title: Shape of rotating rope (lasso problem?)

text: Let's take a wire or a rope. I usually do this with a chain or my scarf. I fixate one end in my hand and apply rotation (by subtle movements of this endpoint like spinning a lasso). The rope gets into rotation and obtains certain bent shape: ![enter image description here](http://i.stack.imgur.com/VO4DP.jpg) Some part is missing because cellphone cameras ain't great for high-speed photos but I hope you can imagine all the scarf. The question is: how can I calculate/predict this shape? Although this problem doesn't seem that bizarre, I have never seen any solution. Nor I have found this question asked anywhere on internet... Must be because I just don't know how to formulate it without pictures. Also by taking longer _rope_ I get more than one bend: ![enter image description here](http://i.stack.imgur.com/TEZcN.jpg) I apoligize for the quality again. It is even harder to rotate this while taking picture. The form is not spiral, it is more like a shape in a plane that's rotating. I'll be thankful for explanations, solutions, links or at least a correct formulation of this problem.


title: Would matter-antimatter annihilation create a fireball or not?

text: There has been a long discussion between me and Anna V on if the products of the annihilation will really cause a fireball to form and we haven't settled it yet. Our point here is that gamma rays from the annihilation are too energetic to be absorbed by the atmosphere in a short time or a little space to actually create a fireball. Also the pions will pass without much probability of hitting a target nucleus and dissipating energy in the air so it won't probably contribute to the explosion. In other words, if no fireball forms, then antimatter annihilation would not look like or have similar effects as nuclear detonations like fireball, blast wave and mushroom cloud but would rather be considered similar to something like a neutron bomb. One last thing, this question is **not** duplicate, I have read the other questions and answers and none of them gave me a useful answer. So what do you think ? is antimatter annihilation so different from nuclear detonations in the effects part ?


title: Show that detF(X,t) is positive in continuum mechanics?

text: I want to show that the determinant of the field $detF$ at the point $X \in B$ is positive, when the following motion. I think that time derivative of Jacobin is positive for $t > 0$. However, I cannot see how this helps. **How to start to show that the $det F(X, t) > 0$?** I think the answer of the other question in the exercise can be found by $X = \phi(x,t) = F(t)^{-1} x$ so probably the components are $F^{-1}_{1} = t/X_{1} -1$, $F_{2}^{-1} = (t - X_{2}) / X_{3}$, and $F_{3}^{-1} = (X_{3} - t) / X_{2}$. I used the normal inverse formula to count those things. **Is there any sense in those inverse components?** **Exericise 4.29 from Gonzalez in Continuum Mechanics:** ![enter image description here](http://i.stack.imgur.com/tSgqj.png)


title: Does a magnetically suspended frog feel weightless?

text: Magnetic levitation has been used to suspend frogs in midair. I was wondering: Does the animal still feel gravitational pull? I mean: Does the frog feel like an astronaut on the ISS, or does he feel like a trapeze artist suspended by a harness?


title: Rocket propelled by a giant monochromatic laser

text: I am preparing for my quals and stumbled across the following problem, and although it only requires undergraduate-level physics, I feel I can't piece everything together. "A rocket of mass $m_0$ is propelled by a giant monochromatic laser mounted on the back of the rocket. The laser emits a beam with a power of $P_0$ watts and a frequency $f_0$, both measured in the rest frame of the rocket. When the beam is turned on, the rocket is driven in the opposite direction by the recoil. (a) At $t=0$, the laser is turned on, with the rocket's velocity initially at rest in the Earth's reference frame. Calculate the instantaneous acceleration of the rocket. (b) If the rocket is moving at velocity $v$, what is the instantaneous beam power as measured in the Earth's reference frame? (c) The laser is kept going until the speed of the rocket reaches $v=0.9c$. What is the rest mass of the rocket at this point?" I work in units where $c=1$. For (a), in the Earth's reference frame, I get a momentum $p(t) = \gamma m_0 v + h f$, where $\gamma = \frac{1}{\sqrt{1-v^2}}$ and $f = f_0 \sqrt{\frac{1-v}{1+v}}$ because of the Doppler shift in frequency as the rocket travels further away. Differentiating w.r.t time, I get $$ \frac{dp}{dt} = \gamma^3 m_0 a - \frac{\gamma a}{(1+v)} h f_0 $$ This is as far as I get. Equaling $dp/dt=0$ for conservation of momentum make the $a$ cancel out, and there isn't any other time dependency to differentiate $v(t)$. For part (b), we can say that in the frame of the rocket, $P_0 = r h f_0$, where $r$ is the rate of emission of photons. In the frame of the Earth, we can therefore write $$ P(t) = \frac{dE}{dt} = \frac{d}{dt} \left( \gamma m_0 + r h f \right) = m_0 \gamma^3 v a - \frac{\gamma a}{(1+v)} P_0 $$ but since I don't have $a$, there's not much I can do. Also, I am not sure if $r$ needs to be corrected with a proper time factor when we boost between frames. Also, for part (c), my first intuition was to say that the rest mass was $m_0$, but now I am thinking that the rocket's total mass decreases because we need to account for the energy lost from the laser. I am slightly confused as to what I need to take into account.


title: Geometric Langlands as a partially defined topological field theory

text: I have heard from several physicists that the Kapustin-Witten topological twist of $N=4$ 4-dimensional Yang-Mills theory ("the Geometric Langlands twist") is not expected to give rise to fully defined topological field theory in the sense that, for example, its partition function on a 4-manifold (without boundary) is not expected to exist (but, for example, its category of boundary conditions attached to a Riemann surface, does indeed exist). Is this really true? If yes, what is the physical argument for that (can you somehow see it from the path integral)? What makes it different from the Vafa-Witten twist, which leads to Donaldson theory and its partition function is, as far as I understand, well defined on most 4-manifolds?


title: reflection at intersection of 2 mirrors

text: Consider 2 mirrors which intersect at an angle. If we draw a light ray intersecting the mirrors at their point of intersection, how will it reflect? What will be the normal at that point? ![enter image description here](http://i.stack.imgur.com/4nnNy.png)


title: Some more questions about the BCFW reduction

text: This question is a continuation of this previous question of mine and I am continuing with the same notation. One claims that one can actually split this $n$-gluon amplitude such that there is just a single gluon propagating between two $n-$point amplitudes and the $p_{n-1}(z)^{-}$ and $p_n(z)$ are on two sides. Define $q_{i,n-1}(z) = p_i + p_{i+1} + ...+ p_{n-1}(z)$ and define $h$ to be the helicity of the gluon when propagating out of the left amplitude. This is summarizied in saying that the following expression holds, $A(1,2,..,n,z) = \sum _{i=1} ^{n-3} \sum _ {h = \pm 1} A^L(p_i,p_{i+1},..,p_{n-1}(z),q^h_{i,n-1}(z)) \frac{1}{q_{i,n-1}(z)^2}A^R(p_n(z),p_1,p_2,...,p_{i-1},q^{-h}_{i,n-1}(z))$   * Is there a "quick" explanation for the above split and why the propagating gluon has to flip helicity? (..it seems to be way of putting in the helicity conservation at high-energies but I can't make it very precise..)   * In the above split shouldn't the sum be from $i=2$ since one can't get lower than $3$-gluon vertices on either side?  Now one can apparently write the momentum squared of the propagator in the following way, $q_{i,n-1}(z)^2 = q^2_{i,n-1} - z[p_{n-1}|\gamma_\mu q^\mu_{i,n-1}|p_n>$, where $q_{i,n-1}(0) = q_{i,n-1}$ and then apparently using the previous expression of $A(1,2,..,n,z) = \sum _{i} \frac{R_i}{(z-z_i)}$ one can re-write the amplitude as, $A(1,2,..,n) = \sum _{i=1} ^{n-3} \sum _ {h = \pm 1} A^L(p_i,p_{i+1},..,p_{n-1}(z_i),q_{i,n-1}^h(z_i)) \frac{1}{q_{i,n-1}^2}A^R(p_n(z_i),p_1,p_2,...,p_{i-1},-q_{i,n-1}^{-h}(z_i))$ where $z_i$ is such that $q_{i,n-1}(z_i)^2 = 0$   * I would like to know how the above expression for $A(1,2,..,n)$ was obtained. (..it looks like Cauchy's residue theorem but I can't make it completely precise..)


title: Multiqubit state tomography by performing measurement in the same basis

text: For a $n$-qubit state $\rho$ we perform all projective measurement consisting of one-particle measurements in the same basis, that is, $$p_{i_1i_2\ldots i_n}(\theta,\varphi) = \text{Tr}\left \\{ \rho [P_{i_1}(\theta,\varphi)\otimes P_{i_2}(\theta,\varphi) \otimes \ldots\otimes P_{i_n}(\theta,\varphi)] \right\\},$$ where $P_0 = |\psi(\theta,\varphi)\rangle\langle \psi(\theta,\varphi) |$ and $P_1=\mathbb{I}-P_0$. Does it suffice to recover the state? If so, how can one do it?


title: Understanding frequency of a signal

text: I'm trying to understand how can one transmit different signals on a same copper wire using different frequencies. How does the electrons actually interact with the frequency? And how does the frequency relates to the speed of a connection?


title: Confused about relating spring force to maximum distance

text: I keep working through this problem and getting the wrong answer: > A spring with spring constant k is positioned vertically and then compressed > from its equilibrium length by a distance Δy. A ball of mass m is placed on > top of the spring, and is launched into the air when the spring is released. > The ball travels to a maximum height hmax. Using the Work-Kinetic Energy > theorem, determine hmax in terms of the given variables and any other > constants you deem necessary. The thought process was ∆K = $K_f - K_i$ = W --> $\frac{mv^2}{2}-\frac{mv_0^2}{2}$ = W $W$ = $W_s - W_g$ where $W_s$ is a spring and $w_g$ is gravity The final velocity is also zero so $K_f$ = 0. Which gives us $\frac{mv_0^2}{2} = -k∆y^2 - mgh_m$ (solve for $h_m$) $h_m = \frac{-k∆y^2}{2mg}+\frac{v_0^2}{2g}$ I might be completely off but I can't figure out what else to do. I would be very grateful is someone walked me through it ![enter image description here](http://i.stack.imgur.com/19aGI.png)


Is it possible to imagine micro blackhole set creating hawking radiation soliton convergence that forms new micro black hole? If so, is there a configuration of micro black hole that would loop?


title: Terminal velocity of two equally shaped/sized objects with different weights?

text: I've tried doing research on this and there are some similar questions. However, they have slightly different scenarios and to make sure I understand things perfectly, I wanted to word it into my own words and confirm that my thought process is correct. I am just getting back into physics after a decade of not doing anything remotely related, so I'd like to make sure I'm not confusing things. We have two objects that are the exact **same size and shape**. Let's call them **A** and **B**.   **A** is **twice the weight** of **B**. Both are dropped from an airplane at the same time. Am I correct to say that since **B** is **half the weight** of **A** , it will reach its terminal velocity much sooner, producing 0 acceleration from there on. **A** , on the other hand, will continue accelerating until the positive force from the air resistance equals its weight, at which point it reaches its terminal velocity and 0 acceleration. **A** will have a **higher terminal velocity** and reach the ground sooner than **B** , because **A** was accelerating for a longer period of time. **EDIT:** Changed from mass to weight, due to the fact that weight is the force pushing those two objects towards the Earth.


