title: Reconstruction of information stored in an evaporating black hole from the emission spectrum?

text: For simple setups, where the radiation field deviates not too far from thermodynamic equilibrium (< 10 %), corrections to the Planckian thermal emission spectrum can be calculated (and measured) from the statistical operator method, using the heat flux and the photon flux as additional constraints in the entropy maximazation procedure: http://einstein.iec.cat/jellebot/documents/articles/PA_99.pdf Now I am courious if similar considerations could be applied to the situation of an evaporating black hole. Can the information, orignally stored in an evaporating black hole, in principle be reconstracted from analyzing it`s nonequilibrium emission spectrum (obtained from including a sufficient larger number of additional relevant parameters as constraints in the statistical operator method)? **Clarification** I`m pondering if one could describe an evaporating black hole applying nonequilibrium statistical mechanics such that the global mean temperature is not sufficient to describe its thermodynamic state and additional (microscopic) degrees of freedom (which are important to store the information of infalling objects too) must be taken into account. My question then is: Are these deviations from thermodynamic equilibrium translated to deviations of the spectrum of the Hawking radiation from Planck s law? And if so, can this in principle be used to reconstract the information stored in the microscopic degrees of freedom of the black hole?


title: How long did inflation take to happen?

text: If inflation happened, it ended about 13.7 billion years ago. But how long was inflation going on before that? Was it just a short burst of time so that it's really okay to say that the age of the universe is 13.7 billion years, or was it going on for a long time before it ended (maybe forever?)?


title: Viscosity coefficients

text: I'm using the 2nd edition of "Transport Phenomena" by Bird and Stewart. I am having trouble with one of the equations: $$\tau_{ij} = \sum_k \sum_l \mu_{ijkl} \frac{\partial v_k}{\partial x_l} $$ Here, $i,j,k=1,2,\:or\:3$. And $x_1,x_2,x_3=x,y,z$ respectively. Similarly $v_1,v_2,v_3=v_x,v_y,v_z$. The derivative is meant to express that the velocity of the fluid towards the $k$ direction varies as you move along the $l$ coordinate. The $\tau_{ij}$ is the viscosity tensor. According to the statement, the viscosity tensor can be expressed as a linear combination of all the velocity gradients. So far I am good. Then it goes on and puts the double series. Which I suppose it is there to express that the velocity of the molecules moving in the $x,y,z$ direction can vary as one moves along the $x,y,\:or\:z$ coordinates. However, I do not understand why there are 81 viscosity coefficients. I cannot imagine a physical interpretation to it. Shouldn't the viscosity coefficients be 9, $\mu_{kl}$? One way that I thought I could rationalize the reason for 81 coefficients instead of 9 was that it is possible that the the direction of the tensor matters. But isn't this information already included on the linear combination of the velocity gradients (without the viscosity coefficient)? Considering this, how and why would the direction of the tensor matter? Why are there 81 coefficients?


title: Poynting's Theorem simplified?

text: How can I apply Poynting's theorem to any system that has a magnetic field & electric field, to state conservation of energy? How does Poynting's theorem state conservation of energy in EM? I struggled to understand with the differential equations. Example: Work being done by the Lorentz force on a conductor by a supplied power source?


title: How do Leptons arise from Lambda decay?

text: I have a question for an assignment: > Use your understanding of the quark model of hadrons and the boson model of > the weak nuclear interaction to explain how leptons can arise from lambda > decay, especially when there is an emission of a muon and a muon > antineutrino. I'm a bit lost as to how to answer this; I'm not looking for the full answer (it is for an assessed assignment so the work needs to be original) but if anyone has any pointers as to where I should be starting I would be very grateful!


title: What effect does shape of dots on a household microwave oven door have on amount of EM radiation that escapes through the door?

text: Why are they dots and not squares, triangles, or some other shape?


title: Binding energy in fission

text: In nuclear fission the total rest mass of the products is less than the mass of the original nucleus. But on the other hand, the binding energy of the products is higher. Doesn't the decrease of the rest mass of the products contribute to the increase of their binding energy? So why then there is enormous energy released?


title: Matter - Antimatter Reactory Practicality

text: With **current technology** , would the energy released by a matter-antimatter annihilation be more than the energy needed to created the antimatter in the first place? Would it be worth it? Just curious, even if was on a small scale, Could one produce a machine that uses annihilation to produce a net energy gain?


title: Path traced out by a point

text: While studying uniform circular motion at school, one of my friends asked a question: "How do I prove that the path traced out by a particle such that an applied force of constant magnitude acts on it perpendicular to its velocity is a circle?" Our physics teacher said it was not exactly a very simple thing to prove. I really wish to know how one can prove it.Thank you!


Wen's question What is spontaneous symmetry breaking in QUANTUM systems? is cute, but here's an even cuter question. What is spontaneous symmetry breaking in QUANTUM GAUGE systems? There are some gauge models where the Higgs phase and the confinement phase are continuously connected, you know. Why are the Higgs phase and the confinement phase identical in Yang-Mills-Higgs systems? You mention cluster decomposition? Sorry, because bilinear operators have to be connected by a Wilson line.


title: Stablising the Higgs without SUSY

text: Should the Higgs be found at the LHC, but no supersymmetry (assuming for the sake of argument that the LHC be capable of eliminating all versions of SUSY that are motivated by solving the hierarchy problem), what then? Are there alternatives for stablising the Higgs scale that don't involve SUSY (other than Randall-Sundrum)?


title: How can we have spinless fermions?

text: I've read that the Jordan-Wigner transformation changes qubits into spinless fermions. What, exactly, are spinless fermions? I'm guessing it doesn't mean spin zero which would be a boson, so what does it mean?


title: Why will clay, when heated to high temperatures,  harden and become waterproof?

text: I have a question > Why will clay, when heated to high temperatures, harden and become > waterproof?


title: Do particles interact with the Higgs field more at higher speed?

text: It is said that interaction with the Higgs field gives rest mass to fundamental particles. When a particle is moving, is it then correct to say that there is more of the same interaction that gives it rest mass, thus accounting for its full relativistic mass?


title: Why Pauli exclusion instead of electrons canceling out?

text: To quote Wikipedia, > _The Pauli exclusion principle is the quantum mechanical principle that no > two identical fermions (particles with half-integer spin) may occupy the > same quantum state simultaneously. A more rigorous statement is that the > total wave function for two identical fermions is anti-symmetric with > respect to exchange of the particles._ As far as I can tell, this means that two identical electrons in identical states have wave functions that cancel each other out. My question is, if this is the case, why do we have the Pauli exclusion principle? It seems that getting two electrons too close should annihilate both of them (or cancel them out, or however you want to say it), but instead we have the Pauli exclusion principle, where somehow the electrons remain separate. Why?


title: Do photons feel gravity of approaching objects only?

text: I have read that photons while travelling near massive objects such as the sun experience gravitational pull which is why we see some stars at different positions than they are when seen towards the sun. Now if photons felt gravity of objects they approach, speed of propagation of gravitational field can be both lesser than or equal to speed of light, but if photons that are receding from objects also feel it's gravity than certainly speed of propagation of gravity would be more than speed of light ! So the question is do photons feel gravity of object they move away from ? If they do is my assertion correct ?


title: Quantum spin in a rotating reference frame

text: Rather predictably, I'm having a hard time understanding quantum spin. I know that it has units of angular momentum and relates to rotation, but that it can't be thought of as the actual spin of a particle with real size. Also, I understand that it's quantized. What I'm wondering is, what happens to quantum spin in a rotating frame of reference? Specifically, are there (rotating) frames of reference in which (for instance) an electron has spin that isn't half-integer? Is there a frame of reference rotating about a single electron such that that one electron acts like a spin-0 particle?


title: One-dimensional (1D) CdSe nanostructures

text: I'm not a physicist, but I'm really interested in Nanotechnology. I've a question, and I hope that I can be provided by an answer. I was reading some articles about the one-dimensional CdSe nanostructures. My question is why it's called one-dimensional? Is it because of their geometrical shapes or something else? I truly appreciate any hint or a resource could lead to the answer. Thanks.


title: Algorithmic approach for applying Kirchhoff's Rules to circuit analysis

text: Working with Kirchhoff's rules, I am attempting to device an algorithmic approach to finding the unknowns of the problems, I am of a Computer Science background and I am finding it difficult to identify proper closed loops within the circuit and I am curious if there is a generalized pice of sudo code applicable to these problems, or perhaps a matlab or sympy script available?


title: Dynamics of moment of inertia

text: I'd like to be able to determine the angular acceleration of a system of two rotating masses, which are connected so as to have a variable mechanical advantage between the two. My background with mechanics went as far as a course in statics, so I'm not sure how to proceed with this. If I have a single mass of some shape and apply a torque to it, I know that the angular acceleration depends on the moment of inertia of that object. But suppose I have a system of two objects, e.g., gears, and apply torque to one of them, and want to know the angular acceleration. I'd assuming that the effective moment of inertia, at the point where I apply the torque, is the moment of the directly driven mass, plus the moment of the secondary mass multiplied by the mechanical advantage between the gears, and that using this 'effective moment of inertia' with the input torque would tell me how fast the input mass accelerates. (The acceleration of the 2nd mass being implied, as there's only 1 dof here) Not sure if this general approach is even correct, and then there's the real problem . Introducing the variable mechanical advantage is what's giving me trouble here. If I take the contribution to the effective moment of inertia from the 2nd mass as just its 'intrinsic' moment times the mechanical advantage at any given instant, am I missing something? Calculus instincts tell me that there ought to be a contributing term from the rate-of-change of the mechanical advantage, too.


title: Object docking in Space

text: How do objects dock in space? There's a recent piece of news that Keppler would dock to the ISS sometime this week. I want to know the operational aspects of it - considering the velocity of the objects involved and also the lack of references.


title: What does it mean, when one says that system has N constants of motion?

text: For example for an isolated system the energy $E$ is conserved. But then any function of energy, (like $E^2,\sin E,\frac{ln|E|}{E^{42}}$ e.t.c.) is conserved too. Therefore one can make up infinitely many conserved quantities just by using the conservation of energy. Why then one can usually hear of "system having N constants of motion"? "System having only one constant of motion"?


title: Using Lagrange's Equations with Generalized forces

text: I am a bit confused on how this works. For instance if I wanted to look at an object moving in 2 dimensions only subject to gravity (and assuming that the potential is just mgy), I get that my Lagrangian is: $$\frac12m(\dot x^2+\dot y^2)-mgy$$ I can easily solve the Euler-Lagrange equations, to see that $\ddot y=-g$ but is there a way I could do this with generalized forces. I feel that this is an awful example. I vaguely remember my professor using generalized forces to find friction force. Any help with understanding generalized forces would be appreciated. The reason I want to know is that I am trying to model a curveball (baseball) and treating the lift and drag as generalized forces then using the E-L equations to solve for the motion.


