title: I would like to know, what would happen if a primordial black hole mass of Earth, would impact with the Earth

text: Sorry for any mistakes, I'm Italian Hello everyone, I'm writing a novel, in which the Earth is devastated by enormous geological cataclysms (something like the end of the world), and to implement my cataclysm, I thought of a primordial black hole with a mass similar to that of the Earth. At first I thought to stabilize the black hole inside the Earth, but it would have destroyed the planet, and I do not want to destroy it; or so thought to pass away from one side to the Earth from the black hole, causing very powerful shock waves, which first pierce the mantle, causing the eruption of every volcano on earth, and then the crust collide, greatly destabilizing the tectonic plates , causing massive earthquakes on a global scale, and triggering the formation of giant tsunami impacting against every continent. such a scenario is likely, or not. ps: I hope that the question is connection to physics.


title: If there was a black hole on earth, what would it look like?

text: If an answer does exist, I'd love to hear it. I'm trying to incorporate a doomed earth story in something I'm writing, and the end of the world I'm going for is a black hole. Let's say the black hole was created in the Philippines (I chose it because it's on the equator). Here's a map for reference. If I was standing 90 degrees of that, in Kenya, or if I was standing 180 degrees of that, in Mexico; what would I see in those two separate instances? What if I was standing in Japan above it, or in Australia below it? Let's also say I was riding a boat towards the source of the black hole. Unlike most imagined scenarios where I am orbiting the black hole and slowly falling into it, this time I'm no longer orbiting it - I'm getting sucked into it instead. I'm imagining that the black hole will "half" be inside Earth and "half" will be in the atmosphere. But as it pulls me in, what will it look like to me (if I was massless, so no tidal forces)? It's probably not going to be a black hemisphere sticking out of the ground. Light will be lensed around it in some way so as to appear like a hole in the air. Also, if the black hole was very small, it probably won't have a strong gravitational pull, and assuming it doesn't disappear, you'd be able to look at it (or at least, its effect on the light in its surroundings). Of course, as you approach it, it would still have an event horizon, so spacetime would still be warped if you're near enough and the surroundings will still experience tidal forces. As the black hole gains mass, it will get larger, and so its event horizon will increase. If I was standing in Kenya and the event horizon "washes over me," (as in a water wave) what will I see? I've so far imagined that, at least when seen from outer space, there will be a hole on the ground and light will be lensed around it. When you are on the ground, though, it becomes a bit harder to imagine. I do hope someone can lend me a hand! **Edit:** Someone had previously answered this question, but I'm not sure why it's been deleted. It was pointed out to me, though, that a black hole cannot stay in place. So an additional question for me is, does that mean it is impossible to be standing still, and the event horizon "washes over you"? **Edit2:** Here's a guiding thought experiment that could answer the question. Imagine that you are in outer space directly above the black hole, looking at the black hole that is on the earth's surface. Can you imagine what it would look like? It would simply be a hole with the surrounding light gravitationally lensed around it (possibly lensing the entire earth if it's big enough - but let's say it's not). Now imagine that you are on the direct opposite of where you were above - you're right behind the black hole this time, looking directly at the black hole but unable to see it because the earth is blocking your view. Now try to imagine, from the second image, rotate the earth into the first image. I find it a bit hard to do. Just as the black hole is appearing as you rotate the earth, how would the light be lensed?


title: Regarding Electromagnetic plane wave and Maxwell's equatins

text: Hello everybody I am kind of struggling with an assignment question related to general relativity. I am just going to type what the question is and explain what I am having trouble with: Consider an electromagnetic plane wave in Minkowski space. We assume that in stationary coordinates $(t,x,y,z)$ the wave propagates in the positive $x$-direction, and that the components of the electric field $E$ and the magnetic field B are only a function of $u = t-x\ ($here $c = 1)$: $E = E_{y}(t-x)\frac{\partial}{\partial{y}}+E_{z}(t-x)\frac{\partial}{\partial{z}}$ $B = B_{y}(t-x)\frac{\partial}{\partial{y}}+B_{z}(t-x)\frac{\partial}{\partial{z}}.$ Moreover, we assume E and B are compactly supported in $u$. (1) Use Maxwell's equations to show that: $B_{y}=-E_{z},$ $\space$ $B_{z}=E_{y}$. First of all sorry if the formatting isn't that great. Now I am having trouble with this because I am not exactly sure what $E$ and $B$ are... by that I mean I know $E$ and $B$ are supposed to be vector fields but in that format they look like scalar fields. Is there another way to represent those two equations, I feel that if I understand the notation it would make doing this question pretty easy. Any help would be greatly appreciated. I just want to understand the notation properly and how to work with it so that I can actually apply Maxwell's equations on them and figure out those identities. Thank you and have a good day!


title: What is curved in Berry Curvature?

text: Can anyone explain to me what is actually "curved" when we speak of a Berry Curvature?


title: Is a quantum system mandatory for generating true random sequence?

text: Is a quantum system necessary if we want to generate true random sequence? The mathematical framework used for classical mechanics doesn't involve any random value. But the mathematical framework of quantum mechanics involves randomness by definition. Can we argue based on these information that a true random number generator must use quantum mechanics? If anyone claims that s/he has a true random number generator and fails to prove that s/he is exploiting quantum mechanics can I discard the claim on the basis that s/he is not using quantum mechanics?


title: Thermodynamics of binary symmetric channels

text: I am reading this very interesting paper: http://m.iopscience.iop.org/1751-8121/41/40/402002/pdf/1751-8121_41_40_402002.pdf about thermodynamics of channels in information theory. More generally, establishing a dictionary between information theory and thermodynamics is an old idea, but it seems to me that we only transpose state parameters (entropy, energy, temperature) and not transformations (adiabatic, isothermic). For instance, mutual information between input and output is related in this paper to internal energy. Suppose we are trying to simulate the BSC using other channels. We would quantify the efficiency of this simulation by looking at the mutual information. The channels having exactly the same mutual information as BSC are indistinguishable from BSC. The dictionary with thermodynamics would lead us to think that we are performing an adiabatic transform in the space of channels. However I can find no rigorous formulation for this intuition, nor any paper on the possible interpretations of thermodynamic transforms for information theory channels.


title: Conical Pendulum -- Can it rotate at 90 degrees?

text: I have a simple question, can you spin a conical pendulum fast enough so that it rotates at 90 degrees? The equation is $\tan(\theta)=v^2/rg$ , but at 90 degrees, $\tan(\theta)=\infty$ ... so what does this mean? Please provide supporting data.


title: What is the mystery of turbulence?

text: One of the great unsolved problems in physics is turbulence but I'm not too clear what the mystery is. Does it mean that the Navier-Stokes equations don't have any turbulent phenomena even if we solve it computationally? Or does it mean we simply don't have a closed-form solution to turbulent phenomena?


title: Would it be possible for geophysicists/geoengineers to develop an artificial way of trapping carbon in the ocean?

text: There's a mechanism by which the southern ocean sequesters carbon from the atmosphere. It happens when strong winds displace a large slab of surface water, accumulating in a specific region where the localised excess water in the surface layer gets injected downward into the ocean's interior. This makes the ocean's absorb about 25% of annual man-made $CO_2$ emissions. Are any geophysicists and engineers working on developing a man-made mechanism by which we can make the ocean's absorb more carbon and therefore reduce climate change? Is it humanly possible? NOTE: I read an article on the phenomenon in Physics World magazine, and it didn't mention anything about scientists looking to make it happen more.


title: Is a 1D vector also a scalar?

text: A vector in one dimension has only one component. Can we consider it as a scalar at the same time? Why time is not a vector, although it can be negative and positive (when solving for time the kinematics equation for example)?


title: which induced current produce due to magnetic field?

text: please i want to ask when induced current produced by changing magnetic field according to faraday's law is this current DC or AC current ?


title: Would a 9v battery be able to produce gas bubbles by water electrolysis at 1000 meters?

text: I'm currently working on an underwater robot and was hoping to use the gas bubbles for buoyancy control .


title: Is the gravitational constant G a minimum value in some sense?

text: Assume a central body of mass $M$, and call $a$ the acceleration of a test body at a distance $r$ due to any interaction whatsoever with the central body. Is is correct to say that the ratio $a r^2/ M$, for any interaction, is always larger than or equal to the gravitational constant $G$?


title: Cosmic Microwave Background and heat flow

text: Can the fact that energy is distributed homogeneously in the universe be explained through heat flow and not the Cosmic Microwave Background(or in other words, can we say CMB is a result of heat flow and not the universe expanding from a singularity)?


title: Extract energy from magnets

text: Is it possible to "extract" energy from a magnet, making it lose its magnetism? Or, to put in another way, is magnetism a form of energy? (I am not talking about potential energy in a magnetic field). Since matter is equivalent to energy, is the property of magnetism equivalent as well?


title: (If and) Why does cold temperature affect semiconductors?

text: I had a college student build an overclocked PC using phase-change technology. (This is essentially an air-conditioning unit with the evaporator attached directly to the motherboard.) He said that cold temperature made the CPU more stable at higher clock speeds (5.5 GHz) by affecting how the atoms and electrons behave. My question is, is this true? If so, in what way does cold temperature affect semiconductors?


title: How to derive the period of spring pendulum?

text: So I wanted to find out how to (simply, if that's possible) derive the formula for a period of spring pendulum: $T=2\pi \sqrt{\frac{m}{k}}$. However, Google doesn't help me here as all I see is the ready-to-bake formula. Could you please point me some directions?


title: Angle rotated by a rod when it's hit by a pendulum

text: Consider a pendulum of length $h$ with a bob of mass $m$ it is held horizontally at and angle of $90^{\circ}$ with the vertical. A rod of mass $M$ and length $h$ is pivoted at its upper end and this pivot coincides with the point of suspension of the pendulum. The rod hangs downwards along the vertical. The pendulum is released. It collides with the rod and sticks there. Find the angle rotated by the rod. ![enter image description here](http://i.stack.imgur.com/kx49m.png) ![enter image description here](http://i.stack.imgur.com/wmAuX.png) This can be easily accomplished using conservation of mechanical energy. Clearly, the initial energy of the system is purely potential energy: $$E_i = U_{bob} = mgh \tag1$$ The final potential energy, when the rod has rotated through $\theta$ is also purely potential energy: $$E_f = U_{bob} + U_{rod}$$ $$U_{bob} = mg(h - h\cos\theta) = mgh(1 - \cos\theta)$$ $$U_{rod} = \int dU = \int dmdhg$$ Consider a small length $dx$ on the rod at a distance $x$ from the top. The potential energy of this length is $dm(h - x\cos\theta)g$. The mass per unit length of the rod is $\frac{M}{h}$, so $dm = \frac{Mdx}{h}$. Putting these into the integral and integrating over all segments on the length of the rod, we get: $$U_{rod} = \int_0^h \left(\frac{Mdx}{h}\right)(h - x\cos\theta)g = \frac{Mg}{h}\int_0^h (h - x\cos\theta)dx$$ $$= \left[Mgx - \frac{Mgx^2\cos\theta}{2h}\right]_0^h = Mgh - \frac{Mgh\cos\theta}{2}$$ $$\therefore E_f = mgh - mgh\cos\theta + Mgh - \frac{Mgh\cos\theta}{2}$$ Equating the above with $(1)$: $$mgh = mgh - mgh\cos\theta + Mgh - \frac{Mgh\cos\theta}{2}$$ $$\implies Mgh = \cos\theta gh\left(m + \frac{M}{2}\right)$$ $$\implies \boxed{\cos\theta = \frac{2M}{2m + M}}$$ But consider the case where the bob has a mass of $100 \ \rm{grams}$ and the rod a mass of $240 \ \rm{grams}$. $$\cos\theta = \frac{2\cdot 0.24}{2\cdot0.1 + 0.24} = \frac{0.48}{0.44} > 1$$ which is clearly not possible. But, from experience, we know that the rod _will_ rotate through some valid angle. Where did I go wrong?