title: What is the best explanation for the fine tuning of the Universe?

text: In my view, this is one of the most profound unsolved mysteries of Science. How to explain the fine tuning of the Universe without recourse to saying that a God/Designer did it. Examples of fine tuning: We exist because of a crucial one part in a billion difference that favors matter over antimatter. If the ratio of the nuclear strong force to the electromagnetic force had differed by 1 part in 10 to the 16th power, no stars would have formed. If the strong force constant were 5% weaker, there would be no stable stars If the strong force constant were 2% stronger, there would be no stable hydrogen, no long-lived stars, no hydrogen containing compounds If the expansion rate of the Universe were larger, no galaxies would form If the expansion rate of the universe were smaller, the universe would collapse even before starts formed If the mass density of the Universe were larger, overabundance of deuterium from the big bang would cause stars to burn rapidly, too rapidly for life to form If the mass density were smaller, insufficient helium from the big bang would result in a shortage of heavy elements I could go on, but lets cut to the chase. How do you account for this: (A) Inflation (B) Multiverse (C) Top Down Cosmology (D) God/Designer (E) Some new innovation you're working on or have that can win a Nobel Prize in Physics Let's be clear on the options. Either we have some law that forces the fine tuning, which does not exist unless you know otherwise, or we put it to chance, which is obvious nonsense given the math, or it's a designer. Unless you have a fourth option?


title: Modelling a matter dominated universe collapsing into a black hole

text: With the FLRW equations we can get solutions for a matter dominated closed universe in which the finale is an ultimate collapse, but this is only in terms of $a$ (the scale factor) and $t$ (time) and some $q_0$ and $\theta$: $a = {\frac {q_0}{2 q_0 -1 } } (cosh (\theta) -1) $, (1) $t = {\frac {q_0}{2 q_0 -1 } } (sinh(\theta) - \theta)$ (2) where $dt=a d\theta$ Unfortunately, the FLRW does not show that the model should collapse into a black hole. So it only tells you when things might end depending on $q_0$. So I will appreciate if someone can give me an opinion of whether the following solution is appropriate for a universe that has collapsed into a black hole. Please feel free to rip it apart if necessary. And secondly, how can the final black hole equation be factored into the FLRW solution to get a complete model? Here it goes. First the energy of the matter is simply: $E_{m}= M c^2$ (3) The gravitational potential energy $U$ of the universe is (see http://arxiv.org/abs/1004.1035): $U = - \frac {3 G M^2} {5 R}$ (4) For a particle away from the event horizon the Newtonian gravity is simply: $g r^2 = G M $ (5) where $r$ is the radial coordinate of the observer. If I assume that $U = E$ once the universe has completely collapsed then I get, $Mc^2 = - \frac {3 G M^2} {5 R}$ (6) Substituting (5), I get: $c^2 = - \frac {3 r^2 g}{5R}$ (7) And finally, $R = - \frac{3}{5} \frac {r^2 g}{c^2}$ (8) Now it turns out that the formula for the Schwarzschild radius in Newtonian gravitational fields is remarkably similar http://en.wikipedia.org/wiki/Schwarzschild_radius): $R_s = 2 \frac {r^2 g} {c^2}$ (9) The obvious issue here is that I got the constant $-3/5$ instead $2$. So the real question now is, are these computations correct and there is a reason for the difference? (for example, a supermassive BH Universe is not the same as your conventional BH). Or is it this a completely wrong approach that yields a completely wrong answer? If wrong, what is the right answer? Note that I tried to avoid solutions with $M$ and $G$ because that results in absurd answers such as those arrived in If the universe were compressed into a super massive black hole, how big would it be? (i.e., the radius of the BH would turn out larger than the current radius of the Universe, etc, etc).


title: Why does blowing on a candle put it out but sucking doesn't?

text: Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air? Ok, so forget the human factor involved in _blowing out candles_. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends. Why does this happen?


title: Bra-ket notation, Bits, & Superposition

text: I am a quantum computing enthusiast, and recently I stumbled upon this the following two propositions: $$ \alpha|1\rangle + \beta|0\rangle$$ What does this mean? My understanding of this is that: the two bits, 1 and 0 are represented in a state of superposition, hence the bra-ket notation (which is commonly used for quantum mechanics), i.e., this is a qubit. Or is there a more concise explanation of this? Also: $$(\alpha|1\rangle + \beta|0\rangle)^N$$ What does it mean to raise this quantity (of superimposed bits, i.e., qubit) to the $N$th degree? If we take $2^N$ where $N$ is the number of qubits then this tells us the number of bits in the desired number of qubits. Is what I have stated in this post, generally correct?


title: Pleiades through a 15x 76 mm telescope

text: How clearly can the stars of the Pleiades cluster be seen through a 15x telescope (for example, FunScope)?


title: Does Mercury have a balmy spot?

text: From Wikipedia: > Although the daylight temperature at the surface of Mercury is generally > extremely high, observations strongly suggest that ice exists on Mercury. Does that mean there could be a spot on Mercury where a person could stand and it would be a balmy 80° F (27° C)? Obviously, let's ignore any radiation issues, etc.


title: Time reversal invariance and boundary conditions in electrodynamics

text: This is really several related questions... The equations of classical electrodynamics are time-reversal invariant. However, when we solve the equations for a particular system of charges it is common to take the solution with only retarded fields. The reason given for this choice is usually so that the solution agrees with our notion of causality, but this choice of boundary conditions is clearly not time-reversal invariant. 1) According to Feynman (Lectures on Physics, Volume II, chapter 20) taking the theory with time-reversal invariant boundary conditions 'in many circumstances does not lead to physically absurd conclusions'... but this implies than in some circumstances it does -- i.e. it should be possible to test via experiment that the choice of retarded fields only is the physically 'correct' one. What would such an experiment look like? 2) If such an experiment exists isn't it therefore wrong to say that the electromagnetic force is time-reversal invariant? 3) Is the time-reversal invariant theory Feynman mentions the same as Feynman- Wheeler absorber theory? 4) Is the same thing true in Quantum Electodymanics? That is, is there a condition equivalent to fixing the boundary conditions in the classical theory where only retarded solutions are considered in the calculation? And if so, does an alternative with time-reversal invariant boundary conditions exist? And if it does, does the distinction lead to experimentally testable differences? Even if you can answer only one or a few of these questions it would be very much appreciated! **Update:** I worry this question is already too long, but I'll try to be more specific about what I mean by time-reversal invariance. Start with some distribution of (classical) charges with their initial velocities and accelerations specified, and assumed to be isolated from the rest of the universe. The Feynman-Wheeler theory predicts their subsequent motion without any reference to fields. Then, at some time $T$ later measure all the new (positions, velocities, accelerations), reverse all velocities, and take this as a new initial condition for a second isolated system. Time-reversal invariance means that the dynamics of the second system should just be exactly the reverse of the first system (so that if I measure the state of the second system a time $T$ later and reverse all velocities again I get back to the initial condition of the first system). I'm pretty sure this is true in the Feynman-Wheeler theory, but if I use the usual theory of electrodynamics and solve for both systems considering only retarded fields I guess it will be false. Therefore, in principle at least it is possible to tell the difference between the two theories experimentally; however, I'm not sure this is doable or has been done in practice.


title: Sound of packing tape dispensation?

text: I am sitting in a cigar lounge, and under the jazz music playing I can hear this strange, rising and falling tone, that sounds similar to one of those _weird tubes_ you can find at a fun fair. You tilt the tube this way, and tip it that way, and a weight moves in the tube and creates a _weird tube_ sound. The pitch seems generally to descend or ascend (with some local irregularities) depending on which direction the pipe is tilted and what it's previous state was. The guy packing boxes in the corner with the **adhesive packing tape** seems to be creating a sound similar to this as he draws out long sections of tape from the roll. I am wondering if there is a physical explanation for the sound made by the dispensing of packing tape, and what this explanation would be?


title: What causes insects to cast large shadows from where their feet are?

text: I recently stumbled upon this interesting image of a wasp, floating on water: ![enter image description here](http://i.stack.imgur.com/Sa7bm.jpg) Assuming this isn't photoshopped, I have a couple of questions:   1. Why do you see its image like that (what's the physical explanation; I'm sure there is an interesting one)?   2. Why are the spots surrounding the wasp's legs circle shaped? Would they be square shaped if the 'feet' of the wasp were square shaped?


title: Does the molten salt used at projects such as Gemasolar need to be replaced after some cycles?

text: The project at Gemasolar uses molten salt to store heat. This heat is then used with water to drive steam turbines. Over time does this kind of repeated heat/cold cycle make the molten salt a less effective medium to store heat? How would the old medium/waste be disposed of?


title: Definition of Free Electrons and Mobile Charges?

text: Could someone please give me a good definition of the following electric terms? Despite what searching I have done, I have not come across a definition that I have found clear for me to understand:   * Free electron   * Mobile charge


title: Nuclear reactor control rods

text: What is the relation between control rods surface exposed into a nuclear reactor and neutron energy? Is it linear? I mean, how do neutron absorbing rate change with the progressive immersion of control rods into the core?


title: Which way do spiral galaxies rotate?

text: Is it known whether spiral galaxies typically (or exclusively?) rotate with the arms trailing or facing? Intuitively it feels weird to think of the arms as facing the direction of rotation, but that's silly, of course -- it seems to have its origin in an instinctive assumption that some sort of friction with the intergalactic vacuum would make it hard for the galaxy to rotate against the barbs, which is nonsense. I assume that for galaxies that we don't view head-on, we can measure rotation curves spectroscopically. But that's not enough; it must also be possible to determine which end of the minor axis is the far one. Dust in front of the bulge can help with that -- but in the images I have seen where it is clear which way the galaxy is tilted, it's not easy to see which way the spiral curls. Still, I suppose there must be some galaxies where both the spiral and the rotation can be determined. What's the verdict for them?


title: Determining cloud cover from observer to near the horizon

text: Does there exist a Clear Sky Chart with the following enhancements?: 1 - Actual Cloud Cover (Offered Visually and not just Colors with a Legend, Over Time/Past & Predictive) 2 - Simulate/Predict Cloud Cover taking into account the direction from Observer to Observed Object and Angle of view \- Close to the horizon (May be helpful to know when you can reasonably start/end tracking something you want to catch that night that is close to the horizon) The reason I'm curious is: A) I wonder if it's just not feasible for any/many reasons. B) It would be great help to know this information. In general, does anyone know of other Earth Weather, Clear Sky Clocks and Charts or anything else that gives more information?...anything related will be helpful. EDIT: I would love to find this lecture "You can do better than Clear Sky Chart" mentioned: http://stjornuskodun.blog.is/blog/stjornuskodun/entry/966941/


title: Axial Tilt and precession rate of exoplanets

text: The Earth's axis is 23.5 degrees away from othogonality to the ecliptic, and it takes about 26 000 years for it to precess fully. I have neither an intuitive sense nor the formula for precession frequency in front of me for how the latter depends on the former (and other things). So... A. What is the general formula for gyroscopic precession? I remember working it out as a function of moment of inertia and torque, but it was fiddly and grad mechanics was seven years ago. B. What is the formula substituting gravity for the force and a reasonable moment of inertia for a planet? C. Is a uniform density sphere a good enough approximation for B? D. Do you know of any empirical measurements of any of the above, for any exoplanets? Particularly, any _really fast_ precessors? E. Any bigger conclusions we can draw from the above? See Is it easier to learn more about the seasonal changes in an exoplanet's atmosphere when the exoplanet orbits a binary star system?