title: Continuously feeding an evaporating micro-black hole?

text: What would happen if you created a micro-black hole and could continuously feed it as quickly as it evaporates? Is it possible that it would remain relatively stable? If so, how might such a thing be continuously fed?


title: Expression for Atmospheric Pressure with Altitude, including Tidal Forces

text: Is this simple to obtain? I've tried to modify the constant-gravity model to use the $GM/r^2$ form instead, but my result was shown to be wrong with some testing. **Constant Gravity Model** You can find this in many places online. Here's one. I'll repeat that work here, jumping over the calculus, and putting it in terms of Earth's radius. r0 will be the surface radius. $$ \frac{ dP }{ dr } \frac{ 1 }{ P(r) } = - \frac{ M_0 g }{ R T } \\\ H \equiv \frac{ R T }{ M_0 g } \approx 7.4 \text{ km} \\\ P(r) = P_0 e^{ \frac{ r_0 - r }{ H } } $$ **Non-constant Gravity Model** So to generalize this (for a small moon with a diffuse atmosphere, for instance), just replace the gravity with its expression $g(r)=GM/r^2$. M0 is still the average formula mass of air, but we add M, which is the mass of the planet. This should work fine for Earth, I see no reason it shouldn't. $$ \frac{ dP }{ dr } \frac{ 1 }{ P(r) } = - \frac{ M_0 G M }{ R T } \frac{1}{r^2} \\\ H' \equiv \frac{ M_0 G M }{R T } \approx 4,739 \text{ km} \\\ P(r) = P_0 e^{ H' \left( \frac{1}{r} - \frac{1}{r_0} \right) } $$ It looks so nice and congruent. But I plot my non-constant gravity model side- by-side with the constant model and it's obviously wrong. So for my question, could somebody do ANY of:   * Find something I did wrong   * Derive or find a correct model for this   * Give a clear reason it can't be obtained in a simple closed form


title: F-Theory on Calabi-Yau manifolds

text: Why F-theory picks Calabi-Yau manifolds as backgrounds? Is there a similar argument like the one in heterotic/IIA,B which singles out Calabi-Yau manifolds based on the requirement of space-time supersymmetry? If there is no 12-dimensional supergravity (hence no 12-dim SUSY variations) then how one can show that the solution of the killing-spinor equations chooses Calabi-Yau manifolds?


title: Is it possible to create only a magnetic wave?

text: I want to know if its possible to create only a magnetic field,which i can direct to a certain chosen object?


title: What is the simplest system that has both, discontinous and continous phase transitions?

text: I am looking the simplest system that has both discontinous phase transition and a continous phase transition between the same phases (you can change one parameter). **discontinous transition:** first order transition where (some) measurable has a discontinuity/non-analyticity as a function of (some) order parameter **continous transition:** measurable is continous/analytical for all values of order parameter. **simplest:** Requires the smallest number of mathematical symbols and is still defined exactly c.f. ising model. **EDIT:** Removed water as an example. Above critical point, water has no phase transition as Alexei pointed out. So, all the transitions of water are first order...


title: What kind of cooling mechanism could be used in outer space?

text: This question arises out of this question on Quora - Apollo 11: 1969 Moon Landing: Did Neil Armstrong really land on the moon? I'm convinced with most of the explanations provided in the first answer to that question. None of these conspiracy theories have bothered me. However one thing that confuses me is the heat. On earth close to 47% of heat is absorbed or reflected by the atmosphere. On moon an object would get struck directly with thermal radiation from sun acquiring about 50% more heat then it would on earth. I am wondering what kind of cooling device could we create for such an environment? A system like an AC or a refrigerator won't work in a vacuum. Off the top of my head the only method I could think of is circulating some liquid with high specific heat in the suits and letting it evaporate. But is this feasible? How much heat do we need to loose? What rate of evaporation would be needed? **What volume of liquid would be needed** to support this rate? Given the high amount of electricity it takes to makes our refrigerators work **how much electricity would such a system need?** Because of very large amount of heat and large surface area (the lunar module, the suits, and the camera casings) I'd assume a large quantity of water (say a few thousand liters) would be needed. How could they carry such a large amount of fluid?


title: Where to go to minimize tidal forces?

text: Suppose you design an experiment where you need to minimize the effects of tidal forces. Where would you go? There are a few possibilities, and the choice depends on how much effort you are willing to make. My reasoning goes like this:   * Maximize the distance from any centers of masses, i.e.     * Climb Mount Everest     * Go in Low Earth Orbit     * Go in even higher orbits   * Use properties of a many-body system where the gravitational field has a minimum     * Equilibrium point between Earth and Moon     * Other Lagrange Points of the Earth/Moon system (Are they better, i.e. have a lower delta a/delta x?) I can think of more places farther out, like where the Jovian Trojans orbit the Sun. Of course, the further away from the Sun you get, the better. But maybe there are closer spots I have overseen where tidal forces are extremely low.


title: An accelerating and shrinking train in special relativity

text: Suppose when a train is at rest, it has a length of $L$. Let the position of the back of the train at any time be $A$, and let the position of the front of the train at any time be $B$. Now assume a stationary observer on the Earth, O, observes point $A$ to be accelerating at a constant rate, $k$. Now as the train accelerates, O will observe the train to continually shrink, thanks to length contraction. Now since $A$ is measured to accelerate at a constant rate, the position of $B$ will depend on A's acceleration, and the rate of shrinkage. So the question is, calculate the acceleration of point $B$, as measured by O, given the acceleration of point $A$, and the length of the train, $L$.


title: How did Einstein derive general relativity?

text: How did Einstein derive general relativity (GR)? Did he use: the equivalence principle? The principle of least action? Anything else? Note, I'm not looking for a _full mathematical derivation_ of GR! But rather, I'd like to know what Einstein's starting points were. Note: I found a _similar_ question that has been asked already, but I couldn't find an answer to _my_ question in the answers.


title: What can make signal triangulation tricky?

text: Usually, when you're trying to find a source of whatever signal, you move around the source and then calculate the position using triangulation. I'm specifically talking about low frequency signals, such as radio waves. I was wondering what can make this process harder or completely impossible (the former seems very unlikely).   If there were two transmitters, would it be easy to tell that there are two and not one? Could the signal be dispersed unevenly in the area, making it appear somewhere else than it is?


title: Is the distance between the sun and the earth increasing?

text: M = mass of the sun m = mass of the earth r = distance between the earth and the sun The sun is converting mass into energy by nuclear fusion. $F = \frac{GMm}{r^2} = \frac{mv^2}{r} \rightarrow r = \frac{GM}{v^2}$ $\Delta E = \Delta M c^2 = (M_{t} - M_{t+\Delta t}) c^2 \rightarrow \Delta M = \Delta E / c^2$ $\rightarrow \frac{\Delta r}{\Delta t} = \frac{G}{v^2 c^2}.\frac{\Delta E}{\Delta t}$ Sun radiates $3.9 × 10^{26} W = \Delta E/\Delta t$ Velocity of the earth $v = 29.8k m/s$ There is nothing that is stopping the earth from moving with the same velocity so for centripetal force to balance gravitational force $r$ must change. Is $r$ increasing? ($\Delta r/ \Delta t = 3.26070717 × 10^{-10} m/s $)


title: Increment of surface tension with increase of temperature

text: What is the reason for the increase in surface tension of molten copper/cadmium with increase in temperature?


title: Witten's constrained S-matrix and Coleman-Mandula Theorem

text: I remember reading somewhere that Witten argued that if the Poincaré symmetry of spacetime were nontrivially combined with internal symmetries, then the S-matrix would be so constrained that the scattering amplitude would be non- vanishing at discrete angles only. I would like to know more about this:   1. At _which_ angles would the scattering amplitude be non-vanishing   2. By crossing symmetry, wouldn't this mean that in the crossed channel, the amplitude be non-vanishing at discrete _energies_?   3. Is there a reference that contains more information on this?


title: Optical constants of noble metals: the Drude model for microwave modelling

text: I have a question regarding the optical constants of noble metals. According to Johnson and Christy's paper Optical Constants of Noble Metals ( _Phys. Rev. B_ **6** , 4370–4379 (1972), doi:10.1103/PhysRevB.6.4370), in the Drude free electron theory the equation for the dielectric permittivity is $$\epsilon(\omega)=1-\frac{\omega_p^2}{\omega(\omega+i/\tau)}=1-\frac{\omega_p^2}{\omega(\omega+i\gamma)},$$ where $\epsilon(\omega)$ is the dielectric constant at frequency $\omega$, $\omega_p$ the plasma frequency, and $\gamma=1/\tau$ the collision frequency. In the nanohub photonics database the constants can be calculated from different theoretical and experimental models. However, in CST Microwave Studio I need to give the epsilon infinity value, which really comes from modified drude model: $$ε=ε(\infty)-\frac{ω_p^2}{ω^2+iγω}.$$ **How do I get the value of $\epsilon(\infty)$ for gold?** Is there any reference by using which we can directly convert the free electron theory Drude model to the modified Drude model? I am specifically searching for the values of $ε(\infty)$, $ω_p$, and $γ$ for gold in the 700nm to 1100 nm range (specifically 830nm).


title: Could an NEO strike the moon with sufficient velocity for the shrapnel to escape Lunar gravity, and be attracted by Earth's?

text: Between the news item of an asteroid giving Earth a close shave, and another news item of the impending GRAIL impact ; I find myself wondering whether a NEO could be a hazard to Earth _via_ the moon. I'm not sure this scenario is realistic. Perhaps such a large body may naturally be captured by Earth gravity instead of Lunar gravity before impact itself... That is to say - Could an NEO be   * large enough &   * have sufficient velocity  to impact on Luna so that fragments would   * escape lunar gravity, and    * make the down-hill run to Earth, and    * remain large enough to cause loss of life/property on Earth?


title: Production of electricity from atmosphere

text: I read the electric field intesity at the surface of earth is 100v/m. Then why can't we keep two metallic sheets at different heights and produce a continuous current by connecting these two sheets via a electric cable?


title: Good introductory text for matrix string theory

text: Where can I find a good introductory text for matrix string theory? Most textbooks don't cover it, or only cover it very superficially. What is the basic idea behind matrix string theory? How can matrices be equivalent to strings?


title: How does the distance between two rails effect the speed of a steel ball bearing?

