title: Photon detection time in NMR rotating frame

text: I think of an NMR experiment, but with a single spin half nucleus initially set to the excited state. When the nucleus finally returns to its ground state, it will emit a photon. An observer in the lab frame will see a photon in the Larmor frequency. An observer in a rotating frame, very close to the Larmor frequency, will see a photon with a very very low frequency. My question is: Because of the frequency difference, will it take the detector of the rotating frame observer longer time to tick (i.e., detect the photon) compared to the lab frame observer? Also, and i know it sounds silly, if there is a different, does this somehow relates to time dilution in general relativity?


title: Continuous spectrum (quantum mechanics)

text: Does a continuous spectrum of an observable always imply that the corresponding eigenvectors will not be normalizable? If yes, how to prove it?


title: Central charges c and topological ground state degeneracy GSD

text: A 2+1D topological field theory (topologically ordered states), implies that the topological ground state degeneracy (GSD) on $T^2$ torus (2D manifold **without boundary** ). For example a level k U(1) Chern-Simons theory implies a GSD$=k$. If we put the topological field theory on a 2D manifold **with 1D boundary** , we expect 1+1D gapless edge states; and there are central charges $c$, which roughly measures the degree of freedom of the gapless edge states. My question is: are there some explicit formula relates: **topological ground state degeneracy GSD** and **central charges c**? Say, $$\text{GSD}=\text{GSD}(c,\dots)$$ and $$c=c(\text{GSD},\dots)$$ here $\dots$ are other possible data. RHS are the desired functions of my questions. It will be better to take some examples of **non-Abelian** topological field theory (topologically ordered states) to test its formula's validity.


title: Which green spectral line(s) are emitted in a Thomson tube?

text: Related: What is meant by boiling off electrons in a heater coil? In the Thomson tube we used in our class to produce an electron beam, the lab manual stated that the tube was filled with a low pressure argon gas. When the electrons collided with the argon gas, electrons in the gas molecules were excited and when they de-excite, green light is emitted. However, when one looks at the atomic spectra of argon, there are two green lines, as shown below. ![enter image description here](http://i.stack.imgur.com/kihYy.jpg) Is one able to distinguished between these two green lines in this experiment or am I actually seeing both of these lines? I assume that because the Thomson tube sends in a range of electron velocities, that maybe its both green lines that I am seeing. However, I don't know for sure.


title: Could the universe be a series of Big Bangs?

text: Imagine an eruption of energy/mass $E$ from a singularity $O$, as in a Big Bang. After the energy/mass $E$ is all at more than a distance $d$ from $O$, is it for some value of $d$ possible that there could be a new eruption of energy/mass from $O$, i.e. a new Big Bang? If yes, is there an upper limit to the number of succeeding Big Bangs?


title: Molecular Dynamics (MD) Simulation: energy fluctuations in NVE ensemble

text: I'm writing my first MD simulation (ever) for liquid Argon. The code is up and running. I am supposed to do the calculations in the NVE ensemble. Having implemented a 4th order symplectic integrator (forest-ruth) the total energy of my system (approx 1000 atoms) oscillates (as expected). Now I am "measuring" the heat capacity of my system, which I do by calculating the mean total energy and it's fluctuations for several distinct simulations (using the same parameters). This is exactly the point that I don't get (conceptually). By definition, the total energy of a system is fixed in the microcanonical ensemble. Am I in the microcanonical or in the canonical ensemble when I do an MD simulation? The literature (I have) is somewhat sloppy when it comes to this.


title: Perpendicular Fields and Equations of Motion

text: I have the following problem to solve: > A particle of mass $m$ and charge $e$ moves in the laboratory in crossed, > static, uniform, electric and magnetic fields. $\mathbf{E}$ is parallel to > the $x$-axis; $\mathbf{B}$ is parallel to the $y$-axis. Find the EOM for > $|\mathbf{E}|<|\mathbf{B}|$ and $|\mathbf{B}|<|\mathbf{E}|$. I was planning on using the following: $$\vec{E}'=\gamma(\vec{E}+\vec{\beta}\times\vec{B})-\frac{\gamma^2}{\gamma +1}\vec{\beta}(\vec{\beta}\cdot \vec{E})$$ $$\vec{B}'=\gamma(\vec{B}-\vec{\beta}\times\vec{E})-\frac{\gamma^2}{\gamma +1}\vec{\beta}(\vec{\beta}\cdot \vec{B})$$ along with $$t'=\gamma (t-\vec{\beta}\cdot\vec{x})$$ $$ \vec{x}'=\vec{x}+\frac{(\gamma -1)}{\beta^2}(\vec{\beta}\cdot \vec{x})\vec{\beta}-\gamma \vec{\beta}t$$ with the addition constraints $$\frac{d\vec{p}}{dt}=e\left[ \vec{E}+\frac{\vec{u}}{c}\times \vec{B}\right]$$ and $$\frac{dU}{dt}=e\vec{u}\cdot \vec{E}$$ To solve this I am going to switch to a frame with $$\vec{\beta}=\frac{E}{B}\hat{z}$$for the first case. With this case $dU/dt=0$ and i can solve the equations of motion to find $\vec{x}(t)$ directly, and then boost back to get the trajectories in the original frame. However for the the second case I was wondering if my procedure is correct. I am going to switch to a frame with $\vec{\beta}=(B/E) \hat{z}$ to remove the magnetic field. Now here it seems that $\vec{u}_0$ is perdepdicular to $\vec{E}$ to start in the new frame, but that it will be accelerated in the $x$ direction and hence $dU/dt\neq 0$ and I can't just straightforwardly solve the EOMs. How would I proceed from here? Thanks,


title: Electromagnetic current-current correlators

text: Let the free electromagnetic current $J_\mu(x)$ be = $:\bar{\psi}(x)\gamma_\mu Q \psi(x):$ where $::$ is the normal ordering.   * In this expression why is $Q$ thought of as a "charge operator" instead of just a number?...its quite pesky to keep track of this operator while doing the current-current OPEs though I don't see anything changing conceptually if I just thought of it as a number... After a lot of (I found it very subtle!) calculations one can show that in the light-cone limit $x^2 \rightarrow 0$ the commutator, $[J_\mu(x),J_\nu (0)]$ has one of its terms (say X), $$X = \frac{iTr[Q^2]}{\pi^3}\\{\frac{2}{3}g_{\mu \nu}\delta''(x^2)\epsilon(x_0) + \frac{1}{6}\partial_\mu \partial_\nu [\delta'(x^2)\epsilon(x_0)]\\}$$ Now one wants to compare the contribution of this term to two different situations,   * The total hadronic cross-section,  $\sigma(e^+e^- \rightarrow hadrons) = \frac{8\pi^2\alpha^2}{3(q^2)^2}\int d^4x e^{iq.x}<0|[J_\mu(x),J^\mu(0)]0>$   * The inclusive hadronic tensor in deep inelastic lepton-nucleon scattering,  $W_{\mu \nu}(p,q) = \frac{1}{M} \sum _{\sigma} \int \frac{d^4x}{2\pi} e^{iq.x}<p,\sigma|[J_\mu(x), J_\nu(0)]|p,\sigma>$ In the derivation/argument for the expression for $W_{\mu \nu}$ it is kind of clear that the initial and final states have to be $|p,\sigma>$ - the initial state of the proton.   * But I am unable to pin down as to exactly why the initial and final states in the first case had to be vacuum $|0>$. It would be great if someone can explain this conceptual point about the difference in the initial and final states.    * Hence if someone can explain as to why the term $X$ contributes (and is infact the leading contributor!) to $\sigma$ but does not contribute to $W_{\mu \nu}$!? (..my vague understanding is that this difference stems from the difference in the initial and final states..but can't make this precise..)   * Though initially $W_{\mu \nu}$ is defined in terms of the correlator $[J_\mu(x), J_\nu(0)]$, often I see that during calculations one is in practice evaluating $[J_\mu(\frac{x}{2}), J_\nu(-\frac{x}{2})]$. Why this change?


title: 2D - Kinematics - Linkage System using Vector Algebra

text: I have this question that I dont know how to solve correctly : ![enter image description here](http://i.stack.imgur.com/HsYdc.jpg) My question is, how do I find $V_B$ ? I will find the angular velocities myself, but I want to know the method to get $V_B$ ? I know I can start by using $V_C=V_B+V_{C/B}$, but then I'm not sure what to do next...is the direction of $V_B$ and $V_{C/B}$ the same ? I'm using vector algebra (cross product with i and j etc). More specifically : $$\vec{V}_C=\vec{V}_B+\vec{V}_{C/B}$$ $$\vec{V}_C = -1 \hat{j}$$ $$\vec{V}_B= ?$$ $$\vec{V}_{C/B}= \omega_{CB} \hat{k} \times \vec{r}_{CB}$$ Shouldn't $\vec{V}_B = \omega_{CB} \hat{k} \times \vec{r}_CB$ ??? If not, then what should it be ? If you need more info please let me know.


title: The notion of an adiabatic process in thermodynamics -vs- quantum mechanics

text: I'm confused about the terminology in the two contexts since I can't figure out if they have a similar motivation. Afaik, the definitions state that quantum processes should be very _slow_ to be called adiabatic while adiabatic thermodynamic processes are supposed to be those that don't lose heat. Based on my current intuition, this would mean that the thermodynamic process is typically _fast_ (not leaving enough time for heat transfer). What gives, why the apparent mismatch?


title: Horizontal Gravitational Force

text: So I was writing a lab report for Physics I, where I was describing projectile motion. I had written > Since gravity on Earth does not have a horizontal force, the object will not > lose horizontal velocity (velocity in the x direction). Then this made me think, is there any gravitational force found in nature that has both vertical _and horizontal_ components?