Are meteorites igneous, sedimentary or metamorphic rocks? Or do we need a new method of rock classification for them?


title: Where does 1/Gamma characteristic angle come from in EM Radiation?

text: Very curious as to where this angle comes from? It describes the peak of radiation for almost all radiation regimes, but I am having a difficult time seeing where it comes from. Also, the physical significance of it. Thanks in advance!


title: Need help with relativistic dynamics

text: I understand the concept, but I'm having a hard time applying the consequences of conservation (energy/momentum). For example: A proton with kinetic energy 437 MeV hits a proton at rest elastically and as result both protons rebound with equal energies. What is the included angle between them? Attempt at a solution: Since the collision is elastic the masses remain the same. I'm also assuming that the velocities of the protons form the same angle with the incident proton in order to preserve momentum in the direction perpendicular to the incident proton. Writing up the four-vectors for the system (particle 2 being the target, primes are post event, $\phi$ is the angle from the incident proton's path, T is the kinetic energy) $$ P_1 = (mc^2+T_1,|\vec{p_1}|,0,0) \hspace{4 mm} P_2 = (mc^2,0,0,0) \\\ P'_1 = (E'/2c,|\vec{p'_1}|\cos{\phi}, |\vec{p'_1}|\sin{\phi},0) \\\ P'_2 = (E'/2c, |\vec{p'_1}|\cos{\phi}, -|\vec{p'_1}|\sin{\phi}, 0) $$ If I postulate that ${P_1 + *{P_2}*= P'_1+P'_2}$ then this yields $$ 2|\vec{p'_1}|\cos{\phi} = |\vec{p_1}| = T_1/c $$ but this is where things go pearshaped (or at least when I realize that they have done so much earlier). (** are edited sections)


title: Are there any systems in physics which can only be formulated as an integral equation?

text: My question is are there any systems in physics that can only be formulated as an integral equation? Or do all integral equations have an equivalent differential equation?


title: Intuition behind Faraday's Law?

text: Faraday's Law seems more like an observation than an explanation. Sure, a changing magnetic current causes emf, but _why_? How does a changing magnetic field cause electrons to move in the direction of a wire? Assume that all you know is Bio-Savart's Law and $\vec{F} = q\vec{v} \times B$ I'm hoping for an explanation similar to how a battery generates a field that causes the electrons to move, which causes surface charge buildup and forces the electrons to move in the direction of the wire.


title: Why is radiation for an ultrarelativistic charge zero on axis?

text: I attribute it to the fact that for an ultrarelativistic charge the field is contracted and essentially there are only fields in the transverse direction and nothing longitudinally (wrt the charges velocity). Any thoughts?


title: Double slit experiments - basic pondering

text: I have some ponderings about the double slit experiment(s) and mainly their interpretations. Some of my questions may exhibit insufficient scholarship so please bear along, or please point out the fallacies wherever they are instead.   1. Why is it commonly said that it is impossible to know in advance which slit any single particle will go through, and all that can be said is that there will be a probability distribution that can describe many particle's aggregate behavior. Isn't it more humbly correct to say that we just don't know how to predict that?   2. Say that the experiment is held with electrons. And photons are used to detect whether an electron passed through one of the slits. And of course, this detection collapses the wave function etc. What however should be the effect of the photons on the electron? is an electron totally and entirely oblivious to having interacted with a photon? is only the photon affected by an electron (in the classical model to start with) and not vice versa?   3. When the particle beam is being shot, or a a single particle is shot at a time, what can be said about the direction of the particles or the angle of the beam? are those of a probabilistic nature, and what would happen if they are not probabilistic but rather if the direction of each particle is known in advance? this is probably the most straightforward to answer of my questions. Thanks! I just find it very interesting, and yet in a sense basic.


title: What is meant when it is said that the universe is homogeneous and isotropic?

text: It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise because of this?


title: Shapes of galaxies

text: 1. I've heard most of galaxies are spiral or ellipsoid shaped. Is it true?    2. If true, then why they form in such shapes?    3. How did arms of the spiral galaxies form?


title: Idea is to make an equipment like torch, which should emit darkness.. Is it possible?

text: Can we have any equipment as such? If light can emit form a hand held device, why not darkness


title: Do all the individual stars that we can see in the night sky belong to Milky Way?

text: I was wondering if all the stars that we can see with the unaided eye as distinct point sources are from our own galaxy? In other words, can we see stars from the Andromeda Galaxy or other galaxies without telescopes?


title: How does the average Mars-Jupiter distance compare with the average Earth-Jupiter distance?

text: One thing is certain: Mars can be further from Jupiter than Earth can ever be (when they're on opposite sides of the solar system), but Mars can also be closer from Jupiter than Earth can ever be. And does this carry any physical implications?


title: Beta-function non-zero at classical level?

text: In Jaume Gomis's lecture 5 on CFT at Perimeter Institute, he says (at 27:40 minute mark) that the beta function, _classically_ , of the $m^2$ parameter in massive $\lambda \phi^4$ theory is $$\beta(m^2) = -2m^2\,.$$ The intuitive reasoning he gives is that since the dimension of $m^2$ is 2, then the beta function should be -2 times m-squared. I always thought that the beta functions for parameters of the Lagrangian only start at one-loop and are classically zero. What is the rigorous definition of a beta function (including the classical part)?


title: What are the minimum telescope requirements to view a nice spiral galaxy?

text: I would like to view some spiral galaxy. However, I would prefer not to spend lots of $$$. What are the minimum specs a telescope must have to resolve detail in a spiral galaxy?


title: If An Object Explodes With A Force, What Force Are Fragments Given?

text: So let's say for example I have an object with 5kg of mass. It explodes with a force of 500N. The object fragments into four fragments: a 0.5kg, a 1kg, a 1.5kg and a 2kg object. **What force does each of those fragments have (and thus what acceleration)?** My assumption is that it will get a percentage of the force relative to the percentage of the mass. So the 1kg fragment is 1/5th of the original mass thus it gets 1/5th of the force?


title: Steady-state adiabatic nozzle, unknown exit velocity & temperature

text: The question is 2.10 taken from: Introductory Chemical Engineering Thermodynamics, 2E by Elliot,Lira > Air at 30ºC and 2MPa flows ata steady state in a horizontal pipeline with a > velocity of 25m/s. It passes through a throttle valve where the pressure is > reduced to 0.3MPa. The pipe is the same diameter upstream and downstream of > the valve. What is the outlet temperature and velocity of the gas? Assume > air is an ideal gas with a temperature-independent Cp=7R/2 and the average > molecular weight is 28.8. Since it is to be discribed as a steady state, and that we may assume that $\Delta E_{pot}=0$ since the pipeline is horizontal we end up with $\dot{m}\left [ \Delta H+\Delta E_{kin} \right ]=0$. Or simply: $$H_{2}-H_{1}+\frac{v_{2}^{2}-v_{1}^{2}}{2g_{c}}=0$$ But since both $T_{2}$ and $v_{2}$ are unknown I do not know how to solve for $H_{2}$. Therefore I would want to find the exit temperate using the following formula: $ T_{2}=(\frac{P_{2}}{P_{1}})^{(\frac{R}{C_{P}})}T_{1} $ to find the exit temperature. And I also know that the Volume and Velocity are closely related, and important in solving the problem. This since the volume will be higher for the same m when exiting compared to entering the nozzle. What would be a reasonable approach to this problem, and have I made any assumptions falsely? Continuation 1: Through $PV=nRT$ we find that $\frac{V_{1}}{n}=\frac{RT_{1}}{P_{1}}=\frac{8.31446JK^{-1}mol^{-1}*303.15K}{2*10^6MPa}=0.0012602642745\frac{m^3}{n}$


title: What free software is there for observing the sky (sky map software)

text: I used until now only `stellarium.org` and I'm curious if there is any other software that is better than stellarium. By better, I mean:   * doesn't have high system requirements   * requires very little power from the battery when running(as the normal setup would be going somewhere outside with the telescope and a laptop.. this may be hard to estimate, but maybe there is here someone who tried out more than one program and has seen a clear difference)    * easy to use   * has extra features (extra star catalogues, telescope control, etc) A big plus would be if that software is free for personal use!


title: Potential on a Charged Sheet

text: I know the field difference across a charged sheet-but what's the potential on a charged sheet? Is it just asymptotic? If we had a thin cylindrical sheet would it be the same? Zero field inside the cylinder, and jump to infinity on the sheet, and q/r outside?


title: If electrons are magnets do they attract each other?

text: It is said every electron is negatively charged and so they repel each other but if electrons are tiny magnets(which are responsible for atomic attraction and how solid magnet works) does it mean opposite electron poles attract each other?


title: Does thermal energy include the kinetic energy of the electrons?

text: Or is the kinetic energy of the electrons not counted towards the thermal energy? (In other words only the energy of the lattice structure is counted)


title: Geodesic curvature and Weyl transformations

text: The geodesic curvature is given by $$k=\pm t^a n_b\nabla_a t^b,$$ where $t^a$ is a unit vector tangent to the boundary of the string worldsheet and $n_a$ is an outward vector orthogonal to $t^a$. I don't understand why under the Weyl transformation $$\gamma_{ab}\rightarrow e^{2\omega}\gamma_{ab},$$ $t^a$ and $n_b$ transform as $$t^a\rightarrow e^{-\omega}t^a,~~~~~n_b\rightarrow e^{\omega}n_a.$$ Is this really as trivial as a normalisation? Also, what do "time-like boundary"$(+)$ and "space-like boundary"$(-)$ mean? I appreciate any discussion related to this. The geodesic curvature was somehow never mentioned in my GR class.


title: Wavy stream of liquid

text: When pouring a liquid into a glass some streams have a wavy shape, like the one in the following photo: ![example](http://i.stack.imgur.com/SpGs5.jpg) (Couldn't find a better picture, sorry.) What causes the stream to be of such a shape?


title: Electric Field and Magnetic field of Guassian beam

text: The center of the gaussian beam in glass has a beam power of 1 mW and a beam radius of 20 μm (micro meter). The wavelength of the light $\lambda_0$ is 1550 nm. How can I find the electric field E and magnetic field H?


title: Abrikosov and his fluxoids (why only one ?)

text: In superconductivity, when we solve the ginzburg-landau equations relative to a 2 dimensional plane immersed in a magnetic field, we find that a periodic lattice of vortices is a solution. When we look at the higher critical magnetic field, we suppose that the plane is covered with vortices each having inside a flux equal to an integer n times the quantum flux $h/e$ and we assume just $n=1$. I can understand intuitively why this is the less energical configuration (it's just easier to make some holes somewhere else), but does someone know how to show it explicitly (actually, I didn't know how to calculate the free energy associate without having the complete solution which is an Infinite serie...) or know some reference that proves why should this state with multi-fluxoids be unstable ? Thanks a lot if you can help !