text: As part of a school science project, I constructed a Rollercoaster using Polyurethane tubing as rails for a steel ball bearing to rest on. In the process of building the coaster I observed that sections of track where the rails were closer together would cause the ball bearing to roll faster, as the cost of stability as the ball bearing was more likely to fall off the track. Increasing the distance between the rails would cause the ball bearing to roll slower, but increased the odds that the bearing stayed on the track. What caused this phenomena?


title: Statistics and macrolocality in string theory

text: Take two identical closed strings, both tracing out exactly the same path in space. These two strings are coincident everywhere. Call this state I. Take a single closed string following exactly the same closed path as in the first case, but not closing just yet. The string goes around exactly the same path once again before closing in on itself. Two cycles around the same closed path. Call this state II. String field theory tells us unambiguously states I and II are distinct. Stretch this closed path to make it much larger than the string scale. Supposedly, stringy nonlocality only happens at the string scale. States I and II still differ. Partition target space into local regions the size of the string scale. The path cuts across a chain of such local regions. If string theory were local, we can reconstruct the state of the entire universe from the restricted states of each subregion if we allow for quantum entanglement between regions. Locally, states I and II ought to be indistinguishable over each local region. For each local region, we always see two string segments passing through it. Thus, states I and II have to be identical? This can't be. Either string theory is inherently nonlocal over scales much larger than the string scale, or it obeys Maxwell-Boltzmann statistics and not Bose-Einstein statistics. This isn't some Aharonov-Bohm effect. Even if we include all the local regions in the "interior" of the closed loop, this doesn't change matters the least bit. PS. Please reread my question more carefully. What you call configuration III is actually my configuration I. PPS: Let me try to understand your explanation. If we have N coincident strings, or a string which winds round the same loop N times or any other combination in between, this can be described by an $S_N$ discrete gauge symmetry. The conjugacy class of the holonomy of this discrete gauge symmetry around the loop distinguishes between the various combinations. Feel free to correct me if I am wrong. This has the flavor of parastatistics, does it not?


title: Extending Solutions To Hamilton's Equations to Whole Time Axis

text: In Arnold's Classical Mechanics book, he says "We assume that every solution to Hamilton's equations can be extended to the whole time axis", and adds that 'For this it is sufficient, for example, that the level sets of $H$ be compact'. How so?


title: what is two species Bose Hubbard Model? what chage it takes on Fock basis representation?

text: The Hamiltonian will look like \begin{equation} \ H= -t^a\sum_{<i,j>}^{}(a_i^\dagger a_i + h.c.)-t^b\sum_{<i,j>}^{}(b_i^\dagger b_i + h.c.)+U^a/2\sum_{i}^{}n_i^a(n_i^a-1)+U^b/2\sum_{i}^{}n_i^b(n_i^b-1)+U^{ab}\sum_{i}^{}n_i^an_i^b \dots(1) \end{equation} How it will operated in Number State representation? or more clearly how number will look? for example consider 3 sites and 2 bosons [1,0,1] this will be a one state if **_H_** have only **_a_** or **_b_** terms. Since **_H_** contains **_a_** and **_b_** so how corresponding state will represented for **_H_**? thanks for help.


title: Time reversal and parity symmetry

text: I was previously under the misapprehension that time $T$ and parity $P$ symmetries in conjunction ($PT$) were a reflection in $(3+1)$-dimensional space-time, where $$P: \vec x \to -\vec x$$ $$T: t \to -t$$ but, in fact, because it has determinant $(-1)^4=1$, $PT$ is merely a rotation (by $\pi$). Is this correct? Shouldn't the symmetry we seek be a reflection in space time, to generalize parity so that it includes time? In other words, if $P$ is a reflection in $3$ spatial dimensions, why isn't $PT$ defined to be a reflection in $(3+1)$-dimensional space-time? My thought is that $P$ is an intuitive symmetry in spatial dimensions. In the spirit of relativity, why don't we generalize $P$ from a reflection in space to a reflection in space-time (called, say, $PT$)? Why are $P$ and $T$ reflections considered separately? It's problematic to me because their combination ($PT$) is not a reflection. I suppose I know that $P$ and $T$ are separate because only $T$ must be anti-unitary etc...


title: Normalization of the real Klein Gordon Field in Peskin and Schroeder chapter 2

text: In Peskin & Schroeder's QFT, how do you get from equation 2.35 to 2.37? (In particular, how does the invariant normalization of the Klein-Gordon real field imply that $U(\Lambda)|p> = |\Lambda p>$ ?) Also, on a more general note, could some explain why for the _real_ Klein- Gordon field we need to make the effort to define invariant normalization? In particular, why do we care if the expression $<q|p>$ is invariant if it is $|<q|p>|^2$ which bears physical meaning?


title: Is it true that quantum mechanics technically allows anything to happen?

text: Maybe this is a silly question (I think it is), but it's a question I'm arguing with some of my friends for a long time. The ultimate question is: **Is everything (in our Universe) possible ?** I've read some books (Penrose, Hawking, Greene, etc.) where it was clearly stated several times that quantum mechanics allows **any situation** to happen ( **e.g. everything is possible** ) - even with absurdly tiny probabilities? Does this mean only physically possible - or "ultimate" possible ? For example (I know this example seems _really absurd_ , but I'm saying this so you get the idea) what my friends clame it is possible:   * If I ask my friend(s): "Is it possible there is a Sun-sized star in my pocket right now?" They will respond: "Yes, there is **very very tiny** possibility this can be true!"   * Etc.


title: How does a Primary Charge Roller work?

text: I have been recently trying to get more deeply into the workings of a common laser printer.   While the basic concept seems to be fairly simple to understand I have been having trouble finding more information on the physics behind it. In this case for example the Primary Charge Roller.   I'm aware of what it does but I haven't found any information on the **how**. How exactly does it create the negative charge?   Is the simple contact of the roller and the other medium enough to transfer it evenly?   What are the restrictions regarding the target material, could you transfer a charge with the same method to glass, wood, metal? I hope someone here can help me out with some information or point me into the right direction.


title: Thrust to Weight ratio in Space with an off set CoM

text: With regards to this thread, Thrust center in space My question is, if the thrust to weight ratio was increased so that it was much higher than the weighted mass of the sphere (ship), would the sphere then not start to follow a straight trajectory along the axis of the thruster.


title: Physics of a burning log of firewood

text: According to my knowledge, heat is nothing but the result of the vibrations of atoms and molecules. I guess this mean that in heating up a gas or liquid, we are increasing the rate at which the container of this gas or liquid vibrates, thereby imparting to the molecules a certain amount of energy. How would this work in case of a burning log in the fireplace? Is it vibrating and thus heating up the molecules in the room? Or is it releasing highly energetic photons that do the required bombarding? What exactly is happening when we light a fire?


title: Newton's gravitational constant $G$, the reduced Planck constant $\hbar$, the speed of light $c$: the Dream Team of moderators?

text: The three great constants of Nature are well known:   * the speed of light $c$ (special relativity),   * the reduced Planck constant $\hbar$ (quantum mechanics),   * Newton's gravitational constant $G$ (general relativity). However, is it possible that they play, in fact, a similar role, a moderator role, in the precise sense of decreasing energy? Consider the following baby model. We consider a particle of radius $r$ and mass $m$. We admit an auto-gravitational potential, and a pseudo auto-quantum potential. The energy of the particle is given by $$E(m,r) = m c^2 - \frac{\hbar^2}{m r^2} - \frac{G m^2}{r}.$$ (I do not care about possible coefficients of order unity.) We demand the physical condition $$E(m,r) \ge0.$$ We may easily observe that, putting $\hbar = 0$, we recover the gravitational limit for the radius of particle when it becomes a black hole, and, by putting $G=0$, we recover the "quantum limit" for the radius (a taste of Heisenberg inequalities). What is the effect of choosing $c^{-1} > 0$ ($c$ not infinite), or choosing $\hbar > 0$, or choosing $G > 0$? This is the same effect, that is, it decreases the energy $E(m,r)$. So the team $(c^{-1}, \hbar, G\,)$, when they are not zero, decreases the energy, it appears as a (dream) team of moderators of energy. Of course, we are working with a baby (particle) model. But do you think that these conclusions could be extended to the real word?


title: Solar energy and optical fiber

text: I'm not really good at science or physics, just wanted to ask about the idea (doesn't have an opportunity to test this). Will this scheme work and what can be its efficiency?(The goal is to make solar energy cheaper) A panel that consists of multiple lenses receives sun light. Each of the lenses is focused on a short optical fiber(1-10 meters) that transmits the light to the 2nd lens. The 2nd lens absorbs the light from all optical fibers of the panel and focuses it on a single solar cell.


title: What applications would room temperature super conductors have?

text: I've heard that a room temperature super conducting material would be a major discovery. How likely is this within the next century and if discovered what would be possible?


title: Coefficients and Parameters for contracted Gaussian basis sets

text: This is a repost from Chemistry.stackexchange in the hopes that someone here will be able to help me. Any help at all would be greatly appreciated. As far as I understand, an STO-NG contracted Gaussian basis function has the following form: $$\phi_{\mu}^{CGF}(\vec{r}) = \sum_{p=1}^{N} d_{p\mu} (x-X_A)^{i_{p\mu}} (y-Y_A)^{j_{p\mu}} (z-Z_A)^{k_{p\mu}} e^{-\alpha_{p\mu}\zeta^{2}|\vec{r}-\vec{R_{A}}|}$$ where the contraction coefficients, $d_{k\mu}$ and exponents, $\alpha_{k\mu}$ are chosen such that $\phi_{\mu}$ provides the 'best fit' to a Slater-type orbital having stater-exponent $\zeta$, $\vec{R_{A}}=(X_A,Y_A,Z_A)$ is a fixed reference centre (usually a nucleus) and $l_{p\mu} = i_{p\mu} + j_{p\mu} + k_{p\mu}$ defines the angular momentum of the primitive Gaussian function: $$g_{p\mu} = (x-X_A)^{i_{p\mu}} (y-Y_A)^{j_{p\mu}} (z-Z_A)^{k_{p\mu}} e^{-\alpha_{p\mu}\zeta^{2}|\vec{r}-\vec{R_{A}}|}$$. I'm having problems interpreting data files documenting contraction coefficients and exponents for various basis sets. For example, please consider the following snippet from the Gaussian 94 STO-3G basis set file documenting contraction parameters for Carbon:               ****     C     0      S   3   1.00          71.6168370              0.15432897                 13.0450960              0.53532814                  3.5305122              0.44463454            SP   3   1.00           2.9412494             -0.09996723             0.15591627                  0.6834831              0.39951283             0.60768372                  0.2222899              0.70011547             0.39195739            ****      I have read in various places that there are enough coefficients here to describe 5 contracted functions, but I see only enough for 3 maximum (if the exponents are the same, but the coefficients change between 2S and 2P). I'd be eternally grateful if someone could explain for me, precisely how one would determine the following information from such a basis set data file: 1) Which EXACT basis functions appear within each contracted function? 2) How many contracted Gaussian basis functions there are? If possible, would you be able to do the same for the following snippet, again for Carbon, Gaussian 94, but this time for the more complicated 6-31++G** basis set?:               ****     C     0      S   6   1.00        3047.5249000              0.0018347                 457.3695100              0.0140373                 103.9486900              0.0688426                  29.2101550              0.2321844                   9.2866630              0.4679413                   3.1639270              0.3623120             SP   3   1.00           7.8682724             -0.1193324              0.0689991                   1.8812885             -0.1608542              0.3164240                   0.5442493              1.1434564              0.7443083             SP   1   1.00           0.1687144              1.0000000              1.0000000             SP   1   1.00           0.0438000              1.0000000              1.0000000             D   1   1.00           0.8000000              1.0000000             ****      Thank-you for reading.


