why galaxies look like discs rather than spheres?

Black holes have much gravitation to hold all stars and nebulas but why they are aligned in disc type shapes rather than spheres because gravitation is everywhere around black holes. Even on upper and lower sides. Even if rotation is the cause why cant they move above black holes in smaller circles.


Why are our planets in the solar system all on the same disc/plane/layer?

I always see pictures of the solar system where our sun is in the middle and the planets surround the sun. All these planets move on orbits on the same layer. Why?


Rotating uniform rod

> A uniform rod of mass 1.2 kg and length 1.8 m is pivoted in the horizontal > position as shown (black point).![enter image description > here](http://i.stack.imgur.com/cGmYy.png) The rod is at rest and then > released. The acceleration due to gravity is $g = 9.8 m/s^2$. 1) Find the > rotational inertia of the rod relative to the axis perpendicular to the > screen and passing through the pivot point. 2) What is the angular speed (in > rad/s, but do not include units) of the rod as it passes through the > vertical position (when end marked B is at the bottom)? My attempt of solution: To find moment of inertia, I use the moment of inertia of a uniform rod respect to it's center of mass (middle in this case), which is $$I_{cm}=mL^{2}/12$$ where $m$ is mass of the rod and $L$ its length. Then I use the parallel axis theorem to find the moment of inertia in this case: $$mL^{2}/12+0.45^{2}m=0.567$$ For the 2º part, since the net torque on the rod is not constant I use potential energy to find first the speed of center of mass at the vertical position. The initial height of center of mass is 0.45 m taking as reference point position of the middle point of the rod at vertical position. So: $$mg0.45=1/2mv_{cm}^{2}+1/2I\omega ^{2}=1/2mv_{cm}^{2}+1/2I(v_{cm}^2/0.45^{2})$$ From here I get $v_{cm}$ and later the angular velocity. Where am I wrong?


Why does the light intensity increase as I approach a distance light source?

Analogy: assume that I have constant rain fall and I have a water bucket to collect this rain. If I am rest relative to the earth, I will catch a certain amount of rain. However, if I now move towards the rain, I will increase the amount of water I collect. Now I want to apply this idea to a source emitting photons instead of rain drops. I image a distance light source (distance star) that is emitting a constant number of photons such that I can ignore the $1/r^2$ fall off of the intensity (that is, the intensity does not change very much over some appreciate distance). So instead of a water bucket, I now have a light collector that measures light intensity. Here is my question: **Does the light intensity measured by my light collector that is moving towards the distant star increase or stay the same when compared to a light collector at rest realtive to the distant star?** My first thought was that the light intensity would increase because the light is blue shifted and higher energy photons will therefore produce a higher intensity. However, I believe that there might be another contribution due to length contraction. Since an observer moving towards a light source has to account for length contraction, does this mean that there is an increase in photon density? If so, this higher photon density in the frame of the light collector will also contribute to increasing the light intensity. Can someone verify or correct my thinking.


Doubt in collision

> _If a ball under the influence of gravity falls straight down from a height > $h$, collides elastically with the floor, at the instant of collision, what > forces does it experience?_ Shouldn't the floor exert a force equal to the weight of the ball, on the ball itself? Then won't the forces cancel out?(since the only to forces on the ball are it's weight and the force of the floor, which are equal and Opposite) I know that the better way to find resulting velocity, etc. Would be to use work-energy or kinematics, but I would like to reconcile those with the force treatment


Has a theory of a small Big Bang in a much larger universe been proposed?

I have a theory I think might explain why the universe has accelerated in a way that doesn't require dark energy. I'm wondering if someone has proposed this theory before (did some research and couldn't find anything). The theory is that the big bang was essentially a huge super-nova-like eruption inside an even larger universe. Just like the visible universe has super novae that happen inside much much larger galaxies and dust clouds, so could the Big Bang have been a huge explosion in a much huger universe. The acceleration we have seen evidence of could, then, have been caused if the big bang was off center in the huge amount of matter in the outer universe - ie if one side exerted more gravity than the other, mass closer to that side would accelerate away from mass closer to the center of the big bang. I imagine a huge black-hole sucking in a giant swath of matter from the greater universe, coalescing into a small area that leaves a large space around it rather empty. Then when it explodes (or perhaps it just emitted large amounts of matter out of it via something like hawking radiation), it fills this space again with what we now see as our universe. This theory doesn't seem to require any sort of bizarre unknown physics, like dark energy, singularities or anything else that causes divide by 0 errors in established physical equations. Has it been proposed before? Also, I'd be interested to know if anyone has any concrete reasons why this theory wouldn't work.


Stellar Viscosity in Galaxies

Is there such as thing as the viscosity of stars in a galaxy, along the lines of gravitational attraction between stars changing the dynamics. If so, how is that put in terms of the Virial Theorem?


Hamilton's characteristic and principle functions and separability

Just hoping for some clarity regarding Hamilton's characteristic function (W). When we take a time independent Hamiltonian we can separate the Principle function (S) up into the characteristic function minus $ht$, yes I know its the Legendre transform but, \begin{equation} W=S+ht \end{equation} Meirovitch in his _Methods of Analytical Dynamics_ p 356 gives $h$ as being the Jacobi energy function $h$ as defined in earlier chapters both his and Goldstien's texts. This is the only time I have seen it being called $h$ rather than the Hamiltonian. I was just wondering if anyone had read through it and perhaps noticed something different in Meirovitch's definition that escapes me. Most authors define these integration constants as the Hamiltonian instead. I know the difference is subtle but it is intriguing as to why he chose $h$ not $H$! Is it just down to how you express the conjugate momenta?


Do the standard cosmology models spontaneously break Lorentz symmetry?

In standard cosmology models (Friedmann equations which your favorite choice of DM and DE), there exists a frame in which the total momenta of any sufficiently large sphere, centered at any point in space, will sum to 0 [1] (this is the reference frame in which the CMB anisotropies are minimal). Is this not a form of spontaneous Lorentz symmetry breaking ? While the underlying laws of nature remain Lorentz invariant, the actual physical system in study (in this case the whole universe) seems to have given special status to a certain frame. I can understand this sort of symmetry breaking for something like say the Higgs field. In that situation, the field rolls down to one specific position and "settles" in a minima of the Mexican hat potential. While the overall potential $V(\phi)$ remains invariant under a $\phi \rightarrow \phi e^{i \theta}$ rotation, none of its solutions exhibit this invariance. Depending on the Higgs model of choice, one can write down this process of symmetry breaking quite rigorously. Does there exist such a formalism that would help elucidate how the universe can "settle" into one frame ? I have trouble imagining this, because in the case of the Higgs the minima exist along a finite path in $\phi$ space, so the spontaneous symmetry breaking can be intuitively understood as $\phi$ settling **randomly** into any value of $\phi$ where $V(\phi)$ is minimal. On the other hand, there seems to me to be no clear way of defining a formalism where the underlying physical system will randomly settle into some frame, as opposed to just some value of $\phi$ in a rotationally symmetric potential. [1] The rigorous way of saying this is : There exists a reference frame S, such that for all points P that are immobile in S (i.e. $\vec{r_P}(t_1) = \vec{r_P}(t_2) \forall (t_1, t_2)$ where $\vec{r_P}(t)$ is the spatial position of P in S at a given time $t$), and any arbitrarily small $\epsilon$, there will exist a sufficiently large radius R such that the sphere of radius R centered on P will have total momenta less than $\epsilon c / E_k$ (where E is the total kinetic energy contained in the sphere). * * * Ben Crowell gave an interesting response that goes somewhat like this : Simply put then : Causally disconnected regions of space did not have this same "momentumless frame" (let's call it that unless you have a better idea), inflation brings them into contact, the boost differences result in violent collisions, the whole system eventually thermalises, and so today we have vast swaths of causally connected regions that share this momentumless frame. Now for my interpretation of what this means. In this view, this seems to indeed be a case of spontaneous symmetry breaking, but only locally speaking, because there should be no reason to expect that a distant causally disconnected volume have this same momentumless frame. In other words the symmetry is spontaneously broken by the random outcome of asking "in what frame is the total momentum of these soon to be causally connected volumes 0?". If I'm understanding you correctly, this answer will be unique to each causally connected volume, which certainly helps explain how volumes can arbitrarily "settle" into one such frame. I'm not sure what the global distribution of boosts would be in this scenario though, and if it would require some sort of fractal distributions to avoid running into the problem again at larger scales (otherwise there would still be some big enough V to satisfy some arbitrarily small total momentum).


1. What is the relation between entropy and mass of black hole?   2. And what is the relation between symmetry of physics operation and entropy?For instance,measuring or doing measure on state of quantum system especially pure state of a system has to lead to increase in entropy of the system,and the measuring is not symmetric operation,but it seems that the statement is tautologic,because that operations make processes symmetric,means nothing has been changed.


What makes a Feynman diagram real or virtual?

Simple question: as the title says, what makes a real Feynman diagram real, and what makes a virtual diagram virtual? Or in other words, how do I tell whether any given diagram is real or virtual? I've never gotten a really satisfying explanation of this. I would imagine it has something to do with virtual particles, but all internal propagators are virtual particles and I know for a fact that having internal lines doesn't make a diagram virtual.


Small oscillations of heavy string

I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points. First of all I wrote down this Lagrangian: $$S=\int dt ds \left[\frac{\rho}{2}(\dot{x}^2+\dot{y}^2)-\rho g y(s,t)+\frac{\lambda(s,t)}{2}\left(\left(\frac{\partial x}{\partial s}\right)^2+\left(\frac{\partial y}{\partial s}\right)^2-1\right)\right]$$ This Lagrangian describes heavy string with fixed ends in gravitational field. Where $\rho$ is density, $g$ is gravitational acceleration, $s$ is natural parameter. So I have 3 equations from Euler-Lagrange equations. $$\rho\ddot{x}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial x}{\partial s }\right)=0$$ $$\rho\ddot{y}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial y}{\partial s }\right)+\rho g=0$$ $$\left(\frac{\partial x}{\partial s }\right)^2+\left(\frac{\partial y}{\partial s }\right)^2=1$$ After that I've found stationary solution ($\frac{\partial x}{\partial t}=\frac{\partial y}{\partial t}=\frac{\partial \lambda}{\partial t}=0$). (I just put $\ddot{x}=\ddot{y}=0$) $$y_0(x)=-\frac{C_1}{\rho g}\cosh\left(\frac{\rho g x}{C_1}+C_2\right)$$ Where $C_1,C_2$ is integration constants (depends on positions of ends of string). And $\cosh(x)$ is hyperbolic cosine. To study small oscillations I've tried to use pertrubation theory. So, I put $$y(s,t)=y_0(s)+\bar{y}(s,t)$$ $$x(s,t)=x_0(s)+\bar{x}(s,t)$$ $$\lambda(s,t)=\lambda_0(s)+\bar{\lambda}(s,t)$$ But after that I get difficult differential equations, which I can't solve. **Maybe someone know the more simplier aproach to solve this problem or know how to solve it in this way?**


