Curvature of Spacetime

I have been exploring for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets. In reading the book "Gravitation" by Misner, Thorne and Wheeler, the authors stress that Riemann came very close to make a decisive connection between gravitation and curvature of space, but he failed to do so, they say, precisely because he thought of SPACE and curvature of SPACE instead of curvature of SPACETIME and this makes the whole difference! Can anybody explain in conceptual terms, as far as possible, why spacetime, unlike SPACE alone, can be seen and understood as curved? I firmly believe that mathematics is only a language, albeit a complex one, which facilitates our understanding of reality, but that the same reality is not hopelessly beyond reach without maths. We should remember after all that Einstein's mathematical formulations of the Special and General Relativity are rooted in thought experiments and in a basic conceptual grasp, which preceded its mathematical formulation.


Ampère's Law for an infinitely long straight wire

In order to find the magnetic field generate by an infinitely long straight wire of radius R, I have to use local Ampère's Law : $\boxed{ \oint_L B\cdot dl = \mu_0 \sum_j I_j}$ If $r > R$ then $ \oint_L B\cdot dl = B(r)\cdot 2\pi r = \mu_0 I \Rightarrow B(r) = \dfrac{\mu_0 I}{2\pi r} $ (I'm okay with this expression) But if $r<R $ I'm supposed to find $ \oint_L B\cdot dl = B(r)\cdot 2\pi r = \mu_0 I \cdot \dfrac{\pi r^2}{\pi R^2} \Rightarrow B(r) = \dfrac{\mu_0 I r}{2\pi R^2} $ I don't understand where come from the $\dfrac{\pi r^2}{\pi R^2}$ factor. Thanks for your help (and sorry for my bad english)


If the net work on a particle is zero, can the speed change?

The following question was on a quiz in physics class: > If the net work done on a particle is zero, which of the following > statements must be true? > > a) The velocity is zero   >  b) The velocity is decreased   >  c) The velocity is unchanged   >  d) The speed is unchanged   >  **e) There is no displacement for the object** The correct answer was e. In what scenario would the speed change? There was an explanation beside the question: > Since the work done is zero, it indicates that the applied force is zero. > Since Force = Mass X Acceleration, and the mass is not zero, this implies > that the acceleration is zero. When asked about this question, the teacher responded: > If there is no displacement, then only work done is zero. If the speed is > unchanged, then there is no acceleration. This will lead to an absence of a > force. Hope it explains the situation.


BPS state and annihilation of SUSY charges

As we all know that anticommutator of one set of supercharges in massive extended supersymmetry is something like $$\\{b_\alpha, b_\beta^\dagger \\} = \delta_{\alpha \beta} (M-\sqrt{2} Z).$$ My question is that everyone says that it is obvious that BPS states annihilate half of the supersymmetric charges. Why is this so? It may be trivial but I don't know how. May be it is just because the annihilation operator acts on the lowest energy state and annihilate BPS States? But why only half? Can anyone clear that up.


Does centrifugal force exist?

Currently in my last year of high school, and I have always been told that centrifugal force does not exist by my physics teachers. Today my girlfriend in the year below asked me what centrifugal force was, I told her it didn't exist, and then she told me her textbook said it did, and defined it as "The apparent force experienced towards the outside of a circle is the centrifugal force and is due to the mass of the object resisting the inward centripetal acceleration that the object is experiencing". I was pretty shocked to hear this after a few years of being told that it does not exist. I did some reading and found out all sorts of things about pseudo forces and reference frames. I was wondering if someone could please explain to me what the hell is going on (in high school student terms), is it wrong to say that centrifugal force does not exist? This has always nagged me a bit as I often wonder that if every force has a reaction force then a centripetal force must have a reaction centrifugal force, but when I asked my teachers about this they told me that centrifugal force does not exist. Now I'm just confused. Help.


How should a physics student study mathematics?

**Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house).** This question is broad, involved, and to some degree subjective. (I started out as a physics-only student, but eventually decided to add a mathematics major. I am greatly interested in mathematics; the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. Naturally, I enjoy mathematics immensely.) This question asks mainly of undergraduate-level study, but feel free to discuss graduate-level study if you like. Please _do not_ rush your answer or try to be comprehensive. I realize the StackOverflow model rewards quick answers, but I would rather wait for a thoughtful, thorough (on a point) answer than get a fast, cluttered one. (As you probably know, revision produces clear, useful writing; and a properly- done comprehensive answer would take more than a reasonable amount of time and effort.) If you think an overview is necessary, that is fine. For a question this large, I think the best thing to do is focus on a specific area in each answer. * * * **Update:** To Sklivvz, Cedric, Noldorin and everyone else: I had to run off before I could finish, but I wanted to say I knew I would regret this; I was cranky and not thinking clearly, mainly from not eating enough during the day. I am sorry for my sharp responses and for not waiting for my reaction to pass. I apologize. **Re: Curricula:** Please note that I am not asking about choosing your own curriculum in college or university. I did not explicitly say that, but several people believed that was my meaning. I will ask more specific questions later, but the main idea is how a physics student should study mathematics (on his or her own, but also by choosing courses if available) to be a competent mathematician with a view to studying physics. _I merely mentioned adding a mathematics major to illustrate my conclusion that physics student need a deeper mathematical grounding than they typically receive._ And now I have to run off again.


To all experienced theoretical physicists out there, what is the step by step process in your math education?

I am not doing a physics degree but an engineering degree but i am planning using my free time to self study all the math in preparing myself to self study subjects in theoretical physics. (I've always wanted to do theoretical physics but there are no scholarships offered in my country for the subject and my family is not rich so I had no choice but to self study) By "self studying" I mean buying books and reading online materials on my own and try to understand everything I can. So far this is my plan, Analytic geometry -> Single Variable Calculus -> Advanced Calculus (basic Linear algebra + multivariable and vector calculus) -> Linear Algebra -> ODEs -> but here I am stuck as to where do I go from here? Well first of all is this plan okay?   If it is : how do I continue? Should I continue to real and complex analysis and do more things from pure math? Or should I just stick to the more computational part of math and learn more advanced techniques in solving differential equations or learn more algebraic techniques? How about geometry? Topology? Group theory and more things from pure Math? If its not : what is the better way of doing it? Should I learn all the basic math first before reading Goldstein's classical mechanics ? Or should I just start with the physics and just pick up the math along the way? And I also have the time issues to worry about how I wish I can just do a physics degree but unfortunately things don't always go your way. Due to imminent time constriants (engineering has lots of projects im sure) i would like my self study and learning to be as efficient as possible. Hence good advice from all experienced physicists would be very much helpful. Thanks in advance.


What are the prerequisites to studying general relativity?

This question recently appeared on Slashdot: > Slashdot posts a fair number of physics stories. Many of us, myself > included, don't have the background to understand them. So I'd like to ask > the Slashdot math/physics community to construct a curriculum that gets me, > an average college grad with two semesters of chemistry, one of calculus, > and maybe 2-3 applied statistics courses, all the way to understanding the > mathematics of general relativity. What would I need to learn, in what > order, and what texts should I use? Before I get killed here, I know this > isn't a weekend project, but it seems like it could be fun to do in my spare > time for the next ... decade. It seems like something that would be a good addition to this site: I think it's specific enough to be answerable but still generally useful. The textbook aspect is covered pretty well by Book recommendations, but beyond that: **What college-level subjects in physics and math are prerequisites to studying general relativity in mathematical detail?**


How should a theoretical physicist study maths?

> **Possible Duplicate:**   >  How should a physics student study mathematics? If some-one wants to do research in string theory for example, Would the Nakahara _Topology, geometry and physics_ book and other geometry and topology books geared at physicists be sufficient for that purpose, or should one read abstract math textbooks e.g. Spivak _Differential geometry._ What about real analysis and functional analysis (not just the introductory functional analysis chapter that's present in quantum mechanics textbooks)?


Mechanics Energy (Calculus)

A particle moves with force $$F(x) = -kx +\frac{kx^3}{A^2}$$ Where k and A are positive constants. if $KE_o$ at x = 0 is $T_0$ what is the total energy of the system? $$ \Delta\ KE(x) + \Delta\ U(x) = 0$$ $$F(x) = -\frac{dU}{dx} = m\frac{dv}{dt} = m v\frac{dv}{dx}$$ Integrating to get U(x) and 1/2mv^2 I get $$\Delta\ U(x) = \frac{kx^2}{2} - \frac{kx^4}{4A^2}$$ $$\Delta\ KE(x) = -\frac{kx^2}{2} + \frac{kx^4}{4A^2}$$ Which Makes sense. But how do I find the function KE(x) where KE(0) = $T_0$? Do I Even need to? The total energy in the system is $T_0$ Correct? Also a kind of side note. What is really confusing me, is when should I add limits of integration and under what circumstances should I just use an indefinite Integral?


Someting almost faster than light traveling on something else almost faster than light

> **Possible Duplicate:**   >  Travelling faster than the speed of light   >  Double light speed Lets say that   * an airplane can fly at 4/5 of the speed of light, and   * I can run at 2/5 of the speed of light, and   * I'm in the airplane. Suddenly I start running towards the cockpit down the middle lane. Now what? From what I know, the speed of light is not the speed of light just because it is. The speed of light is instead the maximum speed limit because light moves as fast as possible and therefore equals the max speed.   But when it's the maximum speed limit, nothing can ever move faster. So, I'm running in the airplane... now what?


Driving car with (almost) the speed of the light and switch the headlights on

I'm curious what will happen if we 'drive' a car with (almost) the speed of light, and all of sudden we switch the car headlights on? Will the car headlights ray have double speed (speed of 600.000km/h) or that ray will never be seen?


Can we travel faster than light?

Consider two cars going in opposite direction one at speed $c/2$ and other just greater than $c/2$, then one bus will appear to other moving at speed more than $c$. How can an object travel at speed more than light?


Relative speed when approaching the speed of light

According to this chart of the Lorentz factor as a function of speed: ![Lorentz factor](http://i.stack.imgur.com/jgzti.gif) If a spacecraft neared (roughly) 0.85c, would it appear to be traveling at 1.7x the speed of light from the perspective of those on board - i.e. covering 0.85c distance, but in half the time amount of time?


What happens to body chemistry at the speed of light?