title: Special relativity paradox and gravitation/acceleration equivalence

text: One of the features of the black hole complementarity is the following : According to an external observer, the infinite time dilation at the horizon itself makes it appear as if it takes an infinite amount of time to reach the horizon, while the free falling observer reach the horizon in a finite amount of time. But, because we cannot differentiate between acceleration and gravitation, it may be equivalent to the following "paradox" in Special Relativity. Le be the plane $z = 0$ be a kind of "horizon". Let be an observer $O_1$ which is at $z = 1 \space m$ (1 meter), at $t = 0$, and which is moving left (towards decreasing $z$), with a constant speed $(-1) \space m/s$. Let be a second observer $O_2$, which has the position $z = 1$ m (1 meter), at $t = 0$, but which is moving right (towards increasing $z$), and which is accelerating, in a precise sense (see below). Divide the initial distance from the observer $O_1$ to the "horizon" $z=0$, into a serie of distance intervals $L_n$, with $L_n = 1/2^n \space m$(meter): $$1/2, 1/4, 1/8,....,..., 1/2^n,...$$ The initial distance (1 meter) between the observer $O_1$ and the horizon is : $$ 1 = \sum^{\infty}_{n=1} L_n$$ For each distance interval $L_n$, the corresponding elapsed time (from observer $O_1$ point of view) needed for observer $O_1$ to reach the end of the interval $L_n$, is $\tau_n = 1/2^n s \space (second)$, because its speed is $-1 \space m/s$. The total time (from observer $O_1$ point of view), necessary for the observer $O_1$ ,to reach the horizon $z = 0$ is then : $$\tau = \sum^{\infty}_{n=1} \tau_n = \sum^{\infty}_{n=1} 1/2^n = 1 s$$ Now, for each interval $L_n$, we may adjust the speed of the accelerating observer $O_2$, such that, due to the time dilatation, the elapsed time necessary for the observer $O_1$ to travel during the interval $L_n$, from the point of view of observer $O_2$, be : $$T_n = a_n \tau_n$$, where $a_n$ is a coefficient > 1 From the point of view of observer $O_2$, the time necessary for observer $O_1$ to reach the horizon $z = 0$ is then : $$ T = \sum^{\infty}_{n=1} T_n = \sum^{\infty}_{n=1} a_n \tau_n = \sum^{\infty}_{n=1} a_n/2^n$$ By choosing, for instance, $a_n = 2^{n+\epsilon}$, where $\epsilon >0$, it is easy to see that, from the observer $O_2$ point of view, the observer $O_1$ needs an infinite amount time to reach the horizon $z=0$, while from the $O_1$ point of view, he reaches the horizon in one second! Did you agree ?


title: Pair Creation and Reference Frames

text: Ok, so I have a question about an example problem in my textbook (its long but not difficult to follow. My question is at the bottom after I outline the problem). Consider the situation below in frame $S$. A gamma ray encounters an electron, disappears and creates an electron-positron pair, then the two electrons and single positron move off together with speed $u = 0.8c$ ![enter image description here](http://i.stack.imgur.com/d6Lw6.png) From conservation of energy and momentum: $$E_i = E_{\gamma} + mc^2 = E_f$$ $$p_i = \frac{E_{\gamma}}{c} = p_f$$ Now consider a frame $S'$ that moves with speed $v = 0.8c$ so that the three particles after the pair creation are at rest. In this frame, we know that the invariant rest energy of the system equals the sum of the rest energies of the constituent particles since the particles are not moving relative to one another in $S'$. My book places a big emphasis on the fact that $$(mc^2)^2 = E^2 - (pc)^2$$ the rest energy in the above formula is not the sum of the rest energies of the particles that form the system. Only when the particles are not moving relative to one another (as in $S'$), can you say that $mc^2$ is $\Sigma_i = m_ic^2$. Anyways, so in $S'$ after the pair creation, $mc^2 = 3mc^2$ since electrons and positrons have the same mass. My book then goes on to write, $$(3mc^2)^2 = E^2 - (pc)^2 \\\ 9(mc^2)^2 = (E_{\gamma} + mc^2)^2 - (\frac{E_{\gamma}c}{c})^2$$ My Question: In the above formula, for $E$ we substituted $E_{\gamma} + mc^2$ from the conservation laws written earlier. However, this energy was determined in frame $S$. Why didn't my book use the transformation: $$E' = \gamma(E - vp'_x)$$ Since we are in the $S'$ frame? I have the same question for momentum. Why didn't we use the Lorentz transformation for momentum? By the way: This was a problem specifically looking at what the energy of the photon $E_{\gamma}$ has to be in order for this particular electron-positron pair to be created. However, my main problem is with the reference frames since summing the rest energies requires us to be in $S'$. Then, $E$ and $p$, in $(mc^2)^2 = E^2 - (pc)^2$ need to be the energy and momentum in $S'$ as well?


title: In what applications are lasers necessary vs. just convenient?

text: I’m trying to understand, given the wide array of current laser applications:   1. Which applications actually require laser characteristics, as opposed to just use lasers because they are the most practical/efficient source of light with characteristics _not unique_ to lasers   2. What alternatives exist, or may exist, to lasers for applications that don’t require laser characteristics My key assumption here is that the essential characteristic of a laser is that it produces temporally coherent light. Often, but not necessarily, lasers produce light with two other useful characteristics:   1. Monochromaticity   2. Collimation But these convenient characteristics can be produced using non-lasers. For example, LEDs produce monochromatic light. And lenses can take diffuse light and produce a beam collimated to near the diffraction limit, right? If this is the case, then the only applications that require lasers are those that depend on interferometry, and that’s pretty much just holography and measurement, right? All of the other applications – directed energy, medical therapies, optical discs, optronics – could be accomplished with non-coherent collimated and/or monochromatic light, right? Presumably lasers are used in these cases only because they are currently the most efficient or practical way of producing light with those characteristics. But now I am wondering: What other technologies could supersede lasers in practice (and efficiency) for applications that require light with high energy, collimation, and/or monochromaticity?


title: Quantum Eraser thought experiment with light photons of distinct color

text: I tried to recreate the Quantum Eraser experiment into a thought experiment with a few changes. It left me a little perplexed as to what outcomes I should expect. Any help would be appreciated. Lets say you have 2 entangled photons so that when one is blue the other is always red. You shoot the 2 particles in different directions without measuring the color of the photon in either direction. Each particle, P1 and P2, hit a corresponding measuring device labeled D1 and D2. D1 only will measure the color of P1 when it hits it. D1 can also be turned on or off. When D1 measures the color of P1, P2 collapses into either red or blue state based on the measurement of P1. I assume prior to measurement from D1, P2 would be in both a red and blue state. Now lets say that D2 has a single slit in front of it (like you would see in the double slit experiment, just with one slit) that has a yellow film over it that could alter the color of the photon passing through it to either orange or green based on if the photon was red or blue respectively . I imagine that if D1 was turned on you would gather data on D2 of either an orange or green mark based on D1's measurement collapsing P2. However when D1 was turned off would the results on the wall at D2 change to brown dots as the photons would pass through the yellow film in both a red and blue state simultaneously? Any help with this is greatly appreciated.


title: What would give us more heat ? infrared or microwaves?

text: As we know that our body is made up mostly of water and the frequency of vibration of water molecules matches that of microwaves which is the working principle of microwave ovens. When we come in contact of sunlight and feels its warmth we say that it's due to the heat waves coming as infrared. Atleast to explain the heat **we** feel shouldn't we give reasoning of microwaves rather than infrared ?


title: Does Ising model belong to the field of strongly correlated systems?

text: How to make a judgement that whether a problem is within the field of strongly correlated systems? Do classical problems (not quantum mechanical) belong to this field?


title: How does inductor store energy in the magnetic field?

text: I learnt from book that magnetic field does no work because the force is proportional to $\vec{v}\times\vec{B}$ where $\vec{v}$ is the particle velocity. That vector cross product is always at right angles to $\vec{v}$, so that $\vec{F}\cdot\vec{v}=0$, i.e. no work is done on the particle. But then how come does an inductor store energy in the magnetic field?


title: Why is it hard to give a lattice definition of string theory?

text: In Polyakov's book, he explains that one possible way to compute the propagator for a point particle is to compute the lattice sum $\sum_{P_{x,x'}}\exp(-m_0L[P_{x,x'}])$, where the sum goes over all paths between $x$ and $x'$. One then needs to compute this sum and choose the bare mass so that there's a good continuum limit. Polyakov then goes on to say that this doesn't work for string theory. I skimmed the literature and couldn't find any explanation of this fact. Naively I would think that in order to find the propagator, you could just compute the sum $\sum_{W_{C,C'}}\exp(-T_0 A[W_{C,C'}])$, where the sum is over worldsheets that end on the curves $C$ and $C'$. What goes wrong? Is this just a hard sum to do?


title: Can internal forces do work?

text: My Mechanics textbook claims that the sum of the work by internal forces is not generally zero. translated to English the paragraph reads: > Notice about the work by internal forces: the work by the internal forces is > in general not equal to zero although the sum of the internal forces is > always equal to zero. Since the work also depends on the relative > displacement of the two pointmasses in the material system, the work > provided by these 2 internal forces will generally not be equal to zero. Other sources claim that the work done by internal forces is always zero. Why does a car engine not do work if the wheels don't slip? (answer by Mark Eichenlaub) > Then I didn't do any work on the system at all since, by definition, only > external forces can do work. http://in.answers.yahoo.com/question/index?qid=20110131130859AAMLJLd > the net work done by internal forces is zero since they do not cause motion > of the object. To me it would seem that these sources contradict eachother. Do they? If so, which one is correct? If not, please explain why they don't contradict eachother.


