title: Can Zeno's Dichotomy Paradox be Resolved with Quantum Mechanics?

text: I would like to start off by saying this is not a philosophical question. I have a specific question pertaining to physics after the following explanation and background information, which I felt was necessary to properly formulate my question. I have done some research on Zeno's Paradoxes as well as some of their modern day mathematical and physical "solutions." I understand that Zeno's dichotomy paradox can be explained mathematically by constructing an argument that makes use of the following infinite series: $$ \sum_{n=1}^\infty {1 \over 2^n} = 1$$ Because this series converges to $1$, then that implies that if physically traveling from some point A to another point B truly does require an infinite amount of actions in which an object travels ${1 \over 2}$ of the distance, then ${1 \over 4}$ then ${1 \over 8}$ then ${1 \over 16}$ and so on, the sum of those actions would eventually result in traveling the full distance. Mathematically, this makes perfect sense. However, knowing that the mathematics we use to describe the physical world is only a model, adopted when there is sufficient evidence to support it, could it be that the solution gained by the above argument comes to the correct conclusion using the wrong path? Using the infinite series provides a solution to the dichotomy paradox if it does indeed require an infinite amount of actions to move from point A to point B. So, then the question becomes, does it truly take an infinite amount of actions to physically move from one point to another (in which case, I believe the mathematical solution is a perfect model), or are there other phenomena at play that cannot be described in this fashion? I have one idea pertaining to a quantum mechanical description of the situation that does not disprove, but provides an alternative to the reasoning that it requires an infinite number of actions to move from point A to point B.   * Suppose we have a box of some macroscopic size (say $1m^3$), and it is in the process of moving with a constant velocity from some point A to another point B. Obviously, the box is made up of atoms, and if we zoom in far enough on the leading face of the box we could see that the motion of said box is really the motion of a very large number of atoms moving in the same direction. According to the uncertainty principle, it is impossible to determine both a particle's momentum and position simultaneously. According to the physics professor who first taught me about this, the reason for this uncertainty does not lie within our measurement methods, but rather, it is an impossibility inherent in the fact that all particles have wavelike characteristics. Thus, their position and momentum are not even clearly defined at a specific moment in time. Could it not be possible then, that if we zoomed in far enough on the leading face of our box, that we would find the face of the box does not have a definite position? If this is true, then wouldn't there be a certain point very close to the destination point, B, where the idea of cutting the remainder of the distance to the destination in half makes no difference to the particles in question? We may say that to complete the trip, we must travel another ${1 \over 2^x}$ portion of the distance we started with, but if the uncertainty in position is larger than this remaining distance, could the particle (and by the same argument, all of the particles that make up the leading face of the box), traverse the final distance to point B without physically having to travel the "infinite" amount of points remaining? Also, I have a second idea that I want to present as an aside, something that I don't believe can be proven as of now, but I wonder if, according to current understanding, it is possible.   * What if, instead of analyzing the situation at hand through the distance traveled, it was analyzed by analyzing the passage of time. Specifically, if time is quantized, would it resolve the dichotomy paradox in a physical sense? I realize that whether or not time is quantized is up for debate (there is even this question here at stack exchange that explores the idea), so I am not going to ask if time is quantized. Based off of my understanding, however, there is nothing known as of now that says time cannot be quantized (please correct me if I am wrong). Therefore, if time _were_ quantized then could we not say that for a given constant velocity there is a minimum distance that can be traveled? A distance that corresponds to $\Delta t_{min} * v$? Would this not imply that as our moving box reached a certain distance, $d$ away from point B, and if $d < \Delta t_{min} * v$, the box would physically not be able to travel such a small distance, and in the next moment that the box moved, it would effectively be across the destination point B? All of that being said, here is my specific question: In my above arguments, is any of my logic faulty? Is there any known law that would disprove either of these explanations for Zeno's dichotomy paradox? If so, is there a better way of physically (not mathematically) resolving the paradox?


title: How to determine n equidistant vectors from point P in three dimensions

text: As an assignment for uni I need to figure out an algorithm that explodes a particle of mass $m$, velocity $v$, into $n$ pieces. For the first part of the assignment, the particle has mass $m$, velocity of $0$, the particle explodes into 6 equal pieces, and is not affected by gravity. The problem I'm having is, how do I determine 6 equidistant unit vectors in three dimensions? Assuming I need an initial vector, a unit vector in the direction of $v$ (the initial particle velocity) will do. In the initial case, where $v$ is $(+0, +0, +0)$, let the initial unit vector be in an arbitrary direction. Also, how do I determine what speed each sub-particle will have? I know that the sum of sub-particle momentum will be equal to the momentum of the initial particle, and because each particle has the same mass they will each have the same speed... Do I need an explosion Force amount or something?


title: Solving for initial velocity required to launch a projectile to a given destination at a different height

text: I need to calculate the initial velocity required to launch a projectile at a given angle from point A to point B. The only force acting on the projectile after launch will be gravity – zero air resistance. The projectile is launched within a simulated, virtual environment; however, I am asking for help with the physics rather than the simulation itself. I have had success (with a different equation) when point A and point B are at the same height; however, **once point B is at a different height** , my calculations become less precise – well, wrong in fact. I researched the following formula for finding the range of a projectile on uneven ground. The parameters are all available to me, except the initial velocity that I need to solve for. > $d$ : range or distance   >  $v_i$ : initial velocity   >  $g$ : gravity   >  $\theta$ : launch angle   >  $y_0$ : launch height   > $$d = \frac{v_i \cos\theta}{g}(v{_i} \sin \theta + \sqrt{(v_i \sin\theta)^2 + 2gy_0})$$ I attempted to solve for initial velocity (eq: **A** ): $$v_i = \sqrt{\frac{d^2g}{2\cos\theta^2(y_0+d \tan\theta)}}$$ Using this equation in my simulation I apply the velocity to a normalised displacement vector and launch the projectile. It gets close to its target but:   * $x_{final}$ is always correct   * $z_{final}$ is always incorrect - it is close to the desired $z$ but always offset by a seemingly proportional amount.  I have spent many hours trying to review the equation but I have been unsuccessful. Any advice would be most appreciated:   * is this the correct equation?   * have I solved for $v_i$ correctly? I have looked for an example equation online but I have not found one structured how I need it, i.e. solving for $v_i$   * I have spent hours researching online - perhaps I have missed some good (but entry-level) resources. Do you know of any reference material that may help me? **Update - I have now named the equation above for referencing below** Following on from the answer by @Pygmalion, which I am still gratefully working to understand: I agree that your derived equation is equivalent to mine (A). Using yours (@Pygmalion's) in my simulation, I therefore observe the same failures: the projectile always lands short of the target. I have simplified the simulation keeping the **launch height and target height the same**. I still encounter the same problem with equation A; however, when passing the same parameters to the following equation the projectile always hits the target precisely (eq: **B** ): $$v_i = \sqrt{\frac{dg}{\sin2\theta}}$$ Given the success of equation B when the launch and target heights are the same and the failure of equation A given the same conditions, I question whether equation A is the correct one to solve the problem since I would expect it to work for any combination of launch|target height. I think the help I need is around why equation B works. Does it cater for conditions that equation A overlooks? If equation A is unsuitable, are you able to recommend an alternative? In short, the $v_i$ calculated by equation A (in my simulations) is always less than $v_i$ from equation B given the same inputs - the projectile therefore always falls short of its target.


title: Did anyone claim that quantum theory meant lasers would never work

text: I've been reading 'How the Hippies saved Physics', which describes a design for a superluminal communication device, of which the crucial part was a laser which duplicated an incoming photon many times. The reason this won't work is what is now known as the no-cloning theorem - a quantum state can't be duplicated in this way. It may appear that a laser can do this, but it can't. The thing is that I have vague memories of reading that when the laser was first talked about, it was claimed that quantum theory forbade such a device. What I'd like to know is whether there were such claims, and if so were they based on the idea that a laser would duplicate a quantum state.


title: Setting up a local-coordinate system in space-time using only a single clock and light beams

text: I have a question to ask about the operationalist view of space-time. I am a mathematician who happens to be interested in physics, so if anyone thinks that my question is a silly or vague one, please feel free to close it. Many physics textbooks that discuss Special Relativity mention that one way of setting up a coordinate system in flat Minkowski space is to imagine that there is an infinite grid of meter rules that pervades the universe, with synchronized clocks located at each node of the grid. In order to obtain more precise measurements of space-time events, one simply divides the grid into more nodes. I feel that this presents problems, because what does it mean to say that ‘all the clocks are synchronized’? Any act of moving a clock from a starting point to its designated node is sure to involve some acceleration, and this is going to upset any synchronization attempt. Also, such a description is not satisfactory because it is impossible to deal with an infinite number of meter rules and clocks. Things are made worse in curved space-time because of the further effect of gravitational time-dilation. Hence, I believe that the only logically consistent way to perform measurements of space-time events is to use a single atomic clock carried by a single observer and to use the world-lines of photons emitted by a light source carried by the observer to ‘carve out’ a local-coordinate system. The exact description of how this can be done is beyond me, so I would like to gather some useful information from anyone in the community. Thank you very much!


title: Conservation of angular momentum in helicopter

text: I have a small RC-controlled toy helicopter with removable tail rotor. Suppose I remove the tail rotor, hold the tail with my hand, start the rotor until it moves with constant angular velocity and then let it go. Suppose that the heli remains on the floor all the time. Then the body of the helicopter begins to turn into the opposite direction of the rotor. My problem is to explain this in terms of angular momentum. At first this seemed to contradict angular momentum conservation because I assumed that the helicopter is a closed system and there is no external torque, but the rotor keeps the same angular velocity while the body starts to turn into the other direction. My second thought was that I have to take the air into account. If I do that it seems plausible to me to explain it as follows: The rotor transfers angular momentum from to the surrounding air. However to keep it running at constant angular velocity, the motor has to transfer angular momentum to the rotor, which is only possible if the body of the helicopter gets the same amount of angular momentum as goes to the rotor, but into the opposite direction. But then I thought, the body should turn faster and faster to infinity which obviously doesn't happen. So I guessed I should also take the friction with the floor into account where the angular momentum of the body is transferred to. This leads me to the following three questions:   1. Is my reasoning described above correct or I am completely wrong?   2. If I would do the experiment on an almost frictionless turntable (which I don't have available at the moment), would the body of the helicopter indeed become faster and faster almost to infinity?   3. If I would do the same thing in a vacuum, what would happen? I guess (see my first thought above), that the body wouldn't turn at all. Is that correct? So it would be great if someone could clarify this in terms of angular momentum. Since the same problem should occur in real helicopters, I guess that there should be some detailed literature about it's physics, but I didn't find any resources about this. So if you have any references for more details or further reading on this problem, it would be also great if you could post it.


