Physics textbooks that distinguish between laws and definitions?

Often when I am learning physics I start to think about whether the laws I'm learning are mere definitions or experimentally determined, and usually the textbook does not make this clear. As Thomas Kuhn writes in _The Structure of Scientific Revolutions_ , > These generalizations look like laws of nature, but their function for group > members is not often that alone. Sometimes it is: for example the Joule- > Lenz Law, $H = RI^2$. When that law was discovered, community members > already knew what $H$, $R$, and $I$ stood for, and these generalizations > simply told them something about the behavior of heat, current, and > resistance that they had not known before. But more often, as discussion > earlier in the book indicates, symbolic generalizations simultaneously > serve a second function, one that is ordinarily sharply separated in > analyses by philosophers of science. Like $f = ma$ or $I = V/R$, they > function in part as laws but also in part as definitions of some of the > symbols they deploy. Furthermore, the balance between their inseparable > legislative and definitional force shifts over time. In another context > these points would repay detailed analysis, for the nature of the > commitment to a law is very different from that of commitment to a > definition. Laws are often corrigible piecemeal, but definitions, being > tautologies, are not. For example, part of what the acceptance of Ohm's Law > demanded was a redefinition of both 'current' and 'resistance'; if those > terms had continued to mean what they had meant before, Ohm's Law could not > have been right; that is why it was so strenuously opposed as, say, the > Joule-Lenz Law was not. Another example is conservation of momentum; is momentum conserved because it is defined as a quantity that is conserved or because it has been observed to be conserved? Are there any physics textbooks (maybe for classical mechanics) that take a more axiomatic approach, clearly distinguishing definitions from laws determined by experiment?


Tensor Operators

**Motivation.** I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. I then skimmed the Wikipedia page on tensor operators, and felt similarly dissatisfied. Here's why In these discussions, one essentially defines an indexed set of operators $T_{i_1\cdots i_k}$ to be a "cartesian" tensor operator of rank $k$ provided $$ U(R)\, T_{i_1\cdots i_k}\, U^\dagger(R) = R_{i_1}^{\phantom{i_1}j_1}\cdots R_{i_1}^{\phantom{i_1}j_1}T_{j_1\cdots j_k} $$ for each rotation $R\in\mathrm{SO}(3)$ where $U$ is some unitary representation of $\mathrm{SO}(3)$ acting on a Hilbert space (usually that of some physical system whose behavior under rotations we with to study). Similarly one defines a "spherical" tensor operator of rank $n$ as an indexed set of operators $T^{(n)}_{q}$ with $-n<q,q'<n$ for which $$ U(R)\,T_q^{(n)}\,U^\dagger(R) = \sum_{q'=-n}^n D_{q'q}^{(n)}(R)T_{q'}^{(n)} $$ where $D^{(n)}$ is the irreducible representation of $\mathrm{SO}(3)$ of dimension $n$. Based on these standard definitions, I would think that one could define something less "coordinate-dependent" and extended to representations of any group, not just $\mathrm{SO}(3)$, as follows. > **Candidate Definition**. Let a group $G$ be given. Let $U$ be a unitary > representation of $G$ on a Hilbert space $\mathcal H$, and let $\rho$ be a > representation of $G$ on a finite-dimensional, real or complex vector space > $V$. A $k$-multilinear, linear operator-valued function $T:V^k\to > \mathrm{Lin}(\mathcal H)$ is called a **tensor operator** relative to the > pair of representations $U$ and $\rho$ provided \begin{align} U(g) T(v_1, > \dots, v_k) U(g)^\dagger = T(\rho(g)v_1, \dots, \rho(g)v_k) \end{align} for > all $g\in G$ and for all $v_1, \dots, v_k\in V$. Notice that if a basis $u_1, \dots, u_N$ for $V$ is given, and if we define the components $T_{i_1,\dots i_k}$ of $T$ in this basis by \begin{align} T_{i_1 \dots i_k} = T(u_{i_1}, \dots, u_{i_k}) \end{align} and if $\rho(g)_i^{\phantom ij}$ denotes the matrix representation of $\rho(g)$ in this basis, then by using multilinearity the defining property of a tensor operator can be written as follows \begin{align} U(g) T_{i_1\cdots i_k} U^\dagger(g) = \rho(g)_{i_1}^{\phantom {i_1}j_1}\cdots \rho(g)_{i_k}^{\phantom {i_k}j_k} T_{j_1\cdots j_k} \end{align} So this definition immediately reproduces the cartesian tensor definition above if we take, $V =\mathbb R^3$, $G=\mathrm{SO}(3)$, and $\rho(R) = R$, and similarly for the spherical tensor definition if we take $V=\mathbb C^{2n+1}$, $G=\mathrm{SO}(3)$, $\rho = D^{(n)}$ and $k=1$. **Question.** Is the sort of object I just defined the "proper" formalization/generalization of the notion of tensor operators used in physics; it seems to contain the notion of tensor operator used in the physics literature? Is there any literature on the sort of object I define here? I would think that the answer would be yes since this sort of thing seems to me like a natural generalization a mathematically-minded physicist might like to study.


Is squared motor gearbox ratio proportional to inertia ratio?

I read an interesting article http://m.machinedesign.com/news/motor-sizing-made-easy It is very interesting, but I can not follow the 2nd last paragraph. I don't understand why it is true. _**Gearboxes offer a significant benefit in that they affect the inertia ratio by a factor of the gearbox ratio squared._** Please help. Thanks


What would earth look like if traveling at 77% the speed of light?

I've been having a bit of trouble wrapping my mind around this. If I were in a ship going 77% the speed of light (enough to experience reasonable time dilation) would I see earth going in fast forward? Would people on earth see my ship moving very slowly? I have this question because I was reading Physics of the Impossible (Micho Kaku) and he stated that if we were to travel this fast, we would get to a nearby star in about 10 years, but on earth it would have been many more years. This leaves me to suspect that if many years passed on earth even though I would experience time as normal, that I would see people in fast forward. Or would they be moving slowly because relative to me _they_ are moving the speed of light?


Acceleration of bouncing ball with air resistance

Here is a strange test question I really don't get: > A ball falls freely from a certain height, and due to air resistance it will > reach a certain terminal velocity (assuming there is a normal gravitational > acceleration going on all the time). Then it hits the ground, and bounces > back without loss of energy. > > Question : what is the acceleration on the ball immediately after it > bounced? The answer is $2g$ (2 times gravitational acceleration), but the explanation is not included. _Note: This is a question from the Physics Olympiad!_


Is reparameterization invariance some kind of gauge symmetry?

On page 116 of this book it is said, that reparameterization invariance of the string action is analogous to the gauge invariance in electrodynamices. Whereas Maxwell's equations are symmetric under a gauge transformation which allows to describe the same $\vec{E}$ and $\vec{B}$ fields by different potentials $A_{\mu}$, reparameterization invariance makes it possible to use different grids on the world sheet to describe the same physical motion of a string. So is reparameterization invariance some kind of gauge invariance too? What would take the role of the gauge potential $A_{\mu}$ and what would the corresponding "gauge invariant" action look like?


Which twin produces more widgets?

Two twins are identical in every way, including their widget production. One twin goes on a long space trip at near the speed of light to produce widgets during the trip. When space-twin returns he has aged less than earth-twin. But, who produced more widgets?


Delta to Star/Y Conversions and vice versa in Electric Ciruits

We all know the basic rules for conversion of $"Delta"$ circuits to $"Star"$ circuits and vice versa. We also know that this is needed for simplification of circuits in complex cases. Can anyone please explain HOW the concept of such conversions came about? To be clearer, can anyone show the derivation of the Conversions? for both Capacitors and Resistors?


Is it possible to extract nitrogen from the air via a centrifuge

I was thinking about how they have cans of compressed nitrogen for filling paint cans, and also that they sometimes will blow nitrogen in to food containers before sealing them, and that got me interested in extracting nitrogen for a nitrogen filled refrigerator. I read up on it and it appears the main way that nitrogen is extracted from the air now is via liquidification of the air and then relying on the different boiling points. I was curious why they didn't use something like a centrifuge to separate the nitrogen out. I'm guessing from what little I know of centrifuges use in nuclear refinement is that it's not energy efficient since the weights are not different enough to cause easy separation. But I was curious if that was the real reason, or if there were other forces that prevented a centrifuge from being used in this kind of gas separation.


Why does one of the extra dimensions of F-Theory have to be a temporal dimension?

F-Theory, as I understand it, is a realisation of Type IIB String Theory as a 12-dimensional theory in such a way that the $SL(2,\mathbb Z)$ symmetry becomes natural because Type IIB String Theory is F-Theory compactified on a torus. _Question 1:_ However, this explains only why F-Theory needs to be 12-dimensional. Why does one of these extra infinitesimal dimensions need to be temporal, though. _Question 2:_ I've also heard that F-Theory isn't exactly a 2-time theory. However, that doesn't make sense to me, as, since the metric signature needs to be $(-1,-1,1,1,1,1,1,1,1,1,1,1)$, i.e. mostly plus, with 2 temporals and 10 spatials. Or am I missing something obvious, here?


Compressilibility and the form of Newton's second law in fluid mechanics

In deriving Euler's equations for fluid mechanics, in particular $$f=\rho \partial_t v +\rho v\cdot \nabla v$$ for some body force $f$ (e.g Landau & Lifschitz 2.3) one assumes the continuity equation $\partial_t\rho=-\nabla\cdot\left(\rho v\right)$ and Newton's second law in the form $F=ma$ so $$f=\rho D_t v$$ where $D_t=\partial_t+v\cdot \nabla$ is the total time derivative (e.g Landau & Lifschitz 2.1). If one instead uses $F=\dot p$ so that $$f = D_t\left(\rho v\right)$$ then in order to get to get our initial equation back one must assume $D_t \rho=0$ but of course $D_t \rho=\partial_t\rho+v\cdot \nabla\rho=\nabla\cdot\rho$ and so we have assumed the flow is incompressible. Landau implies that he has not assumed incompressibility at this point so how is it that one can choose the first form for Newton's second law without loss of generality? _(If your answer would appeal to stress-energy tensors or Navier-Stokes etc., if possible please show how the assumption isn't implicitly made.)_ To be more clear, as I have shown above, if you say that $f = D_t\left(\rho v\right)$ and also that $f=\rho \partial_t v +\rho v\cdot \nabla v$ then it follows that the flow is incompressible. I don't think that this does imply incompressibility so why is the assumption false?


Why do we use spinors for describing fermions?

I.e., what properties of the spinors gives us a reason for using them for describing of wavefunctions of fermions?


