reading in Ashcroft and Mermin about indirect optical transitions: ![AM](http://i.stack.imgur.com/42Htu.png) So, a photon comes in, and it only excites the electron across the indirect band gap if a phonon with the appropriate wave vector can "help" the electron by giving it momentum (they point out that the momentum of the photon is negligible). First of all, does this mean that, if the photon comes in and no phonon happens to help it, it's not absorbed, but most likely reflected? Or would something else almost certainly absorb it? Secondly, when a phonon _is_ there to "help" the electron, does this mean the crystal is now cooler as a result, because there are fewer/less powerful lattice vibrations (because the phonon gave up some momentum)? The crystal has certainly gained energy from the photon, but the energy is now just in the form of... the electron having a higher energy state? (Is that actually where the energy has "gone"? Do we just say that the electron is in a higher energy state, and that's the extent of it?) As opposed to kinetic energy of the crystal, which seems like it has decreased. If this is possible (I'm ready to be told that it's not), could it be scaled up and carefully done to cool certain semiconductors? Thanks!


title: Local real-time lightning tracker site?

text: Is there a site that tracks lightning strikes at a local level in real-time? Google results for "lightning tracker" yield several sites, but none do this. It seems easy enough to set up. Have volunteers report:   * their location    * the time they saw the strike    * the direction of the strike, to the nearest 45 degree (eg, "southwest" or "east")    * the delay between the lightning and the thunder (ie, the distance in fifths of a mile)    * optional: description of the lightning strike  Of course, portions of the report could be automated (eg, geolocated tweets). It would then be easy to map lightning strikes at a given time, watch a storm move, etc, down to the several-hundred-feet level. Since this seems easy to do, I'm sure someone has done it, no?


title: Lagrangian formalism and Contact Bundles

text: In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$: > A line integral makes geometric sense only if it's integrand is a 1-form. Is > $Ldt$ a 1-form? Well, that's the wrong question. The correct question is: On > what space is $Ldt$ a 1-form? It is _not_ a 1-form on configuration space, > the space of positions, because it can have a non-linear dependence on > velocities. A 1-form must be a linear operator on the tangent vectors. The > correct space for $L dt$ is the line-element contact bundle of the > configuration space. Now, why intuitively the correct setting for lagrangian mechanics is on the contact bundle? I understand the contact bundle as pairs $(p,[v])$ where $p$ is a point in configuration space and $[v]$ is an equivalence class of vectors, explicitly $v\sim kv$. Thinking not on all that arguments for selecting the space on which $Ldt$ is a $1$-form, physically, how can we intuit that the contact bundle is useful for that? I mean, is there some observation in classical mechanics that guides us in building the theory on that space?


title: How do we know that internal conversion creates no intermediate photon?

text: I've read, from several sources, that in internal conversion -- an excited electron transferring its energy to another electron which is then emitted -- no intermediate gamma radiation is produced. How can we know that? Given that the distance it would travel is on the Å scale, is there any way to detect (the absence of) such a photon?


title: Line of action force

text: The task is to reduce the two forces into a single net force and then find the equation of its line of action. I chose to find the net force and momentum with respect to the origin. The magnitude of each force is $P$. ![asd](http://draw.to/static/d/3yezUM.png?v=ec211) We can therefore write $F_{net} = (P,0,0) + (0,0,-P) = (P,0,-P)$. The distance vector from origin to $F_1$ is $(0,b,c)$ and from origin to $F_2$ is $(a,0,c)$. The net moment is then $$M_{net} = (0,b,c)\times (P,0,0) + (a,0,c)\times (0,0,-P) = (0, aP + cP, -bP ).$$ What I tried to do was to assume the attacking point of the net force to be $(x,y,z)$. This force must cause the net moment with respect to the origin. So I set up the cross product $$ (x,y,z) \times (P,0,-P) = (0, aP + cP, -bP).$$ but I was unable to determine $x$, $y$ and $z$. So what can we do?


title: What spacetimes satisfy this identity?

text: 1. What spacetimes satisfy $R^{\mu\nu} R_{\mu\nu} =\alpha R^2$, where $R = g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $\alpha$ is some constant?   2. A follow-up question: in what spacetimes does $R^{\mu\nu\sigma\rho} R_{\mu\nu\sigma\rho} =\beta R^2$, for some constant $\beta$?


title: (References) Study of Asymptotically Flat spacetimes

text: I am interested in studying the asymptotic structure of Minkowski spacetime in General Relativity. I believe most of the work in this area concerns the asymptotic structure of Minkowski space at null and spatial infinity. Well known authors who have contributed to these areas are: Null: Bondi, Sachs, Penrose, Newmann, Unti. Spatial: Abott, Deser, Ashtekar. I am wondering if there are any more on the list here. Also, it would be helpful if you can produce a list of papers that one should read for a complete understanding of this.