Assume that I'm traveling at the speed of light in one direction. My brain is also traveling at the speed of light in that direction. Presumably there is at least one receptor site in my brain that is oriented in such a way that particles would need to travel faster than the speed of light to reach their destination receptor. What would happen to those particles? Would my brain cease to function because these particles couldn't reach their destination?


Is there more mass in stars or interstellar medium?

From what I gather the interstellar medium has about about 1 atom per cubic centimeter. But on the other hand, as they say, "Space is big, really really big" So if it is known (or at least theorized about) what percent of the universes matter is in the following: black holes, stars, and interstellar medium?


cause of maximum kinetic energy and elastic potential energy at the same point of a transverse wave in a string

Suppose a string element oscillating transversely in _SHM_ having both kinetic energy and elastic potential energy travels with velocity $u$. Let $y(m)$ be the amplitude. When it reaches $y(0)$, it has max. $K.E.$ and elastic $P.E.$ together at the same time. But in _SHM_ , when the kinetic energy is max. the potential energy is min. but in the string it is contradicted. Does transverse wave violate _SHM_ and if not why at the same time it has both K.E. and P.E. maximum unlike _SHM_?


Does diamond dust conduct heat as well as diamond?

Diamond is one of the best thermal conductors you can get. If the diamond is crushed into dust and spread out over a flat surface, but still held fairly compact (for instance in a small petri dish), would it still conduct as well as before?


Single photons: Is there a 90° offset of the electric to the magnetic component in the direction of propagation? ![Electromagnetic wave](http://i.stack.imgur.com/ksHYn.png)


Number theory in Physics

As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications of Number theory to physics. I have heard Applications of linear algebra and analysis to many branches of physics, but not number theory. Waiting forward in receiving interesting answers!


Time-dependent Schrodinger equation from variational principle

In the paper, "Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997 the authors mentioned that the action $$ A= \int_{t_0}^{t_1} dt \langle \Phi(t) | i \hbar\partial / \partial t - \hat{H}(t) | \Phi(t) \rangle \tag{1} $$ provides the solution of time-dependent Schrodinger equation at its stationary point. Wikipedia  called (1) as the Dirac action without further reference. If I do a variation, indeed the stationary point of action (1) gives $$ i \hbar\partial / \partial t | \Phi(t) \rangle = \hat{H}(t) | \Phi(t) \rangle $$ However, from path-integral point of view, the least action principle is only a limiting case when $\hbar \rightarrow 0$. In general, there is no least action principle in quantum mechanics. My question is, how to reconcile these two aspects? What does vary of action (1) mean?


What does a curved natural log graph suggest?

Sorry if this is rather simple, but I've only just started learning about using logarithms in experimental physics. I did an experiment to test the amount of time it would take for an amount of water to leave a burette. I used the starting volume of water in the burette as a control variable, $50cm^3$. I recorded the time it took for the a given volume of water to be left in the burette. For example, $10cm^3$ left took a time of roughly $71\mbox{s}$; $45\mbox{cm}^3$ left took roughly $6\mbox{s}$, and then many values in between. I would expect this to represent exponential decay, seen as different concentrations and masses of water in the burette would have different effects on the speed of the water leaving the burette. (Correct me if I'm wrong.) So I plotted a graph of volume against time and it showed exponential decay, but it was only very slightly curved, but curved nonetheless. So I decided then to plot a graph of $\ln\left(V/\operatorname{cm}^3\right)$ against time/s. However, this did not produce a straight line. If I were to follow the plotted points with a curve, the gradient of the line would have been negative and increased in negative 'magnitude'. I'm meant to analyse the extent of whether or not my experiment shows exponential decay. I'm quite stuck, because my original graph shows very slight decay, whereas my log graph isn't a straight line. Does the fact that the log graph doesn't produce a straight line show that there isn't exponential decay? Does it not matter? Would it have been straight had there been very few experimental errors/uncertainties (there would have been a lot)? So I guess, fundamentally, my question is: What does the curved line on my natural log graph suggest?


When is the Fermi surface a surface of constant mean curvature?

Fermi surfaces are surfaces of constant energy in reciprocal space. They provide information about the properties of a material in solid state physics. Constant mean curvature surfaces are a superset of minimal surfaces, which minimize area and have zero mean curvature. Soap bubbles are surfaces of constant mean curvature. The two certainly coincide in the case of a sphere, but must they always coincide? If so, why? Is there a sharper constraint that Fermi surfaces must obey as well? I found a few papers that mention the two as connected, but they never establish the exact connection between the two. Preliminary literature review: Mackay, Alan L. Periodic minimal surfaces, 1985. > Minimal surfaces are found, as mentioned above, in soap films . . . The > surfaces of constant energy in reciprocal space, used in solid state physics > for finding the Fermi surface, are very similar. Mackay, Alan L. Periodic minimal surfaces from finite element methods, 1994. > Fermi surfaces, which are surfaces in reciprocal space, are closely related > to nodal surfaces.


Why is the space-time interval squared?

The space-time interval equation is this: $$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$ Where, $\Delta x, \Delta y, \Delta z$ and $\Delta t$ represent the distances along various coordinates according to an observer, and $\Delta s$ is the space-time interval. All observers agree on the space-time interval, it is constant. My question is **why is it squared?** If we had in equation like this: $$\Delta s'=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$ $\Delta s'$ would be constant as well. It would also never be imaginary. It would have units of $[length]^2$ instead of $[length]$ though. Is there a theoretical or practical reason that we define the space-time interval based on squaring, or is it just to make it look similar to Pythagoras' theorem/give it simpler units or something else entirely?


Regarding state of Klein-Gordon field

In regular quantum mechanics of particles, I have the Schrodinger evolution picture for a **general state** $$ i\hbar \frac{d}{dt} \left|\psi(t)\right> = \hat H \left|\psi(t)\right> $$ then we take the inner product with respect to $ \left< x\right| $ to obtain the equation in the position representation. $$ i\hbar \frac{d}{dt} \psi(x,t) = -\frac{\hbar^2}{2m}\nabla^2\psi(x,t) - V(x)\psi(x,t) $$ where $$ \psi(x,t) = \left< x |\psi(t)\right> $$ and $$ \left< x |\hat H|\psi(t)\right> = -\frac{\hbar^2}{2m}\nabla^2\psi(x,t) - V(x)\psi(x,t) $$ In the case of quantum fields, the position and momentum and demoted from operator status to simply paramaters and the field operator (here for KG real field) is given by $$ \hat \psi(\vec x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} \Bigl[\hat a(\vec p) e^{i\vec k. \vec x} + \hat a^\dagger(\vec p) e^{-i\vec k. \vec x} \Bigr] $$ in the Schrodinger picture. The states are generated from vacuum by the operator $$ \hat \psi(x) \left|0\right> = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}e^{-i\vec p.\vec x} \left|\vec p\right>$$ Now what exactly are these states, are they eigenstates of some operator (seems like the field operator) or quite arbitrary ? In what representation is the field operator given here ? EDIT : I think I understood the first part of the question, these states are the eigen-states of Hamiltonian of the field. But the question about the choice of basis of the operator still remains.


Why does the echo of the last chord appear to rise in pitch

I - and others - observe that when a musical performance ends, the echo of the last chord appears to rise in pitch by up to a quarter tone while the echo decays. This effect appears to be independent of the type of performance - orchestral or choral. I am tempted to assume that this is related to what is known in electro-acoustic circles as Space Echo, but I cannot see why this should occur.


Small question about accuracy and precision

Let's say I have a law like this, $$D=\frac{c}{r}$$ where $c$ is a constant, $r$ a distance in meter. my measures of $r$ are [$0.02m$, $0.01m$], then $<r>=0.015m$ and $\delta r = \pm 0.005m$. So now if I want to calculate $D+\delta D$ should I use $+\delta r$ or $- \delta r$ in my equation? because if I use $+\delta r$ I get a smaller value than if I use $-\delta r$ since $r$ divide $c$ edit: in my real problem I have a lot of data, all is fine when I use the minus delta. I just want to be sure...


In interferometry, what is the origin of the name "Airy function"?

In interferometry (specifically, in the domain of Fabry-Perot cavities), the function $$f(\phi) = \frac{1}{1 + F \sin^2 \phi}$$ , which describes the shape of the resonant structure of the cavity, is often called the "Airy function" (for instance, in Wolfram Mathworld). However, it is obviously quite different from the special functions Ai(x) that usually go by that name. This function resembles probability density function of the wrapped Cauchy distribution. **How did it get the name "Airy function"?** I've heard that Fabry and Perot gave it this name in one of their original papers (maybe this one? PDF, in French, which I can't read), in honor of (the same) George Biddell Airy who had earlier considered similar interferometers. It would be great if someone could help ferret out the first reference to that function by this name.


Vanishing of Weyl Tensor Contraction

Within the context of Einstein space-times, we know that the contraction of the Weyl tensor across a set of indices always vanishes, like so : $$C{^{\alpha }}_{\mu \alpha \nu }=0$$ From a purely mathematical standpoint this should be straightforward ( but perhaps tedious ) enough to prove from the definition of the conformal tensor in terms of the Riemann tensor and its contractions. However, I am wondering what the physical and/or geometric meaning and significance - if any - of this vanishing contraction really is ? I am a very visual person and learner, so an intuitive geometric understanding of this would be very helpful to me.


Will accelerating a massive particle generates a blackhole?

I have a naive question about blackhole. If I accelerate a massive particle very close to the speed of light, the particle will have large energy-momentum tensor. Will it become a blackhole?


Relativistic Black Hole?