title: What conserved quantities are in 1D free quantum particle

text: From Laundau & Lifshitz "Quantum Mechanics": > If there are two conserved physical quantities $f$ and $g$ whose operators > do not commute, then the energy levels of the system are in general > degenerate. Let's consider a 1D free particle. Its wavefunction would be $$\left<x\middle|\psi\right>=e^{ikx}$$ For all $k>0$ it's twice-degenerate: $E(k)=E(-k)$. I know there are two conserved quantities here: energy and momentum (wavenumber). But their operators commute. But judging from the quote, there should exist another quantity, which would have operator, which doesn't commute with momentum or energy operator. What is this quantity?


title: Is it possible to see domains in a metal/magnet under microscope

text: I was just curious to know that is it possible to see domains of a magent/metal in motion on being magnetized, under microscope. If there is someone who has access to microscopes, can help me out on this. I was thinking of buying one. Question is lil stupid but I am putting it here because I know no better place !!


title: When is entanglement entropy the same as free energy?

text: I am given the feeling that there exists scenarios when this equality holds. Can anyone state/refer to the situations? * * * One case that I hear of is that for $2+1$ CFTs the entanglement entropy across a spatial disk is the same as the free energy of the CFT system on a $S^2 \times S^1$ (or $S^3$?)   * What is the exact correct statement?    * I am told that this paper http://arxiv.org/pdf/1102.0440v2.pdf "proves" the above but despite reading this paper a few times I can't locate this argument! (...may be I am missing something very badly!...)


title: Can you please help with our water tank and spray system?

text: For our design project we are working with a client in Wisconsin to help them remove snow from their roof-mounted solar panels in the winter. We are designing a system to spray water a vertical distance of 40-50 ft.   * Flow rate coming from the hose in the house measured at 4.9 $\frac{gal}{min}$    * The pressure is being pumped to the house from a series wells at 60 psi  We want to increase this flow rate to 10 $\frac{gal}{min}$ to help melt the snow off the panels more effectively. Our idea was to use a 100 gallon storage tank, we would fill this with the garden hose, and have water leave the tank to get our desired flow rate. From the tank the hose would approximately be 50 ft. in length, and at the end of the hose we were planning to use nozzle designed to give us a solid stream.   If from the tank the water leaves at 10 $\frac{gal}{min}$, using the inner diameter of the hose (0.62992 in.) we calculated the velocity of the water to be 3.1375 $\frac{m}{s}$. Will this velocity stay constant until the end of the hose and be the Vin for the nozzle? Then if we had a nozzle diameter of 0.2 in, using $A_{in}V_{in}=A_{out}V_{out}$ we calculated the water leaving the nozzle to be at a velocity of 31.09 $\frac{m}{s}$. Using the trajectory equations   $$R= Vi^2sin\frac{(2 θ)}{g}$$ and $$h=Vi^2*sin^2\frac{θ}{2g}$$   we found our stream would travel R=92.58 m and h=62.55 m Could we put a valve at the bottom of the tank, and get that flow rate and velocity to shoot the water that distance, or will we have a problem with pressure with this system? If we use a pump to pump out the water at 10 gal/min would the water still be able to shoot that far, or would we have to model our system similarly to a pressure washer and have a pump attached closer to where we would be spraying the water from?


title: Why Negative Energy States are Bad

text: The argument is often given that the early attempts of constructing a relativistic theory of quantum mechanics must not have gotten everything right because they led to the necessity of negative energy states. What's so wrong with that? Why can't we have negative energy states? As I understand it, we know now that these "negative energy states" correspond to antiparticles. So then, what's the difference between a particle with negative energy and an antiparticle with positive energy? It seems to me that there really is no difference, and that the viewpoint you take is simply a matter of taste. Am I missing something here?


title: What is force times angle?

text: I'm looking at an explanation of pendulums, and the following is said: ![enter image description here](http://i.stack.imgur.com/5LhtV.png) What I don't understand is where it says "if restoring force is given by $mg\theta$..." - conceptually, what is a force times an angle? Never being one to simply commit something to memory for the sake of it, I really want to understand this... Any help greatly appreciated! Tim. EDIT: Follow up question here.


title: A draft makes people feel cold. How can that be measured?

text: On some days our office feels very cold to the point at which we find it difficult to type because our hands have gone numb. But our facilities manager insists it is the same temperature as always. We have discovered the reason for feeling cold is that the air conditioning causes a draft of cold air. Whoever is in the draft feels more and more cold the longer it goes on. My question is: Would a thermometer in the path of this draft of air show any change in temperature, and if not, how can this cooling effect be measured so we can provide evidence of our discomfort to our company?


title: Changing the capacitance of an object to trigger a capacitive touchscreen

text: I'm trying to trigger a human touch in a capacitive touchscreen without the 'human' part. I've noticed that the negative pole of a battery triggers it if you are currently holding it. Or if you use the edge - why only the edge? - of a wire touching the battery and your hand. I also attached a wire in a metal bottle and connected the battery to it's end (with the battery touching the screen) and it also worked - but sometimes it also didn't. The question is: How can I trigger a touch on the screen without the human touch, using objects from the hardware store? I have almost no physics knowledge for this, so you guys have to tell me exactly what to do (which material, where to buy and etc). Thank you


title: What are the easiest to get/make LN2 superconductors?

text: I am starting to build multistage Peltier cooler at the moment, and it should be able to reach -100C at least (but if I fail I can always get boring LN2). Doing some experiments with superconductors would be great, but where can I get superconductor samples or which one is easier to make on my own (have access to all chemicals & lab). Especially awesome would be to be able to make thin superconductor film. Any suggestions?


title: Geometric/Visual Interpretation of Virasoro Algebra

text: I've been trying to gain some intuition about Virasoro Algebras, but have failed so far. The Mathematical Definition seems to be clear (as found in http://en.wikipedia.org/wiki/Virasoro_algebra). I just can't seem to gain some intuition about it. As a central extension to Witt Algebras, I was hoping that there has to be some geometric interpretation, as I can imagine Witt Algebras rather well. If anyone has some nice Geometric or Visual Interpretation of Virasoro Algebra, I'd greatly appreciate it!


title: acceleration of rings in aerotrim (human gyroscope)

text: I'm working on a graphic simulation (just for fun, for an open-source screensaver) of an Aerotrim \- a "human gyroscope", one of those exercise/training machines with a human in the middle, perched on a ring that swivels on an axis with respect to another ring, which swivels on an axis perpendicular to the first axis, on another ring. (AFAICT there could be 2 or 3 such axes that turn, each at right angles to the next. If I can simulate 2 axes / 3 rings, that would be sufficient. A general solution for n rings would be lovely. :-) Simulating the thing turning, with each axis rotating at a constant rate, is no problem. But it would "feel" more real if I could apply some realistic acceleration from the interaction between the rings and the mass of the rider. In the Youtube videos, you sometimes see a bystander push on one of the rings, to help get the rider going. Not only does that ring accelerate, but others start spinning too. And the pushed ring does not accelerate smoothly, but undergoes resistance from the other rings, apparently transferring acceleration to them. I am not up on angular momentum, torque, and all that, so answers will need to take my ignorance into account, though I'm obviously willing to learn a few things as necessary. I don't feel like it's necessary to take into account the asymmetry of the person's mass - modeling the person as a point mass in the center of the rings should be fine, I think. In other words I don't plan on modeling the way that the rider accelerates the rings by leaning one way or another. I was thinking that in order to make things a little more interesting, I would have the program occasionally apply a force like a hand pushing on one of the rings. I can figure out what the torque on that ring would be if it were independent of the others, but I don't know to model a set of 3 rings connected on axes. Any thoughts on how I could model the interaction between the rings, each affecting the others? Simplifications are fine. My intent is not to discover new phenomena through accurate physical modeling, but to create a visual display that _looks_ more realistic than just rings turning at constant rates. **Edit:** P.S. Part of my uncertainty is, how much does the Aerotrim really behave like a gyroscope? The gyroscope's main properties are focused on the mass in the middle spinning fast on its axis, and therefore its axis tends not to move. This is obviously different from the aerotrim, where the person is often not rotating much around the inner axis, and the inner axis is certainly not staying still. I wonder if other factors, like the conservation of angular momentum of the _outer_ rings, plays more of a part in the Aerotrim that is negligible with gyroscopes. If it _is_ essentially just a gyroscope... I've looked at the "fundamental equation" at http://en.wikipedia.org/wiki/Gyroscope#Properties, but as a non- physicist I find it difficult to imagine how to turn that into code computing what happens when a force is applied. Pointers to example code for simulation of gyroscopes would be appreciated. Thanks, Lars


title: Effects of parallel superconducting plates

text: Assuming the existence of virtual particle field ( zero point energy field) Casimir force is produced by 2 parallel conducting plates excluding some of the frequencies between the plates, if these 2 plates are superconducting does that change the amount of frequencies being excluded? One theory I've read postulates that it is interaction of matter with the ZPF (=zero point field) that is the origin of mass and inertia, if the parallel plates are suppressing the appearance of some of these virtual particles, would that not in some manner be modifying inertia for matter passing between the plates?


title: Software for physics calculations

text: What is some good **free** software for doing physics calculations? I'm mainly interested in symbolic computation (something like Mathematica, but free).


title: Generalisation of Reversible Equation to Non-Reversible Situations Because it Only Contains 'Properties of the System'

text: I've just been reading through Van Ness' _Understanding Thermodynamics_ , and I'm having a little trouble following his argument at one point. He is deriving the useful differential equation: $$dU=T \ dS -P\ dV$$ By noting that: $$dU=dQ-dW$$ for all processes, and then also noting that, _only for reversible processes_ , $$dQ_{rev}=T\ dS$$ and $$dW_{rev}=P\ dV$$ and substituting these last 2 equation into the first equation. As long as we're talking about reversible processes, I'm fine with this. But then he then goes on to say: _Now we derived this equation for a reversible process, but once derived we see that it contains just properties of the system, and so it must not depend on the kind of process considered. What we have really done is derive an equation for a special case, and then conclude that it must be general._ Maybe this is blindingly obvious, but I'm really struggling to follow his logic through. He's derived an equation for one particular case, and then, just because this equation doesn't explictly refer to anything other than properties of the system, he concludes that it must hold for all cases. I mean, for the reversible case, sure, but I'm struggling to see why, logically, $dU=TdS-PdV$ would hold for a given irreversible case. To give a very rough analogy of my thinking, I feel live I've found an equation like Boyle's law: $$P\propto\frac{1}{V}$$ and can then conclude that, just because this equation (a) holds for one particular system (e.g. low pressure, high volume gas) and (b) only contains 'properties of the system' $(P, V)$, then it must therefore hold for all situations, which is of course nonsense. Anyway, I'd appreciate any help, thanks


title: Why is work in a constant electric field equal to the force times distance?