title: The meaning of scale invariance in power law distribution

text: A function $f(ax)$ that satisfies $$ f(ax)=a^\Delta f(x)\,\,\, (\Delta \in R) $$ is said to be _scale invariant_. The most general function $f(x)$ that satisfies the previous condition is of the form $$ f(x)=C x^{\Delta} $$ If we consider the set of distributions, namely the set of all $f(x)$ such that $\int f(x)dx=1$, is it possible to prove that the only distribution that is invariant under _scale transformation_ is $$ f(x)=C x^{-1}. $$ This means that in general I can build a generic distribution as a power law that still satisfies the scale invariance properties $f(ax)=a^\Delta f(x)$, but this generic distribution will not be invariant under scale transformations. So, what is the physical meaning of the scale invariance property in a generic power law distribution? **First Edit:** For a distribution $f$, if I perform a change of variables, I must preserve the probability, namely $$ |f(x)dx|=|g(y)dy|, $$ so that the new distribution $g$ will be given by $f(x)|\frac{dx}{dy}|=g(y)$. If a distribution is invariant under change of variables, I must have $$ g(y)=f(y). $$ Replacing $x$ with $ax$ and using the invariance of the distribution $f(x)$ under this transformation, I have $f(x)=C x^{-1}$.


title: How does the conformal Ward identity guarantee a vanishing 3-point function in this case?

text: I was looking through some conformal Ward identity related things when I noticed that this paper (arXiv:1212.3788) writes in their equation (33), a 3-point function between a conserved current and two scalars, complex conjugate of each other and having the same conformal dimensions. The equation reads \begin{equation} \partial_\mu\langle j^\mu\mathcal{O}(y_1)\bar{\mathcal{O}}(y_2)\rangle=-iq\langle \mathcal{O}(y_1)\bar{\mathcal{O}}(y_2)\rangle(\delta(x-y_1)-\delta(x-y_2)) \end{equation} They also comment on the next page that this equation shows, Ward identity guarantees a vanishing three point function for scalar operators of unequal conformal dimensions. How to see that? Also, is this a general outcome of Ward identity that the operators inside (in such cases; may be instead of current, some other conserved tensor) the operators have to have equal conformal dimensions to have a non-vanishing correlator? A reference will be excellent. Thanks in advance!


title: FPS to MKS usage issues

text: I'm currently taking a high school physics class and we're doing friction problems now. I've been doing the ones that use MKS perfectly well, but when the problems start to use the FPS system, I'm always getting a relative error of around 12%. This mostly stems from my not knowing how to deal with FPS using the $ F = ma $ formula. I've tried converting FPS to MKS for it, but it still ends up with a large relative error. How do I apply the FPS system to a formula like $ F = ma $ that is defined using MKS units and why? For the why, I'm just wondering if I'm to consider pounds when used in a physics problem at my level as a force unit and my mistakes are just the results of not using proper significant figures, or am I supposed to convert to MKS force units? If I'm to convert to MKS units, is using the rate of 4.44822162 (NIST) lbf to one Newton appropriate? Thank you for reading my question, I hope I'm not skirting on the edge of the rules too much. I tried consulting Google and searching this site for related questions, but I guess the answer is obvious and I can't see it :( **EDIT:**   Here are two problems that display my attempt to convert, then plug in the values, and my attempt to just use the values as-is then plug in. **Problem 1**   "A horizontal force of 6 lbs is applied to a 10 lb block which moves along a horizontal surface. If the coefficient of kinetic friction is 0.4, determine the acceleration of the block. **($a = 6.4 ft/s^2$)** "      **$F_N$** = Normal Force, **$F_k$** = Kinetic Friction Force, **$F_h$** = Forward Force, **$F_w$** = Weight Force         $F_h = 6 lb$   $F_h = 6 lb * 4.44822162$ (conversion factor to Newtons)   $F_h = 26.69 N$      $F_w = 10 lb$   $F_w = 10 lb * 4.44822162$ (conversion factor to Newtons)   $F_w = 44.48 N$      $F_w = mg$   $44.48 N = m * 9.8 m/s^2$   $m = 4.54 kg$      $μ_k = \frac{F_k}{F_N}$      $0.4 = \frac{F_k}{44.48 N}$      $F_k = 17.79N$         $-F_k + F_h = ma$   $-17.79 N + 26.69 N = 4.54 kg * a$   $a = 4.54 m/s^2$      Need to convert now...      **Final:** $a = 14.89 ft/s^2$         There's another problem where I just directly put in the FPS values without converting, but if this method is right, there's no point in posting it. If this method is wrong, I'll write it out here.


title: Modeling galactic colonization

text: I found a really cool sounding order-of-magnitude modeling question but am a bit at a loss on how to approach this: > The Milky Way contains 100 billion stars and has a radius of 250,000 light > years. Suppose that humans set a long term goal to colonize every > inhabitable star system in the Milky Way and that 10% of star systems are > suitable. This colonization might be accomplished by sending out a ship of > settlers to a nearby solar system, establishing a colony, and then waiting > long enough - 500 years? - for the colony to develop enough that it can send > its own settler ships. Do an order of magnitude estimate of how long it > would take to colonize the galaxy, explaining any assumptions you make. My first assumption is to put our solar system into the center of the galaxy with the argument that this makes calculations more symmetric, assuming that it will change the results by a factor of order 1. (I'm shorter away from the "ends" of the solar system but I benefit more from the hopefully exponential growth). Now the question is how to model things further. A crude estimate of the average distance between inhabitable star systems tells me that they are about 5 light years apart from each other. I guess since 5 is much smaller than 250,000 I could move to some sort of continuum description of colonization progress, but then I'm not sure on how to proceed from there. Another important question would be the speed of my colony ships. I guess it should be at least on the order of the escape velocity of the earth-sun- system, so again using crude estimates I'll get something like $5\cdot 10^4\, \mathrm{m}/\mathrm{s}$ which means that a ship needs about 5000 years for a journey. If the time per journey was significantly larger than the time to prepare the next ship (500 years), I guess I could ignore the 500 year delay and assume instant readiness, but that doesn't seem to be appropriate here. And if I don't do that, then the process seems a bit tricky: During the 5000 year journey of the first ship, I can produce 10 additional ships and send them to a journey, i.e., growth of colonized space is not just proportional to the number of fringe systems but depends in a non-obvious complicated way on the number of colonized planets AND when they were colonized. (Similar to "broadcasting in networks" scenarious?) Maybe someone immediately recognizes a commonly used underlying model that is appropriate here? PS: I'm open for suggestions re tags.


title: Gravity duals to Navier Stokes and interpretation of non linear contributions

text: I have been reading the paper The Incompressible Non-Relativistic Navier- Stokes Equation from Gravity. In it they state, > "An instability, if it occurs, must necessarily break a symmetry ... that > the background solution preserves." There is also discussion about the non-linear terms contributing to instabilities. Although I have some thoughts about how this should be interpreted. I wanted to see if there is some standard discussion relating to non linear instabilities in this context. Addendum: In order to provide more specificity, in standard discussion for understanding string theory, weak field approximations for gravity involve linearized fluctuations around a Minkowski metric: $$g_{\mu\nu}(x) = \eta_{\mu\nu}+h_{\mu\nu}(x)$$ In stronger fields, the fluctuations become highly non-linear. The non- linearity is thus understood as being associated with strong gravitational fields. The Navier-Stokes equations can be understood as a composition of two types of equations, the Heat equations and the Euler equations. In a crude sense, the unforced heat equations generally govern the diffusion of energy throughout an object, and the Euler equations govern the conservation of mass. The incompressible Navier-Stokes equations in cartesian coordinates are: $\partial_tu+u\partial_xu+v\partial_yu+w\partial_zu = -\partial_xp + \nu(\partial_{xx}u+\partial_{yy}u+\partial_{zz}u)$ $\partial_tv+u\partial_xv+v\partial_yv+w\partial_zv = -\partial_yp + \nu(\partial_{xx}v+\partial_{yy}v+\partial_{zz}v)$ $\partial_tw+u\partial_xw+v\partial_yw+w\partial_zw = -\partial_zp + \nu(\partial_{xx}w+\partial_{yy}w+\partial_{zz}w)$ $\partial_xu+\partial_yv+\partial_zw=0$ Where one can identify the "heat equation terms" as (noting that in complex form these are the Schrodinger equations): $\partial_tu = \nu(\partial_{xx}u+\partial_{yy}u+\partial_{zz}u)$ $\partial_tv = \nu(\partial_{xx}v+\partial_{yy}v+\partial_{zz}v)$ $\partial_tw = \nu(\partial_{xx}w+\partial_{yy}w+\partial_{zz}w)$ And one can identify the "Eulerian terms" as: $u\partial_xu+v\partial_yu+w\partial_zu = -\partial_xp$ $u\partial_xv+v\partial_yv+w\partial_zv = -\partial_yp$ $u\partial_xw+v\partial_yw+w\partial_zw = -\partial_zp$ In the solution I provided in a previous question, the interesting characteristics is that solution satisfies both sets of equations (sans the pressure term for the heat equation). The pressure identified is also negative, and the solution is unbounded. This doesn't seem to be the type of space identified with the AdS in the above referenced paper or the related paper discussing the gravity-fluid dynamics relationship. There is one other thing that is of interest, and that is the vanishing of the non-linear and pressure terms with the given solution (related to the apparent separability of terms). If I relate the Eulerian terms with gravity, then the solution in my mind suggest that there are possibly some set of solutions that can be shared in common with both the Heat equation terms and the Eulerian terms (and the number of these are unknown). It seems that those solutions would have some sort of privileged status, or perhaps they are completely trivial and not interesting, but I am thinking that answer is context dependent. However, I am somewhat encouraged that those solutions are not trivial, since bifurcation is not generally considered a good thing in equations describing physical phenomenon. So getting back to the original question, in the above referenced paper it is implied that instability at high Reynolds numbers ($\frac{1}{\nu} >> 1$) is associated with symmetry breaking, which is a necessary physical phenomenon that is not well understood. So I am trying to tie these thoughts together and was wondering what misconceptions I might have on this or how these are tied together in academic settings.


title: Combining of iron nuclei

text: Iron is the most stable single element (I don't know about combinations of elements, but oh well, not applicable to question). Stars go through cycles where they change the materials of their cores, culminating in iron for the longest lasting and largest stars. The cores of stars naturally have tremedous forces. My question: Has humanity managed to combine to iron nuclei together? and if yes why can we do it but not stars? Alternatively, where would the progression past iron nuclei go, since no element is more stable?