title: Is it necessary to consume energy to perform computation?

text: As far as I know, today most of the computers are made from semiconductor devices, so the energy consumed all turns into the heat emitted into space. But I wonder, is it necessary to consume energy to perform computation?   * If so, is there a theoretical numerical lower bound of the energy usage ? (I even have no idea about how to measure the amount of "computation")   * If not, is there a physically-practical Turing-complete model that needs no energy? * * * **edit:** Thank @Nathaniel for rapidly answering the question and pointing out it's actually Landauer's principle. Also thank @horchler for referring to the Nature News and the related article. There are lots of useful information in the comments; thank every one! This whole stuff is really interesting!


title: Proving Kepler's 1st Law without differential equations

text: Is there a way to show that motion of Earth around the Sun is elliptical (Kepler's 1st law) from Newton's laws without resorting to the use of differential equations of motion?


title: Electrodynamics and induced EMF question

text: > A very long straight wire carries a current I. A plane rectangular coil of > high resistance, with sides of length $a$ and $b$, is coplanar with the > wire. One of the sides of length $a$ is parallel to the wire and a distance > $D$ from it; the opposite side is further from the wire. The coil is moving > at a speed $v$ in its own plane and away from the wire. > > (a) Find the e.m.f. induced in the coil. > > (b) Let R be the resistance of the coil. Calculate the force needed to move > the coil with speed v as described, and show that the mechanical power used > to move it is equal to the rate at which heat is generated in the coil. I have included my workings/thoughts. I know that i first have to calculate the magnetic field of the wire: $$B(y)=\mu_0 I/2πy $$ Then the emf, $$\mathcal{E}=-\frac{dφ(B)}{dt}= -\frac{d}{dt} \int_D^{D+b}B\cdot ds = -\frac{d}{dt}(a\cdot \int_D^{D+b} B\cdot dy)$$ I have been given the answer of $$\mathcal{E}=\frac{\mu_0Ivab}{2\pi D\left(D+b\right)}$$ Do I have to perameterise the integral? If so does anyone have a guide on how to do this? What I am having trouble with is the intermediate step getting from the integral to the above answer. I know that $P_{mech}=F.v$ and that $P_{heat}= v^2/R = \mathcal{E}/R$, but I do not know how to calculate the force on the coil.


title: Hall effect with similar positive and negative carriers?

text: The Hall effect includes the transverse (to the flow of current) electric field set up by the charges which accumulate on the edges, to counter the magnetic component of the Lorentz force acting on them to move towards the edges. These charges can be both _positive_ holes and negative electrons in semiconductors. If the concentration of any one kind of carriers is very high as compared to other(n-type and p-type), the electric field set up can be understood from the accumulation of charges (negative for n-type and vice versa) at one edge due to the Lorentz force acting on it. But if the intrinsic semiconductors contain the same density of the positive holes and the electrons, both move in opposite directions to contribute to the current but the lorentz force acting on them is in the same direction. And hence their deflection owing to the magnetic field is in the same direction. > **This means, both the positive and negative charges are deflected towards > the same edge and hence cannot produce a net charge density and therefore > should not be able to set up an countering electric field and the Hall > potential. Does this mean that they will continuosly keep being deflected > towards one of the edge and keep annihilating at that edge, never setting up > a counter electric field?** _If the difference in mobility (or the effective mass) makes any difference, what would happen if they have the same mobility and consequently the same effective mass?_


title: Joint probability function for the values of a free field at two different points

text: For a free real field $\phi$ in its ground state, is there a way to find the probability distribution $p(\phi_x,\phi_y)$ for joint measurement of $\phi(x)$ and $\phi(y)$ at two spacelike-separated points $x$, $y$? Presumably the information needed to construct $p(\phi_x,\phi_y)$ would be contained in the expectation values of products of powers of $\phi(x)$ and $\phi(y)$, since we could invert the formula $$ E[\phi(x)^n\phi(y)^m] = \int d\phi_xd\phi_y p(\phi_x,\phi_y)\phi_x^n\phi_y^m $$ but the expectation values are divergent for $n$ or $m$ greater than $1$.