title: Couldn't we measure electrons with good enough technology?

text: I am a bit confused about the Heisenberg's Uncertainty Principle - just read about it in _How to Teach Physics to Your Dog_ , by Chad Orzel. He states that the reason electrons can't be measured is because the photons used to measure electrons, collide with them causing changes in momentum and velocity. Couldn't this simply be fixed by creating a vacuum and using a different method of observation? Maybe something that doesn't exist today? **Edit:** A full disclaimer: I never took a physics course higher than a basic intro to physics. All the answers below are still referring to light (photons) as the method for measuring the momentum/location of the particle. My question is: is it not possible to create a vacuum without any possible light source from entering; using this vacuum we can device a more advanced method which does not affect the momentum or position of the electrons.


title: Magnetic force on capacitor

text: I have this problem: I have a capacitor formed by two parallel square-shaped plates, of side $a$, and with distance $d$ between them. They're asking me out of other things, the magnetic force as a function of some things. I calculate de magnetic force by calculating the gradient of the magnetic energy, given by: $$E=\frac{1}{2}\iiint_VB\cdot H\;dV$$ I have calculated $B$ and $H$, that form, during the charge of the capacitor, loops around the center proportional to $r$, being $r$ the distance from the central axis. The problem is that I have to integrate that to a cubic volume: $a^2d$, one of those variables is independent, but the other two are not, and this problems never involve complicated integrals, such that one: $$\int_0^a\int_0^a rdxdy$$ If the plates were circles it would be trivial, but being squares, I'm sure I'm missing something.


title: How fission and fusion create energy?

text: > **Possible Duplicate:**   >  How to explain $E=mc^2$ mass defect in fission/fusion In order for energy to be released, mass has to be "lost", because mass is a form of energy. In my science class, we are discussing fission and fusion of atoms, and since energy can be retrieved from these processes, I'm wondering where the mass loss is coming from. Can somebody explain where mass is lost in these two processes?


title: twin paradox inferring cosmic speed limit

text: As I understand it, the twin paradox was integral to Einstein's conclusion that nothing can travel faster than the speed of light. Why wasn't there a similar conclusion for a hypothetical scenario where 2 blind musicians play the trumpet whilst moving away from each other at a speed close to the speed of sound. They would obviously hear a slower/lower tone (doppler shift) but at no point did someone suggest that time was slowing down. If a spacecraft left earth accelerating up to 80% of the speed of light and views earth through a telescope and notices both a red shift and an appearance of time slowing down (as you would expect), then on the return journey there is a blue shift and time appears to speed up (as you would expect) why does time actually HAVE to be tied to the blue/red shift effect? Why is it different than the hypothetical scenario with the sound waves? Furthermore the time's apparent slowing down on earth on the journey to the star is "made up" by the time speeding up on the return journey isn't it?


title: How to obtain deconfined theory from an s-confined N=1 susy gauge theory?

text: Is there a systematic procedure for obtaining a deconfined theory from an s-confining theory (as defined in hep-th/9610139 for example)?


title: Finding radius of Earth through observation of Sun's motion

text: The question I'm about to pose is from a physics book I had recently bought. Since I am very interested in physics I am quite keen in understanding how this question can be solved. Before I present this question, I would like to explain the context in which this question was posed. It is a question related to the chapter entitled - Measurement, dealing with the understanding of the basic quantities like mass, time, length etc. Now let me pose the question:      Suppose that while laying down on a beach near the equator of the Earth, you watch the sun setting below the horizon. You start a stopwatch the moment the top of the sun disappears below the horizon. You then stand to a height of 1.70 metres, and stop the watch when the top of the sun disappears again. The time interval is exactly 11.1 seconds. What is the radius of the Earth?      I must inform you that this is not a homework question or any such related assignment, I am simply asking a question whose solution I have been unable to find myself or online. I hope you'll be able to answer this question.


title: Restrictions on defining microstates (Entropy)

text: If we have an **isolated** system $Sb$ with thermodynamic entropy $Eb=X$ (and growing by the 2nd law of thermodynamics), we could define an abstract system $Sa$ (containing the system $Sb$) but define the possible microstates as a single bit, being 1 (one) when $Eb>=X$, and 0 (zero) when $Eb < X$. The microstate will tend to be $1$, so the entropy of the system $Sa$ tends to zero. > po-->0 > > p1-->1 > > Eb = - k [p0 * log (p0) + p1 * log (p1)] --> 0 How could it be possible that an isolated system have a decreasing entropy? Does the growth (or lack of growth) of the entropy depend on the microstates choosen? Finally, if there is a restriction on how to choose the definition of the microstates, then what is that restriction? Thanks.


title: Ricci scalars for space and spacetime, local and global curvature

text: 1. If Ricci scalar describes the full spacetime curvature, then what do we mean by $k=0,+1,-1$ being flat, positive and negative curved space?    2. Is $k$ special version of a constant "3d-Ricci" scalar?    3. What is the difference between the local and global spacetime curvature?


title: Calculating phase difference of sound waves

text: An observer stands 3 m from speaker A and 5 m from speaker B. Both speakers, oscillating in phase, produce waves with a frequency of 250 Hz. The speed of sound in air is 340 m/s. What is the phase difference between the waves from A and B at the observer's location? $$v=fλ\\\ λ=1.36m\\\ \\\ Δr=|{ r }_{ 2 }-{ r }_{ 1 }|\\\ =2$$ I have nothing that relates any of this to phase angle. :(


title: Spectral series' formula of a given atom (other than hydrogen-like)?

text: The hydrogen spectral series is given by the Rydberg formula: > The energy differences between levels in the Bohr model, and hence the > wavelengths of emitted/absorbed photons, is given by the Rydberg formula: > > $${1 \over \lambda} = R \left( {1 \over (n^\prime)^2} - {1 \over n^2} > \right) \qquad \left( R = 1.097373 \times 10^7 \ \mathrm{m}^{-1} \right)$$ > where $n$ is the upper energy level, $n'$ is the lower energy level, and $R$ > is the Rydberg constant. There is a similar formula for every hydrogen-like atom. **Question** : Is there a theoretic formula for the spectral series of a given atom (other than hydrogen-like)? Else, why, what are the difficulties?


title: Superpositions and expectation values in quantum mechanics

text: When the wavefunction of a particle is not an eigenfunction of an operator, the property to which the operator corresponds does not have a definite value, Why? Also consider a linear combination of basis functions: $$\psi=c_1\psi_1+c_2\psi_2+...=\sum_kc_k\psi_k$$ When the momentum is measured, in a single observation one of the eigenvalues corresponding to the $\psi_k$ that contribute to the superposition will be found, why?


title: Why is boiling water loud, then quiet?

text: Water in my electric kettle makes the most noise sixty to ninety seconds before the water comes to a full boil. I have been fooled many times by the noisy kettle, only to discover that the water was not yet hot enough for tea. The kettle is only at a full boil after the noise has _subsided_. I have noticed the same phenomenon with many other kettles, including conventional kettles on kitchen ranges; it is not a peculiarity of this electric kettle. Why does the boiling become quieter as the water reaches full boil?


title: When water is about to boil

text: Have ever noticed? When water is about to boil, no matters the kettle, there is some **sound** I have no idea where it comes from, sometimes long before it boils. Is there any explanation for this phenomena?


title: Could an attachment to cell-phones limit or neutralize electro-magnetic radiation?

text: **Assumption:** Cell-phones generate some degree of electro-magnetic radiation. Whether it's harmful or not is not the question here. **Claim:** Certain devices may "neutralize" this radiation. (Ex: http://www.giawellness.com/2/products/terra-gia/cell-guard/) **Questions:**   * Is such a claim even physically possible with what is known today (in stick-on form, as advertized)?    * Presumably a full case made out of special material could limit it.. What effect would a suitable case have on the call quality and/or wifi connectivity? (Related Question: Could cell-phone radiation cause cancer?)


title: Is there a simple way to define/solve for a null cone for a general spacetime geometry?

text: I'm wondering if there's any simple way to define and solve for a null cone for a general spacetime geometry in $n+1$ dimensions, given its vertex $p^\mu$. I can't seem to find a simple way to do it that doesn't involve solving for null geodesics. It seems like it's something that should be in a lot of GR textbooks, but I can't seem to find a simple alternative (perhaps I'm not looking hard enough, but that's why I'm asking). So far, the best I can do is to define it as the $n$-surface formed by the family of null geodesics $y_{\bf\Omega}^\mu$ passing through some spacetime point $p^\mu$, where $\bf\Omega$ is a combination of $n-1$ angles that distinguishes the curves (at $p^\mu$, it's the collection of angles specifying the direction of the null curve). The problem is, I'm interested in doing a foliation of spacetime using future or past null cones with respect to some timelike curve $\gamma$, and would like to define the null cones in the form of a constraint: $C_{\gamma}(x^\mu)=\tau$ Where $\tau$ is the proper time along the timelike curve $\gamma$. This constraint satisfies: $U(C_{\gamma})=0$ Where $U$ is the vector field defined using the tangent vectors to $y_{\bf\Omega}^\mu$ (this can be defined for all spacetime, if you take the union of all the null cones along $\gamma$). In any case, I still have to solve for the null geodesics, given what I have so far. I'm not a fan of solving for all the null geodesics at every point along the curve, so I'm looking for a simpler way to do it. I have the feeling that there should be a simpler way of defining and solving for null cones--it seems like they've been extensively studied in GR. Is there any way to do this without resorting to solving for the null geodesics? In particular, has anyone written down a general constraint form for the null cone?


title: Lattice geometry and dispersion relation

text: Is there a general theorem which gives some information about which influence have the lattice geometry (for example sub-lattice structure, square lattice, honeycomb lattice, lattice symmetries, ...) to the dispersion relation in the low-energy regime? For example: graphene or magnons in antiferromagnets. Both have a sub-lattice structure and the dispersion relations are linear.


