title: Why does the note played by a flute not change in this case when  the air column is interrupted?

text: I play the flute as a hobby, and I've noticed that when playing middle D or E flat, one can interrupt the air column by releasing a certain key (which is near the middle of the air column), and yet have no effect on the pitch (though the quality changes for the better). I'll be putting a few diagrams here, since it is hard describing the situation in words. The black portions of the diagrams represent closed holes--basically "air cannot escape from here". The gray represents holes which are closed due to lever action, but need not be. Here's a diagram without coloring (all diagrams are click-to-enlarge): ![enter image description here](http://i.stack.imgur.com/Zwe1Um.jpg) The mouthpiece is attached on the left, as marked in the diagram. The second key is just a ghost key connected to the first (and has no hole underneath it), so I'll just remove it from the diagrams: ![enter image description here](http://i.stack.imgur.com/4anOpm.jpg) # A few examples Alright. Normally, when playing consecutive notes, you make the air column shorter by releasing a key. For example, this is F: ![enter image description here](http://i.stack.imgur.com/SP75Im.jpg) This is F#: ![enter image description here](http://i.stack.imgur.com/Mj7dym.jpg) And this is G: ![enter image description here](http://i.stack.imgur.com/AAUeEm.jpg) One can easily see the physics behind this, an unbroken air column is formed from the mouthpiece. # The weird stuff Now, let's look at middle D and E flat: D: ![enter image description here](http://i.stack.imgur.com/d95T2m.jpg) E flat: ![enter image description here](http://i.stack.imgur.com/tG9v3m.jpg) Here, the air column is broken in between. I feel that both should play the same note, that is C#: ![enter image description here](http://i.stack.imgur.com/wGxZ4m.jpg) But they don't. I _can_ close the hole, creating an unbroken air column in both cases, but the sound quality diminishes. # A bit more experimentation (aka "what have you tried?") ## Reading this section is strictly optional, but will probably help I did a lot of experimenting with this key, turning up some interesting results. Hereafter, I'm calling the key "the red key", and marking it as such in the diagrams. When the red key is "closed", no air can escape and it forms part of the air column.   * If I play low D/E flat, I only get a clear note when the red key is _closed_. With it open, I get a note which has extremely bad quality, as well as being off-pitch. This is markedly opposite with what happens on middle D/E flat (mentioned above), there there is no change in pitch, and the difference in quality is _reversed_.  ![enter image description here](http://i.stack.imgur.com/td2fjm.jpg) Pictured: Low E flat (for low D extend the RHS of the black portion a bit more). Note that the fingering, save the red key, is the same for middle D/E flat Actually, this seems to be happening for _all_ of the low notes--each one is affected drastically when the red key is lifted.   * Going on to the notes immediately after E flat     * For E, quality is drastically reduced when the red key is open. The harmonic (second fundamental) of E, which is B, is more prominent than the note itself. One can make the E more prominent by blowing faster, but this reduces quality. Red key closed gives a clear note, as it should. ![enter image description here](http://i.stack.imgur.com/8QopWm.jpg)     * For F, a similar thing happens as with E. With the red key closed, it plays normally. With it open, you hear a medium-quality C (first harmonic of F), and no F at all. Blowing faster just gives a high C. ![enter image description here](http://i.stack.imgur.com/V5GnPm.jpg)   * The notes immediately below D have a fingering starting from no keys pressed (has to happen every octave, obviously). For the first few notes here, lifting the red key gives you C#, as expected. (I'm not explicitly marking the key red here, otherwise it'll get confusing what the correct fingering is)     * In C#, pressing the red key will obviously change the note ![enter image description here](http://i.stack.imgur.com/FJthmm.jpg)     * ....to C: ![enter image description here](http://i.stack.imgur.com/LhuSqm.jpg) Obviously, lifting the red key here will get you back to C#     * One (half) step lower, we have B, which again goes to C# when the red key is lifted ![enter image description here](http://i.stack.imgur.com/1etnlm.jpg)     * It gets interesting again when we play B flat. Lifting the red key here gives a note between C# and C ![enter image description here](http://i.stack.imgur.com/qE7uJm.jpg)     * And if we go down to A, lifting the red key gives us a C ![enter image description here](http://i.stack.imgur.com/SXu3Pm.jpg)   * And a bit of experimentation with the trill keys (the actual holes are on the other side of the flute). Whereas messing with the red key for D and E flat produces no change of pitch, messing with the trill keys (which are the same size as the red key and are furthermore pretty near it) does.     * Hitting the second trill key while playing D gives E flat. One should note that this second trill key opens the hole closest to the mouthpiece. ![enter image description here](http://i.stack.imgur.com/n8fWHm.jpg) Note the visual similarity between this and the situation in the "the weird stuff" section     * Hitting the first trill key while playing D gives a note between D and E (the two trill keys are close to each other, you may have to see the enlarged version to get the difference) ![enter image description here](http://i.stack.imgur.com/T2odhm.jpg)     * Hitting the second trill key while playing E flat gives a note between E and F      * Hitting the first trill key while playing E flat gives E flat (No diagram here, these two are the same as the last two, except that the far right edge of the black portion is closer) * * * # The Question Now, the red key(and the trill keys) are about half the diameter of the other keys. I suspect that this is quite significant here, but I can't explain it myself. My main question is, **why does disturbing the air column as shown in the section "the weird stuff" not change the pitch?** One has added an escape route for air, the column should then vibrate as if the remaining keys were open--that is C#. I suspect that the underlying principle is the same, so I have a few other related questions (optional):   * Why does the red key not change the pitch on D/E flat, but makes it go into the second fundamental/harmonic for E and above?   * Why does the red key change the pitch to notes which are not harmonics, instead _close_ to C#(one of them isn't even part of the chromatic scale--it is between two notes) for B flat and A?    * The red key is pretty similar to the trill keys with respect to size and general location. Yet, using a trill key on D changes the pitch, whereas using the red key doesn't. Why is this so?