title: Gaining symmetry through boundary conditions

text: **Can Hamiltonian gain/lost some symmetry through boundary conditions?** Initial expression may not have symmetry, but after specyfing boundary conditions and predicted form of the wavefunction it will actually acquire this symmetry. Interesting things may happen with time-reversal symmetry and magnetic field in curved geometries I suppose. For example, $$H = -i\hbar v_{F}\left[ \sigma_{x}\partial_{x} + \frac{\sigma_{y}}{R}\left(\partial_{\phi} + i\eta \right)\right]$$ It is not time-reversal symmetric because of non-zero value of $\eta$. If I know tell that this Hamiltonian describes massles Dirac fermion on the surface of a topological nanowire with radius $R$ where wavefunctions has to satisfy antiperiodic boundary conditions in $\phi$ and transaltion in $x$-direction, then You will solve Schrodinger equations through ansatz, $$\Psi = \Phi e^{ikx}e^{il_{n}\phi}$$ with $l_{n} = n - 1/2$, quantized with integer numbers $n$. You can rewritte Hamiltonian, $$H(k,n) = \hbar v_{F}\left[k\sigma_{x} + \frac{\sigma_{y}}{R}\left(l_{n} + \eta\right)\right]$$ Now for $\mathcal{T} = i\sigma_{y}\mathcal{K}$, with complex conjugation operator $\mathcal{K}$ one gets, $$\mathcal{T}H(k,n)\mathcal{T}^{-1} = -H(k,n)$$ but for $\eta=1/2$ one can actually write, $$\mathcal{T}H(k,n)\mathcal{T}^{-1} = H(-k,-n)$$ It implies that the time reversed wavefunction $\mathcal{T}\Psi$ is a solution of the Hamiltonian $H(-k,-n)$ with the same energy $E$, $$H(k,n) \Psi = E \Psi \\\ H(-k,-n) \mathcal{T}\Psi = E \mathcal{T}\Psi$$ You can look at this as a pseudo "particle-hole symmetry", cause for every state $\Psi$ with energy $E$ exists time-reversed state $\mathcal{T}\Psi$ with energy $-E$.


title: Equations of motion with replacing the Lagrangian by irrep diagrams generating functional

text: I have read that equations of motion of ghosts is equal to $$ \tag 1 \frac{\delta \Gamma}{\delta \bar{c}^{a}(x)} = -\partial^{\mu}_{x}\frac{\delta \Gamma}{\delta K^{\mu , a}(x)}, $$ where $\Gamma = W - \int d^{4}x \sum_{i}(J_{i} \cdot \varphi^{i})$ is generating functional for irreducible Feynman diagramms of nonabelian gauge theory given by $$ S = \int \mathrm{d}^{4}x\left( -\frac{1}{4}F_{\mu \nu}^{a}F^{\mu \nu}_{a} - B_{a}f^{a} + \frac{1}{2 \varepsilon}B_{a}B^{a} - \bar{\omega}^{a}M_{ab}\omega^{b} + K_{\mu}^{a}\delta A_{a}^{\mu} + L_{a}\delta \omega^{a} \right), $$ and $K_{\mu}^{a}$ is the source for $\delta A_{\mu}^{a}$, where $\delta \varphi $ is BRST transformations of $\varphi$ field (for other descriptions see here). Why $(1)$ is correct (i.e. why we can replace $L$ to $\Gamma$ in equations of motion)?


title: Trying to make a realistic simulation of 'breaks' (Pool)

text: I'm working on a 2D pool simulation project right now, and I'm trying to create good looking physics that are also performant. After collisions are detected, I update the velocities of the balls. So far, I've assumed collisions are elastic, which gives pretty good results. The formulas for the final velocities of both balls after a 2D elastic collision are easy to find, and I can follow the derivations of them well enough. The problem is that 'breaks' are not realistic: momentum is often passed along the top and bottom of the triangle of balls, and only the top and bottom corners fly off. I was wondering if changing the equations to use partially inelastic collisions would make the break look more realistic. I would use a coefficient of restitution that is close to 1. However, I'm not able to find many examples of 2D partially inelastic collisions that give final velocities. I'm not sure if I can derive these easily myself, and I'm a bit short on time. Would this approach be worthwhile? Will using slightly inelastic collisions give me the results I'm looking for? If so, are there any resources that can help me solve this problem?


title: Aftermath of Plasma Arc

text: You have a chamber with constant amount of gas (say either helium or argon). You create the plasma with an arc and turn it off. How much of the gas used to create the plasma returns back to helium or argon as opposed to dissipating into the chamber or creating new elements?


title: Where I can find a torsional stiffness table for different types of stainless steels?

text: I am trying to comparing the torsional stiffness of different types of stainless steels. I googled the keywords but I could not find any useful information. Would anyone please give me some links which provide those information?


