title: How does upwards acceleration affect tension on whirling mass?

text: Consider the following setup and problem: ![http://i.stack.imgur.com/Oxv08.png](http://i.stack.imgur.com/LG1h0.png) When solving this problem, I had the intuition that the string tension $T_A$ would be equal to $T_B$, which does not agree with the solutions. What would lead you to know the string tensions in both cases are, in fact, different?


title: Some limiting cases of the Heisenberg XXZ model (2/2)

text: _NOTE: Because this was a long question I have split it up in two different questions!_ For a course on quantum integrability I am reading these notes. (Franchini: Notes on Bethe Ansatz Techniques. Lecture notes (2011)) Some questions have arisen to me, concerning the Heisenberg XXZ model. The general idea is that we will solve several versions of this model in class, using the Bethe Ansatz Approach. However, the basics are not yet clear to me. Consider the Hamiltonian: \begin{equation} \hat{H} = - J \sum_{i=1}^N \left(S^x_jS^x_{j+1} + S^y_jS^y_{j+1} + \Delta S^z_jS^z_{j+1}\right) - 2h\sum_{i=1}^NS^z_j, \end{equation} where we have periodic boundary conditions: $S^{\alpha}_{j+N} = S^{\alpha}_j $. In the following I will set $h=0$.   1. Letting $J\Delta\rightarrow\infty$ we retrieve an Ising model with also the ground state $|0> = |\uparrow\uparrow\uparrow\dots\uparrow>$. Does this mean that the Ising model and the XXX model are equivalent? I also noticed that the states $|\uparrow\uparrow\uparrow\downarrow\uparrow\uparrow\uparrow>$ and $|\uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow>$ have the same energy, namely $E=-J\Delta/2$. This means that when a single spin is flipped, it costs no more energy to flip more! Is this the reason that the Ising model in 1D has no phase transition? I know it is related to the fact that in 1D the number of nearest neighbours is equal for a single spin and a domain of spins, but I forgot the exact reasoning I once understood. (Peierls argument?) Finally the notes I mentioned state that domains in this ferromagnetic state have spin $S=1$. For one excited spin I can see this, but for more I don't. How to see this?   2. Letting $J\Delta\rightarrow-\infty$ we get an anti-ferromagnetic ground state. The low-energy excitations are now domain walls. I don't really see why these carry spin $S=1/2$. Again, how to think about this?


title: Normalizing the sum of wavefunctions and calculating probabilty - understanding concepts

text: A state of a particle bounded by infinite potential walls at x=0 and x=L is described by a wave function $\psi = a\phi_1 + b\phi_2 $ where $\phi_i$ are the stationary states. So let's say we want to normalize this wave function. As I understand it the procedure is as follows: The probability of the particle being at any point from 0 to L is 1. So I need to integrate the wave functions squared over that interval. By the superposition principle it is OK to just add them. On top of that, any $\psi$ can also be expressed as $\psi \psi^*$ $\psi = a\phi_1 + b\phi_2 $ $\psi = (a\phi_1\phi_1^* + b\phi_2\phi_2^*)$ We want to integrate $\vert\psi\vert ^2$ $(a\phi_1\phi_1^*)^2 + 2ab\phi_2\phi_2^*\phi_1\phi_1^*+(b\phi_2\phi_2^*)^2 = (a^2 + b^2)$ Since the phi functions are eigenvalues, the ones on the diagonal of the matrix are the only ones not zero, which is why the cross terms in the middle disappear (they are zero) and the end terms $(\phi_i\phi_i^*)$ are equal to 1. So we get $\psi = \int_0^L\vert\psi(x)\vert ^2dx = \int_0^L\vert(a^2 + b^2)\vert^2dx=1 $ and therefore $\int_0^L\vert(a^2 + b^2)\vert^2dx=1 $ and $(a^2+b^2)^2x\vert^L_0 = 1 \rightarrow (a^2+b^2)^2L=1\rightarrow L=1/(a^2+b^2)^2$ The conceptual question I had was that if we have the probability squared here, is it that or the square root of that probability that is your normalization constant? Further, would it also be permissible to treat each of the wavefunctions as $A\sin\frac{n\pi x}{L}$ where $A_1=a$ and $A_2=b$, and try the integration that way? Given that the wave functions are supposedly different that seemed like it would be wrong, but we also know they are stationary states so they go to zero at either end of the potential well and are sinusoidal, correct? I know that this area doesn't always cotton to HW type questions. But this is the kind of thing that I think could help a lot of people get their heads around this concept, because I can't be the only one who is a bit lost on how to actually use these techniques.


title: Why does proximity to a superconductor open a gap in the surface states of topological insulators

text: I have read in many places that the gapless surface states of 3D topological insulators are robust to perturbations which do not break time-reversal symmetry. I have recently also seen many papers (such as http://arxiv.org/abs/1002.0842) on topological insulators in proximity to a superconductor. They all predict that the surface states will get gapped and that superconductivity is induced at the surface. But I am not able to reconcile the two issues. Superconductivity does not break time-reversal symmetry. Then why is a gap produced in the surface states?


title: What is the definition of a "UV-complete" theory?

text: I would like to know (1) what exactly is a UV-complete theory and (2) what is a confirmatory test of that?   * Is asymptotic freedom enough to conclude that a theory is UV-complete?  Does it become conclusive a test if the beta-function is shown to be non- perturbatively negative? If the one-loop beta-function is negative then does supersymmetry (holomorphy) immediately imply that the beta-function is non-perturbatively negative and hence the theory is proven to be UV-complete?   * Or is vanishing of the beta-function enough to conclude that theory is UV-complete?  Again similarly does supersymmetry (holomorphy) guarantee that if the 1-loop beta function is vanishing then it is non-perturbatively so and hence the theory is UV-complete?   * Does superconformal theory necessarily mean UV-complete? We know that there exists pairs of superconformal theories with different gauge groups - related by S-duality - which have the same partition function - what does this say about UV-completeness?    * Does a theory being UV-complete necessarily mean that it has a Maldacena dual? (...and isn't that the meaning of AdS/CFT that every UV-complete theory has a string dual and vice-versa?..)


title: Does a domestic microwave work by emitting an electromagnetic wave at the same frequency as a OH bond in water?

text: I was told once that microwaves work by exciting water molecules in food. Also that this worked because the frequency in the microwave was the same as that in the bond between Oxygen and Hydrogen in the water molecule. This similar frequency match lead to the molecule being energised, releasing heat into the food. My question is - how do microwaves work? Does any of this 'hearsay' have any basis in scientific fact?


title: Material strain from spacetime curvature

text: Let's say that you moved an object made of rigid materials into a place with extreme tidal forces. Materials have a modulus of elasticity and a yield strength. Does the corresponding 3D geometric defect from curvature of spacetime directly cause _material strain_? How could you calculate that? An obvious first step for practical geometries would be to take the Gaussian curvature of Flamm's paraboloid (thus avoiding much more GR specific knowledge), luckily, a book already did that for us. Here is the curvature: $$ K = -\frac{r_s}{2 r^3 } $$ The negative indicates hyperbolic geometry. * * * The problem in concrete terms: Imagine a location where tidal forces are equal to roughly $(10 \mathrm{m}/\mathrm{s}^2)/(2 \mathrm{m})=5 \mathrm{s}^{-2}$. I pick this value for relationship to human experience. An astronaut could wedge themselves between a ceiling and a floor $2 \mathrm{m}$ apart and experience about $\frac{1}{2}$ Earth gravity pulling their feet and hands in opposite directions. Furthermore, imagine this location as the Innermost Bound Circular Orbit of a black hole. These two constraints determine exactly the necessary mass of the black hole, which is 25,000 solar masses. Not uncommon. The distance of the knife-edge orbit from the singularity would be about the radius of the sun. This would also be perfectly accessible due to the orbital physics of a black hole. With conventional rockets you could throw the ISS into that orbit and retrieve it later, given that your initial trajectory had sufficient accuracy. What I want to know is: would the astronauts' bones break due to tidal forces? There is another important implication of this question - whether **spaghettification** is a relevant phenomenon for a rigid object falling into a black hole. Obviously, an unraveled string (oriented in a line intersecting the singularity) falling into a black hole would be spaghettified. But for any object that is roughly spherical (meaning length isn't much greater than width), would the hyperbolicity of space itself exceed the material limits first? Could we construct a criteria that would determine if something is spaghettified or strained to death? We already have the geometric curvature. The object properties of length, width (assuming a coke can shape), modulus of elasticity, yield strain or stress, and black hole mass would seem to be sufficient. Then a radius of spaghettification breaking would follow, along with a radius for the strain breaking. You could then obviously say which one is larger. Perhaps this approach is fundamentally mistaken in separating geometric stress from gravitational forces. But I don't see how. Consider:   * For tidal force breaking (spaghettification), I only need material yield stress   * For geometric defect strain I only need material yield strain By dimensional analysis, these two things must be distinct, but maybe there is some other deep reality of general relativity that isn't obvious to me.


title: The Movement of Light

text: I've asked several other questions like this, and I know it's a dumb question but I really can't understand. When light is emitted from a moving source, it still moves straight in that source's frame. I understand that the path can look different in a different frame, but taking a pond of water as an analogy to the E&M field, I don't understand why a moving source would see emitted light in their frame as still moving straight. Perhaps it's a misunderstanding of the E&M field, I just can only think of some person running parallel to some water, and tapping the water as they run. The disturbances won't move _with_ them, they'd stay behind. Does this mean that when objects move, they _pull_ the E&M field with them? How much would be pulled? Clearly I'm just not understanding something very obvious, but I can't figure it out. I'm sorry for all the reposts.


title: Spherical Waves-Strength at close distances

text: If the amplitude dies off as the radius squared, what happens in areas very close to the source? It would have nearly infinite strength. How is this treated?


title: Elementary question on pion-proton scattering cross-section

text: ![enter image description here](http://i.stack.imgur.com/5zrh3.jpg) Is $E_k$ the energy of the outgoing or incoming pion? The first peak is supposed to be a delta baryon. What does the graph tell us, experimentally? A pion of kinetic energy x comes in, then we look at the graph and find the cross-section, so then we have the probability of _what_ happening? Or is $E_k$ the net loss in kinetic energy of the pion, and this graph tells us the distribution of its energy loss? How is this graph constructed, what is measured?


title: Energy in a wind instrument?

text: My physics teacher said that he saw a guy playing a very large wind instrument on TV, and the guy apparently calculated that the total energy present in the instrument when he was playing was almost as high as the energy of the atomic bombs dropped in WW2. How is it possible to contain and direct so much energy in the human body without destroying it? This is very hard to believe. Thanks!


title: How long can you survive 1 million degrees?