title: Maxwell equation boundary conditions on a conducting sheet

text: I'm having difficulties solving boundary conditions for an infinitely thin conducting layer in a presence of an alternating field. I use the Maxwell equations:   $\nabla \cdot \mathbf B = 0$   $\nabla \cdot \mathbf D = \rho$   $\nabla \times \mathbf E = i \omega \mathbf B$   $\nabla \times \mathbf H = \mathbf j - i \omega \mathbf D$   The problem arises with the last two ones. For stationary fields the terms $i \omega \mathbf B$ and $i \omega \mu \mathbf D$ vanish, and I can use the "textbook" boundary conditions like: $\hat n \times (\mathbf H_1 - \mathbf H_2)=\mathbf j$   Do those equations get an additional term $-i \omega \mathbf D$ in the case of time-varying fields?   If they do, those terms are different above and below the sheet, and I am not sure what value should be used.   The current on the sheet $\mathbf j = \sigma \mathbf E$ turns out to be discontinued as well. What field (current) should be used as the field on the very boundary? I would assume it's the average of the fields directly above and below the boundary, but I'm mostly guessing and don't know an exact proof. I am deeply grateful for any help.


title: Time dilation as an observer in special relativity

text: I've been having a discussion regarding time dilation relating to special relativity and how it should be observed from the FoR (Frame of Reference) of "the person moving" : **I assert** \- If we have a time is slower in the moving FoR than when I observe the stationary frame of reference (earth) then I should perceive events as occurring faster, i.e., if I observe 15 years passing and 90 years pass in the stationary frame then I should observe events on earth happening at a rate of 6:1. So if I observe a 1 min event on earth I watch that entire event occur in 10 seconds. My thought process is similar to the idea that if I was near a massive object and looked out I would observe events further away as moving faster. My teacher seems to believe this line of reasoning is incorrect but will not (or cannot) give me the correct thought process. Hence my question is, "is my logic correct, and if not why?"


title: what is universality (with respect to SUSY and neutral kaon oscillations)?

text: I'm having trouble putting the pieces together. In SM, neutral kaon oscillation is heavily constrained. This means, roughly, that the squark mass matrices have to be diagonal. And this is called universality of soft parameters. What exactly is universality and why do we have universality in this situation? Furthermore, is having diagonal squark mass matrices sufficient? In a lot of SUSY theories, they go one step further and assume the scalar masses are the same for the first and second generations (pMSSM for instance). Why is this additional step necessary?


title: What was the major discovery on gravitational waves made March 17th, 2014, in the BICEP2 experiment?

text: The Harvard-Smithsonian Centre for Astrophysics held a press conference today to announce a major discovery relating to gravitational waves. What was their announcement, and what are the implications? Would this discovery be confirmation of gravitational waves as predicted by general relativity (even though Sean Carroll links to the Nobel website implying that G-waves was detected decades ago, while my book on GR (B. Schutz) says they're still looking...I'm confused) Also, regarding inflation theory, would this discovery confirm inflation or refute it? Or something else, a la string theory?


title: Can a neutron be created from pure energy

text: Is it possible to create a neutron out of pure energy, i.e. not by bringing a bunch of already-existing quarks together? (A quick calculation using E = mc2 shows the energy required would be about 1.5 e-10 J.) Would an equivalent antiparticle always need to be created?


title: Why does connecting a battery's positive terminal to the negative terminal of another battery not create a short circuit?

text: This is a question regarding the physics behind the observation. I have guessed the answer to the question, but I may be wrong, so I want to wait for the responses before posting it. Some major considerations:   1. A battery's positive terminal does have a positive potential. ie, a test positive charge will repel it and a test negative charge will attract it. Vice versa for negative terminal. From the paper below (Section 1.2.1), it seems abundantly clear that the battery will have positive and negative potential on respective terminals.   2. Given 'point 1', above, connecting the positive terminal of battery A to negative terminal of battery B will lead to current flow in the conductor.   3. If the potential difference only comes into effect when the circuit is closed using the terminals of the same battery (due to cell chemistry), then does that mean that 'point 1' will not be valid after this experiment (connecting positive terminal of A to negative terminal of B)?   4. If it is only a tiny quantity of current that flows, then will it ever stop? And what determines this quantity? (Equation would be nice) Related paper: http://www.astrophysik.uni- kiel.de/~hhaertel/Circuit/electric_circuit.pdf


title: Does an antiDeSitter space compensate the vacuum energy?

text: Sometimes I read here or there that a problem with the electroweak vacuum is that it implies a huge positive cosmological constant. On other side, I also read that AdS spaces are not useful anymore for real world observations because we need to get a small positive cosmological constant at the end of the day. I wonder, is it not possible to compensate one problem against the other? This is, having in the Lagrangian an anti-deSitter space with negative radius or order 246 GeV, and then breaking of the electroweak vacuum so that the final result is slightly positive.


title: Finding difficulties in taking continuum limit in nonlinear sigma model

text: I am learning nonlinear sigma model from **Assa Auerbach** 's book " **Interacting Electrons and Quantum Magnetism** " and getting some difficulties in taking continuum limit. I am following chapter 12: The continuum approximation. Let me formulate the problem first. Consider the Heisenberg antiferromagnet on $d$-dimensional cubic lattice: \begin{equation} H[\hat{\Omega}]=\frac{1}{2}S^2 \sum_{ij}^{}J_{ij} \hat{\Omega}_i \cdot \hat{\Omega}_j. \end{equation} where summation runs over first, second, third nearest neighbours. The prefactor $1/2$ is introduced because of over counting. Using Haldane's mapping and taking some approximations (discussed in the book), we get \begin{eqnarray} H \approx \frac{1}{2}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j - \frac{1}{4}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j (x_{ij}^\mu \partial_\mu \hat{n})(x_{ij}^\nu \partial_\nu \hat{n}) + \frac{1}{4}\sum_{ij}^{}J_{ij} \left[2\mathbf{L}_i \cdot \mathbf{L}_j - \eta_i \eta_j \left(\mathbf{L}_i^2 +\mathbf{L}_j^2 \right)\right] \end{eqnarray} or \begin{eqnarray} H \approx \frac{1}{2}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j - \frac{1}{4}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j (x_{ij}^\mu \partial_\mu \hat{n})(x_{ij}^\nu \partial_\nu \hat{n}) + \frac{1}{2}\sum_{ij}^{}J_{ij} \left[\mathbf{L}_i \cdot \mathbf{L}_j - \eta_i \eta_j \mathbf{L}_i^2 \right]. \end{eqnarray} where repeated indices $\mu,\nu=1,\dots,d$ are summed over. The author takes the continuum limit using Equation (12.17) [Page 132 in the book]: \begin{equation} \sum_{i}^{} F_i \; \rightarrow \; a^{-d}\int d^dx \sum_{i}^{}\delta(\mathbf{x}-\mathbf{x}_i)F(\mathbf{x}) = a^{-d}\int d^dx \; F(\mathbf{x}) \quad (12.17) \end{equation} The continuum expression is given in Equation (12.18) [Page 132 in the book]. \begin{equation} H\approx E_0^{cl} + \frac{1}{2}\int d^dx\; \rho_s \sum_{\mu}^{} |\partial_\mu \hat{n}|^2 + \frac{1}{2}\int d^dx \int d^dx^\prime \left( \mathbf{L}_x\chi_{xx^\prime}^{-1}\mathbf{L}_{x^\prime} \right) \quad (12.18) \end{equation} The classical energy \begin{equation} E_0^{cl}=\frac{1}{2} S^2 \sum_{ij}^{}J_{ij}\eta_i\eta_j \quad (12.19) \end{equation} and the "stiffness constant" is \begin{equation} \rho_s = -\frac{S^2}{2dNa^d} \sum_{ij}^{}J_{ij} \eta_i\eta_j |\mathbf{x}_i-\mathbf{x}_j|^2 \quad \quad (12.20) \end{equation} where $N$ is the number of spins (or lattice sites). The inverse uniform susceptibility is \begin{equation} \chi_{\mathbf{x},\mathbf{x}^\prime}^{-1}= \frac{1}{Na^d} \sum_{ij}^{} J_{ij} \left[ \delta(\mathbf{x}-\mathbf{x}_i)\delta(\mathbf{x}^\prime-\mathbf{x}_j)-\delta(\mathbf{x}^\prime-\mathbf{x})\delta(\mathbf{x}-\mathbf{x}_i)\eta_i\eta_j \right]\quad \quad (12.21) \end{equation} I am facing problem in understanding how the continuum limit has been performed to get Equation (12.18). I have following technical and conceptual difficulties:   * In the stiffness and susceptibility, they still have summation over $i$ and $j$. Which lattice sums they have replaced? Do they introduced some other lattice sums using Kronecker delta function.   * How the prefactor in stiffness and susceptibility expressions comes?    * In n-field part, we have one integral so there is one $a^d$ in denominator of stiffness. In L-field part, we have two integrals but we still have one $a^d$ in denominator of susceptibility. How?   * I think the volume of unit cell, $a^d$, is same either we consider first, second or third nearest neighbours. Is it right? I stuck somehow to derive the Equations (12.18), (12.20), and (12.21). I would appreciate very much if someone illustrates by giving intermediate steps.


title: Net torque on a frictionless pulley

text: If we consider a system comprising a massless string over a frictionless pulley,then we write the torque equation as $(T_2-T_1)R=I a$ ,where $T_2$ and $T_1$ are tensions on either side of the pulley. The tangential force acting on the pulley is the friction $F$ between the pulley and the string. How is that the torque applied by friction is equal to the torque applied by the difference in tensions ? In other words how is the friction equal to the difference in the tensions ? If we consider the pulley and the string over it as one system such that the string does not slip then the net force acting is the difference in the tensions and net torque $(T_2-T_1)R$. But when we consider pulley in isolation then the force which applies torque is the friction between the string and pulley. Could someone help me understand mathematically how do we calculate net torque on the pulley by considering pulley as the system i.e how is friction $F =T_2-T_1$ ?


title: Understanding the partial trace and deriving $\langle l|R_{B}|k\rangle = \text{Tr}((\mathbb{I}_{A} \otimes |k\rangle \langle l|)(R_{AB}))$

text: By definition according to the notes I am looking through: The **partial trace** $\text{Tr}_A:L(H_A \otimes H_B) \rightarrow L(H_B)$ is the unique map that satisfies: $$\text{Tr}(L_B \cdot \text{Tr}_A(R_{AB})) = \text{Tr}((\mathbb{I} \otimes L_B)R_{AB})$$ for all $L_B \in L(H_B)$ and $R_{AB} \in L(H_A \otimes H_B)$. Now, according to the Wikipedia definition,$$T \in L(V \otimes W) \mapsto \text{Tr}_W(T) \in L(V).$$ Now if, in trying to relate the two definitions I let $W = H_A$, and $T = R_{AB}$ in the second definition, then I recover $\text{Tr}_A(R_{AB})$ However in the first definition I have the outer trace and $L_B$ - the first part which is $\text{Tr}(L_B \cdots$ Why are these different (at least seemingly)? $\leftarrow$ Question (1) Now I am trying to understand why, as in the subject line: $$\langle l|R_{B}|k\rangle = \text{Tr}((\mathbb{I}_{A} \otimes |k\rangle \langle l|)(R_{AB}))$$ First, I know that $\langle l|R_{B}|k\rangle = \text{Tr}(|k\rangle \langle l|R_{B})$, but I don't see how to get to the result. I think some of the confusion lies in that I don't know how to operate on $(\mathbb{I}_{A} \otimes |k\rangle \langle l|)(R_{AB})$. Since $R_{AB}$ is a matrix I don't know what to do with it in regards to the tensor product (I couldn't find or recognize a similar example with distributing a matrix over a tensor product). Question 2: How do we get $\langle l|R_{B}|k\rangle = \text{Tr}((\mathbb{I}_{A} \otimes |k\rangle \langle l|)(R_{AB}))$?


title: Meaning of dimension

text: I was wondering what dimension can mean in physics? I know it can mean the dimension of the space and time. But there is dimensional analysis. How is this dimension related to and different from the previous one? How is it related to and different from units (e.g. kilometer)?


title: Why is the energy density of gasoline so high?

text: We sometimes play a game in my family whereby we trace the energy for a device back to it's source:   1. The Xbox got power from the wall.   2. The wall got power from the local transformer.   3. The transformer got power from the coal plant via the power lines.   4. The coal plant got power from the turbines. ...etc. On and on until we've reached the core of the sun in dinosaur times. But my son stumped me the other day...what it is it in particular about petroleum (and gasoline) that makes it so high in energy density? Secondarily, I presume that gasoline has other qualities that make it a good fuel, e.g., not exploding spontaneously without significant energy. What are these other qualities? Finally, if energy input were not an economic consideration, and we only cared about energy density and the other Goldilocks factors, is there a better fuel? Or is gasoline not only the best actual fuel, but the best theoretical fuel?