Why does the cosmic censorship conjecture hold so well? Penrose proposed spacelike singularities and closed timelike curves are always hidden behind event horizons in general relativity. His conjecture appears to hold pretty well. But he only assumed some energy condition (null, weak, dominant) for it to hold, and some topological assumptions on the initial conditions? That makes it a mathematical conjecture? If so, why does it hold so well?


title: What is the minimal symmetry required for a spin Hamiltonian to describe a spin-liquid ground state?

text: Let's restrict to the case of spin-1/2 system. As we know, a spin-liquid (SL) state is the ground state of a lattice spin Hamiltonian with no spontaneous broken symmetries (sometime it may spontaneously break time-reversal symmetry and is called a chiral SL), where two _essential_ symmetries of a SL state are **lattice translation** and **spin-rotation** symmetries. Since, traditionally, we usually describe a SL state by using a spin Hamiltonian with the full $SU(2)$ spin-rotation symmetry (e.g., Heisenberg model), and the corresponding SL state is hence also $SU(2)$ symmetric, i.e., a RVB type SL. While, the honeycomb Kitaev model provides us an exact SL ground state with $Q_8$ spin-rotation symmetry, where $Q_8$ is a finite subgroup of $SU(2)$, indicating that the Kitaev SL does NOT belong to the RVB type. Thus, my question is: Generally speaking, what is the **minimal spin-rotation symmetry** required for a spin Hamiltonian to describe a SL ground state? Is $Q_8$ group the minimal one? Thank you very much. [My motivation for this question is that for a spin Hamiltonian without any spin-rotation symmetry, whether or not can it possess a SL ground state? And does the existence of a SL state with some spin-rotation symmetry imply the occurrence of emergent symmetries?]


title: Why is the Heisenberg uncertainty principle stated the way it is?

