title: Coherent State, Unitary Operators, Harmonic Oscillator

text: Consider the operator: $$O = e^{\theta(a^\dagger b - b^\dagger a)}$$ where $\theta$ is a constant. $O$ is a unitary operator. $a$, $a^\dagger$, $b$, and $b^\dagger$ are ladder operators for two harmonic oscillators. A normalized coherent state is defined as: $$\lvert\alpha\rangle = e^{-\lvert\alpha\rvert^2/2} e^{\alpha a^\dagger} \lvert 0\rangle$$ where $\lvert0\rangle$ is the ground state of the harmonic oscillator. I'm trying to see how $O$ acts on the coherent states by calculating $O \lvert\psi\rangle = O\lvert\alpha\rangle\lvert\beta\rangle$ in terms of coherent states. Also, how does $O$ act on $\alpha$ and $\beta$? I'm trying to use $$O a O^\dagger = a \cos(\theta) + b \sin(\theta)$$ and $$O b O^\dagger = -a \sin(\theta) + b \cos(\theta).$$


title: equivalence principle and nontrivial compactifications

text: it is commonly argued that the equivalence principle implies that everything must fall _locally_ in the same direction, because any local variation of accelerations in a small enough neighbourhood is equivalent to an inertial force. Is the purpose of this question is to understand, what are the bounds for a frame to be considered local, when the space might be compactified, and some of the macroscopic asymmetries of fields we see are explainable by symmetries in the inner dimensions. For instance, consider the following argument: _antimatter cannot fall away from matter, otherwise a local frame falling toward matter will not see local antimatter falling inertially_ don't get me wrong, there might be good reasons why in the particular case of anti-matter, it will fall exactly like regular matter (some of those might be found on this answer), however, i'm concerned about this particular argument. In a compactified space, what we mean with _locally_ in the context of macroscopic observations needs to be refined; for instance consider the scenario where antimatter (or for the purpose of avoiding controversy, lets call it $matter \dagger$ ) falls gravitationally away from regular matter; it seems to me that the principle of equivalence still holds if what we mean as a local neighbourhood means actually: $\lbrace$ usual local neighbourhood in macroscopic spacetime $\rbrace$ $\times$ $\lbrace$ the subset of the compactification space where regular matter lives $\rbrace$ in this scenario, $matter \dagger$ falls inertially in its own neighbourhood given by removing the regular matter sector from the local neighbourhood in the whole space (macroscopic plus compactified) I even made a ugly picture of the so called scenario:![two particles in the same macroscopic local neighbourhood see widely different curvatures, which macroscopically seems to contradict EP](http://i.stack.imgur.com/Flz1u.jpg) Question 1: **Are there known reasons why such reinterpretations of the equivalence principle, taking into account the additional dimensions, are flawed?** Question 2: **What does the original Kaluza-Klein theory has to say about how and under what forces the electric negative and positive charges move?** I'm sorry for the rather crude arguments, hope the intent is rather clear though


title: Potential and Kinetic Energy

text: In engineering school you learn the basic swing problem. Essentially that there is a transfer of kinetic energy (as seen in the velocity at the bottom of as swing) to potential energy at the top of the swing's arc. My question: What is the nature of KE and PE and how do they seamlessly "transfer" between the two? How is this explained by classical physics?


title: Space-time hospitality

text: Are there any reasons to assume that organic processes cannot arise in relatively flat spacetime environments?.. An argument for the existence of a critial value or window ..


title: Weyl exponential form of the Canonical Commutation Relations

text: What is the physical meaning of the $c$-numbers $Q, P\in \mathbb{R}$ in the exponent of the Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$? Here $\hat{q},\hat{p}$ are operators of position and momentum with canonical commutation relation $[\hat{q},\hat{p}]=i\hbar$.


title: Shape of Fan Blades

text: How is the shape of the blades of an air fan determined? Trial and error, or is there a theory behind it? What are they trying to maximize, volume of air dislocated per rotation?


title: Noether's theorem and gauge symmetry

text: I'm confused about Noether's theorem applied to gauge symmetry. Say we have $$\mathcal L=-\frac14F_{ab}F^{ab}.$$ Then it's invariant under $A_a\rightarrow A_a+\partial_a\Lambda.$ But can I say that the conserved current here is $$J^a=\frac{\partial\mathcal L}{\partial(\partial_aA_b)}\delta A_b=-\frac12 F^{ab}\partial_b\Lambda~?$$ Why do I never see such a current written? If Noether's theorem doesn't apply here, then is space-time translation symmetry the only candidate to produce Noether currents for this Lagrangian?


title: What happens when a polarized light get reflected by a rough dielectric material?

text: I have a polarized light (lets say p polarized) which happens to undergo scattering when it is obstructed by a rough opaque dielectric material. What will happen to polarization? Will some part of it becomes s polarized or circularly polarized? I am assuming light is either reflected or absorbed and there is no refraction therefore fresnel equation does not apply (or does it?).


title: Magnetic field or EM field around wire with electric current?

text: My question is based on the experiment where iron filings scattered around a piece of paper with a wire carrying an electric current poking through shows magnetic field lines. If you start to vary the current through the wire, perhaps using AC, then the magnetic field would be disturbed. [From what I understand, at this point the wire starts emanating a EM waves]. Is this actually an EM field or is the magnetic field moving up and down like a wave? If it is indeed an EM field would the iron filings then no longer show magnetic field lines? Again if it is indeed an EM field, at what point does the magnetic field stop being a magnetic field and start becoming an EM field? What mechanism causes the magnetic field to transform into an EM wave? [My assumption is that the current through the wire is also producing an electric field with field lines pointing outward from the wire, so when the electric current changes the electric field and magnetic field some how couple together to instead emanate an EM wave.] **Edit** Thinking about this further my thoughts are as such. A current moving through a wire produces an electric field with imaginary field lines moving outward from the wire. The electric field produces a magnetic field seen as circular lines around the wire (perpendicular to the electric field lines) in the iron filings. Further, the magnetic field around the wire is not static but has a velocity away from the wire (ie at the speed of light). When the current is alternated, the electric field and magnetic field, which are already coupled (as in both moving outward from the wire at the speed of light), now oscillates along with the current. Question still remains as to whether the iron filing will still show the magnetic field lines and if not, at what point does the magnetic field stop effecting the iron filings?


title: Taylor expansion of the metric

text: Consider a coordinate change $$ x^a\mapsto \tilde x^a=x^a+\epsilon y^a $$ In the note I am reading, the author calculate the change of metric by $$ g_{ab}(x) = \tilde g_{ab}(\tilde x)=\tilde g_{ab}(x^a+\epsilon y^a)=\tilde g_{ab}(x^a)+\epsilon\mathcal{L}_Y\tilde g_{ab}(x^a)+\cdots $$ My question is, why do we use $\mathcal{L}_Y\tilde g$ rather than $\nabla_Y\tilde g$ in the first order term? * * * Update Following is my newly try. As Prahar pointed out, I should write $$ g_{ab}(x)\mapsto\tilde g_{ab}(\tilde x)=\frac{\partial x^c}{\partial {\tilde x}^a}\frac{\partial x^d}{\partial {\tilde x}^b} g_{cd}(x) $$ Then we have $$ g_{ab}(x)=\tilde g_{ab}(x^a+\epsilon y^a)=\tilde g_{ab}(x^a)+\epsilon\nabla_Y\tilde g_{ab}(x^a)+\cdots $$ And I have to show $$ \nabla_Y\tilde g_{ab}(x^a)=\mathcal{L}_Yg_{ab}(x^a) $$ We have $$ \begin{align} \tilde g_{ab}(x^a)&=\frac{\partial x^c}{\partial {\tilde x}^a}\frac{\partial x^d}{\partial {\tilde x}^b} g_{cd}(x^a)\\\ &=\left(\delta_a^c-\epsilon\frac{y^c}{\tilde x^a}\right)\left(\delta_b^d-\epsilon\frac{y^d}{\tilde x^b}\right)\left.\right|_{x^a-\epsilon y^a}g_{cd}(x^a-\epsilon y^a)\\\ &=g_{ab}(x^a-\epsilon y^a)-\epsilon\left[\frac{y^d}{\tilde x^b}g_{ad}+\frac{y^c}{\tilde x^a}g_{cd}\right]\left.\right|_{x^a-\epsilon y^a}+O(\epsilon^2)\\\ &=g_{ab}(x^a) - \epsilon\nabla_Yg_{ab}(x^a)-\epsilon\left[\frac{y^d}{\tilde x^b}g_{ad}+\frac{y^c}{\tilde x^a}g_{cd}\right]\left.\right|_{x^a} +O(\epsilon^2) \end{align} $$ So we get $$ \nabla_Y\tilde g_{ab}(x^a)=\nabla_Yg_{ab}(x^a)-\epsilon\nabla_Y\left(\nabla_Yg_{ab}+\frac{y^d}{\tilde x^b}g_{ad}+\frac{y^c}{\tilde x^a}g_{cd}\right)\left.\right|_{x^a} + O(\epsilon^2) $$ On the other hand, we have $$ \mathcal{L}_Yg_{ab}(x^a)=\nabla_Yg_ab{x^a}+g_{ac}\nabla_bY^c+g_{cb}\nabla_aY^c $$ I cannot see these two are identical. Is there anything wrong in my deduction?


title: Stability theory

text: I'm studying stability theory recently and met a lot of phrases like linear stability and nonlinear instability. After searching on Google, I became more confused. Thus I wonder if there is any approperiate or famous books discussing these issues of dynamical systems.


title: gravitational field as a spin 2 particle using gauge invariance

text: can someone help me prove that a gravitational field corresponds to a spin 2 particle using gauge invariance. i know about the tensor formulation of GTR and the gauge invariance in electrodynamics corresponds to a spin 1 particle because we can add a gradient of scalar field to the four vector potential. how should i proceed for the same with the potential as g(u,v)


title: Physics engine - collisions

text: I'm working on a 2D physics engine somulator and I want it to be very accurate on a Physics point of view. Currently I'm making some researchs about how rigid body collisions may be calculated, and I've found this page, that presents the following equation: ![enter image description here](http://i.stack.imgur.com/tc4lR.png) Where $w$ is the angular velocity, $j$ is the impulse during the collision, $r$ is the radius and $\hat{n}$ is (I'm not sure but I am guessing) a normalized vector. My question is about what exacly is $I^{-1}$ .It says it's an inertia tensor, but for what I know it is used for 3D objects. Since my engine is 2D, may I simply use this formula: ![enter image description here](http://i.stack.imgur.com/ZrL61.png) Which is presented here as the moment of inertia of a rectangle. If yes, should I still use the (-1) power? Btw, I would also accept a link to some pdf explaining how collisions change angular and linear momentum in 2D... this wikipedia page was the best I could find.


title: What if UV behaviour of gravity was perturbative?

text: I understand that the UV behaviour of gravity ought to be dominated by black hole production and that graviton-graviton scattering ought to blow up above the Planck scale. Suppose, however, that gravity was somehow screened above the Planck scale, so that gravity was of constant strength, rather than ever increasing strength, above the Planck scale, and somehow remained perturbative. Would this imply a maximum curvature? and therefore a minimum radius/mass for a black hole, since small black holes require more curvature. Also, I guess there could be no singularity within a black hole. Could black holes still form, given that arbitrarily small black holes can't exist in this model? Assuming Hawking radiation, a massive black hole would decay until it reached that minimum radius, at which point it would cease to be a black hole. The contents, and information, would be no longer be trapped, and there was no singularity for it to hit. It would just come out, I suppose. Is this scenario at all plausible? Where is there likely to be trouble? Does this violate any experiments? Theoretical principles?