title: Find the bending moment of a pole attached to a moving block

text: I'm having trouble with the following problem. ![Problem](http://i.stack.imgur.com/ed8nE.png) _What I've done so far:_ x-y is the usual coordinate system. $a=\frac{F}{m}=\frac{800}{60}$ and the y component of this is $a_y=a\sin{60^\circ}$. To figure out the force acting on the rod and contributing to the momentum, I take the sum of all forces in the y direction. Note that from now on, I am thinking about the rod as being separated from the rest of the body. The weld is represented by forces ($O_y, O_x$), and the rod is moving upwards with the acceleration calculated above. $\Sigma F_y= O_y-20g = 20a_y\implies O_y=20a_y+20g$ Now, the momentum equation about the center of mass of the rod should be $\Sigma M_G=0.7O_y+\bar{I}\omega_G=\bar{I}\alpha=0$ Because the rod is not rotating. Is this equation right? (Why not?) If it is right, it implies that $\bar{I}\omega_G=-0.7O_y$. Where do I go from here? The things I've tried from here on don't yield the right answer, and I do not have much confidence in those strategies. The correct answer, if anyone cares, is 196 Nm.


title: Hooke's law limitation question

text: Let's consider a spring. I am a strong man(well, lets assume) and I am pulling the spring. the work I do is being stored in the spring in the form of its elastic potential energy. Then suddenly, elastic limit is crossed and the spring reaches to the yielding region. then, at that very moment, i become exhausted and release the spring. then I realise that the spring doesn't budge. the energy i stored is lost. its gone. Nowhere to be found. Mind helping me find it?


title: Why does $c_{-k,-\sigma}$ create a particle with momentum $k$?

text: In Mudelung's book, Introduction to Solid-State Theory, I am confused by the following statement. > For many applications a further simplification is helpful. The concept of > the hole presents us with the confusing situationthat a hole in the state > $\mathbf{k},\sigma$ has a momentum $-ħ\mathbf{k}$, while an electron has a > momentum $ħ\mathbf{k}$. This asymmetry can be avoided by defining new > _quasi-particles_ which have momentum $ħ\mathbf{k}$ inside and outside the > Fermi sphere. When we note that the creation of a particle with > $+ħ\mathbf{k}$ is achieved by the operator $c_{\mathbf{k}\sigma}^+$ when > outside the Fermi sphere and by $c_{-\mathbf{k},-\sigma}$ when inside the > Fermi sphere, it is only a small step to define the following operators: $$ > \begin{align} > \alpha_{\mathbf{k}\sigma}^+=u_\mathbf{k}c_{\mathbf{k}\sigma}^++v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma} > \\\ > \alpha_{\mathbf{k}\sigma}=u_\mathbf{k}c_{\mathbf{k}\sigma}+v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma}^+ > \end{align} $$ with $$ \begin{align} u_\mathbf{k}=1,\; > v_\mathbf{k}=0\quad\text{ for }\quad k>k_\text{F}, \\\ u_\mathbf{k}=0,\; > v_\mathbf{k}=1\quad\text{ for }\quad k<k_\text{F}. \end{align} $$ He showed the motivation of introducing Bogoliubov tansformation. But I don't understand his statement. "Creation of a particle with $+\mathbf k$ is acvieved by the operator $c^{+}_{\mathbf k,\sigma}$ when outside the Fermi shere and by $c_{-\mathbf k,-\sigma}$ when inside the Fermi sphere". $c^{+}_{\mathbf k,\sigma}$ means create a particle with $\mathbf k$. So, why must we distinguish ouside and inside the Fermi sphere? And why does annihilation operator $c_{-\mathbf k,-\sigma}$ inside the Fermi sphere have the same effect as creation operator outside the Fermi sphere?


title: Aharonov–Bohm effect on a non-contractible manifold

text: The Aharonov–Bohm effect shows the vector potential $\mathbf{A}$ is more fundamental than magnetic flux density $\mathbf{B}$. However, vector potential is introduced by $$ \nabla \times \mathbf{A} = \mathbf{B} $$ or $$ \mathrm{d} \mathbf{A} = \mathbf{B} $$ which requires the Poincaré's lemma to ensure the existence of $\mathbf{A}$. If the manifold is not contractible, there is no guarantee that $\mathbf{A}$ exists. On a non-contractible manifold, what would happen for the Aharonov–Bohm effect?


title: Calculate the center of mass of a semicircle

text: How I determine the center of mass of a semicircle using the definition of center of mass? I only know solve this using the Pappus theorem. Consider that the semicircle is centered on the origin and a homogeneous mass distribution.


title: Joule-Thomson effect of Van der Waals gas

text: I'm supposed to calculate the inversion pressure $p_i$ of a Van der Waals gas. The state equation of the Van der Waals gas is: $$(p + \frac{a}{V^2})(V-b) = RT.$$ To get a hold of the inversion temperature, I differentiate this equation with respect to $T$, while keeping $p$ constant. This gives me: $$\frac{(\frac{\partial V}{\partial T})_p \cdot (2ab - aV + pV^3)}{V^3} = R,$$ which with the condition for the inversion curve $T(\frac{\partial V}{\partial T})_p = V$ I get: $$\frac{1}{T_i}(\frac{2ab}{V^2} - \frac{a}{V} + pV) = R,$$ which gives me an easy to solve equation for $T_i$. But to get $p_i$, I'd have to solve the equation for $V$ and put it back into the state equation. But solving this equation for $V$ gets really messy. That's why I'm thinking I went wrong somewhere. Can you guys help me out here?


title: What exactly is meant by the conformal group of Minkowski space?

text: This is sort of a silly question because I'm a total beginner, and I debated whether it was better to ask here or on Math.SE. I decided on here because it's about how physicists use terminology, even though the terminology is intrinsically mathematical and there's not actually any physical content in the question. It's a well-known fact in theoretical physics that the conformal group of Minkowski space is $\operatorname{SO}(4,2)$ (or sometimes its double cover $\operatorname{SU}(2,2)$ is chosen instead). This is infinitesimally generated by the standard Poincare generators $M_{\mu \nu}$ and $P_\mu$ together with scale transformation $D$ and special conformal generators $K_\mu$. I've also seen conformal groups discussed for other manifolds (the case which lead me to this question is Witten's 2003 paper Perturbative Gauge Theory As A String Theory In Twistor Space, which includes the conformal group of $\mathbb{R}^4$ with all possible signatures.) However, the above doesn't seem to agree with the definition of the term "conformal group" which I know from the minimal amount of conformal geometry I know. Specifically, the conformal group of a pseudo-Riemannian manifold $M$ is the group of conformal self-maps on $M$, that is, self-diffeomorphisms for which the pullback metric is conformally equivalent to the original metric. (Two metrics $g$, $h$ are conformally equivalent if there is a positive real- valued function $u$ such that at each point $x \in M$ we have $g_{x, \mu \nu}=u(x)h_{x, \mu \nu}$.) The trouble is that the conformal group of Minkowski space isn't $\operatorname{SO}(4,2)$ with the above definition. The special conformal transformations, given by the expression (for some vector $b_\mu$) $$ x' ^\mu = \frac{x^\mu - x^2 b^\mu}{1-2b \cdot x +b^2 x^2}$$ aren't self-maps of Minkowski space at all. The denominator can become 0 (while the numerator remains finite), and at that particular $x^\mu$ the transformation given by $b^\mu$ is undefined. For an example, take $x=(1,0,0,0)$ and $b=(0,1,0,0)$. So in fact the special conformal transformations are only maps from a subset of Minkowski space back to the full Minkowski space, and shouldn't be included as part of the conformal group if we take the above definition. It's difficult to tell from this formula, but from the fact that $$ \frac{x'^\mu}{x'^2} = \frac{x^\mu}{x^2} - b^\mu$$ we can tell that the inverse transformation is just given by $-b^\mu$, another special conformal transformation, which means these maps aren't surjective either. These problems stem from the fact that the conformal group is typically derived in physics by considering the algebra of vector fields (which we think of as infinitesimal coordinate transformations) which satisfy the conformal Killing equation. There's in general no reason to expect that the corresponding finite transformation to be well-behaved away from the origin. **How, exactly, are conformal groups defined in physics for arbitrary (in particular noncompact) pseudo-Riemannnian manifolds so that we get the expected result for Minkowski space?** To be perfectly clear, I don't want answers of the form "find the Lie algebra, then exponentiate everything". I know that this works, but I want a definition for the conformal group as a group, without referencing algebras. In my opinion, there ought to be a way to define it this way, and it seems a bit important to me since _a priori_ I don't see any reason why the group of conformal self-maps of a manifold needs to be a (possibly infinite- dimensional) Lie group at all. There are probably some high-power theorems (perhaps in PDE) which would imply this trivially, but naturally we need to know an actual group structure to be able to apply them. In the case of the conformal group as I defined it above (so excluding special conformal transformations) I believe I know a way to prove this for the "mathematician" definition, but when we start throwing in maps that are not everywhere defined the approach I came up with fails. Hence, even if this is how most physicists would define it, I'm looking for an alternate global definition to get some better geometric intuition which produces the correct answers. * * * Some thoughts on this (feel free to ignore, especially if I'm on the wrong track): It seems to me possible to remedy this in a fairly trivial way. We can consider all injective conformal maps from dense open sets of Minkowski space to (possibly distinct) dense open sets of Minkowski space. We also need to define compositions to be restricted to the subset of Minkowski space on which they are defined. Furthermore, we impose an equivalence relation that two maps which restrict to the same map on the intersection of their domains are equal. Using these definitions, we can at least get a group which contains all of the conformal transformations it should. It also contains some other less desirable sorts of transformations (for instance, take two rotations and patch them together on the interior and exterior respectively of $\mathbb{R} \times S^2$; incidentally, this is essentially why the proof I have could potentially fail for the "physicist's" conformal groups), but by imposing some conditions on the behavior of the maps in a neighborhood of the boundary of the region on which they are defined it should be possible to get back the get back the ordinary conformal group. (I have not checked this, nor do I know exactly what conditions should be applied, though I could figure it out in the case of Minkowski space based on an alternate model below for the conformal group.) That approach, while easy to justify based on our above assumptions, is not very intuitive. It requires a lot of technical work to get everything to work out properly. We're implicitly using the fact that finite intersections of dense open sets are dense and open. The conditions required as the functions approach the boundary of their domains to get everything to work are also not obvious. More importantly, the whole thing seems to be suggesting an alternate approach which is much more geometrical; that we should view Minkowski space as contained within some slightly larger space on which the conformal group acts more nicely, where we view the complement of Minkowski space as points at $\infty$. Specifically (and at this point I'm just making conjectures because they seem helpful) for any psuedo-Riemannian manifold $M$, there ought to be some unique space $\bar M$ which has the properties that $M \subset \bar M$ is dense (this is based on the above observations, and might not be necessary or desirable in the general case; however, it seems reasonable), the restriction of the metric on $\bar M$ to $M$ produces a conformally equivalent metric to the standard metric on $M$, and such that all the conformal self-maps of $M$ which "should exist" can be legitimately interpreted as conformal self-maps of $\bar M$. I don't know that this actually implies $\bar M$ is compact, but certainly if $M$ is compact then the sorts of problems we run into for Minkowski space can't happen. We can then just define the conformal group of an arbitrary manifold $M$ to be the conformal group, in the "mathematics definition", of $\bar M$. A construction of $\bar M$ would thus be a perfectly reasonable answer to this question, though at this point I'm still not sure that any space $\bar M$ need exist in general. I briefly tried to do this myself, but I was not able to come up with anything general enough to be satisfactory, nor was I able to prove that the few cases I know of this construction are unique. I do know that this approach works for Minkowski space in some sense. Specifically, we can complete Minkowski space in this sense by including it into the projective light cone (i.e. the space of all null lines through the origin) in 6-dimensional Euclidean space with $(4,2)$ signature. The particular embedding is well-known and not so important for my purposes. However, this doesn't generalize particularly well when presented in this way. The reason why it works, at least intuitively for me, is that $\operatorname{SO}(4,2)$ naturally acts on this space, which to me seems like a big coincidence that can't be expected to continue to work if we consider arbitrary spaces. I considered using Penrose diagrams to try to do the equivalent construction for all of the cases I care about, but these only work for asymptotically flat spaces as far as I know and that misses some cases of interest for me (e.g. AdS), plus a general construction would be far more satisfying. I also considered the possibility that physicists may be using the term inconsistently outside the context of Minkowski space. That is, there may be a number of constructions of "conformal groups", all of which give the same $\operatorname{SO}(4,2)$ on Minkowski space. That would actually be more interesting for my purposes if it's true.