Saturation of the Cauchy-Schwarz Inequality

Going to as little details as possible, here is a statement from Wald's text on QFT in curved spacetimes(I am not quoting the book) He considers two vector spaces ${\cal S}$ and ${\cal H}$. > Note - For more details about ${\cal S}$ and ${\cal H}$, read this box. I > believe for most part of the question, the following details are irrelevant, > but I will provide them nonetheless. Otherwise, skip below. > > He starts by considering the solution space of a classical system ${\cal S}$ > with symplectic structure $\Omega$. This has a natural vector space > structure. He complexifies it to ${\cal S}^{\mathbb C}$ and extends $\Omega$ > to ${\cal S}^{\mathbb C}$ by complex linearity on each variable. He then > defines the map $(\cdot, \cdot): {\cal S}^{\mathbb C} \times {\cal > S}^{\mathbb C} \to {\mathbb C}$ on ${\cal S}^{\mathbb C}$ as $$ (y_1, y_2) = > - i \Omega( \overline{y_1}, y_2) $$ This satisfies all the properties of an > inner product except positive-definiteness. He then considers the subspace > ${\cal H}$ of ${\cal S}^{\mathbb C}$ on which the inner product above is > positive-definite. (There are of course many such choices of ${\cal H}$. Any > one of them will do.) He then shows that there is a one-one onto map $K: {\cal S} \to {\cal H}$. He shows that one can define a real inner product $\mu: {\cal S} \times {\cal S} \to {\mathbb R}$ on ${\cal S}$. He then goes on to show that one can use this to define a complex inner product on $\cal H$ as $$ \left( K y_1, K y_2 \right)_{\cal H} = \mu(y_1, y_2) - \frac{i}{2} \Omega(y_1, y_2)~\forall~y_1, y_2 \in {\cal S} $$ where $\Omega: {\cal S} \times {\cal S} \to {\mathbb R}$ is an antisymmetric function on ${\cal S}$, i.e. $\Omega(y_1, y_2) = - \Omega(y_2, y_1)$. He then uses the Cauchy-Schwarz Inequality for ${\cal H}$. This reads $$ \left( K y_1, K y_1 \right)_{\cal H} \left( K y_2, K y_2 \right)_{\cal H} \geq \left| \left( K y_1, K y_2 \right)_{\cal H} \right|^2 \geq \left| \text{Im} \left( K y_1, K y_2 \right)_{\cal H} \right|^2 $$ Expanding it out, he writes $$ \mu(y_1, y_1) \mu(y_2, y_2) \geq \mu(y_1, y_2)^2 + \frac{1}{4} \Omega(y_1, y_2)^2 \geq \frac{1}{4} \Omega(y_1, y_2)^2 $$ More specifically $$ \boxed{ \mu(y_1, y_1) \mu(y_2, y_2)\geq \frac{1}{4} \Omega(y_1, y_2)^2 } $$ Now, here is the statement that confuses me > **Indeed, since $K$ is one-to-one and onto and since the Schwarz inequality > on ${\cal H}$ always can be ``saturated", we obtain the following stronger > version of the last inequality: For each $y_1 \in {\cal S}$ we have** $$ > \mu(y_1, y_1) = \frac{1}{4} \max_{y_2 \neq 0} \frac{ \Omega(y_1, > y_2)^2}{\mu(y_2, y_2)} $$ Here's my question **Q. Where did he get the above > expression from?** He seems to be claiming that the boxed inequality is always saturated for some vector $y_2 \in {\cal S}$. Is that true? Why? PS - I will understand if some people think that this question is more of a math question than a physics one. But, I thought that it might be possible that the answer relies on some of the assumptions we make in physics, so I asked it here. Any comments will be helpful


Relationship between current through a motor and it's load

When a motor, connected to a battery that has a constant voltage, spins without a load it's speed is higher than with load. I'm told that because of back emfs the current is very small when there's no load because of the higher speed. And so when there is a load the back emf is less as the motor spins slower, and so the current is higher. What is a back emf and what is the relationship between that and the speed of the motor (well the coils inside the motor)? Is the output energy of the motor constant, whether it has a load or not? (ignoring friction and electrical resistance) because I don't understand how the current can be higher when the motor is (or seems to be) doing more work due to the load?


Gravitation as the source of redshift of light beams

According to Hubble's law, light and other kinds of electromagnetic radiation emitted from distant objects are redshifted. The more distant the source, the more intense is the redshift. Now, the expansion of the universe is expected to explain the redshift and its nearly linear dependence on distance between source and observer. But isn't there an other source influencing the redshift? We know that a light beam passing the Sun is deflected by the Sun's gravity in accordance with predictions made by Einstein's general theory of relativity. This deflection is dependant on a gravitational interaction between the Sun and the light beam. Thus, the position of the Sun is affected by the light beam, though by such a tiny amount that it is impossible to detect the disturbance of the Sun's position. Now, during its journey to the Earth a light beam, originating from a distant source in the Universe, is passing a certain amount of elementary particles and atoms. If the light beam interacts gravitationally with those elementary particles and atoms, affecting the microscopic mechanical properties of the individual elementary particles and atoms at issue, can this interaction be detected as a redshift of the light beam? If so, could we use this gravity redshift to measure the mean density of matter and energy in space? The beginning of the sentence "If the light beam interacts gravitationally with those elementary particles and atoms..." should be interpreted to say "If the light beam _interacts gravitationally_ with those elementary particles and atoms by way of leaving them in a state of acceleration different from their initial state of acceleration...." This clarification seems to necessitate the additional question: "why would a gravitationally interacting object (a cluster of photons) passing another gravitationally interacting object (a mass) leave that mass in the same state as before the passage?"


How to measure speed of a ceiling fan?

I thought for fun I might measure how my ceiling fan slows down when turned off, and maybe find out what forces act on it when it slows down. My problem is that for that I need to measure its rotational speed when it's turned on. When it's in the lowest speeds it's easy because you can see it rotate, but for higher speeds that's not an option. I thought one way would be to use a strobe light and try to match up the frequency, but I don't have one of those. Can you think of any DIY way to measure the speed of a very fast ceiling fan?


Formalizing Quantum Field Theory

I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or conformal field theories, and of older approaches such as algebraic QFT, and the classic works of Wightman, Streater, etc. etc . I am more interested in more current approaches, in particular such approaches that incorporate the modern understanding of the subject, based on the renormalization group. I know such approaches exists and have had occasions to hear interesting things about them, I'd be interested in a brief overview of what's out there, and perhaps some references. Edit: Thanks for all the references and the answers, lots of food for thought! As followup: it seems to me that much of that is concerned with formalizing perturbative QFT, which inherits its structure from the free theory, and looking at various interesting patterns and structures which appear in perturbation theory. All of which is interesting, but in addition I am wondering about attempts to define QFT non-perturbatively, by formalizing the way physicists think about QFT (in which the RNG is the basic object, rather than a technical tool). I appreciate this is a vague question, thanks everyone for the help.


Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, which in general have no mathematical meaning (at least not yet) ect. My question is a little vague, but i am interested in hearing what is the status of rigor in QFT. What is known to be mathematically rigorous and consistent, what is known to be not rigorous? Any examples and references are welcome. **Added:** Just to clarify by rigorous I meant anything that a mathematician would find satisfactory. Also my question wasn't for books with rigorous (in some sense) approach, although that was welcomed. It was about specific examples of what is considered mathematically satisfactory and what not. For example the quantization of free fields satisfying the Klein-Gordon equation can be done rigorously. There is no mathematical definition in general of the Feynman path integral and so on.


Will a charged capacitor discharge if one lead is connected to ground?

If I charge a capacitor and connect one lead to ground keeping the other lead floating, will the capacitor discharge ?                  G-------||------ open/floating              +q -q      (G for ground)


Does a complete theory of quantum gravity require anthropic post-selection? Certainly the black hole complimentarity and causal patch conjectures highlights the essential role of observers, at least in the asymptotic future of their future timelike trajectories. Does the measure problem in quantum gravity cosmology suggest that a "global god's eye view" of the universe might be an incoherent fiction? However, if observers are essential, don't we have to post-select to those states containing the observer of interest? It might be suggested the S-matrix of string theory in a superselection sector with only a finite energy difference from the vacuum over a BPS background provides a counterexample, but can an asymptotic future state of noninteracting Fock space particles really support the future trajectory of an observer?


Are physics and philosophy compatible?

Lately, there has been a war between physicists and philosophers. Some of the exchanges can be found at http://www.nytimes.com/2012/06/10/opinion/sunday/what-physics-learns-from- philosophy.html?bl, http://opinionator.blogs.nytimes.com/2012/05/10/can- physics-and-philosophy-get-along/?ref=sunday and www.3ammagazine.com/3am/time- lord/. are physics and philosophy compatible? Does physics need philosophy?


Idea of precursors of the electro-magnetic waves

The idea of the material Maxwell equation is almost clear. But I'm curious about the idea that except for material equation the pure Maxwell equation should work, but in harder sense: more currents and charges. There are plenty of vacuum in solid body, so, I think, that there should be some precursors of electro-magnetic waves that propagates through the body: $v = c/n$ the speed of wave in body, but part of wave propagates in vacuum with the speed of $c$. And for **any** frequency some part of amplitude would be observed much earlier. Is there some theory behind this and does this phenomena is observed?


Particle physics plots

I'm having a hard time understanding what some of the plots that are presented by ATLAS/CMS actually show. See for example: http://resonaances.blogspot.com/2011/07/higgs-wont-come-out-of-closet.html What is the y-axis? How would the plot look like if a Higgs was in the given mass range? ![enter image description here](http://i.stack.imgur.com/IbqWV.png)


Torque due to Gravity

I have learned that a torque on an object due to the gravity can be calculated as if the gravity acts on the center of mass of the object when the object is near the surface of the earth. Now I want to prove it... So, I have $$\tau=\int d\tau=\int r(dmg)sin\theta=g\int rsin\theta dm.$$ Then, how do you obtain $$\int rsin\theta dm= mr_{cm}sin(\theta_{cm}),$$ where $r_{cm}$ is the distance between an axis of rotation and the center of mass, and $\theta_{cm}$ is the angle between the two vectors $\vec{r_{cm}}$ and $m\vec{g}$ at the center of mass??? Would $$\int r dm=mr_{cm}$$ be helpful to prove this? Would the integral be dependent on the object I use, or would there be a general way to prove this?


Question about Rayleigh scattering

To quote from Wikipedia on elastic scattering, "In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed." How does this work? The impacted particle will also start moving to conserve momentum. So it'll also carry some energy. How can the photon not lose any energy during the collision? Another possibly unrelated question: how exactly can the photon and the particle interact? The photon essentially carries information about a changing electromagnetic field. How can it influence a particle that carries neither charge nor magnetism?