title: Young elasticity modulus anisotropic media

text: Im studying anisotropic system composed by a elastic matrix (Young modulus $E_m$) filled with oriented rods. Given this filler orientation, the material is elastic-anisotropic, with Young elastic modulus $E_\parallel$ in the direction of the rods orientation and $E_\perp$ in the transversal direction of the oriented rods. If we set the axis $z$ like the direction of the rods orientation, then $E_\parallel = E_z$ and $E_\perp = E_x = E_y$. I know the stress -strain curves for the material in $z$ and $x$ (or $y$) directions. Given the initial position of the rods, how can I obtain the final positions of the rods if a stress $\sigma$ is applied in the $z$ direction?


title: What's the $\ell$ in the Bicep2 paper mean?

text: The BICEP experiment's recent announcement included the preprint of their paper, > BICEP2 I: Detection of $B$-mode polarization at degree angular scales. > BICEP2 Collaboration. To be submitted. BICEP-Keck preprint, arXiv:1403.3985. Gravitational lensing of the CMB’s light by large scale structure at relatively late times produces small deﬂections of the primordial pattern, converting a small portion of E-mode power into B-modes. The lensing B-mode spectrum is similar to a smoothed version of the E-mode spectrum but a factor 100 lower in power, and hence also rises toward sub-degree scales and peaks around $\ell$ = 1000. I think the $\ell$ is this: For example $\ell=10$ corresponds to roughly 10 degrees on the sky, $\ell=100$ corresponds to roughly 1 degree on the sky. (From CMB introduction, by Wayne Hu.) But how does that apply here? When BICEP looks for something with an $\ell$ around 80, does that mean a "multipole moment" which spans 80 degrees across the sky?


title: What's the significance of the definition of T1 and T2 relaxation times

text: In MRI we measure $T_1$ and $T_2$ relaxation times. $T_1$ relaxation time is defined as the time it takes for the longitudinal component of magnetization to recover $1-1/e$ of its original value, and $T_2$ relaxation time is the time it takes for the transverse component of magnetization to recover $1/e$ of its original value. Is there any significance to those two numbers adding to 1? ($1-1/e + 1/e = 1$) As far as I understand, $T_2$ relaxation time is influenced by different atoms falling out of phase with each other. Was the reason for $1-1/e$ and $1/e$ that if the atoms would stay in phase, then $T_1$ would be equal to $T_2$, so that the difference between $T_1$ and $T_2$ relaxation time measures the degree of interaction? In that case, since $T_1$ and $T_2$ measure orthogonal components of a vector, shouldn't it have been $1/e$ and $\sqrt{1-1/e^2}$, so that the length of the vector becomes $1$ instead of the sum of the components becoming $1$?


title: Does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom?

text: The starting point and notations used here are presented in Two puzzles on the Projective Symmetry Group(PSG)?. As we know, Invariant Gauge Group(IGG) is a _normal subgroup_ of Projective Symmetry Group(PSG), but it may not be a _normal subgroup_ of $SU(2)$, like $IGG=U(1)$. But this may results in a trouble: By definition, we can calculate the $IGG$ and $IGG'$ of the $SU(2)$ gauge equivalent mean-field Hamiltonians $H(\psi_i)$ and $H(\widetilde{\psi_i})$, respectively. And it's easy to see that for each site $i$, we have $U_i'=G_iU_iG_i^\dagger$, where $U_i'\in IGG'$ and $U_i\in IGG$, which means that $IGG'=G_i\text{ }IGG \text{ }G_i^\dagger$. Now the trouble is explicit, if $IGG$(like $U(1)$) is not a _normal subgroup_ of $SU(2)$, then $IGG'$ may **not** equal to $IGG$, so does this mean that _two $SU(2)$ gauge equivalent mean-field Hamiltonians $H(\psi_i)$ and $H(\widetilde{\psi_i})$ may have different IGGs_ ? Or in other words, does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom? Thank you very much.


title: Landau Lifshitz energy for uniform rotation

text: Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega \times r.$ This does not make sense to me, since this centrifugal term is negative( I mean, compared to the energy of the rotating system the energy in the inertial system should be HIGHER, since there is an additional kinetic energy of the rotation) as I had exected something like $ E = \frac{m(v + \omega \times r)^2}{2} + U$ in the inertial frame. Then they claim that in the rotating system(in that case they use the notation $v_0$ instead of $v'$ we have $ E = \frac{mv'^2}{2} - m v' \omega \times r+ U,$ ( This looks reasonable to me, as kinetic energy is less than the inertial one), but still,in that case I had expected something like $ E = \frac{m(v' - \omega \times r)^2}{2} + U,$ so where am I wrong(especially about my reasoning in the first place) and why is it wrong, just to plug in the velocity of the transformation.


title: How to interpret vacuum instability of Higgs potential

text: If the Higgs mass is in a certain range, the quartic self-coupling of the Higgs field becomes negative after renormalization group flow to a high energy scale, signalling an instability of the vacuum itself. However, I don't know how to interpret the fact that the vacuum appears to be stable at a certain scale, and not stable at a higher scale. Does it mean the actual collapse of the vacuum will start from the higher momentum modes? In other words, when the vacuum collapses, is it true that the value of the scalar field averaged over large distances remain bounded for an extended period of time, but appears to be diverging quickly when averaged over short distances? Also, is it true that such a collapse only happens via the slow process of quantum tunneling under usual circumstances, but will take place quickly if a particle is produced at a high energy comparable to instability energy scale?


title: Twistors in Curved Spacetime

text: I am looking for good and recent references to constructing twistor space for curved spacetime. This could be a general spacetime, or specific ones (say maximally symmetric spaces different from Minkowski). This could be in he context of the twistor correspondence, or the twistor transform of field equations, either subject generalized to curved spacetime. The references I am familiar with are the standard ones from about 30-40 years ago, where most constructions involve flat spacetime. Some generalizations are mentioned, but my impression is that the community had not settled at the time on a single approach. Many things happened since, and one of the things I am hoping to get is some understanding of the landscape of current approaches to the subject.


title: realization of: CFT generating fuction = AdS partition function

text: An important aspect of the AdS/CFT correspondence is the recipe to compute correlation functions of a boundary operator $\mathcal{O} $ in terms of the supergravity fields in the interior of the $AdS_{n+1}$ (as we approach the boundary). Namely, $\big< \exp \int_{\mathbb{S}^n} \mathcal{O} \phi_{0}\big > = \mathbb{Z}_{s} (\phi \big|_{\partial(AdS)} = \phi_0)$, where $\mathbb{Z}_s$ is the supergravity partition function. The review papers I have found (and Witten's original paper as well) explain how to use the above formula but fail to provide a satisfactory explanation why the formula ought to work, or even how it came about. Can anyone explain if there is a logical (and/or insightful) path that would lead to the above correspondence between the generating function of the $n$-points correlators and supergravity/string theory?


title: Torsion and gauge invariant EM kinetic term

text: Everytime I hear about adding torsion to GR, something struggles me: how do you create a kinetic term for the electromagnetic field that is still gauge- invariant? One of the consequences of torsion is that covariant derivatives no more commute on scalar functions, so by looking at the $F_{\mu\nu} = \nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ (constrained to minimal coupling by the equivalence principle) and we perform a gauge transformation on it $A_{\mu} \to A_{\mu} + \partial_{\mu}\Lambda$ we get $F_{\mu\nu} \to F_{\mu\nu} + T^{\rho}_{\mu\nu}\partial_{\rho}\Lambda$ where $T^{\rho}_{\mu\nu}$ is the torsion tensor. So how can we handle torsion in non pure-gravity?


title: What is a backlit CCD sensor, and why might I prefer it to other kinds?

text: I've heard Astronomers talking about backlit CCD sensors, and talking about how much better they are than other types. What are they, why might I want to get one, and what are the pros/cons of this compared to related technologies?


title: Are the Voyager probes still sending us a signal ?

text: This article reported today, a finding from one of the voyager probes. It talks about magnetic structures at the (far) outer edge of our solar system. Does that mean we're still able to pick up a signal from the Voyager probes ? And how long does it take to receive a transmission from a probe that is 16billion km from earth ?


title: Why don't more rocky planets/moons have appreciable atmospheres?

text: It seems obvious why mercury has no atmosphere, given its proximity to the sun --but yet Venus is also fairly close, and has an extremely dense atmosphere. Titan is a large moon with an atmosphere thicker than Earth's, and then Triton, a more distant large moon, has only a tenuous Nitrogen atmosphere. Why aren't more rocky bodies able to maintain an atmosphere? What factors affect whether a planet or moon is able to hold on to its atmosphere? What factors determine whether a planet or moon develops an atmosphere to begin with?


title: Assuming that the Sun wouldn't evolve into a red giant, how long would the Earth's atmosphere last?

text: It's not going to last forever - Jeans Escape is going to eventually act on the atmosphere after trillions of trillions of years: see http://faculty.washington.edu/dcatling/Catling2009_SciAm.pdf


title: Is our universe the only universe out there or are there also other possible universes?

text: There is the specific universe we live in, the "actual" universe with planet Earth, humans evolving, and us discussing metaphysical debates on physics.se in 2012, etc. There might or might not also be other possible/potential worlds out there, some differing majorly, like in the laws of physics, or minorly, almost like our current world, but maybe I didn't pose this question on physics.se. According to many worlds advocates, the other possible worlds also exist out there in Platonic space. Is this world we find ourselves the only universe out there, or do other possible worlds also exist?


title: Relation between Newtons and Kilograms

text: Work is expressed as `W=F*d`, where the `F` is in Newton, `d` is in meters and result `W` is in **Joules**. For example, if I take `1N` on earth and lift it `1m` up in the air I have done `1J` of work. Kinetic energy is expressed as `E=(1/2)*m*v^2`, where the `m` is mass of the body in kilograms and `v` is the speed of the body in m/s and the result is again in **Joules**. For example, if I have 80kg and run at speed of 5m/s then I have 2000J of kinetic energy. I am very puzzled because I do not see the relation between newtons and kilograms in these two examples. I mean, how can you mix up two different things such are mass and force and get the results in same unit such as Joule - Are those two results are the same Joules???


title: time constant for typical fluorescent lights

text: For a typical flourescent light available in the United States on a a standard 120V 60Hz line, is there information available on the decay time of the light? For example, if I am running the light a full power (ON) and shut off power (instantaneously) how long before the light is completely off (or reduced by $1/e$...)? E.g. if the luminance is given by $$ L = L_0 e^{-t/\tau} $$ where $t$ is the time elapsed after killing the power: what is $\tau$? Order of magnitude estimates would be great. More precision would be even better. I couldn't find this information easily on google.