text: I spent a long time being confused by the Heisenberg uncertainty principle in my quantum chemistry class. It is frequently stated that the "position and momentum of a particle cannot be simultaneously known to arbitrary precision" (or any other observables $[A, B] \neq 0$). This made no sense to me -- why can't you measure both of these? Is my instrument just going to stop working at a certain length scale? The Internet was of little help; Wikipedia describes it this way as well and gets into philosophical arguments on what "position" and "momentum" mean and whether they really exist (in my opinion, irrelevant nonsense that has no effect on our ability to predict things). Eventually it was the equation itself that gave me the most insight: $$\sigma_x \sigma_p \geq \frac{\hbar}{2}$$ Look at that – there's two standard deviations in there! It is _impossible_ by definition to have a standard deviation of one measurement. It requires multiple measurements to have any meaning at all. After some probing and asking around I figured out what this really means: Multiple repeated measurements of identically prepared systems don't give identical results. The distribution of these results is limited by that formula. Wow! So much clearer. Thus $\hat{r}(t)$ and $\hat{p}(t)$ _can_ be known for the same values of $t$ to as much precision as your measuring equipment will allow. But if you repeat the experiment, you won't get identical data. Why doesn't everyone just state it that way? I feel like that would eliminate many a student's confusion. (Unless, of course, I'm still missing something – feel free to enlighten me should that be the case). EDIT: This post was at +1. Who downvoted me? I took a while to write out my question clearly and made sure it followed the guidelines on here.


title: Open String Coupling

text: I'm reading Zwiebach's First Course in String Theory. At present I'm learning about string coupling. Zwiebach says it's possible to prove that $g_o^2=g_c$ where $g_o,\ g_c$ are open and closed string coupling constants respectively. He claims this is due to certain topological properties of world sheets. Why, more precisely, is this true? Could someone explain and/or point me towards a proof? Many thanks!


title: Can experiment distinguish the basis in which a singlet state is represented?

text: Let $\left(|\uparrow\rangle,|\downarrow\rangle\right)$ and $\left(|\nearrow\rangle,|\swarrow\rangle\right)$ be two bases of the $2$-dimensional Hilbert space $H$. Can an experiment distinguish between $\frac 1{\sqrt 2} \left(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle\right)$ and $\frac 1{\sqrt 2} \left(|\nearrow\swarrow\rangle - |\swarrow\nearrow\rangle\right)$? As vectors in the Hilbert space $H\wedge H$, the two clearly coincide (at least up to a phase factor) - simple linear algebra calculation can prove it. But in terms of two $1/2-$spin particles from the Hilbert space $H$, one may think that they are distinct. For example, one may think that one can determine the basis, by an EPR-Bohm experiment. Of course this will not work, since we choose the basis when we choose along which direction of space to measure the spin. But, is there any known effect in which it matters which basis is used in the singlet state? Is there any kind of "gauge-freedom" associated to the choice of this basis? Are there any theoretical speculations about this? * * * **Update** Seeing the comments (for which I am grateful), I think I should add more clarifications. I let the original question unchanged, and hope this comment can help clarifying what I mean. There is no difference between $|\psi\rangle$ and $e^{i\vartheta}|\psi\rangle$. Not in theory, but also not in experiment. Two state vectors which differ by a phase factor are undistinguishable (the state is invariant under the action of the group $U(1)$). But if we can't find the phase, we can find the phase differences. Think at interference, or at the Aharonov-Bohm effect. Now, the singlet state can be seen as being invariant under $SU(2)$. Did anyone try to do something with this "phase"? If it can't be determined, can we at least determine some "smaller" information, similar to the case of the phase? Can this suggest an experimental test for the Fock space in quantum mechanics?


title: Nature of Derivatives of Anticommuting Variables

text: This may be a noob question but I've tried searching about this and haven't been able to put things into the context of what I've been studying. (Dot means the usual derivative w.r.t. time) If $c$ and $\bar{c}$ are independent anti-commuting variables, I want to confirm a few properties: first, will $\dot{c}$ and $\dot{\bar{c}}$ anti- commute with themselves ($\left\\{\dot{c}, \dot{c}\right\\}$ = $\left\\{\dot{\bar{c}}, \dot{\bar{c}}\right\\}$ = 0)? Can we say that $c$ and $\dot{c}$ (or $\bar{c}$ and $\dot{\bar{c}}$) anti- commute? And finally, can $\dot{c}$ and $\dot{\bar{c}}$ be considered as mutually anti- commuting variables? In fact, can they be considered anti-commuting variables by themselves? Thanks in advance. EDIT: Well at some point later in the text (which I'm referring to), the author "uses" $\left\\{\dot{\bar{c}}, \dot{c}\right\\} = 0$. So I tried to come up with an explanation (note that square brackets in the following do NOT represent commutators, but curly braces represent anti-commutators): $\left\\{\frac{d}{dt}(c + \bar{c}), \frac{d}{dt}(c + \bar{c})\right\\} = 2[\frac{d}{dt}(c + \bar{c})][\frac{d}{dt}(c + \bar{c})]$ $= 2[\frac{dc}{dt}\frac{dc}{dt} + \left\\{\frac{dc}{dt},\frac{d \bar{c}}{dt}\right\\} + \frac{d \bar{c}}{dt}\frac{d \bar{c}}{dt}]$ $= \left\\{\dot{c}, \dot{c}\right\\} + 2\left\\{\dot{c}, \dot{\bar{c}}\right\\} + \left\\{\dot{\bar{c}}, \dot{\bar{c}}\right\\}.$ Now the result that I mentioned above (which was "used") makes sense to me only if the total derivatives anti-commute with themselves, BUT I'm not too sure about this.