As the title suggests, Can the effects of a person's mass upon the local gravitational field be detected and measured remotely? I am aware any mass produces and effects gravity but couldn't find anything in my searching if it is possible or theoretically possible to detect this effect remotely.


title: What are the values of effective mass approximation

text: For my research work i am trying to calculate band gap of zinc oxide theoratically and found this paper (Determination of the Particle Size Distribution of Quantum Nanocrystals from Absorbance Spectr). What i am doing is I am making a small matlab programm which will reporduce the results using the same values from the above paper. I used the values given in the paper to make the program ![enter image description here](http://i.stack.imgur.com/v8Fbp.jpg) **Absorbtion onset** $\lambda ^{onset}=\frac{c*h}{E^{*}}$                  Values used for calculating Band gap     8.85418782 ×  10^-12          permittivity of free space     1.60217657 ×  10^-19          charge of an electron     9.11 ×  10^-31                free electron mass     1.05457173 × 10^-34               hcut     0.59                      effective mass of holes     0.26                     effective mass of electrons     8.5                      relative permittivity     3.2                     zno bandgap     c                       velocity of light      But after running the programm i got these results               Size    Absorbtion Onset     1   6.21E-26     1.5 6.21E-26     2   6.21E-26     2.5 6.21E-26     3   6.21E-26     3.5 6.21E-26     4   6.21E-26     4.5 6.21E-26     5   6.21E-26     5.5 6.21E-26     6   6.21E-26     6.5 6.21E-26     7   6.21E-26     7.5 6.21E-26     8   6.21E-26     8.5 6.21E-26     9   6.21E-26     9.5 6.21E-26     10  6.21E-26      The results i got doesnt match with tha paper and I belive that my coding in matlab is correct but i am doubtful about the values i have taken for calculating band gap. Can you help me with this


title: Is it physically meaningful to talk about the 'total temperature' of an object?

text: If I had a semi infinite, 1-D object and a finite 1-D object, both heated at the same constant rate at one end each for the same time period and both begin at the same initial temperature, is it physically meaningful for me to integrate along the length of the object and consider this integral as a function of time and a measure of the 'amount' of heat on the object?


title: Does classical physics predict the effects of shining a laser at a hair?

text: The discussion on this webpage mentions that shining a laser beam at a hair produces an effect like that of the double-slit experiment. Does classical physics predict the effect you observe when you do this (since light is a wave)?


title: Is it possible to have a self-balancing system?

text: I am trying to create a machine that moves on two points (Wheels or legs). Because of the extremely difficult nature of perfectly balancing the parts, I am wondering is there any way to create a mechanical mechanism to balance it. I know that it can be done with many different electric circuits, but I am wondering is it even theoretically possible to create such a system? And if it's possible, does anyone know of any systems that do that? P.S. Could someone retag this appropriately? I'm new to this SE, and I'm not sure how to tag it.


title: Why don't massive water-borne animals crush under their own weight when they're in water?

text: If you take a giant whale out of the water and put it on land for long enough, it will crush itself under its own weight. Why doesn't the animal get crushed under its own weight when it's in water?


title: Can the electric field -- always -- be derived from the potential?

text: After studying the definition (& derivation) of the potential to an electric field and the Poisson equation I'm currently wondering whether the following is possible:   * Can one give an example of a physical setup where the Poission equation fails to provide the electric field? I have tried to come up with an example but failed. What I thought of was, that considering a valid solution of the Poisson equation, the electric field derived from the solution would have to be differentiable (since the second derivative of the potential appears in the differential equation). If one were to find a setup with a discontinuous electric field, would the Poisson equation fail?


title: Is it true that the sun will cause very high temperatures on Earth long before the red-giant-phase?

text: I heard at least three claims about the development of the heat of the sun. In an old book, I read, that nothing dramatically will happen in the next few billion years. Wikipedia states, that the average temperature will hit 30° in a billion years, 100° a billion years later, and in an internet forum, someone claimed that in 500 million years, the oceans would cook. Which of this claims is true? How reliable are the claims about events in such far future?


title: Does Light Experience Length Contraction?

text: Lorentz length contractions states that the length of any moving object gets divided by the Lorentz factor equal to the Lorentz factor for that object (always $\geq 1$), equal to $$ \gamma=\frac{1}{\sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } } $$ However, in massless particles $v=c$, so the Lorentz factor becomes $\infty$, meaning that an object traveling at $c$ will have $0$ length. However, photons and obviously all forms of electromagnetic waves move at c when traveling through a vacuum, such as from a space shuttle to a space station or back to Earth. Does this mean that photons have no length? How does this affect wavelength?


title: Complete Energy Equation for Non-Zero Masses

text: The most simplified version of the energy equation (which is also the most known) is $E=mc^2$   However, I understand that this _only_ applies to objects with non-zero mass and zero velocity. I also read that "relativistic mass" is equal to the "rest mass" multiplied by the Lorentz factor: $$ m=\gamma m_0 =\frac { { m }_{ 0 } }{ \sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } } $$ What exactly _IS_ the "complete" equation for objects with non-zero mass that can be applied to objects with any velocity? My knowledge would suggest that the equation is: $$ E=\sqrt { { (\gamma m{ c }^{ 2 }) }^{ 2 }+{ (\gamma pc) }^{ 2 } } $$ where E is energy, $\gamma$ is the Lorentz factor, $m$ is the mass, $c$ is obviously the speed of light, and $p$ is the momentum of the object (defined as $p=mv$ but multiplied by the Lorentz factor because the mass increases with velocity)


title: Magnetic field in materials with non-constant magnetic susceptibility

text: I'm quite lost what $B$ and $H$ is. It seams to me that most of the texts I read do quite poor job in explaining them properly. They are explained only in cases when magnetic susceptibility is constant in the space. First _law_ says that $M = \chi H$, $M$ is magnetization and to my understanding $H$ is something like external magnetic field (for example Earth magnetic field when I do experiment in some lab). Next _law_ says that magnetic field is $B = \mu_0(M+H) = \mu H$. This has to be wrong when $\chi$ is not constant because than we are not guarantied that divergence of $B$ is zero. $$ \nabla \cdot B = \nabla \cdot (\mu H ) \neq \mu \nabla \cdot H = 0 $$ Can you please direct me to some book when this is explained properly? My aim: I want to calculate magnetic field around paramagnetic material. I cannot use formula $B = \mu H$ because outside paramagnet is $\mu = \mu_0$ so magnetic field would be unchanged.


title: Can anything come out from the big bang?

text: 1. If any configuration of matter can fall into a black hole and hit the singularity, and ditto for the big crunch, and there is time reversal CPT invariance, does it mean anything can pop out of the big bang singularity, and there is unpredictability at the big bang?    2. How do we explain the very low entropy of the big bang then?


title: What is discontinuity in Vector Fields

text: I am reading David J. Griffiths and have a problem understanding the concept of discontinuity for E-field. The E-field has apparently to components. (How does he decompose the vector field into the following?) The tangential and the normal component. The normal component is discontinuous? (Question 1: Why?) But the tangential components are _not_ discontinuous? (Question 2: Why?). My suggestion for question 1: I tried drawing it, and the way I see it, the size of the E-field is continuous, but the direction is not a continuous! My suggestion for question 2: The tangential components has a direction that changes continuously. The understanding of the word Continuous, that I am applying is this: If I can take the vector field, imagine it in a $\mathrm{R}^2$ graph, and if it is continuous (not abruptly stopped), then.. That's it! Can anyone please give me the explanation I am missing or another source? Also why does this have anything to do with ANYTHING: $$ {E}_{above} - E_{below} = \frac{\sigma}{\epsilon_0}\hat{n} $$


title: Aharonov-Bohm vs de Witt

text: dewitt claimed in his paper > Bryce S. DeWitt. Quantum theory without electromagnetic potentials, _Phys. > Rev._ **125** no. 6 (1962), pp. 2189-2191, DOI: 10.1103/PhysRev.125.2189, that the discovery of the Aharonov and Bohm that electromagnetic potentials play primary role in quantum mechanical theory is false. Who won? What are the errors in the argument of the losing side in this battle?


title: Mathematical approaches to atmospheric refraction

text: Understanding atmospheric refraction, particularly of ultraviolet, and into the blue part of the visible spectrum is of great interest to me. Although, I have a strong background in trigonometry and geometry, I am very much interested in how these are applied in the context mentioned. Are there any references (books, papers and the like), that are mathematically rigourous, but also have examples of how the mathematics is applied to atmospheric refraction? Are there any other mathematical knowledge that would be required?


title: Help on unit conversion problem

text: This is a problem from school. I will show my attempt. The question: "The gas constant for dry air R is 287 $\frac{m^2}{s^2*K}$. Assuming the temperature is 330 K and the pressure is 1050 hPa, what is the atmospheric density." The professor said DO NOT produce an answer by finding a formula, but to use the magic of unit conversion to try to solve things. I know density is measured in kg/m^3 or thereabouts so I tried the following: 1050 hPA = 105, 000 Pa 1 Pa = 1 kg/m*s^2 105,000 $\frac{kg}{m*s^2}$ * 330 K * 287 $\frac{m^2}{s^2*K}$. This cancels some units... but not enough...in fact it cancels just K, so far as I understand, far from what I need for my density unit. Any ideas on what Im doing foolishly here?


title: What makes Poynting's theorem consistent for a charge moving in a static electric field?

text: Poynting's theorem is given by $$\frac{\partial}{\partial t}\int_{v}Udv \+ \oint_{A}\vec S\cdot \vec {dA} +\int_{v}\vec E\cdot\vec J dv =0 $$ Where,   * the total electromagnetic energy inside the volume v is $U = \frac 1 2 (\vec E\cdot\vec D+\vec B\cdot\vec H)$   * the Poynting vector $\vec S=\vec E\times \vec H$ This equation is interpreted as the conservation of electromagnetic and mechanical energy for a volume of space with each term representing respectively the rate at which   1. electromagnetic energy changes inside the volume   2. electromagnetic energy crosses the boundary of the enclosing surface   3. mechanical work is done on charges inside the volume  Now take the case of a charge accelerating from rest by a static electric field, and initially at the centre of a spherical volume fixed in space with radius cT where c is the speed of light and T the time taken for electromagnetic fields to propagate from the center to the spherical boundary. For 0 < t < T both the magnetic and mechanical energy inside the volume increases without electromagnetic energy crossing the boundary. So during this time, where does the negative term come from to maintain the RHS = 0 in Poynting's theorem?


title: Virtual Higgs boson?

text: Can particles emit a virtual Higgs boson in a similar manner to the way a virtual photon is emitted?