Does magnetic geometry determine the scaling of a Polywell fusor? Forgive imprecise terminology here - by "magnetic geometry" I mean the configuration of the magnets, the configuration that creates the "Christmas star" geometry of the electron field in the fusor. On to my question... I've read that the rate of fusion scales at $r^5$, with $r$ being the radius of the reaction chamber. Is this because of the aforementioned magnetic geometry? Or is it because of something more essential, like Bremsstrahlung, or something else?


title: What are the consequences in high-energy of the non-interaction of the Higgs Field?

text: At high-energies when the Higgs field won't affect (interact with) particles, when the symmetry breaking won't occur, what would be $\rm W\pm$ or $\rm Z^{0}$ bosons speed if they would then have a $0$ rest mass? What about Fermions?


title: How can I explain why the weak nuclear interaction between individual nucleons is 'weak'?

text: > By considering the energy-time uncertainty principle, estimate the range of > the weak nuclear interaction at low energies. Compare this range to the size > of a typical nucleon (for example, a proton) and therefore explain why the > weak nuclear interaction between individual nucleons is 'weak'. I'm a bit stuck with this question: to estimate the range of weak nuclear interactions do I need to put the energy levels of a quark into the energy- time principle? Beyond doing this I am a bit lost! Any comments would be much appreciated; this is an assignment so the work has to be original.


title: Can I use imaginary time propagation for many-body problems?

text: There are various ways to numerically find the ground state energy and wavefunction of a many-body Hamiltonian. You can diagonalize the Hamiltonian and pick out the lowest eigenstate, or you use Lancoz. **My proposal is that can I use imaginary time propagation for many-body problems?** Make it simple, I multiply a trial wavefunction $$|\psi_0\rangle = \sum_{i\in\text{eigen}} w^0_i |i\rangle$$ by an operator $\exp(-\hat H \tau)$. Then we have $$|\psi_\tau\rangle = \sum_{i\in\text{eigen}} w^\tau_i |i\rangle$$ with $w_i^\tau = w^0_i\exp(-E_i\tau)$. For a sufficiently large $\tau$, we have $\exp(-\hat H \tau)\approx w_g^\tau \hat P_g$. The trial function will be projected to the ground state! Choosing a complete set of basis states, we can numerically calculate the operator by taking Taylor expansion of the exponential and iterate to the $n$-th order, finally we obtain a matrix. Now multiply the matrix on a trial function written in terms of the basis we chose, then normalize it, and we get the ground state wavefunction. **Will it be _accurate, stable and fast_?**


title: Calculate the stable height of a balloon

text: My son has asked me a question, and I'm afraid my high school maths are not up to the task: Is it possible to calculate the height that a given balloon (filled with a known quantity of helium) will attain when released, assuming it won't burst before? My line of reasoning is as follows (and may already be flawed, so please point out my errors):   * The balloon will ascend as long as the mass of the air that it displaces is heavier than the mass of the balloon. It will stop when both these masses are equal, so that's what we need to find out.   * We know the mass of the balloon and assume that it will remain constant throughout (no leaks).   * We know the air pressure on the ground. For simplification, we can assume that air pressure drops linearly (?) when ascending (no inversion weather etc.), so we can calculate the air pressure at any given height.   * The effects of gravity on the balloon are reduced the farther up it goes.   * The balloon expands when ascending because of the reduced air pressure, thereby displacing more air (which adds to its buoyancy), but this air is lighter (which takes away buoyancy). <\-- This part is giving me the most trouble. Are there any other factors that need to be taken into account? Shouldn't it be possible to calculate a height where equilibrium is reached and the height of the balloon remains stable? Is it as complicated as it looks to me?


title: momentum four vector and dirac matrices

text: $$c\left(\alpha _i\right.{\cdot P + \beta mc) \psi = E \psi } $$ From the above dirac equation it can be shown for zero momenta that spin and antimatter are associated with $\beta $. On the other hand, how are each of the momenta $P_x$,$P_y$,$P_z$ associated with $\alpha _1$,$\alpha _2$,$\alpha _3$? I mean does each $\alpha _1$,$\alpha _2$,$\alpha _3$ correspond directly with $P_x$,$P_y$,$P_z$ ? OR are they related to the Four-Momenta $P_4$ of Minkowski space, and can I write $P_4$ as $$ \left( \begin{array}{c} E/c \\\ p_x \\\ p_y \\\ p_z \end{array} \right) $$ I understand they are covariant, which follows naturally from the Dirac equation, but I just can't see how $\alpha _1$,$\alpha _2$,$\alpha _3$ operate to yield the momentum $P_x$,$P_y$,$P_z$.


What is the curvature scalar $\Psi_{4}$? Is it related to the scalar curvature $R$? What does its real and imaginary parts represent?


title: Could a variable capacitor divider replace a Variac?

text: Hmmm... You can definitely drop down the voltage, and ideal capacitors don't dissipate any power. So it seems, at first glance, that you could use a capacitor divider as a lossless voltage step-down device for AC. So you could use a cheaper variable cap divider as a replacement for a Variac? And it would be continuously variable, too, whereas the Variac is only variable in discrete windings. There has to be something wrong with this. What am I not thinking of?


title: Magnetic field when moving into the influence of a ferrous object

text: Suppose I have a magnetometer. I am on the Earth's surface, away from the influence of other magnetic sources. Assuming the magnetometer is aligned with the reference frame, it should read only the Earth's magnetic field vector at that point. Suppose now I walk towards a large ferrous object, for example, a large parked truck. As I approach the truck, what will happen to the magnetic field that I measure? Is the distortion to the magnetic field able to be modelled as additive as a function of position? If not, how could one mathematically describe the distortion? Are there any canonical references on the subject? * * * EDIT: In response to the answer(s) below, here are some additional details:   * Assume that I know what the magnetic field in the local geodetic frame _should_ be to within some desired tolerance, e.g. by using EMM2010 or similar.   * Ultimately, I do want to calculate attitude, but I have other sensors at my disposal (IMU and GPS) where heading is observable under motion. My desire to to "learn" the magnetic deviation on-the-fly to help stabilise the heading during near-stationary periods. My intuition says that the deviation should be observable when yaw is observable. But I need to start with some sort of model as to what the deviation looks like :)


title: What is the gist of the equations on Alejandro Guijarro's "2/14" "Momentum" blackboard?

text: The artist Alejandro Guijarro has an exhibit of photographs of a variety of blackboards with physics equations, drawings and text on them, as covered at Quantum Chaos on Display in Top Physicists’ Chalk Scrawls | Raw File | WIRED What is the one in the first closeup photograph (numbered "2/14") about? The circled text "non SUSY" suggests that at least one of the topics is Supersymmetry (Wikipedia), but what is the gist of it? For clarity, this question is about the blackboard in the first closeup image on the Momentum exhibit web site, labeled "2/14". ![enter image description here](http://i.stack.imgur.com/YhrRI.jpg) Caption: $\uparrow$ Click image to enlarge it. For example, the equation in the upper left starts with > $T_{ds} \sim 1/R$ For extra credit - where is the blackboard / who wrote on it / etc? (Note that I am also interested in the other blackboards, and suggest separate questions here for each of those. There is some commentary on some of the others at The Beautiful Blackboards at Quantum Physics Labs | MetaFilter)


title: Electric field for two adjacent infinite layers of width h charged + and − uniformly

text: I'm trying to figure how to calculate the electric field created by two adjacent infinite layers of width h charged + and − uniformly. I don't understand why isn't the field outside of the layers isn't simply zero. Why aren't the 2 cancelling each other? And how can I calculate the field anywhere between the two using gauss law? Thanks


title: Relativistic Elastic Collision

text: I am having trouble getting my head around the transfer of energy in a relativistic elastic collision. My understanding of a _relativistic elastic collision_ is one in which the total rest mass on each side of the equation is unchanged, i.e: $$\sum_{\text{before}} m_{i}=\sum_{\text{after}} m_{j}$$ If we have a particle of rest mass $M$ and relativistic energy $E$ colliding with a particle at rest with rest mass $m$, then we have by conservation of 4-momentum: $$\mathsf{P}_{1}\cdot\mathsf{P}_{2}=\mathsf{P}_{1}'\cdot\mathsf{P}_{2}'$$ Initially we have: $\mathsf{P}_{1}=(\frac{E}{c},p,0,0)$ and $\mathsf{P}_{2}=(mc,0,0,0)$, and therefore: $$\mathsf{P}_{1}'\cdot\mathsf{P}_{2}'=\frac{E'E_{2}}{c^{2}}-p_{1}'p_{2}=mE$$ However, this leaves us with a lot of unknowns $E'$, $E_{2}$ and $p_{2}$, so I'm not sure how I'd go about reducing this to one unknown $E'$ (assuming it's possible)?