title: Toy helicopters at very high altitude

text: Because air is thin up there, toy helicopters usually have a maximum operational altitude. If the toy brought up by a real aircraft, higher than its intended maximum altitude, would it be able to hover inside the real aircraft? If so, what would happen if the real aircraft would now vanish magically, so that the toy helicopter is hovering in the middle of thin air?


title: How fast would a truck have to go to pull a pedestrian onto the road?

text: Let's suppose a pedestrian **P** is walking or standing next to a highway.   Suppose a truck **T** will drive by the said pedestrian at speed **V** leaving distance **L** between the two. Assuming **L** is a reasonably small finite size how would one calculate the force acting upon the pedestrian from both the general differences in pressure and turbulence? Shortly put, how fast would the truck have to go _in practice_ so that the air flow left in its wake would pull the pedestrian onto the road? ![Sample image](http://i.stack.imgur.com/Q8R1P.png)


title: Why doesn't water start to boil in soils?

text: Consider the following two scenarios: I take a vessel filled with some water and evacuate the air, the water will (or rather can) start to boil; if the pressure in the vessel is << atmospheric pressure. (Say -1e3 hPa) Now, consider a nearly dry porous medium such as an arbitrary soil clod. In this water can move according to cappilary forces from one place to an (energetically) lower place. Which force is commonly measured in "water potential" which can be as low as ~ -1e7 hPa. Why does water boil in the evacuated vessel (-1e3 hPa) and not in the dry soils with water potentials < -1e3 hPa? Where is the error in my thinking?


title: How to practically find the RA-DEC coordinates of a star?

text: How are RA-DEC (Right Ascension & Declination) coordinates of a star measured i.e. if scientist spot a new star in the sky how do they accurately measure it's coordinates?


title: Why are diodes able to amplify signals in groups, but not alone?

text: Single diodes can determine whether a signal passes or not (depending on the biased)(plus they can even rectify a signal), yet why does it take a combination of diodes(like a transistor) to amplify a signal?


title: Is Initial Coherence a Problem in Experiments?

text: In experiments, or optical setups, is the initial coherence of lasers something to be concerned about? You can calculate interference patterns, but those generally rely on the phase shift between the two sources being zero, right? In a laboratory, how do you know two lasers will be in phase?


title: Throwing an object in the air

text: Studying formulas about velocity and acceleration I came up with a question: _if I throw an object in the air with a velocity $v_0$ (suppose i throw it vertically) in how much time its final velocity $v_f$ will reduce to $0$ due to the force go gravity_? Here is how I tried to solve the problem: ## **Calculation of the time** I know that the final velocity of a object that receive an acceleration is: $$v_f=v_0+at$$ where $a$ is the acceleration and $t$ is the time in which the acceleration acts. I supposed that $v_f$ after a negative acceleration (the gravitational acceleration on Earth $g$) will reduce to $0$ and so I set up the following equation: $$0=\vec{v_0}-\vec{g}\cdot t$$ and solving the equation for $t$ I got that \begin{equation} t=\frac{v_0}{g}\tag{1} \end{equation} ## **Calculation of the space** I know that the formula to calculate the space that is made by an object moving with an acceleration is $$S=v_0t+\frac12 at^2$$ But now I can apply $(1)$ to the equation: $$S=v_0\cdot \frac{v_0}{g}-\frac12 g\left(\frac{v_0}{g}\right)^2$$ $$S=\frac{v_0^2}{g}-\frac{v_0^2}{2g}=\frac{v_0^2}{2g}\tag{2}$$ That would be the formula for the space. * * * Reassuming an object thrown in the air with a velocity $v$ will stop moving in the air after a time $t=\frac{v}{g}$ after making a distance $S=\frac{v^2}{2g}$. * * * Is this correct?


title: What happens when you pass a current through a coil made of a ferromagnetic substance?

text: I know when you pass a current through a length of copper coil or a solenoid, there is the induction of a magnetic flux. But what of the coil made of the ferromagnetic material, the permanent type magnetic material? Is there superposition? Is there Interference thus resulting in the cancellation of the overall field?


title: Why is this mass equation giving me mass = weight when clearly it isn't?

text: Convert the weight measured in pounds to the equivalent in Newtons. In the formula for determining mass based on weight, mass is measured in Newtons. Weight is measured in Kilograms, and the acceleration of gravity on Earth is measured as 9.8 meters per second squared. These are metric system unit measurements. To find the equivalent in U.S. units, you perform conversions. One pound is equal to 4.44822162 Newtons. Therefore, to convert pounds to Newtons, multiply the weight in pounds by 4.44822162. For example, to convert 150 lbs to Newtons, calculate as follows: 150 x 4.44822162 = 667 Newtons. Divide the weight in Newtons by the acceleration of gravity to determine the mass of an object measured in Kilograms. On Earth, gravity accelerates at 9.8 miles per second squared (9.8 m/s2). For example, to determine the mass of an object weighing 667 Newtons, calculate as follows: 667 Newtons / 9.8 m/s2 = 68 kilograms. Convert the mass measured in kilograms to the mass in pounds. One kilogram is equal to 2.20462262 pounds. Therefore, to convert kilograms to pounds, multiply the kilogram value by 2.20462262. For example: 68 kilograms x 2.20462262 = 150 pounds. This obviously is not right because 2 things that weigh the same can have a different mass. What is wrong with this conversion?


title: Tricks at manipulating creation/annihilation operators

text: Manipulation of terms in algebras different from the standard one (e.g. boolean algebra) can be a bit unnatural but there are always shortcuts that can help you. I was wondering if there is a list with the standard tricks for manipulating creation and annihilation operators of bosons and fermions, instead of using intensively their commutation relations and having to rediscover the wheel every time. This would prevent me (and many others) from losing a big amount of time at blind guess-checking... Even a good set of exercises can count as an answer.


title: Mathematical model that allows the existence of an infinite number of smallest, but different particles

text: Does there exist a mathematical formalism (model) in particle physics that assumes the existence of an infinite number of different, yet smallest particles (building blocks)?


title: Rod Falling on Frictionless Surface

text: A rigid rod with mass m is initially held at an angle with respect to a frictionless plane by a string. Then the string is cut. What happens to the rod? My intuition suggests that the rod should slide to the left but I cannot figure out how this is supposed to happen when the only forced involved are vertical (reaction force and weight). Moreover, while initially the reaction force should balance the weight, the reaction force causes the rod to rotate about its centre of mass. This would cause the end of the rod in contact with the surface to leave it. But then the reaction force disappears and the weight of the rod places it back on the surface again. I am not sure how to deal with this "jerky motion" either.


title: Quantum physics and black body

text: I'm a high school student, I just read something about black body. So I wanna know if I understand it correctly that black body is an ideal perfect absorber and emitter in sense that   a normal object, which is not under a thermal equilibrium condition, absorbs energy carried by EM radiation and emits them again. The object can't absorb all the energy from EM and neither emit all the energy. However, a black body can absorb and emit energy with highest efficiency. If what I understand is correct, when an object is not under thermal equilibrium condition, it will not emit all the energy it absorbs, so does it just store the energy? In what form? Kinetic energy of an electron?


title: Distribution of charge over a conductor?

text: Excess charge in a conductor is supposed to distribute itself evenly over the surface of the conductor. If the conductor has a charge of $2e$ how would the charge distribute itself? There's no way you can distribute $2$ electrons over the conductor so that the charge is evenly distributed, unless you can have less than $1e$ of a charge. I have absolutely no knowledge of quantum mechanics, so please keep answers intuitive rather than technical.


title: POVM positive density matrix decomposition

text: I have to prove that given a density matrix $\rho$ in a finite d-dimensional Hilbert space $\mathcal{H}_d$, it always exists at least one informationally complete POVM measure $\\{E_i\\}_{i=1}^{d^2}$ such that $$ \rho=\sum_{i=1}^{d^2} \alpha_i E_i, $$ with $\alpha_i\geq 0$. Without the last constraint on the $\alpha_i$ it is obvious, how can I prove it imposing the constraint $\alpha_i\geq 0$? Any idea?


title: Having trouble understanding the concept of current and Ohms law

text: Circuit in series: $10\,\text{V}$ power supply, 2 resistors connected. Each resistor is $10\,\Omega$, total of $20\,\Omega$ resistance. If I put an ammeter at the end of the circuit, will the current be calculated using 20 Ohms or 10? Electrons flow through the first 10 Ohm resistor and are slowed down, yes? Then they flow through the second 10 Ohm resistor. They have passed through a total of 20 Ohms BUT at 10 Ohms per resistor. Will the current be the same as if it had passed through 1 20 Ohm resistor? That violates everything I understand.


title: Why didn't we replace our SI units with a better system?

text: **Intro** It seems to me that the SI units we use today are nothing but the result of a historical 'coincidence'. I recently began researching about natural (absolute) systems of units, which are defined in such a way that selected universal physical constants are normalized to unity. These are very convenient, since almost all equations in physics are simplified. **Planck Units** Take for example the Planck units, where five fundamental physical constants take on the numerical value of 1. To quote from Wikipedia: > Planck units have a profound significance since they elegantly simplify > several recurring algebraic expressions of physical law by > nondimensionalization. They are particularly relevant in research on unified > theories. Planck units are even dubbed "God's units", since Planck units are free of anthropocentric arbitrariness. **Justification** I know that many 'everyday' quantities for a physicist in terms of Planck units would be very small numbers (for instance) but that shouldn't be a problem if we use scientific notation. In fact, it's a great thing since it can give us a better picture about those quantities being now in a unit system which is conceptually linked at a fundamental physical level. Besides, many such 'everyday' quantities are also expressed by very small / very large numbers in SI units. Think Planck's constant. Frank Wilczek even argues that using Planck units would help us re-frame important questions in physics: > We see that the question is not, "Why is gravity so feeble?" but rather, > "Why is the proton's mass so small?" For in [Planck] units, the strength of > gravity simply is what it is, a primary quantity, while the proton's mass is > the tiny number [1/(13 quintillion)]. AI am aware that it would be a big upheaval if we are to change our SI units but if it is to be done it better be done right now before more physicists grow up using the old unit system. The change is not likely to render old written material useless since, for instance, Newton's Principia Mathematica remains a fundamental text in mechanics even though it uses outdated notions such as "quantity of motion" and "quantity of matter". **Question** Given all that I explained, is there a reason we choose to stick with the old and weary unit system? Why didn't we leave the kilogram, defined by an arbitrary French rod, to ladies gossiping about Kim Kardashian's weight and choose something that makes more sense scientifically?


title: Definition of metre

text: We know that 1 meter is the distance travelled by light in vacuum within a time interval of 1/299,792,458 second. My question is why we didn't take a simpler number like 1/300,000.000 or why not just 1?


title: Relation between charge and voltage in spherical capacitors

text: 1. If we had a spherical capacitor and the voltage across it is 1000 V, I need to know the charge on every plate, I know 2 ways to solve this either using C=Q/V or V=Q/r * 1/4*pi*Epsilon. So, which one should I use, and why?   2. Another thing, according to hyperphysics the formula for the capacitance of the spherical capacitor = 4*pi*epsilon / 1/a - 1/b. Does that mean that the dielectric constant of the dielectric has no effect like parallel plates?