text: I asked my Dad this once when I was about 14, and he said that no matter how short the amount of time you were exposed to such a great temperature, you would surely die. The conversation went something like:               Me:  What about a millisecond?     Dad: Of course not.       Me:  A nanosecond?       Dad: Nope.       Me:  10^-247 seconds?       Dad: Death.        15 years later I still think that you could survive for some amount of time even though you'd surely get a few layers cooked. Thus, what's the longest period that you could theoretically withstand 1,000,000 degrees F heat and live to tell about it? Let's assume that you are wearing normal clothes (a t-shirt, jeans, sneakers, sunglasses, and a baseball cap), that the air around you is the only thing that is heated, and that after the time is up your friend sprays you with a fire extinguisher and the air magically returns to a nice 73 degrees F.


title: Velocity of measurement

text: As per to Heisenberg uncertainty we will not be able to calculate the position and momentum at same instant because by the time we calculate the next of the one, it changes (i.e.) the changes are very fast. Suppose my speed of calculation is near to the speed of light or equal to the speed of light. Can I at that instant calculate the accurate position and momentum of a particle


title: Aerodynamics and thermodynamic

text: Why can we refer an object as being aerodynamic but we can't refer an object as being thermodynamic, and if an object is thermodynamic what does it even means?


title: Understanding bending light beam perpendicular to motion

text: I'm just reading a book about gravity. An example it gives is a spaceship accelerating. A beam of light travelling at right angles to the direction of movement of the spaceship enters it via a small pinhole. The book states that the beam of light would appear to bend to an observer within the spaceship. Intuitively this makes sense - if the ship were travelling at $0.5\,c$, for example, and was $10\,m$ across (from the pinhole to the opposing wall) then I'd expect the light to hit the wall $5\,m$ further back down (assuming the ship is accelerating "up") from the pinhole - the wall would have moved 5m further in the time it takes the light beam to cross the interior to the opposing wall I think I'm fundamentally misunderstanding though. The book states this is down to the acceleration of the ship and goes on to talk about how using equivalence the same bending would be a result of gravity. In my description above though, it's the velocity that causes the apparent bending, and the same would be visible if the ship was maintaining a constant $0.5\,c$. I suspect that if this were the case (constant $0.5\,c$) there would be no bending at all, but I'm not understanding why Can anyone enlighten me?


title: Eulerian Angles --- Why three rotations can transform fixed frame into body frame?

text: "In general, if we restrict ourselves to rotations about one of the Cartesian axes, three successive rotations are required to transform the fixed frame into the body frame" The origin of our fixed frame and the body frame are common. The axes of the body frame points in the direction of the principal axes of rotation of the body. I do not understand, how three rotations are all what is needed to transform the two frames.


title: How much smaller will be human body, when we hypothetically get of every space between particles

text: I had an interesting dream, where advanced civilization compress their body with technology, that turn off space betweens atoms/particles. They travel in small spaceship with billions citizens very effectively. Crazy idea, but my question is how much smaller will the human body be, when we hypothetically get of every space between particles.


title: How to calculate the velocity of this body?

text: > _Graph of a train (the body) is provided which starts from rest. What is the > velocity after the train has 10 metres displacement?_ ![enter image description here](http://i.stack.imgur.com/hyLJd.png) Thats the only things provided for the question, please help me out here, I find that more variables will be required but the book states that no more variables are required. The answer key says that the answer is 10 m/s only! I am totally unable to understand how.


title: In how big percentage of gravity core does play role in?

text: If something would take whole core of earth and done this $(((coreearth)-(restofearth)):coreearth)*100=X$ What would `X` be? In short, which part of our planet matters more in gravity, the core, or what's around it? Does the same apply to stars?


title: Schrödinger's Equation and the depth of a finite potential well

text: Before I ask my question, I have to stress: I have absolutely **no** idea what the math is going on. I've read my textbook, several Wikipedia articles, scoured the internet, and don't feel anymore knowledgeable after hours of reading. I am frustrated beyond belief—I apologize if the quality of this post decreases because of that. Here's the question I'm working on (but any illumination of the surrounding concepts is much appreciated; the question itself doesn't have to be answered at all, I'd just like to have some idea of what I'm doing): > A finite square well 1.0 fm wide contains one neutron. How deep must the > well be if there are only two allowed energy levels for the neutron? I know that Schrödinger's Wave Equation is: ![this](http://upload.wikimedia.org/math/2/7/7/27734849d22b4b26bd85be1a1435ba2e.png) Which is all fine and dandy... but what is it? I understand what it is _mathematically_ , but that isn't telling me anything of why it's useful. Did Schrödinger just plug De Broglie's relationships into the wave equation, call it a day, and get famous? What is this a wave of? What's it supposed to _be_? This equation is so out of all context for me that just working with it is made more difficult. My textbook says: > The reasoning followed by Schrödinger is somewhat difficult and not > important for our purposes. But then goes on to throw around some constants and make some assumptions about $\Psi(r,t)$ in order for it to... preform some undisclosed magical wave- matter-related things. From what I've read, all I can gather is that—in solving this problem—I have to construct wave-equations for various potentials inside and outside of some "well"? What is the well? There's something about even and odd parity, which I think refers to there needing to be a whole-number of wavelengths within that 1 fm range (why?), but otherwise I'm pretty much dead in the water. This isn't a Quantum Mechanics class in particular; it's a general "Modern Physics" class. So far, the whole of the class seems to be randomly tossing out equations, constants, and relationships without context, derivation, or motivation—this confusion has become a very unfortunate (and frustrating) standard for the class. (And I _despise_ the textbook. Help me. :l)


title: Could the Sun be partly powered by cosmic rays?

text: The Earth and the Sun has magnetic fields which shields us from cosmic rays, as a charged cosmic ray particle will loose kinetic energy when its direction is perpendicular to the magnetic field. So what happens to the kinetic energy of the cosmic ray particle? According to the first law of thermodynamics it can't just disappear. According to electro-mechanics it will induce an electric current. The particles also get a spiral motion and can induce a magnetic field like a current in a solenoid. Some radiation due to breaking of the particle (bremsstrahlung) may also occur. The vast magnetic field of our sun is bombarded with cosmic rays with far bigger energy intensity than the cosmic rays that hit our shielded Earth. On Earth the most energetic cosmic ray observed is a subatomic particle with kinetic energy equal to that of a baseball (5 ounces or 142 grams) traveling at about 100 kilometers per hour (60 mph). So I wonder if its a possibility that the Sun is partly powered by cosmic rays? How large can the total cosmic ray energy hitting the heliosphere be and where does the energy go?


title: How to find whether the electric field between capacitor plates is above or below the electrical breakdown limit of the insulator between the plates?

text: Let's say the two plates of a capacitor have charges $q_1$ and $q_2$. The separation between these plates is $d$. I know that I can calculate the electric field by dividing the voltage that is applied to the capacitor by the separation distance and comparing the answer with the electrical breakdown limit to see whether it is above or below the electrical breakdown limit. However, what should I do if I know the charges on the plates rather than the voltage? Sould I use $E=kq/d^2$ where d is the separation distance? If I should use it then which q should I use in the equation, $q_1$ or $q_2$? Because the electric fields calculated with $q_1$ and $q_2$ are different which makes me confused.


title: Parallel Transport and covariant derivative

text: I have been trying to understand the notion of parallel transport and covariant derivative. I am unable to see why the change in a vector when it is parallel transported from one point to another shouldn't be a vector. If it is, why isn't the Levi- Cevita connection not a tensor ? Hence my questions are : What is a connection geometrically ? What is parallel transport in a particular coordinate system geometrically ?


title: Local tension and pressure measurements

text: We want to measure tensions and pressures on the surface of a tooth for an experiment in dental research. We want to do local measurements at different points on the tooth. The tooth is not a real one, but a model, so it would be possible to build a device into the tooth. The tensions and pressures we want to measure are of the order of a few kilo Pascal or potentially even less. Is there a reliable way to achieve such a measurement?


title: Matrix operation in dirac matrices

text: If we define $\alpha_i$ and $\beta$ as Dirac matrices which satisfy all of the conditions of spin 1/2 particles , p defines the momentum of the particle, then how can we get the matrix form ? \begin{equation} \alpha_i p_i= \begin{pmatrix} p_z & p_x-ip_y \\\ p_x+ip_y & -p_z \end{pmatrix} . \end{equation}


title: Research on ground state configuration of Ising model

text: I want to do mathematical research (algorithm construction and mathematical analysis) on Ising model ground state configuration. From what I know, the state of art research is using graph theory formalism. Could someone give me some advice on how to learn this subject (say, what books or papers to read) to a research level from a basic undergraduate math background?


title: Total angular momentum operator forms a complete set? (Clebsch-Gordan coefficients)

text: While introducing Clebsch-Gordan coefficients, they state that the operators: $$ \vec{J_1}^2,\vec{J_2}^2,J_{1z},J_{2z}$$ form a complete set of compatible observables. Which means that there is no degeneracy in their common eigenspaces. What I wonder is, how it follows that (if the above operators form indeed a complete set), that also $$\vec{J}^2,\vec{J_1}^2,\vec{J_2}^2,J_z$$ with $$\vec{J} = \vec{J_1}+\vec{J_2}$$ forms a complete set. Which is what they claim in the textbook. I see that the operators in both sets are compatible (i.e. they commute) and that $J_{1z}$ and $J_{2z}$ are not compatible with the second set. Does it follow from this information that the second set of observables is complete, if the first is?


title: Distance of universe's most distant objects in relation to expansion of the universe

text: Was reading this article about the Hubble XDF and it had the quote: > The most distant objects here are over 13 billion light years away, and we > see them when they were only 500 million years old. But if these galaxies were closer to us 500 million years after the big bang (and wouldn't they have to be -- since we've been expanding ever since?) then why did it take so long for the light from where they were 500 million years after the big bang to reach us? Shouldn't that light have hit earth a lot sooner since it was a lot closer back then? What principle am I missing here?


title: which letter to use for a CFT?

text: In math, one says "let $G$ be a group", "let $A$ be an algebra", ... For groups, the typical letters are $G$, $H$, $K$, ... For algebras, the typical letters are $A$, $B$, ... > I want to say things such as "let xxx be a conformal field theory"   >  and "let xxx $\subset$ xxx be a conformal inclusion".   > Which letters should I use?   What is the usual way people go about this? Here, I'm mostly thinking about chiral CFTs, but the question is also relevant for full (modular invariant) CFTs.