What is an electron/hole pocket and what is the significance? I'm trying to get my head around this. I've read what Ashcroft and Mermin have to say on the subject, but it's a little convoluted. They talk about the 1st, 2nd, 3rd, etc Brillouin Zones (BZ), and then show that, as you increase the radius of the free electron Fermi sphere (FS), parts of its surface will lie within certain BZ's. From there, you do this kind of strange process of taking the parts of the FS that lie within the nth BZ, and translate it with reciprocal vectors so they're within the 1st BZ. It seems like (in the example they use of a FCC metal) the "electron pockets" come from the 4th BZ being translated into the first. I understand what they're doing, but I don't really see why. I also don't really see the significance of the pockets, but I keep seeing them mentioned in papers. I have a suspicion -- do they (reduced) BZ's (i.e., after you translate the appropriate part of the nth BZ into the 1st BZ) represent different bands (n = 1,2,etc), and the electron/hole pockets represent the highest band? Any other elucidation would be greatly welcomed as well.


title: How do we show that photons generated by a constant electric current are distributed according to a Poisson distribution?

text: I saw the answer sometimes ago in a book "Quantum Electronics" or similar title. I don't remember the author since I lost the book. The book sets ( I believe so ) a constant electric current $I$ in a cable along the $z$ axis. Then, after a long ( but not straightforward ) calculation it arrives to the result I mentioned above. Edit: A possible analogous calculation that OP refered to, Klein-Gordon particles generated by a classical source: problem 4.1  in these notes(In fact a solution manual to Peskin and Schroeder). If this is not what OP refered to, please delete it.


title: Theoretical or experimental violations of the 2nd Law of Thermodynamics?

text: **Theoretical challenges to the 2nd Law?** What are some the theoretical challenges to the 2nd Law? (cf. Čápek, Vladislav, and Daniel P. Sheehan. _Challenges to the Second Law of Thermodynamics: Theory and Experiment_. Springer, 2005., chapter "Modern Quantum Challenges: Theory.") **Experimental challenges to the 2nd Law?** Is there an experiment that has found violations of the 2nd Law of Thermodynamics? Is there an experiment that shows that entropy decreases for certain systems? thanks


title: How do I derive the Lorentz contraction from the invariant interval?

text: Reviewing some basic special relativity, and I stumbled upon this problem: From the definition of the proper time: $$c^2d\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using $x=vt$: $$c^2d\tau^2=c^2dt^2-v^2dt^2=c^2dt^2\left(1-\frac{v^2}{c^2}\right)\rightarrow d\tau = dt\sqrt{1-\frac{v^2}{c^2}}=t/\gamma$$ Now, I would very much like to be able to derive the length contraction formula in a similar fashion, and feel strongly that this should be possible. The definition of the invariant interval is: $$ds^2=dx^2-c^2dt^2$$ using $t=\frac{x}{v}$ I tried: $$ds^2=dx^2-\frac{c^2}{v^2}dx^2=dx^2\left(1-\frac{c^2}{v^2}\right)\rightarrow ds=dx\sqrt{1-\frac{c^2}{v^2}}$$ This is where I'm stuck: I don't see how this can be converted to a Lorentz factor... Any help that will allow me to arrive at the desired result $ds=\gamma dx$ would be very much appreciated.


title: How to calculate the minimum number of extrinsic dimensions of a metric tensor?

text: # The Question How does one calculate the minimum number of dimensions of an extrinsic space that can be used to define the metric tensor \begin{align} g_{mn} = \dfrac{\partial y^k}{\partial x^m} \dfrac{\partial y_k}{\partial x^n} \end{align} where $y^k = (y^1 , y^2 , y^3 , \dots y^{m-2} , y^{m-1} , y^{m})^k$ is the extrinsic space of $m$ dimensions and $x^k = (x^1 , x^2 , x^3 , \dots x^{m-2} , x^{m-1} , x^{n})^k$ is the embedded space of $n$ dimensions. # Related Question This questions extends from a previous question: How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative?.


title: Confusion With Neutrino Oscillation

text: Neutrinos produced in the weak interaction are eigenstates of the weak interaction hamiltonian or flavor eigenstates. Right? If this is so, then $\nu_e,\nu_\mu,\nu_\tau$ should be the stationary states of weak hamiltonian and weak hamiltonian is diagonal in this basis. Then, why do they oscillate during the propagation from one flavor to another? **_Is the hamiltonian different during propagation (i.e., no longer weak hamiltonian?) and $\nu_e,\nu_\mu,\nu_\tau$ are no longer eigenstates of the new hamiltonian_**?


title: Does Newton's first law state something substantive, or is it merely describing a convention?

text: Newton's first law is often said to _define_ what an inertial frame is - namely, a reference frame in which a body not acted on by a force will move with constant velocity. In other words, a frame where the first law holds. However, this raises the question of what is meant by "not acted on by a force" and more generally how and what we define to be a force. Suppose you are floating in space with just one other object nearby. The object is at rest relative to you, and so you determine that your frame is inertial. Now suppose the object starts jiggling about in a seemingly arbitrary way. You have at least these options :   1. You can say that you are no longer in an inertial frame, but instead being accelerated every which way, and that is why the object appears to move. (Fictitious forces).   2. Or you can say that you are still in fact in an inertial frame, and the object is now moving because of some time dependent forces that started to act on it. Perhaps even after observing the object's motion for a long time, you would be able to describe the motion, and hence these forces, quantitatively. Of course when there is more than one object moving around, it might be simpler to choose 1 over 2, for calculation purposes. But if so, then the determination of whether you are in an inertial frame or not is not an objective fact, but rather a subjective convention that you choose to employ for simplification of your calculations. What then, is the objective content of the first law?


title: Normal force of ball sliding on concave surface

text: Imagine a ball is sliding along a surface shaped like $y=x^2$. Like ![y=sin\(x\)](http://i.stack.imgur.com/7u0dr.png), but please ignore the fact that the center of the ball is on the surface instead of the edge. When the ball is stationary, I can calculate the normal force fairly easily. $F_\text{net}$ in the direction normal to the surface is 0. The normal force is normal to the surface, so it is the force that is exactly strong enough to cancel the normal component of the force of gravity. The normal force comes out to $<\frac{-a}{(1 + a^2)}, \frac{1}{(1 + a^2)}>$, where $a=f_{\text{surface}}'(x)$ (or in this specific case: $a=2x$) I initially expected this to be true even after the ball begins sliding down the surface, but I found that it was not. Intuitively, I understand that a velocity with a non-zero component in the normal direction affects the strength of the normal force. This seems like the same mechanism by which the normal force is able to stop a falling object as it hits the ground, for example. In the falling ball example, the relationship between velocity and normal force seems a bit sticky mathematically. Assuming neither the ground nor the object compress, the normal force would have to be infinite initially, right? It would only be infinite for an infinitesimal amount of time (so the integral would still be finite). However, in that case, the $v(t)$ curve is discontinuous, while it seems like $v(t)$ would be continuous for a ball sliding on $y=x^2$, so hopefully the relationship is less obnoxious in the continuous case. **Is there a mathematical formulation of this relation between normal force and velocity?** Can I add a velocity parameter to my equation for $F_\text{net}(x)$ (making it $F_\text{net}(x, v)$) to take this relationship into account? **Note** : I also see a conservation of energy solution that sidesteps the normal force issue altogether, but I'd like to understand the forces aspect.


title: What creates the chaotic motion on a double pendulum?

text: As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random? I'm just a high school kid. So, try to make answers understandable.


title: Forces on a helical screw?

text: The common screws which we use, are right handed helices, the simplest parametric equations of which are:- $$x(s)=\cos(s),y(s)=\sin(s),z(s)=s$$, with $z$-axis as the axis of the helix. ![enter image description here](http://i.stack.imgur.com/F2LMt.jpg) My question is:- > Why does a screw move forward or backward (forward for clockwise rotation of > a right-handed helix), on applying a pure rotational torque? Even if we provide only torque for rotation, i.e. force perpendicular to the axis, it moves along the axis. What force acting on it propels it forward? if it is a component of the normal reaction of the material it is driven into, what is the action-reaction pair for this normal reaction?


title: Cellular automata rules for quantum mechanics

text: My limited understanding of quantum theory is that a **quantum system is completely described by its wave function** , which **deterministically evolves according to Schrödinger's equation** until wave function collapse. Schrödinger's equation is essentially just a differential equation, which means (as far as I can see) that the **state at every point in space changes depending only on its surroundings** (i.e local rules). Correct me if I'm wrong but it's at the point of wave function collapse that the non-locality comes in, which is what cellular autonoma (CA) models can't deal with very well. I'm not hypothesizing or suggesting that the universe **IS** a cellular autonoma (as Wolfram has suggested), but **what would a CA _model_ of Schrödinger's equation be like**? (Ignoring wave function collapse for the minute) Providing my understanding described above isn't wrong, I want to know, for the CA model:   1. What values describe the state at a given point? (i.e. what information would be in each CA cell?)   2. What rules are followed every discrete time tick to progress these values? I ask because I've gained a good intuition (an "understanding" that I'm happy with at least!) of heat flow, wave propagation and the rules of fluid dynamics through **interpreting their differential equations as local cellular automata rules** , and writing little toy simulation programs to see how they behave and look. I'm a computer science/mathematics undergraduate and I've always said that I don't understand something until I can write a program that simulates it, so being a little too ambitious for my own good I want to write a simulation for a QM system! My background is I've read a few Penrose/Feynman books, and I've got the complete Feynman lectures on physics (not read them yet...), and physics has always been a side hobby, but I'm not too hot yet on the mathematical language that is used for QM.


title: Euler angle: space-fixed vs body-fixed axes

text: I am sooo confused!! Between active and passive, intrinsic and extrinsic, vectors and basis .... Stipulate that we stick to active rotations only. Then Standard derivation of $R(\alpha, \beta,\gamma)=R_{z^{\prime\prime}}(\gamma)R_{y^\prime}(\beta)R_{z}(\alpha)$ uses intermediate frame $(x^\prime,y^\prime,z^\prime)$ in transformation from space-fixed axes $(x,y,z)$ to the body-fixed axes $(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime})$ to derive $$ R(\alpha, \beta,\gamma) = \left(\begin{array}{ccc} ~~\cos{\gamma}&-\sin{\gamma} & 0 \\\ \sin{\gamma}&\cos{\gamma}& 0 \\\ 0 & 0& 1\end{array}\right) \left(\begin{array}{ccc} \cos{\beta} & 0 &\sin{\beta} \\\ 0 &1& 0 \\\ -\sin{\beta}& 0&~~\cos{\beta} \end{array}\right) \left(\begin{array}{ccc} ~~\cos{\alpha}&-\sin{\alpha} & 0 \\\ \sin{\alpha}&\cos{\alpha}& 0 \\\ 0 & 0& 1\end{array}\right) $$ But when rewriting in terms of spaced-fixed axes (Sakurai pg 172, e.g.), fairly straightforward arguments (mathematically, just similarity transformations), take us to $R(\alpha, \beta,\gamma)=R_z(\alpha)R_y(\beta)R_z(\gamma)$. But this does NOT multiply out as the same matrix -- despite the use of = everywhere! So I figured the former applies to the basis, the latter the vector components (since they transform inversely to one another). But the results are not transposes of one another. And even so, what of their purported equality? As you can see, I'm really tied in knots!! Anyone have a sword?


title: Intuition Behind Conservation of Angular Momentum

text: I'm having a fairly hard time understanding the intuition behind Noether's derivation of the conservation of angular momentum from the rotational invariance of the Lagrangian, though I do understand it mathematically, I just don't get it intuitively. I'm saying this because it appears so clearly derivable from the original conservation of energy statement, but I know they're quite different, qualitatively. What am I missing?