title: Understanding the concept of cross section

text: The cross section is a tiny area, measured in barns ($10^{-28} \mathrm{m^2}$), which relates to the probability of a reaction. But I have trouble interpreting that number. For example how can I calculate, when two deuterons collide what's the chance that they fuse? I read in forums that they much more likely scatter.


title: Is there a textile like material that can act as a spring?

text: Imagine some sort of textile (as in very light, not metallic) that is tightly rolled up, can be easily unrolled by applying some force (i.e pulling it) but when released will eventually return to it's rolled up state, over and over again. Does something like this exist?


title: Is there any general position function $x(t)$ that gives the solution to $x''(t) = k/x(t)^2$, where k is a constant?

text: In physics class, I often come across various inverse square law equations like the following:   * $F_G= G\frac{m_1m_2}{r^2}$   * $F_E = k_e\frac{q_1q_2}{r^2}$ Specifically, we are typically given problems involving one fixed body and one body that interacts with that body over a one-dimensional axis (in a manner given by one of the above laws). Thus, the acceleration at any given time of the free body can be modeled by the following differential equation, where the masses/charges/physical constants involved collapse into a single value, k. (This physics stuff is just included for some context.)   * $x''(t)=\frac{k}{x(t)^2}$ However, my exact question is, given any initial conditions for x(0) and x'(0), is there a general solution to the above equation? **In short, tell me about solving the subsequent equations for x(t)** please.   * $x(0) = \mathrm{some\, initial\, position} $   * $x'(0) = \mathrm{some\, initial\, velocity} $   * $x''(t) = \frac{k}{x(t)^2} $, where k is a constant Again -- please help, A Curious Student Who Doesn't Know Much _P.S. Additionally, how does the sign of k -- whether the force is attractive or repulsive -- affect the result?_


title: Number of Goldstone bosons in paramagnetic-to-ferromagnetic phase transitions

text: In paramagnetic-to-ferromagnetic phase transitions, the symmetry spontaneously breaks down from SO(3) to the subgroup SO(2) below $T_\text{crit}$. This implies that there should be two Goldstone modes and not one because SO(3) has three generators and SO(2) has one. How do we distinguish between these two excitations physically?


title: Redshifted photons

text: When a photon is emitted from a far away source and then measured by an observer, there is a loss of energy or redshift which takes place. Why does this happen? I have read this similar post, however the confusion I'm having is due to the invariance of photons, which travel at the speed of light, to non-inertial reference frames. If they are invariant then why would there be a perceived energy loss as it switches from one reference frame to the other?


title: Finite amount of matter and space

text: (Please bear with me.) If the universe included multi-universes, and black holes transferred matter between them, AND yet there was a large, but FINITE amount of matter and space, How would that max-limit change our understanding/perception or theories of the universe? Would it make any difference or change current concepts?


title: Projectile motion problem with upward acceleration and horizontal velocity

text: An electron in a cathode-ray tube is traveling horizontally at 2.10×10^9 cm/s when deflection plates give it an upward acceleration of 5.30×10^17 cm/s^2 . B.) What is its vertical displacement during this time? My work http://i.imgur.com/AqKlTuX.jpg I am asking this because the answer my teacher gave is 2.3cm Thanks in advance


title: Is there a table that sums up the parameters, the assumptions/symmetries, and the predictions of the standard model?

text: The title says it all: is there a table that sums up the parameters, the assumptions/symmetries, and the (most important) predictions of the standard model?


title: Temperature of air effused into tank - different predictions?

text: Consider air from outside at $T_0$ effusing into an evacuated and thermally isolated chamber. By thermodynamic potentials, the temperature inside the tank is given by: $$T_1 = \gamma T_0 = \frac{5}{3} T_0$$ However, by using Maxwell-Boltzmann statistics, the distribution of particles effusing is $\propto v^3 exp(-\alpha v^2)$. Calculating the average KE, we find that $$T_1 = \frac{4}{3}T_0$$ Why is there a difference between the two?


title: Projectile along incline

text: A particle is projected with velocity $u$ from the bottom of an inclined plane whose inclination with the horizontal is $\beta$. If afterwards the projectile strikes the inclined plane perpendicular to the surface, find the height of the point struck (distance from the ground). The angle made by the velocity with the incline is $\alpha$. I got an answer while solving this question that differs from the one in my textbook. I got the correct value for the time period though... This is what I did: ![enter image description here](http://i.stack.imgur.com/1F2MI.jpg) Could someone please try this question? Please help me find the height and also tell me the expression you get for the time taken. My textbook shows ![enter image description here](http://i.stack.imgur.com/7ZqKe.jpg)


title: Possible ambiguity in using the Dirac Delta function

text: When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this constraint. For example, when calculating $$\displaystyle\int f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z$$ subject to the constraint $$g(x,y,z)=0,$$ one may calculate instead $$\displaystyle\int f(x,y,z)\delta\left(g(x,y,z)\right)\mathrm{d}x\mathrm{d}y\mathrm{d}z.$$ The ambiguity here is that, instead of $\delta\left(g(x,y,z)\right)$, one may as well use $\delta\left(g^2(x,y,z)\right)$, $\delta\left(k g(x,y,z)\right)$ where $k$ is a constant, or anything like these to account for the constraint $g(x,y,z)=0$. But this leads to problems, since the result of the integration will surely be changed by using different arguments in the $\delta\text{-function}$. And we all know that the $\delta\text{-function}$ is not dimensionless. My impression is that many physical books use the $\delta\text{-function}$ in a way similar to the above example. The most recent example I came across is in "Physical Kinetics" by Pitaevskii and Lifshitz, the last volume of the Landau series. In their footnote to Equation (1.1) on page 3, there is a term $\delta(M\cos\theta)$ to account for the fact that the angular momentum $\mathbf{M}$ is perpendicular to the molecular axis. But then, why not simply $\delta(\cos\theta)$ instead of $\delta(M\cos\theta)$? One may say that when using $\delta(\cos\theta)$, the dimension of result is incorrect. Though such an argument may be useful in other contexts, here for this specific example the problem is that **it is not clear what dimension should the result have** (the very reason for me to have this question is because I don't quite understand their Equation (1.1), but I am afraid that not many people read this book). To be clear: I am not saying the calculations in this or other books using the $\delta\text{-function}$ in a way similar to what I show above are wrong. I am just puzzled by the ambiguity when invoking the $\delta\text{-function}$. What kind of "guideline" one should follow when **translating** a physical constraint into a $\delta$-function? Note that I am **not** asking about the properties (or rules of transformation) of the $\delta\text{-function}$. **Update** : Since this is a stackexchange for physics, let me first forget about the $\delta\text{-function}$ math and ask, how would you understand and derive equation (1.1) in the book "Physical Kinetics" (please follow this link, which should be viewable by everyone)?


title: A 0.1kg ball of dough is thrown up with a velocity of 15m/s. What is the momentum halfway up?

text: I know that $p=mv$ and (0.1kg)(15m/s)=1.5 kg m/s and the momentum at the vertex is 0, but what is the momentum halfway up?


title: Quantum Philosophy a la John Bell

text: I recently discovered this website http://www.quantumphil.org/ and wondering whether Quantum Philosophy is an actual field, or just an aspect of QM? Apologies if this is in the wrong place.


title: Pressure vs Volume Calculation for air

text: I have 1 litre of air in a sealed container at atmosphere pressure (approx 15 psi absolute). I want to reduce pressure in the container by 1 psi ( 14 psi absolute or -1 psig).   * If I understand correctly that 1 psi reduction is 6.66 % reduction in pressure compared to standard atmospheric pressure,    * Applying the same to volume would it be correct to say that if I increase the volume of the container by 6.66 % the pressure would drop by 1psi for air OR   * If i remove 6.66% volume of air in the container I would still be able to achieve the 1psi drop in pressure? Do some other factors come in to play for pressure / volume calculation...?


title: Force and Newton seeking intuitive explanation

text: Force is explained as a push or pull, feels quite intuitive at first. My question is if an object accelerates over distance of $100$ meter hits a car, and another object of same mass at same acceleration over a distance of $1000$ meters hits the same car, then the force of these two objects is the same, since mass and acceleration of the two objects are the same. Yet, the "push" felt would be very different, because surely the second object will "push" the car much "harder" and "farther" than the first. Applying another physic equation, work, forced over distance, may explain this, but I am left not truly understanding what/where is force in everyday situations - how can force be understood as a "push" if the push felt is different from force of the same magnitude? My second question is the basic math for Newton unit. My understanding of multiplication is a way of counting: unit per group multiplied by number of groups yielding total number of units. How/what's the best way to understand the math of physics composite units, such as force: the product of $\text{kg}$ and $\text{m/s$^2$}$ yielding a composite unit, $\text{N}$? Within the context/definition of multiplication, What am I actually counting with two different mixed units?