title: Elastic collisions in Franck-Hertz experiment

text: Looking at a Franck-Hertz experimental setup, and given a potential difference such as $4.0\ V$ which is too small to excite out the first electron orbital, the electrons moving through the tube will have elastic collisions with the Hg atoms. I'm supposed to show that given some kinetic energy $E$, of a flowing electron, that the maximum recoil of a Hg atom that it hits is approximately $4Em/M$ (where $m$ is the mass of the electron, and $M$ is the mass of the Hg atom). I know to use conservation of energy, and given that the Hg atom was initially at rest, I know the electron is probably bouncing backward (so something like $2\times E-\text{sub - elec} = E\text{ - sub - hg}$). I can't seem to find where the 4 comes from. Is it the $4.0\ V$ potential difference? * * * I tried playing with conservation of energy formula but can't get $\frac{4Em}{M}$. (Taking lower case to be the electron and UPPER CASE TO BE THE ATOM:) [$mv^2 = MV^2 - mv^2$] so [$mv^2 = 0.5MV^2$] so [$2e = E$] but I'm supposed to find that [$E = 4Em/M$]. I know there is something I'm missing here. P.S.: The original problem is #44 here (or as an image here) in case I'm misunderstanding something.


title: Modular invariance for higher genus

text: As far as I understand, there are roughly 2 "common" kinds of 2D conformal field theories:   1. Theories that are defined only on the plane, more precisely, on any surface of vanishing genus. Such a theory can be mathematically described by a vertex operator algebra, or by "usual" axiomatic QFT in 2D with the added requirement of conformal invariance   2. Theories defined on arbitrary surfaces. Such a theory can be described e.g. by Segal's axioms In order for a theory of the 1st category to become a theory of the 2nd category, it must pass the test of modular invariance. In my experience, this term usually means that the theory has to be defined in genus 1, i.e. that the would-be torus partition function $$Tr(q^{L_0-\frac{c}{24}}\bar{q}^{\bar{L_0}-\frac{\bar{c}}{24}}) $$ is invariant under the modular group $SL(2,\mathbb{Z})$ What about higher genus? Is the torus condition sufficient to make the theory well defined there? Or does it impose additional conditions? If so, can they be cast into an anologous elegant algebraic form? Have these conditions been proved for CFTs used in string theory?


title: GPS Working Principle

text: > _Hand-heldGPS units in modern phones identify your location by_ > > _(A) transmitting their location and time to GPS satellites._ > > _(B) receiving location data of GPS satellites._ > > _(C) receiving time data from GPS satellites._ > > _(D) exchanging location and time data with GPS satellites._ I think correct answer is (D), but the key says correct answer is (C). How can GPS work without knowing the location Suppose you don't know the location, how do you know where to send the data to and where to get the data from?


title: Top angular speed of electric motor

text: I recently came across a question asking the following: > If a motor is switched on, it quickly reaches a top speed. Why does it not > just go faster and faster and faster? I thought it might be related to the fact that the electrical power is constant, so I tried approaching with $P=\tau \cdot \omega$. However I still do not have any clue to reaching the conclusion. Is it somehow related to the energy loss (due to friction)? Like the terminal speed of a parachute or something? But I was thinking that friction should be constant ($f=\mu \cdot N$) while air drag is related to the speed, so this may not be the answer. Thank you.


title: When we talk about speeds in relativity theory, where are they measured?

text: I recently asked a question here about if the direction we travel matters in relativity theory: Does it matter in which direction I travel in relativity theory? After I got answers and making more research about the topic, I found out that I was talking about speeds relative to the earth all the time. So, when I for example said that I travel away from earth at 50% of the speed of light, I automatically assumed that this is also the speed I can use to calculate the time dilation. But since the earth already travels at some always varying speed and thus is not standing still in space, I think I also need to put this into the calculation. I mean, if I would travel in the exact opposite direction the earth currently "flies" and at exactly the same speed, this would mean that I'm standing still and that the time for me would actually go as fast as possible, while the time on earth is still slowed down, as it always is due to its speed. So: What I wanted to show you is that it really seems to depend from where I actually measure the speed I travel and what I consider the "root". Moving away from earth at some speed can mean moving faster in space OR moving slower in space and thus also mean slower or faster time flow than the one on the earth. Am I right with this? Or am I totally wrong and it's valid to measure the speed I travel by how much the distance from me to the earth gets bigger? **UPDATE** Actually, I think I have understood the answer of @John Rennie, and it really seems to explain a lot of the issues I ran into before. His answer was, at least to my understanding, the following: If two objects in space stay together, they share the sime timeflow. If the distance between the objects changes over time, we cannot say which one will see the others clock going slower and which one will see the others clock going faster. To say that, we need to go back to the change of velocity that led to the increasing/decreasing distance. Here, we need to determine which of the objects changed it's velocity by the method John mentioned. And this will lead us to the answer. This also explains the twin paradox perfectly. But after doing some thought experiments I came across a scenario that I cannot explain: Imagine we humans have indeed built a death star as a moving base in space. This death star has also a lot of smaller fighter ships on board. Now the death star starts to fly away from earth at, say, 0.1c. According to the people on earth, the time on the death start now goes slightly slower than the time on the earth. After a while, a fighter on the death star begins to fly twice as fast (0.2c relative to the earth, 0.1c to the death star) in the same direction to explore the forthcoming space. According to the earthlings, the time on the fighter goes even slower than the time on the death star. The guys on the death star confirm that and notice that the time on the fighter seems to go slower compared to their time. After a while, the fighter found some enemy and decides to return to the death star to alert them. Here, the fighter has 2 options: 1) since the death star is also on its way, he could just stop and wait until the deathstar comes across. 2) Stopping, and moving into the other direction to inform the death star faster and before he runs into the enemies. The fighter goes for option 2) and flies back at 0.1c. For the earthlings, the fighter is now getting closer at 0.1c, while the death star is still moving away at 0.1c. For the death star, the fighter is getting closer at 0.2c. And that is the point where this starts getting weird: For the earthlings, the fighters speed decreased from 0.2c to 0.1c, which means that the fighters clock is now going faster, while still being slower than that on the earth. Actually, since the death star and the fighter are moving at the same speed relative to the earth, they share the sime timeflow. For the deathstar, the fighters speed increased from 0.1c to 0.2c, which means to them that the fighters clock is now going even slower than that on the death star. Can it really be?


What is the energy required to create mass of m at a height of h above the Earth? Is it $E= m c ^2$ or $E = mc ^ 2 + mgh$ ? Let's reverse the process also. If you convert mass $m$ at $h = 0$ to energy then $$E=mc^2 \tag{1}$$ Now if you raise the mass to a height $h$ and convert it to energy which you are going to measure at the height $h$ then $$E=mc^2 + mgh \tag{2}$$ Is equation (2) correct? If this is correct then If you take a rock of mass $m$ on the Earth to very large distance or provide it with escape velocity so that it escapes the Earth's gravity (ignoring any other gravitational field), What is the energy contained in that rock? Is it equation (1) or $$ E = mc^2 + \dfrac{1}{2}mv^2, \tag{3}$$ where $v$ is the escape velocity? If equation (3) is the accurate one according to the discussion above then once the mass has come out of the gravitational field the only way to store this extra energy will be by an increase in mass. So, $$ dm = mv^2/(2c^2) $$ or $$ dm = mgh/(c^2) $$


title: Can the $\vec H$-field have non-zero divergence? Does an $\vec H$-field monopole exist?

text: We know that, $\vec \nabla \cdot \vec B=0$ but $\vec \nabla \cdot \vec H\neq 0$, if $\vec \nabla \cdot \vec M \neq 0$. Does it mean that, in those cases $\vec H$-field has poles although $\vec B$-field does not?


title: How to find power factor value of parallel RL circuit knowing the power factor value for the series circuit with the same components

text: Because I know the value for the power factor for the series RL circuit I've tried to find the ratio between the series circuit power factor and the parallel circuit power factor, expressed in terms of resistance and impedance, and I found that: $$\frac {\cos_s^2\phi}{\cos_p^2\phi} = \frac {R^2}{X_L^2},$$ which doesn't really help me, because I need an actual value. Is there another way to get the power factor for the parallel circuit? The values of the components are not known, but are the same in the series and parallel circuit.


title: How to generate Red light from a Blue-Dominated Spectrum

text: I have a white LED light source, the spectrum of which looks as follows --- ![White LED Spectrum](http://i.stack.imgur.com/OuC4w.png) I want to filter out red light from this white LED source. But as you can see the spectrum dwindles in the red zone. And therefore, when I put red filter (absorptive filter) in front of my white LED light source, the brightness suddenly drops down to abysmal values. Is there a way in which I could use this spectrum and get red lights without compromising on the brightness? I thought about interference and then I came across dichroic filters. But I'm not sure if that'll do it. I also came across these filters called LEE LED filters which are filters specifically meant for LEDs. But I do not understand the scientific principle they work on. I would be grateful if you can follow the link and explain to me the scientific principle. From the spectrum shown in the link, these filters can give you more than 85% red. But the overall transmission is just 10.1% of the original intensity. So it's kinda bleh. Or maybe I don't understand properly. A red LED light source would be just perfect for me, but a high wattage (50W-100W) red LED light source is hard to come across in the market. Let me know if there are other options you guys can think of, for producing cool red LED light. Right now I use a tungsten lamp with RED absorptive filter, but there is so much heat (infrared) in it that my houseflies get fried up. [ _I work in an insect flight lab and we use red light for high speed videography, since houseflies dont see well in the red zone_ ]. So I want to switch to LEDs.


title: Why does a wave actually diffract?

text: I know that waves diffract around a slit and this is due to the Huygens- Fresnel principle. But I never understand this in an intuitive wave that why does a wave become a spherical wave front at the slit? Huygens principle gives all the math behind this but does not actually explain that why does a wave bend round the edges. Some might say that this happens because this a property of a wave that it forms a spherical wave front and the final wave is a result of superposition of those wave fronts. I know that, but it would be great if anyone could come up with an intuitive approach to this and actually explain why waves diffract or why do they form a different wave front at the slit. This question might be somewhat similar to, why do waves bend around corners. ![enter image description here](http://i.stack.imgur.com/PLzVC.gif) Image source: https://www.ph.utexas.edu/~coker2/index.files/Diffraction.gif


title: Conversion of mass and energy

text: First of all I am not a scientist and all these doubts are coming from my curiosity. When Googling about Einstein's $E = mc^2$. I understand that mass and energy are convertible. What it exactly means? Also can we have greater amount of energy from smaller mass objects? And if this conversion is possible then why we cant convert 1kg of iron to large energy? (I know that the question is really worst. But I am a little bit weak in English and almost all Google searches disappoint me.) Any simple explanation?


title: Does the Renormalization of QFT Contradict Canonical Quantization?

text: Does the renormalization of QFT contradict canonical quantization? In canonical quantization, you take the classical fields and canonical momenta and turn them into operators, and you require that the commutators ( or anti- commutators) equal the Poisson brackets of the original fields. Also, you need to quantize the classical Hamiltonian (of course, since classical fields commute but quantum operators don't, you need to quantize the Hamiltonian according to a well defined prescription, such as normal-ordering). The form of the quantum Hamiltonian will resemble the form of the classical one, but it will be an operator rather than a number. In particular, if a physical constant appears in the classical Hamiltonian, it will also appear in the Quantum one, and have the same value. Now, in renormalization the constants that appear in the (bare) Lagrangian are not the physical ones, and so are different from the constants that appear in the "classical" Lagrangian (and therefore Hamiltonian). So the "bare" quantum Hamiltonian will have a similar form to the classical Hamiltonian, but the constants will be totally (and perhaps infinitely) different! So does the quantization scheme need to be reformulated?