So recently, looking at high energy particles through the lens of General and Special Relativity has peaked my interest. One thing I was considering, using the electron as the first example, is as follows: If gravity is the result of mass (stress-energy tensor), and as particles approach the speed of light, their mass is increased as measured by an outside observer (A) by a factor, $ \frac{1}{\sqrt{(1-(v/c)^{2})}} $. The mass of the electron, $ m_{e} $, is $ 9.109 \times 10^{-31} kg $, and the radius is $ r_{e} = 2.818 \times 10^{-15} m $. Given the Schwarzschild Condition is: $$ r_{s} = \frac{2Gm}{c^{2}} $$. Observer A, due to the Lorentz dilation, will measure a dilated mass term that is increasing monotonically as a function of the electron's velocity. The mass that observer A will then measure is: $$ m_{A} = \frac{m_{e}}{\sqrt{(1-(v/c)^{2})}} $$. Combining the Swarzschild Condition with the mass as measured by A, we attain a relation between the mass that observer A measures with the particle's velocity, accounting for relativistic effects. This allows me to now pose the question that given the Schwarzschild radius is the electron radius, at what velocity does the mass increase to the point where the electron, as seen by observer A, become a black hole? This also leads to the more conceptual question of what are the implications of simultaneity in this situation? If the electron's mass is still $m_{e}$ in it's own frame, then shouldn't the electron not really turn into a black hole? Of course, this whole argument falls apart if the derived speed is greater than $c$. I went own to calculate that. I found that speed to be: $$ v = \sqrt{c^{2}-\frac{4G^{2}m^{2}_{e}}{r^{2}_{e}c^{2}}}$$ The velocity at which an object of a given radius and mass would become a black hole. To my disappointment, this came out to be $8.988 \times 10^{16} m/s $. Over twice the speed of light. I have yet to calculate the velocity for more massive objects, such as stars ( which could theoretically reach relativistic velocities as the result of being flung by a galactic collision ). Either way, if this velocity is attainable for anything, what would simultaneity say about this?


Gravitational atraction of fast object

Let's imagine a asteroid that travels with 0.99999999999999999c. (I know it's impossible). Anyway... Relativistic mass of such object would be almost equal to earth's stationary mass. Now let's imagine that such object passes closely to me (assuming I'm immortal etc.). According to $F_g=GMm/r^2$ (Where M is relativistic mass of this asteroid and m is my mass) such an object would attract me. So would I gravitate after it? Would I gain it's speed constantly falling on it? Or maybe it would just pulled me for a moment, but then I would return to my ol' good non-relativistic speed in "everyday reference system"? Or maybe something else would happen? What happens with gravity of relativistic, NON ACCELERATINNG objects? (Let's assume that there's something that makes this asteroid speed constant).


Can a black hole form due to Lorentz contraction?

> **Possible Duplicate:**   >  If a 1kg mass was accelerated close to the speed of light would it turn > into a black hole? Imagine, a rod of length **L** is moving with velocity approaching the speed of light with respect to a human observer on Earth. Due to Lorentz contraction, the rod will observed to be very short. And since all laws of physic hold true in every frame of references, the gravitation law which state the force acting is inversely proportional to square of distance between them, will be acting between various part of the rod. Now, as velocity approach **c** , **L** will approach 0. This should cause an enormous gravitational force enough to form a black hole. Isn't this suggesting a black hole can be formed when the object velocity comes near the speed of light? According to the rod's frame of reference, it will see itself as stationary and a man who observe it is moving, so according to the rod, the human must be a black hole. Isn't this a paradox?


Has the speed of light changed over time?

Could someone judge my (stoner) hypothesis that the speed of light has changed over time -- ie. as the universe has expanded in volume light has slowed down, perhaps going so far as back to the big bang when it was infinitely fast and there was no time because everything happened at once etc. Thinking that the speed at which information can propagate through the universe is linked to the size of it seems intuitive to me. My question -- is there an easy disproof of this? Would Einstein have to be wrong? Does it violate anything supposedly more fundamental such as quantum or string theories? Do any current experiments invalidate it? If not can you show me in any case why you think its unlikely. # Edit 8/24 I'm accepting Mark M.'s answer but will post this here because there is a character limit on comments @Mark M thanks, good answer but. as someone whos only read some popular physics, and should leave this to the experts, im still muddled in my personal theory. i dont see why you should need two units to measure the speed of light. the thing i have a hard time wrapping my head around is the relation of time and distance. they seem like they could fundamentally be the same thing. if you say time is measured fundamentally by the vibration of so and so quantum object in space....why cant we just measure that vibration distance as the constant... ill repeat myself to try to be clear... there is a certain minimum distance that particles have to go to interact with each other ....... if it wasnt vibrating there wouldnt be time,its what creates the illusion of time...so instead of talking about speed or c as distance/time.... cant we simply talk about that distance a quantum object vibates....I'll lead up to my point.....perhaps there can need be only one constant here, and that is the physical size of the universe. a tiny metal tuning fork doesnt appear to be vibrating at all but if you blew it up to the size of the empire state building the metal rods would move from window to window. perhaps as our universe expanded in size the length of that minimum vibration (perhaps infinite at point zero) would have expanded, and therefore created the illusion of time and the speed of light, which, as the universe expands, will continue slowing down. perhaps we are like a big balloon and we have been blown up and all the fields/particles-without-size are vibrating more and more in that space.. am i missing something obvious here?


Curl of a vector field with two different systems of coordinates

Let $$\mathbf{H} = H_x \mathbf{u}_x + H_y \mathbf{u}_y + H_z \mathbf{u}_z$$ be a vector field whose components are defined with respect to the unit vectors $\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$, so in the $(x,y,z)$ system of coordinates. **Question 1** : If we computed its curl in a _new_ system of coordinates, that is $(x' = x, y' = y, z' = -z)$, how would we do it? $$\nabla \times \mathbf{H} = \mathrm{det} \begin{vmatrix} \mathbf{u}_{x'} & \mathbf{u}_{y'} & \mathbf{u}_{z'}\\\ H_x? & H_y? & H_z?\\\ \displaystyle \frac{\partial}{\partial x'} & \frac{\partial}{\partial y'} & \frac{\partial}{\partial z'} \end{vmatrix} $$ In other words, which quantities should we put in the second line? $H_x$, $H_y$ and $H_z$ are along $\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$ and _not_ $\mathbf{u}_{x'}$, $\mathbf{u}_{y'}$ and $\mathbf{u}_{z'}$. **Question 2** : And if we would like to keep $H_x$, $H_y$ and $H_z$ in the second line, which would be their meaning in this computation? These questions are strictly relative (but not equal) to my previous one.


SR time dilation

Two clocks are located at either end of a two light-hour long pole and motionless relative to the pole. Each clock transmits its time and notes that the other clock shows a reading two hours behind its own. That is, the clocks can be considered synchronised with each other. There is a flashbulb at the midpoint of both clocks. It goes off, and when each clock sees it (one hour later), it starts accelerating toward the other and each at the same rate (applying the same amount of thrust for the same local time). They do this for a short time until reaching a steady speed of 0.4c, relative to the flashbulb. Now according to the most recent generation of relativity experts, each clock should observe the other running more slowly. This needs to be the case because time dilation is based on the square of velocity so direction of travel is unimportant.   As they approach each other, the observed time difference will reduce because it takes less time for the transmitted signal to arrive. But the observed clock rate (after adjusting for doppler shift) will be slower at all times. By logical extension then, when they finally pass alongside or stop adjacent to each other, each clock should observe the timestamp of the other clock to be less than its own. Now obviously that outcome can’t be acceptable. Therefore we must conclude that any time dilation observed during their passage is nothing other than an illusion, and certainly not ‘real’ according to any experimental measurement, since only the final side-by-side comparison counts. It goes without saying then that if the clocks were instead moving apart from each other then any observed time dilation must also be an illusion. Am I correct?


Rotatory motion of uniform disk

Consider a uniform disk rolling without slipping with a certain constant angular velocity.Firstly it is moving in sufficiently rough surface.What will happen if it crosses the rough surface and just enters the smooth frictionless surface in its way?Will it be in the state of pure rotation or attains translatory motion or remains in pure rolling state.Please explain.


How is Special Relativity (SR) shown to NOT be an artifact of perception or measurement

Special Relativity (SR) paradoxes are old-hat. But as I read explanations, they tend to resolve issues of simultaneity by applying the appropriate math... but that seems to me to be proving the theory by positing the theory. Just because the math is valid doesn't mean an argument is sound. All I have to do is believe warping space and time against curves and yes... it all works out. For instance... A space ship is traveling toward an asteroid and away from a stationary body at 60 percent the speed of light while at the same time an asteroid is traveling directly toward the space ship and toward the stationary body at 60 percent the speed of light. So the typical question is, "when does the space ship meet its demise?... distance / (1.2*c)?" _My question is distinctly different._ Why is it we are confident that the conclusion matches reality... i.e. how do we know the space ship doesn't just blow up before it thought it should based on only the perception and measurements available to it which seem to be governed by SR?


Study of Black-body Radiation

Why did scientists study black body radiations from something as complicated as a hollow container rather than the radiation from something simple like a thin solid cylinder?


Water pump with a tap connected at its opening not pumping water when water is drawn from the pump before starting the pump

Please give your explanation in layman terms, without throwing any complex equations at me. We have a 5000 liter tank dug in ground and a water pump connected to this tank that draws water from it and fills a overhead tank. We also have a tap at the opening of the pump. (A T joint is connected to the output of the pump, one goes to overhead tank and the other has a tap attached to it). Now, people draw water from this tap without turning on the pump (We try a lot to prevent this, yet people stubbornly do it). Nowadays when we turn on the pump, unless we pour water into the tap (About a liter of water) after turning on the pump we are not getting water output from the pump. A plumber suggested to us that it is because of Air lock and said to turn the tap upside down, like its a fountain :| Because of some physical constraints we cannot do this and it will be freaking expensive to do it. I am somehow not convinced that it is a solution to the problem. Could someone tell me what is happening here and how to solve this problem ?


Reference Frame and Angular Speed Related?

I am given the following problem: > If an airplane propeller rotates at 2000 rev/min while the airplane flies at > a speed of 480 km/h relative to the ground, what is the linear speed of a > point on the tip of the propeller, at radius 1.5m, as seen by (a) the pilot > and (b) an observer on the ground? The plane’s velocity is parallel to the > propeller’s axis of rotation. I was able to solve part a pretty easily by just using the formulas that relate linear speed with angular speed, but I didn’t get part b correct. I thought that the answers would be the same from both perspectives because the speed of the plane does not contribute to the rotation of the propeller, but I’m assuming that is the wrong way to think about it (because that produces the wrong answer). Can someone explain why this is the wrong approach? The solution to this problem involved noting that "The plane’s velocity $v_p$ and the velocity of the tip $v_t$ (found in the plane’s frame of reference), in any of the tip’s positions, must be perpendicular to each other." How is this relevant to the problem?