text: Let's say, for example, that an infinite plate generates a constant electric field that exerts a force of 6N on a 1C charge. To move the charge from a distance of 5m to 1m from the plate, the work done is given by 6N * 4m = 24Nm. The way this has always been explained is that we apply a 6N force over the 4m distance. However, this doesn't make intuitive sense to me:   * If we apply a constant 6N force to the charge in the opposite direction of the electric field, won't the charge simply remain stationary? The force generated by the field and our applied force in the opposite direction should net to zero.    * If the electric field exerts a force of 6N on the charge at every single infinitesimal point along the path from 5m to 1m, when we apply an opposite force of 6N over that same distance, shouldn't the charge simply stay still?   * I think what's troubling me is that I don't see how applying a 6N force can "overcome" the electric field's force and actually move the charge in the opposite direction. I think the intuition is the same for the force of gravity. A 1kg object that is 1m above the ground is experiencing a force downwards of 10N. If I apply a constant upwards force of 10N, the object won't fall. But how is that if I apply a 10N upwards force over 1m, I can actually move the object upwards? Gravity is exerting a downwards 10N force on the object the entire time I am exerting an upwards 10N force over that 1m. If the forces are equivalent, shouldn't the mass stay still?


title: Collision between electron and proton?

text: What would happen if an electron collided with a proton such that the two do not collapse? Would the two become a unit, or would some force prevent them from bonding thus forcing the electron to orbit around the proton?


title: Time reversal and basis independence

text: It is generally assumed that to time reverse a state, one just takes the complex conjugate of the wave function. This is apparently not basis-independent. For example, if we take $|\psi_0 \rangle $ as a basis vector in some basis, then the time reversal operator $K$ acts on it as $K|\psi_0 \rangle = | \psi_0 \rangle $. Now let us take $|\phi_0 \rangle = e^{-i \theta } |\psi_0 \rangle $, $\theta \in \mathbb{R}$, then $K|\psi_0 \rangle = K (e^{i \theta } | \phi_0 \rangle ) = e^{-i \theta } | \phi_0 \rangle \neq |\psi_0 \rangle $. Therefore, the simple complex conjugate recipe is not basis-independent. How can this be reconciled with the idea that it represents time reversal?


title: Why are the zeroth order terms in degenerate perturbation theory the eigenstates of the perturbing Hamiltonian?

text: I have for quite some time now tried to find a satisfactory answer to this, but I haven't yet. In perturbation theory, with small parameter $\lambda$, we expand the eigenstate as $$| E \rangle=| E^{(0)} \rangle + \lambda | b \rangle + ...$$ Where $| E^{(0)} \rangle$ is an eigenstate of the unperturbed Hamiltonian. The problem is that when there is degeneracy, there is a choice of these eigenstates. I know the answer is that they are chosen to be eigenvectors of the perturbing Hamiltonian, but my question is why.


title: How is the velocity not constant?

text: > A bead is moving along the spoke of a wheel at constant speed u , the wheel > rotates with uniform angular velocity w radians per second about an axis > fixed in space , at t=0 the bead is in the x axis at the origin, find the > velocity in t in POLAR coordinates In the text book it says : first we have $r(t)= ut$? doesn't that mean $v = u$ is constant? How did they obtain this result (it changes in time) if $v$ is constant? $v = u (r) + utw(\theta)$ (($r$) and ($\theta$) are the bases of the polar plane)


title: Effective action for bosonic string theory with enhanced symmetry

text: See these lecture http://members.ift.uam-csic.es/auranga/lect7.pdf page 17. Usually one derives the effective action from the massless states calculating amplitudes, otherwise through beta functions(worldsheet conformal invariance). One obtains a effective field theory containing a metric, a Kalb-ramond field and a dilaton. These came from the $N=1,\bar{N}=1$ sector of the mass spectrum. This is wellknown. If the 25th dimension is compactified with $R=\alpha'^{1/2}$ then another massless 25dimensional fields emerge. This happens in the $N=0,\bar{N}=1$ sector and $N=1,\bar{N}=0$ sector. After analysing the enhanced symmetry from the 25d point of view one can deduce that these are $SU(2)\times SU(2)$ gauge bosons. On page 17 of the lecture the autor says: it is possible to cook up a new 25d effective field theory by including by hand the new massless modes. How this action looks like? Is there any paper or reference which explains the calculations or steps to obtain the action?


title: Mirrored decoupling fermion doublers and a lattice chiral fermion / gauge theory

text: **Nielsen Ninomiya Fermion-doubling problem** has known to be a challenge to construct a **chiral fermion or chiral gauge theory on the lattice**. There is a proposed resolution to use so-called two mirrored worlds, say the original copy (called A) and a duplicated mirrored copy (called B); the two worlds A and B together are non-chiral. And adding some interactions to the mirrored copy(B) to cause the mass-gap to decouple the mirrored sector (B). Thus the left-over sector can be the desired chiral fermion / gauge theory. This is called a **mirrored fermion decoupling appraoch**. One framework is this model. There are several results against the thinkings - mostly are based on **perturbative** arguments: 1\. Absence of Chiral Fermions in the Eichten-- Preskill Model-Maarten Golterman, Don Petcher, Elena Rivas; 2\. Nondecoupling of Heavy Mirror-Fermion - Lee Lin' Here is one for I found for **non-perturbative** arguments: 3\. Decoupling a Fermion Whose Mass Comes from a Yukawa Coupling: Nonperturbative Considerations, T. Banks, A. Dabholkar But those are just **arguments** , not theorems or not any proofs. I read those cited papers beforehand. Some numerical attempts had fallen to see the decoupling: 4\. On the decoupling of mirror fermions. Some of the papers/arguments above are about SO(10) or SU(5) non-Abelian gauge theory. > Here instead let us take a simplest case: U(1) chiral symmetry and a 1+1D > system. > > $\star$ My inquiry is that: **What are the SOLID objections on "there does > not exist any lattice realization of 1+1D chiral fermion model with U(1) > chiral symmetry, using mirrored fermion decoupling appraoch?"** The reader (you, and of course I) should know that: **If we just focus on 1+1D, there is no spontaneous broken global symmetry due to Coleman-Mermin- Wagner theorem. Then one needs not to worry about spontaneous break down the chiral symmetry (such as U(1)).** ps. please feel free correct me if my understanind above has any ambiguity or error.


title: Understanding Triplet And Singlet States

text: We know, $2\otimes 2=3\oplus 1$. Thus we have a spin triplet of states and a spin singlet. Can we regard these states as the spin part of wavefunction for the excited states and the ground state of the deuteron nucleus? What is the difference between the members of a triplet?


title: Integration on a general equation for instantaneous angular acceleration

text: An equation for instantaneous angular acceleration is given as: $$ \alpha \equiv \lim_{\Delta t\to0}\frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt} $$ The text I am reading says writing this equation in differential form $$ dw = \alpha dt $$ and integrating from $t_1$ = 0 to $t_f$ = $t$ gives $$ \omega_f = \omega_i + \alpha t $$ I am not exactly sure how the authors came to this. Any help would be much appreciated.


title: Non-Euclidean spaces in Quantum Mechanics

text: In quantum mechanics, I have been going through basics of the subject. In general the space of quantum states is Hilbert space (which is Euclidean - I presume). Being just curious, are there any instances where Non-Euclidean spaces are used in quantum mechanics ? I have heard that there is something called Topological QFT, is it associated to my question ? I know my question is a little ridiculous, but it just out of sheer curiosity.


title: photon polarization, uncertainty in Energy

text: A beam of red light is sent along the $z$ axis through a polaroid filter that passes only $x$ polarized light. The beam is initially polarized at $30$°, and the total energy is $10$ Joules. Estimate the fluctuations in the energy of the beam. After it passes the polaroid, the quantum state will collapse into $\vert\psi\rangle =\vert H\rangle$. And the energy will be $5\sqrt{3}J$, but how can I find the fluctuation? I have tried to use the uncertainty principle without success. Based on the comments, how is the Poisson distribution for the process? Thanks


title: Connection between the Big Bang and Black Holes

text: So... this is my first question here after a little while of lurking; so be gentle. I've been thinking lately about trying to wrap my head around the concept of spacetime and how it can be affected by gravity and the density of matter in space (i.e. near a black hole). I've been wondering. Is it possible that matter - which is broken down (spaghettified?) into it's constituent parts and into energy - is drawn to a point in spacetime which has already got a massive density? Would it be possible that any black hole, which has been or will ever be, causes energy to be drawn to a single point in space or time? I.E. all black holes are "feeding" into a single, infinitesimal time and location; causing (possibly the wrong word - it seems to me that it would be less cause vs. effect and more that it just _is_ ) the big bang? I tried to find some papers that would touch on the topic but couldn't find anything specific - which likely means I'm being a total dolt and there are many good reasons why this isn't the case :) I would appreciate any information which would correct my thinking and counter my naivety.


title: Current flowing through a square loop - Magnetic Field

text: > _A square conducting loop of wire of side $2a$ carries a uniform current I. > Calculate the magnetic field at the centre of square due to the current in > the loop._


title: How many bits are encoded on the surface of the smallest black hole?

text: Here's my guess ($157\, \textrm{bits}$) and how I got there. Please feel free to disregard completely and give your own answer. My understanding (please correct any wrong assumptions as there may be many): The smallest black hole is a "Planck particle", which is a Planck mass in size, and has a theoretical radius of $5.72947 \times 10^{−35}\, \mathrm{m}$ (or so Wikipedia says). A "Planck area" is the smallest area to encode a (digital) bit of data. In the Holographic principle, it's believed the information content of a black hole is equal to its surface area (or more specifically, it encodes one bit for every "Planck area" which can cover its surface/horizon). Now, assuming a Planck particle is spherical and uses normal euclidean geometry and there aren't relativistic or quantum effects that I've completely ignored, this means you can calculate a Planck particle's surface area (from its radius), and thus its information content. Which I calculate as being $157\, \textrm{bits}$. Is this correct? Does this make any sense at all? Is it possible to calculate the information content of the smallest hypothetical black hole? How would you do it? Is there any meaning to the data capacity of a Planck mass of black hole?


title: Atomic physics - photoelectric emission

text: I understand that when photoelectric emission occurs: $$h\nu = h\nu_o + K.E.$$ Where $\nu_o$ represents the threshold frequency. What I don't understand is what happens if the frequency is just equal to the threshold frequency? Won't the electron have zero kinetic energy, hence zero velocity? Thus how does it escape?


title: nuclear physics homework

text: i have a homework in our nuclear physics class. the question is "An astronaut whose height in the earth is exactly 6ft is lying parallel to the axis of a spacecraft moving at a 0.9C relative to the earth. what is his height as measured (a) by an observer on this earth? (b) by an observer on the same spacecraft? " i used the formula `L=Lo sqrt[1-(v/c)2]` and i got the answer L=2.62ft now i am confused if my answer was from observer from earth or from observer on the same spacecraft. please teach me to solve (b) or (a).. sorry if this is literally an assignment.. please don't vote down.. i will delete or close this question once it is answered.. thank you very much..


title: What is the meaning of $U''(x)=0$?

text: Most potentials with a minimum can be described approximately as a harmonic oscillator. So the procedure is to Taylor expand $U(x)$: $$U(x)=U(0)+U'(0)x+\frac{1}{2}U''(0)x^2 +...$$ If we suppose that the potential is cero at the origin an it has a minimum there, we get: $$U(x)=\frac{1}{2}U''(0)x^2$$ We take $U''(0)$ to be the _spring_ constant $k$. So the angular frecuency is given by: $\omega=\sqrt{\frac{k}{m}}$ But what if $U''(0)=0$ and there is still a minumum at zero, like a potential $U(x)=x^4$? In this case if you blindly apply the formula you get zero frecuency, which is false. Does it just mean that to a small approximation a body will not oscillate?