title: How about destroying a black hole?

text: Can we destroy a black hole? I know it is pretty powerful and hardcore stuff but there should be a way even theoretically as we know know nothing is perfect in this universe as it exist due to imperfection. Then how could a black hole be so perfect, so near to god power?


title: Covariant Derivative with a Torsion Free Metric

text: Where $\triangledown$ is the covariant derivative and we are to assume that the connection is torsion free (that is, we can exchange the lower indices of the connection coefficients), how can I prove the following identity? $$ X^b\partial_bY^a-Y^b\partial_bX^a=X^b(\triangledown_{{\partial}_b}\overrightarrow{Y})^a-Y^b(\triangledown_{{\partial}_b}\overrightarrow{X})^a $$ I am struggling quite a bit with this. If anyone has any input on how to show this, please let me know.


title: What are the technological challenges to building a muon collider?

text: There is a strong physics case for building and running a muon collider (primarily as a Higgs factory). Currently, what is the principal technological hold-up in making the muon collider possible?


title: Ballistic Conduction - Difference between Bosonic and Fermionic Transport

text: Ballistic Conduction is the phenomenon of an ideal conduction environment for quantum particles - for electrons the Ballistic Conduction is not infinity, but is proportional to the difference between Fermi surfaces of the leads on both sides.   One of the reasons, it seems to me, is that electron is a Fermion and cannot occupy the "transport state" that spans the length of the leads - if there is another electron occupying it. Supriyo Dutta also argues for this in terms of chemical potential.   It seems to me then that for bosons like phonons or photons, if there exists even one state spanning the leads, there should be an effective short circuit and the transmission would be the theoretical maximum. Is that true?


title: Fluid mechanics of aircraft at 10000 m

text: ![question](http://i.imgur.com/hXoK3PP.png) ![answers](http://i.imgur.com/9o7hB3h.jpg?1) I have come up with the answers to parts a and b as shown in the image above. However in part c, I can't seem to find the answer, even though it only involves fitting numbers into Bernoulli's equation. $P+\frac{\rho u^2}{2}+\rho g z=const$ Is this because I am using the wrong relative velocity?


title: How to understand movement in expanding universe?

text: I know that universe is expanding equally between every pair of points but it was a single point in it's very past... so I was wondering if we could locate this center point of universe. Now I know that we can't and read (on this site) that whatever direction would we go - it would take us closer and closer to big bang... it makes me wonder - if we will trying to get there, there will be our new starting point so going back should take us even closer to this big bang but well, we should have the opportunity to go back to our 1st starting point and it was "farther" before, not closer at all. Also, as the universe is still expanding and the stars will eventually run out of fuel, would it be possible to try to escape from the situation by flying ahead the big bang and reaching parts of universe that have their state similar to past of "our part of the universe"? Does it mean that the universe is creating itself all the time and dying all the time too?


Is Wick rotation invariant under proper conformal transformations? Why or why not? Does Wick rotation apply to conformal field theories? $(1-i\epsilon )T$ is not invariant under proper conformal transformations.


title: Inclusion of information about external particles to calculate scattering amplitudes

text: In this (schematic) equation to calculate the scattering amplitude A by integrating over all possible world sheets and lifetimes of the bound states $$ A = \int\limits_{\rm{life time}} d\tau \int\limits_{\rm{surfaces}} \exp^{-iS} \Delta X^{\mu} (\sigma,\tau)$$ the information about the incoming and outgoing particles is still missing. It has to be inserted by hand by including additional multiplicative factors (vertex operators) $$ \prod\limits_j e^{ik_{j_\mu} X^{\mu}(z_j)}$$ into the integrand as Lenny Susskind explains at about 1:18 min here. But he does not derive the fact that the information about the external particles can (or has to) be included by these additional multiplicative factors like this, he just writes it down. Of course I see that these factors represent a particle with wave vector $k$, and $z$ is the location of injection (for example on the unit circle when conformally transforming the problem to the unit disk) over which has finally to be integrated too. But I'd like to see a more detailed derivation of these vertex operators (do there exit other ones too that contain additional information about other conserved quantities apart from the energy and the momentum?) and how they go into the calculation of scattering amplitudes, than such "heuristic" arguments.


title: Control cable failure and critical flutter speed margin?

text: I'm a private pilot, and I have some questions to those who have knowledge of the aeroelastic effects and flutter phenomenon. I would like to talk a little about aerodynamic flutter onset speed and flight control malfunction. It is known that freeplay, worn-out control rods or slop in flight control cables might induce flutter. What I'm interested in is how critical flutter speed is affected by those problems. I'm wondering especially about a cable control failure where the surface would be disconnected and freefloating. There are small light sport aircrafts and even some FAR 23 standard certified aircrafts which don't have mass-balanced surfaces, especially ailerons which I guess would be more prone to flutter. How critical flutter speed lowers in a situation like that (and how prone to violently flutter are these ailerons in an emergency disconnected sitation)? Is there a linear drop in flutter speed? Can it reach even lower speeds in the normal operating envelope e.g. below usual cruise speed? Normally, assuming no malfunction, flutter speed is at least 10% above Vne (never exceed speed) or Vdf (max dive test speed used during certification flight test). How do you think these things might change? The same about the others control surfaces e.g. a broken trim tab linkage. I really appreciate your help. Thank you very much!


title: What is known about quantum electrodynamics at finite times?

text: I'm aware that we can describe the time evolution of states/operators (choose your favourite picture) of non interacting quantum fields and that perturbation theory is very effective in computing S matrix elements between free states in the remote past and free states in the remote future. Clearly the non-perturbative description of what's going on at finite times, where the interaction is active is intractible, but my question is - are there simplified toy models (scalar electrodynamics ? reduced numbers of dimensions ?) where we _can_ describe what's happening non perturbatively. Even if nothing like electrodynamics has been treatable like this, any results on the other "textbook" QFTs (like $\phi^4$) would be interesting.


title: Disconnected aileron/surface deflection in terms of speed?

text: Would a disconnected surface, especially aileron, deflect upwards as you slow down due to increased alpha? I figure out it is more likely to deflect upwards as you increase your airspeed, thus having more airflow pressure below it and pushing it upwards.


title: Power dissipated in Series versus Parallel

text: Do two resistors in parallel dissipate more heat per unit time for an applied voltage when compared to two resistors in series?


title: light beams of the sun

text: > We receive sunlight on earth surface. What type of light beams are these? > > Random/Parallel/Converging/Diverging I think it should be Diverging as Sun is radiating these beams away. But in one book, answer is given as Random, in another it's Parallel.


title: Is it possible to have uniform proper acceleration along a large object without breaking it?

text: I'm learning about accelerating reference frames (to eventually get grasp of general relativity too). I've just read about the Rindler coordinates and this one caught my eye > Note that Rindler observers with smaller constant x coordinate are > accelerating harder to keep up! This may seem surprising because in > Newtonian physics, observers who maintain constant relative distance must > share the same acceleration. But in relativistic physics, we see that the > trailing endpoint of a rod which is accelerated by some external force > (parallel to its symmetry axis) must accelerate a bit harder than the > leading endpoint, or else it must ultimately break. This is a manifestation > of Lorentz contraction. As the rod accelerates its velocity increases and > its length decreases. Since it is getting shorter, the back end must > accelerate harder than the front. So does this mean, that in Einstein's elevator (that's used to introduce general relativity, and is supposed to accelerate uniformly) the acceleration felt by the observer on the top is smaller than the acceleration felt by the observer at bottom?


title: What is the Coandă effect? How is it defined, and what causes it?

text: Can anybody explain to me the Coandă effect well? I am finding that many definitions and explanations conflict. I am particularly confused about the following points regarding the Coandă effect:   * whether the Coandă effect is the same as normal boundary layer attachment to a surface   * whether the Coandă effect is only defined for convex surfaces over which a jet (of the same state as the surrounding fluid) is tangentially blown   * what causes the Coandă effect: the jet entraining the ambient fluid or a balance of pressure and centrifugal forces, or something else?


title: What Material for reflective surface of a solar grill/barbecue?

text: I have been planning on building a solar grill for quite a time now, and the one problem, which is also the biggest - I need to shape a mirror into a paraboloid, so it bundles solar energy efficiently. My designs for the whole thing have the focal area of the paraboloid - i.e. the grillage - placed in close proximity of the mirror, depending on the design between 1 and 1.20 meters. I am afraid that the heat radiating from the focal area would heat up the reflective surface too much, especially if I use an easy-to-shape material like acrylic mirror, which would without a doubt deform after a couple of minutes. So long story short, I want to know if there is a material that   1. Has high reflective properties   2. Does not deform at temperatures of 300°C or even higher (400°C to be on the safe side, if the BBQ is going on for hours and to calculate in unexpected risks)   3. Is reasonably easy to shape (it should be possible without having to handle gigantic machinery or overly dangerous tools)   4. Can be paid for by a poor engineering-student, so the amount I need should be below 200$. If possible. (a griller is pointless if I have to eat rice for the rest of the year) Ps.: English is not my first language, so there likely are a lot of mistakes in there. If something is unclear, please comment! I apologize for those mistakes beforehand.


title: Why the heat flux vector at a point must be perpendicular to the temperature isothermal surface? Is it a definition or a deduction?

text: Before the question: I am working on numerical calculation of three dimension parabolic equation that based on Fourier's Law of which I am a little confused. Here comes the law in modern mathematics language. > "The local heat flux is proportional to temperature gradient" $$ > \vec{q}=-k\nabla T, $$ where $k$ is the material's conductivity. How extremely concise it is, but how to understand the Law? I read the book written by Fourier in 1822 but I know neither of the law in modern mathematics language nor in Fourier's language. I found that every statement or formula related to proofing the Law are not done with rigour enough. Here is some statement from a book by YUNUSA.CENGEL on its page 65 chapter 2. > To obtain a general relation for Fourier's law of heat conduction, consider > a medium in which the temperature distribution is three-dimensional. The > figure below shows an isothermal surface in that medium. **_The heat flux > vector at a point $P$ on this surface must be perpendicular to the surface, > and it must point in the direction of decreasing temperature._** If $n$ is > the normal of the isothermal surface at point $P$, the rate of heat > conduction at that point can be expressed by Fourier's law as $$ \dot{Q_n} = > -kA\cfrac{\partial T}{\partial n} $$ ![enter image description here](http://i.stack.imgur.com/Aa8d5.jpg) My questions toward the issue I have mentioned are   * How could heat flux be a vector?   * What's the meaning of the direction of the heat flux?   * Why heat flux at a point is normal to the isothermal surface?   * What's the definition of heat flux vector, not heat flux which is defined as the quantity per second per area? You might say it is true just because of the Second law of thermodynamics. > The heat always flows spontaneously from regions of higher temperature to > regions of **lower temperature** , and never the reverse, unless external > work is performed on the system. It is **lower** but not **fastest decrease** , isn't? If the direction is not through the line in the tangent plane of the isothermal surface it would transfer to a colder palce, isn't? So why choice the normal line to be the heat flux direction since there is infinity line to the colder place. Maybe it is that project works when considering the other line! However, it is human beings not the nature define the direction of heat flux for convenient. Am I right? It may be related with Fick's Law. I am not sure about the proof of three dimension situation.