title: Some complex-number manipulation when calculating coefficients

text: I am going through Sakurai's quantum mechanics, and at one point the solution to a problem says: $(\sin(\beta)\cos(\alpha)-i\sin(\beta)\sin(\alpha))b+\cos(\beta)a=a$ $(\sin(\beta)\cos(\alpha)+i\sin(\beta)\sin(\alpha))a-\cos(\beta)b=b$ $a,b$ are the complex coefficients for some state, so we have: $|a|^2+|b|^2=1$ What I don't understand is that from there he goes to: $a=\cos(\frac{\beta}{2})e^{i\theta_a}$ For some $\theta_a$. I understand that you can write $a$ as $r e^{i\theta}$ for some $r,\theta$, but I dont see how he is able to get $r=\cos(\frac{\beta}{2})$. Where does this come from?


title: Reference request: FLRW with k>0, dust, and positive cosmological constant

text: The exact solution representing a FLRW universe with $k>0$ and dust (p=0), and $\Lambda=0$, is described by a cycloid. What is the exact solution for dust, in the presence of a positive cosmological constant $\Lambda>0$? Could you give references?


title: The physical meaning of electromagnetic wave

text: What, fundamentally, is an electromagnetic wave? As far as I know, all wave phenomena are derivations of an oscillating processes, e.g. particles vibrating in a medium. I can't imagine a wave process which couldn't be explained as moving particles: sound wave - movement of gas molecules, waves on the water - movement of water molecules, earthquake - movement of ground particles (and finally molecules), even phonons - vibrational movement of crystal lattice. Phonons are virtual but the atoms arranged in lattice are real. I tend to think that the reason has to do with the restless nature of the photons along with the uncertainty principle - the impossibility to localize a primary chunk of matter (with momentum) - when we catch the wave (stop the process) the frequency is gone. But again, it looks like it is a particle and we just can't observe it with our methods. So, what is the electromagnetic wave? How can it move vibrating across the vacuum? How can we physically interpret phenomenon of electromagnetic waves? **_Update_** The concept of _field_ is itself very complex phenomenon to understand its physical meaning by our "everyday" mind. The field looks like a math tool to describe and predict nature events. But its physical representation is unclear. I can't think of light as some 3-D helix shaped pulsing of continuous substance :) The world is quantized and it's obvious that the `sine`-shape like objects cannot be elementary in reality.


title: Scattering Amplitude in second Born Approximation for the Yukawa potential

text: Does anyone know where I can find the analytical expressions of the scattering amplitude in second Born Approximation for the Yukawa potential? I need it for the both cases of the method of partial waves and the Born series as defined to solve the Lippmann-Schwinger equation. In general the second term is unimportant being the Born Approximation very good in high-energy physics but it is not my case as I consider collisions in semiconductors. I would not like to waste time to calculate it as I presume that probably someone else has already done.


title: In the Lennard-Jones potential, why does the attractive part (dispersion) have an $r^{-6}$ dependence?

text: The Lennard-Jones potential has the form: $$U(r) = 4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]$$ The (attractive) $r^{-6}$ term describes the dispersion force ( _attraction at long range_ ). Why is the power $-6$ rather than something else? I think that it has to do with dipole-dipole interactions, but I am having difficulty working out the physics. Can you please help me? I know that, at a distance $z$ away, the on-axis electric potential due to a dipole at the origin and aligned on the z-axis is given by: $$V = \frac{1}{4\pi\epsilon_0} \frac{p}{z^2}$$ where $p$ is the (magnitude of the) dipole moment. But if there is an atom nearby, the dipole will induce a dipole on the atom. The induced dipole will in turn induce a dipole on the first dipole. Is it possible to classically derive the $r^{-6}$ dependence of the dispersion force? Thanks for your time.


title: Does an object slow down after acquiring terminal velocity?

text: Once an object reaches terminal velocity in free fall, is there a point in time when the object actually starts slowing down due to resistance, or does it stay constant?


title: Melting ice pattern

text: What causes ice to melt in these patterns? ![Melting Ice](http://i.stack.imgur.com/WlcMR.jpg) And why does this always happen when the ice is covered in snow?


title: Where does the energy of a lightning strike go?

text: Lightning contains a lot of energy, so where does this energy go after lightning has hit the ground? Does it travel all the way to the core? What happens after that?


title: Why is there no aberation of a static electric field, but there is aberation for EM radiation?

text: So consider the case where there is a particle and a moving telescope: ![enter image description here](http://i.stack.imgur.com/HD5xk.jpg) If the particle is located at position 1, and the telescope is moving horizontally as shown, the EM radiation appears to be coming from position 2, from the telescopes perspective. Let us now think of the object 1 as being an electron, producing a static electric field. Now, the telescope will detect the electric field to indeed be at position 1, despite travelling at the horizontal velocity. Why is there such a difference between EM and electric fields.


title: Quantum Chemistry Kronecker Delta formality

text: In semi-empirical quantum chemistry, one frequently encounters the so called _zero differential overlap_ approximation $$\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu\nu}\delta_{\lambda\sigma} \langle \mu \mu | \lambda \lambda \rangle .$$ Why is it rather not written as $$\langle \mu \nu | \lambda \sigma \rangle = \delta_{\mu\nu}\delta_{\lambda\sigma} \langle \mu \nu | \lambda \sigma \rangle = \langle \mu \mu | \lambda \lambda \rangle $$ since on the right hand side of the first equation there are no $\nu$ nor $\sigma$ contained anymore. So either all four variables **plus** the Kronecker Deltas (middle expression of second equation), or only the "remaining" variables after evaluation of the Kronecker deltas (last expression of second equation).


title: Why does an airplane need to climb during a takeoff even if it is in emergency situation?

text: Right after take-off (which means an airplane already exceeded V1) it is recommended that an airplane keeps climbing even when emergency occurs. Beside worries of crashing into houses and buildings by flying low altitude, why should an airplane not stop climbing up? Why should it climb up and then, lowers to some altitude and then land?


title: Tunneling v. Hopping

text: Can someone explain the difference between hopping and tunneling? The context I'm considering is conduction in semiconductors, specifically between impurity states within the bandgap. It's always been my understanding that hopping _is_ tunneling. Variable range hopping and nearest neighbor hopping, as I understand it, are both forms of tunneling between overlapping states (for example, see http://igitur- archive.library.uu.nl/dissertations/2002-0806-101243/c4.pdf). However, in papers such as DJ Thouless 1974 _Electrons in Disordered Systems and the Theory of Localization_ , hopping and tunneling are described as two different processes. I guess I don't exactly understand what "hopping" is supposed to mean, with respect to charge transport.


title: What does the wave principle of light actually represent?

text: Light has a dual nature, one of photons and the other of waves. But energy doesn't really travel in waves. So what do the wave represent?


title: Two related questions about double-slit experiments moving at a relativistic speed

text: I was wondering as how would appear the interference pattern of a double-slit experiment moving at a relativistic speed v, 1) in the case of light and, 2) in the case wave matter (i.e. electrons for instance) as seen in comparison between rest and moving reference frames (respectively S and S’). ![enter image description here](http://i.stack.imgur.com/BzhTC.jpg) ![enter image description here](http://i.stack.imgur.com/8yogk.jpg) More precisely, I am considering the spacing between fringes and their nature. In terms of the spatial distribution of the individual speckles observed on the screen, my first impression was that it must be exactly of the same nature, i.e. fringes being centered at same screen positions (the y axis being transversal to the relative movement). Doing the usual simple math (see details below) to obtain fringes positions and spacing, and using the longitudinal length contraction L’ = L / G(v) (where G(v) is the usual gamma factor G(v) = 1/(1 – (v/c)**2)**0,5) gives me finally an equation relating the respective wavelengths in S’ and S : Lambda’ = G(v) Lambda **And then, I have the following two questions:** **1) In the case of light** (with Lambda = c/f), this would imply that f’ = f / G(v) (see details below). Then is it true that a red shift must always be taken into account due to a Doppler effect acting only transversally? **2) Since for a matter wave** , the de Broglie wavelength is Lambda = h/p, the observed pattern spacing would be maintained if p’ = p / G(v) (see details below). But I doubt seriously that there is such a simple relativistic transformation between the respective linear momenta of rest and moving frame (the correct composition of momenta is different). What are the errors then? Thanks for your patience. I would clearly appreciate the help of someone. **Details of the simple math:** Starting in the frame S with the walk difference delta(m) for a maximum of intensity observed at fringe of order m on the screen (along the y axis): delta(m) = r2 – r1 = m Lambda and d/L -> epsilon   => y(m)/L = tan theta -> sin theta = delta(m) / d = m Lambda / d => position of fringe m: y(m) = m Lambda L / d   and spacing : Dy = y(m+1) – y(m) = Lambda L / d The same shall be true in the frame S’: position of fringe m: y’(m) = m Lambda’ L’ / d and spacing Dy’ = Lambda’ L’ / d. Then, the length contraction, together with the requirement that the spacing must remain the same give: L’ = L / G(v) with Dy = Dy’ => Lambda L / d = Lambda’ L’ / d = Lambda’ L / (G(v) d) => Lambda’ = G(v) Lambda **1) In the case of light** having a constant speed c = Lambda f = Lambda’ f’ = G(v) Lambda f’ => f’ = f / G(v). which is the transversal Doppler Effect with the resulting red shift of the fringe colors. This seems a priori reasonable since the moving experimental setup is observed transversally as it moves (from left to right), thus, there is no need to have a longitudinal Doppler Effect. However, this appears somehow in contradiction with the fact that the walk difference delta’(m) = m Lambda’ is drawn along a diagonal (at angle theta’) and has also a longitudinal component. What does that mean in terms of possible x and y components of (longitudinal and transversal) Doppler effects? What is missing or wrong here (either in the calculation or the interpretation)? **2) In the case of wave matter** (electrons) having the de Broglie wavelength Lambda = h/p, things become more complicated since one should suppose for the electron beam a speed u in the reference frame S and a speed w in S’ being related through the relative speed v of S’ by the subtractive relativistic composition of speeds (with u > v): w = (u – v)/(1 – uv/(c**2)) with the well-known geometric identity   w G(w) = (u – v) G(u)G(v) Then, the relation between wavelengths leads to a relation between p = p(u) = G(u) m0 u and p’ = p(w) = G(w) m0 w (where m0 stands for the electron rest mass and G for the relativistic gamma factor): Lambda’ = G(v) Lambda => h/p(w) = G(v) h/p(u) => p(w) = p(u)/G(v) which is surprising and possibly too simple to be true. Indeed, using the geometric identity w G(w) = (u – v) G(u)G(v), one can see that the correct composition of momenta for p’ = p(w) should read as : p(w) = G(w) m0 w = G(w) m0 (u – v) G(u)G(v)/G(w) = m0 (u – v) G(u)G(v) = G(v) G(u) m0 u – G(u) G(v) m0 v = G(v) p(u) – G(u) p(v) And again, as it was the case for the light situation, one should observe that these momenta are not necessarily longitudinal since they are related to the walk difference delta’(m) = m Lambda’ which is drawn along a diagonal (at angle theta’). So, what is wrong in this logic and in these probably too simple calculations? Thanks for your help! Wilj


title: Why doesn't diamond glow when hot?