title: Why are galactic centers always brighter than the edges?

text: As you can see this image http://i.stack.imgur.com/YOt8C.jpg and other galaxy images, the centers generally much brighter. Why is that? Is there a very big star? A very big gravitational field?


title: Is it possible to create an entangled pair of photons if they originate initially from operations at two separate sites?

text: My question is whether or not it would be possible to create an entangled state between two photons that do not share the same initial photon source and their respective sources are separated by an arbitrary distance in space. This is a curiosity of mine and I have a very basic understanding of this phenomena at present, though I am attempting to learn more. If it is the case that this is possible, I would also appreciate an explanation of why or at least have someone point me to a resource explaining this. Thank you.


title: What is wrong with this downforce calculcation?

text: I tried calculating the downforce of a spoiler I need for a robot. I used this formula. $$F = \frac{1}{2}Ac_a\rho v^2$$ Where $\rho$ is air density, $v$ is the speed of my robot, $c_a$ is the lift coefficient and $A$ is the windage of the spoiler which I calculate by multiplying its width, length and angle of attack in radians. $$A = w l \sin(\alpha\frac{\pi}{180})$$ I used the following values for calculation. $w = 0.07m, l = 0.04m, \alpha=45^°$ $c_a =1$, because I don't know a better approximation. $\rho = 1.2041\frac{kg}{m^3}$ $v = 3.5\frac{m}{s}$ And I got as result, that the downforce is around 0,0146 Newton which seems a lot too small. What is wrong with my calculation, or is it actually correct? (I need to compensate about 50N in small turns.)


title: Read-off particle from (projected) Dynkin labels

text: In the review of Slanksy "Group theory for unified model building" in chapter 6: How do one relate the projected Dynkin diagrams from for example $\overline{5}+10$ of $su(5)$ to the corresponding representation in of the subgroup $su(2)\times su(3)$? An example is $P( 0 0 1)=(0)(01)$, how do one see that this must be a charge $1/3$ antiquark singlet? Reference: http://cds.cern.ch/record/134739/files/198109187.pdf?origin=publication_detail


title: What errors would one learn from Eliezer Yudkowsky's introduction to quantum physics?

text: Eliezer Yudkowsky wrote an introduction to quantum physics from a strictly realist standpoint. However, he has no qualifications in the subject and it is not his specialty. Does it paint an accurate picture overall? What mistaken ideas about QM might someone who read only this introduction come away with?


title: Distance along projectile motion

text: I was doing kinematics these days, projectile motion. And I did the basic concepts: range, max height, time of flight, projectile on inclined plane. I am curious to know how we would solve a question if it asks us to find the distance not range traveled by the particle undergoing projectile motion. Is it by finding area under the curve or what? Also what is the average speed then.


title: How do I calculate the experimental uncertainty in a function of two measured quantities

text: I am performing an experiment where I'm measuring two variables, say $x$ and $y$, but I'm actually interested in a third variable which I calculate from those two, $$z=f(x,y).$$ In my experiment, of course, both $x$ and $y$ have experimental uncertainties, which are given by the resolution of my measurement apparatus among other considerations. I am also considering doing multiple runs of measurement to obtain good statistics on my measurement of $x$ and $y$, and therefore on $z$. I don't really know how the statistical spread will compare to my calculated (resolution-induced) uncertainty, though. I would like to know what the final uncertainty for $z$ should be, and I am not very familiar with the error propagation procedures for this.   * What are the usual ways to combine the experimental uncertainties in measured quantities?   * When should I use the different approaches?   * How do I include statistical uncertainties when they are present?   * What happens if the statistical spread of a variable is comparable to the instrument's resolution, so that I can't neglect either contribution?   * What are good references where I can read further about this type of problem? I would also appreciate answers to cite their sources - and particularly to use 'official' ones - where possible.