What does the Rayleigh Phase Function tell us

I am working on some radiative transfer equations, and struggling as I'm fairly new to this field. I have read about the Rayleigh Phase Function which is: $P(\theta) = \frac{3}{4}(1 + cos^2 \theta)$ I can plot this function, and generate values from this function in my computer program (which is what I need to do) - but what is it actually telling me? If I know I have a ray of incident light at a certain angle, will this function tell me what angle it will scatter at? Or is it more of a statistical generalisation where the areas where the function is higher are the angles at which the scattering is more likely to happen? Furthermore, when I see the function plotted it is always from 0-180, rather than 0-360 as I would expect for the scattering. This makes me think that the function may be giving the difference between the incoming angle and the outgoing angle rather than the absolute angle it is scattered at. Is that correct?


Continuous vs. Discrete Spectra in various materials

I read that the reason solids emit continuous spectra is that they don't have time to let their electrons decay-they are too close together. Given that electrons decay on the order of 100 nanoseconds I find this difficult to believe. Also, do electromagnetic waves move the electrons, or the atom, or both? If it is simply exciting the electrons, I don't know why is should also give way to the vibration of the atoms. If it does give way to vibration, then shouldn't gases _also_ give way to continuous spectra?


Non-relativistic Kepler orbits

Consider the Newtonian gravitational potential at a distance of Sun: $$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$ I write the classical Lagrangian in spherical coordinates for a planet with mass $m$: $$L ~=~ \frac{1}{2}m (\dot{r}^{2} + r^{2}\dot{\theta ^{2}} + r^{2}\dot{\phi ^{2}}\sin^{2}\theta ) + \frac{GM}{r},$$ and find that the canonical momentum $p_{\phi }$ is a constant of motion, because: $$\dot{p_{\phi }}~=~ \frac{\partial L}{\partial \phi} ~=~ 0.$$   1. What is the physical interpretation of the canonical momentum?   2. How can we from the Lagrangian see that it is a constant of motion?


Killing vector fields

I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.


Euler-Lagrange equations and friction forces

We can derive Lagrange equations supposing that the virtual work of a system is zero. $$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i (\mathbf{F}^{(a)}_i+\mathbf{f}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=0$$ Where $\mathbf{f}_i$ are the constrainded forces and are supposed to do no work, which it's true in most cases. Quoting Goldstein: > [The principle of virtual work] is no longer true if sliding friction forces > are present [in the tally of constraint forces], ... So I understand that we should **exclude friction forces** of our treatmeant. After some manipulations we arrive to: $$\frac{d}{dt}\frac {\partial T}{\partial \dot q_i}-\frac{\partial T}{\partial q_i}=Q_i$$ Further in the book, the Rayleigh dissipation function is introduced to **include friction forces**. So given that $Q_i=-\frac {\partial \mathcal{F}}{\partial \dot q_i}$ and $L=T-U$, we get: $$\frac{d}{dt}\frac {\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i}+\frac {\partial \mathcal{F}}{\partial \dot q_i}=0$$ **Question:** Isn't this an inconsistency of our proof, how do we know the equation holds? Or is it just an educated guess which turns out to be true?


What advantages does action-at-a-distance description have over the field view of forces?

It is written in Jackson (page 3) : > In fact, though there are **recurring attempts** to eliminate explicit > reference to the fields in favor of action-at-a-distance descriptions of the > interaction of charged particles, the concept of the electromagnetic field > is one of the most fruitful ideas of physics, both classically and quantum > mechanically. What does he mean? **What advantages can a action-at-a-distance description have over the field view of forces, that makes (at least some) people try toward that description?**


Gas pressure and centrifugal force

I think about a rotating torus (simplified tire) filled with ideal gas. Mass of gas is $m$ and molar mass is $M$. Pressure in non rotating torus is $p_0$. Temperature is constant $T$. Inner radius of torus is $r$ and outer radius is $R$. Then the cylinder begins to rotate with angular velocity $\omega$ How can I derive difference of pressures near the wall between inner wall and outer wall? Do I need more data (for a simple model)? If someone knows how to make it numerically, It would be also interesting for me. I would also appreciate if someone could tell which book should I read to know more about solving such problems.


Physics of every-day life: rotating bag of tea

Whilst studying for my physics courses, I like to drink tea. Today, I noticed that if you pull a bag of tea out of a hot cup of water, it gradually starts to rotate, picking up speed as time progresses and reaching a asymptotic speed after a while. My first question is: why does it start rotating? Of course it has something to do with the liquid water entering the gas phase just above the cup of tea, but how does that give the bag of tea a directed motion to one side? I presume that the momentum of the gaseous water is quite randomly directed vertically, so I don't see a reason for a horizontal rotation. Second question, how is the direction determined? I haven't tried, but might the direction of rotation be equally probable to be left-handed and right-handed? Third question: might the shape of a tea bag have an influence on what we observe? If we take a pyramid tea bag, for example, does the behavior change? I will perform some of these experiments while making my cup of teas and report back to you :)


Numerical simulation of sound propagation in air

How does one go about accurately modeling sound propagation in a room (with reflections, absorption, and diffusion characteristics) from the motion of a loud speaker? More specifically what are the governing equations that are needed? Obviously Naiver-Stokes but this is too general? Is there an easier way?


30kHz solenoid and power amplifier design

I have some difficulties designing my experimental apparatus: The goal is creating a Solenoid in air (or iron powdered core) able to produce at least 50G of peak B field on his axis at a distance of 1". I would like to drive this solenoid in such a way that the B field varies sinusoidally from 0G to 50G (hopefully even higher) with a frequency ranging from DC to 30kHz. Probably the easiest way is to use a function generator as an input for a power amplifier (I have trouble choosing it, I don't know if I need a four quadrant amplifier or a traditional one) Since I need a fast varying B field I suppose that I have to try to keep inductance as low as possible -> so maybe I'll put not so many turns but more current..inductance will probably also create problem in the correct impedance matching with the power amplifier (I don't have a strong background on this topic.)


What invariant counting process derives the Minkowski metric?

Starting with Euclidean space, suppose I make a number of copies of a coordinate system all coincident at the origin, together with copies of a standard unit length. For any space interval common to all the coordinate systems, all the observers will count the same minimum number of times their unit length can be laid end to end to one another between this interval. And from this invariant counting process we derive the Euclidean metric in terms of its orthogonal components. What invariant counting process do we carry out to derive the Minkowski metric in terms of its orthogonal components?


Integrating the generator of the infinitesimal special conformal transformation

(c.f Di Francesco, Conformal Field Theory chapters 2 and 4). The expression for the full generator, $G_a$, of a transformation is $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta \omega_{a}} \partial_{\mu} \Phi - \frac{\delta F}{\delta \omega_a}$$ For an infinitesimal special conformal transformation (SCT), the coordinates transform like $$x'^{\mu} = x^{\mu} + 2(x \cdot b)x^{\mu} - b^{\mu}x^2$$ If we now suppose the field transforms trivially under a SCT across the entire space, then $\delta F/\delta \omega_a = 0$. Geometrically, a SCT comprises of a inversion, translation and then a further inversion. An inversion of a point in space just looks like a translation of the point. So the constant vector $b^{\mu}$ parametrises the SCT. Then $$\frac{\delta x^{\mu}}{\delta b^{\nu}} = \frac{\delta x^{\mu}}{\delta (x^{\rho}b_{\rho})} \frac{\delta (x^{\gamma}b_{\gamma})}{\delta b^{\nu}} = 2 x^{\mu}x_{\nu} - x^2 \delta_{\nu}^{\mu}.$$ Now moving on to my question: Di Francesco makes a point of not showing how the finite transformation of the SCT comes from but just states it. $$x'^{\mu} = \frac{x^{\mu} - b^{\mu}x^2}{1-2x\cdot b + b^2 x^2}$$ I was wondering if somebody could point me to a link or explain the derivation. Is the reason for its non appearance due to complication or by being tedious? I am also wondering how, from either of the infinitesimal or finite forms, we may express the SCT as $$\frac{x'^{\mu}}{x'^2} = \frac{x^{\mu}}{x^2} - b^{\mu},$$ which is to say the SCT is an inversion $(1/x^2)$ a translation $-b^{\mu}$ and then a further inversion $(1/x'^2)$ which then gives $x'^{\mu}$, i.e the transformed coordinate.


Strömgren Sphere of Sun

I have a homework problem: > The Sun emits $ \sim5 x 10^{23}$ photons per second with $hν > 13.6$ $eV$. > If the density of hydrogen atoms in interplanetary space is $n =$ $109 > m^{-3}$, what is the size of the Stromgren sphere? Assume a recombination > coefficient $α = 2.6 x 10^{-19} m^3s^{-1}$. From Wikipedia, I was able to get to $$ R_S = \left( \frac{3S_*}{4\pi n^2 \beta_2} \right)^\frac{1}{3} $$ And I know $\beta_2 = \frac{\alpha}{T}$, but I have no idea what the $S_*$ is. Wikipedia describes it as a source of flux, which is obviously the sun, but I cannot figure out anything else about it. I'm actually quite sure that the professor would have given us a different equation, but he never went over it, so I don't know. Anyone know how to solve this?


Clouds in closed hydrosphere

Is possible estimate the needed size of an geodesic dome (like in the Eden project) for creating an real hydrosphere - especially clouds (and rain)? With other words, under what circumstances can happen clouds formation in an closed space -isolated from the outer atmosphere?


Modification of de Donder gauge

The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \bar g_{ab} + \gamma_{ab}$, then the de Donder gauge reads   $\nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma) = 0$. The partial differential equation for the gauge transformation vector $v^a$ is $ \nabla^b\nabla_b v_a + R_a^b v_b = \nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma)$. In chapter 7.5 of Wald, I read that this equation can always be solved because it is of the form $g^{ab}\nabla_a\nabla_b \phi_i + \sum_j (A_{ij})^a\nabla_a \phi_j + \sum_j B_{ij}\phi_j + C_i$.   Theorem 10.1.2 of Wald says that in a globally hyperbolic spacetime this equation has a well posed initial value formulation on any spacelike Cauchy surface. In stead of de Donder gauge, I want to use a similar gauge:   $\nabla^a(\gamma_{ab} - n \bar g_{ab}\gamma) = 0$.   The partial differential equation changes to   $ \nabla^b\nabla_b v_a + (1 -2n)\nabla_a\nabla_b v^b + R_a^b v_b = \nabla^a(\gamma_{ab} - n\bar g_{ab}\gamma)$. This equation is not covered by theorem 10.1.2 of Wald. My question is: is the existence of a solution for this equation guaranteed in an AdS background when $n=1$?