text: In an answer to this SE question, the respondent explains that heating a perfect diamond will not cause it to glow with thermal blackbody radiation. I don't quite follow his explanation. I think it comes down to: there is no mechanism for diamond to generate light in the visible region of the spectrum. He mentions that interband transitions are well out of the visual range, so there will be no contribution from that. He mentions that the Debye temperature for diamond is > 2000 K. I presume that the argument here is that optical phonons will be frozen out, too. (But diamond doesn't have infrared-active phonons, does it?) So is that why hot diamond doesn't glow? I suppose that if one considers real (not ideal) crystals, imperfections, impurities, and the existence of surfaces lead to the possibility of emission mechanisms, and thus glow. In fact it might be the case that a finite but otherwise perfect crystal might have an extremely faint glow. Is this basically the reason that hot diamond does not glow? Further elucidation welcome.


title: Force on a test charge placed at the center of a 13-sided regular polygon?

text: Assume a test charge with charge $Q$ is placed at the center of a 13-sided regular polygon, the vertices of which are charges with equal charge $q$. What is the electrical force on the center charge? To start we write this as our electrical force... $$ \vec{F}_{Q} = \sum^{13}_{n} \frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{R^{2}} \left(cos\left(\frac{2n\pi}{13}\right)\hat{i} , sin\left(\frac{2n\pi}{13}\right)\hat{j}\right) $$ and with some simplification we get... $$ \vec{F}_{Q} = \frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{R^{2}} \left\\{1-2cos\left(\frac{\pi}{13}\right) + 2cos\left(\frac{2\pi}{13}\right)-2cos\left(\frac{3\pi}{13}\right) + 2sin\left(\frac{\pi}{26}\right)-2sin\left(\frac{3\pi}{26}\right)+2sin\left(\frac{5\pi}{26}\right), 0\right\\}$$ ...more simplification... $$ \vec{F}_{Q} = \frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{R^{2}} \left\\{ 1.11022\times10^{-16}, 0 \right\\} $$ Now we have an electrical force on the test charge, however, in a ring of equidistant charges such as the one mentioned above, shouldn't the $\hat{i}$ component be equal to zero? Does anyone have any ideas as to why I'm getting a non-zero number for my $\hat{i}$ component?


title: How does a freezer slow freeze food?

text: I have been told that the a freezer will freeze food slower if: a) the product load is greater then the capacity b) the capacity is greater then then the load. In a) I think this means if a freezer can freeze 2kg of food in 12 hours, using more food will cause the freezer to work slower and so freeze slower. Is this the case? In b) surely if the capacity is greater then the load that means it will freeze fast or at a normal rate? Or do you think it is referring to a much larger capacity i.e. if 1kg of food is placed inside a deep freezer, then it will freeze slower then if 1kg of food was placed inside a smaller freezer. If this is the case, why? Not sure what is right. Can somebody please explain. Thanks


title: What materials have an electroluminescent threshold in the millivolt or microvolt range?

text: Does anyone know any materials that have an electroluminescent threshold in the millivolt or microvolt range? Specifically, a stable solid state material. I hope to deposited the material as a thin film. Thank you for the help!


title: Lifespan of particles

text: After reading a very informative tutorial on elementary particles physics over at http://www.particleadventure.org, I have a question I can't figure out. I understand the need to accelerate a particle to high energy level, as it allows to view "smaller" stuff when you bombard a target with it. But the resulting particles that fly out into detectors have very short life as they quickly decay. Does it mean that those particles don't occur in nature because they decay very rapidly? And if so, what is the usefulness of studying those particles?


title: What is Verlinde's statistical description of gravity as an entropic force?

text: What is Verlinde's statistical description of gravity as an entropic force leads to the correct inverse square distance law of attraction between classical bodies? ![inverse square distance law of attraction between classical bodies](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/200px- NewtonsLawOfUniversalGravitation.svg.png)


title: Deriving the curve of a cantilever

text: Essentially, there is a beam of length L and negligible mass sticking out of a wall with a mass Mg hanging at the end of it. We are given an equation for elastic energy (which I don't think needs to be modified), and we have to come up with the potential energy contribution from the mass, and then hopefully be able to minimize the energy using variational techniques. The elastic energy per unit length can be approximated to be $U[y]=\int_0^L \frac{1}{2} EI (y'')^2 dz$ If I can properly define the potential energy of the mass, I think I can do the rest, but I can't figure out the right term. Thanks for reading, and please comment if there is more I should add, including my work involved getting the (incorrect) gravity terms.


title: What are the implications of the possibility that the BICEP2 results are caused by a self ordering scalar field transition?

text: I've found this interesting paper that mentions another possible way to interpret the recent BICEP2 results, and that hadn't been ruled out yet 1. As interesting as the possibility that the BICEP2 team has observed CMB polarization due to gravitational waves resulting from a self ordering scalar field transition sounds, I would like to know what its implications/significance would be!


title: What is the physical intepretation of harmonic coordinates?

text: > When I see harmonic coordinates used somewhere, what should my association > be? > > Is there some general use or need to consider the harmonic cooridnate > condition? I don't really see what's behind all the things said in the wikipedia article and I've seen these coordinates used in some papers but don't really understan their significance, other than that one can consequently algebraically use the relation $\Delta x^\mu=0$ in computations. I don't see what that means though, do the full set of these coordinates have some general geometrically visible properties, maybe characterized with the associated base of distributions? Moreover, do these cooridnates have some special role in geometric quantization?


title: Does the potential energy related to a particle determines its rest mass?

text: Would it be possible to determine the rest mass of a particle by computing the potential energy related to the presence (existence) of the particle, if this potential energy could be determined accurately enough? I noticed from the answers to a recent question that I always assumed this to be true, without even thinking about it. However, it occurred to me that this concept was at least unfamiliar to the people who answered and commented on that question, and that it's even unclear whether this concept is true or meaningful at all. Let me explain this concept for an idealized situation. Consider an idealized classical spherical particle with a charge $q$ and a radius $r$ at the origin. Assume that the particle generates an electrostatic field identical to the one of a point charge $q$ in the region outside of radius $r$ and vanishing inside the region of radius $r$. Now let's use a point charge $-q$ and move it to the origin in order to cancel this field in the region outside of radius $r$. Moving the point charge to the origin will generate a certain amount of energy, and that would be the energy which I mean by the potential energy related to the presence (existence) of this idealized classical spherical particle. I'm well aware that really computing the potential energy related to the presence (existence) of any real particle is not practically feasible for a variety of reasons, but that never worried me with respect to this concept. What worries me now is whether this notion of potential energy is even well defined at all, and even if it is, whether it really accounts for the entire rest mass (not explained by other sources of kinetic, internal or potential energy) of a particle. After all, the rest mass of a particle might simply be greater than the mass explained by any sort of potential energy.


title: What are the best astronomy blogs and podcasts?

text: For someone who likes learning about astronomy what are the blogs and/or podcasts that I should be following?


title: Are orbits interior to Jupiter's orbit less stable than orbits exterior to Jupiter's orbit?

text: Or in other words, are there differences in average Lyapunov timescale between orbits interior to Jupiter and orbits exterior to Jupiter? I'm trying to answer a question at http://www.quora.com/Why-does-Pluto-have-so-many- satellites/answer/Alex-K-Chen but I'm not totally sure if the last part of my answer is right. I'll quote the last part of it: > If the 2nd theory is true, then it's harder to answer this. One thing for > sure though: Jupiter is much farther away, so its tug on the system is a > much smaller factor than it is for the inner planets (where it can be a > major source of instability over the solar system's lifetime) > > In fact - I suspect that another factor is that from Pluto's perspective, > Jupiter is practically in the center of the solar system anyways, so you're > unlikely to see periods of time where Jupiter is in such a position where > its constant gravitational tugs (over several Jupiter orbits) can accumulate > and tug a satellite into an unstable orbit (which is what can happen with > planets that orbit the Sun at distances interior to Jupiter [1]). Anyways I > don't fully know the physics on this (yet) so some of my details could be > wrong - what I do know is that it could happen to both asteroid belt objects > and to Mercury's orbit - > http://en.wikipedia.org/wiki/Stability_of_the_Solar_System#Mercury.E2.80.93Jupiter_1:1_resonance > \- perhaps because there are positions where Jupiter's pull on interior > bodies is in a direction opposite to that of the Sun's pull (we never hear > about the Sun creating any orbital instabilities). And Mercury is much > closer to Jupiter than Pluto is. > > Of course, Pluto is vulnerable to Neptune's influence, but Pluto and Neptune > have a 3:2 orbital resonance so it's relatively safe from collisions with > Neptune (although the resonance may not be constant over the solar system's > lifetime) > > Anyways, we might finally know more once New Horizons reaches Pluto in a few > years > > [1] http://www.alpheratz.net/murison/papers/Lyapunov/LFM.pdf says that there > is something special about asteroids with orbits interior to Jupiter - I'll > try to find more information on this.


title: time dependent current/ magnetic field

text: Is there a general way to calculate the magnetic field for a time dependent current of a long thing wire? For ex: If the current is $$ I(t)=I\sin wt, $$ is there a general method to use in order to calculate the magnetic field? I know for time-independent currents we can use $$ \int \vec B\cdot d\vec l=\mu_0 I_{enc} $$ if the symmetry of the system is nice, or we can use Biot- Savart Law for other cases. Thanks...


title: What is the composition of the universe's population of neutrinos?

text: I believe earth-based detectors measure mainly solar neutrinos, which have energies on the MeV scale of nuclear physics, are directed from the sun, and have flavors determined by the sun's nuclear reactions plus neutrino oscillations. In addition to this, I recently learned from a physics.SE answer that the universe contains neutrinos that are thermalized with respect to the 3 K background temperature of the universe. I assume these latter can't be detected with present technology...? What is our best present knowledge of the universe's population of neutrinos? Energy spectrum? Flavors? Directions of motion? (Are they isotropic?) Spatial distribution? Are there antineutrinos in the mix? Is the neutrino contribution to the cosmological stress-energy tensor primarily from one component of this population?


title: "Blue Bumper" Stars

text: I was recently overviewing various massive compact halo object studies (the Anglo-Australian MACHO collaboration and the French I/II EROS collaboration), and they frequently reference "blue bumper stars," irregular variable stars which produce light curves very similar to gravitational microlensing event. Further searches for more information about them was mostly fruitless, producing a few conference proceedings:   * The MACHO Project LMC variable star inventory: Aperiodic blue variables (code _1995llnl.reptR....P_ )   * The MACHO Project LMC Variable Star Inventory: Aperiodic Blue Variables (code _1995AAS...18710202P_ ),  but little else. Is there any more information on this type of variable star, especially a more rigorous and targeted study of them? Also, how is it possible to get the full text for either of those articles?