title: Spacelike and timelike photons

text: I know the definition of space-like and time-like intervals. How do you show that in some reaction the virtual photon(s) is spacelike/timelike? **On page 191 Peskin and Schroeder, it says "since $q^2 <0$ for a scattering process..." Why is this statement true? I don't see it straight away** For example in Bahba scattering discussed on this page.


title: Why Does Passing Light Through a Slit Make it Coherent?

text: Related: Why incoherent source of waves have to be passed though a single slit for double slit interference pattern to occur? If you have a an incoherent light source, why does passing it through a single slit make it coherent? Surely it makes a wavefront, which will be a coherent wave front, but there are still multiple wavelengths coming through still. Lastly, passing the now coherent wave front through a double slit will merely result in an interference pattern, correct? It wouldn't result in any other useful characteristics, would it?


title: On-shell action in asymptotically AdS space

text: Consider a field theory coupled with gravity described by the action: $S=\int d^Dx \sqrt{-g} \left( \mathcal{R}-\Lambda+\mathcal{L}_m[\phi] \right)$, with the requirement that g must be asymptotically AdS. Usually in AdS/CFT correspondence the action evaluated on-shell reduces to a purely boundary term. Is there a general formula to express this boundary term?


title: The Soda Can and air pressure experiment

text: Yesterday, I made an experiment. I filled an empty (but contains air) soda can with a little bit of water. Then I turn on fire at the sides of the soda can so that the water starts boiling. Next, I prepare bucket filled with normal temperature water. And when the water inside the soda can starts evaporating, I quickly turn the SC upside down inside the bucket and then, CRANK! The air pressure made it's effect on the soda can. However, I don't know why. I explained it by saying: "The water vapour eliminates the air in the soda can, when we reverse it into the bucket, the water vapour quickly becomes liquid and we have a vacuum now in the soda so the effect of air pressure will be important." But is it true?


title: Density of States of Free Particle in One Dimensions

text: I am using _Quantum Mechanics_ by David H. McIntyre, chapter. This is problem 15.7: Find the density of states $g(E)$ for the case of a free particle in one dimension; further, show that the density of states $g(E)$ for the free particle dispersion relation in two dimensions is a constant (which?). I am not sure what background information is necessary, but I can provide it. In particular, I do not know if any potentials are involved, but if they are, I would guess that we are dealing with a periodic potential of $N \in \mathbb{N}$ identical finite square wells. Basically, I am not sure what to make of this problem, what it means, or how to handle it. I would prefer detailed guidance and rather hand-holding hints. Thank you!


title: Angular momentum Bohr's model

text: I have been trying to derive speed, radius etc. in hydrogen atom using Bohr's postulates and **not** neglecting the coulombic attraction on proton. I know that they will be revolving around their centre of mass with same angular speed. But, I have this one doubt. Do we write $$L=\frac{nh}{2\pi}$$ of electron with respect to their centre of mass or wrt proton(nucleus)?


title: How does electron capture occurs?

text: Electron capture is a kind of decay by which a nuclear transformation takes place. Below is an example of it. $$ _{29} ^{64} \text{Cu} + e^- \rightarrow _{28}^{64} \text{Ni} $$ Of course, nucleus being positively charged and electron being negatively charged, there is an attraction between them, but electron generally don't fall in the nucleus. I am unable to understand that exactly how does nucleus is able to capture electron, using which interaction or force? Uncertainty principle won't allow electron in nucleus, though it is being captured. Exactly how?


title: Band structure and band index

text: Please let me know If my understanding is right. For a given $\vec{k}$, $H$ is a function of $\vec{k}$ the energies vary discretely for $n$ ie.,the band index. For a given $n$, we choose all the vectors to lie in the first Brilouin Zone (FBZ), $E_k$ varies continuously, that is the variation of energy for a particular band. So from these energy eigenvalues and eigenstates we could get the number of electronic states available in the band? I know am going in circles here. Please help! Thank you in advance.


title: Do Positrons Bend The Same Direction As Electrons In A Magnetic Field?

text: Electrons obey the right hand rule when a magnetic field bends their path. According to the right-hand rule, will positrons bend in the same direction?


title: Does wrapping a wet paper towel around a glass bottle really speed up the cooling process?

text: There are claims like this one that you can improve the cooling speed of beverages when you put them wrapped in a wet paper towel inside the refrigerator/freezer. ![enter image description here](http://i.stack.imgur.com/qqdOM.jpg) I've just tried it by myself and it seems it does not work as expected, although I might have confused freezer with refrigerator. **My question:** Is this kind of an urban legend or does it actually help?