title: Getting diffeomorphisms from boundary conditions in $AdS_3$

text: As usual I'm asking a question about boundary conditions for AdS${}_3$, based on the thesis by Porfyriadis. He is solving equations $\mathcal{L}_\xi g_{\mu\nu}$ for AdS${}_3$ metric, with a given boundary conditions that are basically a fall off conditions in $r$, since we're interested in the asymptotic behavior (as $r\to\infty$). I've finally gotten to the part where he, using the ansatz for diffeomorphism $\xi^\mu=\sum\limits_n \xi^\mu_n(t,\phi)r^n$ ($\mu=t,r,\phi$), gets a set of 6 equations for coefficients. I'll write the component of metric that was used to obtain the equation next to the equation, for clarification: $$(tt)\qquad \xi^r_{n-1}+l^2\xi^t_{n,t}+\xi^t_{n-2,t}=0,\ n\ge 2$$ $$(tr)\qquad l^4(n+1)\xi^t_{n+1}-l^4\xi^r_{n,t}+3l^2(n-1)\xi^t_{n-1}+2(n-3)\xi^t_{n-3}=0,\ n\ge 3$$ $$(t\phi)\qquad l^2\xi^t_{n,\phi}+\xi^t_{n-2,\phi}-l^2\xi^\phi_{n-2,t}=0,\ n\ge 1$$ $$(rr)\qquad l^2(n+1)\xi^r_{n+1}+(n-2)\xi^r_{n-1}=0,\ n\ge 2$$ $$(r\phi)\qquad l^2(n+1)\xi^r_{n+1}+(n-3)\xi^\phi_{n-3}+l^2\xi^r_{n,\phi}=0,\ n\ge 3$$ $$(\phi\phi)\qquad \xi^r_{n-1}+\xi^\phi_{n-2,\phi}=0,\ n\ge 2$$ I got that, and I understand how the author got that. But how did he find the $\xi^r,\ \xi^t$, and $\xi^\phi$ I don't get :\ He said: for ($rr$) equation, using backwards induction, since for large $n$ the series for $\xi^r$ must truncate, we get that the components $\xi^r_{2m}=0,\ m\ge 1$, and $\xi^r_{2m+1}=0,\ m\ge 1$, so the most general form of $\xi^r$ is $$\xi^r=\xi^r_1(t,\phi)r+\xi^r_0(t,\phi)+\mathcal{O}(r^{-1})\quad (\star)$$ How did he get that? I mean, I tried putting from n=10 towards n=2, and for n=2 I get $$3l^2\xi^r_3+0\cdot\xi^r_1=0$$ And that means that for n=even my odd terms are 0, if $n\ge 2$? I could kinda relate this to the fact that the $\xi^r=\sum\limits_n \xi^r_n r^n$, and for n=2 I'll get 0, so only n that are less than that will contribute, since I am making a expansion around infintiy. But I don't know if I'm right about this. And how did he make other equations? I tried using the same 'reasoning' but cannot get what he gets. In equations ($tr$) and ($r\phi$) he just drops off the $\xi^r_n$ terms. Why? :\ And then all of a sudden he gets $$\xi^t=\xi^t_0(t,\phi)+\xi^t_{-1}(t,\phi)\frac{1}{r}+\mathcal{O}(r^{-2})$$ $$\xi^\phi=\xi^\phi_0(t,\phi)+\xi^\phi_{-1}(t,\phi)\frac{1}{r}+\mathcal{O}(r^{-2})$$ How?! :( I'm desperate :(


title: Understanding Weinberg's soft-photon theorem

text: The soft-photon theorem is the following statement due to Weinberg: > Consider an amplitude ${\cal M}$ involving some incoming and some outgoing > particles. Now, consider the same amplitude with an additional soft-photon > ($\omega_{\text{photon}} \to 0$) coupled to one of the particles. Call this > amplitude ${\cal M}'$. The two amplitudes are related by $$ {\cal M}' = > {\cal M} \frac{\eta q p \cdot \epsilon}{p \cdot p_\gamma - i \eta > \varepsilon} $$ where $p$ is the momentum of the particle that the photon > couples to, $\epsilon$ is the polarization of the photon and $p_\gamma$ is > the momentum of the soft-photon. $\eta = 1$ for outgoing particles and $\eta > = -1$ for incoming ones. Finally, $q$ is the charge of the particle. The most striking thing about this theorem (to me) is the fact that the proportionality factor relating ${\cal M}$ and ${\cal M}'$ is independent of the type of particle that the photon couples to. It seems quite amazing to me that even though the coupling of photons to scalars, spinors, etc. takes such a different form, you still end up getting the same coupling above. While I can show that this is indeed true for all the special cases of interest, my question is: **Is there a general proof (or understanding) that describes this universal coupling of soft-photons?**


title: Is the spin-singlet state also a Resonating-Valence-Bond(RVB) state?

text: The _spin-singlet state_ of a lattice spin-1/2 system is defined as $S_x\Psi=S_y\Psi=S_z\Psi=0$, where $S_\alpha=\sum S_i^\alpha(\alpha=x,y,z)$ are the total spin operators, in other words, a _spin-singlet state_ is a spin state with _gobal $SU(2)$ spin-rotation symmetry_. On the other hand, a RVB state is the superposition of various configurations of singlet-product states (no need for equal weight superposition here), thus, a RVB state is of course a _spin-singlet state_ according to the above definition. For the reversed statement, is the spin-singlet state also a RVB state? How to prove it? For the extreme 2-spin system, it's direct to show the equivalence between the spin-singlet state and the RVB state.


title: Topological phase

text: Can anybody tell me, if generically any system, which is solely described by a topological field theory, resides in a topological phase? I cant find any clear notion of topological phase. Only topological phase of matter, but I mean any kind of system. Thanks for your help.


title: Why the dissolution of hydrophobic compounds decreases the entropy of water molecules in the vicinity of the solute?

text: The following is a quote from Lehninger's Principles of Biochemistry, 4th edition, pg.52: > (...) dissolving hydrophobic compounds in water produces a measurable > decrease in entropy. Water molecules in the immediate vicinity of a nonpolar > solute are constrained in their possible orientations as they form a highly > ordered cagelike shell around each solute molecule. Why is this so? My intuition tells me that dissolving polar molecules (hydrophilic) would cause a higher ordering of water molecules in the vicinity of the solute than the ordering caused by dissolving nonpolar compounds. This is because I think that the polar solutes should align with the dipoles of the water molecules, thus constraining the alignment of water molecules in the vicinity of the hydrophilic solute. Is there a simple explanation for the opposite behavior? **Edit:** I want to understand, at least intuitively, the **physical mechanism** behind the entropy decrease of the water molecules around an hydrophobic molecule.


title: How to prove $(\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0$?

text: Studying the basics of spin-$\frac{1}{2}$ QFT, I encountered the gamma matrices. One important property is $(\gamma^5)^\dagger=\gamma^5$, the hermicity of $\gamma^5$. After some searching, I stumbled upon this interesting Phys.SE answer to an earlier question on this forum. Specifically, I am interested in the formula \begin{equation} (\gamma^\mu)^\dagger=\gamma^0\gamma^\mu\gamma^0 \end{equation} which is mentioned but not proven. After consulting a faculty member of my university, I pieced together that the proof must rely somehow on the fact that the $(\gamma^\mu)^\dagger$ also obey the Clifford algebra: $$\\{(\gamma^\mu)^\dagger,(\gamma^\nu)^\dagger\\}=-2\eta^{\mu\nu}$$ $$\\{\gamma^\mu,\gamma^\nu\\}=-2\eta^{\mu\nu}$$ (for clarity, I am using $- + + +$ signature for the Minkowski metric). This should imply that there is some similarity transformation relating the two, but I am not well-versed in group theory. I guess that it should somehow turn out that the matrix that acts to transform the two representations of this algebra into each other is $\gamma^0$, which is equal to its inverse $\gamma^0=(\gamma^0)^{-1}$, as can be seen immediately from taking $\mu=\nu=0$ in the Clifford algebra. Then, the similarity transform is in the right form: $$ (\gamma^\mu)^\dagger=S\gamma^\mu S^{-1}=\gamma^0\gamma^\mu\gamma^0 $$ I have the feeling I've got most of the necessary ingredients. However, I can't seem to be able to make this argument explicit and clear (due to my lack of proper knowledge of group theory). Could someone help me out? It would be much appreciated. EDIT: I am looking for an answer that does not rely on using a particular representation of the gamma matrices.


title: What is the nucleon axial charge?

text: Can someone point me to a short definition of what the nucleon axial charge is?


title: Physical Quantities

text: What are the physical quantities which most relate to their definition? Could you give some examples? The definition I found is: " it is any property which can be measured". In the case of mass the given example is "when we go buy apples we purchase two kilograms of apples. Please give another example.


title: How to check the ohmic contact to the film?

text: I have a thin semiconductor film deposited on an isolating substrate. I would like to check different metals to find out do they form the ohmic contact or Schottky barrier. What is the best way to do that? I know that ohmic contacts show a linear I-V curve. But how to measure it? Should I deposit small dots of different metal on top of the film and measure the I-V curve? How large should be the dots? And what distance between them? Or should I deposit my semiconductor film on a conductive substrate and measure the I-V through the film? In most of the papers people just write – we measured the I-V curve of this film and such metal, but what design do they use?


title: The Universe is a Hologram

text: First my background: no physics, but Ph.D. in math. I was reading this article on Slashdot. It links to this Wikipedia page. The content of the Slashdot article is that some physicists believe the universe is a hologram, describable by a two dimensional system and some researchers have proposed a method to test this. The Wikipedia page says this is an outcome of some string theories. Now, I've been told that what many view as a failing of string theories is that they are not testable within sane energy levels. My question is whether this proposed experiment would now make string theories testable within reason. Thanks.


title: Grassmann paradox weirdness

text: I'm running into an annoying problem I am unable to resolve, although a friend has given me some guidance as to how the resolution might come about. Hopefully someone on here knows the answer. It is known that a superfunction (as a function of space-time and Grassmann coordinates) is to be viewed as an analytic series in the Grassmann variables which terminates. _e.g._ with two Grassmann coordinates $\theta$ and $\theta^*$, the expansion for the superfunction $F(x,\theta,\theta^*)$ is $$F(x,\theta)=f(x)+g(x)\theta+h(x)\theta^*+q(x)\theta^*\theta.$$ The product of two Grassmann-valued quatities is a commuting number _e.g._ $\theta^*\theta$ is a commuting object. One confusion my friend cleared up for me is that this product need not be real or complex-valued, but rather, some element of a 'ring' (I don't know what that really means, but whatever). Otherwise, from $(\theta^*\theta)(\theta^*\theta)=0$, I would conclude necessarily $\theta^*\theta=0$ unless that product is in that ring. But now I'm superconfused (excuse the pun). If Dirac fields $\psi$ and $\bar\psi$ appearing the QED Lagrangian $$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ are anticommuting (Grassmann- valued) objects, whose product need not be real/complex-valued, then is the Lagrangian no longer a real-valued quantity, but rather takes a value which belongs in my friend's _ring_??? I refuse to believe that!!


title: Curvature and edge state

text: If the boundary of quantum hall fluid has non-constant curvature, how will it affect the edge state which is usually described in chiral Luttinger fluid?


title: How do I find the expectation value given only the eigen energy?