title: Find net force on a mass centered between four other masses

text: I'm not going to post the full question, I just want a general idea of how I should go about solving this type of problem. There is a square with 4 charged masses on each of the corners, in the center of the square, there is a fifth charged mass. I am supposed to find the net force acting on the center mass ($m_{5}$). Now I could find the gravitational forces between the center mass and each of the corner masses, split that into x and y components, and then repeat the same process for the electrostatic forces. I'm pretty sure there is an easier, less cumbersome way to do this. Any ideas?               m2--------m1     |          |     |    m5    |     |          |     m3--------m4      None of the charges/masses directly cancel each other out, however the electrostatic ones seem to be small enough to be trivial in the final result.


title: Question about electromagnetic Spectrum

text: I have question related to electromagnetic Spectrum. If energy of photon is $E=m_e c^2$, to which part of the electromagnetic spectrum does it belong ?


title: Gregory-Laflamme Instability of Black Strings and $p$-Branes

text: In a paper by Gregory and Laflamme (http://arxiv.org/abs/hep-th/9301052) in 1993, it was demonstrated that black strings and $p$-branes which were solutions to certain low energy string theories were unstable by searching for solutions to the perturbation equations, $$\delta R_{ab}=-\frac{1}{2}\Delta_L h_{ab}=0$$ where $h_{ab}$ is a linear perturbation, and $\Delta_L$ the Lichnerowicz operator,$^{\dagger}$ computed with respect to the background solution, e.g. a black string. In addition, they formulated an _ansatz_ , $$h^{\mu\nu}= e^{\Omega t} e^{i\mu_i x^i}\left( \begin{matrix} H^{tt} & H^{tr} & 0 & 0 & \dots \\\ H^{tr} & H^{rr} & 0 & 0 & \dots \\\ 0 & 0 & K & 0 & \dots \\\ 0 & 0 & 0 & K/\sin^2 \theta & \dots \\\ \dots & \dots & \dots & \dots & \dots \end{matrix} \right)$$ My questions are the following:   1. What is the motivation for the _ansatz_ above?   2. Why are solutions to the perturbation equations which maintain the Ricci curvature (and hence energy-momentum tensor) deemed instabilities?   3. Is the general approach outlined above applied to most solutions to determine instabilities? Are there alternate methods (excluding the entropic argument by comparing solutions' entropies)? * * * $\dagger$ In de Donder gauge, the operator is given by $\Delta_L h_{ab}=\left[\delta^c_a \delta^d_b (\nabla_e \nabla^e) +2R^{cd}_{ab}\right]h_{cd}$.


title: What problems with Electromagnetism led Einstein to the Special Theory of Relativity?

text: I have often heard it said that several problems in the theory of electromagnetism as described by Maxwell's equations led Einstein to his theory of Special Relativity. What exactly were these problems that Einstein had in mind, and how does Special Relativity solve them?


title: How do I place point charges when using the method of images with a sphere?

text: > We have a point charge $q>0$ in vacuum at a distance $a$ from the center $M$ > of an isolated metal sphere that has a total charge $-q$ and radius $R$. > ($a>R$) > > A) Use the method of images to find a charge configuration that has the same > boundary conditions. (hint: you need 3 charges) For question A I know that my point charge $q$ should stay in the same place and that I need to place 2 other point charges inside the sphere. But I can't figure out what the charge and place of these two charges should be. My book (introduction to Electrodynamics, Griffiths) has an example where the sphere isn't charged but they seem to magically find the answer. They place a new charge $q'=-\frac{R}{a}q$ at a distance $b=\frac{R^2}{a}$ but they don't explain how this is a logical configuration. Can someone please explain this to me, as I would like to know how to place my point charges for different situations and not just one.


title: From power of a laser beam to electric field amplitude

text: In my experiment, I use a laser beam with wavelength $\lambda=894 \text{nm}$ for some magnetic resonance experiment. Right now, I'm doing some calculation using Quantum Mechanics, which requires the amplitude of the electric field to be inserted to my Hamiltonian. **My question is basically: How do I go from a power, measured in $\mu W$ to an Electric Field amplitude $E_0$ to use it in a QM's Hamiltonian?** So I think that I need to use the Poynting vector, which is defined as: $$\vec{S}=\frac{1}{\mu_0} \vec{E}\times \vec{B}$$ where I'm using $\mu_0$ because the interactions happen in a vacated, dilute gas cell. If I take the magnitude of the last equation I get $$S=\frac{EB}{\mu_0}=\frac{E^2}{c \mu_0}$$ where I used the fact that $E_0=c B_0$ according to Maxwell equations. However, the Poynting vector is energy per unit area per unit time, or power per unit area. This means that I have to know the area of my laser beam, **and that's my problem now**. The laser beam in my experiment comes out of a multi-mode fiber, and then it goes to a linear polarizer, and then a quarter wave plate, where we use the latter to get circularly polarized light, and then a collimator lens. And after that it interacts with my atoms. I know the core-diameter of my multi-mode fiber, let's call it $r_F$, but I don't know what transverse modes are produced in it. How can I know the effective area of my laser beam (call it $A_L$), so that I could substitute that in my Poynting vector and calculate the electric field amplitude of my laser beam? I imagine that eventually I should be able to calculate $A_L$, and I can use the following relation to get my electric field amplitude, which I get from the Poynting vector definition $$E=\sqrt{\frac{\mu_0 c S}{B}}=\sqrt{\frac{\mu_0 c P}{B A_L}}$$ where $P$ is the power of my beam. Is my reasoning correct? Please advise. If I'm missing something in the problem, let me know. Please note: this is a real experiment, so all practical approximations are allowed.


title: Obtain the same Maxwell's equation after a change of coordinates

text: In the usual $(x,y,z)$ system of coordinates, if we expand the Maxwell's curls equations for phasors $$\nabla \times \mathbf{E} = - \mathbf{J}_m - j \omega \mu \mathbf{H}$$ $$\nabla \times \mathbf{H} = \mathbf{J} + j \omega \epsilon \mathbf{E}$$ we obtain equations like the following: $$\displaystyle \frac{\partial H_y}{\partial z} - \frac{\partial H_z}{\partial y} = J_{x} + j \omega \epsilon E_x\tag{1}$$ Now, let $\mathbf{E}'$, $\mathbf{H}'$, $\mathbf{J}'$, $\mathbf{J}_m'$ be a new field (with its new sources). In particular we have $$H'_y = H_y$$ $$H'_z = - H_z$$ $$J'_x = - J_x$$ $$E'_x = - E_x$$ The equation (1) for this field becomes $$\displaystyle \frac{\partial H'_y}{\partial z} - \frac{\partial H'_z}{\partial y} = J'_{x} + j \omega \epsilon E'_x\tag{2}$$ If we define $x' = x$, $y' = y$, $z' = -z$, (2) becomes $$\displaystyle -\frac{\partial H'_y}{\partial z'} - \frac{\partial H'_z}{\partial y'} = J'_{x} + j \omega \epsilon E'_x$$ that is, substituting the primed quantities with the equivalent unprimed ones, $$\displaystyle - \frac{\partial H_y}{\partial z'} + \frac{\partial H_z}{\partial y'} = -J_{x} - j \omega \epsilon E_x$$ $$\displaystyle \frac{\partial H_y}{\partial z'} - \frac{\partial H_z}{\partial y'} = J_{x} + j \omega \epsilon E_x\tag{3}$$ The equation (2) can be written in the form (3): so, we have obtained the same equation as (1), but with respect to the primed variables! **Questions** : can (3) still be considered a _Maxwell's equation_ as well as (1)? Why? My question arises because (3) is weird: it involves components of fields along the unprimed unit vectors ($\mathbf{u}_x$, $\mathbf{u}_y$ and $\mathbf{u}_z$), and derivation with respect to the primed variables ($x'$, $y'$ and $z'$). In particular, $H_z$ has a different behaviour according to the system of coordinates, because $\mathbf{u}_z = - \mathbf{u}_{z'}$. So, how can this be taken into account while evaluating (3)? This is what I can't understand. This demonstration is used to derive the fields in the Method of the Images in the Electromagnetic theory (the primed field is the image field).


title: Electromagnetic field and continuous and differentiable vector fields

text: We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields. We know electrostatic and magneto static fields aren't actually well behaved. They blow up at the sources, have discontinuities and yet we use the same mathematical formulations for them as we would have done for continuous and differentiable vector field. Why is this done ? Why are laws of electromagnetism(maxwell's equations) expressed in the so called differential forms when clearly that mathematical theory is not perfectly consistent with the electromagnetic field. Why not use a new mathematical structure ? Is there a source/book, which can help me overcome these issues without handwaving at particular instances when the methods seem to give wrong results. **EDIT** : Also one of the major concerns is that, given a charge distributions, the maxwell equations in differential form, will always give a nicely behaved continuous and differentiable vector field solution. But the integral form (alone, not satisfying the differential form) can give a discontinuous solution as well. Leading to two different answers for the same configuration of charges. hence there is an inconsistency. Like there is an discontinuous solution for the boundary condition of 2D surface, the perpendicular component of the electric field is discontinuous. ( May be it is just an approximation) and actually the field is continuous but due to not being able to solve the differential equation we give such an approximation, but this isn't mentioned in the textbooks.


title: Can sound produce electricity?

text: Energies of wind and water can produce electricity. But, can sound energy also be used to produce electricity?


title: Perturbation metric problem

text: I know this is an already answered question, but I couldn't make head or tail of it, and it's bugging me. I know I'm probably asking a silly question, but please bear with me as I'm 14 and this is my first post. I saw this perturbation while studying up a bit of linearised gravity: $$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} \tag{1}$$ and then the next line says: " you raise the indices to get": $$g^{\mu\nu} = \eta^{\mu\nu} - h^{\mu\nu}. \tag{2}$$ I have done a bit of tensor calculus, but I still couldn't understand how the obvious fact $g_{\mu\nu}\, g^{\mu\rho} = \delta^{\rho}_{\nu}$ leads to the second equation. I tried everything I could, but all I get is the first equation with raised indices different from the given ones, and a persistent + sign. Please explain and derive the second equation step by step, as if to an idiot.


title: Landau Levels and Magnetic length

text: I am looking into Landau Levels and keep coming across a magnetic length defined as follows for a 2D system: $l_B=\sqrt{\frac{\hbar c}{e B}}$ I have seen numerous sources say this is: $l_B\approx 26nm \sqrt{B[Tesla]} $ But by plugging those numbers in I don't see how this is the case - the units are just plain wrong. The thing is, I cannot for the life of me figure out what type of theorist units they used to 1) get this expression in the first place, and 2) calculate it for something realistic. The only hint I have is that it was done in the "Landau Gauge" - googling this yields nothing useful. What units are being used, and how did they calculate this number?