title: The far $\vec{\mathrm{E}}(\vec{\mathrm{r}},t)$ field of an arbitrary antenna given an arbitrary input current $I(t)$

text: I was wondering if the following expression is reasonable to express the far $\vec{\mathrm{E}}\left(\vec{\mathrm{r}},t\right)$ field for an arbitrary antenna that is fed with an arbitrary time-varying current $I\left(t\right)$ where the functions $\vec{\mathrm{a}}_i\left(\hat{\mathrm{r}}\right)$ and the integer $n$ depend **only** on the **antenna geometry**? $$ \vec{\mathrm{E}}\left(\vec{\mathrm{r}},t\right) = \frac{1}{4\pi \epsilon_0 c^2 r} \; \sum_{i=1}^{n} \left[ \vec{\mathrm{a}}_i\left(\hat{\mathrm{r}}\right) \; \frac {d^i}{dt^i}I\left(t - \frac {r} {c}\right) \right], \qquad \vec{\mathrm{a}}_i\left(\hat{\mathrm{r}}\right) \cdot \hat{\mathrm{r}} = 0 $$ I've built this house of cards based on the fact that for a current $I_0 \cos{\omega t}$ passing through an antenna,   * There's no radiation from any antenna when the current is DC ($\omega = 0$)   * The far electric field for a small dipole is $\propto \omega \;\sin{\omega t_r}$   * The far electric field for a small loop is $\propto \omega^2 \cos{\omega t_r}$ Antenna parameters as commonly used all depend on the antenna geometry as well as the excitation frequency (and thus the wavelength $\lambda$):   * radiation resistance and self reactance,    * gain/directivity, radiation pattern and effective aperture    * polarization  I was playing with this idea to see if a _frequency independent_ characterization is possible. I guess this won't actually be very _useful_ , but if correct it may be of some help in understanding antennas generally. I have reasoned as follows. If all points of non-zero charge density and current density lie within a sphere of radius $\mathcal{R}$ centered on the origin $\vec{0}$, it can be shown that $$ \vec{\mathrm{E}} \left(\vec{\mathrm{r}},t\right) \approx \frac {\left(\vec{\mathcal{Q}} \left(\hat{\mathrm{r}},t_r \right)\times \hat{\mathrm{r}}\right) \times \hat{\mathrm{r}}} {4\pi\epsilon_0 c^2 r} $$ ... where $\vec{\mathrm{r}} \equiv r \hat{\mathrm{r}},\quad r \gg \mathcal{R}, \quad t_r \equiv t - \dfrac r c$, and ... $$ \vec{\mathcal{Q}} \left(\hat{\mathrm{r}},t\right) = \iiint_{\mathbb{B}\left(\vec{0},\mathcal{R}\right)} \left[ \frac {\partial} {\partial t} \vec{\mathrm{J}} \left( \vec{\mathrm{s}}, t + \dfrac {\vec{\mathrm{s}} \cdot \hat{\mathrm{r}}} {c} \right) \right] \space dV\left(\vec{\mathrm{s}}\right) $$ Now the current density field $\vec{\mathrm{J}}\left(\vec{\mathrm{s}},t\right)$ would be derived from the input current $I\left(t\right)$ somehow. For a 'small' dipole it is easy to see how $\vec{\mathcal{Q}} \left(\hat{\mathrm{r}},t\right) \propto \frac{d I\left(t\right)}{dt}$ and for a 'small' loop how $\vec{\mathcal{Q}} \left(\hat{\mathrm{r}},t\right) \propto \frac{2\mathcal{R}}{c}\frac{d^2 I\left(t\right)}{dt^2}$. But can this line of thinking be generalized when these structures are no longer 'small'?


title: Superposition and the Winning Jackpot Numbers

text: Let's say I buy myself a lottery ticket (Mega-Millions). I have $\frac{1}{175,711,536}$ chance of winning. Before I tune on the tv/radio and listen to the winning numbers (i.e. make an observation), is it correct to say that the winning numbers are in some kind of 'superposition' of states? And the act of watching the numbers come out of the drawing machine is somehow 'collapsing' the wave function? Could this possibly affect the way I think about quantum mechanics? EDIT: I found the following article very similar to what I've proposed: http://iopscience.iop.org/0031-9120/38/2/606/pdf/pe32m6.pdf


title: Does a scale measure weight or mass?

text: What does a measuring scale measure: mass or weight? I mean, when we stand on a weighing scale the reading we get is in $\mathrm{kg}$'s. What does this mean?


title: Are there any known (closed form even if approximate) solutions to problems in relativistic elasticity?

text: There are several useful known solutions to the EFE with relatively simple / trivial stress-energy-momentum tensor, such as the Schwarzschild solution. Despite the idealizations made therein they are enormously important and insightful. I am curious to know 1) Do any such solutions exist for the case of relativistic elastic solids? 2) If not, what are the principal obstacles in constructing solutions, even approximate (say linearized) ones? 3) If I understand correctly, various competing versions of a "theory" of relativistic elasticity have been around for several decades. Is there a reason why there are multiple theories? 4) Also, IIUC, matters of existence and uniqueness for the Cauchy problems arising from such theories have also been dealt with. I came across this paper by Beig and Schmidt, in which the authors deal with questions of local existence and uniqueness for a theory they describe as a "relativistic version of classical elasticity". Is this a generally accepted state of affairs or are there caveats? 5) There seem to be several disparate sources of information about this topic, is there a nice "unified" resource to get started? Thanks.


title: One problem on Classical Mechanics

text: ![enter image description here](http://i.stack.imgur.com/L1h39.jpg)![I'm wondering why in this problem I can't apply newton second law? The mass m of the moving part will be the mass solved in the problem.](http://i.stack.imgur.com/HXxjS.jpg) I'm wondering why in this problem I can't apply newton second law? The mass $m$ of the moving part will be the mass solved in the problem.


title: One proof of linear momentum of the system

text: ![enter image description here](http://i.stack.imgur.com/lkF4l.jpg) ![enter image description here](http://i.stack.imgur.com/FtuXZ.jpg) ![enter image description here](http://i.stack.imgur.com/WxU2A.jpg) I copy and paste all the proof here, but I'm confused about the last step, which is that the summation of the internal force and be written as two other summations.


title: About photons and mirrors

text: If a photon hits a 'perfect' mirror (with no environment interference) would the mirror move a bit?


title: Revisiting the microscopic concept of Touching with some more questions

text: This question is regarding the amazing answer given by Terry Bollinger at this Phys.SE post. I think this answer is very helpful but i do have some standing questions. He says > Once the bonding has happened, however - once the atoms are in arrangements > that don't leave unhappy electrons sitting around wanting to engage in close > bonds - then the territorial aspect of electrons comes to the forefront: > They begin defending their turf fiercely. Does this "Turf" extend past the electron? Or is it the space occupied by the electron? And what exactly are they defending it with? Is it like an electric field surrounding the electron? And is this field any less _physical_ then the electron its self? Another question is when he stated that they move _Close enough_ to not join but to still make "Contact" how close is close enough? Is it like two balls coming together and, instead of passing through each other, elastically deform? Is it this deforming, as opposed to merging, what is referred to a repelling? If the electrons themselves don't "repel" (In the sense i outlined above) then is it some sort of field or force that repels? Is this field or force the _Electron Cloud_? If not, is this field or force present in the atom at all times, and by this logic, can this be thought off as the outer layer of the atom? I also read here: http://users.hal-pc.org/~clement/Whats%20in%20a%20Touch.pdf That electrons clouds overlap all the time. that your feet and shoes overlap the electron clouds of the atom all the time. Is this correct?


title: About the speed of light

text: If Mr. E is aboard a spaceship traveling near the speed of light the usual reason for the spaceship not going faster than $c$ is the (relativistic) mass of the ship increases without bound, I think. Yet Mr. E says out loud what about the mass of the fuel? When the fuel's (relativistic) mass increases would its potential energy increase enough to compensate for the ship's (relativistic) mass increase?


title: Why is the stress-energy tensor symmetric?

text: The relativistic stress-energy tensor $T$ is important in both special and general relativity. Why is it symmetric, with $T_{\mu\nu}=T_{\nu\mu}$? As a secondary question, how does this relate to the symmetry of the nonrelativistic Cauchy stress tensor of a material? This is apparently interpreted as being due to conservation of angular momentum, which doesn't seem connected to the reasons for the relativistic quantity's symmetry.


title: Powercounting in String Frame?

text: if I consider the low energy effective action of type IIB string theory in the string frame, i.e. with an $e^\phi$ prefactor, is it possible to do standard powercounting with this action? I.e. how do I read of the mass dimensions of the fields? Do I need to transform it to Einstein frame first? In other words, how do I do powercounting for the action $$S=\int d^{26}x e^{\phi}\Big(R+(\partial\phi)^2-\frac{1}{12}H^2\Big)$$ with $H_{ijk}=3\partial_{[i}b_{jk]}$. $\phi$ is a dilaton, $b$ a Kalb-Ramond field and $R$ the Ricci scalar. What's the mass dimension of the exponential? And than the other fields? Should'nt there be Newton's constant somewhere, too? Thanks


title: adhesion(bonding etc.) of materials and frictional force

text: I am studying material physics. I heard in the class that every adhesion or bonding can occur only with frictional force. Today, I tried to find a proper reference for this statement but I could not. Is there any reference that every adhesion or bonding(esp. Injective moldings) can occur only with frictional force? Or is there any way that can happen without frictional force? For example, in the process of molding, when melted materials harden, why does it stick to the other material? How does the frictional force acts in this case? I want some references regarding these. Thank you!


title: Non Classical mechanic answer to : length of time a thrown object spends in rest before falling down?

text: When an object is thrown upwards, when it eventually comes to rest and starts falling, for how long is it stationary? What about an particle in electric field having an initial velocity towards it's same charge? That too would come to rest and reverse velocity, the question is for how long is it at full stop with rest? Classical physics gives a time of 0, but is that correct? Is it really at rest for 0 seconds? That answer seems a bit counter intuitive.


title: Does this relation about direction of particles make sense?

text: Maybe I've just stared at this statement too long and I've missed something obvious. Nevertheless, here's the problem: Landau-Lifshitz vol. 1§16, problem 1. > Consider (classical) collision of two particles in center of mass > coordinates. Before the collision, the particles are just traveling towards > each other and in CM coordinates then the direction of the velocities of two > particles should be opposite to each other, i.e. $$ \theta_1 = \theta_2 + > \pi , $$ where $\theta_i$ is the angle between the velocity of particle $i$ > and the coordinate axis. However, in the solution in Landau-Lifshitz, they state that "In the C system, the corresponding angles are related by $\theta_1 = \pi - \theta_2$." Is there a mistake in L-L? If not, can you explain me the relation above?


title: Does infrared rays pass through polarized glass?

text: Actually I had asked in another post that "Does infrared rays pass through active shutter glass" but someone just commented that infrared rays dont pass through polarized glass. If infrared rays doesnt pass through polarized glass can someone explain the reason or give reference link to look through.


title: How to liquefy Hydrogen?

text: I have got a science project and my teacher has recommended me to do "Liquefying Hydrogen". I have been continuously thinking about that but I have not come to a solution. Can anyone please tell me how to liquefy and store hydrogen??


title: Analogy between magnetic bottle and Van Allen's radiation belt

text: A magnetic bottle is an arrangement that permits to confine charged particles. Here you can find a review for charged particle rotating in a magnetic field and at the bottom of the page a description of the magnetic bottle: Motion of a charged particle in magnetic field Let's use this image since it's more clear ![enter image description here](http://i.stack.imgur.com/y45wc.jpg) What results from this picture is that a particle inside a magnetic bottle makes a circular motion following magnetic field lines which have a sort of cylindrical symmetry as it's expressed by this simulation: Magnetic Bottle 1 So you can imagine to put the earth in the middle of the picture: in this way the two rings are to the two magnetic poles. In the case of a Van Allen's radiation belt motion simulation (Radiation Belt as a Magnetic Bottle), particles move around a magnetic field lines instead of moving around the earth. But from this simulation: Magnetic Bottle 2 you can see that a motion around a magnetic line is possible but i can't understand heuristically why. So the questions are:   1. how a particle can move around a magnetic line if there is closed to it another one? so how can a particle choose around which line move (losing in this way the cylindrical symmetry)?   2. why the two type of motions are in disagreement? Is the motion in the first picture just an approximation perhaps?