Kähler potential vs full effective potential

In evaluating the vacuum structure of quantum field theories you need to find the minima of the effective potential including perturbative and nonperturbative corrections where possible. In supersymmetric theories, you often see the claim that the Kähler potential is the suitable quantity of interest (as the superpotential does not receive quantum corrections). For simplicity, let's consider just the case of a single chiral superfield: $\Phi(x,\theta)=\phi(x)+\theta^\alpha\psi_\alpha(x) + \theta^2 f(x)$ and its complex conjugate. The low-energy action functional that includes the Kähler and superpotential is $$ S[\bar\Phi,\Phi] = \int\\!\\!\\!\mathrm{d}^8z\;K(\bar\Phi,\Phi) \+ \int\\!\\!\\!\mathrm{d}^6z\;W(\Phi) + \int\\!\\!\\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi) $$ Keeping only the scalar fields and no spacetime derivatives, the components are $$\begin{align} S[\bar\Phi,\Phi]\big|_{\text{eff.pot.}} = &\int\\!\\!\\!\mathrm{d}^4x\Big(\bar{f}f\,\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}} + f\,W'(\phi) + \bar{f}\, W(\phi)\Big) \\\ \xrightarrow{f\to f(\phi)} -\\!&\int\\!\\!\\!\mathrm{d}^4x\Big(\frac{\partial^2K(\bar\phi,\phi)}{\partial\phi\partial{\bar\phi}}\Big)^{-1}|W'(\phi)|^2 =: -\\!\int\\!\\!\\!\mathrm{d}^4x \ V(\bar\phi,\phi) \end{align}$$ where in the second line we solve the (simple) equations of motion for the auxiliary field. The vacua are then the minuma of the effective potential $V(\bar\phi,\phi)$. **However** , if you read the old (up to mid 80s) literature on supersymmetry they calculate the effective potential using all of the scalars in the theory, i.e. the Coleman-Weinberg type effective potential using the background/external fields $\Phi(x,\theta)=\phi(x) + \theta^2 f(x)$. This leads to an effective potential $U(\bar\phi,\phi,\bar{f},f)$ which is more than quadratic in the auxiliary fields, so clearly not equivalent to calculating just the Kähler potential. The equivalent superfield object is the _Kähler potential + auxiliary fields' potential_ , as defined in "Supersymmetric effective potential: Superfield approach" (or here). It can be written as $$ S[\bar\Phi,\Phi] = \int\\!\\!\\!\mathrm{d}^8z\;\big(K(\bar\Phi,\Phi) + F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})\big) \+ \int\\!\\!\\!\mathrm{d}^6z\;W(\Phi) + \int\\!\\!\\!\mathrm{d}^6\bar{z}\;\bar{W}(\bar\Phi) $$ where $F(\bar\Phi,\Phi,D^2\Phi,\bar{D}^2\bar{\Phi})$ is at least cubic in $D^2\Phi,\bar{D}^2\bar{\Phi}$. The projection to low-energy scalar components of the above gives the effective potential $U(\bar\phi,\phi,\bar{f},f)$ that is in general non-polynomial in the auxiliary fields and so clearly harder to calculate and work with than the quadratic result given above. * * * **So my question is** : when did this shift to calculating only the Kähler potential happen and is there a good reason you can ignore the corrections of higher order in the auxiliary fields?


Areas of computer science required for quantum computing

What knowledge of computer science should I have, to be able to pursue research in quantum computing. I am a Physics undergrad and would take three core courses in QM, before the completion of my degree. SO I guess necessary QM would be done. What about computer science?


Introduction to differential forms in thermodynamics

I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics. I'm looking for a small reference, to learn familiar concepts of (equilibrium ?) thermodynamics formulated through differential forms. Once again, it shouldn't be a complete book, a chapter at max, or an article. **UPD** Although I've accepted David's answer, have a look at the Nick's one and my comment on it.


How to learn physics effectively and efficiently

How do you effectively study physics? How does one read a physics book instead or just staring at it for hours? (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)


How to study physics as a first year student?

I firstly apologise if this question is off the topic. I am a first year undergraduate student and the required textbook for the course Advanced Physics is Fundemental of physics by Halliday. This textbook is huge and to be honest I was scared when I saw the size of the textbook. There are too much text to read and there are many exercises at the end of each chapter and I do not think that I would have time to do every single problem at the end of each chapter. So with this information, what is the effective way of studying physics? what to read and what not to read? what exercises should I do? should I try to do all exercises? any other advice that I should do to be successful in physics is also appreciated. If you need also some information about my background: I am a first year combined bachelor of electrical engineering and bachelor of mathematics and I have good knowledge of math but did not do any physics in highschool.


What is an effective and efficient way to read research papers?

I will be a grad student in condensed matter theory starting this fall. As an undergrad, I did the basic physics and math courses as well as a few grad classes (qft, analysis, solid state physics etc.) When I start reading research papers, I often feel overwhelmed because there is so much that I don't know and I find it hard to decide which points to gloss over and which points to spend time on and understand more thoroughly (which in my case, would probably require supplementary reading of textbooks or related papers) What are some things to keep in mind while reading a paper so that:   1. I get a general overview of the paper and I more useful insights into parts   2. I can do the above reasonably fast (say, finish reading at least 1 paper a week for a start) You don't have to be specific to condensed matter theory papers when you answer.


Good ways for learning and cramming formulas?

I got an exam coming up and its numerical based (its pre-university level exam) but really tough. I want to know about various ways and methods to learn formulas. I know how can I derive them but its time consuming. I want to know how you people learnt these formulas?


Effects of a very large magnetic field on the human body

Ever since reading about the NHMFL I have always wondered about this and asked several people without getting a good satisfactory answer. My question is, considering the simplest case let's say a uniform magnetic field with a very high magnitude constant both in time and space permeating a large room, what would happen to my body in such a field like if I was to just walk through it? How large would the magnitude need to be before I "feel" anything? One tesla is fairly large, would I feel anything? I imagine as we crank up the magnitude, I would feel queasy and sick. What if I crank it up to fifty teslas or a hundred teslas? Would it hurt? Would it mess up my synaptic potentials? Would I go crazy with a dysfunctional brain? Would I pass out or go into a coma? At what point will the damage become irreversible? What magnitude will cause death? With a high enough magnitude would molecules in my body start falling apart? What happens at hundreds of teslas? Thousands of teslas or millions of teslas? The closest thing here I found was here in the accepted answer "a magnetar that would be 1000 miles away would kill us due to diamagnetism of water in our cells". This is the kind of stuff I am looking for, the magnitude of the B-field and then its effect on a human body. Like "at 100T, your body would **___ _** because **_ _**and at 1000T, your body would **___ ___ __** because **___ ___**" and so on. ~~If we allow changes in the magnetic field in time, does the induced electric field "hasten" the effects somehow? Would a strong enough E-field cause a shock within us burning our organs and killing us even if the average B-field is small but the db/dt is "large" for example?~~ I don't know if any experiments on biological samples have been done because artificially creating magnetic fields beyond a 100T or so hasn't been quite done yet much less sustaining them and observing the effects on a biological tissue. But if there are any cool references even if they are on theoretical grounds, that would be interesting. Thanks. * * * Edit: Just to make the question more answerable, I'll focus only on static fields. Anyone know of any references/experiments regarding the effects of large magnetic fields on biological tissues?


Why is there a $\pi$ phase difference when light goes from a rarer medium to a denser medium?

I want the basic reason. I want the description of answer graphically also.


Why is a $\pi$ phase added on reflection and why do things change with absorption?

1. Mathematically, how does a $\pi$ phase shift appear upon reflection of light off a optically denser medium?   2. Why is it always $\pi$?   3. If the medium is absorptive it is no longer $\pi$?


Why does the echo for soundwaves hitting a vacuum come back out of phase?

> **Possible Duplicate:**   >  Phase shift of 180 degrees on reflection from optically denser medium I've read in a physics book for musicians that, when a soundwave hits a near- solid object, it generates an almost completely in-phase echo. However, when the wave hits a near-vacuum, the echo generated is almost completely out of phase echo. Can someone explain, in detail, why this happens (or point me at a detailed explanation)? (My math-fu is fairly strong, but my physics-fu is not).


why does the phase change when wave reflects from rigid boundary

when the wave is reflected from open end there is no phase change but when it reflects from the rigid surface its phase changes. in open surface its phase change is 0 and in rigid surface its phase change is pi. How is it possible. Is is possible to reflect from open end.


Bouncing ball simulation computer science

In my Computer science class I was given a problem where I have to simulate a bouncing ball using "real physics". I have been trying to find a equation that will simulate the height of the bounce given a gravity and an arbitrary mass. And I will need to calculate the next bounce and it's height. A lot of the equations I've found require a time. But I don't really care about a time, all I want is to get the next height after the previous bounce until it finally hits height of 0 or close to it. I haven't taken a physics class since high school( 5 years ago) and that was basic physics.


Do the stars imaged by a telescope even exist at present?

I know that we now have telescopes which can capture the images of the stars and galaxies millions of light-years away from us. Does the telescope capture the past image of the star, i.e. the light which it emitted centuries ago? What guarantee is there that the star is still alive? What basis do organizations like NASA plan missions for evading such stars?


The equation of state for a perfect fluid is that $p=\omega \rho c^{2}$, where $p$ is the pressure, $\rho$ is the density, $c$ is the vacuum speed of light, and $\omega$ is called the equation of state parameter. $\omega$ may be constant or varying in time. I'm looking for broad answers (and references if possible) to the following:   * Why is it important that the equation of state parameter of dark energy is measured?   * What will it tell us?    * What are the implications?


Does the photon-phonon interaction always rotates the photon polarization of 90°?

I'm reading about the acousto-optic effect and on the Acousto-Optical Tunable Filters on particular and wanted to understand the physics under its working. I found this paper http://dx.doi.org/10.1016/S0079-6727(03)00083-1 where they talk about the interactions of polarized light within a crystal. They say > "For instance, if the incident radiation is an ordinary ray (o-ray), it will > be converted into an extraordinary ray (e-ray) upon interaction with the > acoustic wave." and a few paragraphs later > "Considering the entering o-ray [...], photons interact with phonons [...] > and are converted to e-rays by a 90° polarization rotation." They do not explain the origins of this polarization rotation (and I imagine that it's beyond my knowledge) but is this rotation of 90° always true and why of 90° particularly?


Question on Newton's law

I have heard my friends referring to quantity ma as the "force of acceleration" whereas my teacher told us it can't be referred as so. Is it correct to refer that quantity a force? If not, how can we better describe this quantity?


How does a wave packet acts rigidly?