First, pardons to all because I don't have a physics background and I'm probably using laymans terms. What is the rebound angle of a circle striking a concave surface? Assume the simplest case:   * the circle (red) is not spinning   * the circle does not compress   * the arc (blue) does not deform   * there's no friction   * there's no energy lost in the impact Here's my attempt at an illustration with a red circle moving in a blue circle traveling towards the blue circumference along the gold path and rebounding on an unknown green angle. ![enter image description here](http://i.stack.imgur.com/tt3gB.png)


title: What happens in electron-electron collisions?

text: What are the results of high energy electron electron collisions? Are other particles created?


title: Why does increasing the temperature of a thermistor decrease it's resistance?

text: Surely, upon an increase in temperature, the atoms within the thermistor would vibrate with more energy and therefore more vigorously, hence making the electrons flowing through the electric circuit more likely to collide with one of the atoms, so increasing resistance. However, the effect of temperature on a thermistor is contrary to this. I can't understand how it can be. It's analogous to running across a playground: if everyone is still you're less likely to collide with someone, however if everyone is constantly moving from left to right then a collision is more likely. So why does an increase in temperature decrease the resistance of a thermistor?


title: Horizon problem in the standard cosmological model

text: The horizon problem is usually stated as follows: Causally disconnected parts of the universe that we see today have the same temperature, so how could that be so if no information travels faster than the speed of light. But (without assuming inflation) wasn't the universe causally connected right after the big bag? And that those parts that are disconnected today only became disconnected due to expansion? Also, if temperature irregularities existed but to a very minute level, then why do we expect that the universe without inflation must look different than the homogeneous universe (on large scales) we see today?


title: Clarity in Electric field Definition?

text: The electric field at a point is defined as q/$r^2$. How does this definition take the nature of source and test charge into consideration. If I bring any positive/negative charge around the source charge, it brings a change in the distances involved due to forces of attraction or repulsion.how can I use this definition for practical purposes?


title: antiferromagnetic spin wave

text: I have a hamiltonian that is derived from a spin wave energy dispersion calculation for a nearest neighbor interacting cubic antiferromagnet. After a Holstein-Primakoff transformation and making a substitution of -q for q in one place, I get the following hamiltonian (J is the exchange energy, S is the magntiude of the spin on each sublattice, z=6 is the number of nearest neighbors of each site, and $\mathop \gamma \nolimits_q = \mathop e\nolimits^{i*\mathop q\nolimits_x } + \mathop e\nolimits^{ - i*\mathop q\nolimits_x } + \mathop e\nolimits^{i*\mathop q\nolimits_y } + \mathop e\nolimits^{ - i*\mathop q\nolimits_y } + \mathop e\nolimits^{i*\mathop q\nolimits_z } + \mathop e\nolimits^{ - i*\mathop q\nolimits_z } $ : $$ \sum\limits_q {\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q^\dagger }&{\mathop b\nolimits_q^\dagger }&{\mathop a\nolimits_{ - q}^\dagger }&{\mathop b\nolimits_{ - q}^\dagger }&{\mathop a\nolimits_q }&{\mathop b\nolimits_q }&{\mathop a\nolimits_{ - q} }&{\mathop b\nolimits_{ - q} } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {JzS/2}&0&0&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}\\\ 0&{JzS/2}&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}&0\\\ 0&0&{JzS/2}&0&0&{JS\mathop \gamma \nolimits_q /2}&0&0\\\ 0&0&0&{JzS/2}&{JS\mathop \gamma \nolimits_q /2}&0&0&0\\\ 0&0&0&{JS\mathop \gamma \nolimits_q /2}&{JzS/2}&0&0&0\\\ 0&0&{JS\mathop \gamma \nolimits_q /2}&0&0&{JzS/2}&0&0\\\ 0&{JS\mathop \gamma \nolimits_q /2}&0&0&0&0&{JzS/2}&0\\\ {JS\mathop \gamma \nolimits_q /2}&0&0&0&0&0&0&{JzS/2} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q }\\\ {\mathop b\nolimits_q }\\\ {\mathop a\nolimits_{ - q} }\\\ {\mathop b\nolimits_{ - q} }\\\ {\mathop a\nolimits_q^\dagger }\\\ {\mathop b\nolimits_q^\dagger }\\\ {\mathop a\nolimits_{ - q}^\dagger }\\\ {\mathop b\nolimits_{ - q}^\dagger } \end{array}} \right)} $$ I then, as outlined in this paper (http://jmp.aip.org/resource/1/jmapaq/v19/i1/p277_s1) solve the equation det(H-x*{{1,0},{0,-1}})=0 for x, where H is the 8x8 matrix above and 1 is the 4x4 identity matrix. This procedure gets the right answer for the energy x, as expected. However, if I transform sums over $ \mathop b\nolimits_{ - q}^\dagger \mathop b\nolimits_{ - q} $ to $ \mathop b\nolimits_q^\dagger \mathop b\nolimits_q $ and similarly for a_q, I get this hamiltonian: $$ \sum\limits_q {\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q^\dagger }&{\mathop b\nolimits_q^\dagger }&{\mathop a\nolimits_{ - q}^\dagger }&{\mathop b\nolimits_{ - q}^\dagger }&{\mathop a\nolimits_q }&{\mathop b\nolimits_q }&{\mathop a\nolimits_{ - q} }&{\mathop b\nolimits_{ - q} } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {JzS}&0&0&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}\\\ 0&{JzS}&0&0&0&0&{JS\mathop \gamma \nolimits_q /2}&0\\\ 0&0&{0}&0&0&{JS\mathop \gamma \nolimits_q /2}&0&0\\\ 0&0&0&{0}&{JS\mathop \gamma \nolimits_q /2}&0&0&0\\\ 0&0&0&{JS\mathop \gamma \nolimits_q /2}&{JzS}&0&0&0\\\ 0&0&{JS\mathop \gamma \nolimits_q /2}&0&0&{JzS}&0&0\\\ 0&{JS\mathop \gamma \nolimits_q /2}&0&0&0&0&{0}&0\\\ {JS\mathop \gamma \nolimits_q /2}&0&0&0&0&0&0&{0} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\mathop a\nolimits_q }\\\ {\mathop b\nolimits_q }\\\ {\mathop a\nolimits_{ - q} }\\\ {\mathop b\nolimits_{ - q} }\\\ {\mathop a\nolimits_q^\dagger }\\\ {\mathop b\nolimits_q^\dagger }\\\ {\mathop a\nolimits_{ - q}^\dagger }\\\ {\mathop b\nolimits_{ - q}^\dagger } \end{array}} \right)} $$ But solving the equation det(H-x*{{1,0},{0,-1}})=0 (where H is the new 8x8 matrix above and 1 represents a 4x4 identity matrix) does not give me the same answer. I'd really like to know what is going on. Thanks!


title: Very short ($\ell \ll \lambda$) dipole as a receiving antenna

text: I'm trying to understand how a very short dipole of length $\ell \ll \lambda$ works for _receiving_ radiation. (It is center-fed, and has two thin perfectly conducting arms each of length $\ell/2$ separated by a negligible distance.) Say I have such a dipole oriented along the direction of the spatially uniform incident electric field $\vec{E}\left(t\right)$, and at all times in the region where the antenna is places, we have $$\vec{B}\left(t\right) \perp \vec{E}\left(t\right)$$ $$\left|\vec{B}\left(t\right)\right| = \dfrac{\left|\vec{E}\left(t\right)\right|} {c}$$ What will be the open circuit voltage $V\left(t\right)$ seen across the dipole? Do I need to specify $\vec{E}\left(t\right) = \vec{E}_0 \cos \omega t$ for a meaningful answer, or will a general $\vec{E}\left(t\right)$ do? My first guess would be $\displaystyle V\left(t\right)=\int_{\left(0,-\ell/2\right)}^{\left(0,+\ell/2\right)} \vec{E}\left(t\right)\cdot d\vec{r} = \ell\;\left|\vec{E}\left(t\right)\right|$, but I have two confusions:   1. The integral above would be the _potential difference_ between the tips of the dipole, but given that the electric and magnetic fields are time varying, I understand that the scalar potential itself becomes meaningless. Is this expression still usable for EMF anyway? Or does the fact that $\ell \ll \lambda$ make the potential difference still usable--and if so, why?   2. Given that the arms of the dipole are perfect conductors, is the potential difference between the two ends of any arm zero? Should I therefore look at the potential difference between the _midpoints_ of the two arms rather than the _tips_? I will then get $$\int_{\left(0,-\ell/4\right)}^{\left(0,+\ell/4\right)} \vec{E}\left(t\right)\cdot d\vec{r} = \frac{\ell} 2 \;\left|\vec{E}\left(t\right)\right|$$ Or does the radiation resistance of the arms come into play somehow? Thanks ...


title: Magnetic field in a cavity

text: We are given an infinitely long cylinder of radius $b$ with an empty cylinder (not coaxial) cut out of it, of radius $a$. The system carries a steady current (direction along the cylinders) of size $I$. I am trying to find the magnetic field at a point in the hollow. I am told that the answer is that the magnetic field is uniform throughout the cavity. and is proportional to $d\over b^2-a^2$ where $d$ is the distance between the centers of the cylinders. _**Attempt_** : I have found by using Ampere's law that the magnetic field at a point at distance r from the axis in a cylinder of radius $R$ carrying a steady current, $I$, is given by $\mu_0 I r\over 2\pi R^2$. So I thought I would use superposition. But what I get is ${\mu_0 I \sqrt{(x-d)^2+y^2}\over 2\pi b^2}-{\mu_0 I \sqrt{(x)^2+y^2}\over 2\pi a^2}$. However this is not the given answer!