title: What exactly is the difference between radiation and convection?

text: Okay, so everywhere I've read, I hear the main difference is the requirement of a medium. But for example, if you take the case of heat 'radiating' from a red-hot iron, isn't that actually convection and not radiation? I mean, isn't the temperature difference between the surrounding air and the iron causing the air to gain heat?


title: How to express the heat capacity in terms of heat?

text: The first law of thermodynamics divides the internal energy change into contributions of heat and work. $$\text dU=\omega_Q-\omega_W,$$ Here I chose the notation to emphasise that the two parts are in general not exact differentials, i.e. that there is no heat counting function. > What would be the proper way to express the heat capacity > > $$C_V=\left(\frac{\text d U}{\text d T}\right)_V$$ > > in terms of $\omega_Q$?


title: Does the efficiency of radiation change in any way in the presence of a medium?

text: I've read that radiation doesn't really require a medium. But if you're taking, for example, the sun's light, then does its efficiency of transmission increase or decrease once it reaches the earth's atmosphere? (Even though the speed of light is taken as a constant, does that necessarily mean the presence or absence of medium is irrelevant to it's speed?)


title: Why do the space time get curved around a massive object?What problems do we face if we do not consider the space time to be curved?

text: As far as I have the knowledge of GTR that a mass bends the space time around it.But why does this bend occur?The example from real life that when a mass is placed on a net then the net bends but it us very difficult for me to visualise the situation of bending of spacetime due to a mass.What is actually happening?What is the physical basis behind this bend?I want to say that the books which I have studied till now just say that the space time "just bends" around an object and even my teacher told me that it just bends without any explanation so I just want to know the reason for the occurrence of this bend.What is the need for spacetime to bend around an object?What physical problem we can encounter if we do not the bending of space time.


title: Do you age at different rate at different speed?

text: This is probably answered many times, but my confusion is the following. In special relativity it is said that there is no preferred coordinate system. Does that mean that for the people at rest (any rest frame of reference) things happen at same rate ? But just are seen from other system as delayed. So that people live to same age or particles decay at same life times in any rest frame. The GPS satellite clocks run at different rate as seen from earth. But the atomic clock should run in space at same rate as the one on earth. (not taking into account the gravity effects) With all of the above said my question is simply: can we conclude from having same physics in all rest frames (that are non accelerating) that it does not matter with what speed we travel we should always have same time rates for particle decays, life span.. Yes, no answer will be perfect.


title: What forces of stress really represent?

text: I'm studying fluid mechanics and the book says the following: > For any continuum, forces acting on a piece of material are of two types. > First there are forces of stress, whereby the piece of material is acted on > by forces across its surface by the rest of the continuum. Now, I can't get the significance of this. Since he says "forces across its surface" it seems that the piece of the material must be three dimensional. Is this right? So, forces of stress are always exterted on three dimensional pieces of a material? What I mean by this question is: suppose $D\subset \mathbb{R}^3$ is a three dimensional region filled with a fluid, then forces of stress are always exerted on three dimensional subsets of $D$? Also, what they represent? As I understand, they represent the force that the rest of the fluid exerts on that piece of it, but why they exert such forces? Is it some sort of force exerted at molecular level that has this macroscopic effects?


title: In what direction does a frustrated magnetic moment get aligned?

text: Consider 3 layers of Ferromagnetic materials stacked on top of each other with appropriate spacer layers in between. Let the top and bottom layers be pinned to layers of Anti Ferromagnets adjacent to them such that the magnetic moments of the top and bottom layers are anti parallel. The two spacer layers in between top & middle layer and bottom & middle layer have thicknesses such that the magnetic moment of the middle layer is anti parallel to the top and bottom layer mediated by the RKKY interaction. Since top and bottom layers are pinned to be anti parallel to each other, the middle layer is frustrated as it cannot be anti parallel to both the layers simultaneously. So in what direction does the magnetic moment of the middle layer get aligned?


title: rate of change of spring potential energy $\frac{dU}{dt}$

text: Suppose we have a setup like this. In orange are two wooden sticks sort of things, and they are attached to the block of mass $m$(as usual) at a joint which is hinge type something. A similar connection is with spring and with the ground.The block is moving off to the right with velocity $v$. The spring is already extended by a length $l$. I want to find the rate of change of spring potential energy at the instant. If the block moves towards right then I think the spring is going to move a little bit to the right, not staying perfectly vertical and in that case there would be some sort of angle $\phi$ between force and displacement. But that case is difficult to imagine. Any other ideas/hints? Thanks in advance. EDIT: I've decided to solve this thing the Hamiltonian way.(as suggested by Bernhard). Even though I don't know much of it, this is my attempt.![enter image description here](http://i.stack.imgur.com/wP2oF.png) I've read here basic intro(I didn't understand much of Wikipedia, as it was too technical). So now I know I have to make a variable $H=T+V$(the hamiltonian) and do something. Now the question is about $\frac{dU}{dt}$ at any time $t$ where at $t=0$, $\gamma=45^o$ (for ease of coordinates). $$H=T+V=\frac{1}{2}m\dot x^2+\frac{k}{2}(\sqrt{(Rsin\gamma-\frac{R}{\sqrt{2}}-l-l_0)^2+(Rcos\gamma - \frac{R}{\sqrt{2}})^2}-l_0)^2$$ I've assumed sticks, spring as massless. $l_0$ is the natural length of spring. What should I do ahead? Please try this problem in simplest language, as I don't know if this involves something difficult like tensors etc. I don't know if this is the place to ask something like this, or if this problem can be solved in Hamiltonian at all. Thanks a lot.


title: CFT and the conformal group

text: Equations 2-7 on page 21 of these notes, http://www.math.ias.edu/QFT/fall/NewGaw.ps seems to give a fairly compact definition of what a CFT is. But I have two questions,   * This definition is specific to 2 dimensions. Is there an analogue of this definition for higher dimensions?    * I want to know if there is an equivalent definition of a CFT in terms of the conformal group in the specific dimension.  Like I would like to know if I can make precise a statement like this (for any dimension), "CFT is a QFT such that its Hilbert space splits into Verma modules and the correlation function of its primary fields is invariant under the conformal group in that dimension." We do have odd-dimensional CFTs and there I don't know what is a "conformal group"!


title: Jet taking off from an aircraft carrier

text: If a jet takes off from an aircraft carrier, should it not already be traveling at the speed of the carrier. Also, does the jet require more energy to get off the carrier than from a stationary base?


title: Modeling with First Order Equations

text: A ball with mass 015kg is thrown upward with initial velocity 20m/s from the roof of a building 30m high. There is air resistance of magnitude v^2/1325 directed opposite to the velocity , where the velocity is measured in m/s. a. Find the max height above the ground that the ball reaches. b. Find the time that the ball hits the ground. I am unsure how to set this one up.


title: Four momentum of a photon squared, trouble with combining wave-particle duality and energy-frequency relation

text: Since $p^2=E^2-\vec{p}^2=m^2$ and $E=h\nu=\frac{hc}{\lambda}$ and $|\vec{p}|=\frac{h}{\lambda}$ we have that $p^2=\frac{h^2c^2}{\lambda^2}-\frac{h^2}{\lambda ^2}$ If I go to Planck-units ($c=1,h=1$), this becomes zero. Is this a correct thing to do? It doesn't feel right because it depends on the units (SI or Planck) I use.


title: Origin of the Higgs field

text: Are there any attempts in the literature at addressing the origin of the Higgs field? And, which lines of research that find it inevitable to address this question?


title: Should any theory of physics respect the principle of conservation of angular momentum or linear momentum?

text: Is it possible that a theory that can describe the universe at the planck scale can violate things that we now consider fundamental in nature?For example can it violate rotational and translational invariance and subsequently momentum wouldn't be conserved ?Should one consider these invariance principles to be fundamental that we must choose the lagrangian to respect them ?


title: What forces act on a person sitting in a chair with wheels when they push off another stationary object

text: 1. What is slowing me down when I push off my desk?   2. What are the "Major" contributors? and   3. Does how hard I push with one hand (or with two hands) make a significant difference?


Might a planet perform figure-8 orbits around two stars? I'm thinking that if the two stars were equal mass (and not orbiting each other) then a planet that were to go right between them would continue in a straight line, with no preference for either star. But since the two stars would in fact be orbiting each other, the system would be rotating and thus there would be a Coriolis preference for one of the stars. Might that preference be made to alternate stars? Another possibility would be if each star were in turn orbited closely by another planet, which would perform three orbits for each orbit of our planet of interest. Then things could be timed where on one pass star A's inner planet were aligned right to pull the planet of interest into an orbit, and on another pass star B's inner planet would be aligned right to pull the planet of interest into an orbit. So we have a system of five bodies, two massive (stars) each orbited by a minor, and one minor performing figure-8s. Is this at least plausible if contrived?


title: Are Thomas Breuer's subjective decoherence and Scott Aaronson's freebits with knightian freedom the same things in essence?