title: Wind Turbine Impact

text: OK, I'm not sure this is not a silly question, but here goes. I've often wondered what the impact is of wind turbines on the weather. My question is, since wind turbines transform wind energy into electricity, this means that the winds lose in power. I'm well aware that for the effect to be noticeable, you'd need a pretty large wind turbine park. But I wondered how large? How would the effect scale with the size of the park? Bonus question: what about tidal turbines? Most closely related questions: Is it possible that windmills can consume the wind? What does the quantification of causes and effect look like for clouds in offshore wind turbine wakes?


title: Compressibility of solids (quadratic approximation)

text: (according to Steven H. Simon's "The Oxford Solid State Basics") A well-known approach to describe a one-dimensional chain of atoms is to approximate the potential of each of the atoms quadratically: ![potential well](http://i.stack.imgur.com/zx81v.png) Therefore, $V(x) \simeq V(x_{eq}) + \frac{\kappa}{2}(x-x_{eq})^2$ and hence $F=-\kappa (x-x_{eq})$. We can use this approximation to describe compressibility whose coefficient is defined as: $$\beta = -\frac{1}{V} \left (\frac{\partial V}{\partial P} \right )_T$$ In 1D this becomes (I'll skip the T subscript): $$\beta = -\frac{1}{L}\frac{\partial L}{\partial F}=\frac{1}{\kappa x_{eq}}$$ I don't understand how this result was obtained. Isn't $\frac{\partial F}{\partial x}$ simply equal to $-\kappa$? What exactly are we differentiating with respect to? Also, am I right in thinking that the $P \rightarrow F$ transition may be justified because pressure "becomes" force in 1D?


title: Can the length of the closed string larger?

text: Can the length of the closed strings becomes larger than the plank length ? With that , does the string describe the higgs particles ?


title: Does the material of the slits affect the pattern of the electrons in the double-slit experiment?

text: The question comes from thinking about the way how electrons act on the slits. Do material of the slits affect electrons motion? Assume the radius or length of the slits are set in this case.


title: How Was the Speed of Light Discovered?

text: The title pretty much says everything; I just want to know how the speed of light was discovered.


title: Solving a Young Laplace equation for a meniscus against a flat plate

text: This is more of a math question and one, furthermore, that I know the final answer to. What I am asking is more of a "how do I get there" question as this question was generated during a self study situation. So, for a flat plate with a meniscus on it at some contact angle the form is: a y''$/(1+(y')^2)^{3/2}$ + b g y = 0 where a is surface tension, b is density, and g is the usual 9.8 m$/s^2$. Now, I know that when you "A first integration, together with the boundary condition dy/dx=y=0 as x goes to infinity yields: 1$/(1+y'^2)^{1/2}$=1-(bg/2a)$y^2$ " \--From de Gennes on Menisci My problem is with the integration. How do I integrate something like the 1st equation? I find myself running in circles. Worse, I know that this type of integration is something I've run across numerous times, but I can't seem to find the methodology in the current stack of books. So I beg, can someone either tell me where to look to find the appropriate methodology to attack this problem or take me through the process? I would be grateful for as long as it would be useful for you. Thanks, Sam


title: Newton's law of gravitation in complex form

text: In an ebook about elementary complex analysis I came across Newton's law of universal gravitation with a complex valued function in place of $r(t)$. Can somebody please explain the intuition about how a real valued function gets translated into a complex function $z(t)$? I don't understand how this was done. Thanks! (I think this might be a pretty illustrative way to connect real and complex functions, so I thought I'd put it up.) ![enter image description here](http://i.stack.imgur.com/vEFqw.jpg)


title: Why are the alkali and alkaline earth metals present in the ashes of burnt wood?

text: The alkali earth metals form the two left-most column of the periodic table of the elements, other than hydrogen. See wikipedia articles:   http://en.wikipedia.org/wiki/Alkali_metal   http://en.wikipedia.org/wiki/Alkaline_earth_metal Now "alkali" means that these are in a sense the opposite of "acid". One of the ancient uses of alkali materials is in the conversion of corn into hominy (or, for those in the southern US, grits similar to posole, polenta or farina ). The conversion of corn to hominy is of ancient origin. From a nutrition point of view it is useful because it increases the availability of the lysine and tryptophan proteins in the corn. See http://www.practicallyedible.com/edible.nsf/pages/hominy This helps prevent the nutritional disease pellagra http://en.wikipedia.org/wiki/Pellagra In ancient times, hominy was manufactured by soaking corn in a mixture of water and ashes. This is equivalent to the modern method, which is to soak the corn in lye. So my question is this: why is it that ashes are alkali?


title: Why does Joule-Thomson expansion of a gas do no (external) work?

text: I understand one of the benefits of the Hampson-Linde cycle is that there are no cold-side moving parts, but isn't one losing an awful lot of energy in the throttling process? There must be something basic I'm missing here, because it looks almost like this cycle destroys energy: pressure drops while volume increases, and temperature drops at the same time. The expansion is isentropic (How does that differ from an adiabatic expansion? I thought the condition that dS = 0 constrained a gas to a unique trajectory - an adiabat - along which it does work. But unless I'm mistaken, a Joule-Thomson expansion does no external work, despite not being an expansion into an evacuated space. Where did the work "go"? Maybe an example might illustrate. If I let air out of my car's tyres, is that a Joule-Thomson expansion? If so, doesn't the expansion do work in lifting the atmosphere by 20 attometers? If not, what _is_ an everydayish example?


title: How fast can fire spread?

text: You can build a home-made flamethrower by using a spray and a lighter: ![enter image description here](http://i.stack.imgur.com/uxv6S.png) (from this YouTube video) But this can be dangerous if the fire gets in the can and the can explodes. So I guess the speed at which the particles come out of the box has to be high enough to prevent this. > **NOTE** : I don't want to do this at home, because I think it's too > dangerous. This is just a thought experiment! I guess the question " **how fast can fire spread?** " highly depends on the details (like: What kind of exothermic reaction is going on; how big is the pressure at which the spray exits the can; temperature around the can). I would like one theoretical answer (choose everything to make it as fast as possible / slow as possible) and a practical answer (make a reasonable assumption of what could go on with the lighter / spray stuff).


title: Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

text: I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch what I tried until now and why it is not working: Analogous to the mentioned publications I introduced $$b_n:= \frac{1}{2}Exp{(\frac{q_n-q_{n+1}}{2})}\\\ a_n:= -\frac{p_n}{2} $$ where it follows directly with $\frac{\partial H}{\partial q_i}=-p_i$ and $\frac{\partial H}{\partial p_i}=q_i$: $$\dot{b_n} = (a_{n+1} - a_n)b_n \\\ \dot{a_n} = 2 (b_{n}^2 - b_{n-1}^2)$$ When now using the Lax Pair $L$,$B$: $$ L f_n = b_n f_{n+1} +b_{n-1} f_{n-1} + a_n f_{n}$$ $$ B f_n = b_n f_{n+1} - b_{n-1} f_{n-1} $$ it can be shown that $\partial_t L=[B,L]$. My problem arises in defining the border conditions of my couple $q_1$ and $q_2$ in the 2d lattice above, since one needs to shift to the 3d representation $\\{b_0,b_1,b_2\\}$ in order to satisfy the periodic conditions (One mutual coordinate $q_3 = 0$ coupled to the others). Since it can be shown easily that $\dot{\lambda} = 0$ (where $\lambda$ is an eigenvalue $Lv=\lambda v$) the constants of motion reduce to the calculation of the eigenvalues. But in this case the eigenvalues of $L$ dont seem to simplify, in fact it doesnt seem to be a solution, which was my inital goal. In general this approach seems to be at overkill for the 2d problem since it solves the n-dimensional Toda lattice.   1. Anyone knows of an easier approach to the 2d problem?   2. The Matrix $L$ seems to yield the wrong solution: $$ L = \begin{pmatrix} a_0 & b_0 & 0 \\\ b_0 & a_1 & b_1\\\ 0 & b_1 & a_2 \end{pmatrix}$$ The Matrix does neigher solve $\partial_t L = [B,L]$ (with $B=L_+ - L_-$) nor are eigenvalues constants of motion. Was has gone wrong?   3. Since the inverse scattering method can be applied here, I tried to get the scattering data, but actually I was not able to do the task. Any literature?


title: Baryogenesis - P and CP Violation

text: There are 4 requirements for baryogenesis to happen: 1\. A process that violates baryonnumber conservation 2\. The universe has to be out of equilibrium 3\. P has to be violated 4\. CP has to be violated Why do we need the CP violation and why do we need P violation?


title: ${f=ma}$: a duality between F-theory and M-theory?

text: $$F = M \Big|_{A(T^2) \to 0}$$ The above equation is the duality equation between F-theory and M-Theory on a vanishing 2-torus. What's the explanation for this equation? Is there anything similar to this equation with M-theory and type IIB theory, and how can ones explain it?


title: Why do we only have complete particle generations?

text: There are 3 generations of fermions in the standard model. I know that there is a theorem that states, that only complete generations are allowed. This means that there have to be quarks with three colors. With this, if we sum over all leptons and quarks in a generation the overall charge is zero. Why is it assumed that this must hold?


title: Non-conservative field?

text: A conservative field is a function of position/configuration, what about a non-conservative field? It's a function dependent on what?


title: Entropy as an arrow of time

text: From what I understand, entropy is a concept defined by the experimentalist due to his ignorance of the exact microstate of a system. To say the number of accessible microstates $W$ of the universe is constantly increasing is nothing more than saying 'ignorance begets ignorance'. I have often encountered the argument of ever increasing entropy for the presence of an inherent time asymmetry, most prominently in the works of Penrose. It just doesn't seem to make sense. Let us imagine alien beings who are experiencing time in reverse. For them, increasing ignorance is in the direction of our decreasing ignorance. So how can perpetual increase in entropy indicate an 'arrow of time'? One possible explanation (I thought of) for this argument was the fact that there may never be a mechanism to reduce ignorance. I will pose this as a question: An observer determines the number of possible microstates of system+observer to be $W_0$ at time $t_0$. After improving his measurements, can he (at a later time) measure $W'$ (where $W_0>W'$) as the number of possible microstates? Assume that all the microstates are equally probable in this case. Unless the above in false in general, how else can anyone claim that entropy reveals an arrow of time? **EDIT:** I am performing this discussion for isolated systems (the universe or it's parts if relevant). The means I propose to reduce entropy in the present, is by redefinition of macroscopic variables and of microscopic models used to count the number of distinguishable micro-states (that obviously yield the same macro-state). This is essentially an argument against the robustness of entropy via redefinition.


title: What is a capacitive screen sensing?

text: What should be a properties of a body so a contact with it can be detected by a capacitive touchscreen or a touchpad?