title: Derivation of the oil drop experiment

text: The oil drop experiment performed by Robert Millikan in 1909 enables us to calculate the elementary charge. The following forces have effect on the droplet:   1. $F_\mathrm{G} = m g = \frac{4}{3} \, \pi r^3 \rho_\mathrm{Oil} g$   2. $F_\mathrm{A} = \frac{4}{3} \, \pi r^3 \rho_\mathrm{Air} g$   3. $F_\mathrm{E} = q E = \frac{qU}{d}$   4. $F_\mathrm{R} = 6\,\pi\eta r v$ However, which steps of calculation are necessary to finally calculate the elementary electric charge (charge of the electron) or at least a multiple of it? $$q = \frac{9 \pi d}{2 U} \sqrt{\frac{\eta^3 (v_1 - v_2)}{\rho^* g}} (v_1 + v_2)$$


title: What is Time like Quantum Entanglement?

text: I came across a New Study at : http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.2565v1.pdf . Which talks about Time like quantum entanglement. What does that mean? **Comment added by L.Motl** : The same preprint has been discussed in this question and will be discussed in this question in the future.


title: What is a microstate, macrostate and thermodynamic probability in statistical mechanics?

text: Currently I am learning Maxwell-Boltsmann distribution (MBD) and in that I am learning about microstate, macrostate and thermodynamic probability (TDP). I understood the derivation of MBD but I am getting hard time (and also confusing) to visualise these terms (microstate, macrostate and TDP) with MBD. Can any one explain (possibly with pictures) what is microstate, macrostate and TDP and how it is used to derive MBD.


title: Inverting the laws of physics

text: Are there any good theories on inverting the law of physics? To make physics work the opposite way of what it does. Please ask question if this is unclear! I'll answer as well as I can. Ok, I'll try to explain better. I was taking a smoke break and started thinking about time travel, and somehow I concluded that to travel in time itself is impossible. But then I thought that, what if we instead of travel in time, could reverse all processes in the world, or in an area for example. To make all particles do all what they have already done, but in reversed order. i.ex. A candlelight produces heat and light emitting from the flame, the candle itself melts and turns into vapor. If we reverse that, all the heat, vapor and light would be absorbed by its origin point and reassemble the candle the way it was. Damn this is hard to explain. And there are so many holes in what I'm thinking here.


title: Magnetic induction outside of the finite length solenoid

text: I'm programing simulation software for solenoid and hemholtz coils, where I have to calculate **B(x,y,z)**. I have found dozens of variations of Biot- savart law for determining B on axis. But how about **outside of the solenoid**? This is said to be outside of solenoid: $$ B(z) = \frac{\mu_0}{4\pi}\cdot\frac{iR^2}{2(R^2+z^2)^{\frac{3}{2}}} $$   But why only z coordinate? What about length of solenoid? In order to build simulation, I have to count with finite-length solenoid. There are hints, that I should apply line integral (curve integral?) to the biot-savart law. But what is that curve? The electric wire? So I have to use eliptic integral to have something close to the solenoid? I'm quite lost in all of this.


title: What is the necessary amount of energy to propel a 100g from ground to orbit ? (Slingatron)

text: I would like to know what is the energy needed **considering the air resistance** to shoot from ground to the orbit a 100g object (even if it vaporizes in the process). **Variables to make approx calculations:** Lets assume it has a **pointy bullet shape** (best know shape) with **5cm diameter x 10cm length** and the orbit is the lower possible, lets assume the **ISS orbit 150km above ground** , **we are at equator latitude** shooting to the **east** ( **Earth spining thrust, 1500km/h** ) The first thing that crossed my mind was the russian meteor entry and its tremendous bright and shock wave (dissipated energy) According to a news on physicsworld.com


title: How to mathematically formulate the Two Slit Experiment in Special Relativity?

text: How to mathematically formulate the Two Slit Experiment in a Lorentz invariant framework. Is there a paper about this?


title: Maximum acceleration for wheel-driven vehicle from standstill

text: Once I discussed cars and there was a question about rate at which the car gains its speed ($100\:\mathrm{km/h}$) from standstill (i.e. acceleration). At some point of conversation it came up that cars cannot gain their $100\:\mathrm{km/h}$ speed for such tiny time interval I proposed (less than $3$ seconds), because _it can't go with acceleration more than $g$_ i.e. _free fall accel_ on Earth (we talked about wheel-driven vehicles - not a rocket- driven - and all we know that at certain acceleration value the wheels lose their grip with ground and vehicle is not moving forward it's burning tyres istead). I was disagree, I feel intuitively that the critical acceleration for the wheel traction exists but it couldn't be equal to $g$ or approaching to, because it should look like a some deep law of the nature (between friction and acceleration) and this phenomenon would be widely known (but I couldn't find anything). We started searching for contra-evidence. I found `Bugatti Veyron` which achieves $100\:\mathrm{km/h}$ speed roughly for $2.5\:s$ and some `Formula 1` cars. But the sources of acceleration values were considered as non-reliable and not proven. I know that the friction and cohesion proccesses are very untrivial, it brings electromagnetic interactions between atoms and molecules on microscopic level as well as surface geometry on macrolevel - that was another my argument against the statement that _acceleration can't be more than $g$ without traction loss_ \- because such complex and heterogeneous process as wheel traction can't have such a simple solution and equal to $g$. So, could someone prove it or provide a formula or methods to show that _critical acceleration of wheel-driven body without traction loss could be more than $g$_ (or less)?


title: Tensor Product of Hilbert spaces

text: This question is regarding a definition of Tensor product of Hilbert spaces that I found in Wald's book on QFT in curved space time. Let's first get some notation straight. Let $(V,+,*)$ denote a set $V$, together with $+$ and $*$ being the addition and multiplication maps on $V$ that satisfy the vector space axioms. We define the complex conjugate multiplication ${\overline *}:{\mathbb C} \times V \to V$ as $$ c {\overline *} \Psi = {\overline c} * \Psi,~~\forall~~\Psi \in V $$ The vector space formed by $(V,+,{\overline *})$ is called the complex conjugate vector space and is denoted by ${\overline V}$. Given two Hilbert spaces ${\cal H}_1$ and ${\cal H}_2$ and a bounded linear map $A: {\cal H}_1 \to {\cal H}_2$, we define the adjoint of this map $A^\dagger: {\cal H}_2 \to {\cal H}_1$ as $$ \left< \Psi_2, A \Psi_1 \right>_{{\cal H}_2} = \left< A^\dagger \Psi_2 , \Psi_1 \right>_{{\cal H}_1} $$ where $\left< ~, ~ \right>_{{\cal H}_1}$ is the inner product as defined on ${\cal H}_1$ (similarly for ${\cal H}_2$) and $\Psi_1 \in {\cal H}_1,~\Psi_2 \in {\cal H}_2$. That such map always exists can be proved using the Riesz lemma. Here the word "bounded" simply means that there exists some $C \in {\mathbb R}$ such that $$\left\| A(\Psi_1) \right\|_{{\cal H}_2} \leq C \left\| \Psi_1 \right\|_{{\cal H}_1}$$ for all $\Psi_1 \in {\cal H}_1$ and where $\left\| ~~ \right\|_{{\cal H}_1}$ is the norm as defined on ${\cal H}_1$ (similarly for ${\cal H}_2$) Great! Now for the statement. Here it is. > The tensor product, ${\cal H}_1 \otimes {\cal H}_2$, of two Hilbert spaces, > ${\cal H}_1$ and ${\cal H}_2$, may be defined as follows. Let $V$ denote the > set of linear maps $A: {\overline {\cal H}}_1 \to {\cal H}_2$, which have > finite rank, i.e. such that the range of $A$ is a finite dimensional > subspace of ${\cal H}_2$. The $V$ has a natural vector space structure. > Define the inner product on $V$ by $$ \left< A, B \right>_V = > \text{tr}\left( A^\dagger B \right) $$ (The right side of the above equation > is well defined, since $A^\dagger B: {\overline {\cal H}}_1 \to {\overline > {\cal H}}_1$ has a finite rank). We define ${\cal H}_1 \otimes {\cal H}_2$ > to be the Hilbert space completion of $V$. It follows that ${\cal H}_1 > \otimes {\cal H}_2$ consists of all linear maps $A: {\overline {\cal H}}_1 > \to {\cal H}_2$ that satisfy the Hilbert-Schmidt condition $\text{tr}\left( > A^\dagger A \right) < \infty$. My question is **1\. How does this definition of the Tensor product of Hilbert spaces match up with the one we are familiar with when dealing with tensors in General relativity?** PS - I also have a similar problem with Wald's definition of a Direct Sum of Hilbert spaces. I have decided to put that into a separate question. If you could answer this one, please consider checking out that one too. It can be found here. Thanks!


title: Number operator and Dirac field (with anticommutation relations)

text: Before using anticommutation relatives the energy, momentum, charge and number operators of the Dirac field have following expressions: $$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {\mathbf P} = \int \mathbf p \left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) - \hat {b}_{s}(\mathbf p )\hat {b}_{s}^{+}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {Q} = \int \left( \hat {a}^{+}_{s}(\mathbf p )\hat{a}_{s}(\mathbf p ) + \hat{b}_{s}(\mathbf p )\hat{b}^{+}_{s}( \mathbf p )\right)d^{3}\mathbf p, $$ $$ \hat {N} = \int \left( \hat {a}^{+}_{s}(\mathbf p )\hat{a}_{s}(\mathbf p ) + \hat{b}_{s}(\mathbf p )\hat{b}^{+}_{s}( \mathbf p )\right)d^{3}\mathbf p. $$ After using anticommutation relations and "neglecting" of infinite constant $\delta (0)$ (as we can do in a case of free field) in an expressions of energy and momentum operators, they transformed into $$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) + \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p ) \right)d^{3}\mathbf p, $$ $$ \hat {\mathbf P} = \int \mathbf p \left( \hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) + \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p ) \right)d^{3}\mathbf p . $$ But I have a couple of questions in connection with number and charge operators.   1. What can I do with infinite constant into espressions of charge and number operators? For example, $$ \hat {N} = \int (\hat {a}^{+}_{s}(\mathbf p )\hat {a}_{s}(\mathbf p ) + \delta (0) - \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p ))d^{3} \mathbf p . $$ Can I neglect summand with infinite constant in a reason that it is connected with infinite "vacuum" energy of field?   2. (Under the condition of neglecting infinite constant). When I acted on one-antiparticle state $|E_{\mathbf k }\rangle = \hat {b}^{+}_{s'}(\mathbf k )| \rangle$ by number operator, I got minus: $$ \hat {N}{b}^{+}_{s'}(\mathbf k )| \rangle = -\int \hat {b}^{+}_{s}(\mathbf p )\hat {b}_{s}(\mathbf p )\hat {b}^{+}_{s'}(\mathbf k)d^{3}\mathbf p | \rangle = -| E_{\mathbf k } \rangle . $$ But $\hat {b}^{+}_{s}(\mathbf p )$ is creation operator (i.e. it was the creation operator before using anticommutation relations; the relations changed this (?)). Did I make the mistake?


title: How do they measure the rotational speed of cricket balls?