Why did they used to make the mill chimneys so tall? This question was asked in an Engineering Interview at Cambridge University.


title: Relativistic beaming - which power of the Doppler factor

text: The context is that I'm building a simulation of a starfield, as seen from a relativistic spaceship. (EDIT: the simulation can now be found here.) One reference that I'm using is this paper by John M. McKinley (1980) "Relativistic transformation of solid angle": http://adsabs.harvard.edu/abs/1980AmJPh..48..612M http://cartan.e-moka.net/content/download/248/1479/file/Astronave%20relativistica.pdf There are two formulas, one with D^2, and one with D^4 (D being the Doppler factor). The logic stated by the above paper seems to be the following: Use D^2 for:   * comparing two detectors in relative motion   * plane waves   * stars Use D^4 for:   * comparing a moving source of radiation, relative to a fixed source   * extended sources: galaxies, cosmic microwave background, astrophysical jets, and so on Question: is the above breakdown of when to use D^2 versus D^4 correct? Here is the last paragraph of the paper in its entirety: > It is hoped that this presentation can clarify the role of the solid-angle > transformation. Because it is a necessary consequence of aberration, there > is always a change in the solid angle associated with light when one changes > reference frame (except for the case of idealized plane waves). Its effect > effect upon the transformation of intensity must be included in the study of > cosmic microwave background radiation, where the angle is associated with > convergence into the detector of radiation from different parts of the > primordial fireball. It must also be included when a moving source is > compared to an equivalent fixed source, both at the same apparent distance > measured in the reference frame of the detector. In this case the angle > represents divergence from the compact source to the detector. The change in > solid angle does not contribute to the transformation of intensity measured > at the same event point by detectors in relative motion. Note particularly > that the amount of divergence of the light, or equivalently the curvature of > the wavefronts, does not determine which intensity transformation is > correct. Thus, in considering the appearance of stars from a moving > spaceship, a typical source distance is 1-1000 pc but the intensity > transformation is equivalent to that for plane waves. On the other hand, > typical receding galaxies are 10^6 times further away (so their light can is > actually more nearly plane), but their light cannot be treated as plane > waves in finding its intensity transformation.


title: Physics of batteries (volts vs charge)

text: Car batteries are usually 12 V. What is the difference between buying a car battery and hooking up a bunch of cheap household batteries in series? Both would register at 12 V. I assume that cars need much more current to start an engine then regular household things like lamps and toys. Does that mean that a car battery holds more charge within it? If we think of a simple cell battery, where there are two electrodes and an electrolyte. One electrode eventually, through chemical reaction, becomes positive and the other becomes negative. Thus in a car battery, does that mean the electrodes (if we can reduce a car battery to a primitive cell), have more charge separated on each electrode? But if that was the case, wouldn't a greater charge separation mean that the voltage would be greater between the terminals as well? In general, what is the relation between charge and voltage? I know the equation V = U/q, just like E = F/q (similar form, in the limit that the test charge is small as to not effect the PE or electric field. V = U/q is sorta useless too, since differences matter, but nevertheless that is how we defined it). If charges are separated further, does that mean greater voltage? If more charge is separated, does that mean greater voltage? Lastly, what is the difference b/w the charge in a 12 V car battery, and the charge in a 12 V makeshift, series strung battery from home?


title: The universe's lack of an 'edge', and how that relates to the multiverse?

text: So, we're fairly sure that the universe is infinitely large. We're also fairly sure that it is expanding. My question is how do these two facts relate to the multiverse? My favorite interpretation was given by Michio Kaku, who stated that in the multiverse each individual universe is like a bubble, and all of these bubbles are floating around in some space (and sometimes interact?). How can there be an infinitely large object in a space with other presumably infinitely large objects? Is this in like math where there are different "sizes" of infinities, where some are larger than others? What is the space that the universe is expanding into? Does that have a boundary, or is it also some infinity?


title: How does the movement of electrons produce radio waves?

text: I'm mostly wondering about radio frequencies. I understand that voltage is the movement of electrons, and that the antenna acts as a light bulb, emitting at radio frequencies, following the reverse square law, some materials are opaque, some are transparent. Yet, at the receiver end, it's almost the same as having the two antennas connected, except with a voltage drop. Are photons and electrons the same thing? (It is called the electromagnetic spectrum). It's obviously not quite the same as electrons moving through the air, as this creates lightning, or something similar. Is anything I've said incorrect? What happens when an electron goes through an antenna?


title: What decides fission products?

text: I am learning about nuclear fissions and learned about the fission fragment distribution. It was interesting to see that the fission fragments have unequal masses. I was wondering as to what governs which fission products are generated in each fission? What parameters decide which fission products will have higher yield?


title: What frequency of sound waves produces the most vibration

text: The name of the question is rather contradictory and counter-intuitive since sound is _produced_ by vibration. However, very low frequencies around 32Hz and receding are bass. From what I have read online, you can feel frequencies below 20Hz which is what we normally hear, yet you can hear as low as 12Hz while amplifying very high frequencies (above 20k Hz? I'm not an expert). So again, which frequency(ies) should I amplify in my equalizer to achieve this effect?


title: Total energy of the Universe

text: In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, "because the positive energy of matter is cancelled out by the negative energy of the gravitational field". But I can't find anything concrete to substantiate this claim. As a first check, I did a calculation to compute the gravitational potential energy of a sphere of uniform density of radius $R$ using Newton's Laws and threw in $E=mc^2$ for energy of the sphere, and it was by no means obvious that the answer is zero! So, my questions:   1. What is the basis for the claim - does one require General Relativity, or can one get it from Newtonian gravity?   2. What conditions do you require in the model, in order for this to work?   3. Could someone please refer me to a good paper about this?


Why is gravitational potential energy negative? How is it different from other forms of energy? I recently saw a video by Dr. Michio Kaku, he said that the total energy content of the universe is zero, since gravitational potential energy is negative, thus balancing all the positive energy. Can anyone help me out here?


title: Total Energy of the Universe?

text: I've heard the total energy is zero, but I've also heard it cannot be said to be zero since there's so much unknown stuff in the universe. Is that true?


title: The Univere's mass-energy and uncertainty

text: A virtual creation with total mass-energy = $E$ is allowed as long as that virtual creation doesn’t last longer than $E/h$. Can the uncertainty principle also be used to estimate the mass-energy in the spontaneous creation of a universe - a spontaneous creation that has now lasted $13.6$ billion years? If so, the principle would require that universe to have a mass-energy less than $1.54\times 10^{-51}$ Joules. Is there a flaw in this?


title: Why is gravitational potential energy negative, and what does that mean?

text: I usually think of gravitational potential energy as representing just what it sounds like: the energy that we could potentially gain, using gravity. However, the equation for it (derived by integrating Newton's law of gravitational force)... $PE_1 = -\frac{GMm}{r}$ ..has me thrown for a loop, especially after this answer.   * If potential energy really meant what I thought it did, then it would always have to be non-negative... but this equation is _always_ negative. So what does "negative potential energy" mean!?   * If $KE + PE$ is always a constant, but PE is not only negative but becomes **more** negative as the particles attract, doesn't that mean the kinetic energy will become arbitrarily large? Shouldn't this mean all particles increase to infinite KE before a collision?   * If we are near the surface of the earth, we can estimate PE as $PE_2 = mgh$ by treating Earth as a flat gravitational plane. However, `h` in this equation plays exactly the same role as `r` in the first equation, doesn't it?      * So why is $PE_1$ negative while $PE_2$ is positive? Why does one increase with `h` while the other increases inversely with `r`?     * Do they both represent the same "form" of energy? Since $PE_2$ is just an approximation of $PE_1$, we should get nearly the same answer using either equation, if we were near Earth's surface and knew our distance to its center-of-mass. However, the two equations give _completely_ different answers! What gives!? Can anyone help clear up my confusion?


title: What is the total observational energy of the universe?

text: Based on what is measurable, how much energy is in the universe if the initial time is the big bang and the final time is the present?


Is the total mass of the universe constant in time?


title: total amount of energy in the universe

text: What is the total amount of energy in the universe? is there the same amount of energy as negative energy, cancelling each other out? Or is it something different?


title: Is the principle of Conservation of Energy empirically verifiable?

text: Before I am inundated by myriad and vociferous claims that conservation of energy is the single most well-attested and experimentally verified principle in all of science, let me say that I am well aware of the ever-growing body of evidence which seemingly bears this principle out. However there seems to be, in my view at least, an abiding problem inherent to a principle of conservation with respect to how we could ever truly verify such a thing empirically. That no humanly constructed device can be perfectly accurate or precise is an incontrovertible and obvious truth. As a result, I am unsure as to how we could ever be sure that energy (or any other conserved quantity) is not being lost or gained (in defiant violation of physical principles) at length and/or time scales orders of magnitude below the resolution of our best instruments. I mean, how could we ever know? Ontologically speaking, it's almost inconceivable that energy could not be conserved. But epistemologically speaking, I am at a loss as to how we might ever verify—via experimentation or measurement—that energy is in fact being conserved to the precision guaranteed by a conservation law.


title: Reduced density matrix of coupled oscillators and entanglement entropy

text: This paper describes a way to find the entanglement entropy of $N$ entangled harmonic oscillators, after tracing out the first $n$. A few statement made within have royally confused me, and I haven't been able to find a solution. It starts at equation (10): I understand that $\Omega$ has to be a symmetric matrix with positive eigenvalues. I suppose it's useful to split it into parts, based on which oscillators are traced over and which aren't. I think this is the reason for the split into $A$ and $C$. Is this all that goes into the construction of $\Omega$, or am I missing something? Then equation (11): comparison with equation (3) shows a lot of similarities, with $x$ and $x'$ now vectors and $\beta$ and $\gamma$ now matrices. However, I do not understand by what method the integrals are 'carried out explicitly', or how the matrices $\beta$ and $\gamma$ are constructed from the entries of $\Omega$. At the top of page four, the author writes: > To ﬁnd [the eigenvalues], we note that the appropriate generalization of Eq. > (4) implies that (det $G$) $\rho_{out}(Gx, Gx')$ has the same eigenvalues as > $\rho_{out}(x, x')$, where $G$ is any nonsingular matrix. Equation (4) is the eigenvalue equation for the case of two oscillators: $$\int_{-\infty}^{+\infty} dx' \rho_{out}(x, x') f_n(x') = p_n f_n(x)$$ where each possible state is indexed by $n$. I have no clue what this appropriate generalisation is, nor how the quoted statement follows. I first thought it was simply subsituting $x$ and $x'$ for their $N-n$ vectors, but I can't even begin to visualise what $\rho_{out}$ looks like if that's the case. I have interpreted all the matrix transforms between here and equation (13) as just a way to eventually write the new reduced density matrix (of $N-n$ oscillators) as a product of reduced density matrices of 2 oscillators. If that is the case, then I understand this part. My last question relates to the block of text directly beneath equation (13), specifically the entropy. The statement is that since the new density matrix can be written as a product, the entropy is simply the sum of the entropies of its parts. I've heard this statement before, but I run into trouble when I try to derive it. The statement is that $$S = \sum_i S(\xi_i)$$ So lets say we look at the simple case where the total reduced density matrix $\rho$ is just the product of two reduced density matrices $\rho_1$ and $\rho_2$. $$S = Tr[\rho \; log(\rho)] = Tr[\rho_1 \rho_2 \; log(\rho_1 \rho_2)] = Tr[\rho_1 \rho_2 \; log(\rho_1) + \rho_1 \rho_2 \; log(\rho_2)] \neq Tr[\rho_1 \; log(\rho_1)] + Tr[\rho_2 \; log(\rho_2)] = S_1 + S_2$$ Is this incorrect? I realise this is a lot of work to answer, since you'd probably need to read the paper as well. Hence, I would be grateful for an answer to any of my questions.