Potential energy graphs of chemical systems

![enter image description here](http://i.stack.imgur.com/ggTmN.png) I really wish someone could dissect this for me in the language of physics. The system is that of two atoms that approximate each other along 'r' (or distance themselves apart). My concerns are these:   * What does negative potential energy mean here?   * Where is the "zero" reference defined?   * Why is there reference to repulsion and attraction in the PE graph?   * If $F=-dU/dr$, shouldn't the graph of the attractive force be flipped upside down? (Specifically they say the Energy of attraction is $-1.436/r$ and the energy of repulsion is $7.32x10^-6/r^8$.)   * My textbook makes reference to $U=\int F\,dr$. Why isn't it defined as the negative work in this system?


Asymmetry of induced e.m.f due to a bar magnet falling through a coil

It is well known that as a bar magnet falls freely through a coil, an electromotive force is induced in the coil by Lenz's Law. However, it appears that the peak e.m.f induced (ignoring the direction) is higher after the passing through a coil, than before it passes through the coil. Why so? My explanation was that after passing through the coil, gravity causes it to accelerate after the coil in the same direction in which the magnetic flux linkage through the coil was changing. By Faraday's Law, e.m.f induced will be greater. However, under this hypothesis, gravity would also have the same effect on the bar magnet before it passes through the coil. Is there any theory that can explain this asymmetry well?


Hot water freezing faster than cold water

This question has puzzled me for a long time. There is already a question like this on Physics.SE. John's answer to the question seems quite satisfying. But when I googled the cause I found this and this explanation. They maybe wrong but I think How Stuff Works is a reliable source. And here's the original paper. I am quite confused now reading the different explanations. Can anyone please shed some light on the issue?


Capillary action meniscus height in a tube fitted inside another tube?

I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved). Height of liquid column: ![enter image description here](http://i.stack.imgur.com/P5s6S.png) where: $\gamma$ = liquid-air surface tension $\theta$ = contact angle $\rho$ = density of liquid $g$ = gravity acceleration $r$ = radius I tried my best to draw the examples I'm interested in order to help my explanation. I didn't consider the capillarity inside the smaller tube in both example #2 and #3 because I'd like to assume that $a/2$ in example #1 is close to $c$ in example #2 and #3 (drawings not to scale). ![enter image description here](http://i.stack.imgur.com/Au1cp.png) Since from what I understand the column height is given, among other things (most of which can't be changed, like liquid-air surface tension, contact angle, density of liquid and gravity acceleration), by the tube radius, I'd like to know if $c$ in example #2 can be considered as $a/2$ in example #1 to calculate column height using above formula. Also I'd like to know how having beads of slightly smaller diameter than $c$ between the two tubes (example #3) would affect the column height. If said beads were less dense than water, could they still improve column height or would they just form a floating mat on top of 1 unit thickness? What'd be the column height of example #2 and #3 assuming $c$ as 1mm? I'm quite sure that given the same reached height $h$ in example #2 and #3, $c$ of #2 has to be smaller than $c$ in #3.


Do interaction-free measurements require a physical collapse or splitting in order to be truly interaction free?

Interaction-free quantum experiments like Renninger's experiment or the Elitzur-Vaidman bomb tester are often taken to be examples of interaction-free measurements of a system. Unfortunately, such assumptions presuppose the ability to post-select in the future just to make sense. Interpretationally speaking, it is hard to see how post-selection can possibly be made without some form of physical collapse of the wave function or a preferred physical splitting of the wave function into branches. Without either a physical collapse or splitting, is it possible to gain information about a system of which we are totally ignorant about the preparation of its properties we are interested in without an actual interaction with it? Basically, does the idea of interaction-free measurements only make sense within some interpretations of quantum mechanics and not others? Is there a philosophical reading of the two-state formalism which does not presuppose a collapse or a splitting? In the two-state formalism, does the necessity of normalizing the overall probability factor to 1 entail the ontological reality of the other outcomes because we have to sum up over their probabilities to get the rescaling factor? The other outcomes where the bomb did in fact go off?


How do two rigid bodies with different 3rd moment of inertia rotate differently?

If rigid bodies $R_1$ and $R_2$ has exactly same total mass $M$, central of mass, and rotational inertia $I$, but different third moment of inertia $M_3$, how would they move/rotate differently? What's the observable phenomenon of the difference in their third moment of inertia?


What are some publications which continue Schrödinger's "What is life?" discussion?

I'm looking for modern publications on the physical nature of life in which the primary reference is to the discussion started by Schrödinger in 1944 in his book "What is life". For example, publications that suggest amendments (which have been generally accepted) to his ideas, similar books (not text books) that take into consideration new discoveries and offer a modern follow up, compiled works, reviews, and the like.


Find Inductance in a Coil

Question: An inductive coil has an internal resistance of 20 $\Omega$. When an AC Voltage source with a frequency of 100 Hz is connected to the coil, the current lags the voltage by 30 degrees. What is the value of the inductance L. My Work: I am unsure where to begin here. If we know that $\omega_d=\frac{1}{\sqrt{LC}}$ and we also know that $f=2\pi \omega_d$, then we can just find the capacitance and solve for L in the first equation and we will have our answer. However, I am not sure how to find capacitance. Any help is appreciated.


Is synchrotron radiation considered incandescent?

Synchrotron radiation is produced via the acceleration of charged particles, much like incandescence. However, all information I've seen state that incandescent light is produced exclusively from the thermal motion of the charges. Is there therefore a better term to describe the light produced via synchrotrons? Thanks!


What force does bathroom scale measure?

When you stand on a bathroom scale, I know that the force displayed is the normal force. Since it is the normal force, then technically the force displayed would actually be Fg= (mass)(acceleration) right? If this is true, then why is it that your normal force changes in an elevator? (i.e. if you travel down an elevator, why would the normal force decrease?) Please explain this to me, as I am really confused! It would be best if you could relate this to the Newton laws. Thanks!


How does carbon dioxide or water vapour absorb thermal infra red radiation from the sun?

We are all told at school water vapour and carbon dioxide are the top two greenhouse gases, and that they absorb thermal infra red radiation, trap heat and warm up the Earth. My question is how do they do that? Why can't Oxygen or Nitrogen or any other gas not absorb infra red radiation as well as water vapour or CO2?


Trying to combine red, green and blue to produce white

I tried to mimic the mechanism of typical screens to produce white color out of red, green and blue. What I did is displayed the attached image on the screen, and moved far away as to let the diffraction effects take place, so that the three colors appear as if they're coming from the same point. Nonetheless, quite paradoxically, what I have seen was black instead of white. I don't know if this question fits this place, so excuse me. ![enter image description here](http://i.stack.imgur.com/EHzCq.png)


Why does motion help you balance on ice skates?

It's almost impossible to balance on a single ice skate if you're standing still. But give yourself just a little forward motion—it doesn't take very much—and it suddenly becomes easy. You can stand there on one leg and glide effortlessly half way across the rink. Friction will cause you to gradually slow down, and it's only when you've slowed almost to a complete stop that balancing becomes hard again. Why? This seems suspiciously similar to the question of why it's easier to balance on a moving bicycle. But the standard answer to that involves the angular momentum of the wheels. There's nothing rotating here. In fact, other than the very slight deceleration from friction, moving should make no difference. In both cases, you're standing still in a nearly inertial reference frame. Does that slight deceleration somehow matter? Or does the interaction between the ice and a moving skate somehow help you to balance?


Can you explain Fermat's Principle to me?

The textbook(F.A.Jenkins and H.E.White **_Fundamentals of Optics_** ) states that the Fermat's principle is that > _the path taken by a light ray in going from one point to another through > any set of media is such as to render its optical path equal, in the first > approximation, to other paths closely adjacent to the actual one._ To be honest, I can't understand that description. But I know that > _the actual ray path between two points is the one for which the optical > path length is stationary with respect to variations of the path._ Are the two descriptions same?


What is the physics behind explosion of Stars?

![enter image description here](http://i.stack.imgur.com/mjhTo.jpg) Looking at the above photo, you will soon find it is impossible to explain what actually happens (with today's physics), in one word star is like a nuclear fusion bomb which its pressure is in balance against gravitational force.   1. A supernova (or a black hole) is triggered by an imbalance between them, so isn't supernova a natural pure fusion bomb?    2. Then what equations are able to simulate a virtual supernova explosion?


Why are interactions always mediated via bosons (as opposed to fermions)?

Weak, Strong and electromagnetic force are mediated via bosons. I've come across the Walecka-model where the exchange particles are $\sigma$ and $\omega$ mesons; bosons again. why are interactions always mediated via bosons (not fermions)?


How do spiral arms form?

Why aren't all spinning galaxies shaped as discs as my young mind would expect? I understand how the innermost parts of a galaxy spin faster than the outer parts, and that could explain why some galaxies are more spiraled than others based on age. Though, this doesn't explain how the arms came into existence in the first place. Might it have something to do with an imperfect distribution of mass, and therefore an imperfect distribution of gravity, causing a split in the disc from which point gravity, centripetal force, and inertia could take over? Or does something happen earlier in a galaxy's life?


Question about Komar integral derivation in Wald

I have a question about derivation 11.2.10 in Wald (page 289). Here is a screenshot of the relevant passage: ![Wald p289](http://i.stack.imgur.com/UI69w.png) I don't get the step $$-\frac{1}{4\pi}\int _{\Sigma}R^{d}{}{}_{f}\xi^{f}\epsilon_{deab} = \frac{1}{4\pi}\int _{\Sigma}R_{ab}n^{a}\xi^{b}dV.$$ Could someone explain to me how he gets that equality? Thank you very much in advance.


Dimensional aspects of the imaginary unit $i$ in physics

From a real world perspective each dimension in the 3-D Cartesian System can be represented by an axis that is perpendicular to 2 other axes. I read somewhere else that the effect of ${\sqrt {-1}}$ is to reorient data 90 degrees on the "imaginary" axis. I guess my question is this: What role ( if any) does ${\sqrt {-1}}$ serve in everyday or even quantum physics? ${Edit}$ To clarify, I am more concerned with the dimensional aspects of ${\sqrt {-1}}$. If for instance you were to consider a 3-D location to be a "point" with length width and height in a complex graph representation with three additional mutually perpendicular axes? PS. I know this thread is getting a bit hair brained. But in my defense, I did look for a "specualation" tag before I posted this thread here.


In classical mechanics, are complex numbers unphysical?

> **Possible Duplicate:**   >  Physics math without $\sqrt{-1}$ When I produce a complex final solution to a problem that began without complex coefficients at all, I have so far (with my limited expertise) tended to discard it as unphysical, unless I had to tinker some more, in which case the $i$s may square and real-ity be restored. Is this acceptable? I understand that the Wessel plane is useful in modelling planetary motion, and many other things, but the fact (?) that you could do it all (albeit more awkwardly) without them seems to be indicative that they are a little contrived. In short, is there a counter-example?; a physical theory ( _not_ a particular, neat, mathematical model of a physical theory) in classical mechanics that could not be formulated without complex numbers?