title: Two spinor tensors and Maxwell's equations

text: Let's have two symmetric (by the indices) spinor tensors $F_{ab}, F_{\dot {a}\dot {b}}$ and conditions $$ F_{ab}, \partial^{\dot {a} a}F_{ab} = 0, \quad F_{\dot {a}\dot {b}}, \partial^{\dot {a}a}F_{\dot {a}\dot {b}} = 0, \qquad (1) $$ $$ \partial^{a \dot {a}} = (\tilde {\sigma}_{\mu})^{\dot {a} a}\partial^{\mu}, \quad (\tilde {\sigma}_{\mu})^{\dot {a} a} = \varepsilon^{\dot {a}\dot {b}}\varepsilon^{a b}(\sigma_{\mu})_{b \dot {b}} = (\sigma_{0}, -\sigma_{i}). $$ by which these tensors may represent massless field with helicity $1, -1$ respectively. If our field is real, we must take the direct sum of these representations. It may be described through antisymmetrical 4-tensor rank 2. I know that there is the correspondence between an arbitrary antisymmetric tensor $F_{\mu \nu}$ and two symmetric spinor tensors $F_{ab} , F_{\dot {a} \dot {b}}$: $$ F_{\mu \nu} = \frac{1}{2}(\sigma_{\mu \nu})^{a b}F_{ab} - \frac{1}{2}(\tilde \sigma_{\mu \nu})^{\dot a \dot b}F_{\dot {a} \dot {b}}, $$ $$ F_{ab} = (\sigma^{\mu \nu})_{ab}F_{\mu \nu}, \quad F_{\dot {a} \dot {b}} = (\tilde {\sigma}^{\mu \nu})_{\dot {a}\dot {b}}F_{\mu \nu}. \qquad (2) $$ Then it obvious that I must substitute $(2)$ into $(1)$ and get Maxwell's equations. But I can't do it, because I have the problem with convolution of sigma-matrices. Can you help me?


title: Conductive glue

text: I would like to use some conductive glue to contact a silicon chip's back gate. Any suggestions on what to use? Can I use thermal silver paste? I would be operating at cryogenic temperatures below 4K.


title: Blocking an IR camera

text: What kind of material would be best to block an IR camera? Would Silicon work? (e.g the Silicon typically used in iPhone cases) Rather than covering the object/subject, I'm interested in fully blocking the actual camera, so it is unable to receive any IR information. Thanks!


title: Information content of the electrostatic Maxwell equations vs Coulomb's Law vs Poisson's Equation

text: In electrostatics, we have Maxwell's equations: $\nabla \cdot E = \rho$ $\nabla \times E = 0$ These four equations (the second line standing for three equations) can also be written in terms of the electrostatic potential: $ -\nabla^2 V=\rho $ $ E = -\nabla V $ Now if we know the positions of every charge in our system, we can find the electrostatic field (completely and entirely, with no additional information required) using Coulomb's law: $ E(x) = -\nabla \int \frac{\rho(x')}{4\pi|x-x'|} \mathrm{d}^3 x' $ My question is: to what extent can we do the same with Maxwell's Equations? For instance, whenever Coulomb's Law is derived from Maxwell's equations, an appeal needs to be made to spherical symmetry. Must we do this? Can we not use the vanishing curl somehow to reach the same conclusion? Similarly, if we derive Coulomb's Law from Poisson's equation, we must specify boundary conditions. We must specify that the potential is some constant, say zero, at infinity. It appears that the information content is less. I have read a little bit about the Helmholtz decomposition, and it appears that the curl and divergence of a vector field (E, in this case) do completely determine the vector field, providing certain restrictions are placed on the smoothness and decay of the field at infinity. In other words, it appears that in fact Maxwell's Equations (in the context of electrostatics) do have less information content than Coulomb's Law.


title: Impressions of Topological field theories in mathematics

text: There have been recent results in mathematics regarding Conformal field theories and topological field theories. I am curious about the reaction to such results in the physics community. So I guess a starting place for a question would be something like: Did Atiyah's axiomatization of TQFT capture what physicists feel are important in QFT's? How many more parameters are there to what a physicist would call a QFT or FT in general? Perhaps I should mention that mathematicians (the ones who dont know any phsyics) think about various Field Theories as things that assign invariants to manifolds that should be sensitive to certain data (depending on the word proceeding field). A lot has been done when you only care about the basic topology of the Manifold you are evalutating your field theory at. Have mathematicians gotten rid of too much for physicists to care? What types of things do physicists want a Field theory to keep track of, what kind of structure on the manifold that is?


title: Energy conservation in diamagnetic levitation

text: How do we account for conservation of energy in diamagnetic levitation when no external energy is required?


title: Noether's Theorem For Functionals of Several Variables

text: My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Does Noether's theorem, for functionals of a single independent variable, just say that, because $\mathcal{L}$ is invariant, we have $$\mathcal{L}(x,y_i,y_i')dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*?$$ It is usually stated by saying that $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} - \left[\sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_j}{dx})} \tfrac{\partial y_j }{\partial x} - \mathcal{L}\right]\frac{\partial x^*}{\partial \varepsilon}$$ is conserved, so is this equivalent to what I've written above (I've hopefully justified all of this below)?. (If there is any ambiguity in this question, I've offered a hopefully unnecessary explanation of the details below) The only problem with it I see is that it works fine for conservation of linear momentum, but it seems to show that conservation of energy and angular momentum are always zero, since $dx^* = 0$, unless I add a constant to the above, i.e. $$\mathcal{L}(x,y_i,y_i')dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* + C$$ which seems fine as adding constants to Lagrangians do not affect the EOM, so that's the only problem I see with it. I like the above expression, it's great for remembering Noether's theorem, it's more general than what almost every physics book presents as Noether's theorem (it's not just conservation of energy!), explains what Landau's additivity of integrals of motion comment means mathematically etc... **Can we generalize it to Lagrangian densities?** The statement of the multivariable Noether I know is that, for $$\mathcal{L} = \mathcal{L}(x_i,u_j,\frac{\partial u_j}{\partial x_i})$$ we have that $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\left(\frac{\partial u_j^*}{\partial \varepsilon_k} - \sum_{i=1}^n\frac{\partial u_j}{\partial x_i}\frac{\partial x_i^*}{\partial \varepsilon _k}\right) + \mathcal{L}\frac{\partial x_i^*}{\partial \varepsilon _k}\right] = 0$$ I can hardly remember this, and as I've indexed it I can't turn it into anything involving what I _think_ is the Hamiltonian for a functional of several independent variables $$\mathcal{H} = \sum_{j=1}^np_{ij}\frac{\partial u_j}{\partial x_i} - \mathcal{L} = \sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\frac{\partial u_j}{\partial x_i} - \mathcal{L}$$ (I have no idea why Jackson adds a $g^{\alpha \beta}$) Can this be turned into something similar to my main equation, perhaps using $\delta_{ij}$s or $g_{\mu \nu}$s or something? An attempt: $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\left(\frac{\partial u_j^*}{\partial \varepsilon_k} - \sum_{i=1}^n\frac{\partial u_j}{\partial x_i}\frac{\partial x_i^*}{\partial \varepsilon _k}\right) + \mathcal{L}\frac{\partial x_i^*}{\partial \varepsilon _k}\right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}\left(d u_j^* - \sum_{i=1}^n\frac{\partial u_j}{\partial x_i}d x_i^* \right) + \mathcal{L}d x_i^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{k=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_k}{\partial x_i})}\sum_{l=1}^n\frac{\partial u_k}{\partial x_l}d x_l^* + \mathcal{L}d x_i^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{k=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_k}{\partial x_i})}\sum_{l=1}^n\frac{\partial u_k}{\partial x_l}d x_l^* + \sum_{l=1}^n\delta^l_i \mathcal{L}d x_l^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{l=1}^n (\sum_{k=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_k}{\partial x_i})}\frac{\partial u_k}{\partial x_l} - \delta^l_i \mathcal{L})d x_l^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m \frac{\partial \mathcal{L}}{\partial (\tfrac{\partial u_j}{\partial x_i})}d u_j^* - \sum_{l=1}^n \mathcal{H}d x_l^* \right] = 0$$ $$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left[\sum_{j=1}^m p_{ij}d u_j^* - \sum_{l=1}^n \mathcal{H}d x_l^* \right] = 0.$$ I don't know if this is right. References:   1. Gelfand, Calculus of Variations - Ch. 3, 4 & 7   2. Jackson, Classical Electrodynamics - Ch. 12 Sec 10 * * * Noether's theorem for functionals of a single independent variables states that, given a continuous one-parameter family of infinitesimal transformations of the form $$T(x,y_i,\varepsilon) = (x^*(x,y_i,\varepsilon),y_i^* (x,y_i,\varepsilon))= (x^*,y_i^*)= (x+\frac{\partial x^*}{\partial \varepsilon}\varepsilon,y_i+\frac{\partial y_i^*}{\partial \varepsilon}\varepsilon),$$ if a Lagrangian $$\mathcal{L} = \mathcal{L}(x,y_i,\frac{dy_i}{dx})$$ is invariant $$\mathcal{L}(x,y_i,\frac{dy_i}{dx}) = \mathcal{L}(x^*,y_i^*,\frac{dy_i^*}{dx})$$ under $T$ then the scary-looking quantity $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} - [\sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_j}{dx})} \tfrac{\partial y_j }{\partial x} - \mathcal{L}]\frac{\partial x^*}{\partial \varepsilon}$$ is conserved (more precisely, you do it in terms of the functional but that is irrelevant for this discussion). Examining this quantity for two seconds let's us notice we can write this conserved quantity as $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_i}{dx})} \frac{\partial y_i^*}{\partial x} - \mathcal{H}\frac{\partial x^*}{\partial \varepsilon}$$ in terms of the Hamiltonian $$\mathcal{H} = \sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \tfrac{d y_j}{dx})} \tfrac{\partial y_j }{\partial x} - \mathcal{L} = \sum_{j=1}^n p_i \tfrac{\partial y_j }{\partial x} - \mathcal{L} $$ or, taking away the parameter, as $$\sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*$$ Notice the similarity between the quantity in Noether's theorem and this: $$\mathcal{H} = \sum_{j=1}^n p_i \tfrac{\partial y_j }{\partial x} - \mathcal{L} \rightarrow \mathcal{L} dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx$$ Does Noether's theorem just say that, because $\mathcal{L}$ is invariant we have, solving for $\mathcal{L}$, that $$\mathcal{L}(x,y_i,y_i')dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*?$$


title: Gravitational Chern-Simons theory for bosons and fermions

text: > Q1: What is the difference of boson and fermions for their **Gravitational > Chern-Simons theory**? I suppose in general if the metric is not flat, we have vierbein ${e_{\hat{b}}}^{\nu}$, with $$ g_{\mu\nu} {e_{\hat{a}}}^{\mu} {e_{\hat{b}}}^{\nu} = \eta_{\hat{a}\hat{b}}, $$ where $g_{\mu\nu}$ is curved and $\eta_{\hat{a}\hat{b}}$ is Lorentzian flat. With the spin-connection is $$ \omega^{\hat{c}}_{\hat{b}\mu} = {e^{\hat{c}}}_{\nu}\partial_\mu {e_{\hat{b}}}^{\nu} + {\Gamma^\nu}_{\sigma\mu} {e^{\hat{c}}}_{\nu} {e_{\hat{b}}}^{\sigma}, $$ where ${\Gamma^\nu}_{\sigma\mu}$ are the Christoffel symbols. **SET-UP: Now let us imagine there are some matter fields bosons $\phi$ or fermions $\psi$ coupling to the spacetime metric, and we integrate out the bosons $\phi$ or fermions $\psi$ to get the effective actions involving gravitational Chern-Simons action in 2+1D.** So a 2+1D gravitational Chern-Simons action can be (spin connection $\omega$): $$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{1} $$ I am sure this works for fermions. > Q2: however, if bosons have spin 0 or spin 1, do we still have spin- > connection? I suppose still yes? > > Q3: or, do we have a 2+1D gravitational Chern-Simons action for bosons to be > ( the connection $\Gamma$): $$ S=\int\Gamma\wedge\mathrm{d}\Gamma + > \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{2} $$ > > Q4: what are the differences between Eq.1 and Eq.2 in terms of background > matter fields? (suppose the metric $g_{\mu\nu}$ is coupled to matters, > either bosons or fermions.) Please feel free give Reference. And make sure that your answer is to my point Q1-4.