title: How is the theory of partial coherent light related to quantum-mechanics?

text: ## Background Let me start this question by a long introduction, because I assume that only few readers will be familiar with the theory of partial coherent light and concepts like a mutual coherence function or a mutual intensity. The coherency matrix and Stokes parameters descriptions of partially polarized light are related concepts which are more widely known. Correct treatment of partial coherent light is important for an appropriate modeling of optical pattern transfer in computer simulations of proximity and projection lithography as currently used by the semiconductor manufacturing industry. When I came to this industry, my previous optics "training" was insufficient in this area. I found chapter X "Interference and diffraction with partially coherent light" in _Principles of Optics_ by Max Born and Emil Wolf most helpful for filling my gaps in this area. Later, I also "browsed" through "Statistical Optics" by Joseph W. Goodman, which has a nice paragraph in the introduction explaining why insufficient familiarity with statistical optics is so common: > Surely the preferred way to solve a problem must be the deterministic way, > with statistics entering only as a sign of our own weakness or limitations. > Partially as a consequence of this viewpoint, the subject of statistical > optics is usually left for the more advanced students, particularly those > with a mathematical flair. The interesting thing is that Hermitian matrices and eigenvalue decompositions like the Karhunen-Loève expansion are used quite routinely in this field, and they somehow feel quite similar to modeling of coherence and decoherence in quantum-mechanics. I know that there are important obvious (physical) difference between the two fields, but my actual question is what they have in common. ## Question Some elementary experiments like the double slit experiment are often used to illustrate the particle wave duality of light. However, the theory of partially coherent light is completely sufficient to describe and predict the outcome of these experiments. There are no particles at all in the theory of partially coherent light, only waves, statistics and uncertainty. The global phase is an unobservable parameter in both theories, but the amplitude of a wave function is only important for the theory of partial coherent light and is commonly normalized away in quantum-mechanics. This leads to a crucial difference with respect to the possible transformations treated by the respective theories. But is this really a fundamental difference, or just a difference in the common practices of the respective theories? How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes? * * * ## More information on what I would actually like to learn One reason for this question is to find out how much familiarity with partial coherence can be assumed when asking questions here. Therefore it explains why this familiarity cannot be taken for granted, and is written in a style to allow quite general answers. However, it also contains specific questions, indicated by question marks: >   * How is the theory of partial coherent light related to quantum- > mechanics? >   * ... the amplitude of a wave function ... But is this really a > fundamental difference, or just a difference in the common practices of the > respective theories? >   * How much of the strange phenomena of quantum-mechanics can be explained > by the theory of partial coherent light alone, without any reference to > particles or measurement processes? > Don't be distracted by my remark about the double slit experiment. Using it to illustrate the particle wave duality of light seemed kind of cheating to me long before I had to cope with partial coherence. I could effortlessly predict the outcome of all these supposedly counter-intuitive experiments without even being familiar with the formalism of quantum-mechanics. Still, the outcome of these experiments is predicted correctly by quantum-mechanics, and independently by the theory of partial coherent light. So these two theories do share some common parts. An interesting aspect of the theory of partial coherent light is that things like the mutual intensity or the Stokes parameters can in principle be observed. A simple analogy to the density matrix in quantum-mechanics is the coherency matrix description of is partial polarization. It can be computed in terms of the Stokes parameters $$J=\begin{bmatrix} E(u_{x}u_{x}^{\ast})&E(u_{x}u_{y}^{\ast})\\\ E(u_{y}u_{x}^{\ast})&E(u_{y}u_{y}^{\ast}) \end{bmatrix}=\frac12\begin{bmatrix} S_0+S_1&S_2+iS_3\\\ S_2-iS_3&S_0-S_1 \end{bmatrix} $$ and hence can in principle be observed. But can the density matrix in quantum-mechanics in principle be observed? Well, the measurement process of the Stokes parameters can be described by the following Hermitian matrices $\hat{S}_0=\begin{bmatrix}1&0\\\0&1\end{bmatrix}$, $\hat{S}_1=\begin{bmatrix}1&0\\\0&-1\end{bmatrix}$, $\hat{S}_2=\begin{bmatrix}0&1\\\1&0\end{bmatrix}$ and $\hat{S}_3=\begin{bmatrix}0&i\\\\-i&0\end{bmatrix}$. Only $\hat{S}_0$ commutes with all other Hermitian matrices, which somehow means that each individual part of the density matrix can be observed in isolation, but the entire density matrix itself is not observable. But we don't measure all Stokes parameters simultaneous either, or at least that's not what we mean when we say that the Stokes parameters can be measured in principle. Also note the relation of the fact that $\hat{S}_0$ commutes with all other Hermitian matrices and the fact that the amplitude of a wave function is commonly normalized away in quantum-mechanics. But the related question is really a serious question for me, because the Mueller calculus for Stokes parameters allows (slightly unintuitive) transformations which seem to be ruled out for quantum-mechanics.


title: Physical properties of cigarette smoke

text: I've been trying to find papers that list various properties of idealized cigarette smoke( ex. the molar mass, gas compressibility factor, density of smoke exiting a cigarette) but I have had very little luck. Does anyone know what these properties are or where I should be able to find them? I'm building a volumetric measurement tool for use in psychological studies in case anyone wonders why I'm looking for this information. I posted this question in chemistry.stackexchange first but I think physics is a better forum for it.


title: Physics classics

text: I would like some help to find good and detailed books on the history of physics. Which are the classics in this domain? Which are your favorite?


title: How is capacitance of a cylindrical capacitor calculated?

text: I've just begun learning capacitance, and my lecture notes have a section on calculating capacitance for capacitors in vacuum of various shapes, e.g. two parallel plates and concentric spherical shells. ![the cylindrical capacitor](http://i.stack.imgur.com/0s7wd.png) For a cylindrical capacitor, comprising of a _long cylindrical conductor_ with radius $r_a$ and linear charge density $+\lambda$, and a _coaxial cylindrical conducting shell_ with radius $r_b$ and linear charge density $-\lambda$, to calculate its _capacitance per unit length_ , the notes lay out the steps as follows:   * A point outside a long line of charge a distance r from axis has potential $V = \frac{\lambda}{2\pi\epsilon_0}ln\frac{r_0}{r}(1)$. Holds here also because charge on outer cylinder doesn't contribute to field between cylinders.   * Then $V_{ab} = V_a - V_b = \frac{\lambda}{2\pi\epsilon_0}ln\frac{r_b}{r_a}$   * Total charge $Q = \lambda L$ in a length $L$, so $C = \frac{Q}{V_{ab}} = \frac{2\pi\epsilon_0L}{ln(r_b/r_a)}$   * Therefore $\frac{C}{L} = \frac{2\pi\epsilon_0}{ln(r_b/r_a)}$ There are a couple of things in here which I don't understand:   * The first bullet mentions "charge on outer cylinder doesn't contribute to field between cylinders". Is this true? Shouldn't the electric field strength double since there is both a positive charge $+\lambda$ and a negative charge $-\lambda$?   * The second bullet assigns $r_b$ as $r_0$ and $r_a$ as $r$ in equation $(1)$. Why is this so? As far as I understand, $r_0$ denotes the arbitrary distance where $V_b = 0$, which suggests that the potential on the coaxial shell is 0. Why is this not, say, the other way around? Why don't we take the potential at the edge of the inner cylinder as 0? Would that cause $r_b$ and $r_a$ to switch places in the fraction? Why is it possible to assign the potential on the coaxial shell as 0 in the first place?


title: Why does Coulomb's constant have units?

text: I think of Coulomb's constant as a conversion factor (not sure if this is correct). Kind of like how you would do calculations in kg and then times it by the conversion constant to convert your answer to pounds. The conversion factor would be $2.2\: \mathrm{lbs/kg}$. Since the units for Coulomb's constant is $\mathrm{N \cdot m^2/C^2}$, would it make sense to define the Newton as: $1\:\text{Newton} = \frac{1}{1/1\: \mathrm{meter^2} \cdot 1\: \mathrm{Coulomb^2}}$ Would the above definition be valid? **EDIT:** So if $k$ is not a conversion factor since the above definition for a Newton is invalid and $k$ is not just a scaling factor, since it has units, then what is it? If its just a proportionality constant to adjust the magnitude then why does it have units? Shouldn't it be a unit less constant? **EDIT:** So $k$ is not just a scaling factor (since it has units) and its not a conversion factor since a Newton can't be expressed as the other units. So if its unit just exists so that things cancel out "nicely" doesn't this make dimensional analysis useless since you can add in random constants and units to cancel out whatever you want? **My question is not about the meaning of $k$. Its about its units.**


title: Approximating evolution as occurring in a two-dimensional subspace

text: Suppose you have a quantum system with a Hamiltonian having some number (greater than 2, possibly infinite) of eigenfunctions, and that the system is prepared in the ground state. When can you approximate it as by two-level system (using just the ground state and first excited state)? Is there some property that will make it better approximated by a two-level system (e.g., something like a bigger energy gap between the second and third energy levels)?


title: Calculating semi-major axis of binary stars from velocity, position and mass

text: I'm trying to calculate the 'instantaneous' semi-major axis of a binary system with two equal (known) mass stars for an $N$-body simulation. I know their velocities and positions at a given time, but am unsure how best to calculate the semi-major axis. I tried using the vis-viva equation $$v^2 ~=~ G(M_1 + M_2)\cdot(2/r - 1/a) $$ However I'm not sure if $r$ should be the distance between the two masses or the distance to the centre of mass and whether the $M$'s should be the masses or the sum and the reduced mass? Whatever I try the semi-major axis seems to oscillate over the orbit at around the correct value, but surely it should be constant? Any advice would be brilliant, thanks. edit: Not allowed to answer my own question yet, but I'm an idiot and wasn't doing relative velocities, thanks guys.


title: Autocorrelation Functions <---> Pair Correlation Functions

text: Are there any ways to convert an autocorrelation function to a pair correlation function, and vice versa?


title: How does pressurized gas constantly push?

text: If a gas, such as hydrogen, is pressurized into an air tight container, a force in terms of pascals (or whatever unit you want to use) is exerted, correct? That is what pushes against every surface within the container. But what I don't understand is how the gas can constantly push against the walls without being supplied more energy. Does the force of pressure not need energy, or am I missing something? What about when the force is used to move something, such as in a hydraulics system?