What is the relation between N=2 super Yang-Mills and its twist

My question is what is the relation between N=2 super Yang-Mills and its twisted version topological field theory? After twisting N=2 super Yang-Mills, i.e. diagonally embedding $SU(2)'_R$ into $SU(2)_R \times SU(2)_I$, we get a topological field theory. My question is since N=2 SYM and TQFT are different i.e. one is physical and the other is topological. Why can we use TQFT to calculate partition of N=2 SYM? What are the same for these two different theories? **Update** : From the second paper of Trimok, the authors claim that SYM under twist are just redefination. How to understand it?


What is the sense of introducing generating functional to the summands of expansion of S-matrix?

Let's have generating functional $Z(J)$: $$ Z(J) = \langle 0|\hat {T}e^{i \int d^{4}x (L_{Int}(\varphi (x)) + J(x) \varphi (x))}|0 \rangle , \qquad (1) $$ where $J(x)$ is the functional argument (source), $\hat {T}$ is the chronological operator, $\varphi (x)$ - some field. I want to understand the reasons for its introduction for the summands of expansion of S-matrix. As I read in the books, it helps to consider only the vacuum expectation values, forgetting about in- and out-states. But in $(1)$ appear summands like $\int \frac{J(p)dp}{p^2 - m^2 + i0}$ instead of the contributions from external lines. It may refer to the internal lines. So what to do with them and are there some other reasons to introducing $(1)$ except written by me?


How can two time theories be compactified to 3+1 without any Kaluza-Klein remnants

I have recently been looking into the two-time theories and the implied concepts. For me this seems slightly hard to grasp. How can I see the basic concept in this theory in a fundamental way based on its implied interaction with normal 3+1 dimension? I am interested specifically in how gauge symmetries that effectively reduce 2T-physics in 4+2 dimensions to 1T-physics in 3+1 dimensions without any Kaluza-Klein remnants.


On Bell's Inequality (Classical Intuition) and Quantum Mechanical Counter Intuition

This posting is directly related to the issue in The System and the Measuring Gadget. The QM expectation is given by: $$\langle\sigma_{1}.\vec{a}{\;}\sigma_{2}.\vec b\rangle=-\vec a.\vec b$$ In the above relation we are considering _the measured value_ of spin which is the outcome between the value of some property of the system itself and the measuring gadget The "classical" formula for evaluating the expectation with the hidden variable is as follows: $$P(\vec a.\vec b)=\int d\lambda\,\rho (\lambda) A(\vec a)B(\vec b)$$ Now some property of the system may depend on the value of $\lambda$ and the probability distribution $\rho (\lambda)$. Is the effect of measurement being fully accounted for by the the hidden variable $\lambda$ and the pdf $\rho (\lambda)$, especially in view of the fact that the process of measurement modifies the wave function itself. Would it be possible remove the contradiction between QM and commonsense intuition,expressed through Bell's Inequality, by considering the above factors? Reference for Bell's Original Paper.


Projectile motion equations @ very (!) high starting velocity

I've already searched Physics StackExchange for some similar question but I didn't find anything about this. Assumptions:   * Earth is a perfect sphere with it's core (X,Y,Z) -> (0,0,0) as a reference-frame center   * Air resistance can be ignored   * Earth rotation can be ignored   * Moon gravity-effect can be ignored And if I know (projectile starting properties):   * Current Earth GPS-coordinates   * Starting angle   * Starting direction (relative to (0,0,0))   * Starting velocity (can be larger than Earth-escaping speed)   * (mass of the projectile is irrelevant (I guess) when we know starting velocity of the projectile) How could I calculate aprox. coordinates of a projectile landing somewhere around the Earth globe (OR detect that projectile will "leave" the Earth)? EDIT: I've found tons of links on the net about projectile trajectories when starting velocities are quite small (& Earth can be considered like a flat plane), but non about upper situation.


Why there is added a partial time derivative in formula for time derivative of potential energy?

In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from? ![enter image description here](http://i.stack.imgur.com/xkrx5.png)


What is difference between $\frac {dr}{dt}$ and $\frac {\partial r}{\partial t}$?

What is difference in **physical meaning** of partial time derivative and ordinary derivative of $r$? $$\frac {\partial r}{\partial t}\quad\text{and}\quad \frac {dr}{dt}.$$ I know that ordinary time derivative is velocity.


What is the common difference between partial time derivative and ordinary time derivative?

What is difference between partial and ordinary time derivative? for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$? where the $v$ is velocity.


Entropy exchange of a free fall

I have a problem in which the tell me that you drop a bag of 50 kg of sand from 10 meters high, and you have to caltulate the entropy difference of the sand, asuming that the speific heat of the sand is so high that its temperature (298K) doesn't change. My result is 16.5 J/K, and my problem is with the sign. The book doensn't give an explanation but just an answer and it's positive, but I'm thinking that in the impact the heat must flow out of the sand, as it's loosing its kinetic energy, so the entropy would decrease, and the surroundings entropy would increase; what am I missing, or what is really going on here?


Misunderstanding of Special Relativity

Person A in reference frame A watches person B travel from Star 1 to Star 2 (a distance of d). Of course, from person B's reference frame, he is at rest and is watching Star 2 traveling to him. Now we know from the principle of relativity, each one will measure the other one’s clock as running slower than his own. Let’s say that Person A measures Person B’s speed to be v, and that Person A measures 10 years for person B to make it to Star 2. Let’s also say that person B is moving at the speed so that the Gamma Factor is 2. This means person A observes person’s B’s clock to have elapsed a time of 5 years. Now let’s look at this from Person B’s perspective: Person B observes Star 2 approaching (and Star 1 receding from) him also at speed v. Since the two stars are moving, the distance between them is length contracted (after all, if there were a ruler in between the stars, the moving ruler would be contracted) by a factor of 2. Since person B measures the initial distance to Star 2 to be d/2 and its speed v, he calculates the time to Star 2’s arrival to be 5 years. Since he observes person A’s clock as running slow (since Person A is moving also at speed v), when Star 2 arrives, he measures Person A’s clock to have elapsed a time of 2.5 years. Do you see why I’m confused? Person A measures Person B’s elapsed time to be the same as Person B measures Person B’s elapsed time (both 5 years), but Person B does not measure Person A’s elapsed time to be the same as Person A measures Person A’s elapsed time (Person B get’s a measurement of 2.5 years while Person A measured 10 years). This is asymmetrical, which probably means it is wrong. But I’m not sure what the error is. I suspect if I had done this correctly, each person should measure his own elapsed time to be 10 years and measure the other’s elapsed time to be 5 years. This would be symmetrical and would make the most sense, but again, I can’t seem to justify how person B wouldn’t measure his trip time to be 5 years. What's my mistake?


Why can different batteries with the same  voltage send different currents through the same object?

According to an answer in this thread on Skeptics: > If you take one of the little 12V garage door opener batteries and short out > (directly connect) the two terminals with a piece of wire or something else. > You'll get a light current flow through the wire or metal. It may get a > little warm. This battery is only capable of supplying a small amount of > current. > > If you take a 12V car battery and short out the two terminals (don't do it, > it's not fun), you will be met with a huge current arc that will likely > leave a burn mark on whatever was used to short it. This is because the car > battery is capable of discharging a large amount of current in a very short > period of time. I'm not sure how this could work given Ohm's law V=IR. If we assume the resistance of toucher is constant, then we'd expect the current to be the same as well.   1. Could it be that a car battery has less internal resistance than a garage battery?   2. Does it have anything to do with contact area? If it does, then how would you model it? Generally resistances add in series, but if I only half touch a contact, then neither the battery nor I change, so you'd expect our resistances to stay the same, but somehow our total resistance changes. So which objects resistance would change - mine or the batteries?


Are black holes really that special?

~~Science and science fiction alike~~ Science fiction describes black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time. However, are they really that special? My thought is that all matter has event horizons. However, most of it isn't dense enough for that event horizon to be large enough to affect the way the matter interacts with the rest of the universe. So, a black hole is simply just a dense star that is simply dark. And, if black holes _must_ have a singularity at their center because of general relativity, can't we conclude from that that all matter creates singularities, since the gravitational field intensifies into infinity the closer you get to the point-mass? What is the difference being really close to a point-mass and being very close to a black hole?


recommendation for a physics history/non-fiction book

I know that there are a lot theses being published on lives of physicists. Is there a history/non-fiction book that tracks the development of a problem chronologically? Like pieces of a puzzle. I would like it to be mathematical and trying to get into the heads of people trying to solve that problem. Something like a case study.


Is it possible for wind to break the sound barrier?

I understand that in nature wind would never get high enough, but I am just curious as to whether physics would allow this to occur or not.


Definition of Information in Information Theory

I am not sure in which SE site I have to put this question. But since I have learnt Shannon Entropy in the context of Statistical Physics, I am putting this question here. In the case of Shannon Information theory, he defines information $I$ for an $i^{th}$ event as, $$ I_i = -\ln P_i \qquad \qquad \forall i=1,...n. $$ Based on this definition we further define Shannon Entropy as average information, $$ S_\text{Shannon} =\langle I\rangle = -\sum\limits_{i=1}^n P_i\log_2P_i .$$ My question is what is the motivation behind defining entropy as some function that is inversely related to probability? I was told by my professor that lesser the probability of an event more information it possesses, although am still not convinced about this fact. Secondly, what is the reason in choosing the logarithmic function in this definition? Are there places where this definition of information is forfeited?


Are all classically impossible quantum possibilities entangled?

Any entangled state represents a quantum possibility that is classically impossible.   Is the converse true?   That is, are all states that are quantum mechanically possible but classically impossible entangled in some way?   If so, can you give a proof, or a reference to a proof?   If not, can you give a counterexample?


Resultant vector problem

$\newcommand{\v}[1]{\vec #1}\newcommand{\i}{\hat i}\newcommand{\j}{\hat j}$ Problem statement (1,2) > A shopper at the supermarket follows the path indicated by vectors $\v A, \v > B, \v C, \v D$ in the figure. Given that the vectors have magnitudes > $A=51\:\mathrm{ft}, B=45\:\mathrm{ft}, C=35\:\mathrm{ft}, > D=13\:\mathrm{ft}$, find the total displacement of the shopper using (a) the > graphical method and (b) the component method of vector addition. Give the > direction of displacement relative to $\v A$ > > ![enter image description here](http://i.stack.imgur.com/iJbOUm.jpg) My work: ![enter image description here](http://i.stack.imgur.com/RCyp8m.jpg) $$\begin{align}A &=&0&\i + &51&\j \\\B &=&45&\i + &51&\j\\\C &= &10&\i + &-35 &\j\\\ D &= &10&\i + &-13 &\j \\\\\text{Resultant} &=&54&\i+&65&\j\end{align}$$ Essentially I am adding all the components to get the resultant vector but it is not leading me to the right answer. What am I doing wrong here?


Does altitude affect sound pitch?