title: Weibull distribution

text: The Weibull distribution probability density is given by: $$f(w,k,\lambda)=\begin{cases}\frac{k}{\lambda}\left(\frac{w}{\lambda}\right)^{k-1}e^{-\left(w/\lambda\right)^k} &w\geq0 \\\ 0 & w<0\end{cases}$$ How can I modify the Weibull distribution to apply in a location where the wind is dead calm ($w = 0$) a fraction $\nu$ of the time.


title: What's wrong with this QFT thought experiment?

text: In quantum field theory, the propagator $D(x-y)$ doesn't vanish for space-like separation. In Zee's book, he claims that this means a particle can leak out of the light-cone. Feynman also gives this interpretation. What is wrong, then, with this thought experiment: Bob and Alice synchronize their watches and space-like separate themselves by some distance. Bob tells Alice that at exactly 3 o'clock, he intends to test his new particle oven, which claims to make a gazillion particles. Since Bob's oven makes so many particles, there is a pretty good chance that Alice will detect a particle right at 3 o'clock. If she detects such a particle, Bob will have transmitted a piece of information (whether or not the oven works) instantaneously - way faster than the speed of light.


title: Work done by vibrating string

text: ![enter image description here](http://i.stack.imgur.com/qBLOQ.png) figure from http://webusers.physics.illinois.edu/~m-stone5/mma/notes/amaster.pdf The string has fixed ends, a mass per unit length of $\rho$, and is under tension $T$. This source claims that "the rate that a segment of string is doing work on its neighbour to the right" is $$ -T\dfrac{\mbox{d} y}{\mbox{d} x}\dfrac{\mbox{d}y}{\mbox{d} t} $$ without any justification. I have no idea how this is derived. I would like to see a simple (if possible) derivation. I wonder in what way can the a segment of the string $dx$ can do work to the neighboring segment on the right since in the context of this problem there is no movement in the x-axis (right?).


title: Repost with correct format: Potential Energy of an Electron in a Dielectric Container

text: This is a repost of one of my formerly-stated questions, now with correct tags and format: In reference to this problem: > An electron is placed into a dielectric container with capacitors at each > end. The side of the container is of length l. A small tunnel is hollowed > out in the middle, which is where the electron is contained. The whole > container has charge Q and the capacitors supply a potential difference > $\phi$ across the material. Find the total potential energy (not just > electric potential) of the electron if we measure x from the center of the > tunnel. In lecture, the Professor said that the equation for the potential is $$\frac{1}{2}Kx^2-\frac{|e|\phi}{l}x$$ where $|e|$ is the magnitude of the electron's charge. However, he neglected to state the derivation as well as what exactly K was in the equation. Can someone please help me understand?


title: Elastic vs Inelastic vs isospin violating scattering particle physics models

text: I'm looking for a nice paper that explains the difference between three particle physics models for spin-independent dark matter interaction with nuclei: elastic, inelastic and isospin violating scattering. I've found a nice paper that is giving a nice summation of the current results in direct dark matter searches ( http://arxiv.org/abs/1210.4011 ) and want to read up some more. Cheers, Adnan


title: Conserved quantities and total derivatives?

text: I am having a bit of a crisis in understanding of the physical meanings of total derivatives. When a quantity $\rho$ (be it a _vector_ or a _scalar_ ) is said to be **_conserved_** , then (mathematically) $$\frac{d\rho}{dt} = 0$$ (right??) Now, if I just integrate both sides with respect to time I get $\rho$ = constant. But the total derivative can be written as (via the chain rule) $$ \frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \mathbf{u}\cdot\nabla\rho$$ where I am not even sure what $\mathbf{u}$ is (the speed of a moving reference frame?). Given these equations, if the former has $\rho$ = constant as a solution, then what is the point of partial derivative and gradients? They are all going to be zero anyway? And what exactly is the **physical meaning** of the total derivative? In _fluid dynamics_ and _plasma physics_ I have been told that it describes how a quantity changes when observed from the frame "moving with the fluid"...


title: Does sound gets faster when air bubble is supend in water?

text: Does sound gets faster when air bubble is suspend in water? c = sqrt(K/P) c = speed K = bulk module P = density When air bubbles is homogenized into water the density is lower, so should sound gets faster? Thank you.


title: Eckert number in incompressible flow

text: Most of the time, temperature evolution for incompressible flow is computed assuming the Eckert number is negligible, that is the viscous dissipation and work of pressure variation is negligible compared to convection and diffusion. However, if a flow is assumed isothermal, work of pressure fluctuation and viscous dissipation are the only terms remaining in the energy conservation equation. In such a case, the isothermal assumption leads to an equilibrium between both and the pressure is ruled by a transport equation where viscous dissipation is a source term. Most of the incompressible numerical solver compute the pressure as the constraint to ensure $\nabla\cdot\mathbf{u}=0$. How compatible is pressure as a incompressibility constraint with pressure in equilibrium with viscous dissipation?


title: Can He-4 atoms create black holes?

text: Suppose that in the intergalactic space far from any significant gravitational attractors there is a relatively small concentration of He-4 atoms. Due to gravitational attraction fermions in this case would form a sphere (gas planet), but as bosons He-4 atoms aren't affected by exclusion prinicple, so what's holding them back from collapsing into singularity?


title: Why is the charge naming convention wrong?

text: I recently came to know about the Conventional Current vs. Electron Flow issue.   Doing some search I found that the reason for this is that Benjamin Franklin made a mistake when naming positive and negative charges.   There is even this little comic about that http://xkcd.com/567/ My question is, how can a naming convention be wrong??   Perhaps I don't understand what is the actual mistake here, I don't know. But I don't see how a naming convention could be wrong or right. There is no right and wrong about that. It could've been any other name, like charge A and charge B. I'll appreciate anyone can help me understand what is wrong in all this.


title: Why are electrons defined to have negative charge?

text: We normally think of the "default" or "root" state of things as being on the positive side of the spectrum. For example, we don't normally use a `+` symbol to indicate the sign of positive numbers, whereas negative numbers are invariably preceded with `-` to indicate their negative sign. Similarly, positive temperatures (in °C or in F) are those that we tend to encounter more typically. Why, then, do we define the basic unit of _electricity_ as something with a negative charge?


title: Who (and Why) started the "electrons are negative, protons are positive" convention?

text: For some reason everyone labels electrons using a minus sign and protons using a positive sign, even though the opposite seems more intuitive: ![enter image description here](http://i.stack.imgur.com/Palaa.png) ![enter image description here](http://i.stack.imgur.com/RTgX7.png) Who started the convention that electrons should be "negative" and protons should be "positive"? What's the reason for choosing so?


title: positive charge , current and electron flow in a simple circuit …confusion

text: my question is that in a simple circuit one wire attached with battery cell ,and then electrons start flowing from lower potential to higher and as we know in metal wire only electron is the thing which is carrying charge ,,, then why we say there is current and it flow opposite to electron ??? even there is no other thing , please dont go in other stuff yeah i know positive charge also moves some where like alpha particals in some cases but here in simple circuit why they dont just say electron flow ? why current and why its opposite ???


title: Topological Mass Generation Mechanism

text: 1. What is the topological mass generation mechanism?    2. And what is its relation with the Higgs mechanism?    3. Can we say that after the discovery of Higgs boson, the topological mass generation mechanism is ruled out?


title: Does the Earth's revolution around the Sun affect radioactive decay?

text: **Premises:** >   1. The radioactivity is either hastened or slowed inside a fast moving > aircraft. >   2. Speed of fastest aircraft: _3,529.6 km/h_. >   3. The earth's revolution is: _107278.87 km/h_. >   4. The earth's revolution is significantly higher than the speed of moving > aircraft. > **Questions:** >   1. Are all the premises correct and if not, which part is not correct? >   2. Does this mean that the earth's revolution affects modern dating > methods? > **Sources:** > Radiometric Dating, Creationism, Aircraft Speed, Revolution Speed


title: Total momentum of the Universe

text: What is the total momentum of the whole Universe in reference to the point in space where the Big Bang took place? According to my reasoning (and a bit elementary knowledge) it should be exactly equal to 0 since the 'explosion' and scattering of the matter throught the space would not change the total momentum in any way.


title: Relationship between the fall time and projected area

text: I am investigating the relationship between the fall time of a paper tray and its projected area. In this investigation I have been using the following variables:   * Controlled Variable: The air density, the height from which we drop the tray, the shape of the tray, its mass and the material it is made from (paper).   * Independent Variable: The cross-sectional area of the paper tray.   * Dependent Variable: The time it takes for the tray to fall on the ground * * * I found the relationship for when the paper tray is traveling at terminal velocity. This relationship I devised by using the formula for the speed of a object traveling at terminal velocity: ![Terminal Velocity](http://i.stack.imgur.com/YvjyR.png) Terminal velocity can be considered uniform motion (acceleration = 0) and so the above formula can be rewritten in the following way: ![Time Squared vs Projected Area at Terminal Velocity](http://i.stack.imgur.com/47JVL.png) * * * But all this excludes the acceleration period before a object reaches terminal velocity. I suppose that because the paper tray is only 0.016 kg and the paper tray has a fairly high drag coefficient 1.31 that the acceleration period can be ignored since the tray reaches terminal velocity quite soon, but then again the trays are being dropped from only 2 meters. * * * Can I ignore the acceleration period? Is there a better equation for the relationship? Can you suggest further reading?