title: Program for radiation and toxic hazards

text: I worked in my masters thesis with $^{87}Rb$ and $^{40}K$, really small beta emitters. But there are so many other things around in the lab, that I want to keep track on all the things I might get in contact with. Is there any computer program to calculate the dose of the whole decay chain to get a picture of the artificial radiation and supports logging. I don't want to look up all the individual numbers and calculate it manually. Also in my apparatus various clusters (Cr, Ni, Co, Cu, Ag, Pd, Ca....) are produced. E.g. I know chromium(VI) is carcinogen, but the pure metal is not. But in the nanoworld things may change. So is there a database around where I can lookup the toxity of various substances, with an emphasize on nanoparticles? I use nitrile rubber gloves and try to do not inhale something if I clean the apparatus. But this might be not enough precaution. The laser dyes are not healthy too.


title: independence of the bare parameters on μ for beta function

text: So I know re-normalization has bean "beaten to death". I want to understand something a bit specific which might seem trivial. Independence of the bare parameters on $\mu$ and relevance to the beta function derivation. This seems to be an important assumption or rather fact that is built upon in making further renormalization and renormalization group arguments. In fact, even explicit statements about locality trace to this assumption. Again there is a chance this is an extremely lousy question, but can someone give me a precise mathematical argument describing this independence. I am also looking to put together some reading lists comprising of the original papers treating the subject at a pedagogic level($\phi ^{\text{something}}$ theories). Not those directly addressing the QED problems yet still assuming ignorance of the reasons behind the divergences. Papers making explicit and hopefully proven statements accounting for some of the assumptions taken. I want to know the deal with independence of the bare parameters.


title: Electromagnetism Permittivity simple question

text: Just a simple question regarding $\epsilon$, the absolute permittivity. I read that it measures the resistance to of certain medium to 'permit' the formation of an electrical field. Does this mean that given a electrical field $E$, one needs more energy to produce it in a medium with a higher $\epsilon$ ? Thanks in advance.


title: Why do atoms stick together?

text: The atoms in my table "stick together" to form a rectangle. Why? What makes them stick together? I know about ionic/covalent bonding etc., but consider a sheet of pure iron. Just atoms of one element. The atoms still stick together to form a sheet instead of being "everywhere".But if I place ten balls on the ground beside each other they are completely independent of each other. So what "connects" the atoms?


title: Objects made up of electrons?

text: Say you have a neutral rod, and you bring a positively charged rod beside it (call the side the charged rod is brought near side A and the other side side B). The electrons from the side B will start moving towards side A and the positively charged nuclei in side A will start moving to side B. After a large part of side A consists of only of electrons, the electrons would start repelling each other and the movement of charges will stop. But at this point a very large portion of side A consists of electrons and a large part of side B consists only of positively charged nuclei. The system is at equilibrium. But the rod has now been mostly split into a electron side and a positively charged nuclei side, however it doesn't look different at all? Why? Shouldn't a rod made of atoms look and be completely different than a rod that is mostly made up of electrons on one side and nuclei on the other? I know that electrons are very small, and since side A mostly consists of electrons, shouldn't it be almost invisible?


title: Interpretation of Stiffness Matrix and Mass Matrix in Finite Element Method

text: I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in FEM. For instance if we are solving a linear elasticity problem and we modelize the relation between a node $i$ and a node $j$ as a spring system, then $K_{i,j}$ (where $K$ is the stiffness matrix of the system) can be seen as the stiffness constant of the virtual spring between the two nodes. But does there exist a more general interpretation? Perhaps in terms of internal work? Another similar question is: What could be an interpretation of the coefficients $M_{i,j}$ of the mass matrix $M$?


title: Does Kaluza-Klein Theory Require an Additional Scalar Field?

text: I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2.   1. In one, there is a constant as well as an additional scalar field introduced: $$\tilde{g}_{AB}=\begin{pmatrix} g_{\mu \nu}+k_1^2\phi^2 A_\mu A_\nu & k_1\phi^2 A_\mu \\\ k_1\phi^2 A_\nu & \phi^2 \end{pmatrix}.$$   2. In the other, only a constant is introduced: $$\tilde{g}_{AB}=\begin{pmatrix} g_{\mu \nu}+k_2A_\mu A_\nu & k_2A_\mu \\\ k_2A_\nu & k_2 \end{pmatrix}.$$ Doesn't the second take care of any problems associated with an unobserved scalar field? Or is there some reason why the first is preferred? References:   1. William O. Straub, _Kaluza-Klein Theory,_ Lecture notes, Pasadena, California, 2008. The pdf file is available here.   2. Victor I. Piercey, _Kaluza-Klein Gravity,_ Lecture notes for PHYS 569, University of Arizona, 2008. The pdf file is available here.


title: From Manifold to Manifold?

text: Tensor equations are supposed to stay invariant in _form_ wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor equations is consistent with the fact that the individual components of the tensor may change on passing from one frame to another.[Incidentally,Preservation of the metric implies preservation of norm, angles etc.] But in General Relativity the tensor equations (examples: the geodesic equation, Maxwell's equations in covariant form) are considered to be invariant in form when we pass from one manifold to another. The metric is not preserved in such situations. Preservation of the value of the line element is consistent with the fact that $g_{\mu\nu}$ may be considered as a covariant tensor of second rank: $ds'^2=ds^2$ ${=>}g'_{\mu\nu}dx'^{\mu}dx'^{\nu}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}$ ${=>}g'_{\mu\nu}=g_{\alpha\beta}\frac{dx^{\alpha}}{dx'^{\mu}}\frac{dx^{\beta}}{dx'^{\nu}}$ Rigorous Calculations: $ds'^2=g'_{\mu\nu}dx'^\alpha dx'^\beta$ $=g'_{\mu\nu}\frac{\partial x'^\mu}{\partial x^\alpha}{d x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}{d x^\beta}$ $=g'_{\mu\nu}\frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}{d x^\alpha}{d x^\beta}$ $=>g_{\alpha\beta}=\frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}g'_{\mu\nu}$ Therefore $g_{\mu\nu}$ is a covariant tensor of rank two. But in the above proof we have assumed the value of $ds^2$ as invariant wrt to our transformation.This not true when different types of manifolds are in consideration. Non-conservation of the value of $ds^2$ will result in dismissing $g_{\mu\nu}$ as a second rank tensor of covariant type. This will be the situation if we pass from one manifold to another.*It is important to emphasize the fact that the problem will remain even if when we pass from an arbitrary manifold to flat spacetime in particular to the local inertial frame.*Differential considerations are not improving matters as indicated in the above calculation. The very concept of a tensor gets upset by considering different/distinct manifold. What is the mathematical foundation for the invariance of form of the tensor equations in such applications where we consider different/distinct manifolds?


title: How does a geodesic equation on an n-manifold deal with singularities?

text: My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds and then find applications to theoretical physics by:   * Minimalizing the change in the fundamental shape of the vertices (I want a convex polytope to stay a convex polytope).   * Describing the dynamics of the transformation. (Possible applications to theoretical physics)   * Finding constraint conditions on what can and cannot occur when the space in which the transformation is occurring is not "regular". This includes not-everywhere-differentiable manifolds, singularities, and other pathological cases. Well, what an ambitious project. But in any case I have made some progress for the n-dimensional case with n-vertices where the space is "regular" and I haven't introduced advanced mathematics yet. I'm mostly playing around in configuration space for vertices in euclidean space or something similar. I have defined a functional in configuration space that gives the length of all possible paths in configuration space between two distinct sets of vertices in $\mathbb{R^n}$ as follows: $$T = \int\limits_{\lambda_{1}}^{\lambda_{2}} \sqrt{\sum_{I=1}^{n} \sum_{i=1}^{d} \left(\frac{d}{d\lambda}\left(\sum_{j=1}^{d} s(\lambda)R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda))\right)\right)^2} d\lambda$$ Where $\lambda_{1}$ and $\lambda_{2}$ correspond to the start-point and end- point of the transformation. The members of the gauge group are $s$ (dilatation), $R$ (rotation), and $a$ (translation). $q_{I}^{j}$ corresponds to a point in configuration space representing $I$ vertices and it lives in $j$ dimensions. ($i$ is also a dimension index) You can think of the equation as finding the straight line between two points (in configuration space) given all of the varied paths when you solve for constraint conditions in the Euler-Lagrange equation (recall): $$\frac{\partial{f}}{\partial{x}} - \frac{d}{d \lambda} \left(\frac{\partial{f}}{\partial{\dot{x}}}\right) = 0$$ This is represented in the figure below when the space has no curvature. ![Transformative Norm Equation Figure](http://i.stack.imgur.com/JFxCH.png) Hopefully, you have a general idea about what I'm talking about, this is at about page 5 in the paper where this is introduced, there is a lot of background that I'm not going to discuss. I've constructed an explicit example with tetrahedra where a tetrahedron is approximated to another tetrahedron in a minimalized manner. Here's a figure I made: ![Tetrahedral Approximation](http://i.stack.imgur.com/xvn4u.png) that may help you understand what I'm trying to work with... in any case I'll ask my questions now. - _[Optional for the intrigued reader]_ : How can I begin to generalize this geodesic equation so that it can comment on spaces with different metrics, or irregular behaviour? As an example of what I am trying to reach for, I would be interested in seeing how a set of vertices could geodesically traverse a singularity in a smooth 2-manifold, and if I could define a general functional for any behaviour on any manifold (I would attempt to restrict it to 2-dimensions to start) eventually. There are two main types of singularities I would be interested in investigating, the first would be the "black hole type" in which a diffeomorphism in the manifold causes "a hole at the bottom". This means that there would be an exponential increase in curvature around the vicinity of the singularity, and I want to find out how a set of vertices deals with those imposed conditions. The other would be for a 2-manifold with no curvature and I know that for a particular configuration of vertices, with the geodesic (based on the equation above) lying along a certain line. If I create a hole (simply remove a point in the same way you can create a hole in a single-variable piecewise defined real function), what happens to the geodesic? How can I encode this information, so it either pushes right through the hole somehow, or defines a new path around it? - **Main Question:** For a functional (assuming I could derive one) that gives the path from a distinct set of vertices to another in an n-dimensional smooth manifold with shape preserving characteristics. When a singularity is introduced into the manifold, will the geodesic equation:   * Somehow push through the singularity/ignore it is present.   * Define an alternate path around the singularity (this is what I hope, so I can investigate further with applications to black hole dynamics)   * Become undefined and nothing useful results


title: Fourier Methods in General Relativity

text: I am looking for some references which discuss Fourier transform methods in GR. Specifically supposing you have a metric $g_{\mu \nu}(x)$ and its Fourier transform $\tilde{g}_{\mu \nu}(k)$, what does this tell you about the Fourier transform of the inverse metric $\tilde{g}^{\mu \nu}(k)$ or the Riemann tensor $\tilde{R}^{\mu}{}_{\nu \rho \sigma}(k)$. There are some obvious identities you can derive and I am looking for a references which discusses these and says if they are useful or not. An example of what I mean is the following identity: $g^{\mu \alpha}(x)g_{\alpha \nu}(x) = \delta^{\mu}_{\nu} \implies (\tilde{g}^{\mu \alpha} \ast \tilde{g}_{\alpha \nu})(k) = \delta^{\mu}_{\nu} \delta^{4}(k)$. (This can be made sensible on a compact manifold or for metrics which are asymptotically flat etc.)