Particles can be represented as wave packet. So how do particles get scattered? Waves superimpose on one another, they don't bounce off of on one another.


Is time dilation an illusion?

It is said that we can verify time dilation by flying a very accurate clock on a fast jet or spaceship and prove that it registers less time than the clocks on earth. However, the clocks on earth would be moving relative to the clock on the spaceship, and since time always dilates and never goes faster regardless of the direction of relative motion, the clocks on earth should register less time than the clock on the spaceship. Is this true? Whenever there is a fast-moving object such as a rocket do all clocks on earth really become slow? If the rocket with the clock landed after moving at relativistic speed, would its clock and the earth's clock again show the same time since during its travel both appeared slow to each other? Or is all this just an illusion, ie. the clocks just appear to be slow to each other but in actually run at normal speed, and neither is behind when the rocket actually lands?


Calculate time dilation with Lorentz transformation

It is possible to calculate the time dilation with Lorentz transformation? So with the equation $$t'=\gamma(t-\frac{x v}{c^2}) \tag{1}$$ in which $\gamma$ is the Lorentz-Factor? In an exercise I have succeeded. So why can be calculated with the equation $(1)$ the time dilation? What is the reason?


Is time dilation real?

If a body with an accurate clock is moving away from A which is stationary, then the time in B would be slower than that in A. Since relative to B A would have an equal velocity time in A would be slower than that in B. This results in a contradiction. This is the argument given in this website. Can someone say how this argument is wrong from a deeper study of SR.?


Is aging independent of time progression when relativistic effects are taken into account?

I have been trying to understand Einstein's principle of relativity for quite some time. I have at least I believe, understood the notion that time is relative and can actually 'change' relative to some frame of reference close to c. My question is, time is actually a variable that we invented based on how we thought about it; it's the progression of events, life and existence itself. So when, we say in Twin Paradox, one of the twin when he comes back to Earth is younger, how can that be possible? Again, it is true time would slow for him during his travel in space but that's because of the way we defined time and consequently its limitations close to c. How can that affect his physical structure (aging process)? Time should be independent of his personal physique.


Is time dilation an illusion? Variation on the twins paradox

Consider the twins paradox with a slight variation: Twins A and B are in separate space ships both capable of going at the speed of light instantly (i.e. without any acceleration). Both ships are stationary relative to each other in intergalactic space facing in opposite directions. They synchronize their clocks. Then Twin A sees ship B zooms off to the "right" at the speed of light, and ship B travels a round trip of 8 years. Twin A sees ship B recede away from him at the speed of light (in fact, ship A just disappears). When ship B returns, Twin B's clock will show he has been gone 8 years. But from Twin B's perspective, it's ship A that zooms off to the "left" at the speed of light, does an 8 light year round trip, and according to Twin A's clock, he has also been gone 8 years. So they both would agree that one has been away from the other 8 years, and both have aged the same amount of time. It would seem clocks do not actually run slower as they move closer to the speed of light. So is time dilatation just an illusion?


The real meaning of time dilation

Is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then to B it APPEARS that A's clock moving slower, but A sees his own clock moving at normal speed. Similarly, to A it APPEARS that B's clock is moving slower, but B sees his own clock moving at normal speed. If the above is true, then both A seeing his own clock moving at normal speed and B seeing his own clock moving at normal speed means that in reality both clocks are moving at normal speed, and neither has slowed down, whereas the other person's clock APPEARING to move slowly is merely an illusion. Now is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then A sees his own clock move slowly (compared to the speed of the clock when A was at rest with respect to B) and B sees his own clock move slowly, so that in reality both clocks are moving slowly, but they still remain synchronized (since both are slow by the same amount) If the above statement is not true, then why do muons decay slowly when moving fast? [it could only be possible if the muon saw its own clock as moving slowly. If we saw the muon's clock moving slowly, but the muon saw its own clock moving at the normal rate, then the muon would decay at the normal rate, and not slowly] Can anyone please explain where I went wrong?


How does a snowflake "know" to form symmetrically?

> **Possible Duplicate:**   >  Why are snowflakes symmetrical? Under ideal situations, a snowflake forms into near perfect hexagonal symmetry. How? For instance, when a water molecule moves towards the edge of the snowflake, how does it "know" where to place itself so that the crystal formed matches the other five points?


Could ball lightening be a form of plasma?

With regard to the recent arXiv article: J. D. Shelton, _Eddy Current Model of Ball Lightening_   http://arxiv.org/abs/1102.1224 I wonder if this is a reasonable explanation of ball lightening, or if there is such an explanation. The paper is somewhat technical and E&M is one of my worst subjects. Please feel free to edit this question to one better suited, or if you don't have the rep, add a comment suggesting changes.


Why does measured pressure change over time in closed hose with temperature gradient

I have a 4' hose that is closed at one end and connected to a Airdata Test Set (precise control of pressure) and a high accuracy pressure monitor on the other end with a T and valve. The valve allows the Airdata Test Set connection to be closed off resulting in a hose connected to the pressure monitor and closed at the other end. The valve is a high quality needle valve. The pressure monitor is a a Druck DPI 142. Half the length of the hose is in a temperature chamber controlled to 70 C. The Druck connected end is outside the chamber at roughly 22 C. When the airdata test set is commanded to a pressure of 1300 mb and allowed to settle for 30 seconds or so, then the valve closed, the pressure reported by the Druck drops over 20 minutes or so with a decreasing rate of change. The airdata test set draws air from the room when operating. The hose is ~0.190" ID neoprene, Saint-Gobain P/N 06404-15. The temp chamber, hose, etc, are given 1 hour to thermally stabilize prior to commanding the pressure to 1300 mb. The difference between initial pressure and stable pressure is ~18 mb. Why does the pressure take 20 minutes to stabilize?


Value of G, gravitational constant

if the universe is a 0 energy universe, could the value of G be worked out through summing up the strong, weak and electromagnetic force strengths for an elementary particle up to an infinite distance, and take the value of the negative energy gravity provides as that value in reverse? Or at the very least the value of the sum of the force of gravity and the cosmological constant?


Why do we use operators in quantum mechanics?

In classical mechanics, physical quantities, such as, e.g. the coordinates of position, velocity, momentum, energy, etc, are real numbers, but in quantum mechanics they become operators. Why is this so?


Quantum Field Theory without LSZ, how is it possible?

Most modern texts spend some time deriving the LSZ reduction formula that connects S matrix elements to time ordered field correlation functions. It seems essential, and really helps clear up what you are calculating. Yet some earlier texts and even some modern texts (e.g. "Student Friendly Quantum Field Theory" by R. Klauber) seem to skip right past this, working everything out in the "interaction" picture. It seems there must be something going wrong with this latter procedure, but I am not quite able to put it together.


$\mathcal N=2$ Weyl multiplet and chiral superfields in 4d

It is more or less known that a given antisymmetric tensor $F$ in two indices can be written in terms of spinorial indices, splitting into self-dual and anti-self-dual parts $$ F_{\mu\nu} = F_{\alpha\beta} (\sigma_\mu)^{\alpha\dot\alpha}(\sigma_\nu)^{\beta\dot\beta} \varepsilon_{\dot\alpha\dot\beta} + F_{\dot\alpha\dot\beta} (\sigma_\mu)^{\alpha\dot\alpha}(\sigma_\nu)^{\beta\dot\beta} \varepsilon_{\alpha\beta} .$$ My question is the following: start from (equations 5.3 and 5.4) $$ W_{\mu\nu}^{ij} = T_{\mu\nu}^{ij} - R_{\mu\nu\lambda\rho}\theta^i\sigma_{\lambda\rho}\theta^j + \cdots $$ which is self-dual in Lorentz indices $\mu\nu$ and antisymmetric in $\mathrm{SU}(2)$ indices $ij$. Then it is squared to $$W^2= \varepsilon_{ij}\varepsilon_{kl} W_{\mu\nu}^{ij}W_{\mu\nu}^{kl}.$$ How do I see (possibly using the first equation above) that this is the same as a tensor (formula 4.1 and below) $W_{\alpha\beta}$ with self-dual $T$ as top component, where $\alpha$, $\beta$ denote symmetric spinor indices and $$W^2=W_{\alpha\beta}W_{\alpha'\beta'} \varepsilon^{\alpha\alpha'}\varepsilon^{\beta\beta'} ?$$


AC through a pure inductor

_I've studied the AC circuit for an ideal inductor in many physics books. After deriving the final equation for current the integration constant $C$ is assumed to be $0$ by giving inadequate reasons. In this question I seek for an adequate reasons._ Suppose an ideal AC voltage source is connected across a pure inductor as shown: ![i](http://i.imgur.com/aevwKxN.jpg) The voltage source is$$V=V_0\sin(\omega t).$$ From Kirchhoff’s loop rule, a pure inductor obeys $$V_0 \sin(\omega t)=L\frac{di}{dt},$$ so $$\frac{di}{dt}=\frac{V_0}{L} \sin(\omega t)$$ whose solution is $$i=\frac{-V_0}{\omega L}\cos(\omega t)+C$$ Consider the (hypothetical) case where the voltage source has zero resistance and zero impedance. > In most of elementary physics books $C$ is taken to be $0$ for the case of > an ideal inductor. $$\text{Can we assume that } C \neq 0?$$ >   1. (To me this is one of the inadequate reasons). This integration > constant has dimensions of current and is independent of time. Since source > has an emf which oscillates symmetrically about zero, the current it sustain > also oscillates symmetrically about zero, so there is no time independent > component of current that exists. Thus constant $C=0$. > >   2. (Boundary condition) there might exist a finite DC current through a > loop of wire having $0$ resistance without any Electric field. Hence a DC > current is assumed to flow through the closed circuit having ideal Voltage > source and ideal inductor in series if the voltage source is acting like a > short circuit like a AC generator which is not rotating to produce any > voltage. When the generator starts, it causes the current through the > circuit to oscillate around $C$ in accordance with the above written > equations. > >


Relativistic centripetal force

The thought randomly occurred to me that a circular particle accelerator would have to exert a lot of force in order to _maintain the curvature of the trajectory_. Many accelerators move particles at fully relativistic speeds, and I want to ask how that affects things. Why does this matter? Well, if I understand correctly, a particle in the LHC moving at $0.999 c$ would be _dramatically_ more difficult to keep moving in the circle than a particle moving at $0.99 c$. Reading Wikipedia, I was delighted to find a fully specified problem within a single paragraph. http://en.wikipedia.org/wiki/Large_Hadron_Collider > The LHC lies in a tunnel **27 kilometres** (17 mi) in circumference, as deep > as 175 metres (574 ft) beneath the Franco-Swiss border near Geneva, > Switzerland. Its synchrotron is designed to collide opposing particle beams > of either **protons at up to 7 teraelectronvolts** (7 TeV or 1.12 > microjoules) per nucleon, or lead nuclei at an energy of 574 TeV (92.0 µJ) > per nucleus (2.76 TeV per nucleon). I can (or Google can) calculate the proton case to come to $0.999999991 c$. In order to correctly calculate the force that the LHC must exert on it, do I need to start from $F=dp/dt$, or can I get by with $v^2/r$ times the relativistic mass? I'm not quite sure how to do the former. **Question:** Say I have 2 protons, both moving so fast they're going almost the speed of light but one has twice the energy of the other. They move side- by-side and enter either an electric or magnetic field that causes them to accelerate perpendicular to the velocity vector. Is the radius of curvature mostly the same for the two, or is it different by a major factor (like 0.5x, 1x, or 2x)? I want to read some comments from people who understand these physics well. The particles beyond a certain energy are all going _almost exactly_ the same speed. Is it still okay for me to say they have the same "acceleration" too? They just happen to have absurdly huge inertia compared to the same particle at a more modest fraction of the speed of light. Is this the correct perspective?