Due to differences in air pressure, temperature, and other factors, the speed of sound varies with altitude on Earth. Does this affect the pitch of the sound in any meaningful way? For example, if I had a tuning fork that vibrates at around 262 Hz, would I hear the same "Middle C" from it while standing on the shores of the Dead Sea as I would while standing at the peak of Mount Everest - an altitude difference of 9,271 meters or 30,417 feet? Would there be any difference at all, and would it be perceptible to the human ear at close range? Would the altitude difference affect how the sound is heard more at a distance than it would nearby? Or, am I not quite understanding properly how sound works?


What is the experimental uncertainty of an ensemble measurement?

Let's say you measure the time it takes for 10 oscillations of a mass undergoing simple harmonic motion to within ± 0.01s, what is the uncertainty of the period of one oscillation?


Cheat sheet of elementary particles

I am trying to teach myself some particle physics. There are too many particles and its too much for me. I hated biology just because of this sort of stuff. Too many names and it was all Greek to me. Is there a good cheat sheet/ reference sheet of elementary particles? It will also be very helpful if you share how you people manage to remember these things. This is the first time I have ever hated studying physics. :(


I am trying to understand Rømer's determination of the speed of light ($c$). The geometry of the situation is shown in the image below. The determination involves measuring apparent fluctuations in the orbital period of Io. (Jupiter's moon) ![Geometry of the problem](http://i.stack.imgur.com/J5dyj.png) The Earth starts from point A. $r(t)$ is the distance between the Earth and Jupiter. $r_e$ is the radius of the (assumed) circular orbit of the Earth around the Sun, while $r_0$ is the same for Jupiter. $T$ is the period of the Earth's orbit. Under the assumption that the Jupiter-Io system is stationary, $r(t)$ can be expressed as $$r(t) = \sqrt{r_E^2 + r_0^2 -2r_0 r_E \cos \left(\frac{2\pi t}{T}\right)}$$ If we further assume that the period of Io's orbit around Jupiter, $\Delta t$ is much smaller that $T$, then it can be shown that the distance the Earth moves, $\Delta r$ when Io completes one orbit is: $$\Delta r = \frac{2\pi r_E \Delta t}{T} \sin\left( \frac{2\pi t}{T} \right)$$ The point I am stuck is about why is there an apparent fluctuation in Io's orbit as observed on the Earth? And how can we derive the observed delay using these expressions?


Cancel out Earth's Magnetic field

Is there any tools except helmholtz coil to cancel out earth's magnetic field to calibrate magnetometers in practice.


What's the exact gravitational force between spherically symmetric masses?

Consider spherical symmetric$^1$ masses of radii $R_1$ and $R_2$, with spherical symmetric density distributions $\rho_1(r_1)$ and $\rho_2(r_2)$, and with a distance between the centers of the spheres $d$. What is the exact force between them? I know point masses are a good approximation, but I'm looking for an exact formula. This would be useful for a gravity-simulation toy software. \-- $^1$ Assume for simplicity the idealization where tidal or centrifugal forces do not deform the spherical symmetric, i.e., the various mass parts are held in place by infinitely strong and rigid bonds.


Why did my windshield freeze instantaneously?

It was very cold outside, this morning, when I took the car that slept in the snow, with a simple cloth on the windshield. I entered the vehicle, drove a kilometer or so. The air inside was so cold I could see my breath (or maybe I forgot to brush my teeth). Warm air was blowing on the windshield from the inside and suddenly… The glass in front of my became opaque, starting from the bottom (the place where the hot air was blowing) and freezing up, up, up until the whole screen was filled in about 5 seconds. It reminded my a little bit of how some baterias spread. When I tried to remove the mist, I realised it was ice **in** the car. My windshield instant froze. I know the theory behind supercooling: a very cold and still liquid can freeze when moved. I'm not sure what happened here, but there were some pretty big turns in the road before it froze. It really looked like the hot air froze the place and not the movement. Any idea about what happened ?


Chemical potential

This is something probably very basic but I was led back to this issue while listening to a recent seminar by Allan Adams on holographic superconductors. He seemed very worried to have a theory at hand where the chemical potential is negative. (why?)   * For fermions, isn't the sign of the chemical potential a matter of definition? The way we normally write our equations for the Fermi-Dirac distribution the chemical potential happens to that value of energy at which the corresponding state has a occupation probability of half. And within this definition the holes in a semiconductor have a negative chemical potential.   * It would be helpful if someone can help make a statement about the chemical potential which is independent of any convention. {Like one argues that negative temperature is a sign of instability of the system.}   * Also isn't it possible for fermions in an interacting theory to have a negative chemical potential?   * Also if there is a "physical argument" as to why bosons can't have a positive chemical potential? (Again, can an interacting theory of bosons make a difference to the scenario?)   * And how do these issues change when thinking in the framework of QFT? (No one draws the QCD phase diagram with the chemical potential on the negative X-axis!)   * In QFT does the chemical potential get some intrinsic meaning since relativistically there is a finite lower bound of the energy of any particle given by its rest mass?


On the distinction of past and future: could one theoretically reverse direction of particles and cause time to appear to go backwards?

Based on my understanding of physics after seeing _The Distinction of Past and Future_ on Project Tuva, there is no distinction between past and future on a fundamental level- all particle interactions can occur in reverse. So my question is whether or not one could theoretically reverse the direction of all particles in the observable universe relative to each other and have time essentially go backwards indefinitely. If you think about it, things could "fall" upwards because the air resistance would be much lower due to the way the air was moving when it fell, and the velocity from the gravity downwards would be reversed as well as air under the ball pushing up (again due to the way the air was moving previous to the switch). I don't see why this same logic couldn't be applied to a more complex system. Does this logic make sense? If not, where is the flaw? What other constraints would need to be added to make time essentially go backwards other than reversing direction, if it is possible at all, in theory?


Universe with ZERO Fundamental Forces

Warning: I am not a physicist so please excuse my naivety! As you all know, physicists think that there exist four fundamental forces. Would a universe with zero fundamental forces be possible, at least in principle? Would elementary particles be able to exist in such a universe? On a larger scale, what would such a universe look like?


Does pulling or pushing take more effort?

In which case is more effort needed: when you push or when you pull? We have a common experience that man tends to first push anything. How it is justified and what are the reasons behind this?


Derivation of Electric Force between Parallel Plates

So the electric field between two parallel plates is given by $E = V/d.$ How do you derive this?


> **Possible Duplicate:**   >  Choice and identification of vacuums in AdS/CFT I think I know how do we define a vacuum in flat space QFT and also in a curved space QFT. But, can somebody tell me how do the choice of vacuum state in say the CFT side of AdS/CFT changes the choice of vacuum state in gravity side? Let me ask the other way. I mean if we pick a vacuum (say in bulk side, because it may not be unique), how does it reflect on the CFT vacuum (and vice versa)? The CFT side is conformally flat. So, there is no reason in the CFT side that there will be a unique vacuum, is there? So, how does this choice reflect on both sides and how do we generally make the identification? Thanks.


Off-shell corrections to massive vector boson propagator in polarization form

As an exercise for myself, I have been working on rewriting the massive vector boson propagator (unitary gauge). I have run into a problem interpreting some of the terms that stick around when the propagator is rewritten this way. Here's what I have: I've taken the unitary gauge vector propagator $$ D_{\mu\nu}(q) = \frac{i}{q^2 - M_W^2 + i \varepsilon} \left( -g_{\mu \nu} + \frac{q_\mu q_\nu}{M_W^2} \right) $$ and projected it into its helicity components. The convention I am using is that $$ \epsilon^1_\mu(\vec q) = (0,1,0,0) \\\ \epsilon^2_\mu(\vec q) = (0,0,1,0) \\\ \epsilon^0_\mu(\vec q) = \frac{1}{M}(|q|,0,0,E_q) \\\ \epsilon^s_\mu(\vec q) = \frac{1}{M}(E_q,0,0,|q|) $$ where $E_q = \sqrt{M^2+|q|^2}$. These are an orthonormal set, where $$ \epsilon^{\lambda}_\mu \epsilon^{\lambda' \mu} = - \eta_{\lambda} \delta_{\lambda \lambda'} $$ where $\eta_\lambda = 1$ for $\lambda = \pm,0$ and $-1$ for $\lambda = s$, the scalar polarization. So, it's easy to show that $$ X_{\mu \nu} = \sum_{\lambda,\lambda'} X_{\lambda,\lambda'} \epsilon^{\lambda}_\mu \epsilon^{\lambda' }_{\nu} \Rightarrow X_{\mu \nu} \epsilon^{\lambda \mu} \epsilon^{\lambda' \nu} = \eta_\lambda \eta_{\lambda'} X_{\lambda \lambda'} $$ In particular, $$ \- g_{\mu \nu} \Rightarrow g_{\lambda \lambda'} = \frac{\eta_\lambda' \delta_{\lambda \lambda'}}{\eta_{\lambda} \eta_{\lambda'}} = \frac{\delta_{\lambda \lambda'}}{\eta_{\lambda}} = \eta_{\lambda}\delta_{\lambda \lambda'} $$ Where I've run into difficulty is breaking up the transverse term. $$ q_\mu \epsilon^{1\mu}(\vec q) = 0 \\\ q_\mu \epsilon^{2\mu}(\vec q) = 0 \\\ q_\mu \epsilon^{0\mu}(\vec q) = \frac{|q|}{M}(q_0-E_q) \\\ q_\mu \epsilon^{s\mu}(\vec q) = \frac{1}{M}(q_0 E_q - |q|^2) $$ The last of these can be rewritten $$ q_\mu \epsilon^{s\mu}(\vec q) = M + \frac{E_q}{M}(q_0-E_q) $$ So, the helicity components of $$ T_{\mu \nu} = \frac{q_\mu q_\nu}{M^2} $$ are $$ T_{1\lambda} = T_{\lambda1} = T_{2\lambda} = T_{\lambda2} = 0 $$ $$ T_{00} = \frac{|q|^2}{M^2} \left( \frac{q_0-E_q}{M} \right)^2 $$ $$ T_{ss} = 1 + \frac{q_0^2-E_q^2}{M^2} + \frac{|q|^2}{M^2} \left( \frac{q_0-E_q}{M} \right)^2 $$ $$ T_{0s} = T_{s0} = - \frac{|q|}{M} \left( \frac{q_0-E_q}{M} \right) - \frac{E_q|q|}{M^2}\left(\frac{q_0-E_q}{M}\right)^2 $$ The term equal to $1$ in the scalar polarization term cancels out the corresponding scalar polarization term $g_{ss}$. Furthermore, all of the terms proportional to $(q_0-E_q)^2$ I understand. They cancel out the pole in the propagator, since $$ \frac{1}{q^2 - M^2 + i \epsilon} = \frac{1}{q_0^2 - E_q^2 + i \epsilon} \sim \frac{1}{q_0 - E_q + i \epsilon} \frac{1}{q_0 + E_q} $$ and after canceling out the pole they retain a factor $q_0 - E_q$ which forces them to be 0 while on-shell. These are explicitly off-shell corrections. However, I'm not sure how to interpret the terms $$ T_{ss} \ni \frac{q_0^2-E_q^2}{M^2} = \frac{q_0+E_q}{M} \frac{q_0-E_q}{M} $$ and $$ T_{0s} = T_{s0} \ni - \frac{|q|}{M} \left( \frac{q_0-E_q}{M} \right) $$ Naively, it appears to me that these cancel out the pole at $q_0 = E_q$, but the remaining portion does not vanish at at $q_0 = E_q$ (or at least as at $q_0$ approaches $E_q$) I think that a careful analysis of the behavior of these terms the pole might shed light on this, or that maybe it is a gauge artifact, but I am stuck.