title: Similarities between laminar-turbulence transition and others like BCS-BEC crossover, quark-hadron transition etc

text: From my limited readings on fluid dynamics, my understanding is that as the system changes from near-laminar flows to full turbulence, the dimensionless Reynolds number changes from $ R << 1$ to $R >> 1$. How is the perturbation expansion parameter in field theory formalisms of turbulence (See bottom of page-10 and top of page-11 in Gawedzki's chao- dyn/9610003 and Cardy's lecture in Non-equilibrium Statistical Mechanics and Turbulence) related to Reynolds number? If there is no direct relation, does the expansion parameter change values similar to $R$ as the system changes from laminar to fully developed turbulence? Is there a conceptual similarity between this field theory formalisms of fluid dynamics (as the system changes from laminar to turbulence) and systems like BCS-BEC crossover and/or quark-hadron transitions, where too some expansion parameter (usually dimensionless coupling strengths) change from $<<1$ to $>>1$? Am I right in understanding onset of turbulence as closer to a phase transition than a cross-over? Also, just like in these other systems (BCS/BEC and Quark/Hadron), does the region of fluid dynamics, most difficult to model, lie somewhere in-between (ie. $R \sim 1 $)? note: I understand that these other systems are fermionic while turbulence is not, and that these other systems are quantum mechanical while turbulence is classical. I am not asking for an exact dictionary, but only a higher-level similarity between them.


title: A rigid rotating rod that breaks in two pieces

text: Suppose we have a rigid rod of lenght $L$ and homegenous mass density. One of its extreme points, say $P$, is fixed so that the rod can rotate around the axis passing in it. Initially the rod is held horizontally, then it is left and when it reaches the vertical position it breaks, without generating any impulsive force, in two equal pieces. I'm wondering about how to find the angular velocity of the two fragments immediately after the rod breaks. I tried but without succes, here is what I did. I suppose that energy is conserved, that is: $$ E_{rod}= E_{1} + E_{2} $$ where $1$ labels the upper fragment and $2$ the lower fragment of the rod. We have: $$ E_{rod}=\frac{1}{2}I_{rod}\omega_{rod}^{2} $$ $$ E_{1}= \frac{1}{2}I_{1}\omega_{1}^{2} $$ $$ E_{2}=\frac{1}{2}I_{2}\omega_{2}^{2} + \frac{1}{2}\frac{m}{2}V^{2}_{2} $$ where $V_{2}$ is the velocity of the center of mass of the lower fragment and the moment of inertia with respect to P are: $$ I_{rod}=\frac{mL^{2}}{3} $$ $$ I_{1}=\frac{mL^{2}}{24} $$ $$ I_{2}=\frac{7mL^{2}}{24} $$ **EDIT** According to V.Moretti's comments we know that energy and angular momentum are conserved, thus we have the system: $$ \left\\{\begin{array}{c} E_{rod}= E_{1} + E_{2} \\\ \vec{L}_{rod}=\vec{L}_{1} + \vec{L}_{2}\end{array}\right. $$ I think it is: $$ \vec{L}_{tot}=I_{rod}\vec{\omega}_{rod} $$ $$ \vec{L}_{1}=I_{1}\vec{\omega}_{1} $$ $$ \vec{L}_{2}=I_{2}\vec{\omega}_{2} + \frac{m}{2}\vec{r}\wedge\vec{V}_{2} $$ Moreover I think that $\vec{V}_{2}$ can be simply calculated considering the center of mass of the lower fragment as a point of the rod and then use the well-known relation between angular velocity and velocity: $$ \vec{V}_{2}=\vec{\omega}_{rod}\wedge \vec{r} $$ and thus, in a right-handed coordinate system with the z-axis exiting the screen, it is: $$ \vec{V}_{2}=(-\omega_{rod}\hat{z})\wedge(-\frac{3}{4}L\hat{y})=-\frac{3}{4}L\omega_{rod}\hat{x} $$ and: $$ \vec{r}\wedge\vec{V}_{2}=(-\frac{3}{4}L\hat{y})\wedge(-\frac{3}{4}L\omega_{rod}\hat{x})=-\frac{9}{16}L^{2}\omega_{rod}\hat{z} $$ Now there is nothing that indicates motion in the x-z and y-z planes, thus the conservation of the angular momentum should be a scalar equation, namely: $$ L_{tot}=I_{rod}\omega_{rod}=I_{1}\omega_{1} + I_{2}\omega_{2} + \frac{m}{2}\vec{r}\wedge\vec{V}_{2}= $$ $$ =I_{1}\omega_{1} + I_{2}\omega_{2} -\frac{9}{32}mL^{2}\omega_{rod}=L_{1} + L_{2} $$ Now I am trying to solve the system: $$ \left\\{\begin{array}{c} \frac{1}{2}I_{rod}\omega_{rod}^{2}= \frac{1}{2}I_{1}\omega_{1}^{2} + \frac{1}{2}I_{2}\omega_{2}^{2} + \frac{9mL^{2}\omega_{rod}^{2}}{64}\\\ I_{rod}\omega_{rod}=I_{1}\omega_{1} + I_{2}\omega_{2} -\frac{9}{32}mL^{2}\omega_{rod}\end{array}\right. $$ but, quite surprisingly, I am not having good results. What am I doing wrong?


title: Can the multiverse theory be tested by gravity waves?

text: I was wondering if the presence of multiverse could be detected by the fact that there is probably matter disturbing space-time that would cross boundaries?


title: Conservation of energy with Huygens-Fresnel principle

text: I am currently experimenting with Huygens-Fresnel principle. I am trying to simulate the propagation of a beam, emerging from an aperture slot of width w. I assume the slot to be long and therefore work in two dimension. My elementary wave is given by $$ A_E(x,x_0)=\frac{\exp\left(2\pi i/\lambda 2\sqrt{(x-x_0)^2+a^2} \right)}{\left(2\pi\sqrt{(x-x_0)^2+a^2}\right)^{(1/2)}} $$ where a is the distance to the screen, x is the position on the screen and {x0,0} is the center of the elementary wave. This wave should carry an intensity of 1. According to the Huygens-Fresnel principle, I should be able to calculate the amplitude in any place {x,a} behind the slot via $$ A(x)=\int_{-w/2}^{w/2} A(x,x_0)\text{d}x_0 $$ $$ \approx\sum_{n=0}^{w/\Delta w} A(x,n*\Delta w) \Delta w $$ This seems to work quite well (result does not change notably for smaller step width) for $\Delta w<\lambda/5$. I am using in Mathematica in 2 dimensions. This should numerically calculate the image on a 180° screen in a distance of 10 from the center of the aperture:               r = 10     L = .1     SlotRange = {-1, 1}     SlotStepW = L/10     ImageRange = {-Pi/2,Pi/2}     ImageStepW = (ImageRange[[2]] - ImageRange[[1]])/1000     beam = Table[        Sum[         N[           Exp[             I (2 Pi/L Sqrt[(r Sin[Phi] -                      x0)^2 + (r Cos[Phi])^2])]/(2 Pi Sqrt[(r Sin[Phi] -                     x0)^2 + (r Cos[Phi])^2])^(1/2)           ]*SlotStepW,         {x0, SlotRange[[1]], SlotRange[[2]], SlotStepW}         ],        {Phi, ImageRange[[1]], ImageRange[[2]], ImageStepW}        ];     ListPlot[Abs[#]^2 & /@ beam, DataRange -> ImageRange, PlotRange -> All]     energy = (Plus @@ (Abs[#]^2 & /@ beam))*ImageStepW*r      Basically, this works quite well, as said. For smaller distances to the screen, the intensity distribution has a (oscillating) flat top, and in the distance it becomes roughly Gauss shaped. The full energy is always around 0.032. I would expect, however, something like 1 (elemental waves with intensity 1 over a range of length 2, gives two, but I am just getting half the intensity, as the situation is symmetrical around the aperture slot, so there is a backwards beam with another intensity 1). Probably I am just doing some small mistake, but I cannot see it. However, I need proper amplitudes, as in the next step I would like to start new waves from the screen, continuing the beam from there. I initially thought I could just multiply the amplitudes on the screen to the elemental wave above, to get an intensity and phase adjusted new wave, but that doesn't seem to be the case ...


title: Linear quantization in quantum electrodynamics?

text: This is a continuation of this question. What would be an example of linear quantization used on quantum electrodynamics? I ask this because QED is a nonlinear theory.


title: When does not Newton's 3rd law apply?

text: Is Newton's 3rd law valid in non-inertial frames? If so, then are there other cases for which Newton's 3rd law is not applicable?


title: Alpha Decay or Fission

text: In a quiz contest, I came across the following question: What is the term used to describe the splitting of a heavy nucleus into two lighter nuclei? Two options provided were: a) Alpha Decay b) Fission. Apparently, the correct answer is b) Fission. But even in Alpha Decay, a Helium nucleus is formed (i.e., the alpha particle), along with the daughter nucleus; both of them are obviously lighter than the parent nucleus, and so, doesn't this also fit the required criteria?


title: How to learn celestial mechanics?

text: I'm a PhD student in math and am really excited about celestial mechanics. I was wondering if anyone could give me a roadmap for learning this subject. The amount of information about it on the internet is overwhelming, and I honestly don't know where I should start. Any help would be greatly appreciated.


title: What will be the direction of the force required on the ball to move the ball up the stairs?

text: Is this diagram correct? The point of force A is at the farthest distance from the fixed centre O ,so the moment arm CD is greater and therefore,maximum torque would be produced.Is my concept correct?So,is the required direction of force AB?![enter image description here](http://i.stack.imgur.com/ULcea.png)


title: Tensor product decomposition of SU(2)

text: I have a rather trivial question. I am looking for the decomposition of $1/2\otimes 1/2\otimes 1/2$. It should give, $0,1/2$ and $3/2$. I thought one must get as the overall dimension of this space 8, but counting, I just get 7. Does one have 2 singlets?


title: A question regarding particle trajectories in the symplectic manifold formalism

text: How to solve a free particle on a 2-sphere using symplectic manifold formalism of classical mechanics ? Is there a way to get coriolis effect directly, without going into Newton mechanics? And is there a good textbook dealing with classical mechanics problems using symplectic manifolds?


title: some hints/introductions/textbooks for LBM(Lattice Boltzmann methods) fluid simulation?

text: I come from a computer science background and I want to learn details about LBM in fluid simulation. I have been searching around for a concise tutorial that tells me how LBM is used to solve a certain fluid problem(a PDE?). Unfortunately I was depressed with the large number of advanced studies and papers that assumed prior knowledge. I understand there exist quite a few papers on LBM fluid simulation written by computer scientists. However I am not satisfied with those work concentrating on implementation techniques on a specific device(like GPU) only. I teach myself some basic knowledge on real analysis/ODE/PDE. And I think maybe I need some extra learning on Statistical Mechanics/Lebesgue Measure theory... I am totally unfamiliar with these fields. I need your hints on how to choose the right direction towards understanding Lattice Boltzmann Methods. Any suggestions appreciated. Thank you for your time.


title: How can one imagine curled up dimensions?

text: Actually I'm learning String Theory, and one of its proposals is that there are actually 25+1 dimensions of which only 3+1 are visible to us-- and the remaining are curled up. However, superstring theory says that there are 9+1 dimensions; and M theory says 10+1 dimensions. I'm having trouble imagining "curled up dimensions" Is it that we go out of the universe to see a 4th dimension? (This was suggested by a friend of mine)


title: Where is the flaw in deriving Gauss's law in its differential form?