title: Semileptonic decays of the $B_c$ meson

text: I am struggling with calculating the exclusive semileptonic $B_c^+\rightarrow J/\psi l^+\nu_l$ decay. I learnt that the amplitude is given by a product of the leptonic current $L^{\mu}$ and the hadronic current $H^{\mu}$ $$ \mathcal{M}(B_c\rightarrow J/\psi l^+\nu_l)=\frac{G_F}{\sqrt{2}}V_{cb}L^{\mu}H_{\mu} $$ where $V_{cb}$ is the CKM parameter, $L^{\mu}$ and $H^{\mu}$ are expressed as $$ L^{\mu}=\bar{u}_l\gamma^{\mu}(1-\gamma^5)v_{\nu},\quad H^{\mu}=\langle J/\psi|J^{\mu}(0)|B_c\rangle $$ where $J^{\mu}$ is the $V$-$A$ weak current. However, I did not know how this result can be derived. Could anyone provide some help? There is a second problem. On the tree-level, we have the following Feynman diagram ![enter image description here](http://i.stack.imgur.com/l84vp.png) If we calculate $\bar{b}\rightarrow\bar{c}l^+\nu_l$ as a three-body decay in the electroweak theory (not the four-fermion approximation adopted above), how does it relate to $B_c^+\rightarrow J/\psi l^+\nu_l$?


title: How does one determine black hole mass based on the curvature of a photon?

text: 1. How much mass would a black hole need to create a Schwarzschild radius that would trap a photon, whereby the photon would (to an outside observer) be continually curved 0.004km/s at the horizon? (assume a non-spinning black hole, then contrast that with a photon traveling in either direction along the equator of a spinning black hole) Eg. [![Example](https://drive.google.com/uc?export=download&id=0B7P471Xpxoc1N2FoN1FtRmttV2c)]


title: What is meant by incompressible flow?

text: What is meant by incompressible flow?   1. The density of the fluid is a constant, $\rho = constant$   2. The density of a fluid has a spatial dependence but remains constant in time, $\rho = \rho(\mathbf{r})$ In both cases $\rho$ should satisfy $\frac{d\rho}{dt} = 0$ right? (In case 1 is true, how do you call a flow with the properties of 2? How would it be mathematically different?)


title: Why isn't the $\lvert\phi(x)=0\rangle$ Fock eigenstate the vacuum state?

text: If, in a QFT of a scalar field $\phi$, a Fock space $n$-particle position eigenstate $\lvert x_1\cdots x_n\rangle $ is given by $$ \lvert x_1\cdots x_n\rangle =\hat\phi^\dagger(x_1)\cdots\hat\phi^\dagger(x_n)\lvert 0\rangle \,, $$ where $\lvert 0\rangle $ is the vacuum state, then we have $$ \langle x_1\cdots x_n\lvert\phi\rangle =\phi(x_1)\cdots\phi(x_n)\langle 0\lvert\phi\rangle \,, $$ with $\hat\phi(x)\lvert\phi\rangle =\phi(x)\lvert\phi\rangle $. Now, the question is what is the value of $\langle 0\lvert\phi\rangle $? If there's a vacuum configuration state $\lvert\phi_0\rangle = \lvert0\rangle$, then $\lvert 0\rangle $ is orthogonal to any other configuration in the basis, and we would have $$ \langle x_1\cdots x_n\lvert\phi\rangle = 0 $$ except for the case $\langle 0\lvert 0\rangle $ and thus a joined basis $\\{\lvert x_1\cdots x_n\rangle ,\lvert\phi_0\rangle \\}$ for a space of higher dimension than the Fock space. So, I conclude there's no vacuum configuration. But the zero configuration, $\phi(x)=0$, is anyway orthogonal to every Fock space state $\lvert x_1\cdots x_n\rangle $ different from $\lvert 0\rangle $, and thus excludes the presence of any particle. In other words, $$ \lvert\phi(x)=0\rangle=\lvert0\rangle\langle0\lvert\phi(x)=0\rangle $$ So, the zero field configuration corresponds to the vacuum state, a contradiction; what am I missing? What's wrong in all this?


I have a simple question: Does gravity slow down a horizontally thrown baseball? Assuming when a baseball is thrown it has a vertical velocity as well, does it slow does the ball? Any help is much appreciated.


title: Question on conserved quantities and Noether's theorem

text: I have a question about Noether's theorem in the context of QM, which I'll state in the context of the weak interaction but the basic point could be generalized. According to Noether's theorem, given an $n$-dimensional Lie group there will be $n$ conserved quantities. $SU(2)$ is 3-dimensional, so that we'd expect 3 such quantities. However, elsewhere the conserved quantities are defined as the 'good quantum numbers', where these are defined as the eigenvalues of the maximal number of commuting generators in the group. In this case, there is just one such generator, and so it seems only one conserved quantity. Can any one tell me where I'm going wrong?


title: Finding the tangential force experienced by a bob of mass m on a simple pendulum via the gradient/nabla operator)

text: The problem was posed as follows. Given a pendulum of length $L$ with a mass $m$ attached to it, which forms an angle $\theta$ from the y-axis to the direction of swinging. First we had to find the potential energy as a function of the angle, which was trivial enough. $$U = mgL(1 - cos(\theta))$$ But the next question was to develop the gradient for this potential energy function via the arc length $s = L\cdot\theta$, where $s$ is the arc length. I calculated the gradient by using the trigonometric definition of $\cos(\theta)$, but I developed the gradient by changing the variable of my function $U(\theta)$ to $U(x,y)$ after which I calculated it further, and found that $F_{tan} = -mg\sin(\theta)$, as desired. But is there an alternative way to calculate it, by using the gradient, and by utilizing the arc length formula above? The question explicitly asks to develop the gradient via the arc length rather than $x$ and $y$ components.


title: Correct terminology for combined kinematic and dynamic state

text: The _kinematic_ state is defined as the position and orientation in space. The _dynamic_ state is defined as the associated velocities. What is the correct terminology for the combined kinematic and dynamic state? Can I call it the _kinetic_ state?


title: What is the largest diameter an optical fiber can have? - for nonimaging visible light transmission

text: Is there any theoretical limitation on the size of the diameter of an Plastic optical fiber? I would like to transmit visible sunlight through it. I see bundles of small cores available but wouldn't it be more efficient to make a very large single core?


title: Non-local Lagrangian contact interaction

text: Conside a contact interaction given by a delta function on their worldlines. Use a gauge fixed Lagrangian for two point particles in terms of their proper times $t$ and $t^{\prime}$. **Is it possible to find proper equations of motion for this system?** For e.g. $$ S=\int dt \; \dot {x}^{2}+\iint\\! dt \;dt^{\prime}\; \dot{x}(t)\cdot\dot{y}(t^{\prime})~ \delta^{D}(x(t)−y(t^{\prime}))+\int dt^{\prime} \; \dot{y}^{2} $$ working in, say, $D$ target space dimensions. This is written as a simplification of the relativistic / curved space system to illustrate the point. I think it will be possible to integrate the delta function out in $D=1$, but not in higher target space dim. The problem I have is in how to do the variation of the delta function term. Physically it is producing an interaction every time the worldlines of the particles intersect and I've tried writing this as a sum over such points - where $x(t_{0})=y(t^{\prime})$ - of $\frac{\delta(t−t_{0})}{\dot{x}(t_{0})}$ but this is valid only in $D = 1$ and I'm still not sure I can get the variation correct.


title: Can't understand the principle of least action

text: I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone please explain it to me easily? Comment: for example if I have in _énoncé_ , light travel between point $a$ and point $b$ in time $s$, we should formulate the equation of motion where the path is the shortest such that it takes $s$ time to travel from point $a$ and point $b$ by that time? Also how does this relate to Newtonian mechanics and how is it better?


title: What is the name of this principle?

text: What is the name of this principle? > If the speed of the centre of mass $\vec{v}_{CM}$ of a solid is constant > ($cte$), then the sum of the exterior forces that exerts into this solid > $\vec{F}_{ext}$ is zero, and the opposite is true, namely that: > > $$\vec{v}_{CM}=cte\Leftrightarrow\sum\vec{F}_{ext}=\vec{0}$$


title: Navier-Stokes equations: conservation of momentum

text: The first Navier-Stokes equation (conservation of mass) says: $\vec \nabla \cdot \vec v=0$ For a stationary flow, the l.h.s of the second equation is (conservation of momentum): $\rho \frac{D\vec v}{Dt}=\rho (\underbrace{\frac{\partial \vec v}{\partial t}}_{=0} + (\vec v\cdot \vec \nabla) \vec v)=\rho ( \underbrace{(\vec \nabla \cdot \vec v)}_{=0??} \vec v)\stackrel{??}{=}0$ I find that the l.h.s of the conservation of momentum equation is always equal to zero for a stationary field. I know this isn't true but where am I wrong in this reasoning ?


title: Resonance force with multiple emitters

text: Will I get a stronger resonance if I put multiple frequency generators around the same object? example image


title: Infinitesimal transformations for a relativistic particle

text: The action of a free relativistic particles can be given by $$S=\frac{1}{2}\int d\tau \left(e^{-1}(\tau)g_{\mu\nu}(X)X^\mu(\tau)X^\nu(\tau)-e(\tau)m^2\right).$$ If we then make an infinitesimal transformation of the parametrization parameter $\tau$ this would be $$\tau\to\tau^\prime=\tau-\eta(\tau),$$ for an infinitesimal parameter $\eta(\tau)$. Of course we can describe the system as it pleases us so we know that $$X^{\mu^\prime}(\tau^\prime)=X^\mu(\tau).$$ From this relation we see that $X^{\mu^\prime}(\tau)$ must be $$X^{\mu^\prime}(\tau) \approxeq X^{\mu^\prime}(\tau^\prime+\eta(\tau)) \approxeq X^{\mu^\prime}(\tau^\prime)+\eta(\tau)\frac{d X^{\mu^\prime}(\tau^\prime)}{d\tau^\prime} \approxeq X^{\mu}(\tau)+\eta(\tau)\frac{d X^{\mu}(\tau)}{d\tau}$$ Which all can be summarized as $$\delta X^{\mu}(\tau)=X^{\mu^\prime}(\tau)-X^{\mu}(\tau)=\eta(\tau)\frac{d X^{\mu}(\tau)}{d\tau}.$$ Now this is all well I hope. But if one does the same argument for $e(\tau)$ one gets the wrong transformation. It is a scalar function so it has to obey $$e^\prime(\tau^\prime)=e(\tau).$$ Which would give the same transformation. The right transformation are written in David Tong's lectures on string theory on page 13, eq. 1.10. The transformation is $$\delta e(\tau)=\frac{d}{d\tau}\left(\eta(\tau) e(\tau)\right).$$ Could someone show me how this is done and elaborate a little on how one knows how different object transforms?