title: Instantons, anomalies, and 1-loop effects

text: A symmetry is anomalous when the path-integral measure does not respect it. One way this manifests itself is in the inability to regularize certain diagrams containing fermion loops in a way compatible with the symmetry. Specifically, it seems that the effect is completely determined by studying 1-loop diagrams. Can someone give a heuristic explanation as to why this is the case? And is there a more rigorous derivation that "I just can't find any good way to regularize this thing."? An alternative approach, due to to Fujikawa, is to study the path integral of the fermions in an instanton background. Then one sees that the zero modes are not balanced with respect to their transformation under the symmetries, leading to an anomalous transformation of the measure under this symmetry. Specifically, the violation is proportional to the instanton number, and thus one finds the non-conservation of the current is proportional to the instanton density. This is also found by the perturbative method above. My question, which is a little heuristic, is how is it that the effect seems perturbative (and exact at 1-loop) on the one hand, and yet related to instantons, which are non-perturbative, on the other?


title: What is the state-of-the-art on spacelike singularities in string theory?

text: What lessons do we have from string theory regarding the fate of singularities in general relativity? What happens to black hole singularities? What happens to cosmological singularities? Which points of view on string theory yielded results in this respect? String field theory? AdS/CFT? Matrix theory? I suppose perturbative string theory is not applicable in the vicinity of singularities.


title: Paradox Regarding Time Dilation at the Speed of Light

text: Consider the following scenario: On earth, I pulse a laser focused at a far away mirror such that the time it takes for the light to reflect off the mirror and arrive back at me is at least a few seconds. Immediately after the laser pulse, I place a second mirror where the reflected beam will arrive. In the photon's frame, it would appear that the universe is moving around passed the photon at the speed of light. Furthermore, time on the moving universe would be infinitely dilated. Thus, the photon would never see the second mirror being raised. So how does it reflect off of it? * * * EDIT: To be clear, I don't think moving frame velocity being equal to the speed of light here is necessary. I've written a similar example to my question below: Consider an electron accelerated to 0.9c and then released in some arbitrary direction. After some amount of time, let the electron encounter an infinitely strong potential such that it reflects back to its release point. Meanwhile, an experimentalist calculates when they expect the electron to arrive and turns on an electric field right before it arrives that will curve the electron's trajectory so that it collides into a detector. In the electron's frame, the universe's time is dilated so that it never sees the electric field being turned on. Thus, how does it interact with the field and get detected?


title: Photons and phonons

text: A few months ago I asked about phonons. I got some very good answers but I still have difficulty getting an intuition for phonons, while somehow photons, which in many ways are similar and which I realize I hardly understand anything about, seem more accessible to intuition. In Ron Maimon's answer to the question "What exactly is a quantum of light?" he asserts that > A quantum of light of wavelength λ is the minimum amount of energy which can > be stored in an electromagnetic wave at that wavelength and > the classical wave is a superposition of a large number of photons Translating this to vibrations in a crystal lattice, could we say that a phonon is the minimal amount of energy which can be stored in an lattice vibration in a given mode and that a classical vibration is a superposition of a large number of phonons? I hope I am correct when I say that the electromagnetic field can interact with matter through the absorption of a photon, and it is this interaction that makes the photon into something particle-like. Do we have the same for phonon-interactions? I.e. that when a crystal vibration interacts with matter it does so by the creation/destruction of whole phonons at a time, which may also get absorbed at more or less precise locations, e.g. the energy of a single phonon is absorbed by a localized electron. Finally I would like to understand how phonon exchange can effectively establish an attractive force between electrons, but I cannot say I have any intuition for how photons mediate the electromagnetic force either. I am afraid that for the moment this is beyond the scope of my background.


title: States versus ensembles in quantum mechanics

text: In quantum mechanics, we talk about (1) vectors, (2) states, and (3) ensembles (e.g., a beam in a particle accelerator). Suppose we want to translate this into mathematical definitions. If I'd never heard of the von Neumann density matrix, I'd approach this problem as follows. Two vectors can represent the same state if they differ only by a phase, so we should define states as equivalence classes of vectors that differ by a phase. However, I would not see any reason to go a second step and define a further level of equivalence- classing, in which a hydrogen atom in its ground state is considered to be equivalent to a beam of hydrogen atoms in their ground states. Von Neumann is obviously a lot smarter than I am, and his notion of a density matrix appears to be universally accepted as the right way to describe a state. We use the same density matrix to describe one hydrogen atom or a beam of them. Can anyone offer any insight into why there seems to be no useful notion of state that works the way I'd have thought, rather than the way von Neumann did it? Does it matter whether we're talking about classical QM or QFT? Do we not want to distinguish states from ensembles because in QFT particles can be created and annihilated, so fixing the particle number is not really what we want to do in defining the notion of a state? Related: http://mathoverflow.net/q/117125/21349


title: Why does nonlinearity in quantum mechanics lead to superluminal signaling?

text: I recently came across two nice papers on the foundations of quantum mechancis, Aaronson 2004 and Hardy 2001. Aaronson makes the statement, which was new to me, that nonlinearity in QM leads to superluminal signaling (as well as the solvability of hard problems in computer science by a nonlinear quantum computer). Can anyone offer an argument with crayons for why this should be so? It seems strange to me that a principle so fundamental and important can be violated simply by having some nonlinearity. When it comes to mechanical waves, we're used to thinking of a linear wave equation as an approximation that is _always_ violated at some level. Does even the teensiest bit of nonlinearity in QM bring causality to its knees, or can the damage be limited in some sense? Does all of this have any implications for quantum gravity -- e.g., does it help to explain why it's hard to make a theory of quantum gravity, since it's not obvious that quantum gravity can be unitary and linear? > S. Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," 2004, > arXiv:quant-ph/0401062. > > L. Hardy, "Quantum theory from five reasonable axioms," 2001, arXiv:quant- > ph/0101012.


title: What is the significance of Einsteins postulate on speed of light?

text: Einstein postulated that the speed of light in free space is the same for all observers, regardless of their motion relative to the light source, where we may think of an observer as an _imaginary entity_ with a sophisticated set of measurement devices, _at rest with respect to itself_ , that perfectly records the positions and times of all events in space and time. It is really exciting to understand Einsteins thoughts. I have a doubt with respect Einsteins above postulate,   * Is there any significance in saying an _observer_ as an _imaginary entity_? I thought light could be called as some thing imaginary, rather than calling observer as an imaginary entity.    * What does it mean _one being at rest with respect to one self_? I think it mean that, observer is not under motion due to its own virtue, but the observer may be under motion, if considered inside a moving frame of reference. I don't know whether I am right with respect to this, if I have misunderstood, please explain.    * Why is _speed of light considered same for all observers (imaginary entity which is at rest with respect to itself) regardless of their motion relative to the light source_? I thought, if we consider observer inside a moving frame of reference, speed of light should be different for the observer, due to relative motion with respect to light.


title: Why does light diffract only through slits?

text: We can see diffraction of light if we allow light to pass through a slit, but why doesn't diffraction occur if we obstruct light using some other object, say a block? Why are shadows formed? Why doesn't light diffract around the obstruction as it does around the slit?


title: How mirror equation can explain farsightedness correction?

text: I have a friend who has just show me his medical prescription for hyperopia (farsightedness) correction and he needs glasses with 4,25 diopters for that, which seemed to be weird for me because I had learned, from the mirror equation, that the maximum correction possible for hyperopia is 4 diopters: $$ \frac{1}{f} = \frac{1}{p} + \frac{1}{p'} $$ If we have $0.25m$ for the normal eye distant point and more than $0.25m$ for the farsighted eye distant point (negative sign, because it's a virtual image), then we would have: $$ \frac{1}{f} = \frac{1}{0.25} + \frac{1}{p'} = 4 - \frac{1}{|p'|} \in\quad ]0,4[, \quad\text{since}\quad |p'| \geq 0.25m \quad\text{and}\quad p'<0 $$ I did some google search and find out that, indeed, hyperopia can reach values even greater, such as 20 diopters, but I can't find pages where doctors explain that with equations or physics teachers explain how things really work in ophthalmology. Either I am doing some terrible mistake, or doctors are doing some terrible mistake, or this equation just don't apply to hyperopia at all... Which one is true?


title: Why do electrons not bump into impurities in a superconductor?

text: Just a simple question. Why is it, that when a material becomes superconducting, and by that gets zero resistivity, the electrons don't hit impurities in the material? For the material to have zero resistivity, that means that the electrons can just flow without any disturbance at all? Is it because of the Cooper pair creation? In that case, why exactly?


title: Scale factor and age of the universe

text: Why is the cosmological scale factor (expansion rate of the universe) not simply the time $t$, i.e. the age of the universe?


title: Why does a doppler red shift prove expansion?

text: My question boils down to this. If the Universe was contracting the stars closer to the center would move faster to the middle than stars that were further away from the center. That would also produce a red doppler shift. So why is it that the red shift is always linked to expansion? Human optimism? Thanks, Andrew


title: What is the exact cause of flow separation in a viscous fluid?

text: I recently got into a lengthy debate about the exact nature of boundary layer separation. In common parlance, we have a tendency to talk about certain geometries as being too "sharp" for a viscous flow to remain attached to them. The flow can't "turn the corner" so to speak, and so it separates from the body. While I think this way of thinking can properly predict in which situations a flow might separate, I think it gets the underlying Physics completely wrong. From my understanding, what's happeneing is the adverse streamwise pressure gradient precludes the boundary layer from progressing downstream past a certain point, and the upstream flow subsequently has nowhere to go but up and off of the body. This is a very different causal relationship from the first explanation, where the flow lacks a sufficient streamwise-normal pressure gradient to overcome the centrifugal forces of a curved streamline. But which is correct? Considering that normal shockwaves can produce extreme adverse pressure gradients (even along a streamline that is not curved), I figured that shock- induced flow separation might be a way to settle this matter. Any thoughts?


title: Why can certain functions be absorbed into the Schwarzschild metric, while others can't?