Complex variables in classical mechanics

In quantum mechanics complex numbers are absolutely essential because of the relation $$[\hat q_i,\hat p_j]=i\hbar\delta_{ij}.$$ But is complex number also essential anywhere in the formalism of classical physics, except for some convenience of mathematical calculations? Is there any example from classical physics where there is no other way out except to introduce complex variables? Is there any strong reason by which I can say that complex number are unavoidable in classical physics too?


Gauge fields in 2d spacetime

I believe it is only a technical question. However I cannot realize it. It is said in 2d spacetime the gauge fields $A_\mu$ can be rewritten in lightcone coordinates as $A_+=ig\partial_+g^{-1}$ and $A_-=ih\partial_-h^{-1}$, where $g$ and $h$ are the elements of the gauge group. Anyone could give me a simple and transparent explanation on it? Thanks!


What is a quark condensate?

1. What is a quark condensate? is it a bound state between 2 quarks? can we have 3(or more)-quarks condensate?   2. What mediates the interaction between the constituents of the condensate?   3. Are the constituents of the condensate always fermions?


Calculate acceleration and time given initial speed, final speed, and travelling distance?

A motorcycle is known to accelerate from rest to 190km/h in 402m. Considering the rate of acceleration is constant, how should I go about calculating the acceleration rate and the time it took the bike to complete the distance?


Motion with damping involving both $-v^2$ and $-v$ terms

I'm trying to solve for the motion of a particle, accounting for both viscous and drag forces. (There is no potential) The total resistance by medium is modeled by: $\vec{F}=-(\mu_1v + \mu_2v^2)\hat{v} $ Where $v$ is the velocity; There is no potential. I need to set up the Lagrangian and describe motion under the following conditions: $\frac{\mu_1}{\mu_2} = 10^{-5}$ $\frac{\mu_1}{\mu_2} = 1$ $\frac{\mu_1}{\mu_2} = 500$ I have set up the Lagrangian as follows: $L = \frac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2)$ $\frac{d}{dt} \frac{\partial L}{\partial \dot{x}} -\frac{\partial L}{\partial x} =- (\mu_1 \dot{x} + \mu_2\dot{x}\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2})$ and similarly for z and y. Equations of motion turn out to be: $m\ddot{x}= - (\mu_1 \dot{x} + \mu_2\dot{x}\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2})$ Similarly for z and y. I'm at a total loss to how these should be solved. Is there even a way of analytically solving such coupled equations? However, If I orient the x axis along initial direction of velocity, I should get: $m\ddot{x}= - (\mu_1 \dot{x} + \mu_2\dot{x}^2)$ $y=0$ $z=0$ I'm stuck at this point. Wolfram Alpha suggests: ![](http://puu.sh/bmj5K/8d1edff9ab.png) Where $\frac{\mu_1}{m}=a$ and $\frac{\mu_2}{m}=b$ I'm not aware of the techniques used to get to this. Would appreciate some help/pointing to resources on that. After this, I'm not really sure of how to apply limits (Since we have the additional constraint of the domain of Log functions), and how to interpret the motion in the 3 regimes mentioned above. Can someone give me some pointers? Also what really happens when there is a potential (Say, gravity)? (which really is the real world scenario)


How do we deal with irrationals in Physics?

This question is really very basic: how do we deal with irrationals in Physics? If for instance, in some meaningful calculation we get a length of $\pi$ meters, or a force of $\pi$ Newtons, how should we understand this? One could say: "well, we understand it as a rational approximation", but then there comes another question: which rational approximation? We can approximate $\pi$ by $3,14$ or $3,1415$ and so forth, increasing the precision, but yet never getting really to $\pi$. Another problem is: well, the equations gave us $\pi$, and this is not a physically meaningful thing, so is there something wrong with the equation? Because, in nature I think there really isn't irrationals lurking arround, so that this becomes a little complicated. So: how do we deal with irrationals in Physics?


The earth's magnetic field

This might sound like a silly question. Is it possible for the earth's magnetic field to actually destroy or harm earth? (implosion, crushing etc.)


How the value of permitivity of free space is determined?

when electric charges are placed on the medium then force of interaction between them decreases. We say this factor permitivity of the medium. The force of interaction between the charges is max if the charges are at space or if there is absence of medium. We know the electric permittivity of free space is $8.85 * 10^{-12} F/m$. My question is how this value is determined. Is there any experiment held?


Do multiple electrons exist during superposition?

Wikipedia says: > _Quantum superposition is a fundamental principle of quantum mechanics that > holds that a physical system—such as an electron—exists partly in all its > particular, theoretically possible states (or, configuration of its > properties) simultaneously;[...]_ What interests me is what happens when location is the state that is superposed. If an electron exists in two locations at once, as a result of quantum superposition, does that mean that the mass of the universe has increased so long as the superposition exists? So in other words, does it follow that effectively two electrons exist during superposition? Why or why not?


Critics of Mannheim's Conformal Gravity Theory?

I'm looking for more articles/reactions/critiques/support for Philip Mannheim's recent conformal gravity theory. See here: http://arxiv.org/abs/1101.2186v1 Any ideas on where to start?


Peak deconvolution

I need some help understanding how much information I can pull out of this data. I have a sample made up of two materials. Materials A and material B. Then I took a picture of the sample. The two materials mix quite well, but not perfectly, so on my image I can see that some areas are mostly material A, some are mostly material B, and most of the areas are a mix between. I also took an image of a sample consisting only of material A. On my images material A will look white, and material B will look black. If I make a histogram of the image of material A (and only A) it has the center around 6 [A.U] which fits my theory. The histogram also has a FWHM of 0.55 [A.U.] The same histogram of material A & B is centered around 7 [A.U] which also fit my theory since it's a 50:50 mix of A & B and A is centered at 6 [A.U] and B should be centered at 8 [A.U]. The FWHM of this peak 0.64 [A.U] - thus only slightly larger than the image with pure A. Here's my problem, since the peak with the mix only is slightly wider than the peak for the pure material it means that my resolution i not good enough to distinguish the two materials from each other (if I could see areas with pure material A or B it would be a camel/double-peak, if I could see areas with mostly material A or B I would have a very wide peak - here I only have a slightly wider peak). However, it is still wider, so there must be some kind of information I can subtract. I'm not really sure what - if any. EDIT: If anything is unclear, please ask!


A strange audio phenomenon, could there be a physical interpretation to it?

http://mathoverflow.net/q/165038/14414 Motivation : Here is a motivation as to why this problem is so important. Let $f(t)$ be an audio signal. We can safely asume it to be bandlimited to 0-20kHz as we cannot hear anything above that. Capture this signal in digital computer with appropriate sampling frequency and denote it as $f[n]$. Now take Discrete Hilbert transform of $f[n]$ to get $f_h[n]$, (using the code $f_h$ = imag(hilbert(f)); in Matlab). Compute the signal $f_{\theta}[n] = f[n]\cos\theta + f_h[n]\sin\theta$ for any value of $\theta$, then listen to the signal with different values for $\theta$. They all sound **exactly identical**. Similarly our $MI_{\omega_0,\omega_1}(t)$ is same for all $f_{\theta} = f\cos\theta + f_h\sin\theta$, for any value of $\theta$. **Question :** just try it. $<f,f_h> = 0$, they why do they produce same effect in the listner? Is it some quantum mechanical effect gone wrong? **Added :** Also see this metric space : metric space I've recently filed a patent using this metric with a slight change, instead of arccos i used sqrt(2(1-cos(theta))), which makes it a Hilbertian metric. I had then embedded this metric space into an Hilbert space isometrically, to model using vectors. **MATLAB code :** [f,fs] = wavread('audio_file.wav'); fh = imag(hilbert(f)); theta = pi/4; f_tht = f*cos(theta) + fh*sin(theta); wavplay(f,fs); wavplay(f_tht,fs);


Why is the summer, in the temperate latitudes, in average, hotter that the spring?

It is common knowledge that the transition from the Spring to the Summer season occurs in the Summer Solstice when the "Sun reaches its highest excursion relative to the celestial equator on the celestial sphere" (as stated in Wikipedia). It is also stated in Wikipedia' Summer page: "Days continue to lengthen from equinox to solstice and summer days progressively shorten after the solstice, so meteorological summer encompasses the build-up to the longest day and a diminishing thereafter, with summer having many more hours of daylight than spring." My question is: why is the summer, in the temperate latitudes, in average, hotter that the spring?


Based on note, how fast is a stringed instrument's string oscillating?

to be quite honest, I have no idea where to start on this problem mathematically. However, it struck my curiosity and would love to know how it works on a mathematical level. # the problem You have a guitar. All of the strings start at a resting position. When you play a string, it is now oscillating between two points (as far as I can tell). The speed at which it oscillates is correlating to the frequency, and the distance between the points is how hard the note was struck or the volume of the note. You play the E string, open, and let it ring. Then you play the 7th fret of the same string, and let it ring. Now how does one determine how quickly the string is oscillating between the two points in this scenario, in a unit of measurable distance? All I really have so far is that E and B are a perfect fifth apart, making the ratio of their frequencies 3:2. If one were to play the 12th fret of the same string, that would be a 1:2 relationship. Therefore it follows that the speed they oscillate should follow the same proportions. I, however, am not sure how to get an initial distance. ### Edit for some clarity; the application to put this question in context, I am trying to programmatically work with a three-dimensional model of a guitar as it is played by a guitar player and make it visually convincing. As such, when he plays an open string, it would move more slowly. If he put his finger on a fret on that string and played it, nothing from his finger to the nut would vibrate, but the rest would vibrate FASTER. I'm trying to discern a reasonable level of accuracy of this visual effect. Thank you


Alternative derivation for the capacitor energy equation

I hope this is the right place for this kind of post. A friend is trying to derive the equation for the energy stored in a capacitor by analysing the change in potential on one plate when the capacitor is shorted, however I don't think this valid. Could anyone explain why this is right or wrong? He says that when the capacitor is shorted the potential at each plate changes by V/2 (from 0 to V/2 on one side and from V to V/2 on the other), so by substituting in V/2 into the equation he gets E = 1/2 QV, the capacitor energy equation. ![Diagram](http://i.imgur.com/Q7H7oXk.png)


Characteristics of bloch electron in a priodic potential

Effective mass of a Bloch electron in a periodic potential is negative why ?


Can two single particles interfere with each other?

Groups of particles can interfere with one another; In the double slit experiment when measuring single photons at the screen each one arrives at the screen in a random manner and they only show the interference pattern once several particles are detected. Obviously two waves can obviously interfere with one another, but can two single particles interfere with one another? Cohen-Tannoudji writes that > light simultaneously behaves like a wave and like a flux of particles But do two particles constitute a flux of particles? I doubt that this could be tested experimentally but if it were so would this constitute a measurement for each particle? **Here was my thought** (Disclaimer:I do not have a good idea of what the interference of two single particle states is): To have interference of two single particles you would have to know something about there position to be able to describe their interference pattern hence the measurement.