Is the existence of electromagnetic standing waves dependent on the observers reference frame?

If I take two plane EM waves travelling in opposite direction e.g. $E = E_0 \sin(kx-\omega t)$ and $E=E_o \sin (kx + \omega t)$, they sum to give a standing wave with a time-averaged Poynting vector of zero. If I use the appropriate special relativistic transformations to derive how these fields appear to an observer travelling at $v$ along the x-axis, I find that one E-field is diminished, one is boosted, whilst at the same time one wave is blue-shifted and the other red-shifted. These waves do not sum to give a standing wave in the moving frame of reference and have a non-zero time- averaged Poynting vector. So, is the phenomenon of a standing wave dependent on the frame of reference of the observer?


Viability of Dysons bladeless fans?

Do Dyson's bladeless fans produce enough air pressure to cool say an air conditioning unit and do they produce a vortex pressure like a bladed fan?


Does boundedness of observables in the Haag-Kastler axiomatization rule out interactions?

In an interacting theory I expect there to be caustics, resonances, and other situations in which _some_ observables would give an infinite experimental result. Of course, these are idealized states and observables -- if a real device's measurement results are quite accurately modeled by such an observable, and we created a state in which its expected value is large enough, the device would be destroyed. An idealized world in which models never predict infinite results seems somehow different from the world we live in. Even though experiments never return the measurement result "infinity", they do return the measurement result "I'm so sorry, I'm breaking now, it's bigger than you thought it could be". The Wightman axioms do not require operators to be bounded, and to me seem much better for it. QFT as it's used in practice isn't constructed within the Wightman axioms, but it's much less constructed within the Haag-Kastler axioms, and, it seems to me, particularly for this reason. I take boundedness to mean that all the eigenvalues of an operator in its action on a Hilbert space of idealized Physical states are required to be finite. This is much stronger than requiring the expected values of an operator to be finite for a dense subset of the Hilbert space (or other, relatively weaker, requirements). It's certainly mathematically more convenient to use bounded operators (because we don't have to keep track of for which states we get a finite result for a given measurement, because they all do), but is that enough? At least, is this acceptable as part of a major attempt to axiomatize theoretical Physics? Is it obvious enough to be an axiom? I'm prompted to ask this question in this way by a comment in Doplicher's "The principle of locality: Effectiveness, fate, and challenges" J.Math.Phys. 51, 015218 (2010), http://arxiv.org/abs/0911.5136, which I'm reading this morning, where he sets out on the first page (2nd page in the arXiv) that "In quantum mechanics the observables are given as bounded operators on a fixed Hilbert space", which seems a specially sanitized version of QM, insofar as it rules out position, momentum, and energy observables. Finally, this question is asking, as always, have I got something (very) wrong?


In a large city how much hotter on average is it outside due to the air conditioning of all the buildings?

Title pretty much states the question. How much hotter do air conditioning units make it outside in a large city like NYC, Chicago, etc?


Proof that 4-potential exists from Gauss-Faraday field equation

This is a problem concerning covariant formulation of electromagnetism. Given $$\partial^{[\alpha} F^{\beta\gamma]}= 0 $$ how does one prove that $F$ can be obtained from a 4-potential $A$ such that $$F^{\alpha \beta}=\partial^{\alpha} A^{\beta} - \partial^{\beta} A^{\alpha} $$


What is the linearized form of relativistic hydrodynamics?

**What I'm looking for:** Let $\vec{W}$ be the vector of conserved variables for a 1-dimensional, adiabatic, (special) relativistic, electrically neutral fluid. (Yes, something that simple!) I'm looking for a paper that derives the form of the matrix $A$ that linearizes the evolution equation. That is, $$ \partial_t \vec{W} \approx A \partial_x \vec{W}. $$ Alternatively, the matrix $B$ that does the same for the conserved variables will work: $$ \partial_t \vec{U} \approx B \partial_x \vec{U}. $$ These matrices are useful in fluid computations for several reasons. I need more than just the eigenvalues (used in certain solvers) - I also need the eigenfunctions. I've found plenty of papers that treat the nonrelativistic case (for one of many, many examples, see the appendices of Stone et al. 2008, ApJS 178 137) both with and without magnetism. I've found some papers that just quote a few eigenfunctions for relativistic MHD, but these are often the ones that are interesting only with magnetism in play. I'm looking for whatever paper derives these matrices in the rather simple case I'm dealing with. An answer that gives the derivation would be nice, but I'm also trying to locate the relevant literature. In particular, I would like a paper addresses the physical reliability/usefulness of the linear approximation. **Background:** There are three primitive variables defining my fluid: rest-mass density $\rho$, velocity $v$, and pressure $p$. By convention, these variables are combined into a vector $\vec{W} = (\rho, v, p)^\mathrm{T}$. Many approaches to evolving fluids deal with the equations in flux- conservative form. Here, the conserved variables are \begin{align} D & = \gamma\rho && \text{(lab-frame density),} \\\ M & = Dh\gamma v && \text{(relativistic momentum),} \\\ E & = Dh\gamma - p && \text{(relativistic energy).} \end{align} Here I define \begin{align} \gamma & = \frac{1}{\sqrt{1-v^2}} && \text{(standard Lorentz factor),} \\\ h & = 1 + \frac{\Gamma}{\Gamma-1} \left(\frac{p}{\rho}\right) && \text{(enthalpy),} \end{align} where the ratio of specific heats $\Gamma$ is assumed to be constant. These are often combined as $\vec{U} = (D, M, E)^\mathrm{T}$. Along with the vector of conserved quantities, we can define the vector of fluxes $$ \vec{F} = \begin{pmatrix} Dv \\\ Mv + p \\\ M \end{pmatrix}. $$ Then we have the relation $$ \partial_t \vec{U} + \partial_x \vec{F} = 0, $$ which is used as the basis for most Riemann solvers and many fluid codes in general.


What would be the effective resistance of the ladder of resistors having n steps

I'm a tutor. This is a high school level problem. In high school, every one have might have solved a problem of effective resistance of a ladder of resistors having infinite steps. Now the problem is little different. what if it has `n` steps instead of infinite steps. How to calculate effective resistance in that case?![enter image description here](http://i.stack.imgur.com/n1eIX.jpg)


Any liquid metallic alloys which are safe to handle with bare hands?

http://en.wikipedia.org/wiki/Liquid_metal describes some alloys which are liquid at room temperature containing gallium, and sodium-potassium. We are advised not to handle them, and mercury, with unprotected skin. Are there any which are safe (as safe as, say, alcohol or glycerine)?


How does ultrasonic horn produce a convection current in the water?

When I was using ultrasonic horn in a beaker, I notice that there are convection currents in the beaker and stir up my substance. I don't understand why it produce water current, I thought that it will just vibrate like the ultrasonic bath. So why does ultrasonic horn create water currents? Thank you.


Lasers can demagnetize ferromagnets?

How is it possible to demagnetize a magnet with a laser? Source: http://www.helmholtz- berlin.de/pubbin/news_seite?nid=13657&sprache=en&typoid And the paper: http://prb.aps.org/abstract/PRB/v88/i21/e214404 How does this work?


Effect of waters changing specific gravity on objects apparent weight placed in liquid

My goal is to monitor the change in specific gravity of a liquid over a period of time. My question is: What are the appropriate formula for determining expected apparent weight of an object immersed in a liquid where the liquids specific gravity g/ml is expected to change? EG. If I were to take an object who's density is 2.6 (average for glass) weighing 100 grams and plunk it into distilled water I believe I should expect an apparent weight should be roughly 61.53 grams. Please let me know if I am just horridly wrong. So then if that distilled waters density/specific gravity were to change say to 1.010, would my new apparent weight of the object be 61.15 grams? My math is not solid in this. I'm basically using ratios in order to produce these answers. Please for the sake of simplicity if you are to choose to answer leave out extenuating circumstances such as temperature of the liquid/object and possible compression of the object due to pressure. If you do chose to add extenuating circumstances I would ask to add those concepts as tertiary answers. I'm sure that my question is probably very basic, but grasping the concepts has proven perplexing to me. I am probably not using the correct search. Your help in this simple question is greatly appreciated.


Why do nearsighted people see better with their glasses *rotated*?