text: In his remarkable works (1,2 and their recent development 3) Thomas Breuer proves by diagonalization the phenomenon that the observer cannot distinguish all phase space states of a system where he is contained, a theorem paralleling the famous incompleteness theorem by Gödel. This is applicable for both classical and quantum case, but for a quantum system the result is stronger. He thus calls the phenomenon subjective decoherence in case of quantum mechanics. He underlines that for quantum mechanics the phenomenon bears statistical character. In (3) he gives the following illustration: ![enter image description here](http://i.stack.imgur.com/dwIUI.png) More fundamentally, Breuer concludes that neither deterministic, nor probabilistic universally valid theories are possible: no theory can predict the future of the system where the observer is properly included. > Non-self-predictability implies that, even in a deterministic theory, > accurate predictions about subsystems of the universe may at most be > feasible for some observer, namely for one outside the system whose > behaviour is to be predicted. But there will be no observer able to predict > everything accurately. Every deterministic theory must admit the existence > of unpredictable events when a predictor applies it to himself. Even in > classical mechanics with a deterministic time evolution we have this kind of > unpredictability. > > I believe the conclusions of non-self-predictability and non-self- > measurability are correct. In this chapter, they will follow in a more > general framework from the fact that no observer can obtain or store > information sufficient to distinguish all states of a system in which he or > she is contained. This is mentioned in Stanford encyclopedy of Philosophy along a similar result by Marisa Dalla Chiara. But recently I have encountered the essay by Scott Aaronson "The ghost in the quantum Turing machine" where he discussed the problem of free will. In his essay he references the famous argument by Peter van Inwagen who argued that neither determinism, nor randomness are compatible with free will (the part about determinism is known as the consequence argument, the full argument can be found in his monograph "Metaphysics"). Since Peter van Invagen does not consider other theories than probabilistic and deterministic, he concludes that free will is impossible (quite a surprising conclusion for a Catholic professior). In section 3.3 of his essay, Aaronson introduces an idea of "freebits". A freebit in his words, is simply a qubit for which the most complete physical description possible involves Knightian uncertainty. Scott Aaronson introduces freebits as follows: > Thus, by the freebit picture, I mean the picture of the world according to > which > >   * (i) due to Knightian uncertainty about the universe’s initial quantum > state $|\psi\rangle$, at least some of the qubits found in nature are > regarded as freebits, and >   * (ii) the presence of these freebits makes predicting certain future > events—possibly including some human decisions—physically impossible, even > probabilistically and even with arbitrarily- advanced future technology. > The Knightian uncertainty is a term borrowed from economics to describe systems whose states have **uncertain probability**. A formal way of dealing with such variables is described by Dempster-Shafer theory, and in the appendix Aaronson gives a formal description of freebits. In this light it is worth mentioning the idea that Breuer's self-referential uncertainty can also be traced to the initial state of the universe. In his article Ignorance of the Own Past Breuer proves that for an observer the past of a system where he is properly contained (such as the universe) is uncertain due to self-reference. The bad side in this argument is that the proof relies on the explicit assumption of determinism. It should be noted though that the idea that free will may have self- referential nature is rejected by Aaronson on the grounds that in his view it amounts to solipsism. As such, he strives to ascribe free will to all animated beings, a position which seems to me unjustified. At best it looks as an attempt to artificially stretch a physical theory to fit certain philosophical belief. On the other hand, it seems, subjective decoherence perfectly fits the definition of freebits. Since not all states of a quantum system (represented by a wave function, that is, probability) can be discriminated from the inside, such system seems to possess Knightinan uncertainty, and as such, the freebits. It should be noted that a theory dealing with uncertain probability, such as Knightian uncertauinty (some call such theories "possibilistic" contrary the "probabilistic" theories, you can learn more about such theories in the comprehensive monograph on Generalized Information Theory by George Klir), would overcome the objections rised by both Breuer and Peter van Inwagen. Thus a universally-valid _possiblistic_ (rather than _probabilistic_ ) theory would not be impossible even if Breuer's argument is correct. It also will not contradict the existence of free will as understood by Inwagen. A drawback of such theory would be even smaller predictive power than that of a probabilistic theory. That said, I wonder whether the idea of freebits and Dempster-Shafer theory in general, applicable to the description of the phenomenon of subjective decoherence.


title: Do asteroids create sonic boom while entering the atmosphere?

text: Because of the lack of any materials to propagate sound, it can't spread in space. There's the air in the atmosphere, but its volume mass increases constantly (either linear or exponential, I don't know) - it means that atmosphere doesn't have a sharp edge that separates it from space directly. What happens, when an asteroid moves faster than the speed of sound (I suggest it happens quite often, but I'm not sure) and it enters to the atmosphere of Earth? Does it create a sonic boom? Definitely, what's the level of atmospheric pressure that is necessary to the exist of sound (and consequently, sonic bangs) and how far is it from Earth surface?


title: Rocket hovers- and then what?

text: If we have a rocket, using conservation of momentum we derived in my classical mechanics course $$m\dot{v}=-\dot{m}v_{ex}+F^{EXT}$$ $m$ is the total mass of the rocket and fuel still on the rocket $\dot{v}$ is the acceleration of the rocket relative to a stationary reference frame $\dot{m}$ is the time derivative of the total mass, i.e. the rate at which the fuel is ejected to accelerate the rocket and create thrust $v_{ex}$ is the velocity of every bit of mass $dm$ ejected from the rocket $F^{EXT}=-mg$ is gravity in this case Knowing that the rocket and fuel have a combined mass of $2m_r$, with each the rocket and the initial fuel onboard weighing $m_r$, I found the exhaust velocity $v_{ex}$ relative to the rocket that is required for the rocket to hover ($\dot{v}=0$) for $T$ seconds. In order to solve the problem, I introduced a parameter $\alpha$ that is the portion of the fuel you want to burn. Solving for $v_{ex}$, I found that $$v_{ex}=\frac{-gT}{ln(1-\frac{\alpha}{2})}$$ My question is whether it is possible to construct a function for $\dot{m}$ such that after $T$ seconds, the rocket will accelerate vertically ($\dot{v}>0$) as opposed to returning to earth (implying that the fuel would run out, or $\alpha=1$). Or, by definition of $\alpha<1$, is $\dot{v}$ going to be greater than $0$ for some $t>T$?


title: Double-slit expirement fundamentals (half-silvered mirror version)

text: In the double-slit experiment variation in which 2 half-silvered mirrors and 2 mirrors are used to illustrate the interference of a stream of photons or single photons at a given time step, how is it when the experiment is comprised of all 4 mirrors that the photon always knows to go to Camera 1/A in every illustration of the experiment I've ever heard of? Why doesn't it ALWAYS go to Camera 2/B instead with this setup? Seems so arbitrary and I have never come across the reason. Diagram to refresh you: (Assume all distances are equal in diagram.)                                         [CAM 1/A]                                   |        regular mirror             |             //--------------------\     [CAM 2/B]  (notice there is no beam to CAM 2/B)             |                     | half-silvered mirror #2             |                     |             |                     |             |                     |      [XXX>--\---------------------//  regular mirror             ^half-silvered mirror #1      [XXX> is the photon emitter/laser. The backslash to the right of the photon emitter represents one of the two half-silvered mirrors that I've designated as half-silvered mirror #1. So... why is it when interference occurs at half-silvered mirror #2 that the photon decides it prefers the camera designated CAM 1/A when it could easily just go to the other CAM every single time it interferes?


title: Given wave function at $t=0$, what is the process of deriving time dependent wave equation?

text: Suppose $$\Psi (x, t=0)=Ae^{i\alpha _1}\psi _1(x)+Be^{i\alpha _2}\psi_2(x)+Ce^{i\alpha _3}\psi_3(x).$$ If $\psi _n$ are the energy eigenfunctions how would I derive $\Psi (x,t)$? I am having trouble with the $\psi_n$ for which I don't know how to deal with when I use Fourier transform $$\phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \left(Ae^{i\alpha _1}\psi _1(x)+Be^{i\alpha _2}\psi_2(x)+Ce^{i\alpha _3}\psi_3(x)\right)\, dx $$ Any hint would be appreciated.


title: How to calculate air velocity in a fan?

text: How to calculate air velocity in a fan and what data should I know ? So if I have a fan motor, fan blade and a battery, what data should I get from these items ? I think I will need to know **the number of turns per second**. How can I determine it ?


title: Which textbook of differential geometry will introduce conformal transformation?

text: Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ \delta^k_i\partial_j\varphi + \delta^k_j\partial_i\varphi-g_{ij}\nabla^k\varphi $$ $$\tilde R_{ijkl} = e^{2\varphi}\left( R_{ijkl} - \left[ g {~\wedge\\!\\!\\!\\!\\!\\!\bigcirc~} \left( \nabla\partial\varphi - \partial\varphi\partial\varphi + \frac{1}{2}\|\nabla\varphi\|^2g \right)\right]_{ijkl} \right)$$ $$\tilde R = e^{-2\varphi}\left[R + \frac{4(n-1)}{(n-2)}e^{-(n-2)\varphi/2}\triangle\left( e^{(n-2)\varphi/2} \right) \right] $$ I've read many textbooks about differential geometry, such as Do Carmo, Kobayshi, Novikov and so on. But I never found these formulas. Who can give me a reference about these formulas.


title: Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the CF and obtain a summation

text: Applying theorem of residues to a fermionic reservoir correlation function in order to solve the integral in the correlation function and obtain a summation.


title: Diffraction of sound

text: The sound waves, by the virtue of it being a wave, shows diffraction and interference. But in diffraction, I learnt that if the wave is allowed to enter through a small aperture, there is a central maxima, along with several minima (zero intensity) on the sides. > **Why then in everyday life do we not encounter a situation in which we are > at one of those minima and fail to here any sound? For example,in a > soundproof studio room, suppose the narrow open door is the only entry point > for sound, can we then find a spot where we hear nothing going on outside > but by shifting to either side, we can hear it (although with different > intensities)?** Why aren't such phenomenon encountered in daily life, not being able to hear a source because of diffraction (or interference)? My question also relates to **radio waves** whose wavelengths is comparable to that of sound waves.


title: What events lead to quantum decoherence?

text: Is there a very specific definition for all types of events where quantum decoherence occurs? Is it merely any event that is "thermodynamically irreversible" and/or "causes entropy to increase"? Is it mathematically defined? Most importantly, is it possible to list some examples of events where quantum decoherence occurs?


title: Galileo's dictum and how light cannot violate it

text: Okay. So I've been told that the speed of light is constant and cannot violate Galileo's dictum, but even if it weren't constant (in a vacuum), how would it violate it anyway? Say you are on a train and you are stationary. Another train is traveling towards you with velocity v and shoots a bullet with a velocity j. The apparent speed from your point of view would be v+j. If you were instead traveling towards the train, at a velocity v, and they shot the bullet at velocity j, the perceived velocity of the bullet would still be v+j. Thus, it must be impossible to tell if you are moving towards something or if something is moving towards you, even if the bullet's speed from an observer's point of view is different (in one example, it is v+j, in the other it is just j) So if you replace the bullet with light, it would not have to be measured as being the same speed by an observer for the people on the trains to not be able to tell whether they were moving. Can someone explain why it has to be constant not to violate Galileo's Dictum?