Self Inductance and Mutual Inductance

Systems have mutual inductance and self inductance-however won't the effects of these two effect each other as well? Do systems undergo current and magnetic field behavior that are, in this way, dictated by differential equations?


What's the physical significance of using fourier transform for diffraction?

I am studying some basic idea of diffraction and there mention in far field, the diffraction pattern could be understood by Fourier transform. But I just don't understand what's the physical fact for that. And why cannot use Fourier transform for the the near-field case? Also, when I am trying to understand the theory of diffraction, it ends up with some complicate math (integrals). I want to learn that but the books I am reading are not easy to understand. Anyone recommends some good books or video lectures (more theoretical but to explain most of the math in plain way)?


Change in time period of a simple pendulum

There's a question in my book which goes as: A pendulum has a bob of a hollow sphere filled with a liquid and having a hole at the base. If the liquid is allowed to flow out, what would happen to the time period of the pendulum? My thought: Since the time period is independent of the mass of the bob, the time period would remain unchanged. What my books says: The time period would initially decrease, and then gets restored. This is because as the liquid gets lowered, the center of gravity of the bob gets lowered, which increases the effective length of the pendulum string slightly. When the liquid is completely drained out, the center of gravity again comes to the center of the sphere, restoring the time period. I don't understand why the 'effective length' would increase due to the shift of cog. And would the change in time period would be so significant?


Time period of simple pendulum with varying mass

How do you find time period as a function of time for a simple pendulum that is in the form of a hollow sphere that is filled with mercury and there is a hole in the bottom through which the mercury is constantly falling at a fixed rate? I tried creating a function for the time period to check how it varies with the mass of mercury, but I had too many variables. EDIT: This question was a concept based multiple choice question in my homework book. The option were time period becomes erratic, time period increases, time period decreases, time period first increases then decreases.


Where to find experimental/theoretical value of (charge) radius of particles?

Where to find experimental/theoretical value of (charge) radius of particles like proton, neutron, $pi^+$, etc. ?


Resources for current thought on time/spacetime?

Are any of the big-name physicists associated with the time in the same way that Hawking and Penrose are associated with black holes? I'm interested in some good books that focus on the topic.


At what temperature does air become liquid at a pressure of X bar?

... with X = 6 in my particular case. I was trying to get a phase diagram for air, only got one for nitrogen and was slightly puzzled. Here's why: I was wondering about a particular detail in industrial air liquification. The original Linde process used 200 bar pressure and cycled the air though several cycles to condense it, as one decompression only gives around 40 K reduction in temperature. This process was replaced rather early by a modified process (called the Linde-Fränkle process) that compresses the air to only 6 bar and then runs it through a pre-cooled regenerator, where it liquifies in the first pass, proceeding to the distillation column immediately. That process was in use until the 1990s, so it probably had decent performance. Now, I was wondering to what temperature the regenerator has been precooled. From the phase diagram of nitrogen, I'm estimating to somewhere not abive 150 K, most probably lower. Since the pre cooling is done with the gaseous $N_2$ and $O_2$ boiling off the distillation column, it sounds like quite close to the liquification temperature at atmospheric pressure. This puzzles me because it makes me wonder why such little compression is sufficient to make the whole process economic. Also, I can't quite see at what point the Joule-Thompson effect comes into play here. The air is liquefied entirely without decompression, so where dies the cooling come from? I was also wondering if I might be confusing liquid and supercritical here, since I'm told pressurized gases in typical containers at room temperature are really supercritical already, even though laymen like me still call them liquid.


A physical understanding of fractionalization

all! Is there a physical understanding of fractionalization in condensed matter physics? The textbook approach is theoretical, not physical. I'm thinking of spin-charge separation for electrons, the fractional quantum hall effect, and things like that. The theoretical approach is to introduce an auxiliary gauge field with no kinetic term at the bare level so that it is apparently confining and nondynamical at the bare level, but somehow, dynamics intervenes, and it becomes deconfining, and somehow, there's some mixture between the emergent gauge symmetry and the original symmetries, and somehow, fractionalization comes in in the diagonalization. What is the physical interpretation without introducing theoretical auxiliary gauge symmetries right from the onset?


What exactly is a mole?

I have some trouble understanding exactly what a mole represents. As I understand, one unit mole is 1/12 of the mass of an atom of carbon-12 (thus it is the mass of one nucleon?). What is a mole, then?


Spring system - 3 DoF system and its properties while changing stiffness

I was given the attached 3 degree of freedom spring system with the purpose of analyzing it. ![system](http://i.imgur.com/pK0p9kG.jpg) I came up with the following equation of motion ![equation](http://i.imgur.com/vr2QWQr.jpg) and then I ran Matlab to calculate the corresponding natural frequencies and mode shapes using eigenvalues and eigenvectors; I was asked to see what happens when value of stiffness $k_{12}$ is changed. This is the plot of the value $k_{12}$ against the natural frequencies. ![plot](http://i.imgur.com/fdwweRr.jpg) The problem is that I do not know WHY values of natural frequencies are insensitive at low values of $k_{12}$ and why both the 1st and 2nd natural frequencies are insensitive to changes in $k_{12}$ when values are large (first two level off and the third one seems to go to infinity). I assume it has something to do with the equation for force due to a spring between 2 masses but I cannot figure it out. That is why I ask for your help - thanks in advance.


Showing Lorenz gauge is satisfied in retarded potential - vector calculus

I am trying to show that $\nabla\cdot \vec{A}=-\mu_0 \epsilon_0 \frac{\partial V}{\partial t}$ $V=\frac{1}{4\pi\epsilon_0}\int \frac{\rho(\vec{r}',t_r)}{r}d\tau'$ $\vec{A}=\frac{\mu_0}{4\pi}\int \frac{J(\vec{r}',t_r)}{r}d\tau'$ Where $d\tau'=dxdydz$, $t_r=t-\frac{r}{c}$ is the retarded time, $\vec{r}$ is the position vector of the charge and $\vec{r}'$ is the position of the observer (us) and $r=|\vec{r}-\vec{r}'|$ the distance from us to the source. Probably best to start with $\nabla\cdot \vec{A}$: $\nabla\cdot \vec{A} =\frac{1}{4\pi \epsilon_0}\int \nabla \cdot \left(\frac{\vec{J}(\vec{r}',t_r)}{r}\right)d\tau'$ Apparently, $\nabla\cdot ({\vec{J}\over r})=\frac{1}{r}(\nabla\cdot \vec{J})+\frac{1}{r}(\nabla'\cdot \vec{J})-\nabla'\cdot ({\vec{J}\over r})$ This is the first thing I need to do to get any further but I cannot do this. $\nabla$ is differentiation with respect to $r$ and $\nabla'$ is with respect to $r'$. I thought that $\nabla\cdot ({\vec{J}\over r})=\frac{1}{r}(\nabla\cdot\vec{J})+\vec{J}\cdot (\nabla(\frac{1}{r}))$


Problem involving Dirac's equation

I'm stuck in an equation derivation of Ryder's QFT book. Starting with Dirac's equation: $$(i\gamma^\mu\partial_\mu-m)\psi=0$$ If I multiply by $i\gamma^\nu\partial_\nu$, I get: $$((\gamma^\nu\partial_\nu)(\gamma^\mu\partial_\mu)+i\gamma^\nu\partial_\nu m)\psi=0$$ I should get: $$(\gamma^\nu\gamma^\mu\partial_\nu\partial_\mu+m^2)\psi=0$$ I suppose that this means: $$i\gamma^\nu\partial_\nu=m$$ But I don't know why. ¿Could anyone show me the way to prove this property? Thanks.


Is it possible to orbit the sun next to Earth?

With Geostationary orbit your limited to the equator and ~36000km. Would it be possible to orbit the sun at the same speed as Earth, and then in basically the same place so that the distance to the Sun is slightly less or more than Earth, and the inclination is slightly less or more, in order to achieve a geostationary effect over any point at any height. Wouldn't the Earth's gravity interfere. (The reason I ask is, would this be a way of creating a space elevator?)


What do scientists believe about existence in dimensions?

I couldn't really think of a suitable question title, I'm not sure if it's completely related or not. But this is as far as I know (well, I thought it all up last night and it seemed extremely logical so I think it's probably what scientists believe): There are 11 possible dimensions, the 1st, 2nd, 3rd, 4th, 5th and so on and each dimension consists of all the dimensions below it, e.g. the 6th dimension consists of the 1st, 2nd, 3rd, 4th, 5th and 6th, the 4th of which is time, which allows for existence because of course there must be time for existence as you need time to forward to exist. So this means that the space-time (the 4th dimension) must be the lowest dimension with possible existence as that is the lowest dimension to include a dimension of time. So we, as humans, must exist in the 4th dimension because we need time to exist and because all the gravity, space and space curvature around us is governed by the extra dimension of time, creating space-time/4th dimension, but time isn't a spacial dimension so our view of the universe doesn't change because of the extra dimension of time, which is why we only see it as 3-dimensional, because they are the only 3 spacial dimensions that we can actually see/visualise. Are my views correct and a physicist would agree with me or are they completely wrong? If they are wrong, could someone explain what I have explained, but correctly please.


2D quantum well energy spectrum (analytical vs numerical)

I am trying to understand the energy spectrum difference between the analytical and the approximated solution for a quantum well. The particle is inside a box with domain $\Omega=(0,0)$X$(1,1)$. For this I have $\hbar = m = 1$ and the energy is given analytically by $E_{m,n}=\frac{\pi^2}{2}(n^2 + m^2)$ My approximation is done using finite differences with a grid of $60$X$60$. The eigenvalues that I'm getting with the exact solution are always positive whereas the eigenvalues I'm getting with the approximated solution are always negative. I am not familiar with quantum mechanics. That said, can you help me understand this? What can be happening, is it a big error due to the approximation? Where or what can I think of to try to understand these results? I hope the question and the problem are well stated if not just tell me, probably I missed some important data and/or assumption. **UPDATE** Here I paste the code used to do this little simulation               clc;     xO = 0;     xL = 1;     N = 60;     %First build the matrix for a 1D-mesh     h = (xL - xO) / (N-1);     H = diag( (-2/h^2)*ones(1,N), 0 ) + ...         diag( (1/h^2)*ones(1,N-1), 1 ) + ...         diag( (1/h^2)*ones(1,N-1), -1 );     %Using the tensor product build the matrix for the 2D-mesh     H = kron(H, eye(N)) + kron(eye(N), H);     H = (-1/2) .* H;          %Compute the eigenvalues E and eigenvectors Psi     [Pcomp, Ecomp] = eig(H);          fEexac = @(m,n) ((pi^2)/2) * (n^2 + m^2);     Eexac = [];     for n=1:N         for m=1:N             Eexac((n-1)*N+m,(n-1)*N+m) = fEexac(n,m);         end     end          %Plot energy from analytical vs computed     figure(1); hold on;     plot(1:N^2, sort(diag(Eexac)), 'b', 1:N^2, sort(diag(Ecomp)), 'r');          %Plot discretized energy spectrum     figure(2); hold on;     plot(1:40, diag(Ecomp(1:40,1:40)), 'r');          %Compare lowest 300 eigenvalues (exact and computed)     figure(3); hold on;     x = 1:300;     plot(1:300, sort(diag(Eexac(1:300,1:300))), 'b', ...         1:300, sort(diag(Ecomp(1:300,1:300))), 'r');


What does the quantum state of a system tell us about itself?