text: Another question about the Schwarzschild solution of General Relativity: In the derivation (shown below) of the Schwarzschild metric from the vacuum Einstein Equation, at the step marked "HERE," we have the freedom to rescale the time coordinate from $dt \rightarrow e^{-g(t)}dt$. However, later in the derivation, when we pick up an integration constant at the step marked "HERE2," we are unable to absorb that constant into the metric. Actually this constant becomes a fundamental part of the metric itself. Why can't we incorporate this constant into the differentials of the metric as well? Here's the derivation: $$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} = 0$$ Assume the most general spherically symmetric metric solution to the above: $$g_{\mu\nu} = \begin{pmatrix} -e^{2\alpha(r,t)} & 0 & 0 & 0\\\ 0 & e^{2\beta(r,t)}&0&0\\\0&0&r^2 & 0\\\0 & 0 &0 & r^2\sin^2(\theta) \end{pmatrix}$$ Take the trace of both sides: $$R + 2R = 0\\\ \Rightarrow R = 0\\\ R_{\mu\nu} - \frac{1}{2}(0)= 0\\\ R_{\mu\nu}=0$$ We are free to say: $$R_{01} = \frac{2}{r}\dot{\beta}=0\\\ \Rightarrow \dot{\beta} = 0\\\ \therefore \beta = \beta(r) $$ Now, we can take a time derivative of $R_{22}$, $$\dot{R_{22}} = \frac{d\left(e^{-2\beta}[r(\beta'-\alpha') - 1] + 1\right)}{dt} = 0 = e^{2\beta}r\dot{\alpha}'$$ (Because $\dot{\beta} = 0$, $r$ is just a spacetime coordinate (and is therefore independent of other spacetime coordinates), and $\dot{\beta}' = \partial_r\partial_t\beta = \partial_r(0)$.) $$\dot{\alpha}' = 0\\\ \therefore \alpha = f(r) + g(t)$$ because this is the only functional form for which $\partial_r\partial_t\alpha = \partial_t\partial_r\alpha = 0$. Now, the metric becomes: $$g_{\mu\nu} = \begin{pmatrix} -e^{2f(r)}e^{2g(t)} & 0 & 0 & 0\\\ 0 & e^{2\beta(r)}&0&0\\\0&0&r^2 & 0\\\0 & 0 &0 & r^2\sin^2(\theta) \end{pmatrix}$$ **HERE, at this point, we can rescale time: $dt \rightarrow e^{-g(t)}dt$, which makes the metric** : $$g_{\mu\nu} = \begin{pmatrix} -e^{2f(r)} & 0 & 0 & 0\\\ 0 & e^{2\beta(r)}&0&0\\\0&0&r^2 & 0\\\0 & 0 &0 & r^2\sin^2(\theta) \end{pmatrix}$$ Setting $R_{11} = R_{00}$, we find: $$e^{2(\beta-\alpha)}R_{00} + R_{11} = 0\\\ \frac{2}{r}\alpha' + \frac{2}{r}\beta' = 0\\\ \Rightarrow \alpha = -\beta + const.$$ **This constant is absorbed into $dr^2$.** **HERE2,** Now comes the crucial part, which I don't understand: looking back at $R_{22} = 0$, we have $$(re^{2\alpha})' = 1\\\ \int (re^{2\alpha})'= \int 1\\\ re^{2\alpha} = r + \underbrace{\mu}_{\text{Constant of integration}}\\\ e^{2\alpha} = 1 + \frac{\mu}{r}$$ So the metric finally ends up being: $$ds^2 = -(1 + \frac{\mu}{r})dt^2 + (1 + \frac{\mu}{r})^{-1}dr^2 + r^2 d\Omega^2.$$ **Why can't $\frac{\mu}{r}$ be absorbed into $dt^2$ and $dr^2$ like $g(t)$ and the other integration constant in this derivation were?? I don't understand the distinction at all, any help would be greatly appreciated!!**


title: Is there a formal definition of a macroscopic variable in statistical mechanics?

text: Intuitively it's easy to accept that the usual variables like temperature, internal energy, etc. are 'macroscopic', but does there exist a formal definition of a macroscopic variable? In other words, is there a clear way to separate the set of all observables (and functions of observables) on a system into ones we would describe as 'macroscopic' and ones we would not? EDIT: Since apparently the answer is not completely straightforward, I'm interested in hearing any definitions which have appeared in literature, even if they are only conventions. I'm also interested in any necessary or sufficient conditions.


title: Hamiltonian and the space-time structure

text: I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian. Space-time structure dictates the form of Hamiltonian. Indeed, we know how the free particle should move in inertial frame of references (straight line) so Hamiltonian should respect this. I know how the form of the free particle Lagrangian can be derived from Galileo transform (see Landau's mechanics). I'm looking for a text that presents a rigorous incorporation of space-time structure into Hamiltonian mechanics. I'm not interested in Lagrangian or Newtonian approach, only Hamiltonian. The level of the abstraction should correspond to the one in Arnolds' book (symplectic manifolds, etc). Basically, I want to be able to answer the following question: "Given certain metrics, find the form of kinetic energy".


title: Sigma Models on Riemann Surfaces

text: I'm interested in knowing whether sigma models with an $n$-sheeted Riemann surface as the target space have been considered in the literature. To be explicit, these would have the action \begin{align*} S=\frac{1}{2}\int d^2x\, \left(\partial_a R\partial^a R+R^2\partial_a \theta {\partial}^a\theta\right), \end{align*} where $R$ and $\theta$ represent radial and angular coordinates on the target space respectively. Also, $\theta\sim \theta+2\pi n$ for an $n$-sheeted Riemann surface. Has anyone seen anything like this? One thing that I would be particularly happy to see is a computation of the partition function.


title: Entropy of the Sun

text: * Is it possible to measure or calculate the total entropy of the Sun?   * Assuming it changes over time, what are its current first and second derivatives w.r.t. time?    * What is our prediction on its asymptotic behavior (barring possible collisions with other bodies)?


title: In what way are the Mathematical universe hypothesis and A New Kind of Science connected

text: The Mathematical universe hypothesis, mainly by Max Tegmark and A new Kind of Science, mainly by Stephen Wolfram both claim (as least as I understand it) that at its innermost core reality _is_ mathematics. Can this statement be made more precise, i.e. what is the exact relationship between these two hypotheses?


title: Feynman Diagrams in 2 component notation

text: When using two component notation people often prefer to refrain from using arrows in Feynman diagrams to denote charge flow as is done in four-component notation. Instead, if understand correctly, they use arrows to denote chirality. I'd like to know what is the prescription to draw out the diagrams. I have read here (pg. 39) that > arrows indicate the spinor index structure, with fields of undotted indices > flowing into any vertex and field of dotted indices flowing out of any > vertices (see the reference above for many examples). However, trying this out on Majorana and Dirac mass terms, this doesn't seem to be correct. A Majorana mass term, $\psi ^\alpha \psi_\alpha +h.c.$, is thus composed only of undotted indices. With the reasoning above, it should have two arrows pointing into the vertex, ![enter image description here](http://i.stack.imgur.com/8B0z7.png) However, I'm pretty sure that this is a Dirac mass, and not a Majorana mass. What am I missing?


title: Shape of the "flow head" in Laminar flow in pipe

text: ![enter image description here](http://i.stack.imgur.com/6UfAY.jpg) Laminar flow is a streamlined steady flow with a uniform gradient of velocity across the diameter of the pipe. I am familiar with the elementary treatment of laminar flow, like basic velocity profile, shear stresses etc. But my question is:- > The velocity of the fluid particles at the edges is zero while it is maximum > at the centre. This means, that in a specified interval of time, the centre > bulges out more than the progress of the surrounding regions, while the > fluid at the edges will remain stationary. **Then, with passage of time, the > centre should continue to bulge out relative to the rest of the flowing > fluid, causing a very steep spike in the middle of the cylindrical flow. > Does this really happen?** It seems very counterintuitive and against > everyday observations, if the centre does bulge out creating a steep > "pyramidal" (parabolic, to be rigourous) shape of the leading surface. Also, > **after a definite bulge, should not gravity effects distort the shape?**


title: Complex scalar fields conserved charges

text: I'm currently studying field theory and I'm having some trouble with conserved charge given in field components. If we have a complex scalar action of a field $\phi=(\phi_1,\phi_2)^T$ that is $$S[\phi]={\int}\left[ \partial^\mu\phi^\dagger\partial_\mu\phi-m^2\phi^\dagger\phi-\frac{1}{2}\lambda(\phi^\dagger\phi)^2\right]d^4x.$$ In a global $SU(2)$-symmetry when $\phi$ belongs to $SU(2)$ two dimensional representation $$\phi\rightarrow\phi'=g\phi,\quad g=e^{i\chi_iT^i},$$ where $T^i$ is the generator chosen as $T^i=\frac{1}{2}\tau^i$ and $\tau^i$ are Pauli matrices. Now I've gotten that the conserved charges are $$Q_i=-i\int d^3x\:\phi^\dagger \overset{\leftrightarrow}{\partial^0}(T^i\phi).$$ Now the next part says that written with the components of the field these should produce $$Q_1=\mathrm{Im}\int d^3x\:\phi_2^\dagger \overset{\leftrightarrow}{\partial^0}\phi_1,\quad Q_2=\mathrm{Re}\int d^3x\:\phi_2^\dagger \overset{\leftrightarrow}{\partial^0}\phi_1, \quad Q_1=\frac{i}{2}\int d^3x\:(\phi_2^\dagger \overset{\leftrightarrow}{\partial^0}\phi_2-\phi_1^\dagger \overset{\leftrightarrow}{\partial^0}\phi_1)$$ Now I have no idea on how to get these. Why does the generator $T^i$ vanish for example?


title: How seriously do string theorists take the "landscape"

text: The string theory landscape seems to this outside observer to be an intermediate step in the intellectual progress toward a more robust theory that explains why our one universe has the particular properties that it has. is this the majority opinion or do most string theorists view the landscape as a plausibly being included in the final form of the theory?