title: Supercharge definition in Becker-Becker-Schwarz

text: How does one interpret equation (4.20) in Becker-Becker-Schwarz "String Theory and M-Theory", particularly the $ \frac{\partial}{\partial \bar{\theta}^A} $ term where the $\theta$ is the Grassmann coordinates of the super-world-sheet {$\sigma^\alpha,\theta_A$} and A is the spinor index {+,-}? The link to the text is: http://www.cambridge.org/us/academic/subjects/physics/theoretical-physics-and- mathematical-physics/string-theory-and-m-theory-modern-introduction?format=HB


title: Finding a 1-Dimensional Mass Distribution

text: ## Past Discussion There have been a lot of questions and discussions regarding determining the mass distribution of various bodies through non-destructive purposes. Here are a few: Determination of mass density distribution of an object How can I determine whether the mass of an object is evenly distributed? The general conclusion that I reached after reading questions like these is that by mechanics alone, there is no way to positively determine the mass distribution of a 3-Dimensional object. The simplest arguments typically involved spherical masses and newton's shell theorem. ## My Problem My problem is a simplified case of this general problem that I hope can be solved more easily. I have a stick of balsa wood with dimensions 1/8" x 1/8" x 36", and I would like to know its mass distribution. Because it is so slender, it can be safely treated like a 1-Dimensional object with a density function $\rho(x)$ that depends only on variable. I can find the center of mass very easily just by balancing it, but beyond that I am stuck. I would really like to find an estimation for $\rho(x)$ that is accurate to about the 1" or 1/2" scale. ## Solving It One assumption that can be made is that the young's modulus, $E$, of the wood varies linearly with density. This might be useful for solutions that measure the deflection of the wood under different forces. First, I am looking for solutions using the simplest methods and tools. I have a scale accurate to 0.01g (the whole stick weighs about 2.60g) and a micrometer that measures to 0.001". If a solution is possible that uses only these tools and other common and accessible ones, that would be best. If that is not possible, I would still like to hear other more complex approaches that might require advanced equipment or might simply be unrealistic in practice. EDIT: The reason I need to solve this problem is that I am building an 18g, 8" tall structure for a competition that must hold over 600 pounds. I know the mass distributions of each specific stick can vary greatly and be quite unpredictable. Since the structure is subject to such extreme forces, small imperfections can significantly affect performance. If one of my 8" columns has 60% of its mass distributed to its lower half, then it will be weaker in the top half. For this reason it is beneficial for me to select wood that is nearly homogenous, and statistical methods (finding variance, etc...) are not particularly useful in this case because I need to be 100% sure that the wood I select is evenly distributed.


title: How to explain in simple terms why Entanglement is more than just complicated hidden variables

text: I haven't taken a graduate level physics course on quantum mechanics so I get lost in the strange looking equasions. It's hard for me too see in any of the explanations of how quantum computers and entanglement work without thinking that it looks like a big parlor trick. It's like a magician taking a black and white marble shuffling them in a cup then taking one in each hand without anyone seeing which is in which hand and then revealing a white marble in his left hand and declaring: >   1. I can predict with absolute certainty that the marble in my right hand > is black. >   2. Until I had actually observed the white marble in my left hand, neither > marble actually had a definite color. They were just a superposition of two > possibilities. > A magic trick isn't very convincing unless the audience can understand it and not explain in simpler terms. Are there experimental results that the average person can understand and not explain using classical intuition?


title: Expected value $<\hat{x}>$ of: $\Phi(x,t)=Ne^{-a[(Mx^2/\hbar)+it]}$ is infinite, why?

text: The problem says: A particle of mass $M$ is described by the wave function: $$\Phi(x,t)=Ne^{-a[(Mx^2/\hbar)+it]}$$ where a is a positive constant. Asked to determine such things as the normalization constant $N$, which is quite simple. However, after requests to obtain the expected value of $<\hat{x}>$, even though I've tried is hand, always came to the integral: $$<\hat{x}>=cte*\int_{-\infty}^{\infty}y e^{-y^2}dy$$ according to me, the integral can not be calculated with these limits should not be infinite integrla that? Hoping someone will tell me how to interpret this, or if there is way to fix this.


title: Local Lorentz invariance or local Poincaré invariance?

text: Fast question. I know that the group of all the isometries in Minkowski's space-time is the Poincaré group that is just Lorentz's group (rotations and boosts) and translations in space-time. Now, in general relativity texts I often read _local Lorentz invariance_. I figure that in GR since we have that locally SR is fulfilled what is really meant here is local Poincaré invariance. So, is this terminology abuse or is it really local Lorentz? And if it is Local Lorentz, is it the full Lorentz group, just the proper orthocronous or any other thing?


Do there exist objects that absorb energy only (and do not give it away)? Magnets have been mentioned in this post, do we know more examples? Extract energy from magnets And how is this capability of giving away energy or absorbing energy clustered and theoretically modeled?


title: What is "A" in AGeV?

text: AGeV means GeV per nucleon. But why A letter is used for such a short cut? Why not NGeV, for example?


Could Jupiter's gravity destabilize Earth's artificial satellites over a long timescale? After all, it can destabilize Mercury's orbit, and it can also destabilize orbits in the asteroid belt.


title: Relation between isophotal radius and virial radius in spiral galaxies?

text: Is there any (proposed) relation between the B-band isophotal radius of a spiral galaxy and its virial radius (r_200)? If you know of such a relation, please post a reference paper.


title: How is it that some nebulae are rotating and others are not?

text: First, I should make it clear that this isn't a question about angular momentum (unless I may have completely missed something). It is my understanding that a nebula must have some inherent initial spin before collapse to form a solar system (and if it doesn't have that spin, only a star is produced). Then, basically, Conservation of Angular Momentum takes hold (and this is where the rotation is more noticeable), thus producing the accretion disc around the star, then planetary bodies, etc. Now, what determines the initial rotation of a nebula before collapse? Why is it that some nebulae do not have an initial rotation before they collapse?


title: Lack of exoplanet missions in the decadal survey

text: In recent exoplanet meeting "The Next 40 Years of Exoplanets", there was much discussion of the inablity of the community to agree on whether to support coronagraph missions or interferometer missions, with white papers for the decadal survey contradicting each other, saying only their preferred method would work and the other method wouldn't work, with the result that both SIM and TPF have been cancelled. Geoff Marcy said he was unhappy about the last ten years and unhappy about the next ten years and said the title of the Astro2010: The Astronomy and Astrophysics Decadal Survey which is "New Worlds, New Horizons in Astronomy and Astrophysics" is disingenuous in the use of "New Worlds" because SIM and TPF were cancelled. Has the community since then reached a consensus as to which method is preferable and what are the deciding factors ?


title: Can Gauss' Law in differential form apply to surface charges?

text: I'm calculating the electric field outside a coaxial cable using only Gauss' Law in differential form. The charge density on the interior solid conducting cylinder is exactly cancelled by the surface charge of the exterior shell conducting cylinder. The problem is, Gauss' Law in differential form is: $$ \nabla \cdot E = \frac{\rho}{\epsilon_0} $$ How do I turn this into an equation regarding $$ \sigma $$ the surface charge, instead? Every time I try to do it, I wind up with the wrong units. I though perhaps it might involve the Delta function, but couldn't get that to work. (I've been working in cylindrical units)


title: Lorentz force, must there be two magnets?

text: ![enter image description here](http://i.stack.imgur.com/VzJUP.png) The diagram above demonstrates the Lorentz force, Yet I wonder... must there be two magnets/electromagnets? If the two magnets form a magnetic field of $1$ $Tesla$ acting on the wire, why not use 1 magnet and make the wire really really close to it. So that the gap would be 0. Will $B$ still be valid based on one magnet?


title: Distance Between Two Photons Calculated in Different Inertial Frames

text: I am a self-studier. This is a question from a text I am studying: > The distance between two photons traveling along the $x$-axis of an inertial > frame, $S$, is always $l$. Show that in a second inertial frame, $S'$, > moving at constant speed $V = \beta c$ along the $x$-axis, the separation > between them is given by: $$\Delta x' = l\left(\frac{1 + \beta}{1 - > \beta}\right)^{1/2}$$ I would appreciate help solving this. I was also wondering what $l$ is in the frame of the photons. Also I was wondering how to deal with the length $l$ in the $S$ frame, as it seems to be moving at the speed of light. Thanks for your help and patience with an enthusiastic beginner.


title: A four-dimensional integral in Peskin & Schroeder

text: The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660: $$\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2)^2}e^{ik\cdot\epsilon}=\frac{i}{(4\pi)^2}\log\frac{1}{\epsilon^2},\quad \epsilon\rightarrow 0$$ I can't figure out why it holds. Could someone provide a method to prove this? Many thanks in advace.


title: Does measurement prevent tunneling?

text: Does observation collapse the wave function, thus preventing an object from tunneling to a classically forbidden region? If I understand correctly, observation causes objects to collapse into the state in which they were observed, so there will no longer be a probability of finding them elsewhere. Then if I am somehow measuring the position of an object, does that mean it can't tunnel as long as I continue to observe it? For example, can my hand tunnel to the moon, even though I am looking at it right now, thus measuring its position? How does measurement affect the possibility of tunneling?


title: Two disks on frictionless surface, demonstration

text: > Consider a system composed of two plane disks, which we will designate as > $D$ and $d$. Their radii are $R$ and $r$ respectively $(R>r)$. The disk $d$ > is fixed over the disk $D$ and a distance $b$ from the center, as in the > figure. The disk $D$ can spin and move freely in a friction-free platform, > while the other disk $d$ is fixed on a point of disk $D$, but can spin > freely without friction. No external forces act on the system. The masses of > the disks are $M$ and $m$ respectively. ![Two Disks](https://dl.dropboxusercontent.com/u/38822094/TwoDisks.JPG) So let's just say the problem asks me in one of its to calculate the angular momentum of the system with respect to the center of $D$, this is a non inertial system of reference, but this does not influence a lot if I do not do forces yet right? In other words, the angular momentum will just be?... $ \mathbf{L_{D}}=\frac{1}{2}\left(\frac{1}{2}MR^{2}\right)\mathbf{\dot{\theta}}\hat{k}+\mathbf{b}\times m\left(\mathbf{v_{d}-v_{D}}\right)+\frac{1}{2}\left(\frac{1}{2}mr^{2}\right)\dot{\mathbf{\varphi}}\hat{k} $ Where $v_{d}$ and $v_{D}$ are the velocities of the center of mass of the small disk and big disk respectively. Bold are vectors. And if from there I wanted the total momentum do I just add $\mathbf{L}=\mathbf{R_{D}}\times(M\mathbf{v_{D}})$? Where $R_{D}$ stands for the vector that goes from the origin to the center of the disk $D$


title: What's in my box?

text: Let's say that I have a box which is 100% empty. I fly into the vacuum of space, open the box and close it after a certain time. Then I go back to earth and my question is.. What's in my box? (particles/atoms/molecules?)


title: How's the energy of particle collisions calculated?

text: Physicists often refer to the energy of collisions between different particles. My question is: how is that energy calculated? Is that kinetic energy? Also, related to this question, I know that the aim is to have higher and higher energy collisions (e.g to test for Higgs Boson). My understanding is that to have higher energy you can either accelerate them more, or use particles with higher mass. Is this correct?


title: Partition functions in $\phi^{4}$ theory

text: The partition function in a $\phi^{4}$ theory is written \begin{equation}Z[J]=\int D\phi \, e^{-\int d^{4}x \left(\frac{1}{2}\left[(\nabla \phi)^{2}+m^{2}\phi^{2}\right]+\frac{\lambda}{4!}\phi^{4}+J\phi\right)}\end{equation} I'm confused on how the Fourier transform of this is done. According to the Wikipedia page the quartic term becomes \begin{equation}\int d^{4}pd^{4}p_{1}d^{4}p_{2}d^{4}p_{3} \delta^{4}(p-p_{1}-p_{2}-p_{3})\tilde\phi(p)\tilde\phi(p_{1})\tilde\phi(p_{2})\tilde\phi(p_{3})\end{equation} but I don't understand why it isn't just \begin{equation}\int d^{4}p\, \tilde\phi^{4}(p)\end{equation} I get that the first one enforces momentum conservation, but is there a "cleaner" way to do it?