> _In quantum mechanics, quantum state refers to the state of a quantum > system. A quantum state is given as a vector in a vector space, called the > state vector. The state vector theoretically contains statistical > information about the quantum system._ The article in Wikipedia about quantum state suggests that the quantum state of a system is indeed its state vector which contains statistical information about it. But what are those statistical information? The particle's location, momentum, wavefunction and.. energy level? When I measure the momentum and position of an electron, the quantum superposition principle suggests that I'm only getting a result corresponding to one of its possible state because of it exhibits both wave and partile behaviors. But what exactly is a possible state?


Similarity between the Coulomb force and Newton's gravitational force

Coulomb force and gravitational force has the same governing equation. So they should be same in nature. A moving electric charge creates magnetic field, so a moving mass should create some force which will be analogous to magnetic force.


What are the ideal centers of mass and rotation for a steadicam?

For those who don't know what a steadicam is, I'll explain it in two points to give a _basic_ idea. If you know what a steadicam is, skip to point 3. 1) A steadicam is a stabilizing mount for a motion picture camera, which mechanically isolates the operator's movement from the camera, allowing a very smooth shot even when the operator is moving quickly over an uneven surface. 2) How does it work (image courtesy of wsclater): ![Steadicam](http://i.stack.imgur.com/jMrJJ.jpg) Red dots = control masses, Blue dot = center of rotation The idea behind a steadicam is that the center of the mass is taken **out of the camera** itself. Therefore, when moving / panning the whole system (steadicam + camera), the motion of the camera itself is more fluid. This can be seen in action in this amazing youtube example. 3) The question: How do we determine the ideal distance of the center of rotation from the camera? How do we determine the ideal distance of the center of mass from the center of rotation? What's the ideal length of the handle (the point where we hold the system), i.e. how far should we hold the system from its center of rotation? _Disclaimer: This is neither a matter of profit, nor a matter of homework. I'm just trying to build some of those for my own personal use, especially to find out how it really works, as I'm kind of amazed from this._


Irreducible representation in physics

I am confused about something. Group theory books written for physicists say that any reducible representation can be decomposed in terms of irreducible representations (so correct me if I am wrong, to me irreducible representations are like the unit vectors i j k in terms of which any 3D vector can be expanded, or they are like sines and cosines in terms of which any periodic function can be Fourier expanded.) Now at the same time they say that any bigger representation of a group can be built out of irreducible ones. What is unclear to me is the physical motivation for each direction. Of course those books contain physical applications but the big picture is never obvious (they get to applications after 200-300 pages of abstract details). If someone could answer the following questions I would be really appreciated: 1-what is the physical motivation to write a representation in terms of irreducible ones? 2-what is the physical motivation to build bigger representations using irreducible ones?


Derrick’s theorem

Consider a theory in D spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form $$L= \frac{1}{2} G_{ab}(\phi) \partial_\mu \phi_a \partial^\mu \phi_b- V(\phi)$$ where the eigenvalues of G are all positive definite for any value of $\phi$, and V = 0 at its minima. Any finite energy static solution of the field equations is a stationary point of the potential energy $$E = I_K + I_V ,$$ where $$I_K[\phi]= \frac{1}{2} \int d^Dx G_{ab}(\phi) \partial_j \phi_a \partial_j\phi_b$$ and $$I_V = \int d^Dx V(\phi)$$ are both positive. Since the solution is a stationary point among all configura- tions, it must, a fortiori, also be a stationary point among any subset of these ̄ configurations to which it belongs. Therefore, given a solution $\phi(x)$, consider the one-parameter family of configurations, $$f_\lambda(x)= \bar{\phi}(\lambda x)$$ that are obtained from the solution by rescaling lengths. The potential energy of these configurations is given by \begin{align} E_\lambda &= I_K(f_\lambda) + I_V(f_\lambda)\\\ &=\lambda^{2-D} I_K[\bar\phi]+\lambda^{-D} I_V[\bar\phi]\tag{1} \end{align} > $\lambda = 1$ must be a stationary point of $E(\lambda)$, which implies that > $$0=(D-2) I_K[\bar\phi]+D I_V[\bar\phi] \tag{2}$$ **My problem is, how they got the equation(2) from (1)?**


What videos should everyone watch?

Similar to this cstheory question "What videos should everyone watch?" What conferences, workshops or lectures have recordings that researchers in theoretical physics should watch?


Lorentz transformation in light cone coordinates in string theory

What is the explicit form of the Lorentz transformation changing the light cone coordinates in the light cone gauge in string theory? The extended nature of the strings complicate matters, especially with interactions turned on. String modes tend to be mixed up.


$$ \DeclareMathOperator{\dif}{d \\!} \newcommand{\ramuno}{\mathrm{i}} \newcommand{\exponent}{\mathrm{e}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\braket}[2]{\langle{#1}|{#2}\rangle} \newcommand{\bracket}[3]{\langle{#1}|{#2}|{#3}\rangle} \newcommand{\linop}[1]{\hat{#1}} \newcommand{\dpd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dod}[2]{\frac{\dif{#1}}{\dif{#2}}} $$ We start from an abstract state vector $\ket{\Psi}$ as a description of a state of a system and the Schrödinger equation in the following form $$ \ramuno \hbar \dod{}{t} \ket{\Psi(t)} = \hat{H} \ket{\Psi(t)} \, . \quad (1) $$ Now if we move to position representation of the state vector what will happen to the Schrödinger equation? In _Mathematics for Quantum Mechanics: An Introductory Survey of Operators, Eigenvalues And Linear Vector Spaces_ by John David Jackson I found the following information (pp. 77-78). By taking an inner product of both sides of (1) with $\ket{x}$ and using the resolution of identity $\linop{I} = \int\nolimits_{-\infty}^{+\infty} \ket{x'} \bra{x'} \dif{x'}$ on the right side we obtain $$ \ramuno \hbar \dod{}{t} \braket{x}{\Psi(t)} = \int\limits_{-\infty}^{+\infty} \bracket{ x }{ \linop{H} }{ x' } \braket{ x' }{ \Psi(t) } \dif{x'} \, . $$ We then introduce a wave function $\Psi(x,t) \equiv \braket{x}{\Psi(t)}$ and if I understand correctly $\bracket{ x }{ \linop{H} }{ x' }$ is also replaced by a function $h(x, x')$ which will lead us to $$ \ramuno \hbar \dod{}{t} \Psi(x, t) = \int\limits_{-\infty}^{+\infty} h(x, x') \Psi(x', t) \dif{x'} \, . $$ Now we are one step away from the familiar Schrödinger equation in position representation: we need Hamiltonian operator in position representation $\linop{H}(x, \frac{\hbar}{\ramuno} \dod{}{x})$ to be given by $$ \linop{H}(x, \frac{\hbar}{\ramuno} \dod{}{x}) \Psi(x, t) = \int\limits_{-\infty}^{+\infty} h(x, x') \Psi(x', t) \dif{x'} \, . $$ Author claims (p. 44) that > For our purposes the general linear operator $K$ can be written in the > explicit form $$ g = K f \rightarrow g(x) = \int\limits_{a}^{b} k(x,x') > f(x') \dif{x'} \quad(2) $$ The function $k(x,x')$ is called the _kernel of > the operator_ $K$. It is not that I do not trust the author, but since my knowledge of mathematics is not great and I've never seen something like (2) before, I'm confused with this "for our purposes". What does it actually mean? Is (2) true for _any_ linear operator or for a particular kind of linear operators, say, for self-adjoint linear operators on Hilbert spaces?


Can massive fermions have zero modes?

I'm confused about fermion zero modes in relation to instantons. I understand that instantons can create fermion zero modes, but it's not clear to me when a fermion has a zero mode. For example, consider QCD with nonzero fermion masses: i.e. there are explicit mass terms in the Lagrangian. Do these fermions have zero modes? Are they created by the QCD instanton? Is it generically possible to have a zero mode when the field has an explicit mass term, and if so, do those zero modes behave differently in an instanton background than the zero modes of massless fermions? Thanks.


Direct nuclear reaction in nuclear physics

Time taken to occur a direct nuclear reaction is very low $10^{-22}$sec . I want to know the Importance of direct nuclear reactions.


NOT Universal Operator and Computational basis

This is the relationship between density operator and Bloch vector: $$\rho= \frac{1}{2}({\bf{\hat{1}}}+{\bf{b}}.\boldsymbol{\hat{\sigma}})$$ We define the NOT Universal Operator in the following way: $$U: {\bf{b}}\to -{\bf{b}}/3$$ My question is - How does NOT Universal Operator act on the elements of the computational basis: $|0\rangle, |1\rangle$?


Gravity and Magnetism on Stars

I am familiar with the fact that Magnetars have a really strong magnetic field and an ordinary star such as the Sun has a very strong gravitational field. But what about the opposite? What is the strength of the _gravitational field_ of a Magnetar and how strong is the _magnetic field_ of an ordinary star?


Current string-theory version of Big Bang

I always wondered what the description of Big Bang would be like in string theory. How is it different from QFT version of Big Bang? Thanks.


What is meant by boiling off electrons in a heater coil?

In my electricity and magnetism course, we used a Thompson tube to produce an electron beam. There is a heating element at the back of the tube and the lab manual claims that "electrons are boiled off." Can someone explain what is meant by this? Boiled off seems an odd term to use but maybe if I understand it meaning, it won't seem so odd.


Smallest force to move a brick

Having a brick lying on a table, I can exert horizontal force equal to $\mu m g$ to a middle of it's side, and it will start moving (assume $\mu$ is the friction coefficient). However, can I make the brick moving with less horizontal force? May be, applying it not to a middle of a side can help? I have no idea of how to calculate or estimate it, but it would be interesting to know.


How does the friedmon solution to Einstein's equations resolve paradox of bounded infinities?