title: Whether $m$ in $E=mc^{2}$ and $F=ma$ are both relativistic mass?

text: I know that $m$ in $E=mc^{2}$ is the relativistic mass, but can $m$ in $F=ma$ can also be relativistic? If the answer is yes, then can you tell me whether this equation is valid $E=\frac{F}{a}c^{2}$? If not, can you tell why this is not valid? Advance thanks for your help and please forgive me my english as it is my second language.


title: Escape velocity to intersection of two gravitational fields

text: > Find the minimal velocity needed for a meteorite of mass $m$ to get to earth > from the moon. Hint: the distance between the center of earth and the center > of moon is $\approx 60 R_E$, and the meteorite should reach a certain point > $O$ on that distance, where the gravitational forces of the moon and the > earth are canceling each other out. It is located $\approx 6 R_E$ from the > center of the moon. I thought that the best way to solve this question is to find the difference between the initial enegy of the body and the final one at $O$. Beyond that point, the gravitational pull of the earth will 'overcome' the gravitational pull of the moon, and the object will gather speed itself by reducing the GPE in the earth's gravitational field. Therefore: $U_{G,i}=-G \frac{M_{moon}m}{R_{moon}}$ $U_{G,f}=-G \frac{M_{moon}m}{6R_E}-G \frac{M_E m}{54R_E}$ $U_{G,f}-U_{G,i} \approx m \cdot 1.51 \cdot 10^6 \text{J}$ (there's no mistake in the calculation) $E_k=m \frac{v^2}{2}=m \cdot 1.51 \cdot 10^6$ Therefore $v \approx 1739.97 \text{m/s}$ However this answer is wrong. The correct magnitude of the speed should be somewhere around $2.26 \text{km/s}$ which is close to the escape velocity from the moon ($\approx 2.37 \text{km/s}$). Where am I wrong? Why the difference in GPE is not satisfying the problem?


title: What should be the equation satisfied by The Momentum commutators in a curved background?

text: This may be obvious but I have limited experience in physics , The generators of Spatial translation symmetry commutes with each other i.e [P(i),P(j)] = 0 but if Spacetime is a curved manifolds then the value of the commutator should not be zero but some invariant property related to curvature i.e a Function of the curvature tensor If this is false then what should the commutator be like e.g in the vicinity of a gravitational source according to GR , I'm sorry I do not know much in relativity nor differential geometry


title: Is there a relativity-compatible thermodynamics?

text: I am just wondering that laws in thermodynamics are not Lorentz invariant, it only involves the $T^{00}$ component. Tolman gave a formalism in his book. For example, the first law is replaced by the conservation of energy-momentum tensor. But what will be the physical meaning of entropy, heat and temperature in the setting of relativity? What should an invariant Stephan-Boltzmann's law for radiation take shape? And what should be the distribution function? I am not seeking "mathematical" answers. Wick rotation, if just a trick, can not satisfy my question. I hope that there should be some deep reason of the relation between statistical mechanics and field theory. In curved spacetime, effects like particle production seems very strange to me, since they originate from the ambiguity of vacuum state which reflects the defects of the formalism. The understanding of relativistic thermodynamics should help us understand the high energy astrophysical phenomena like GRB and cosmic rays.


title: projective measurement & POVM

text: Let us consider the following completely positive map $\mathcal{B}(\mathbb{C}^n)\ni\rho\mapsto L\rho L^\dagger$, where $L\in\mathcal{B}(\mathbb{C}^n)$ is any arbitrary operator (and can have rank $>1$). The above map is a completely positive Neumark's dilation theorem this can be extended to a projective measurement. Consider $A_1=L L^\dagger$ (and assume $A_1\leq1$), and choose suitable positive semidefinite operators $A_j$'s such that $\sum_jA_j=I$ (identity matrix). By Naimark, we shall get a projective measurement given by a projection operator $P_j\in\mathcal{B}(\mathbb{C}^n\otimes\mathcal{H}_{aux})$, where $\mathcal{H}_{aux}$ is the auxiliary space of minimal dimension for such extension. The relation is given (with respect to a fixed auxiliary state $\rho_{aux}$ as: \begin{equation} \langle m|A_j|n\rangle=\sum_{r,s}\langle m,r|P_j|n,s\rangle\langle s|\rho_{aux}|r\rangle \end{equation} (The above formula is taken from Quantum Theory: Concepts and Methods by Asher Peres) $P_j$'s are not unique. They depend on the $A_j$'s we have chosen and also on the auxiliary $\rho_{aux}$. My question is the following: What will be the extension $\tilde{\rho}\in\mathcal{B}(\mathbb{C}^n\otimes\mathcal{H}_{aux})$ (of $\rho\in\mathcal{B}(\mathbb{C}^n)$) should be so that \begin{equation} Tr_2(P\tilde{\rho}P)=L'\rho L'^\dagger,\quad \text{where} \quad L'L'^\dagger=A_1. \end{equation} $Tr_2$ is the partial trace with respect to the second (auxiliary) system. (It will be nice if $L=L'$, but I guess it is too much to ask). My initial guess was $\tilde{\rho}=\rho\otimes\rho_{aux}$. But this does not seem to be true from a few examples. Please tell me, whether I am making any mistake as well. UPDATE/OBSERVATION: If we put $\rho=I$ (the identity matrix) then the second equation should give us $A_1$.


title: Are gravitational time dilation and the time dilation in special relativity independent?

text: There are two kinds of time dilation:   * One because the other clock moves fast relative to me (special relativity).   * Another one because the other clock is in a stronger gravitational field (general relativity), or accelerating rapidly (equivalence principle). So are these two effects are totally independent? Is it possible to derive the general relativity case from the special relativity case?


title: Why relative positions of stars and other objects in a galaxy do not change over the year

text: Though the identifiable stars in a constellation(say andromeda) and galaxy(say andromeda galaxy) are situated light years apart, why isn't the galaxy appearing at different positions(with respect to the stars) at different time of the year?


title: Newton's third law and General relativity

text: Is Newton's third law valid at the General Relativity? Newton's second law, the force exerted by body 2 on body 1 is: $$F_{12}$$ The force exerted by body 1 on body 2 is: $$F_{21}$$ According to Newton's third law, the force that body 2 exerts on body 1 is equal and opposite to the force that body 1 exerts on body 2: $$F_{12}=-F_{21}$$


title: Searching for a collaborator for a physics simulation of multi-party elections

text: I have completed a series of physics simulations on Matlab to find the equilibrium positions of parties in 2-D or 3-D voting space and compared them to the optimal positions that would provide the maximum average voter satisfaction. The physics aspects are in the use of forces that attract parties to voters. Would anyone suggest a physicist or other academician who would be interested in providing guidance and contributing to a paper? Some leading authors on sociophysics have turned me down because multi-party politics is a field dominated by political scientists, but I think this simulation would be something novel.


title: Calculating the electric relative permittivity of fluid or medium?

text: I'm unsure of how to calculate the permittivity of a fluid. Permittivity differs from one fluid to another: $$\varepsilon=\varepsilon_r\varepsilon_0$$ Since it is an electrical property combined with an electrical capacity, it is possible to measure it indirectly in a capacitive sensor. I have used a capacitive sensor to measure electric relative permittivity factor of a dielectric medium can be expressed as a ration of capacity, $C_x$ of capacitor, which space between and around the electrodes is completely filled with the medium, to capacity $C_0$ of the same electrodes in vacuum. $$\varepsilon_r=\frac{C_x}{C_0} $$ I know and measured $C_0$ value. I am facing problems with calculating $C_x$ of capacitor, which space between and around the electrodes is completely filled with the medium. I am using method as described below. I am applying an **AC** signal ($125\: \mathrm{kHz}$) to the capacitive sensor which is filled with some fluid, in response I am getting **AC** signal with some phase difference. I am able to measure the amplitudes of sensor input, sensor output signals and their phase difference also. I am trying to make equation that will give $C_x$ value from above known values ($V_g$ (input), $V_r$ (output), $\phi$ (phase difference)).


title: 1/r Counting Rate for Radiation Experiment

text: In an experiment performed in the lab, I want to justify that, when the width of the Geiger counter window is approximately the same size as the distance between the window and a mildly radioactive source of Sr-90, we have a $1/r$ dependence of the counting rate. I know that we should have $1/r^2$ dependence by the inverse square law, but how do I show that, if r is relatively small, we obtain $1/r$ instead? Thank you in advance.


title: Quantum Computation

text: Is there any rule or technique so that one can design quantum gate operator from matrix operator? Suppose, what will be the quantum gate operator for this matrix operator : $$ \left( \begin{array}{c c c} -1+2/8& 2/8& 2/8\\\ 2/8 & -1+(2/8) & 2/8\\\ 2/8 & 2/8 & -1+(2/8) \end{array} \right) $$


title: What kind of energy does superfluidity use?

text: Liquid helium (and other similar fluids) can "climb up" the walls of their containers. Who does the work in this case, and what kind of energy does it use? I'm sure we can't make a perpetuum mobile out of this, so I guess some kind of energy must somehow be expended to make the fluid "climb up" the wall.


title: some questions regarding Doppler shifting versus absorption-emission

text: I just got in what I thought was a silly exchange where a self-identified physicist states that the difference between "red-shifting" (in the Doppler sense) and the re-emission of light at longer wavelengths following absorption are only "semantically" different. I'm bad at GR and I expect my (classical) understanding of the Doppler Effect cannot _really_ account for the the red shift of light as its emitter moves away from the observer. But maybe it can? After reading this I think it makes sense to say that as a photon red shifts it doesn't really lose energy at all; the appearance of the energy loss illustrates that energy conservation only holds within an inertial frame. This also means that an ensemble of photons maintains any phase coherences indefinitely as they propagate through space (ignoring vacuum fluctuations.) I'm also bad at QED, so can someone explain whether the notion of a Feynmann diagram for the Doppler effect is trivial? I would figure that if there are no interactions -- the photon is simply propagating in spacetime -- then a Feynmann diagram would be featureless. Just a squiggly line on some vector? Not even a real QED problem? When matter absorbs a photon and then emits one at a longer wavelength, this Feynmann diagram would have to look more interesting than the Doppler shift diagram, right? The absorbed and emitted photons have different identities -- different squiggles on the diagram?