If you are nearsighted (like me), you may have noticed that if you _tilt_ your glasses, you can see distant objects more clear than with normally-positioned glasses. **If you already see completely clear, you can distance your glasses a little more from your eyes and then do it.** To do so, rotate the temples while keeping the nosepads fixed on your nose, as is shown in the figures. As I said, starting with your glasses farther than normal from your eyes, you can observe the effect for near objects too. (By _distant_ , I mean more than 10 meters and by _near_ I mean where you can't see clear _without glasses_ ) **Note that if you rotate more than enough, it will distort the light completely.** Start from a small $\theta$ and increase it until you see blurry, distant objects more clear. (You should be able to observe this at $\theta\approx20^\circ $ or maybe a little more) When looking at distant objects, light rays that encounter lenses are parallel, and it seems the effect happens because of oblique incidence of light with lenses: ![enter image description here](http://i.stack.imgur.com/QpPys.png) The optical effect of oblique incidence **for convex lenses** is called coma, and is shown here (from Wikipedia): ![enter image description here](http://i.stack.imgur.com/lIUw6.png) I am looking for an explanation of how this effect **for concave lenses** (that are used for nearsightedness) causes to _see_ better. One last point: It seems they use plano-concave or convexo-concave lenses (yellowed lenses below) for glasses instead of biconcave ones. ![enter image description here](http://i.stack.imgur.com/s3HdJ.png)


Black Hole evaporation through virtual couples at the horizon

I read about vacuum energy. It explains the Hawking radiation, the black hole necessary radiation: > Physical insight into the process may be gained by imagining that particle- > antiparticle radiation is emitted from just beyond the event horizon. Vacuum > fluctuations are always created as particle–antiparticle pairs. The creation > of these virtual particles near the event horizon of a black hole has been > hypothesized by physicist Stephen Hawking to be a mechanism for the eventual > "evaporation" of black holes. Is this "insight" supposed to make us to think that   1. the virtual pair created is a part of black hole so that when half flies away then the BH mass is reduced? The article on vacuum energy says that the vacuum energy is an underlying background energy that exists in space throughout the entire Universe. So, it is not related to the black holes, though one particle of the couple may create the salute that you may consider as BH evaporation, it actually covers up the fact that the **second particle flies into the BH, increasing its mass**.   2. Why do we believe that the receding half of the couple does not fall back to the Black Hole? Yes, it has escaped the horizon but the (pretty strong) BH gravitation is still in action, and it is only few neutons less than $\infty$ because we are still almost at the horizon initially. This means that to actually escape and not to fall back into BH, the speed of the receding particle must be virtually infinite. I doubt that it is likely that you will have many such virtual particles right at the event horizon. It is much much (Almost surely) more likely that **"escaped" half of the couple will also eventually fall onto the black hole, increasing its mass**. So, considering the virtual particles of vacuum energy, we find two ways to add mass to the BH. Why do they call it BH (mass) evaporation?


Is there a general term for the situation where an improperly chosen measurement range results in a bias?

For example, consider the following measurement: A sensor can measure a specific physical quantity, and has a range of $0$ to $100$. All values above $100$ will be shown as 100. We now take the following measurements: $78$, $100$, $82$, $94$, $100$, and we conclude that the average is $90.8$. However, the true values were $78$, $112$, $92$, $94$, $105$, resulting in the true average of $94.2$. Both end results are in the valid measurement rage, but individual measurements are not. Can we call it saturation? It has a similar meaning in photography, but in mathematics it seems to have a completely different meaning.


The thermodynamics of heating water in a rigid container

When the pressure on the liquid surface is less than the vapor pressure of the liquid at a given temperature, the liquid will start to evaporate. This is common sense. The problem is more difficult when the liquid and its vapor are heated inside a rigid container, with the specific volume of the mixture less than the critical specific volume: ![enter image description here](http://i.stack.imgur.com/ESHNv.png) According to my professor's notes, the level of the liquid in the container would fall. Why? What is the physics (or the thermodynamics) behind this? If a mixture is heated in a constant volume what happens to it? I know that it will not boil completely because it would attain equilibrium pressure with its vapor soon enough, but how do we fix the state when it would stop boiling? Why is it that if the specific volume of the mixture is less than the critical specific volume, the level of the liquid will fall when heated?


Centre of mass, integral

I was answering a question on proving the parallel axis thereom for angular momentum and came across this: $$\int Yy'dm=Y\int y' dm=0$$ Where the position of the center of mass of an object is given by $(X,Y,Z)$, $(x',y',z')$ is a position relative to the centre of mass and m is the mass of the object. My text book (Introduction to classical mechanics) says that this is due to the definition of the centre of mass. There are two things that I don't understand firstly why is $Y$ independent of mass whilst $y'$ is not? and secondly please can you explain what definition of the centre of mass they are using to get the reslut above? I really have no idea to the answer for either of these questions?


Rotate a long bar in space and get close to (or even beyond) the speed of light $c$

Imagine a bar spinning like a helicopter propeller, At $\omega$ rad/s because the extremes of the bar goes at speed $$V = \omega * r$$ then we can reach near $c$ (speed of light) applying some **finite** amount of energy just doing $$\omega = V / r$$ The bar should be long, low density, strong to minimize the amount of energy needed For example a $2000\,\mathrm{m}$ bar $$\omega = 300 000 \frac{\mathrm{rad}}{\mathrm{s}} = 2864789\,\mathrm{rpm}$$ (a dental drill can commonly rotate at $400000\,\mathrm{rpm}$) $V$ (with dental drill) = 14% of speed of light. Then I say this experiment can be really made and bar extremes could approach $c$. What do you say? **EDIT** : Our planet is orbiting at sun and it's orbiting milky way, and who knows what else, then any Earth point have a speed of 500 km/s or more agains CMB. I wonder if we are orbiting something at that speed then there would be detectable relativist effect in different direction of measurements, simply extending a long bar or any directional mass in different galactic directions we should measure mass change due to relativity, simply because $V = \omega * r$ What do you think?


Reaching speed of light

> **Possible Duplicate:**   >  Rotate a long bar in space and reach c Sorry this is very naive, but it's bugging me. If you had a straight solid stick attached on one end and rotating around that attachment at a certain rpm, there would be a length at which the end of the stick would theoretically reach, with that rpm, the speed of light. Well, doesn't seem possible - what specifically would be the limitations that would prevent the end of the stick to reach the speed of light? What would happen?


Why is this thought experiment flawed: A vast lever rotating faster than the speed of light

If there were a vast lever floating in free space, a rigid body with length greater than the width of a galaxy, made of a hypothetical material that could endure unlimited internal stress, and this lever began to rotate about its middle like a propeller so that a person looking at the universe would simply see it spinning at a gentle pace like a windmill, would not its ends be moving many, many times the speed of light? Or am I making so many errors in my thought process that the whole question is absurd? I'm trying to establish why, in essence, a sufficiently large mechanical device (large beyond reason) could not exceed the speed of light.


What prevents an orbiting object from getting a speed which is greater than $c$?

Consider an object orbiting around a point with radius $r$ and angular velocity $\omega$. Here its linear velocity is $v=\omega r$. If we choose a large enough $r$ and reasonable $\omega$, $v$ might be greater than $c$. If this fact is impossible, what prevents it from happening?


Knotted token-ring network

Suppose we have a rigid token-ring network. An observer at any node can seemingly determine the angular momentum of the network by measuring the time it takes for a packet to travel around the ring in each of the two directions. Is it possible by any means for an observer to determine whether the network is knotted?


What allows the modified Urca process to work at lower density than direct Urca in neutron star cooling?

> **Possible Duplicate:**   >  What allows the modified Urca process to work at lower density than direct > Urca in neutron star cooling? This comes from an unanswered question over at physics.se: The dominant method of neutron star cooling is neutrino emission. There are two regimes usually presented, the "direct Urca" and "modified Urca" processes, each of which are sequences of neutron decay and inverse reactions. The direct Urca looks like this: $$n\rightarrow p+l+\overline{\nu_l},\quad p + l \rightarrow n + \nu_l$$ where $l$ is a lepton - either an electron or a muon. These processes cause continuous emission of neutrinos which cools a neutron star relatively quickly. But below a density of $\rho\approx 10^{15}\mathrm{\,g\,cm^{-3}}$ (about three times the nuclear density) this process is suppressed, which means that the direct Urca process only occurs in the core. This is the reason according to a review of neutron star cooling from Pethick and Yakovlev (2004): > The process can occur only if the proton concentration is sufficiently high. > The reason for this is that, in degenerate matter, only particles with > energies within ~$k_BT$ of the Fermi surface can participate in reactions, > since other processes are blocked by the Pauli exclusion principle. If the > proton and electron Fermi momenta are too small compared with the neutron > Fermi momenta, the process is forbidden because it is impossible to satisfy > conservation of momentum. Under typical conditions one finds that the ratio > of the number density of protons to that of nucleons must exceed about 0.1 > for the process to be allowed. This makes some sense. But what surprises me is that this process can still work with a slight modification at lower densities. The _modified_ Urca process can cool the star $$n+N\rightarrow p+N+l+\overline{\nu_l},\quad p + N + l \rightarrow n + N + \nu_l$$ where $N$ is a nucleon - a proton or a neutron. This process, I'm told, can work at much lower densities, but produces 7 orders of magnitude less emissivity. As a result, it's the dominant process in the superfluid outer core. My question is why does the additional nucleon permit lower densities? How does an additional neutron or proton get us out of the conservation of momentum problem with the direct Urca process?


Where do electrons get the energy to remain in orbit?

As we know electrons continuously revolve around the nuclus without falling in it at a high velocity beating it's force of attraction. My question is where do electrons get energy to revolve around the nucleus and withstand its force of attraction.


What is the source for magnetic energy?

_"The electron around the nucleus is in a quantized energy level and can change it only if an external interaction intervenes."_ That is OK but when there is a magnet, it has energy of attracting iron particles for months and years (natural magnet) just by right order of atoms. Where this energy come from? In nature we have to spend one sort of energy to gain another form, but in a magnet what is spending? Thanks


Commutator evolution operator and position operator

Let $H= \frac{p^2}{2m}$, then I am supposed to calculate $[x,e^{-iHt}]$. My idea was to use $[x,p^n]=i \hbar n p^{n-1}$ and so I ended up by using the series for the exponential function with $-\frac{t \hbar}{m} e^{-iHt}$. Could anybody tell me, whether this result is correct?


What will happen to matter if there is a Higgs metastability decay?