This article talks about a potential explanation of dark matter based on something called the "friedmon." I have no interest in the dark matter question, but the article has made me curious about this friedmon thing. Googling "friedmon" turns up very little. The article says the friedmon is a solution to Einstein's equations of GR wherein an infinite expanse (field or space-time, I guess you would call it) sits as an object within another infinite expanse. Well, let me just quote the article: "To an external observer, a friedmon looks like a micro blackhole with an electric charge the same as that of the electron. However, the friedmon's interior can be macroscopically large — up to the size of the known universe." I believe the notion of an infinite universe sitting as a finite object in another infinite field is also central to the multiverse theory. I'm just curious as to how you express this idea mathematically. At face value there appears to be a contradiction in the idea of an infinite universe which is also a finite object situated in some broader expanse. I would also appreciate if someone can point out a paper or something which shows the details of the friedmon solution to Einstein's equations of GR.


Does speed in space really matter?

This is about validating the science from a science fiction novel. My understanding is that speed is all relative. So while I am in a car on the freeway my speed relative to another passenger is zero, relative to a street sign is 75, relative to a car travelling the opposite direction is 150, and relative to some point off the planet it is much greater. Now two cars traveling at the same speed and direction would have a relative speed of 0, but because of imperfect roads and wind resistance, the speed relative to earth would still impact the car's interactions. For example, walking between a board connecting the two cars would be much easier at 10 km/h than at 100 km/h. But the relative speed of the cars compared to a distant star has no measable impact. So how does this translate to two spaceships? Is a space ship traveling at 1000 km/h relative to a star in the solar system any more likely to strike debris at a faster speed, than one that is stationary relative to the star? Likewise, is it more difficult for two ships traveling at 1000 km/h relative to a star (both with the same heading, speed, etc.) to dock, than two ships stationary relative to that star? It would seem like acceleration would be the only influence, unless solar winds have a bigger impact then I imagine. Without resistance, it would require just as much effort from the manuvering thrusters to maintain a specific speed as it would be to hold still. Theoretically, no mater how fast, or in what direction I am traveling, there is some point, at some distance, in space, to which I am stationary (relatively).


Does a spinning part affect the moment of inertia of a composite object?

I have been going back through some Kleppner problems and have a doubt concerning problem 6.18. It states: > Find the period of a pendulum consisting of a disk of mass $M$ and radius > $R$ fixed to the end of a rod of length $l$ and mass $m$. How does the > period change if the disk is mounted to the rod by a frictionless bearing so > that it is perfectly free to spin? The first part (with the disk not free to spin) was reasonably straightforward. My only doubt was that I assumed the moments of inertia of the disk and rod could be added; this seems reasonable, but I don't quite know how to justify it rigorously (side-note: if anybody could give a hint on this, it would be highly appreciated). My result was: $$T=2\pi\sqrt{\frac{MR^2/2+Ml^2+ml^2/3}{gl(M+m/2)}}$$ For the second part, the issue I had was mainly conceptual... So when the disk is free to spin, it's no longer part of the rigid body; so it won't contribute to the moment of inertia, right? The problem comes here: earlier, to calculate the torque on the rigid body about the pivot, I said that: $$\tau=R_{CM}\times W$$ Where $R_{CM}$ is the center of mass of the rigid body. This formula comes from a summation over the torques on every small mass in the rigid body, so I figured that the torque on the rigid body, once the disk was no longer a part of it, would depend only on the center of mass of the rod (the disk would no longer affect the 'effective' center of mass that the torque acts on). This gives $$T=2\pi\sqrt{\frac{2l}{3g}}$$ I checked my answer with this website (pages 5-7) afterwards, and the first part agreed but the second part was in disagreement; the problem was that in that site, $R_{CM}$ was still 'affected' by the spinning disk. Why is this so? (I explained above why I think that $R_{CM}$ should not contribute.)


proof of radius of convergence of perturbation series in quantum electrodynamics zero

Can anyone show detailed proof of why radius of convergence of perturbation series in quantum electrodynamics is zero? And how is perturbation series constructed? So, as this argument requires imaginary electric charge of bispinor field (negative coupling constant), how is this argument considered valid?


why  water molecules in vapor contributes to the green house effect and water condensed in drops not?

I am a teacher planning a unit on climate change. I came across this statement: "Clouds are water vapor (green house gas) and water molecules will reflect heat into space… actually clouds are condensed water and the water molecule absorbs the heat, lowering the temperature of the atmosphere around the cloud." I don't understand why water vapor is a green house gas whereas the water molecule is not. I also don't understand why the water molecule absorbs heat and does not reemit it.... Please explain!


Calculating the speed of an object moving relative to another moving Object (Frame of reference)

> **Possible Duplicate:**   >  How to deduce the theorem of addition of velocities? Let's say that you are in a rocket speeding at 90% the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90% the speed of light. According to Newtonian physics, we add both velocities. Thus, the bullet should be going at 180% the speed of light. But we **now** know that according to Einstein the sum of these velocities is actually close to 99% the speed of light, because length contracts and time slows down. But how can you determine mathematically, that bullet is going at 99% the speed of light?   Additionally, can you also calculate, how much time is slowed down, if you know the velocity? Is there any formula?


Speed of light and relativity

Suppose there are two observers $A$ & $B$ both are in motion, now $A$ sees $B$ is moving with speed $'u'$. A says that another object $'d'$ is moving with speed $c+u$ w.r.t. A in the same direction as $B$. What stops him from saying this ? The formula $\frac {u+v}{1+\frac{uv}{c^2}}$ is such that $'v'$ is speed of the object $'d'$ w.r.t.$ B$ and not $A.$ and $u$ is the speed of $B$ w.r.t. $A$, now even though $B$ also sees that the object $'d'$ moves faster than the speed of light. He can still always see light to travel with the speed of light and also he can be an inertial observer? What is the problem here ?


Can a revolving body self-gravitate?

If a body is revolving around a point at radius R with tangential velocity V, does General Relativity predict that at some tangential speed, the body will revolve around the point without any external forces acting on it? Since General Relativity models gravity as a curvature of spacetime, I was thinking that an observer at the center of rotation of the revolving body would measure some curvature of space as a result of the revolution. This is because the body would be contracted in the tangential direction relative to the observer, such that C<2*PI*R where 'C' is the circumference of the revolving body measured by the observer (the revolving body is thought of as a ring of particles). This implies a curvature of space, which would be equivalent to a gravitational force.


Baryon asymmetry

Baryon asymmetry refers to the observation that apparently there is matter in the Universe but not much antimatter. We don't see galaxies made of antimatter or observe gamma rays that would be produced if large chunks of antimatter would annihilate with matter. Hence at early times, when both were present, there must have been a little bit more matter than antimatter. This is quantified using the asymmetry parameter $\eta = \frac{n_{baryon} - n_{antibaryon}}{n_{photon}}$ From cosmological measurements such as WMAP, $\eta \approx (6 \pm 0.25) \times 10^{-10}$ However, the source of baryon asymmetry is said to be one of the Big Problems of Physics. What is currently the state of the art regarding this puzzle? What's the best fit we can get from the Standard Model? What do we get from lattice simulations?


Why does the second law of thermodynmics only occur in the time dimension?

If I break an egg, how broken it is depends on "when" I am not "where" I am. Why is the time dimension special?


Does gas spread out equally everywhere?

An excerpt from this page: > Gases can fill a container of any size or shape. It doesn't even matter how > big the container is. The molecules still spread out to fill the whole space > equally. That is one of their physical characteristics. If a fixed quantity of gas is let out in a limited space, will it spread out equally and maintain a fixed gas distribution throughout the space? Or, does it go where the gravitational pull is maximum? And what factors affect the distribution of gas in a given area?


Why does magnetic field increase when the number of coil turns increase?

Since we all know that to increase the magnetic field we can do these of the following things: (i) Increase the number of turns in a coil, (ii) Increase the current following through conductor, and there would be more but I know only these (upto my standard - Xth ) now if I use the first method to increase the magnetic field then the length of conductor will also increase because I am increasing the number of turns; say I am increasing the length of conductor by $x$ meters then as we all know that: $R = \rho l/a$ you know resistance is equal to resistivity multiplied by length of conductor and then divided by the area of cross section. According to this $R$ is directly propotional to length of conductor and by Ohm's law $R$ is inverously proportional to current Therefore if I increase the length by $x$ m of conductor then the resistance will also increase at a rate of $x \Omega $ and current will decrease at $x$ Ampere, then how will the magnetic field increase if the current is being reduced?


Is there any difference between a proper photon and a photon of thermal radiation?

Does a thermal photon radiate spherically whereas a photon radiates in a circle/beam? How do you tell a thermal photon of, say 3 eV, from a photon of light?


Simple Projectile Motion Question

A volcano erupts 50m below the sea level. A rock leaves the crater at 20 m/s at an angle 30 deg with the vertical line. The rock has a mass of 15kg. IGNORE WATER RESISTANCE. It gets out of the water, reaches a max height and falls back it. When does it reach 30 below sea for the first and second times? $a = <0, -9.8>$ $v = <20sin(30), -9.8t+20cos(30)> = <10, -9.8t+17.32>$ $s = <10t, -4.9t^2+17.32t>$ For the object to be 30m below sea level, y=20m, so $s_y=20 \implies -4.9t^2+17.32t-20=0$. However, no real solutions exist to this. This is so simple, I apologize for asking, but what am I overlooking?


Inflation of universe and Information stored therein

In an inflationary model of our universe, the space-time expands within a finite boundary. Some other models are also proposed where the universe is not explained to have been expanding rather the masses of the objects are presented to be expanding with time. If so, what happens to the information scattered in space? Does it inflate too? And in literal sense, what is the meaning of inflation of information?


Why doesn't amplitude affect the speed of sound?

I understand why amplitude doesn't affect the speed of the sound AFTER the 'leading compression'. The extra force provided on one stage of the cycle is countered on the other stage. But shouldn't the 'leading compression' travel faster if amplitude is greater? Because wave motion hasn't been set up yet shouldn't all compressions face the same amount of resistance while moving forward?


Observed angular velocity and motion parallax

I would be highly obliged if someone would guide me in the right direction regarding this. I am trying to understand and mathematically explain the relation between angular velocities as observed by a moving observer of an otherwise, static object. For example- A tree passes by, as we travel in a bus. So what is the relation of the angular velocity of the tree as observed by an observer in the bus with the velocity of the bus and the distance, so involved? Also please point me towards any published material on this topic. Thanks.