text: The 2013 Ashes series (a cricket thing between England and Australia) are underway as of today and it seems they have a new (to me) gimmick. It appears that they are now able to measure almost instantaneously (within a few seconds) the rotational speed of balls bowled by spin bowlers. ![enter image description here](http://i.stack.imgur.com/lSivF.jpg) On the (golf) driving range, I was able to get similar readings, but they were from a machine only a few feet away from the tee. I don't quite understand that either, but it seems at least easier. Although there, they were even able to determine the rotational axis as well. ![enter image description here](http://i.stack.imgur.com/tpu8i.jpg) But in cricket they also have a rotational axis along the direction of the ball, which is even more difficult to measure (unless they have a third radar somewhere perpendicular- _ish_ to the pitch). _I think_ I now see that they placed very small black boxes between two of the stumps on either end of the pitch. I guess that must be the first two radars. (One of which will be useless because the batsman is standing in front of it.) ![enter image description here](http://i.stack.imgur.com/KaMpw.jpg) So, essentially: How do they do it? And, how accurate can it be, given that the reflection of the "sides" of the ball may be poor? Or are they using (visible-light) camera images, as suggested by the comments of udiboy and Deer Hunter below? (I'm fairly confident the golf system doesn't do that, but I don't _know_. Anyways, to make sure, the question is primarily about the cricket.) * * * _Added later._ I found that they measure spin rates in baseball as well: http://sportsillustrated.cnn.com/2011/writers/tom_verducci/04/12/fastballs.trackman/index.html. The http://trackman.dk site refers to these technologies and (patent/patent application) numbers: >   * Measuring spin rate of sports balls by radar using multiple harmonic > spectrum traces. (US2009/0075744, EP1698380, DE602006009719.0, > GB/EP/1698380, ZL200680006869.0, JP2008/538085A and KR10/0947898) > >   * Measuring spin axis orientation of sport balls from trajectory > measurements by radar. (US2009/0075744, DE602006015036.9, GB/EP/1853362, > ZL200680006869.0, JP2008/538085A and KR10/0947898) > >


title: Direct Sum of Hilbert spaces

text: I am a physicist who is not that well-versed in mathematical rigour (a shame, I know! But I'm working on it.) In Wald's book on QFT in Curved spacetimes, I found the following definitions of the direct sum of Hilbert spaces. He says - > Next, we define the direct sum of Hilbert spaces. Let $\\{ {\cal H}_\alpha > \\}$ be an arbitrary collection of Hilbert spaces, indexed by $\alpha$ (We > will be interested only with the case where there are at most a countable > number of Hilbert spaces, but no such restriction need be made for this > construction). The elements of the Cartesian product $\times_\alpha {\cal > H}_\alpha$ consist of the collection of vectors $\\{\Psi_\alpha\\}$ for each > $\Psi_\alpha \in {\cal H}_\alpha$. Consider, now, the subset, $V \subset > \times_\alpha {\cal H}_\alpha$, composed of elements for which all but > finitely many of the $\Psi_\alpha$ vanish. Then $V$ has the natural > structure of an inner product space. We define the _direct sum Hilbert > space_ $\bigoplus\limits_\alpha {\cal H}_\alpha$ to be the Hilbert space > completion of $V$. It follows that in the case of a countable infinite > collection of Hilbert spaces $\\{{\cal H}_i\\}$ each $\Psi \in > \bigoplus\limits_i {\cal H}_i$ consists of arbitrary sequences > $\\{\Psi_i\\}$ such that each $\Psi_i \in {\cal H}_i$ and $\sum\limits_i > \left\| \Psi_i \right\|_i^2 < \infty$. Here $\left\| ~~\right\|_i$ is the norm defined in ${\cal H}_i$. Also, Hilbert space completion of an inner product vector space $V$ is a space ${\cal H}$ such that $V \subset {\cal H}$ and ${\cal H}$ is complete in the associated norm. It is constructed from $V$ by taking equivalence classes of Cauchy sequences in $V$. Now the questions - **1\. Why does $V$ have the structure of an inner product space?** **2\. How does he conclude that $\sum\limits_i \left\| \Psi_i \right\|_i^2 < \infty$?** **3\. How does this definition of the direct sum match up with the usual things we see when looking at tensors in general relativity or in representations of Lie algebras, etc.?** PS - I also have a similar problem with Wald's definition of a Tensor Product of Hilbert spaces. I have decided to put that into a separate question. If you could answer this one, please consider checking out that one too. It can be found here. Thanks!


title: state for a classical particle

text: > **Possible Duplicate:**   >  Why are coordinates and velocities sufficient to completely determine the > state and determine the subsequent motion of a mechanical system? In classical mechanics, if all the generalized coordinates and the generalized velocities are simultaneously specified, the state of the system is completely determined and one can calculate its subsequent motion. That also means if we know generalized coordinate $q$ and generalized velocity $\dot{q}$, then we can calculate acceleration $\ddot{q}$. How to prove this?


title: Why doublons and holons are not bounded in spin-1/2 Hubbard chain?

text: The Hubbard model reads $$H = -t \sum_{\langle ij \rangle, \sigma} c_{j\sigma}^\dagger c_{i\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow} $$ In the large $U$ limit and at half-filling, the Hubbard model tranforms to $t$-$J$ model $$H = -t \sum_{\langle ij \rangle, \sigma} c_{j\sigma}^\dagger c_{i\sigma} + J\sum_{\langle ij \rangle}S_i\cdot S_j$$ where $J=4t^2/U$. Obviously, ground state favors anti-parallel spin configurations but it is not obvious to see whether it is the Neel state $(\uparrow\downarrow\uparrow\downarrow...) - (\downarrow\uparrow\downarrow\uparrow...)$ or the dimerized state $(\uparrow\downarrow)(\uparrow\downarrow)...$ where $(\uparrow\downarrow) = \uparrow\downarrow - \downarrow\uparrow$. Calculation shows, in 1-D, dimerized state has lower energy (I'm not very clear about the calculation.) It is also known that holon and doublon are not bounded in 1D (absence of Mott transition, Lieb-Wu's analytic result + VMC simulations). While in higher dimensions, Mott transitions do occur and dimerization is suppressed. It seems that the occurance of dimerization is correlated with the absence of Mott transition. So are there any pictures that show why Neel states lead to bounded doublon-holon?


title: What math is needed to understand the Schrödinger equation?

text: If I now see the Schrödinger equation, I just see a bunch of weird symbols, but I want to know what it actually means. So I'm taking a course of Linear Algebra and I'm planning on starting with PDE's next month. What 'other math' is needed other than Linear Algebra and (Partial) Differential Equations to gain a full understanding of this equation?


title: what could generate a high-pitched whine in electronics where the frequency depends on the current?

text: So I was operating a Laue machine recently which generates x-rays and fires them at a sample, and I noticed that there was a high-pitched whine coming from the instrument (nothing abnormal, just something I noticed). Both the current and the voltage could be set separately; typically values are 10 mA and 20 kV. I noticed that if you change the current, the frequency increases or decreases with it. A frequency spectrum app on my phone showed that the frequency was roughly proportional to the current being applied, so, for example, at 5 mA there'd be a 2000 Hz sound and at 10 mA it'd be 4000 Hz. (I don't remember the exact frequencies. I do remember there were lots of harmonics excited.) The voltage, on the other hand, only caused a small change of the frequency even if doubled. My question is how would such a sound be generated? My best guess is that the vacuum tube where the electrons are accelerated has its current adjusted by moving either the anode or cathode, and that forms a variable capacitor which happens to modulate the resonance frequency of the circuit such that the frequency and current increase together. Is this reasonable?


title: How bright was the AGN eruption at the center of the Milky Way ~2 Ma ago?

text: The black hole at the center of the Milky Way is believed to have erupted as an active galactic nucleus (AGN) approximately 2 million years ago, at which time it would have been visible to early humans. How visibly bright would such an eruption have been, and how sudden would its onset have been?


title: Betti multiplets in Kaluza Klein compactifications

text: It is well known that if the compactification manifold of a supergravity theory has non-zero Betti numbers, this may lead to the so called Betti multiplets in the spectrum of the low dimensional theory. A famous example is compactification of IIB supergravity on $T^{1,1}$, where a Betti multiplet shows up because of the nonzero second Betti number of $T^{1,1}$. My question is this: is it the $L^2$-Betti numbers that necessitate Betti multiplets in the low dimensional theory, or just normal Betti numbers? In particular, do Betti numbers generated by smooth (fixed point free) discrete identification (orbifolding) of trivial manifolds lead to Betti multiplets? (I am actually not even sure if smooth orbifolding of trivial topologies can yield non-zero Betti numbers.) Is there a good reference I can look into for that?


title: Boundary condition for a floating electrostatic potential

text: I have a (probably) simple question regarding boundary conditions. In electrostatic simulations, the relevant Maxwell equation is $\nabla \cdot \mathbf{D}=\rho$ where $\mathbf{E}=-\nabla V$, and $\mathbf{D}$ is related to $\mathbf{E}$ through any applicable constitutive relations. When a floating- potential metal object is embedded inside a domain, I've read that the relevant boundary condition for the metal surface is $$\int_{\partial\Omega}\mathbf{D}\cdot\mathbf{n}dS=0.$$ For example, in this picture the red square is at potential 1V, the blue square is at 0V and the bottom edge is an electrically-isolated metal surface: ![enter image description here](http://i.stack.imgur.com/pXMWv.png) Is there a simple explanation for why the integral boundary condition indeed describes a floating equipotential metal surface? The source I found it in (COMSOL's documentation) does not provide a derivation. Apologies if this is intro E/M stuff that I've just forgotten. I keep thinking of using the divergence theorem, but the boundary $\partial\Omega$ does not necessarily enclose a space over which a volume integration may be performed.


title: what is wrong with the following argument about stokes law in compact universes?

text: I want to understand what is wrong with the following argument: _in a topologically compact spacetime, a closed 3D boundary separates the spacetime in two connected components, because of this continuous symmetry, either one of the two things must happen_ 1) _the stokes theorem is not valid_ 2) _the integral of the derivative of any form field in each connected component must be equal in magnitude and opposite in sign, since both must be equal to the surface integral of the form field, and they must be all identical to zero at all times_ In layman terms, if the space is compact, a gauss surface does not separate the space neatly in a interior region and an external region, rather, both connected components are topologically equivalent, so one is led to the startling conclusion that the validity of the stokes theorem (which naively looks of a local, differential nature) depends for its validity on the asymptotic boundary conditions of the space


title: Can the Klein-Gordon Equation represent Particles with non-zero spin?

text: Every Solution of the Dirac Equation is also a solution of the Klein-Gordon equation. So the K-G equation does not necessarily represent particles with non-zero spin. Would it be incorrect to conclude that KG equation relates solely to zero-spin particles?


title: Why do we test electric fields with positive charges and not negative ones?

text: Is there any difference between using a positive versus a negative charge to test an electric field?