Trouble evaluating an integral arising from particle collision

Assume we have two charged particles colliding. He have particle 1 with mass $m_1$, charge $Z_1 \cdot e$ which travels in $x$-Direction passing by a STATIONARY particle 2 (mass $m_2$, charge $Z_2 \cdot e$) at distance $b$ (impact parameter). I want to calculate the change in momentum in $y$-Direction (as the change in momentum in x-Direction vanishes). $\Delta p_y = Z_1 e \int_{-\infty}^\infty F_y dt = Z_1 e \int_{-\infty}^\infty F_y \frac{1}{v}dx $ (as $dx/dt = v)$ in the script im trying to follow they now multiply by $1$ as follows: $\Delta p_y = \frac{Z_1 e}{2 \pi b v} \int_{-\infty}^\infty 2 \pi b F_y dx$ They now immediately evaluate this theorem using Gauss-Theorem to: $\frac{Z_1 e}{2 \pi b v} \frac{Z_2 e}{\epsilon_0}$ Generally the Gauss-Theorem relates the integral over the surface to an integral over the space that surface encloses. I have a bit of a problem seeing over which surface/space area im integrating here, since its just an integral along the x-axis. Thanks in advance for any help! Cheers


plasma frequency in non-neutral plasma

Is it possible to see the oscillations with plasma frequency in a gas of particles of the same charge (not mixture of positive and negative charges)?


Calculating Phase Diagrams (Calphad)

I am calculating the phase diagrams for Li-Mg binary alloy with reference to the following text: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5600701 In the equation used in the above paper, i am confused about whether G0 for a component is a function of temperature? ![enter image description here](http://i.stack.imgur.com/V78Oq.png)


Question regarding the Bohm interpretation

I tryed to understand the Bohm interpretation and this is what picture appeared to me. Please tell me if I understood something incorrectly.   * All particles have definite positions and follow deterministic rules of dynamics   * Any future configuration of an isolated subsystem is only dependent on initial conditions   * Even slight difference in initial conditions may result in huge differences in the result. The problem is that those initial conditions, are inherently unknown. This is fundamental: even if an observer manages to measure the whole Bohm state of the entire universe, he still would not know the Bohm state of himself. This is like making predictions bout future states of a three-body system based on Newtonian mechanics with initial coordinates known only with finite precision. Due to apparent chaoticity of the solution the possible results may be dramatically different. Correct me if I am wrong.


How do I extend the Lorentz transformation metric to dimensions>4?

How do I extend the general Lorentz transformation matrix (not just a boost along an axis, but in directions where the dx1/dt, dx2/dt, dx3/dt, components are all not zero. For eg. as on the Wikipedia page) to dimensions greater than 4? Thanks


Gradient of the electric potential

I was wondering if for a point-like charged object, does the gradient of the electric potential point in the direction of maximum increase or maximum decrease of the function $V$?


What should I call an n>4 dimensional Minkowski metric?

I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be + or - , depending on the model. (I am not trying to describe physics with this metric). Can I call this metric a Minkowski metric? Or what should I call it?


Langreth rules and Keldysh formalism

I am trying to confirm the proof of Langreth's theorem / rules as seen in http://www.iue.tuwien.ac.at/phd/pourfath/node52.html . My problem is equation 3.55. I would do it like this: $\int_{C_{1}} \mathrm{d}\tau A(t,\tau) B(\tau,t') = \int_{-\infty}^{t} \mathrm{d}t_1 A^{>}(t,t_1) B^{<}(t_1,t') + \int_{t}^{-\infty} \mathrm{d}t_1 A^{<}(t, t_1) B^<(t_1, t')$ And now I flip the integration boundaries on the right and replace $A^r = A^>-A^<$: $ = \int_{-\infty}^{t} \mathrm{d}t_1 (A^{>}(t,t_1) - A^<(t,t_1) ) B^{<}(t_1,t') = \int_{-\infty}^{t} \mathrm{d}t_1 A^r(t,t_1) B^<(t_1,t)$ And thus my integration does not go from $-\infty$ to $\infty$ as shown in the source. What am I missing? * * * Alright, I got it. It works because of the definition of the retarded function with a theta function: $$ A_r(t,t') = \theta(t-t') [ A^>(t,t') - A^<(t,t')] $$ So the retarded function is always zero when $t' > t$. One can therefore extend the integration to $\infty$. I leave this here so it might help someone else later.


light entering a black hole's singularity

I already understand that light cannot escape a black hole after passing the event horizon, so please do not explain that to me. What I would like to know is this: a well known fact about light (a photon specifically) is that it travels at the speed of light, and at no other speed, which means that it has no rest mass, as an example, as it does not stop. As a photon aproached a black hole, it would begin to spiral around it as it got ever closer to the singularity at the centre. The closer the photon got to the singularity, the shorter the amount of time it would take to go once around the singularity, as it remained at its constant speed. However, on reaching the perfect centre, it would stop moving completely relative to the blackhole, so would no longer be travelling at the speed of light. Can you explain why this happens or (more likely) where I have gone wrong?


In what form does proton give energy to electron?

According to Bohr, electron revolves around the nucleus because of force of attraction between electron and proton. This force of attraction gives energy to the electron. So my question is this that- In which form does this electron get the energy?


'ting' 'ting' sound from a tube-light?

Whenever I light up a tube-light it makes 'ting' 'ting' sound every-time it blinks. I am talking about this tube-light   ![enter image description here](http://i.stack.imgur.com/j5j97.jpg) Why is it so? I think its because of sparking(inside glass tube) similar sound in the 6th box hover and listen here


Collision of Phobos

Mars has two moons: Phobos and Deimos. Both are irregular and are believed to have been captured from the nearby asteroid belt. Phobos always shows the same face to Mars because of tidal forces exerted by the planet on its satellite. These same forces causes Phobos to drift increasingly closer to Mars, a situation that will cause their collision in about 50 to 100 million years. How I can calculate, given appropriate data, the estimated time at which Phobos will collide with Mars?


Is the moon a planet?

Can our moon qualify as a planet? With regard or without regard to the exact definition of the planet, can the moon be considered as planet as Mercury, Venus and Earth etc. not as the satellite of the planet Earth.


Semi-conductor band-gap and deformation potential

Submitting a semi-conductor to stress leads to a deformation in the energy- bands, roughly described by:$$H_{ij} = {\cal{D}}_{ij}^{\alpha\beta}\;\epsilon_{\alpha\beta}$$ $\epsilon$ being the strain (linked to the stress by Hooke's law), $H$ the perturbation Hamiltonian to the Hamiltonian describing a stress-free semiconductor, $i,j$ being indexing the energy level of the previous "free" Hamiltonian. Still, I lack intuition regarding the apparition of this term, it seems that compression enlarges the band-gap whereas dilation tightens it. Why do we have this behaviour? I have tried thinking on electrostatic arguments, the potential decreasing as $r^{-1}$ we do have an increase of energies in $r \rightarrow \alpha r$ for $\alpha < 1$ or also seeing the dilatation as a renormalization group transformation, basically going to a coarser grain (although there probably other pertinent length scales in an atomic lattice (spreading of the electronic orbitals, ...) which would make this argument wobbly). Cutting to the point, what is your hand-waved way of seeing it? Books on the subject of deformation potential don't really seem to offer intuition on it, more numerical values for specific materials.


Are electrodynamics problems in the complex plane relevant to real life?

This is a question I asked in Maths SE, and it was suggested I ask it here. This is a direct copy of that question. I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of the book deals almost entirely with physics, using symmetries of conformal mappings to generalise the famous method of images technique in electrodynamics. The method of images is used in finding the electric field due to a charge when a grounded surface (such as a sphere or plane) is nearby. (See e.g. Wikipedia.) However, the problems seem to have very little "real life" applications to me, the main problem being that the complex plane is two dimensional, whereas we live in a 3 dimensional world. To see this problem concretely, the electrostatic force is goes like $F\sim \frac{1}{r^2}$ because the surface area of a ball of radius $r$ centred at the charge is proportional to $r^2$. However since the complex plane is 2 dimensional, a charge in the complex plane produces a field which goes like $\frac1r$. So any solution we find to a problem of this kind in the complex plane isn't relevant in 3d. And this is my question, is there any physical application of this technique? Or is it completely irrelevant?


Why does a ballerina speed up when she pulls in her arms?

My friend thinks it's because she has less air resistance but I'm not sure.


Huge confusion with Fermions and Bosons and how they relate to total spin of atom

I am supremely confused when something has spin or when it does not. For example, atomic Hydrogen has 4 fermions, three quarks to make a proton, and 1 electron. There is an even number of fermions, and each fermion has a 1/2 spin. Since there are an even number of fermions, the total spin value is an integer. This spin number is the "intrinsic" spin number that cannot be changed, but its orientation "up" or "down" can be changed. For atomic Hydrogen, it is a Boson because it has integer spin, however it also has a single electron. I read on physics forums, http://www.physicsforums.com/showthread.php?t=69992, that the spin of atom comes from the electrons and not its nucleus. I also read on here, How to find that a molecule has zero spin?, that the spin of atomic Hydrogen is 1/2! The answer says atomic Hydrogen has spin 1/2 because it ignores the nuclear spin. This is one thing that is confusing me. Shouldn't atomic Hydrogen have an integer spin because of the nuclear component? So does atomic Hydrogen have spin and is affected by a magnetic field? Nuclear spins are affected by magnetic fields, but they aren't as affected as electrons according to the discussion on physics forums. Why do we ignore nuclear spin sometimes? Also, can someone help me out here with all the possibilities? Is there a Boson with an half integer spin value? (Surely, there must not be) However atomic Hydrogen is one of those cases! (It seems...) (Why don't we cancel out the nuclear spin with the electron spin?) Say we have another atom that is a Boson, It has unpaired electrons in different orbitals, so what determines whether or not electrons fill in orbitals as spin up or down? Does spin down nuclear spin cancel out a electron up spin?