A question about Poincare invariance of Polyakov action

I have a question the variation of the Polyakov action, related to this Phys.SE post. For Polyakov action $$ S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \gamma^{ab} \partial_a X^{\mu} \partial_b X_{\mu}.$$ Consider the variation (p. 14 in Polchinski's string theory book) $$ \delta S_p~=~ \frac{1}{2\pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \delta X_{\mu} \nabla^2X^{\mu}$$ $$ - \frac{1}{2\pi \alpha'} \int_{-\infty}^{\infty} d \tau (-\gamma)^{1/2} \delta X_{\mu} \partial^{\sigma} X^{\mu} |^{\sigma=l}_{\sigma=0} \tag{1.2.27} $$ The open-string and closed-string boundary conditions $$\partial^{\sigma} X^{\mu}(\tau,0)=\partial^{\sigma} X^{\mu}(\tau,l)=0 \tag{1.2.28} $$ $$ X^{\mu}(\tau,l) = X^{\mu}(\tau,0), \partial^{\sigma} X^{\mu}(\tau,0)=\partial^{\sigma} X^{\mu}(\tau,l), \gamma_{ab}(\tau,l)= \gamma_{ab}(\tau,0) \tag{1.2.30} $$ will make the "surface term" (the second term in the RHS of Eq. (1.2.27)) vanishes. It is said (p. 14-15 of Polchinski) > _The open-string boundary condition (1.2.28) and closed string boundary > condition (1.2.30) are the only possibilities consistent with > $D-$dimensional Poincare invariance and the equation of motion._ The boundary condition (1.2.28)/(1.2.30) will make the surface term vanish. The resulting equation of motion is tensor equation.   $$(-\gamma)^{1/2} \nabla^2 X^{\mu}=0$$ My question is, why non-vanishing surface term will break the Poincare invariance? E.g. a Lorentz transformation $X_{\mu} \rightarrow \Lambda_{\mu}^{\nu} X_{\nu}$ will not affect $\sigma,\tau$ space


Significance of the the Lagrange multipliers in statistical mechanics

In classic thermodynamics one can derive the Maxwell Boltzmann statistics by solving a Lagrange multipliers equation. In this process a new parameter $\beta$ is introduced to take account of the total energy constraint, $$L = \beta(E-\sum_i p_i \epsilon_i)+...$$ and it is later connected to $\frac{1}{k_B T}$ by inserting our knowledge about perfect gases. On a side note, about every book around seems to skip the passage with a frustrating "it can be shown that". My problem is that, given similar Lagrange multipliers equation, I'm not sure whether one can in general relate $E$ and $\beta$: it is evident that $\beta$ has no physical significance by itself, and it's always bounded to some parameter in the constraints that it is multiplying. For example, say that I have an array of cells that can be either occupied by 0, 1 or 2 dots. I have some constraints, one of which is the density of the dots $\rho$. The Lagrange multipliers equation has the form $$L = \gamma(\rho-p_1-2p_2)+...$$ where $p_i$ is the probability of having a cell with $i$ dots. Is there some way to eliminate the $\gamma$ parameter in the final explicit formula for $p_1$, and use $\rho$ instead?


A Book about the Bohr-Einstein debate?

A book about the Bohr-Einstein debate? Is there any book that details the correspondence between the two? The only books I could find are popular science books, I wonder if there is a book that lists the correspondence in a more 'raw' form.


Calculating the Center of Mass

We have a homogeneous body that looks like this: ![Homogeneous body](http://i.stack.imgur.com/j9fuR.jpg) I have tried dividing the body into different parts using the following definition: **R** g * **A** = **R** 1 * **A** 1 + ... **R** n * **A** n I was thinking I could first divide it into two parts using the x axis, and then add those two together using the definition above. But how can I calculate the coordinates for the center of mass for the lower part?


What does velocity dispersion (sigma) reveal about a galaxy?

I'm getting hung up on this term. In studying SMBHs, I see that velocity dispersion strongly correlates with mass. Just what is the velocity dispersion? How can the velocity dispersion of the galaxy be expressed in one figure (sigma) if it has to be measured all over the galaxy? I can imagine the velocity dispersion changes with radius, so "which" point is used? Why exactly is it that higher velocity dispersion is correlated with higher Mass? And is all this referring to the bulge only or to the entire galaxy? So to get velocity dispersion, is "one slit" at the center enough or do you scan across? I hope someone can answer this without relying heavily on the maths. I am really just trying to understand the concept, because I think that velocity dispersion is real, not just a mathematical construct.


Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?

Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\varphi$ with $\varphi=constant$. If we choose a coordinate in which $x^0=\varphi$, how to prove that the metric components $g_{0i}=0$ for $i=1,2,3$? My try is as follows. Because $W$ is perpendicular to the surface $x^0=c$, let $U=(c,0,0,0)$, then for every vector $V=(c,x^1,x^2,x^3)$ inside $x^0=c$, we must have $$ W_a(V^a-U^a)=0 $$ which leads to $$ W_i=0,\;\;i=1,2,3 $$ Now because $W$ is HSO, we must have $$ W_{[a}\nabla_bW_{c]}=0 $$ Then I am stuck. Since when $W_i=0,i=1,2,3$, the above equation holds automatically, thus can give me nothing. Could anyone help?


Galaxies seen from Earth

If we observe two galaxies from the Earth that are diametrically opposed and each 10,000 light-years from the Earth, will the separation distance between the galaxies be 20,000 light-years? Really, the question is: if the galaxies instead were separated from Earth by 10 billion light-years, would the separation distance between the galaxies then be 20 billion light-years?


Are telescopes with a concave lens useful for astronomy?

Are telescopes with a concave lens (instead of convex ones) ever useful for astronomy? And if so, where are they used? Do they ever affect resolving power?


Will changing amplitude change the frequency?

Will changing the amplitude change the frequency of a wave, or is it possible for a specific frequency (50 Hz. for example) to generate from shifting amplitude patterns?


Time evolution of wave spectrum

A useful way of thinking (not only) oceanic waves is to consider them as a superimposition of linear modes: the elevation η of the sea surface is given by: 1: $\eta({\bf x}, t) = \int_{-\infty}^{\infty} \hat{\eta}({\bf k}) e^{i({\bf k \cdot x} - \omega t)} d{\bf k}$ where $\omega$ and ${\bf k}$ are related by a dispersion relation, and $\hat{\eta}({\bf k})$ is a sort of time independent Fourier transform. $\hat{\eta}$ is given by: 2: $\hat{\eta}({\bf k}) = const \int_{-\infty}^{\infty} e^{-i{\bf k \cdot x}}(\eta({\bf x}, 0) + \frac{i}{\omega({\bf k})} \frac{\partial \eta({\bf x}, 0)}{\partial t}) d{\bf x}$ meaning that if we know $\eta({\bf x}, 0)$ and its time derivative we can determine the amplitude of all the normal modes, and neglecting non linear interactions between the modes we know the evolution of the wave. Clear and smart. What's the simples way to obtain relation 2 from relation 1? Thanks!


Ring theory in physics

Surely group theory is a very handy tool in the problems dealing with symmetry. But is there any application for ring theory in physics? If not, what's this that makes rings not applicable in physics problems?


Snell's law starting from qft?

Can one "interpret" Snell's law in terms of QED and the photon picture? How would one justifiy this interpretation with some degree of mathematical rigour? At the end I would like to have a direct path from qft to snell's law as an approximation which is mathematically exact to some degree and gives a deeper physical insight (i.e. from a microscopic = qft perspective) to Snell's law.


Vapor Pressure and Container

The following comes from my text and concerns vapor pressure: Consider a liquid in a closed, cylindrical container that is half full. The molecules at the surface of the container and along the edges have the fewest neighbors; therefore, they will evaporate more quickly than other molecules. This intuitively makes sense. The text also says that vapor pressure is independent of the shape and volume of a container. Now consider the above situation in two cylinders of the same volume. The first has a much smaller radius than the second. Since the smaller radius cylinder has a greater proportion of its molecules along the edges, I would think that it would have a slightly higher vapor pressure. Is this correct? Perhaps I'm just over-complicating things. Any help is appreciated!


Why does an armature of a generator rotate? a) like poles repel b) unlike poles attract c) current flows up d) an outside force turns it I am thinking that (d) is the correct answer but still not sure of it


If the universe has a shape, what is the surface?

Since the idea of the universe was that it was expanding, is there a way to know the surface area of the universe like the growing 3D object it is illustrated? The idea was that the universe was expanding, and then I think that the surface would expand, but my concept is limited about such a metric. I think I read about the mass of the universe, but is there an agreed measure of the surface of the (bounds of the) universe? Since the universe is assumed to have a shape, then what the surface be?


Rate of Cooling

I was given the following problem: > Two objects have a temperature difference of 20 degrees C at time 0. Two > minutes later, they have a temperature difference of 18 degrees. How long > does it take for them to have a difference of 10 degrees. Initially, I tried to approach the problem with Newton's Law of Cooling. However, I believe my solution is flawed: $$\frac{dT}{dt} = -kT$$ letting $T$ be the temperature difference between the two objects. The solution to the above equation is $T(t) = ce^{-kt}$. Inserting the provided values, I found that $c = T(0) = 20$ and $T(2) = 18 = 20e^{-2k}$, so $k = 0.05268$. Setting $20e^{-0.05268t} = 10$, I found that $t = 13.2$ minutes. **Is Newton's Law of Cooling applicable if I simply let $T$ be the difference in temperatures between two objects?** Also, the problem implies that both objects are changing temperature. **What do I need to do to accommodate for both objects changing temperatures?**


Tensors in special relativity

I'm trying to understand tensors, but I've come across the following question: >   1. Let $T^{\mu\nu}$ by a $(2,0)$ tensor. Give the definitions of > $T_\mu^{\,\nu}$, $T_{\mu\nu}$, and $T^{\mu}_{\,\nu}$. Then give explicit > formulas for the following components in terms of components of > $T^{\mu\nu}$: > $T_{00},\,T_{10},\,T_{12},\,T_0^{\,1},\,T_1^{\,1},\,T_0^{\,0},\,T^1_{\,1}$   >  $$T_\mu^{\,\nu}\equiv\eta_{\mu\sigma}T^{\sigma\nu} \\\ > T_{\mu\nu}\equiv\eta_{\sigma\mu}\eta_{\rho\nu}T^{\sigma\rho}\\\ > T^\mu_{\,\nu}\equiv\eta_{\sigma\nu}T^{\mu\sigma}$$ > I'm finding it hard to understand these definitions. I really don't know what these tensors represent or what it means when both the indices are raised? Also are these standard definitions in special relativity or is it more likely that they are just made up by my professor for this assignment? I've always thought that things with two indices are matrices in general.