title: Big Bang and Spherical Symmetry

text: If the universe did indeed start with the big bang why is the universe not spherically symmetric? As per Wikipedia entry on Big Bang, (and my understanding as well) big bang is the best explanation as of now. > The Big Bang is the scientific theory that is most consistent with > observations of the past and present states of the universe, and it is > widely accepted within the scientific community. Basically considering the universe as a sphere (all points equidistant from a center) for any given state at any given point, there should have existed an identical state at the other end (line passing through the center) of the sphere. Also, all points equidistant from the center should have had the same 'state'. What is the explanation for the 'asymmetry' that we have today?


title: Differences between Goldstone bosons and fermions

text: I have been looking into basic SUSY and SUGRA theory and have a question relating to Goldstinos (particles giving gravitinos mass). Simply are these analogous to Goldstone bosons produced in the Goldstone model which is due to symmetries, not super-symmetries being broken. So they are like the same mechanism but in a Super-symmetric sense, why are they needed because surely if SS is broken (and they are produced) there wouldn't need to be a pair of fermion to boson particles or would there?


title: Question about the derivation of an equation in full replica symmetry breaking solution

text: Using replica method and saddle point method, the free energy of a magnetic system can be expressed as $$-\beta[f]=\lim_{n\to0}\\{\frac{-\beta^2J^2}{4n}\sum_{a\ne b}q_{\alpha\beta}^2-\frac{\beta J_0}{2n}\sum_\alpha m_\alpha^2+\frac{\beta^2J^2}{4}+\frac{1}{n}\log\operatorname{Tr}e^L\\}$$ $$L=\beta^2J^2\sum_{\alpha<\beta}q_{\alpha\beta}S^\alpha S^\beta+\beta\sum_\alpha(J_0m_\alpha+h)S^\alpha$$ $J$ is the negative interaction energy, $J_0$ is the mean of $J$, $q_{\alpha\beta}$ is $[<S_i^\alpha S_i^\beta>]$, $S$ is the magnetic spin, $m_\alpha$ is $[<S^\alpha>]$, $h$ is the external magnetic field. What I don't understand is that if we assume $J_0, h, m_\alpha$ all equal to 0, then expand the above equation to fourth order, we can get the following equation: $$\beta[f]=\lim_{n\to0}\frac{1}{n}\\{\frac{1}{4}(\frac{T^2}{T_f^2}-1)\operatorname{Tr}Q^2-\frac{1}{6}\operatorname{Tr}Q^3-\frac{1}{8}\operatorname{Tr}Q^4+\frac{1}{4}\sum_{\alpha\ne\beta\ne\gamma}Q_{\alpha\beta}^2Q_{\alpha\gamma}^2-\frac{1}{12}\sum_{\alpha\ne\beta}Q_{\alpha\beta}^4\\}$$ $T_f$ is the critical temperature, it equals to $J$. I just can't see how this equation is derived.


title: Rocket drive and conservation of momentum

text: I am currently reading through some lecture notes of Physics 1 and in a chapter about the dynamics of the mass point, there is an example covering the rocket drive. Let $v$ be the velocity of the rocket relative to some external observer, $u$ be the speed of output of the gas relative to the rocket and $M(t)$ be the mass of the rocket at time $t$. The total momentum of this system after a short instance is $$p(t+\mathrm dt) = (M-\mathrm dm)(v + \mathrm dv) + \mathrm dm(v-u)\\\ = Mv + M \, \mathrm dv - v \, \mathrm dm - \mathrm dm \, \mathrm dv + v \, \mathrm dm - u \, \mathrm dm\\\ \approx Mv + M \, \mathrm dv - u \, \mathrm dm, $$ where $\mathrm dm$ is the mass of the ejected gas, and due to conservation of momentum we have $$p(t+\mathrm dt) - p(t) \approx Mv + M \, \mathrm dv - u \, \mathrm dm - M v = M \, \mathrm dv - u \, \mathrm dm = 0,$$ i.e. $M \, \mathrm dv = u \, \mathrm dm$ or $M \frac{\mathrm dv}{\mathrm dt} = u \frac{\mathrm dm}{\mathrm dt}.$ By integrating, we arrive at $$\int_{t_0}^t \frac{\mathrm dv}{\mathrm dt'} \, \mathrm dt' = \int_{t_0}^t \frac{u}{M(t')} \frac{\mathrm dm}{\mathrm dt'} \, \mathrm dt'$$ and in order to solve the integral on the right hand side, the writer uses $\mathrm dm = - \mathrm d M$. **Question:** Why is there a negative sign and not just $\mathrm dm = \mathrm dM$? How does it matter? Up to now, I always understood the notion $\mathrm dm$ to be a very tiny change of mass which would always be positive. Is that wrong? If so, how else do I have to interpret that notion? Thank you very much in advance for any help.


title: Direction of rotation of proton in magnetic field--opposite to a dipole

text: > Chatroom created by @pcr for discussing this: > http://chat.stackexchange.com/rooms/2824/direction-of-rotation-of-proton-in- > magnetic-field Here's a small paradoxical question I was asked a long time ago (and have been asked twice since). I think I do know the answer, but I though it would be fun to ask it here. Let's take a proton and fling it into a magnetic field coming out of the plane of the paper/screen ($\vec{B}=B_0\odot$) Now, looking from above the plane, the proton goes clockwise. Alright. lets take our right hand and find the direction of the dipole moment. It's a proton going clockwise, so it's a clockwise current. This is a downwards ($\vec{M}=M_0\otimes$) dipole moment. But, potential energy of a dipole is $U=-\vec{M}\cdot\vec{B}$. If they are antiparallel, then the dot product is negative, so we get $U=-M_0B_0(\odot\cdot\otimes)=M_0B_0$, and is positive. Compare that with the case where $\vec{M}\parallel\vec{B}$, we get a negative value of potential energy. As we all know, a system tends to reduce its potential energy. Then why, in this case, does a proton deliberately choose the direction of rotation with the _maximum_ potential energy? ## Reason for bounty Multiple things. The bountybox does not provide the ability to overlap reasons, unfortunately ;)   * I have multiple conflicting answers, and while each one is individually convincing, when brought together the whole situation becomes a jumble   * I need more people to take a look at this, upvote answers they agree with, comment, and/or add more answers.   * The answers could be clearer   * It would help if the answers explained the paradox for various levels of understanding.


title: Is there a recognised standard for typesetting quantum mechanical operators?

text: Firstly, I wasn't sure exactly where to put this. It's a typesetting query but the scope is greater than $\TeX$; however it's specific also to physics and even more specific to this site. I've recently been reading a style guide for scientific publications (based on ISO 31-11), however there was no mention of quantum mechanical operators. I've seen them written a few ways and was wondering if there was a decision handed down from "up above" that any particular way is best.   * $H$ -- I see this most commonly but I suspect it's mostly due to (mild) laziness to not distinguish it from a variable.   * $\hat{H}$ -- This is nicer to me because it makes the distinction between operator and variable. From what I understand of the ISO the italic means it's subject to change, which is true of the form of an operator, but not really its meaning? So I'm not totally sure if that's appropriate here.   * $\mathrm{H}$ -- Roman lettering is used for functions e.g. $\sin{x}$, $\mathrm{erf}(x)$, and even the differential operator (as in $\frac{\mathrm{d}}{\mathrm{d}x}$) so this seems to me like the most suitable category to put operators in.   * $\hat{\mathrm{H}}$ -- Probably the least ambiguous but may also be redundant. Which would be the best to use? Am I being too pedantic?


title: Magnetic Fields in water

text: I have a question for all of the persons talking about magnetism. What are the differences in any of the properties, laws, etc. of a magnetic field totally submerged in water compared to a magnetic field not submerged in water?


title: Would one actually find their doppleganger in a "Googolplex Universe"?

text: > Related: Infinite universe - Jumping to pointless conclusions I've recently become a fan of Numberphile, and today I happened to watch their video regarding Googol and Googolplex. In the video, I found a rather confusing claim that I'm hoping someone here can help me sort out. It's first stated in the introduction, and then explained in more detail at around 4:10. The claim can be summed up as this: > In a universe which is "a Googolplex meters across", if you would travel far > enough, you would expect to eventually begin finding duplicates of yourself. Getting deeper into the detail of this, Tony Padilla explains that this is because there is a finite number of possible quantum states which can represent the volume of space in which your body resides. That volume is given as roughly one cubic meter, and the number of possible states for that volume is estimated at $10^{10^{70}}$. This is obviously much less than the possible number of quantum states that could be represented within each cubic meter of a Googolplex Universe, and so the idea does make some sense. But I believe the idea relies on a false premise. That premise would be that the universe as a whole is entirely comprised of random bits of matter. We can easily see that this is not true. The vast majority of our universe is occupied by the near-vacuum of space, and those volumes which are not empty are occupied by some fairly organized objects which all interact according to certain laws and patterns. So, I'm curious to know two things:   1. Who originally proposed this idea? Is it something that perhaps Tony just came up with to include in the video, or is there a noted physicist or mathematician who actually wrote about this at some point?   2. Given the possibility that a universe similar to ours could exist and be one Googolplex across, would this actually be probable? Or, would the natural order and organization of the universe prevent this from being as likely as it might be in a more random universe? * * * EDIT (To resolve some comments) A note regarding the size of the universe, in respect to this question. In the linked video, the following constraints are mentioned:   * The number of particles in the universe is $10^{80}$      * Stated at 1:38 in the video.   * The number of the grains of sand that could fit within the universe is $10^{90}$.      * Stated at 1:55 in the video.   * The number of Planck volumes in the universe is $10^{130}$      * Stated at 2:22 in the video.   * The size of the universe is $(10^{26}m.)^{3}$      * Stated at 6:36 in the video. A few times in the video, Tony does qualify his statements by referring specifically to the _observable_ universe. However, sometimes it's not quite so clear. So, to simplify the problem for the purposes of this question, let's assume:   * _Our_ universe is finite, and much less than a Googolplex in diameter.   * The "Googolplex Universe" suggested in the video, and in question here, is also finite.


title: holographic projection in thin air?

text: from where I come from ,they taught us in high school that it is possible to project holograms in thin air simply by illuminating the hologram with the "correct light" , and having a semi transparent medium in the path of the reflecting light "e.g. Water vapor or smoke" resulting in something like this image ![enter image description here](http://i.stack.imgur.com/XB7YL.jpg) -is this true & if not is there any technology that can project such 3dImages in thin air ?