title: Fluid dynamics, flow of a 2D jet from a narrow slit

text: I'm working through some exam problems, and I came across this one - the solution of which baffles me considerably. > A two-dimensional jet emerges from a narrow slit in a wall into fluid which > is at rest. If the jet is thin, so that velocity $\vec u = (u, v)$ varies > much more rapidly across the jet than along it, the fluid equation becomes:   >  $u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} = \nu > \frac{\partial^2 u}{\partial y^2}$   >  where constant $\nu$ is the viscosity coefficient. The bounday conditions > are that the velocity and its derivatives tend to zero as we leave the jet > (that is as $|y|\rightarrow\infty$) and $\frac{\partial u}{\partial y}$ at > $y = 0$, as the motion is symmetrical about the x-axis. The first question involves integrating across the jet to show that $\int u^2 dy$ is $x$ independent. So you end up with three integrals (subscripts denoting derivatives), 1) $\int u u_x dy$   2) $\int v u_y dy$   3) $\nu\int u_{yy} dy$   Somehow, for 1), you can write $\int u u_x dy = \frac{1}{2}\partial_x\int u^2 dy$ - is this an identity? Secondly for 2) the solution states that: $\int v u_y dy = -\int v_y u dy$ from which you use the incompressibility condition to give $\int u_x u dy = \frac{1}{2}\partial_x\int u^2 dy$. Where does $vu_y=-v_yu$ come from, and again, is there an identity used in the final step? And thirdly, this is a smaller issue but again, one that confused me a bit, the second question supposes that the streamfunction is self-similar and takes the form: $\psi = x^a f(\eta)$, $\eta = yx^b$. Which is fine, sub in for $u^2 = \psi_y^2$ and equate the power of the factor of $x$ that comes out to be zero. However, in the solution, it comes out as $2a = -b$ and I don't see how that works. Suppose you take: $\eta = yx^b$, differentiate to get $\frac{d}{d\eta}\eta = \frac{dy}{d\eta}x^b$ so that $dy = d\eta x^{-b}$. When that's substituted into the integral, the factor that comes out is $x^{2a-b}$. I have noticed some errors in the solutions, so just checking it's me, not them! I'm sure this is a simple problem once you see the trick, thanks very much!


title: uncertainty principle of spin in (super)string theory

text: In quantum mechanics, uncertainty principle states that we can only measure the quantity of spin in one axis but not others. Then what about in superstring theory? As quantum mechanics is basically three-dimensional world, this does make sense, but superstring theory adopts more than three-dimensional space...


title: Does an Increase of Force affect the Speed of an object if the acceleration stays constant?

text: If I had a mass of $100\:\rm{kg}$ accelerating due to gravity, using $F=ma$: $F = 100\:\rm{kg} \times 9.8\:\rm{m/s^2}$ $F = 980 \:\rm N$... If I increased the mass to 200kg, the force would be 1960 N: $F = 200\:\rm{kg} \times 9.8\:\rm{m/s^2}$ $F = 1960 \:\rm{N}$ Now, finally getting to my question: Does this increase in force (which is supposed to be a push/pull) mean that the object would fall faster when it weighs more?


title: Spontaneous radiation

text: The usual explanation of spontaneous radiation is that the energy eigenstates are perturbed by QED interaction, so that the eigenstates obtained from single-particle QM are no longer eigenstates of the full Hamiltonian, and in turn different (original) eigenstates can mix after some time-evolution. However I'm not quite convinced, why in the first place the states must be in single-particle Hamiltonian eigenstates instead of full Hamiltonian? EDIT:I need to rephrase it a bit since the answers reflect that my statement has been misleading. I'm not questioning the fundamental principle of superposition, I'm more concerned about the phenomenological fact that spontaneous emissions always happen experimentally after you excite an atom(if I'm not mistaken). But isn't it possible to excite the atom to an QED eigenstate such that it stays there forever?


title: Are all superalgebra's clifford algebra's

text: I believe the answer to be yes, but I realize that sometimes physicists place additional constraints that might not be obvious. If superalgebras are clifford algebras, why make a literary distinction?


title: Does the collapse of the wave function increase entropy of the atomic system itself?

text: Does wave-function collapse cause the entropy of the atom (ie. the sub-atomic particle system that makes up the atom) to increase?


title: Finding the force of friction of a moving object and its change when it accelerates to a constant speed

text: I have been searching for a straight forward answer to this question for ages now and it is driving me crazy. Here is what I know: If an object is moving at a constant speed the force of friction must equal the applied force (Please assume the applied force is horizontal) and for it to be accelerating or decelerating, the force of friction and the applied force must be uneven. I know that FK = uK * FN I have the applied force, the normal force and the coefficient of friction. This is what I do not understand: If in order to accelerate the applied force must be greater than the friction, how would that work? because surely the object would continue accelerating to infinity? Or does the friction force change to level out and equal the applied force to change the objects state from accelerating to constant velocity? If so, how do i work out the rate of the change in the friction force, surely it does not instantly change otherwise acceleration cannot occur. Here is a sample situation that may help me understand it better if it is answered Say I have a box that has a mass of 10 kg and I apply a force horizontally to the box of 50N and the coefficient of kinetic friction is 0.5. How long does it take for the box to finish accelerating and reach a constant velocity? And what is the rate of friction force change? Then say I suddenly increase the force I am applying to 60N what is the rate of change of the friction force and how long does it take to reach constant velocity? To recap my questions are: 1\. Does the object accelerate to infinity or does the friction force change to equal the applied force? 2\. If the friction force does change, how do I work out the rate of change and how long it takes? 3\. In the situation above, both times a new force is applied how do I work out change in friction force and how long it takes? Could you also show working so I can understand it properly. 4\. If the above is way off and the friction does not change how do I work out the change of velocity over time using friction depending on a change of the applied force? Sorry for the detailed question I just want to make sure I understand the answer properly :) PS A simple solution is all I need as I am using this in a simple physics engine for a game. Thanks, Arch


title: Hours of light per day based on latitude/longitude formula

text: I'm looking for a formula that will return the number of hours per day given a specific location. I was thinking that can be calculated as a difference of sunrise and sunset, but I see that there are some other ways, like in this topic. What is the best, fast and correct way to calculate this?


title: Is there an Algorithm to find the time when the sun is X degrees above the horizon for a given latitude B at date C

text: Is there an accurate algorithm / method to determine the precise time of day/night when the sun is X degrees above (or below) the horizon for a given latitude Y at date Z? Is this the same question here? Reverse Sun position algorithm? I'm having a hard to plugging in specifics.


title: Calculating the length of day at any time of the year?

text: Is it possible to calculate the length of the day at various location (distances) from the sun? I hear agricultural scientists are able to do that. To make calculations easier assume there is no axis tilt, measurements are made at the equator, and would like to know how it would change if there is a tilt.


title: gravity cancelling dark energy

text: Say there were 2 objects with certain masses (e.g. $m_1$ and $m_2$). If they were close together gravity would attract the 2 objects. If they were a large distance apart the expansion of the universe (dark energy) would pull them apart. What is the distance apart that they must be (I assume that it is relative to their masses) in order for gravitational attraction and dark energy repulsion to cancel out, so that the objects remain motionless?


title: Is the spacetime generated by isolated system always asymptotically flat?

text: I read a saying in wiki of asymptotically flat spacetime http://en.wikipedia.org/wiki/Asymptotically_flat_spacetime "In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object. Therefore, such a spacetime can be considered as an isolated system: a system in which exterior influences can be neglected." So my question is :"Is the spacetime generated by isolated system always asymptotic flat? " I remember that there exist vaccum solutions that are not asymptotically flat. So can an isolated system generate a vaccum solution outside the system that is not asymptotically flat? Is it possible that although two objects are far enough away from each other, the gravitational effect cannot be neglected? For example, $$ds^2=-2xydt^2+2dtdz+dx^2+dy^2$$ the determinant is $-1$ for all $t,x,y,z$, the Riemann Tensor is constant but not zero in all spacetime. And the Ricci tensor is zero in all spacetime. So this is a vaccum solution even in global spacetime, while it is still not a asymptotically flat spacetime. So why we can say that an object far away from other object can be treated as an isolated system, i.e. can neglect the influence by others?


title: Proving Lorentz invariance of Maxwell equations

text: I've read somewhere that one does not need to prove Lorentz invariance of the Maxwell equations $F_{\mu\nu,\sigma}+F_{\nu\sigma,\mu}+F_{\sigma\mu,\nu}=0$ because it is "manifestly Lorentz invariant" or "because they are tensor equations"? What is meant by that? I've read that this could mean that space and time are treated "on equal footing". How can this replace a mathematical proof?


title: Why are electrons consider waves?

text: I know the wave nature of electrons was evoked to explain why atoms are stable but I thought waves could be put in the same state like photons yet electrons can not exist in the same state.


title: What is the origin of nonconservative force?

text: My understanding about conservative force is a force that its work is independent of path such that we can construct another form of the work called potential to make our life easier. For friction, if I start from microscopic point of view, it should be the macroscopic effect of the electric force or gravity which are both conservative force. Why do we initially have description by conservative forces (electric/gravity force) but end up with a macroscopic description of nonconservative force(friction)?


title: Variable speed of light in cosmology

text: In this paper, D. H. Coule argues that warp drive metrics, like the one proposed by Alcubierre, require the exotic matter to be laid beforehand on the travel path by conventional travel. At section 5 of this paper "Alteration of the light-cone structure" he basically goes for the same trick that Alcubierre did: write some metric up from his sleeve with the desired properties, and see what properties the energy-stress tensor needs to have in order to satisfy Einstein equations. The metric he writes describes a spacetime with a bigger effective speed of light in the $x$ axis. He explains that this demands a $T_{\mu \nu}$ that violates the dominant energy condition. Now, if you head to the variable speed of light wikipedia page and head to the cosmology section, none of the papers by Magueijo or Albrecht rely on a mechanism of this sort to obtain variable speed of light; They go instead by the arguably harder route of proposing alternative lagrangians or theories. I'm wondering if either there is something so blatantly wrong the Coule metric alteration that the VLS authors didn't dignify it with a rebuttal, or if they simply ignore this simple mechanism to stretch light cones? We know that for explaining cosmic inflation, we need a scalar field that violates the dominant energy condition, but if this scalar had important rotational anisotropies, such a model (if correct) would imply that far-away parts of the primordial universe might have been causally connected after all (at least in some directions that would change from place to place) which might explain a lot of the long scale uniformity we are seeing? isn't this a better start for a simpler explanation that uses well-known gravitational theory?


title: kinetic energy of the stone

text: Suppose we have a man traveling in an open car (roof open) with speed $v$ towards right (man faces right). He throws a stone (mass $m$) towards right, in his frame-forward with speed $V$. In the car's frame, the total energy imparted to it was $$E=E_f- E_i=\frac{1}{2}mV^2$$   In the ground frame, the total energy given to it was $$E=E_f- E_i=\frac{1}{2}m((v+V)^2-v^2)=\frac{1}{2}m(V^2+2Vv)$$ What part am I getting wrong on? Is it okay to get such difference?


title: Alternative methods to derive the static potential in the NR limit of QED

text: In QED, one can relate the two-particle scattering amplitude to a static potential in the non-relativistic limit using the Born approximation. E.g. in Peskin and Schroeder pg. 125, the tree-level scattering amplitude for electron-electron scattering is computed, and in the non-relativistic limit one finds the Coulomb potential. If one allows for 1/c^2 effects in the non- relativistic expansion, one also finds spin-dependent interactions (e.g. spin- orbit, see Berestetskii, Lifshitz, Pitaevskii pg. 337). Are there any alternative methods for calculating a two-particle non- relativistic potential?