Torque and angular acceleration with bicycle wheel

This might be a simple problem for many of you. However, please help me understand it too. I have been looking trough a lot of materials online, and I still have the following questions, that would help me a lot. I saw this video on youtube: see wheel in action If we consider the following model: ![enter image description here](http://i.stack.imgur.com/bGYsu.gif) Please clarify for me the following questions:   1. I believe the gravity is not pulling the wheel down because the torque resulting from multiplying gravity by the l vector is pointing sideways. In other word, the down-pointing gravity vector is transformed into a vector pointing sideways and spinning the wheel around the string. (Is this how force multiplication works? do the multiplied factors transform into the resulting force?)   2. If the wheel would keep a constant angular velocity(assuming no friction) the wheel would spin indefinitely without rising or falling. What would happen if the angular velocity would be increased while spinning? would the wheel rise (ie the free floating end would rise)?   3. Mathematically what causes the wheel to fall when the angular velocity decreases? The only changing quantity is L, which depends on the speed of the spinning wheel. But the direction of L is not changing. And from (1) above, the gravity should not be pointing downwards. So why does the wheel fall when L gets smaller(by wheel falling I mean the free floating end goes down, until it is under the string)


Energy fluctuations in quantum canonical ensemble

How would you go about showing that in the quantum canonical ensemble (that is, in the density matrix and operator formulation), the energy fluctuations, namely $\langle H^2\rangle - \langle H\rangle^2$ is the same as in the classical case, i.e. $k_b T^2 C_V$? I've seen a lot of books asserting this, but can't find a formal derivation similar to the classical one, as given here, for example.


What do I need to know to calculate the price per km for plug-in hybrid or electric cars?

Knowing the per kWh price of electricity, what other factors are needed to calculate how much would one pay per km traveled on electric power compared to gasoline power?   EDIT: For example, the Chevrolet Volt reports a 35 kWh per 100 miles. Supposing the price for electricity is 11 cents per kWh, so this mean one can drive 100 miles for $3.85, or a mile for 3.8 cents?


Why does the fluid inside a cup not spin when the cup is spinning, but starts to spin when the cup stops spinning?

How come when I spin a cup with water in it, the water does not spin, but the moment I stop spinning the cup, the water starts spinning the other way?


Dark matter detection

In the detection of weakly interacting massive particles (WIMPs), which is the basis of dark matter, what is the use of the tank filled with liquid xenon? I mean, how does the releasing of photons contribute in the detection of WIMPs?   To relate to what I'm talking about, you can follow this link:   http://www.universetoday.com/105943/new-dark-matter-detector-draws-a-blank-in- first-test-round/


Does coffee cool faster than tap water?

A friend of mine, based on casual observation, believes that coffee will cool faster than ordinary hot tap water. Being curious about this I have tried to investigate it myself, but I'm not well versed in physics and thermodnamics. Also, I know that there are many variables to this, so I will attempt to ask the question as specifically as possible: I have 8oz of water (we can assume that it is impure, but we can not know what those impurities are) that is heated to 200 degrees F in a microwave (let's say we have a really precise microwave that can do that easily) and that is then placed in a standard, throw-away styrofoam coffee cup. I have an exact duplicate scenario, except that this time we will also brew coffee in the water for several minutes, filter out the coffee using a standard coffee filter (we can assume some coffee particles remain in the water, but we can not know anything about those coffee particles) and then re- heat the water to 200 degrees F before adding it to the foam coffee cup. If each of these coffee cups are then placed on separate tables in a room- temperature room (let's say 70 degrees F) and allowed to sit, undisturbed, will the liquids cool at roughly equivalent speeds (i.e., will the water and the coffee reach 180 degrees F at roughly the same time) or will one cool faster than the other? For either scenario, why? If there's any clarification or wording choices that should be changed, please let me know as I am genuinely curious. My guess is that there's something to do with the specific heats begin different, and that the surface tension of the coffee mixture will be disrupted due to the coffee particles. But, again, I don't really know :-)


Room temperature and fan orientation

So I'm in a tiny dorm room and I normally point my fan blowing outside the window to cool my room off. I've been in some debates on blowing air out or in is more effective, so I'm hoping to get some empirical data to back up my claim. Things I have   * A Heater with only low, medium, high settings.   * The Fan.   * I do _not_ have a thermometer. So the best I could think of was melting ice cubes. I don't care about the exact temperature, just which method is more effective.   I've outline the basis of my experiment: Shut the windows and turn the heat on high, 15 minutes Open the windows, turn off the heat, turn on the fan on high, 15 minutes Leave $n$ ice cubes out and take the average of how long it takes them to melt. All the ice cubes are from the same tray. Can anyone devise a better experiment with no tools...


Current when the slide wire attains a constant velocity

In a slide wire generator, when the slide wire attains a final constant velocity, is the current in the circuit 0? It does not make sense if it is 0 as the area keeps increasing, allowing more flux to pass, meaning more current would need to be generated to counteract the increasing flux.


Does the cell phone make dipole particles in our body dance the same way Microwave oven does in food

I am a non-physics person trying to wrap my head around some EM radiation facts. Please help me with this. I have read about Dielectric Heating on Wikipedia. And watched quite a few videos on internet on how Microwave oven works. It all boils down to the following facts. Please correct me if I am wrong on any of them.   1. The Microwaves are absorbed by Water molecule and the Negative charged Hydrogen atoms want to align with negative radiation and Positive Oxygen atom wants to face positive radiation which makes the water molecule dance.    2. Microwave oven Magnetron changes the negative-positive polarity in charge as often as 2.4 billion times in a second which makes Water molecules dance really really fast.   3. This dance makes the water molecules heated. Now coming to my question, I learned that Cell Phones use UHF frequencies such as 800 MHz and 400 MHz. Does that mean water molecules in our body are dancing at the rate of 800/400 steps per seconds ? Why don't they heat up then ?


Would electromagnetic radiation impart a pressure on a surface of neutrons?

In my physics textbook, it says that a qualitative way to envision pressure from EM waves is as follows: the electric field drives charges in the $x$ direction, and the magnetic field then exerts on them a force $q\vec{v}\times\vec{B}$ in the $z$ direction. The net force on all the charges in the surface produces the pressure. But if this is the case, then does that mean that EM waves wouldn't impart any pressure on a surface of neutrons, given there are no charges to move? Let me know if this is just an analogy that doesn't hold... but in this case, is there another intuitive way to understand why there's a pressure associated with the waves? I understand mathematically, but not physically.


Can cos(x) or sin(x) be the function of stationary action?

Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?


Shouldn't the change in kinetic energy be more in a moving elevator from a stationary frame of reference?

Consider an elevator moving down with uniform velocity. A person standing inside watches an object fall from the ceiling of the elevator to the floor. Say the height of the elevator is $h$. Then the work done by gravity in that frame of reference should be $mgh$. But consider this same event being watched by someone else in the stationary frame of reference. In his reference frame, the object travels a larger distance as it falls from the ceiling to the floor of the elevator because the floor itself is moving downwards (one can calculate this extra distance covered to be $u \sqrt{\frac{2h}{g}} $) and hence the change in kinetic energy should be more in that frame than in the moving frame! I just can't seem to figure out where I'm going wrong here. I'm probably missing something very obvious. So I would be very grateful if anyone could explain this to me. **Edit:** Okay, let's say the object is a clay ball and it collides with the floor inelastically such that it's kinetic energy is converted into heat. In the moving frame of reference the heat would be simply equal to $\frac{1}{2}mv^2$ which is equal to $mgh$. In the stationary frame of reference it would be equal to $\frac{1}{2}mv^2-\frac{1}{2}mu^2$ since the ball after colliding is moving with speed $u$. This can be calculated to be equal to $mgu\sqrt{\frac{2h}{g}} + mgh$ which is clearly greater than the heat produced in the frame attached to the elevator and this is a contradiction because the heat measured in any frame should be the same.


Does black hole evaporation respect quantum superpositions?

I've confused myself about the following scenario: Suppose you make a black hole out of states with spin aligned into one direction, say the positive x-direction, and let's call this "up". Then the so-created black hole has an non-zero angular momentum because the expectation value of the spin is nonzero. (Let's assume there's no other net contribution to the angular momentum, just for simplicity. I don't think this matters.) The so-created black hole evaporates in the semi-classical limit with an energy spectrum that depends on this expectation value of the total angular momentum. (Forget about what happens in the final stage where the semi-classical approximation breaks down, I don't think this matters.) Next you make a black hole out of the same initial state with the only difference that all spins are down. Same thing: the emission spectrum - the distribution of outgoing particles over energy - depends on the expectation value of the angular momentum, at least in the semi-classical limit we're pretty sure of that. Finally, you make a black hole out of a superposition of the two initial states. The expectation value of the total angular momentum is now zero, correspondingly the angular momentum in the semi-classical Kerr-background is zero. What is the decay spectrum (in the early radiation) in this case? Is the outgoing radiation a) a superposition of the radiation that one obtained from the purely up and purely down case? But then it would depend on the angular momentum which the black hole doesn't have. Or does it b) not depend on the angular momentum of the superposed states (note that what is and what isn't a superposition is ambiguous anyway) but then the evolution doesn't seem to be linear? (In the sense that |psi_in> + |psi'_in> doesn't result in |psi_out> + |psi'_out> when |psi_in> resulted in |psi_out> and same for the dashed case).


The $U(1)$ charge of a representation

My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\times SU(2) \times U(1)$. For example the weights of the 10 dimensional representation of SU(5) are ![enter image description here](http://i.stack.imgur.com/SlEAK.png) One can identify the irreps of the subgroup by regrouping the dynkin labels into $((a_3 a_4) ,(a_1), a_2)$ such that (denoting $-1$ by $\bar{1}$): $$ (1,1)_{Y} \rightarrow \left\\{ \begin{array}{l l} (0 0,0,1 ) \end{array} \right. $$ $$ (\overline{3},1)_{Y} \rightarrow \left\\{ \begin{array}{l l} (0 1,(0),\bar{1}) \\\ (1 \bar{1},(0),\bar{1})\\\ (\bar{1}0,(0),0) \end{array} \right. $$ $$ (3,2)_{Y} \rightarrow \left\\{ \begin{array}{l l} (1 0,1,\bar{1}) \\\ (\bar{1} 1,\bar{1},1)\\\ (0\bar{1},\bar{1},1)\\\ (1 0,\bar{1},0)\\\ (\bar{1}1,1,0)\\\ (0\bar{1},1,0) \end{array} \right. $$ My problem is: how can I derive the $Y$ charge of the $U(1)$ factor for each of these from the Dynkin labels? * * * **Edit** The metrictensor for SU(5) is thus $$G= \frac{1}{5}\left( \begin{array}{cccc} 4 & 3 & 2 & 1 \\\ 3 & 6 & 4 & 2 \\\ 2 & 4 & 6 & 3 \\\ 1 & 2 & 3 & 4 \end{array} \right). $$ However in the reference, Slansky, on page 84 the same exercise is done but the axis have negative values... $$\tilde{Y}^W = \frac{1}{3} [-2 \;1\, -1\; 2]. $$ How come they do not agree?


Why must the final state be stationary?

I faced the following sentences: > We consider a gravitational collapse taking place in this spacetime. The > singularity theorems assure us that a singularity will form. The assumption > that the cosmic censorship conjecture is valid will then imply that the > collapse must produce a black hole which will settle down to a stationary > final state. Can anyone explain to me why **The assumption that the cosmic censorship conjecture is valid will then imply that the collapse must produce a black hole which will settle down to a stationary final state**? Why must the final state be stationary?