text: From the divergence theorem for any vector field E, $\displaystyle\oint E\cdot da=\int (\nabla\cdot E) ~d\tau$ and from Gauss's law $\displaystyle\oint E\cdot da=\frac{Q_{enclosed}}{\epsilon_0}=\int \frac{\rho}{\epsilon_0}~d\tau$ Hence, $\displaystyle\int\frac{\rho}{\epsilon_0}d\tau=\int (\nabla\cdot E)~d\tau$ Textbooks conclude from the last equation that $\displaystyle \nabla\cdot E=\frac{\rho}{\epsilon_0}$ My question is how can we conclude that the integrands are the same? Because I can think of the following counter example, assume $\displaystyle \int_{-a}^a f(x)~dx=\displaystyle \int_{-a}^a [f(x)+g(x)]~dx$ where $g(x)$ is an odd function. Obviously the 2 integrals are equal but we cannot conclude that $f(x)$ is equal to $f(x)+g(x)$ so where is the flaw?


title: The Double Slit Experiment and the changing of electron behaviour

text: As you will all know, when one tries to detect which slit an electron has gone through with close up observation, it changes from behaving like a wave and producing an interference pattern to behaving like a particle and producing two lines on the screen. Is there a full explanation as to why the electron changes its behaviour when observed?


title: Coordinate transformations of the metric tensor

text: Let's have metric (it describes the space-time of uniformly accelerating observer in Minkowski space-time) $$ ds^2 = v^2du^2 - dv^2. \qquad (.0) $$ I need to find expressions for $u = f(x, t), v = g(x, t)$, which leads to $$ ds^2 = dt^2 - dx^2, \quad u = f(x, t), \quad v = g(x, t). $$ How to get $f, g$? My attempt. I substituted $u = f(x, t), v = g(x, t)$ in $ds^2 = u^2dv^2 - du^2$ and then, by equating it to $dt^2 - dx^2$, got system of PDE: $$ g^2 (\partial_{x}f)(\partial_{t}f) - (\partial_{x}g) (\partial_{t}g) = 0, \quad g^{2}(\partial_{x}f)^{2} - (\partial_{x}g)^{2} = -1 , \quad g^{2}(\partial_{t}f)^{2} - (\partial_{t}g)^{2} = 1 . $$ It may be "simplified" to $$ (\partial_{x}g)^{2} - (\partial_{t}g)^{2} = 1 , \quad g^{2}\left((\partial_{t}f)^{2} - (\partial_{x}f)^{2}\right) = 1. \qquad (.1) $$ How to solve it? Or maybe there is more simple method to get an expressions for $f, g$? **Edit**. I got an equation $$ \partial_{tt}f - \partial_{xx}f = 0 . \qquad (.2) $$ But I can't get some conditions, which can help me to choose some partial solution. By the other words, I can't use $(.1)$ in the right way. So, what conditions may I use to solve $(.2)$? **Edit**. I got the solution by getting a solution for transition from Minkowski space-time to original space-time which metric is given by $(.0)$. I used method of separation of variables.


title: Precessional motion of active galactic nuclei

text: I want to set a simulation for jet which has a precessional motion. The symmetry axis of jet is $z$ axis, i.e. jet is propagating along $z$ direction making angle $\theta$. I set the velocity component of jet as \begin{eqnarray*} v_z &=& v\cos \theta \\\ v_x &=& -v\sin \theta \cos \phi \\\ v_y&=& v\sin \theta \sin \phi \end{eqnarray*} As the jet has precessional motion, it will rotate in $x-y$ plane. For this reason I have set the $-$ sign to the $v_x$ component to rotate the jet anti- clockwise(according to the Lissajous figures). But some one is arguing that $-ve$ sign will not appear in the $v_x$ component. Where $\phi (t)=\omega t$. Could you please explain this matter ?


title: Does diffraction contribute to the Black Drop effect?

text: Two reasons are given to explain the black drop effect here, but I think I came up with a third. Consider a two-dimensional cross section of the situation just after second contact and just before third contact in a stellar transit. It has many similarities to the classical single-slit experiment:                                           a   b     ============================|---|====================|-----------------------             Empty Space         ↑       Disk of Planet            Disk of Star                                 |                        Edge of Disk of Star      Our slit begins at point `a` and ends at `b`. Consider that the diffraction pattern of the single-slit experiment spreads out the total energy of the light which is incident on the range of the slit over some angle. This indicates that the incident intensity of the light behind a slit is less than the intensity would be if no slit was present (and no diffraction occurs). Could this possibly help account for the Black Drop phenomenon? Or must the "slit" be a lot smaller, such that this could not _possibly_ contribute in any significant way? After all, the aperture in this case would be on the order of hundreds of kilometers. Another thing that just occurred to me is that the left-edge of the star is not being obscured by anything, only the right side is. There would only be one-sided diffraction affecting our incident light intensity.


title: Fundamental invariants of the electromagnetic field

text: It is a standard exercise in relativistic electrodynamics to show that the electromagnetic field tensor $F_{\mu\nu}$, whose components equal the electric $E^i=cF^{i0}$ and magnetic $B_i=-\frac12\epsilon_{ijk}F^{jk}$ fields in the taken frame of reference, has two Lorentz invariant quantities, $$\frac12F^{\mu\nu}F_{\mu\nu}=\mathbf{B}^2-\mathbf{E}^2$$ and $$\frac14F_{\mu\nu}\ {}^\ast F^{\mu\nu}=\frac14\epsilon^{\mu\nu\alpha\beta }F_{\mu\nu}F_{\alpha\beta}=\mathbf{B}\cdot\mathbf{E}.$$ There is, however, a further Wikipedia article which states that these two quantities are _fundamental_ , in the sense that any other invariant of this tensor must be a function of these two. While I find this plausible, I have never seen a proof of this fact, and it is absent from e.g. Jackson. **Is there a simple proof of this fact?** I'm particularly interested in higher- order invariants, but I would also like answers to include a proof that these are the only two bilinears. To be more precise, I would like to see a proof that > Any function $I:F_{\mu\nu}\mapsto I(F)\in\mathbb{R}$ that takes > electromagnetic field tensors to real scalars and which is Lorentz invariant > (i.e. $I(\Lambda_{\mu}^\alpha \Lambda_{\nu}^\beta F_{\alpha\beta})= > I(F_{\mu\nu})$ for all Lorentz transformations) must be a function > $I(F)=I'(F^{\mu\nu}F_{\mu\nu},F_{\mu\nu}\ {}^\ast F^{\mu\nu})$ of the two > fundamental invariants described above. If there are multiple ways to arrive at this result, I would also appreciate comments on how they relate to each other.


title: Are there any exact solutions in the Standard Model of particle physics?

text: I wonder if there are any exact solutions to any equations of the Standard Model (SM)? Do we always have to use perturbation methods to solve anything within the SM? Are there any simple cases where exact solutions are available?


title: Relation between canonical forms and volume of polytopes

text: Références: Ref $1$ : Henriette Elvang, Yu-tin Huang : Scattering Amplitudes Ref $2$ : Jaroslav Trnka : The Amplituhedron [For simplicity, the notations of the $2$ refs have been merged] The area of a triangle in $CP_2$, can be expressed, using dual space complex 3-dimensional quantities $Z_i$, as (Ref $1$, page $157$, formula $10.17$): $$A = \frac{1}{2} \frac{\langle123\rangle^2}{\langle012\rangle\langle023\rangle\langle031\rangle}$$ where $i$ is for $Z_i$, $Z_0 = ^t(0,0,1)$, and $\langle abc \rangle=det(abc)$ On the other way, there is a "canonical form" (Ref $2$, page $26$): $$\Omega_p= \frac{\langle Y dY dY\rangle \langle123\rangle^2}{\langle Y12\rangle\langle Y23\rangle\langle Y31\rangle}$$, where $Y$ represents a point in the interior of the triangle. The relation between the $2$, (see Ref 2, pages $31,32$), seems to be an integration : $A =\int \delta(Y -Z_0) \Omega_p$ If the elements above are correct, that I don't understand is the utility of the canonical form, because in the integration, we keep only one point $Y = Z_0$, so the "integration" is somewhat "trivial", so why is used this presentation with the canonical form (which is linked to the grassmanian) ?.


title: Quark space tensor product Vs Angular momentum space tensor product

text: For two triplet angular momenta states, say $J=1$ and $I=1$, if we wanna look at it in the coupled basis $F=I+J$, we use the regular Angular Momentum rules: $$|I-J|\leq F\leq I+J,$$ and from that you get $$|0\rangle,|1\rangle,|2\rangle,$$ which are $1\oplus 3\oplus 5$, because if we project this on $m$ (magnetic number) space $$|0\rangle \rightarrow \\{|0,0\rangle\\},$$ $$|1\rangle \rightarrow\\{|1,-1\rangle\,|1,0\rangle\,|1,1\rangle\\},$$ $$|2\rangle \rightarrow\\{|2,-2\rangle,|2,-1\rangle,|2,0\rangle,|2,1\rangle,|2,2\rangle\\}.$$ So we can write $$3\otimes3=1\oplus 3\oplus 5.$$ However, what we see in QCD when talking about quark color-mixing is $$3\otimes3=1\oplus 8.$$ Why is this the case in QCD? Why is it different? Don't we just use the same coupling? I thought people still use the same Clebsch-Gordan coefficients with exactly the same procedure. Thanks for any efforts. EDIT: Michio Kaku's book (Modern intro to QFT) says that there's no equivalence between SU(2) and SU(3)... so the question is still open, how can you use Clebsch-Gordan coefficients there? ![enter image description here](http://i.stack.imgur.com/OH9yP.png)


title: Does radiation change entropy?

text: Are there entropy changes associated with the transmission of energy from the sun to the earth? Does radiation differ from other modes of heat transfer with respect to the entropy changes? Are the earth and sun in thermal equilibrium?


title: How to transform mechanical work into electrical energy without using piezoelectricity?

text: can someone help me with the following issue. I need a method for transforming mechanical work into electrical energy without using piezoelectricity. I have such kind of mechanical forces (like on the picture in the link) and I want to transform this mechanical force into electrical energy. mechanical energy


title: Eigenvalues of the square of Pauli-Lubanski operator

text: Let's have Pauli-Lunanski 4-operator: $$ \hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma}, $$ which easy transforms to $$ \hat {W}^{\nu} = \frac{1}{2}\varepsilon^{\nu \alpha \beta \gamma}\hat {S}_{\alpha \beta}\hat {P}_{\gamma}, $$ where $\hat {S}_{\alpha \beta }$ is the spin-tensor. For the square of this operator it's easy to get $$ \hat {W}_{\nu}\hat {W}^{\nu} = \hat {S}_{\alpha \beta}\hat {P}^{\beta}\hat {S}^{\alpha \gamma}\hat {P}_{\gamma} - \frac{1}{2}\hat {P}_{\delta}\hat {P}^{\delta}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}. \qquad (.1) $$ In the case of the rest frame $\hat {P}_{\gamma}\psi = 0$, so $$ \hat {W}_{\nu}\hat {W}^{\nu} \psi = -\frac{1}{2}\hat {P}_{\delta}\hat {P}^{\delta}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\psi = -\frac{m^{2}}{2}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\psi . $$ In the special case of 4-tensor I know, that $$ \hat {S}_{\alpha \beta}\hat {S}^{\alpha \beta} = 2(\hat {S}_{1}^{2} - \hat {S}_{2}^{2}), $$ where $\hat {S}_{1}$ is the spin-vector 3-operator. But what is $\hat {S}_{2}$? And what it do with eigenvalue? Without it $\hat {W}_{\nu}\hat {W}^{\nu} \psi = -m^{2}s(s + 1) \psi $, as it must be. **Addition**. Maybe, I understand my mistake. I ignored the first term in $(.1)$. At the rest frame in a case of diagonal operators of $\hat {P}_{i}$ it has structure $\hat {P}_{i} = (\hat {P}_{0}, 0, 0, 0) = (m, 0, 0, 0)$, so $$ \hat {W}_{\nu}\hat {W}^{\nu}\psi_{\mathbf p = 0} = \left(\hat {S}_{\alpha \beta}\hat {P}^{\beta}\hat {S}^{\alpha \gamma}\hat {P}_{\gamma} - \frac{1}{2}\hat {P}_{\delta}\hat {P}^{\delta}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\right)\psi = \left(\hat {S}_{\alpha 0}\hat {P}^{0}\hat {S}^{\alpha 0}\hat {P}_{0} - \frac{1}{2}\hat {P}_{0}\hat {P}^{0}\hat {S}_{\rho \varepsilon}\hat {S}^{\rho \varepsilon}\right) \psi_{\mathbf p = 0} = $$ $$ =m^{2}\left( - \hat {\mathbf S}_{2}^{2} - \frac{1}{2}2 \left( \hat {\mathbf S}_{1}^{2} - \hat {\mathbf S}_{2}^{2}\right)\right) \psi_{\mathbf p = 0} = -m^{2}s(s + 1)\psi_{\mathbf p = 0 }. $$ But, anyway, what is $\hat {\mathbf S}_{2}$?