How can systematic errors be calculated?

Usually, it is said that systematic errors can not be handled in a well- defined way, unlike statistical errors. My question(s): A) How can systematical errors be calculated for any experimental device or experiment? Maybe does it involve a "calibration" of experimental devices? B) I have heard that bayesian methods provide THE tool to estimate and calculate systematical errors. How is it done?


Addition of a neutral electrolyte to water-- how can it increase conductivity?

> Sparked off by Is sea water more conductive than pure water because > "electrical current is transported by the ions in solution"? This question really belongs on chemistry.SE, which is still in area 51. While answering this question, I realised that there was a flaw in the standard logic for these situations. Let's take $NaNO_3$ solution in water, and compare it with pure water (same size cell). (I'm not taking $NaCl$ for a reason*). In both cases, we have nearly the same(~7) PH and pOH, right? So concentrations of $H^+$ and $OH^-$ are the same, and thus contribution of these ions to the overall conductance/conductivity is the same. These ions both migrate to the electrodes, get reduces/oxidised, and emit/absorb electrons, facilitating passage of current. OK. Now, let's consider the $Na^+$ and $NO_3^-$ ions. Yes, they migrate as well. But, they don't get redoxed (the water ions are preferentially redoxed). So, all I see happening here is a buildup of charges on either electrode, which will stop once equilibrium is attained. This buildup of charges cannot migrate to the outside circuit like a capacitor, as it cannot translate to electrons. But, these ions still make a significant contribution to the net conductance by Kohlrausch law. **So how does a general good neutral electrolyte help water conduct electricity?** I feel that it should conduct electricity at the same rate as water; but by Kohlrausch law, it clearly doesn't. And anyways, I've always heard that impure water conducts electricity better. *$NaCl$ has the issue of overvoltage of $O_2$, causing oxidation to of $Cl_2$, which complicates the situation. The $Cl^-$, being able to get oxidised, facilitate current and thus don't serve as a good example here.


Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT

This maybe a very naive question. I have just started studying CFT, and I am confused by why we have two separate parts of everything in CFT (operator algebras and hilbert space), the holomorphic and anti-holomorphic, which are decoupled from each other. We initially introduced $z$ and $\bar{z}$ as two independent variables instead of say $t$ and $x$ in two dimensions. But now, we have got two isomorphic parts, the holomorphic and antiholorphic (they may given by $z \to \bar{z}$ and $h \to \bar{h}$), then what extra info. does the anti-holomorphic parts provide? And how is all the information contained in just one independent coordinate $z$? Also a physical theory should be the tensor product of the verma modules? Why do we need both the parts, and what is the physical significance of each.


Basic Concepts of Water Pressure for Plumbers

I am learning how to install water pipes and I was told that to check if a pipe is leaking water I need to check the water pressure. But I don't understanding how water pressure works inside the pipe. For example, a water pump is connected to a valve and to a tap like this: pump ----> valve ---->tap the water pressure is caused by the pump. Now if I close the valve and blocking the water from the pump to the tap, what will be the water pressue in the segment of pipe between the valve and the tap? i.e. pump ----> X ----> tap My understanding is that the pressure should drop to zero as this segment is disconnected from any "source of water pressure" (the pump). But If I connect a meter to the tap before closing the valve: pump ----> X ----> meter The reading on the meter doesn't drop to zero. Instead, if there is a water pressure drop, it indicates a leakage. But what causes the pressure when the valve is closed? I know air pressure inside a sealed system is caused by the Brownian motion of the air molecule does the same thing applies to water?


Critical point vs Gibbs phase rule

Why is the critical point for the phase diagram of pure water degrees of freedom equal to 0? Maybe, you know what is the mathematical explanation for the fact that the number of degrees of freedom at the critical point is 0? What else is affected by the lack of degrees of freedom in addition to the Gibbs phase rule?


Why is $\left.\frac{\partial C_V}{\partial V}\right|_T$ different in these derivations?

I want to show that $\delta q$ is not an exact differential. Starting from $dE = \delta q - pdV$ and because $E := E(V, T)$ is a state function, which allows to express the exact differential as $$ dE = \left.\frac{\partial E}{\partial V}\right|_T dV +\left.\frac{\partial E}{\partial T}\right|_V dT, $$ the two expressions can be set equal giving after rearrangement $$ \delta q = \left[ \left.\frac{\partial E}{\partial V}\right|_T + p \right]dV + \left.\frac{\partial E}{\partial T}\right|_V dT $$ and therefore also $$ \delta q = \left[ \left.\frac{\partial E}{\partial V}\right|_T + p \right]dV + C_V dT. $$ Now, by definition $\partial E/\partial T|_V = C_V$, and so $$ \left.\frac{\partial C_V}{\partial V}\right|_T =\left[\frac{\partial}{\partial V}\left.\frac{\partial E}{\partial T}\right|_V\right]_T $$ and because $E$ is a state function, the sequence of partial derivatives can be exchanged (according to Schwarz' theorem, while I don't understand how it works), allowing to write $$ \left.\frac{\partial C_V}{\partial V}\right|_T =\left[\frac{\partial}{\partial T}\left.\frac{\partial E}{\partial V}\right|_T\right]_V. (*) $$ Also, multiplying the above expression for $\delta q$ by $1/\partial V$ at constant $T$, I obtain $$ \left.\frac{\delta q}{\partial V}\right|_T = \left[ \left.\frac{\partial E}{\partial V}\right|_T + p \right]\left.\frac{\partial V}{\partial V}\right|_T + C_V \left.\frac{\partial T}{\partial V}\right|_T $$ where $\partial V/\partial V = 1$ and the second term on the right hand side equals $0$ because $\partial T = 0$ at constant temperature. Multiplying the remaining equation by $\partial / \partial T$ at constant $V$ gives $$ \left[\frac{\partial}{\partial T}\left.\frac{\delta q}{\partial V}\right|_T\right]_V = \left[ \frac{\partial}{\partial T} \left(\left.\frac{\partial E}{\partial V}\right|_T + p \right)\right]_V. $$ Now _assuming_ $\delta q$ were exact, again the sequence of partial derivatives would not matter and I could write $$ \left[\frac{\partial}{\partial V}\left.\frac{\delta q}{\partial T}\right|_V\right]_T = \left[ \frac{\partial}{\partial T} \left(\left.\frac{\partial E}{\partial V}\right|_T + p \right)\right]_V $$ and by using $q=E$ since the "inner" differential on the left side is evaluated at constant volume, $$ \left[\frac{\partial}{\partial V}\left.\frac{\partial E}{\partial T}\right|_V\right]_T = \left.\frac{\partial}{\partial V} C_V\right|_T = \left[ \frac{\partial}{\partial T} \left(\left.\frac{\partial E}{\partial V}\right|_T + p \right)\right]_V. (**) $$ From this we find that $(*)$ and $(**)$ are different and thus the assumption must be wrong and therefore $\delta q$ is not an exact differential. Does this make any sense? * * * Exact wording from book: > Starting with $dE = \delta q - pV$, show that > > a) $\delta q = C_V dT + [P+(\partial E/\partial V)_T] dV$ > > b) $\left(\frac{\partial C_V}{\partial V}\right)_T = > \left[\frac{\partial}{\partial T} \left(\frac{\partial E}{\partial > V}\right)_T\right]_V$ > > c) $\delta q$ is not an exact differential. For c), the book states > If $\delta q$ were an exact differential, then by solution to a), $(\partial > C_V/\partial V)_T$ would have to be equal to $[\partial /\partial > T(P+(\partial E/\partial V)_T)]_V$ but it is not according to solution of > b), hence $\delta q$ is not exact.


Ratio of position error of orbiter to the size of the orbit

This is an assignment question for online AP physics. As if it isn't already tough. This question is killing me. I even asked my physics teacher from last year and my calculus teacher. It stumped them, too. I would show some kind of attempt at solving the question, but I've got nothing. I don't even understand how to begin solving this. The question is: > The space shuttle tracking system predicts the position of the shuttle > orbiter with an accuracy that varies between 30 m and 100 m. Its orbital > radius is slightly larger than the radius of the Earth (average orbit > altitude is 340 km). What, approximately, is the ratio of the position error > to the size of the shuttle’s orbit? The orbiter is approximately 37 m long > and 17 high. ($R_{Earth} = 6378$ km)` Any help with this would be greatly appreciated!


Free-particle solution to Schrödinger Equation

The free particle solution in stationary state (with definite energy) to the Schrödinger equation is $$\psi(x,t) =Ae^{i(kx-\omega t)} + Be^{-i(kx+\omega t)}$$ Since the energy is definite, and hence the momentum is definite, the uncertainty in position must be infinite. How is this reflected by the probability distribution function: $$\Psi = |\psi(x,t)\psi^*(x,t)| $$ The book that I am using just look at the first term of the solution, and derive that the probability distribution function is $A^2$. However, I do not understand why we can do that? Does it imply that if wave function is made up of n terms such that each individual term has a constant probability distribution function, the whole wave function also has a constant probability distribution function? If so, how can I prove it? I know my question might be very vague but that is precisely the problem I am facing now, I don't even know how to ask about the things that I don't understand.


Query into the cumulative velocity of mounted platforms

Consider throwing a stone at an object from rest, it travels at Vms-1. Now throw that stone whilst running at Ums-1. It seems in the latter scenario the total speed of stone is V + U. Now imagine Running at Ums-1, throwing a stone at Vms-1 whilst on a moving train with speed Wms-1 - total stone speed would be V+U+W. Let's extrapolate this to the case where you have a stack of moving platforms, the bottom platform begins to accelerate, once reaching top speed, the platform on top begins to accelerate, and so on and so forth. In a vacuum could it be theoretically possible to reach near infinite projectile velocities using these cumulative platform velocities? ![enter image description here](http://i.stack.imgur.com/G5wtQ.jpg)


Curvature of spacetime in only required to explain tidal forces?