title: S-Matrix Elements in Path Integral Formalism

text: I have a question related to the connection between the S-Matrix elements and the path integral formalism. In order to formulate the question, I will just work with a scalar field theory for simplicity. Let us assume that we are given an action $S[\phi]$. In the path integral formalism, we can now define the generating functional \begin{equation} Z[J] \propto \int \mathcal{D}\phi ~ e^{i S[\phi] + \int d^4x~ \phi(x) J(x)} \end{equation} and calculate arbitrary vacuum expectation values \begin{equation} \left<0| \phi(x_1) \ldots \phi(x_n) | 0 \right> \end{equation} using functional derivatives with respect to the source $J$. I also know how to calculate vacuum expectation values in the "canonical quantization formalism" (Wick's theorem etc.). So far so good. Usually, we are not interested in vevs but rather in S-matrix elements such as $\left<p_1, \ldots, p_n|q_1, \ldots, q_m \right>$ where $p_i$ and $q_j$ are outgoing and ingoing particle momenta. Furthermoe, the transition between $S$-matrix elements and vevs is also clear to me: this is just given by the LSZ reduction formula. So in principle, we are now good to go: we can calculate everything in the path integral formalism and eventually relate this to actual matrix elements using the LSZ formula. Now come my actual questions:   1. It seems that there is a more direct relation between the S-matrix elements and the path integral formalism. In fact, on the Wikischolar article about the Slavnov-Taylor identities (written by Dr. Slavnov himself) it is stated that the $S$ matrix can be written as $S = Z[0]$. Where does this come from and how is it to interpret? I am confused because I thought that $S$ was rather a matrix (whose entries, i.e. matrix elements are numbers) and $Z[0]$ is just a number (an evaluated integral). So to me, thsi reads like "matrix = number"... Furthermore, if this equation holds true, how can we obtain the $S$ matrix elements from there?    2. Even more confusingly, there seems to be another relation to the $S$-matrix element. I have found this in Weinberg Vol. II, chapter 15.7 around equation (15.7.27). There, we have an action that is of the form $I + \delta I$ (the context is here that $I$ is the gauge fixed action of a non-Abelian gauge theory and $\delta I$ is the change due to a small variation in the gauge-fixing condition, but this does not really matter here). It says then: It is a fundamental physical requirement that matrix elements between physical states should be independent of our choice of the gauge-fixing condition, or in other words, of $\delta I$. The change in any matrix element $\langle\alpha|\beta\rangle$ due to a change $\delta I$ in $I$ is \begin{equation} \delta \langle\alpha|\beta\rangle ~\propto ~\langle\alpha|\delta I|\beta\rangle. \end{equation} So now, there seems to be even a relation between the action and the $S$-matrix elements. How does this fit into the entire picture?  My QFT exam is coming up, so thanks a lot for your answers!


title: How makes an equation to Flipping motion

text: Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of flipping which satisfies this constraint. ![enter image description here](https://dl.dropboxusercontent.com/u/55765722/0.fawaed/flipping.gif) Indeed, This equation depends on other equation which is direction of hand . If we will consider 3 axis (x,y,z) to get the behavior of page , we agree that `Z = 0` for all particles of the page at default state. So What about the equation of each point (x,y,z) throughout flipping(browsing) ? Besides, The position of the particle at a time $t$ is given by, $$S(t)=\frac{V_0}{a}(1-e^{-at})$$ How to use this equation in this context ? Or is not useful for this context ? There are :   * rectilinear motion   * uniform rectilinear motion   * sinusoidal motion   * circular motion   * ... However, I do not know where I should classify the flipping motion to build the suitable equation which depends on: `a`(acceleration),`V`(velocity), `t`,[`X`,`Y` , `Z`] AND/OR [$$\theta$$]


title: What is the average distance between objects in our asteroid belt?

text: We've all seen sci-fi movies with asteroid belts that require "great skill" to fly through, but how dense is the asteroid belt really? How much of the belt could you see from the surface of a given asteroid? Is it uneven, with dense and sparse patches?


title: Proton mass and Weizsäcker's formula

text: I'm _so_ confused in the use of nuclear masses and atomic masses. I have two questions. From the book "Outline of Modern Physics" by Ronald, I understand that the _semiempirical mass formula_ (Weizsäcker's formula) is $$M=Zm_{p}+(A-Z)m_{n}-b_{1}A+b_{2}A^{2/3}+b_{3}Z^{2}A^{-1/3}+b_{4}(A-2Z)^{2}A^{-1}+b_{5}A^{-3/4} \qquad (1)$$ and this formula is for the dependence of the mass of a nucleus on $A$ (mass number) and $Z$ (atomic number), i.e. is the formula that gives you (approximately) the mass of a _nucleus_. **1st question** $M$ is the nuclear mass or the atomic mass? (I understand that the difference between these two masses are the masses of the electrons. Obviously it's small, but is nonzero.) Then, he defines the _average binding energy per nucleon_ like $$BE=[Zm_{p}+(A-Z)m_{n}-M]c^{2}/A \qquad (2)$$ So far so good. Then, the book tries to do an example, and calculate the binding energy per nucleon for $_{42}^{98}Mo$ and writes $$BE=[Zm_{p}+(A-Z)m_{n}-M_{nuc}]c^{2}/A \qquad (3)$$ and says: "where the atomic masses are used for $m_p$ and $M_{nuc}$ (so that the electron masses cancel)." And insert the numbers. $$BE=[(42(1.007825u)+56(1.008665u)-97.905409u)/98] \times 931.5\frac{MeV}{u} \qquad (4)$$ But, according to Wikipedia the isotopic mass of $_{42}^{98}Mo$ ( _atomic_ mass) is 97.9054082 u, and the mass of the proton is 1.007276466812 u. So my **2nd question** is: why does the book uses the value 1.007825u for the mass of a proton instead of the value 1.007276466812 u, and why does it uses the atomic mass instead of the nuclear mass in $M_{nuc}$? **Note** The book uses the value 1.007825u for the mass of a proton and the atomic mass instead of the nuclear mass in $M$, in other examples. Why?


title: What has $E = mc^2$ to do with nuclear powerplants?

text: In life, when you talk about nuclear energy, there always happens to be a guy who says that famous Einstein's equation. " _Yeah, they just convert mass to energy, $E = mc^2$ ya know?_ " When I think about that, all I learned about nuclear power resembles dominoes arrangements. You _tonk_ a block and it falls. On its way, it _tonks_ other dominoes and when it falls it releases energy (sound waves).   Quite same in the nuclear physics. You send slow neutron to a core. The core absorbs it, breaks and sends another neutrons and energy (electromagnetic waves). So in the end, I see no domino blocks disappearing in this game. All we do is, that we _tonk_ domino arrangements that has been built by old stars long time ago. So why is this equation related to nuclear power? What mass disappears in nuclear power plants?


title: Why increase pressure to go from liquid to steam?

text: Assume the following phase diagram $T-H$ of water ![enter image description here](http://i.stack.imgur.com/gU12G.png) At a nuclear engineering course, I was told that in order to increase the performance of a pressurized water reactor, one has to increase the pressure. How such a reactor works is seen on the next image ![enter image description here](http://i.stack.imgur.com/zrWF9.gif) In the steam generator the is saturated water. As fusion takes place, very hot water moves towards the steam generator where it transfers its thermal energy to the saturated water. The saturated water becomes hotter and finally it vaporizes. The steam moves the turbine and electric energy is produced. I cannot understand the following: Why the increment of pressure in the reactor vessel, will transfer more energy in the steam generator making the thermal energy of the steam larger?


title: Alpha decay, why does it occur?

text: I was reading about alpha decay and why it happens. The strong force holds protons and neutrons together, but I don't get why does an atom emit helium nucleus when it has too many protons&neutrons? I mean why isn't the strong force holding on to it?


title: Changing Coordinate Systems in Two-Loop Integrals

text: Suppose we have the two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. In the first integration with respect to $k _ {1}$, I take $k _ {2}$ to be the z-axis. Then $k _ {2} = k _ {1} \cos \omega$ and the four-dimensional spherical volume element is $\mathrm{d} V = |k _ {1}| ^ {3} \sin ^ 2 \omega \sin \theta \, \mathrm{d} |k _ {1}| \, \mathrm{d} \omega \, \mathrm{d} \theta \, \mathrm{d} \phi$, where $|k _ {1}|$ is the Euclidean norm of $k _ {1}$. When we perform the second integration $\int \mathrm{d} ^ 4 k _ {2} \, g (k _ {2})$, where $g (k _ {2})$ is the result of the first integration, is it allowed to change coordinate systems and take $k _ {2}$ to be the radius in $S ^ 3$? That is, can we write the spherical volume element for the second integration as $2 \pi ^ 2 |k _ {2}| ^ {3}$, where $|k _ {2}|$ is the Euclidean norm of $k _ {2}$?


title: At an instant, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges?

text: **Question:** In principle, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges? _(At a single instance in time?)_ **Additional Notes:** I am aware that the evolution of this system would not behave the same because orbits cannot manifest in a simple system of like charges due to reasons regarding entropy; Just because it is entropically desirable to evolve from the big-bang state to the current universal state, it is not entropically desirable to evolve from the current universal state to the big-bang state just because we reversed the flow of time. (I intend to reverse the flow of time so that the fields reverse but not the global increase of entropy, this is why I specify _At a single instance in time?_ ). I would ask that SE users to consider that the argument of playing a tape backwards is an ill conceived method for generating an answer since it always admits an unnatural evolution of state of the system (One that would never play out if entropy was increasing, I only reverse time for the sake of reversing the fields). For example if I tape an ink drop falling in water and watch it in reverse, it becomes immediately become apparent to me that it is being played in reverse because the system evolves in a way which violates the second law of thermodynamics. Even though this is true, what I can say with confidence is that every electric field of every particle will reverse in sign and the same can be said about the gravitational field. That is the true purpose of the question. With that being said, I specify " _(At a single instance in time?)_ " because I am more concerned with this idea on a fundamental level _(i.e. The physical properties of the fields)_.


title: At same level do these two pipe lines give same pressure of water?

text: Provided that the two pipe lines are of same length, same material and in the same level, is the water pressure in both the layouts same or different?      ![enter image description here](http://i.stack.imgur.com/XVSrc.jpg) PS: In 1st pipeline the turns are not "upward-downward" turns but "sideward- turns" in same level(height).


title: When do one-point functions vanish?

text: I have read in many places that one point functions, like the one below: $$\langle \Omega|\phi(x) |\Omega \rangle$$ are equal to zero ( $|\Omega \rangle$ is the vacuum of some interacting theory, $\phi$ is the field operator - scalar, spinorial, etc...) Peskin's book, for instance, says (page 212) this is USUALLY zero by symmetry in the case of a scalar field ("usually" probably means $\lambda \phi^4$ theory) and by Lorentz invariance for higher spins . How can I see that? In a more general question: can someone point out a counterexample? A case when this functions are not zero?