title: What type of electromagnetic radiation strikes the Earth's surface the most?

text: If you can could you lists the types of light from the greatest amount to the least amount (Ex: Visible, Infrared, Violet).


title: Cassini: What information could be gained from the 2017 impact with Saturn?

text: First of all:   * Will Cassini be operational in 2017 (pending no unforeseen equipment failures)?     * Power     * Communications     * ...   * What information could we gain?     * Properties of Saturn      * How to build more resistant spacecraft for future atmospheric descents and gas planet impacts


title: Are there trends in the number of meteors in a shower year on year?

text: Some years the Leonids are amazing, other years there are very few - are these predictable at all, following trends or at least reasonably accurate predictions?


title: 2012: Is there some astronomical event happening?

text: The news media has publicized that 2012 is the end of the Mayan calendar, and that all the planets will be aligned. Is there any truth to this?


title: Is there a way to increase photon energy by decreasing its wavelength?

text: Can I decrease a photon's wavelength by a medium or a vacuum? Are there other ways of decreasing the wavelength?


title: Fermion vs. Bosons and particle vs. wave: is there a link?

text: I'm puzzled since several years on this basic aspect of quantum mechanics. Quantum theory is supposed to describe particle-wave symmetry of our world. It also describes our universe in term of bosons and fermions. Now the experimental facts:   * It seems rather easy to observe wave particle of bosons (think about photon of course) whereas they particle behaviour is pretty complicated to observe. You need stringent experimental condition to observe the particle behaviour of photons for instance.   * It seems rather easy to observe particle behaviour of fermions (think about electron of course) whereas they wave behaviour is pretty complicated to observe. You need stringent experimental condition to observe the wave behaviour of electrons for instance. I was just wondering if some people figure out the wave-particle behaviour in terms of fermion vs. boson classification, or if the above experimental facts are just pure coincidence. I admit it would require a less stringent dichotomy boson/fermion, perhaps like in supersymmetry theories, that I do not know enough. Any comment are welcome.


title: What are the advantages in using 2 identical capacitors?

text: What advantage might there be in using two identical capacitors in parallel connected in series with another identical parallel pair, rather than using a single capacitor?


title: Which is more dangerous upon contact, series or parallel capacitors?

text: If a pair of capacitors were connected up in series, and an identical pair was connected in parallel which pair would be more dangerous to handle if connected to the same voltage source?


title: Connection between $\Delta x \Delta p \geq \frac{\hbar}{2}$ and $\Delta E \Delta t \geq \frac{\hbar}{2}$

text: Is there a way to derive second equation from the first one? I mean is there a connection between those two uncertainty relations? \begin{align} \Delta x \Delta p &\geq \frac{\hbar}{2}\\\ \Delta E \Delta t &\geq \frac{\hbar}{2} \end{align}


title: How to deal with the product of distributions by Renormalization or similar?

text: how can we deal for example with the product of distributions in physics ?? is there any mean to define with physics $ \delta ^{2}(x) $ or to treat a product of two distributions within the Renormalization framework ? another question let be a metric with a dirac delta derivative $ ds^{2}= g_{a,b}dx^{a}dx^{b} $ and for example $ g_{x,x}= \delta (x-y) $ and $ g_{x,y}= H(x-y) $ (heaviside function) how could we compute and define the Einstein Equations ?? $ R_ {a,b}=0 $


title: Why is neutrino beam not considered light?

text: I am trying to understand why physicists consider a neutrino beam not to be light. Here's the research that I did to find the answer: https://docs.google.com/document/d/12t3WL7q_FMU_Zx30tF4kGE6DC44GN6MbwkpgIkGff7Q/edit According to the reasoning in that research neutrino beam must be light beam. I am looking for answers from experts explaining why this is not the case and where I go wrong in my reasoning.


title: How did the inflaton field "add" energy to the universe?

text: How did inflation add energy to the universe? What mechanism did this occur by? In other words, where did that energy come from? Was it due to the quantum fluctuation (or that scalar field rolling down a potential energy hill) to a lower, more stable energy state? This energy then manifested as exponential expansion, and then switched from expansion into heating (by filling up the universe with hot quark-gluon plasma)? Do we know the mechanism for this expansion->reheating switch?


title: Is work done by sound wave on air particles?

text: Is it possible for sound wave to do net work on air particles? As in can a sound wave make the air move in one direction so that it can for example move a sail boat ? I think since molecules gyrate about a mean position even though they are in the direction of wave propagation no net work is done but I want to confirm this idea.


title: Massless neutrinos and Chirality

text: The massless neutrinos can be represented by two component Weyl spinors. Then how does one say that it is an eigenstate of the chirality operator $\gamma^5$, which is a $4\times 4$ matrix and can act on $4\times 1$ column?


title: Is $\overline{\psi_{L}^{c}}\psi_{R}^{c}=\overline{\psi_{R}}\psi_{L}$ true for two different spin 1/2 fermions?

text: In the context of seesaw mechanism or Dirac and Majorana mass terms, one often see the following identity $$ \overline{\psi_{L}^{c}}\psi_{R}^{c}=\overline{\psi_{R}}\psi_{L}. $$ Here, I am using 4 components notation in the chiral basis. The convention for the charge conjugation is $\psi^{c}=-i\gamma^{2}\psi^{*}$, and $\psi_{L}^{c}=\left(\psi_{L}\right)^{c}$. The following is my effort of proving it. $$ \overline{\psi_{L}^{c}}\psi_{R}^{c}=\overline{i\gamma^{2}\psi_{L}^{*}}i\gamma^{2}\psi_{R}^{*}=\left(i\gamma^{2}\psi_{L}^{*}\right)^{+}\gamma^{0}i\gamma^{2}\psi_{R}^{*}=\psi_{L}^{T}i\gamma^{2}\gamma^{0}i\gamma^{2}\psi_{R}^{*} $$ $$ =-\psi_{L}^{T}\gamma^{2}\gamma^{0}\gamma^{2}\psi_{R}^{*}=\psi_{L}^{T}\gamma^{2}\gamma^{2}\gamma^{0}\psi_{R}^{*}=-\psi_{L}^{T}\gamma^{0}\psi_{R}^{*}=-\psi_{L,i}\gamma_{ij}^{0}\psi_{R,j}^{*}. $$ Now if $\psi_{L,i}$ and $\psi_{R,i}$ are anticommuting, then one have $$ -\psi_{L,i}\gamma_{ij}^{0}\psi_{R,j}^{*}=\psi_{R,j}^{*}\gamma_{ji}^{0}\psi_{L,i}=\overline{\psi_{R}}\psi_{L}. $$ Question: Is the anticommuting assumption still true if $\psi_{R}$ and $\psi_{L}$ are two different species of fermion? (For example, $\psi_{L}=\chi_{L}$) Do we assume any two fermions are anticommuting even if they are two different fields in QFT?


title: What is the meaning of space-time curvature?

text: What is the difference between the Space-time curvature and Space curvature?


title: What does energy represent in $E= mc^2$?

text: Due to energy-mass equivalence, any object with mass can be said to have a corresponding amount of energy. So in a 5kg object there are 450,000,000,000 joules. A joule as i understand it is the ability to do work or apply force, aka energy. But im struggling to understand where exactly the potential energy is stored in mass. I can conceptualize the classical example of a ball on a platform in the air representing gravitational potential, because if the platform is removed, the ball falls. But i dont understand how one could measure or define force on an atomic level. Is it a function of the force vectors being applied by fundamental forces on all particles in the system? For example, if it werent for the strong nuclear force, electrons would go crashing into the nucleus because an attractive electromagnetic force exists between protons and electrons. So does the force vector of the electron being pulled towards the nucleus count as potential energy that an atom has?


title: Kronecker delta in inertial tensor

text: I feel confused in (11.9) how does the book prove the following identity: $$\sum\limits_{i} w_{i}x_{\alpha,i} \sum\limits_{j} w_{j}x_{\alpha,j} = \sum\limits_{i,j}w_{i}w_{j}x_{\alpha,i}x_{\alpha,j}$$ ![enter image description here](http://i.stack.imgur.com/w8u1I.jpg) I tried to derive it myself: $$\sum\limits_{i} w_{i}x_{\alpha,i} \sum\limits_{j} w_{j}x_{\alpha,j} = \sum\limits_{i}\sum\limits_{k}w_{k}\delta_{ik}x_{\alpha,i} \sum\limits_{j}\sum\limits_{m}w_{m}\delta_{jm}x_{\alpha,j} $$ Exactly from here, I got stuck, but I'm still not very much convincing about the result that the book got, becasuse it does not contain kroncker delta function for the part I tried to derive. Can anyone give me a mathematical proof?


title: What is the physics concept in sponge filter

text: **I really need help I want to ask what is the physics concept in Sponge Filter?**![enter image description here](http://i.stack.imgur.com/39hkq.jpg)


title: Setting up an equation for calculating how long it takes a body to change temperature in its sorroundings

text: The United States has just recently been hit by a massive vortex of Arctic air. These unusually bitter temperatures have sparked my interests to ask the following rheotical question: How much time would it take for a naked human standing outside at the south pole to reach a temp. of 77 F (hypothermic death) given a wind speed of 40 mph, a resultant wind chill temp. of ~ -200 F (derived from the world record low at Vostok,) and various other variables such as body temp., surface area, etc., that I will estimate. So, how would I set up an equation to calculate the time it takes the body to go from T1 to T2 given (I assume) the bodies' radiated power _P_?


title: Energy dispersion in graphene

text: Given that graphene has linear energy dispersion near the Fermi level and the dispersion is given by $E=\hbar \nu_F|\vec{K}|$, I would like to determine the density of states. I think it is equal to $$g(E)=\frac{E}{2\pi\hbar^2\nu_F^2},$$ but how could I show that? In fact, all I am asking is an explanation why the following relation holds: $$N=\text{total number of states}=\frac{A}{2\pi}\int_0^{k(E)}d\mathbf{k}$$ in the vicinity of the dirac points. Also, what are the units of $g(E)$?


title: Does the Earth rotate vertically too?

text: We know that the earth has seen several Ice Ages. And we know the Earth's magnetic poles seem to flip once in a while. Could all of this be explained by a once every 250 Million Years or so the Earth actually rotates head over heels, so to speak?