I'm a bit confused about the equivalence principle in GR. I'm quoting from Wikipedia: > An observer in an accelerated reference frame must introduce what physicists > call fictitious forces to account for the acceleration experienced by > himself and objects around him. One example, the force pressing the driver > of an accelerating car into his or her seat, has already been mentioned; > another is the force you can feel pulling your arms up and out if you > attempt to spin around like a top. Einstein's master insight was that the > constant, familiar pull of the Earth's gravitational field is fundamentally > the same as these fictitious forces Later it is written: > The equivalence between gravitational and inertial effects does not > constitute a complete theory of gravity. When it comes to explaining gravity > near our own location on the Earth's surface, noting that our reference > frame is not in free fall, so that fictitious forces are to be expected, > provides a suitable explanation. But a freely falling reference frame on one > side of the Earth cannot explain why the people on the opposite side of the > Earth experience a gravitational pull in the opposite direction Here are some things I hope I understand correctly:   * A particle in free fall is in an inertial frame of reference   * Curvature of spacetime in only required in order to explain tidal forces, as long as you ignore tidal forces, you can explain gravity without curvature.   * Gravity is a fictious force experienced in a non-inertial reference frame My Questions (2 very related questions)   * 1) The statement that curvature of spacetime in only required to explain tidal forces seems weird to me. In the case that there is no curvature of spacetime, what explains gravity? I mean, if gravity is a "fictitious-force", what is the "real cause" of it? (Again this question stems from the statement that curvature is only needed to explain tidal forces, and not all of gravity). Last example from Wikipedia: > For gravitational fields, the absence or presence of tidal forces determines > whether or not the influence of gravity can be eliminated by choosing a > freely falling reference frame   * 2) If I'm in outer space and I'm freely falling towards earth, let's say I'm very small and I don't experience tidal forces, both me and earth are freely falling and thus in inertial reference frames, and yet I see the earth accelerating towards me, in my frame is it said that "gravity is eliminated"? just because I feel no tidal forces?


Difference b/w Kinetics & Kinematics w/concrete example

(I know whether I understand this or not doesn't matter much to my work & study but am just curious.) I still can't differentiate in my head kinetics and kinematics (similar thread is found but doesn't explicitly answer to my question yet What is the difference between "kinematics" and "dynamics"?). Some websites out there say (ex.) explain that force is only considered in kinematics. Does this mean for example Newton-Euler method is in kinetics and Lagrangian is in kinematics? I also prefer concrete examples in both category.


Can a force in an explicitly time dependent classical system be conservative?

> If I consider equations of motion derived from the pinciple of least action > for an _explicilty time dependend_ Lagrangian > > $$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$ > > under what circumstances (i.e. which explicit functional $t$-dependence) is > the force conservative? By force I understand here the term on the right hand side of the equation, if I shove everything to the right except the expression $mq''(t)$. * * * As a sidenote, besides the technical answer I'd be interested here in some words about the physical motivations involved. I'm somewhat unhappy with a formal $\text{curl}[F]=0$ condition, since it seems to be to easy to fulfill (namely we have to consider closed circles only at single points in time, respecively). The physical motivation behind conservative forces is the conservation of energy on closed paths, where any parametrization $q(s)$ of curves can be considered. But practically, only loops tracked in finite time are physically realizable, i.e. we would move in a circle while t changes. I guess as soon as one computes the r.h.s. for the equations of motion, one would also be able to define a more physical alternaltive to the above stated idea of conservative forces in this case. I.e. a ask-if-the-forces-integrate- to-zero-on-a-closed-loop functional for a rout between two points in time $t_1$ and $t_2$. This would be an integral where the momentarily force along the point in the path I'm taking would be taken into account. It wouldn't be path independend of course. (We could then even construct another optimization problem on its own, by asking for path with the smallest energy difference, which really would be a sensible question if friction is involved.)


Lunar twilight and sixth magnitude stars

Summary: when the Moon is x degrees below the horizon, it interferes with stargazing the same as astronomical twilight would. What is x (as a function of the Moon's phase)? We define civil, nautical, and astronomical twilight as when the sun is 0-6, 6-12, and 12-18 degrees below the horizon respectively. This corresponds roughly to what most people call twilight, the ability to distinguish a horizon at sea, and the ability to see 6th magnitude stars at the zenith. However, even when below the horizon, the Moon shines brightly enough to interfere with stargazing. What are the equivalent twilight angles for the Moon? I realize that even the full Moon overhead isn't bright enough for civil twilight, so my real interest is in astronomical twilight. Of course, this will vary greatly with the Moon's phase, and slightly with Moon's distance.


Commutator of scalar fields

So, in the calculation of $ D(t,r) = \left[ \phi(x) , \phi(y) \right] $, where $ t= x^0 - y^0,~ \vec{r} = \vec{x} - \vec{y} $ you need to calculate the following integral $$ D(t,r) = \frac{1}{2\pi^2 r} \int\limits_0^\infty dp \frac{ p \sin(p r) \sin \left[(p^2 + m^2)^{1/2} t \right]} { (p^2 + m^2 )^{1/2}} $$ For $m=0$, the integral is simple. We get $$ D(t,r) = \frac{1}{4\pi r} \left[ \delta(t - r) - \delta(t + r) \right] $$ I even know what the answer for $ m \neq 0 $. I have no idea how to calculate it though. Any help?


Energy realized in fusion and electrons?

I was doing this question about the energy released in a fusion reaction: ![enter image description here](http://i.stack.imgur.com/Cogxc.jpg) In the mark scheme it included the mass of the electrons (for part cii) on the left and just used the mass of protons for the H. Do the H in this case not contain electrons and if so why do we not include them in the calculation but do the once on the left??


How does an electron adjust itself to fit in an excited state that is completely filled?

According to quantum mechanics each state has a specific shape. So, how does the electron get into that shape of the orbital?


decoherence free subspace of a single photon

Take the state vector for a single photon as $\psi = \int \gamma_{\omega} | \omega \rangle \otimes (\alpha |H \rangle + \beta | V \rangle )d \omega$ $H, V, \omega$ are the horizontal polarization, vertical polarization and frequency components of the photon. We have a nice, big tensor product space of states for this photon. In particular, we have entangled single photon states! The entanglement of the single photon comes from entangling the polarization with its own frequency. Is it possible to use this effect to create decoherence free subspaces that would be useful in transmitting information within the polarization state of single photons?


Non-associative operators in Physics

Are non-associative operators (or other kind of elements) used in Physics? For example, in QM I'm looking for something like this: $A(BC)|\psi\rangle \ne (AB)C|\psi\rangle$ NOTE: I think that this question does not make much sense, in that case I will close it.


Gauge-invariance of pole mass using Ward Identity

I am able to explicitly verify to one-loop order that pole masses are independent of the choice of gauge paramter. But how do I use the Ward-Identity/Taylor-Slavnov identity show that the position of the poles in Greens functions are gauge-independent to all orders? The difficulty I'm running into is in using the identities to make statements about the analytic structure of Green's functions.


Can we "safely" assume that quantum computing systems will be finite-dimensional?

This is a common assumption in the study of quantum computation to assume that the quantum systems involved are finite-dimensional, since qubits lives in the two-dimensional Hilbert space. According to Roman Gielerak and Marek Sawerwain, the quantum registers corresponding to a quantum computing machine with coherent pulses of light must have an infinite-dimensional character. I am looking for arguments for and against the hypothesis that finite- dimensional structures are sufficient to study quantum computation.


B physics and CP violation

So I am studying CP violation in SM. Experimentalists are trying to study B meson decays now a days. B meson systems involve quarks from the third generation and hence B physics gives more information than the Kaon system. Is there any other reason?


Zeolite-based oxygen concentrators

I wonder, what is content of output of zeolite-based pressure swing adsorption oxygen concentrators (both oxygen output, and exhaust output)? Yes, they can produce 95% oxygen. But what's the remaining 5%? Is it just argon and other noble gases, or some air contaminants could also be concentrated (like CO2, NO2, CO)? Any references to quantitative gas analysis results would be extremely useful (wasn't able to find any, probably bad google skills).


Is this paragraph on probabilities of sub atomic partials accurate?

I am working on a concept for something and i want to make sure i understand something clearly before i start on everything else. Note, my project is more about the interactions of elements of complex systems rather than physics. Im just using this paragraph as an example, not doing a project on quantum mechanics. Is this paragraph accurate: The universe is built on probabilities. In the whole scope of the universe, absolute mathematics and absolute certainty do not exist. Its incorrect to say 1 + 1 = 2, the more accurate way to say it is 1 + 1 probably equals 2, but not always. It is all based on the probability that what you expect to happen, will. Its quantum probabilities that dictate that a group of sub atomic particles, at this exact moment in time, line up precisely in a certain way that allow an atom to exist. That atom lives in an environment (dictated by other probabilities) that has a calculable probability to grab onto another atom and form an object with mass. That object was hammered into a shape by a craftsman trusting that the probabilities dictating the actions of the subatomic particles allow it to not crack, or disintegrate, or break the tool. That object is now sitting on your desk holding your soft drink in the form of a can. The subatomic particles that are almost insignificantly small form a thread of interaction that leads up to that can sitting on your desk. If that thread breaks at any point, that can ceases to exist as you know it and your soda is running all over your desk. Thanks everyone, im not sure if this falls in the scope of this site, but i can't think of any other place to ask this question.


Casimir effect as an entropic force

When I first learned about the depletion interaction, my initial reaction was that it looks very similar to the Casimir effect. On making this remark to the professor, he replied somewhat mystically: "It **is** the Casimir effect." No further detail was supplied, however. Nevertheless, it really looks like both problems can be understood in terms of degrees of freedom maximising their positional entropy by reducing the volume of some "forbidden region" between two objects. For the depletion interaction, the fluctuating degrees of freedom are small particles, while in the Casimir effect, these degrees of freedom are long-wavelength modes of the free radiation field. However, a key difference is that in the first case the fluctuations arise thermally, while in the second case they are unavoidable quantum fluctuations of the vacuum. > Is it possible to derive the Casimir force between two conducting plates > from entropic considerations alone?


Relation between (super)integrability and closed orbits

Inspired by this recent question, I would like to understand from a more general and mathematical perspective why closed orbits are only found for the Kepler ($V(r) \sim 1/r$) or harmonic ($V(r) \sim r^2$) potential problems, as follows from Bertrand's theorem. There are two aspects that make these problems special, which I suspect may be related to the closed-orbit property. First, both problems are superintegrable. This property sits intuitively well with the idea that phase- space orbits should close "as quickly as possible", thus implying that real- space orbits close after a single revolution. Second, each problem possesses an additional "unexpected" conserved quantity, due to a larger symmetry of the problem than the obvious $O(3)$. For the Kepler problem, this is the Runge- Lenz vector, related to the $O(4)$ symmetry of the Hamiltonian. Meanwhile, the harmonic oscillator Hamiltonian conserves the Fradkin tensor: $$ F_{ij} = \frac{p_i p_j}{m\omega^2} + m\omega^2 q_i q_j, $$ which is related to an $SU(3)$ symmetry. In fact these symmetries and corresponding conserved quantities exist for _any_ central field problem (D. M. Fradkin, Prog. Theor. Phys. **37** (1967), p.798). However the conserved quantities only take a "nice" form for the Kepler and harmonic problems, which also allows the corresponding quantum problems to be diagonalised exactly by symmetry arguments alone. These considerations motivate the following question: > What specific physical/mathematical feature(s) do these two problems share > that gives them the property of closed orbits? Does this feature bear > relevance to the quantum counterpart?


Scalar two loop diagram in $\varphi^4$ theory

Could someone explain how, or at least show me a link that explicitly shows the calculation of a two-loop corrections to scalar’s two-point function in $\varphi^4$ theory in the massless limit.