Previously, on Save us from swallowing baby universes, please!, I pointed out the dangers associated with false vacuum decay. I wish to be more specific here. Suppose the Standard Model remains a good description of our universe even at higher energies. The observed value of the Higgs mass predicts that our current phase is metastable. If our phase decays in a bubble, its interior will have a different much higher value for the Higgs field. What happens to matter as it encounters the expanding bubble wall? Coleman-de Luccia assumed the exterior is initially in the vacuum state. They did not take the presence of matter into account. A changing value for the Higgs field is unlikely to affect the electromagnetic field much. But we expect the electromagnetic coupling strength to change inside the bubble due to a different running of the coupling constants under the renormalization group in the new phase. How much will this affect photons, and what is its frequency dependence? The leptonic and quark masses due to the Yukawa Higgs coupling will jump up dramatically in the new phase. Will this lead to a partial reflection and a partial transmission of leptons as the bubble passes through? What about confined bound states of quarks in hadrons? How will hadrons be affected? Leptons and quarks also happen to be dressed states, and the dressing will differ between the two phases. Might this cause a particle shower if these particles are transmitted? The masses of the W and Z bosons will also jump up. What will happen to them? Will neutrinos be unaffected? What if neutrino oscillations are also taken into account? This leads us to the ultimate question. Suppose we have some lump of matter, like a cyborg, which is a complicated bound state of leptons, hadrons and photons. What will happen to this cyborg as the bubble expands and hits it?


Does light bend in a vacuum?

I'm familiar with gravitational lensing but still I'm wondering if there is experiments (conducted here on Earth) which show that light bends due to gravity. For example mirrors setup to hold the light or something like that. My question is inspired by this bounty question.


Mach-Zehnder with PBS = Bit-Flip?

Is it true that a Mach-Zehnder interferometer with two polarizing beam splitters (PBS) is nothing but a bit flip for the polarisation degree of freedom? Say the PBSs reflect vertical polarized light and transmit horizontal polarized light. If we send in a state like $|+\rangle = \frac{1}{\sqrt 2}(|H\rangle+|V\rangle)$ the horizontal part gets reflected only once (by the mirror in the interferometer) but the vertical part gets reflected three times (first PBS, mirror, second PBS). At the end we have a relative Phase of $i^2=-1$ between $|H\rangle$ and $|V\rangle$ which means that we end up with the state $|-\rangle = \frac{1}{\sqrt 2}(|H\rangle-|V\rangle)$. Thus it is a bit flip of the polarisation. Right?


Neutron to antiproton decay

Would it be possible for a neutron to lose a positron and become an antiproton? Or would would it need to be the decay of a antineutron to antiproton instead?


What does "P-wave" mean when referring to a particle?

In scattering theory, P wave means $l=1$, where $l$ is the azimuthal quantum number. However, what does P wave mean when referring to particle states? For example, in this paper (arXiv link), the authors are talking about _P-wave charmonia states_. What does that mean? More specifically, I understand that in some sort of potential model, solved using Schroedinger equation for example, there will be states that may be labeled by $n$=something, $l=1$. But here, the article says P-''wave'' charmonia! What is this wave?


What are the ways of finding the spin of a particle

How many ways are there for this mission? Any article about that on the internet. I heard about TODAI made a reprogrammable quantum computer. Any more ideas are welcome


How do you measure proton's spin?

I've probably read it somewhere in Sakurai but I cannot recall it at the moment. So how does one really measure the proton's spin? I mean the proton's spin and not its constituents. Do you measure it using a Stern-Gerlach type of setup? How about atoms and other stuff, how is their spin measured?


Complex part of the solution for physical values

What's a physical meaning of, for example, complex part of the solution for coordinate change of the anharmonic oscillator? Why after substitute (for diff. equation solve) for real x we can earn $x = Re(z) + iIm(z)$? It's because of substitute?


Some questions about the large-N Gross-Neveu-Yukawa model

Consider the following action with a fermionic field $\psi$ and a scalar field $\sigma$, $S = \int d^dx \\{ -\bar{\psi}(\gamma^\mu \partial_\mu +\sigma )\psi + \Lambda^{d-4}[ \frac{(\partial_\mu \sigma )^2 + m^2\sigma^2 }{2g^2 } + \frac{\lambda \sigma^4 }{4!g^4 } ] \\} - (N'-1)Trln(\gamma^\mu \partial_\mu + \sigma )$ Assuming that this has a large-N saddle with uniform $\sigma$ one gets the large-N free energy density as, $E(\sigma) = \Lambda^{d-4}(\frac{m^2\sigma^2}{2g^2} + \frac{\lambda \sigma^4 }{4!g^4 } ) - \frac{N}{2}\int^\Lambda \frac{d^dq}{(2\pi)^d}ln [\frac{q^2 + \sigma^2 }{q^2 } ]$ And the large-N saddle value of $\sigma$ is determined by the large-N gap equation, $E'(\sigma)=0$ Now from here how do the following conclusions come?   * Firstly that a non-trivial solution to the gap equation exists only when, $\frac{m^2}{g^2} < N\Lambda^{4-d}(\frac{1}{(2\pi)^d} \int^\Lambda \frac{d^dk }{ k^2 } ) $  How does this one come?   * Secondly from this apparently follows that the inverse $\sigma$ propagator in the massive phase is,  $\Delta_\sigma^{-1}(p) = \Lambda^{d-4}(\frac{p^2}{g^2} + \frac{\lambda \sigma^2}{3g^4} ) + \frac{N(p^2+4\sigma^2) } {2(2\pi)^d}\int^\Lambda \frac{d^dq }{(q^2+\sigma^2)((p+q)^2 + \sigma^2)} $ How does this equation come?   * Now from this one can show that $\Delta_\sigma \sim \frac{2}{N b(d) p^{d-2} }$ From the above it follows that the canonical dimension of $\sigma$ is 1. How does one understand that the mass dimension of the field $\sigma$ does not depend on the space-time dimension?   * Now I don't understand this argument which says that now since $[\sigma] =1$, both the terms $(\partial_\mu \sigma)^2$ and $\sigma^4$ are of dimension $4$ and hence for $2\leq d \leq 4$ these terms vanish in the IR critical theory? For this argument to work was it necessary that the IR theory was critical?


Inertial navigation system: am I doing it wrong?

I'm trying to develop an inertial navigation system. I can access data from an accelerometer sensor (acceleration on three axes) and gyroscope sensor (angular velocity on three axes). First of all, I integrate my angular velocity data with respect of time, and get angles on all three axes at every moment ($x \to \phi, y \to \theta, z \to \psi$). Then I feed the angles to this rotation matrix ![Rotation matrix](http://i.stack.imgur.com/USq1f.png) and I use it to rotate my acceleration vector at that time, thus taking all my acceleration values on the same reference frame. Finally, I integrate acceleration to get space travelled, using the simple formula $$s(t) = \frac12 a(t) t^2 + v(t - 1)t + s(t - 1)$$ My method seems to work fine with fake data, but performs really bad when I plug in the real data, the output is almost meaningless. Am I doing something wrong with the math or I have to search the problem in my implementation?


What causes different decays?

Nuclei spontaneously decay according to a certain decay rate. There are however different kinds of decay, alpha, beta, gamma... What causes then the nuclei, when they decay, to do so in one way of another? Is there a different decay rate for each kind of decay?


Accelerating particles to speeds infinitesimally close to the speed of light?

I'm in a freshmen level physics class now, so I don't know much, but something I heard today intrigued me. My TA was talking about how at the research facility he worked at, they were able to accelerate some certain particle to "99.99% the speed of light". I said why not 100%, and I didn't quite understand his explanation, but he said it wasn't possible. This confused me. Since the speed of light is a finite number, why can we go so close to its speed but not quite? Edit: I read all the answers, and I _think_ I'm sort of understanding it. Another silly question though: If we are getting this particle to 99.99% the speed of light by giving it some sort of finite acceleration, and increasing it more and more, why cant we increase it just a _little more_? Sorry I know this is a silly question. I totally accept the fact we cant reach 100%, but I'm just trying to break it down. If we've gotten so close by giving it larger and larger acceleration every time, why cant we just supply it with more acceleration? And how much of a difference is there between 99.99% the speed of light, and _the_ speed of light? (I'm not really sure if "difference" is a good word to use, but hopefully you get what I'm asking).


What is so special about speed of light?

I will try to be as explanatory as possible with my question. Please also note that I have done my share of googling and I am looking for simple language preferable with some example so that I can get some insight in this subject. My question is what is so special about $c$? Why only $c$. Its like chicken and egg puzzle for me. Does Einstein reached to $c$ observing light or does he got to light using some number which turned out equal to $c$. Why is $c$ not relative. If something has zero rest mass like a photon why they only travel at $c$ in vacuum and not with $c+1$ or $c-1$?


The LHC - Proton Speed Limit

I'm originally from stack overflow, so this is my first foray into this particular community. I've read in numerous places that the Large Hadron Collider is capable of accelerating protons at 0.999999991 _c_ , which mathematically works out to being 3 metres per second slower than the speed of light. That seems so incredibly close to the speed of light, that it's hard for me to understand why we can't quite get all the way there. **My Main Question :** What is preventing the LHC from achieving 1 _c_ , in terms of how fast it can propel these protons? **Side Question :** If we were able to accelerate the protons to the speed of light, what would the results of their collisions be?


Is it possible to build an instrument which can travel faster than light?

> **Possible Duplicate:**   >  Accelerating particles to the speed of light I have heard about atomic rockets in novels which have the capability to travel faster than light. I have also heard about fictional stories where objects capable to travel faster than light. My question is: > Is it scientifically possible to build an instrument which can travel faster > than light? And atlast i like to clarify that whether this question suitable to your site or maybe asked in anyother stackexchange site's,usually downvote will be awarded when there is lack of research ,i am seeking answer where the research be-ginned for it,am i wrong ? if my question is off the topic here please let me to know where can i ask this question?


About accelerating particles

Can a particle moving below the speed of light be accelerated more and more until it is travelling at c? IF so does it behave like electro-magnetic radiation?


Why does light travel at finite speed?

We have known for quite some time now that light travels at a finite speed of 3x10^8m/s (approx) through vacuum. But why is that? Why can't the speed be infinite? Or at least higher than the current value? What is limiting its speed? I mean, the speed of sound is limited by the density of the medium. Then, what limits the speed of light in vacuum? Does finiteness of speed of light mean the possibility of existence of a omnipresent medium like aether, which, may be is intangible (or doesn't interact with the particles we know of currently)? * * * I do know that light is propagated by photons. That only makes me more curious as to why the speed is fixed to the current value? Especially as to WHAT is limiting the speed of a particle with no inertia?


Why is the speed of light arbitrarily the limit?

I know Einstein was great and all. Why is it that exactly at the speed of light is where infinite energy is required to accelerate any object with mass? Is it simply because the math of relativity checks out and explains most of everything? Are there any physicists who disagree with Einstein's theory?