title: Anticommutatorrelation in Bogoliubov-de Gennes Hamiltonian

text: I almost solved the problem Equivalence of Bogoliubov-de Gennes Hamiltonian for nanowire. In the next steps I used the notation by arXiv:0707.1692: $$ \Psi^{\dagger} = \left(\left(\psi_{\uparrow}^{\dagger}, \psi_{\downarrow}^{\dagger}\right), \left(\psi_{\downarrow}, -\psi_{\uparrow}\right)\right) $$ and $$ \Psi = \left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), \left(\psi_{\downarrow}^{\dagger}, -\psi_{\uparrow}^{\dagger}\right)\right)^{T}\text{.} $$ I'm trying to show that the Hamiltonian for a nanowire with proximity-induced superconductivity $$ \hat{H} = \int dx \text{ } \left[\sum_{\sigma\epsilon\\{\uparrow,\downarrow\\}}\psi_{\sigma}^{\dagger}\left(\xi_{p} + \alpha p\sigma_{y} + B\sigma_{z}\right)\psi_{\sigma} + \Delta\left(\psi_{\downarrow}^{\dagger}\psi_{\uparrow}^{\dagger} + \psi_{\uparrow}\psi_{\downarrow}\right)\right]\text{,} $$ can be written as $$ \hat{H} = \frac{1}{2}\int dx \text{ } \Psi^{\dagger}\mathcal{H}\Psi $$ with $\mathcal{H} = \xi_{p} 1\otimes \tau_{z} + \alpha p \sigma_{y}\otimes\tau_{z} + B\sigma_{z}\otimes 1 + \Delta 1\otimes\tau_{x}$ (here $\tau_{i}$ are the Pauli matrix for the particle-hole space and $\otimes$ means the Kronecker product). Here I calculate as example the first und third term of $\Psi^{\dagger}\mathcal{H}\Psi$. $$ \tau_{z}\Psi = \left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), -\left(\psi_{\downarrow}^{\dagger}, -\psi_{\uparrow}^{\dagger}\right)\right)^{T} = \left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), \left(-\psi_{\downarrow}^{\dagger}, \psi_{\uparrow}^{\dagger}\right)\right)^{T} $$ $$ \Rightarrow \left(\left(\psi_{\uparrow}^{\dagger}, \psi_{\downarrow}^{\dagger}\right), \left(\psi_{\downarrow}, -\psi_{\uparrow}\right)\right)\xi_{p}\left(\left(\psi_{\uparrow}, \psi_{\downarrow}\right), \left(-\psi_{\downarrow}^{\dagger}, \psi_{\uparrow}^{\dagger}\right)\right)^{T} = \left(\psi_{\uparrow}^{\dagger}, \psi_{\downarrow}^{\dagger}\right)\xi_{p}\left(\psi_{\uparrow}, \psi_{\downarrow}\right)^{T} + \left(\psi_{\downarrow}, -\psi_{\uparrow}\right)\xi_{p}\left(-\psi_{\downarrow}^{\dagger}, \psi_{\uparrow}^{\dagger}\right)^{T} = \psi_{\uparrow}^{\dagger}\xi_{p}\psi_{\uparrow} + \psi_{\downarrow}^{\dagger}\xi_{p}\psi_{\downarrow} - \psi_{\downarrow}\xi_{p}\psi_{\downarrow}^{\dagger} -\psi_{\uparrow}\xi_{p}\psi_{\uparrow}^{\dagger} $$ Now I use the anticommutatorrelation $\\{\psi_{\sigma}, \psi^{\dagger}_{\sigma^{\prime}}\\} = \delta_{\sigma,\sigma^{\prime}} \Leftrightarrow \psi_{\sigma}\psi^{\dagger}_{\sigma} = 1 - \psi_{\sigma}^{\dagger}\psi_{\sigma}$ $$ \Leftrightarrow 2\psi_{\uparrow}^{\dagger}\xi_{p}\psi_{\uparrow} + 2\psi_{\downarrow}^{\dagger}\xi_{p}\psi_{\downarrow} - 2\xi_{p} $$ However, the term $-2\xi_{p}$ here are wrong. For the third term I obtain $$ \psi_{\uparrow}^{\dagger}B\sigma_{z}\psi_{\uparrow} + \psi_{\downarrow}^{\dagger}B\sigma_{z}\psi_{\downarrow} + \psi_{\downarrow}B\sigma_{z}\psi_{\downarrow}^{\dagger} +\psi_{\uparrow}B\sigma_{z}\psi_{\uparrow}^{\dagger} = \psi_{\uparrow}^{\dagger}B\sigma_{z}\psi_{\uparrow} + \psi_{\downarrow}^{\dagger}B\sigma_{z}\psi_{\downarrow} - \psi_{\downarrow}^{\dagger}B\sigma_{z}\psi_{\downarrow} -\psi_{\uparrow}^{\dagger}B\sigma_{z}\psi_{\uparrow} + 2B\sigma_{z} = 2B\sigma_{z} $$ Does anybody see my mistake?


title: Simple rope tension problem

text: I have this really simple rope tension problem that can be solved for seconds using simple logic. I know what the answer is, I know why it is, but I cannot write it down and explain it mathematically. The problem is the following: An object with mass 6 kg hangs from a rope. The rope is pulled with acceleration of 1 m/s*s. What is the tension in the rope? Clearly the answer is 66 N, but I fail when try to explain it. If I choose downward for positive then tension T = 60 N - 6 N = 54 N. Where am I wrong? How is actually tension in a rope measured?


title: Non-relativistic limit of complex scalar field

text: In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. And for that he uses the condition $\partial_{t} \Psi \ll m \Psi$, which in fact I suppose he means $|\partial_{t} \tilde{\Psi}| \ll |m \tilde{\Psi}|$, otherwise I don't get it. In any case, starting with the Lagrangian: $$\mathcal{L}=\partial^{\mu}\tilde{\psi} \partial_{\mu} \tilde{\psi}^{*} -m^{2}\tilde{\psi}\tilde{\psi}^{*}$$ Using the inequation I think it's correct, I can only get to: $$\mathcal{L}=-\nabla\tilde{\psi} \nabla \tilde{\psi}^{*} -m^{2}\tilde{\psi}\tilde{\psi}^{*}$$ And from that I've tried relating $\tilde{\psi}$ or $\psi$ (as we can write the above Lagrangian with both, as it's invariant under multiplying by a pure phase), to $\dot{\psi}$


title: Will macroscopic object change its angular velocity after absorbing electron?

text: Consider macroscopic object (a ball for instance) which has angular momentum equals 0. Now single electron hits the ball and is absorbed by it. Let's assume it hits in direction perpendicular to tanget plane to the point of hit. As an electron is quantum of angular momentum, the total angular momentum of composite system (ball and electron) must change due to conservation of angular momentum. As angular velocity of macroscpic object is proportional to its angular momentum the ball should start rotating (although very slowly due to small amount of momentum carried by electron and large moment of inertia of ball). Is this true? And if so, how is it possible to rotate an object using non- rotating particle (electron)? This question is perhaps naive and maybe I miss something, but I would be grateful for explaining this.


title: Understanding wave functions

text: I'm currently writing an essay on the measurement problem, and I'm not quite certain that I've fully understood the purpose of the wave function, in that does the following sentence make sense with respect to the wave function: > _Although the wave function itself is never physically observed, we are > given resultant macroscopic events which we can interpret, allowing us to > formulate a conclusion._ What I mean to say is that although we don't actually see the wave function, we can see the kind of after-math of what happens in the form of a physical macroscopic event..is that right?


title: Coffee filter experiment: time as a function of mass

text: In an experiment I performed, I dropped $n$ amount of coffee filters from a constant height and recorded the time $t$ taken for the $n$ coffee filters to reach the ground. In the experiment, the value of $n$ was kept as small as possible. I assumed that the coffee filters reached their terminal velocity directly (for simplicity). I tried to find a relation between the no. of coffee filters and the time taken for these coffee filters to reach the ground. I am very certain that $t(n) \propto n^{k}$, however, I am not sure which value the constant $k$ should have. The closest I got was $k=-1/4$. Does anyone know if my $k$ has a reasonable value? **EDIT:** the equation I get in MS Excel when I plot the data directly is: $t(n)=3.2108n^{-0.295}$, where $k=-0.295$. I think I need to find what fraction this value is close to. **EDIT2:** I discovered that this value is close to $1/3$. The $R^2$ value is closer to $1$. * * * The mass of a coffee filter was $2g$, so time can be expressed in terms of mass also.


title: Does halving the intensity of a gamma ray mean absorbing half its energy?

text: Say a 1 MeV gamma ray is shielded by 1 cm (halving thickness) of lead. Now, what does "havling the intensity" mean ? Like will the new gamma ray exit with energy of 0.5 MeV ? and by that we can say that the material absorbed 0.5 MeV ? or it will be halved in another way ? Sorry if the question seem too amateur, I am just not good at particle physics.


title: How do you weigh a box on a scale whose limit is too low?

text: As you will see I know nothing about physics and after being asked to solve a physics problem in a recent interview wanted to ask it of professionals and see what the response would be: I have a set of domestic scales (actually just one scale) that weigh up to the maximum weight of 5kg and a large package that weighs more than 5kgs but less than 10kgs. How can I tell the exact weight using the inadequate scales. The package is a long item akin to a steal girder but not in weight obviously. I know that stack exchange has rules about this type of question so please treat it as light entertainment from someone in awe of your intellect.


title: Casimir Effect and polarization of photons

text: I have read Casimir's derivation of the Casimir fore between 2 parallel plates and have been told that in reality, the Casimir force should be twice as large due to the 2 polarization states of photons. I don't understand how/why the 2 polarization states affects the force. Could anyone kindly explain?


title: Change of hue near shadows

text: My brother is an artist and he told me his paintings look more natural when he slightly change the hue (color) of the surface near the transition between shadow and light. Here's three examples: (1) without this effect, looks artificial, (2) with this effect, looks natural, (3) with overuse of this effect, looks strange. Assume pure white light (this hue change is only caused by surface properties). ![Hue change example](http://i.stack.imgur.com/9WXaN.jpg) What would be the physical explanation for this effect? The hypothesis I got is that when an object is painted (semi-transparent surface over pure reflective surface), if the rays reach the surface in a near parallel direction, then we get some refraction. If the reflectance of the surface is not a pure wavelength, then we can see some hue drift due to that refraction. Does this make sense? Is there a simpler explanation?


title: Wave theory of light

text: how come light is a wave and how do you visualize it?Why waves are thought to be like sine or cosine function?Also my most important question what is the basic difference between interference and diffraction?


title: How to distinguish 4D and 3D vectors in handwriting?

text: Usually vectors are denoted with bold font in printbooks and with arrows above in handwriting. In Thorn's e al. Gravitation, 4D vectors are denoted with bold and 3D vectors with bold italic. How to implement similar distinguishing in handwriting?