text: Let $|n \rangle$ denote the $n^{th}$ stationary state of the harmonic oscillator, with energy $E_n = \hbar \omega(n+\frac{1}{2})$ How would I find $\langle x\rangle$ and $\sigma_x$ I know that $$\langle x\rangle = \langle \alpha|x\alpha \rangle = \sqrt{\frac{\hbar}{2m\omega}}\langle \alpha|(a_+ + a_-)\alpha \rangle=\sqrt{\frac{\hbar}{2m\omega}}(\alpha + \alpha^*)$$ but that is just from the definition of ladder operator, what does it have to do with energy eigenstate? Any help would be appreciated


title: Why is fundamental physics taught in terms of particles?

text: According to this paper, there can be no relativistic quantum theory of localizeable particles ("relativity plus quantum mechanics exclusively requires a field ontology"). Sean Caroll has also argued that fundamental physics should be conceptualized in terms of fields, not particles. Here is another paper on the subject as well, and two other related questions on this website. My question is primarily about learning and conceptualization: Quantum fields, not particles, are the fundamental constituents of nature (at least as far as we know), yet particles are often the conceptual tool used to teach field theory/particle physics. What concepts to particles help explain where fields fail? And conversely, where do fields succeed while particles fail?


title: Is a superposition of (anti)symmetric states (anti)symmetric?

text: Let's say we have the following wavefunction of two identical particles, $A$ and $B$: $$\frac{1}{2}[(\chi(A)\psi(B)\pm\psi(A)\chi(B))+(\phi(A)\eta(B)\pm\eta(A)\phi(B))]$$ Is this properly (anti)symmetric? i.e. can it be put in the following form? $$\frac{1}{\sqrt2}(f_1(A)f_2(B)\pm f_2(A)f_1(B))$$


title: Quantifier problems of equations in physics

text: Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. I am wondering shouldn't it be written rigorously as something like: For all $E$ in a given range of values and all $m$ in a given range of values, we have $E =mc^2$? If not, then logically the equation $E = mc^2$ is sometimes valid and sometimes invalid as $E, m$ run through their given ranges, so that one cannot use "$E = mc^2$" to express what it is intended to mean. Then how does one incorporate this phys-mathematics writing style, as illustrated in the beginning, with the authentic mathematical writing style, as illustrated in the second paragraph, in the sense of logical rigor?


title: Are Both figures correct?

text: Actually I am in high school I just want to confirm that whether two figures of this question possible: Question is : A car is moving along a straight line OP.It moves from O to P in 18 seconds and returns from P to Q in 6 seconds , where OP = 360 metres and OQ = 240 metres.What are the average velocity and average speed of the car is going (a) From O to P and back to Q? My physics teacher made this figure from the question: ![by My teacher](http://i.stack.imgur.com/SCZjb.jpg) and I this figure is also right: ![by me](http://i.stack.imgur.com/URVXD.jpg) But When I solve the above Question by first figure I got Average speed of 20 m/s and 40 m/s from 2nd figure. Which one is correct? (When I ask my Question to my physics teacher , He said "First one is correct,Second one doesn't make any sense")


title: List of the most fundamental equations of all Physics

text: On chapter 18 of "The Feynman Lectures on Physics Volume II: Mainly Electromagnetism and Matter", Richard Feynman presents a table which he says has "all that was known of fundamental classical physics, that is, the physics that was known by 1905." And he adds "Here it is all, in one table. With these equations we can understand the complete realm of physics." Even though this sounds too reductionist to me, I think it is quite interesting and I was wondering whether it is possible to imagine and create a second table, containing the most fundamental equations of all current Physics. I understand that a mere list of formulas is useless if one doesn't get the underling concepts. But, even for aesthetic reasons, what I am trying to do is just to ask: What would a more or less brief list of the supposedly most fundamental equations of Classical and Modern Physics look like? ![Classical Physics according to Feynman.](http://i.stack.imgur.com/psO8e.png)


title: Where does the energy flow differ between a forward and reverse process?

text: ![PV diagram of cycle](http://i.stack.imgur.com/Yn1d3.png) Mathematically ($W=\int PdV$) and by the First Law, I understand that $1 \rightarrow 4 \rightarrow 3 \rightarrow 2 \rightarrow 1$ and $1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1$ are not symmetric; the first requires work, whilst the latter gives it to the environment. What I am having trouble with is bellyfeeling _why_. A physical example: imagine a machine that works thusly: > **(1-2)** A piston loaded with 3 bricks in a hot reservoir isothermally > expands when 1 brick is removed. > > **(2-3)** Then, removed from contact from any reservoir, it adiabatically > expands when the 2nd brick is removed, cooling to the temperature of the > cold reservoir, where > > **(3-4)** the 2nd brick is replaced, isothermal compression takes place, > then > > **(4-1)** away from reservoir the 3rd is added, the temperature rising to > the intial hot temperature. **Given that the two possible cycles are asymmetric, _at what point (1,2,3,4) does the internal energy of the gas or the potential energy of the bricks differ for the two different cycles?_**


title: Finite quantum gravity?

text: I'm working through an article that has some questionable assertions. The article is by Frank Tipler, "The structure of the world from pure numbers". (I'm going to ignore the fact that some of Tipler's ideas are appropriately, I think, labeled _pseudoscience_.) There's a (questionable, I think) claim that "requiring the joint mathematical consistency of the Standard Model of particle physics and the DeWitt–Feynman–Weinberg theory of quantum gravity can resolve the horizon, flatness and isotropy problems of cosmology." I'm not worried about that right now. (But I imagine that won't prevent folks from commenting.) My goal is less ambitious for now. I'll just analyze, with your help hopefully, whether the following claim is correct. In section 5 of the Rept. Prog. Phys. article cited above, he claims (quoting verbatim): > So basic quantum field theory quickly forces upon us the general invariant > action \begin{align} S &= \int d^4\\!x\sqrt{-g}\Big[\Lambda+\frac{1}{8\pi > G}R+c_1^2R^2+c_1^3R^3+\cdots+ \nonumber \\\ > &+c_2^2R_{\mu\nu}R^{\mu\nu}+\cdots > +c_1^3R_{\mu\nu;\alpha}R^{\mu\nu;\alpha}+\cdots \end{align} > > This is the qualitatively unique gravitational Lagrangian picked out by > quantum mechanics. > > Physicists do not like it because (1) it has an infinite number of > (renormalizable) constants $c_j$ , all of which must be determined by > experiment and (2) it will it not yield second order differential equations > which all physicists know and love. But the countable number of constants > are in effect axioms of the theory, and I pointed out in an earlier section > that the Lowenhein–Skolum theorem suggests there is no real difference > between a theory with a countable number of axioms and a theory with a > finite number of axioms. The finite case is just easier for humans to deal > with, provided the ‘finite’ number is a small number. Further, as Weinberg > (1995, pp 499, 518–519) has emphasized, this Lagranian generates a quantum > theory of gravity that is just as renormalizable as QED and the SM. The first thing I note is that, I think, the renormalization constants (we call them "low-energy constants" in hadron theory) should be _different_ for each term. For example, as $\cdots + c_1R^2 + c_2 R^3 + \cdots$, etc. The next thing I note and would like to focus on for this question is the claim that a countably infinite number renormalization constants, the $c_j$, isn't a problem. (I don't know what the Lowenhein–Skolum theorem is but I don't think it's crucial to this discussion.) The counterargument is that an infinite number of constants would require progressively more experimental observables to fix them as the energy scale of the probe increases. I believe that what saves the day in hadronic effective field theories is that there is a power-counting that orders the sizes of the terms in the Lagrangian density. And that this power-counting requires the determination of a finite number of low-energy constants. Now the curvature scalar, $R = R_\mu^\mu$, is a sum of terms, the highest derivative order of which are second order in the derivatives of the metric, $g_{\mu\nu}$. And powers of $R$, say $R^n$ will have products of highest order that are $n$ factors of second derivatives of the metric. And the other terms, like $R_{\mu\nu}R^{\mu\nu}$ just have higher order derivatives. Presumably, the same power-counting logic applied above can be applied to these terms. I believe this argument indicates that the quantum theory of gravity is predictive when viewed as an effective field theory. So my questions are: 1) Have I made the correct argument regarding the power-counting of the theory? 2) Is this argument predictive?


title: does the background spacetime of a black hole affects its thermodynamic properties?

text: The question is this: will the thermodynamic properties of a black hole (Hawking radiation spectra and temperature, entropy, area, etc.) depend if the black hole sits in a DeSitter or an Anti-DeSitter space-time? why and how?


title: Is temperature a Lorentz invariant in relativity?

text: If an observer starts moving at relativistic speeds will he observe the temperature of objects to change as compared to their rest temperatures? Suppose the rest temperature measured is $T$ and the observer starts moving with speed $v$. What will be the new temperature observed by him?


title: Angular momentum of a black hole

text: I recently read this Phys.SE post and, since I didn't know that black holes had a spin, a question came to my mind: how can I calculate the spin velocity of a black hole? Does mass or radius affects it? I googled it but I couldn't understand much, all I found was about Orbital Velocity of a planet...


title: How do people historically have come to use the Yang-Mills theory in physics?

text: There are many books, in which Yang-Mills theory is introduced "just like that". But I didn't find some book with set of historical arguments, which had led people to using it in quantum field theory. Can you tell me about this? Maybe, my question leads to the next question: how did people guess that they need to expand the group of local gauge invariance for describing, for example, quarks?


title: How does band narrowing (heavily doping the emitter) affect the alpha F and alpha R in the Ebers-Moll model of a BJT?

text: I think that when the emitter is heavily doped, the bandgap is reduced which increases the reference forward current(current flowing through the E-B junction ) but the alphas themselves do not change.


title: what forces are present when slicing an object.

text: I found here:What happens when we cut objects? that it is due to fracture wood splits open when using a wedge. Does it only apply when the wedge is already in the wood? How about the moment before a crack is made, do you need to overcome the compressive and shear stress of the wood to start a crack? I want to compute for the forces needed to cut the wood, along the grain, using a wedge. but Im confused whether shear and compressive strength or fracture initiation is involved to start the cut. and when the wedge in the the wood should I compute for fracture propagation or the cosine of the normal force on the wedge that is pushing the grains apart. I hope I'm making sense.


title: what determines the wavelength of waves on the open ocean?

text: Looking at the picture below, you can totally see that these are tiny boats.   The water is practically washing over the hull of these boat models. But the water has boundaries that are very far away, and even if the water is very deep, it would (at least in my mind) not make a big difference to the size of the waves after just a few inches of depth. What dictates the size waves in the open water? Is it the molecular properties of the water? ![enter image description here](http://i.stack.imgur.com/zEneX.jpg)


title: How do you calculate the position of the antinodes on a wave?

text: if for example i have a waveform with the formula $y=\sin(1.2x)+\sin(1.8x)$. The first 6 antinodes on my graph come up at around: $$ x=\begin{cases} 1.01\\\ 2.97\\\ 4.65\\\ 5.80\\\ 7.48\\\ 9.46\\\ \end{cases} $$ but I can't find a way to find the exact positions.


title: Ice cube in a pool

text: Imagine this situation: Large ice cube placed at the bottom of an empty pool starts to dissolve. I was wondering if it is possible for cube to start float in the water? If yes, what fraction of the cube needs to dissolve?


title: Does measuring destroy entanglement

text: Before measuring a quantum particle(photon) it exists in a superposition state, once we observe(measure) it, it settles in one of the possible states(destroying superposition). For entangled particles, does measuring destroy entanglement as well?