title: Confused about Impulse

text: Encountered a problem that involves impulse while studying for my exam and I'm not sure how to even approach it. I know that momentum is conserved, but I'm not sure how to relate that to avg force. Maybe someone can help point me in the right direction? I know that it's in quadrant III, through intuition, but I can't come up with a provable explanation Relevant equation: $J=F_{avg}\Delta T$ ![Couldn't copy and paste the problem, description in link](http://i.stack.imgur.com/dDlRS.png)


title: Why is no EMF being induced in this ring passing through a magnetic field?

text: ![enter image description here](http://i.stack.imgur.com/B1KNO.png) The book's logic is that there is no induced EMF because flux is constant as it passes through the magnetic field. Which makes sense, but this seems counter-intuitive to what I previously learned. If i'm not mistaken, if an electron is relatively moving perpendicular to a magnetic field, it WILL experience a force. The electrons in the ring are moving perpendicularly relative to the stationary field, so why aren't they experiencing a force?


title: Boundary conditions of Navier-Cauchy equation

text: I'm having difficulties with Neumann boundary conditions in Navier-Cauchy equations (a.k.a. the elastostatic equations). The trouble is that if I rotate a body then Neumann boundary condition should be satisfied with zero force. In math language: if deformation is given by $$u_i ~=~ a_{ij}x_j - x_i.$$ Where $a_{ij}$ is rotational matrix. Then this $$\mu n_j ( u_{i,j} + u_{j,i}) + \lambda n_i u_{k,k} ~=~ 0 $$ (Neumann boundary condition) should hold everywhere and for any vector $n_i$ (basically it doesn't matter how the body looks like). But if I substitute for $u_i$ I get $$2 \mu n_j(a_{ij} - \delta_{ij}) + \lambda n_i ( a_{jj} -3 ), $$ which is not zero. Because first term rotates with $n$ and the rest two just scale $n$. So I cannot get a zero for every $n$. Can someone see what am I doing wrong? I would be most grateful for any help. Tom


title: How would one begin to try to identify the properties of the mathematical structure corresponding to physical reality?

text: I was thinking about the Mathematical universe hypothesis and a natural question popped into my mind: _Assuming that the universe (by universe I mean the complete physical reality) is really isomorphic to some conceivable, mathematically constructible structure, how would one begin to narrow down the possibilities? How would one identify its properties without necessarily finding the structure itself?_ My first guess is that one should look at QFT and GR and assume that the mathematical structure we want would have to be **consistent** with those two theories at least in the appropriate approximations/limits and that we could somehow find all the familiar symmetries in some form on that structure. But those are just words, I don't understand what would one have to do rigorously to rule out some of the structures, I would really appreciate some non-handwavy guidelines if it's possible. **P.S. I'll gladly elaborate further and edit my question if something is particularly unclear.** **EDIT 1:** I'm not asking how theoretical physics should proceed in general and how should the scientific method be used in order to understand the world. I'm interested in how much can we say about the **"final, true and complete"** theory (assuming it exists) without actually having it. I'm interested in the mathematical structure associated with that theory, what are the most general statements about it that are almost certainly true?


title: At what speed (km/h) are the ride-goers (harnessed at opposite ends of ends of "propeller") travelling?

text: _The Revelation is one of the fastest and most exhilarating extreme rides in the world! Designed like a giant airplane propeller, the Revelation holds two to four riders comfortably in the cockpit seats at each end of its 160-foot arm. Riders take off quickly spinning up to 100 kph and experience the same intense "G" forces as fighter pilots! The combination of the rotation, tremendous height, high speed and the rush of wind will definitely reveal the intensity of the ride. Not for the faint of heart, the Revelation is for the ultimate thrillseeker! - See more at:_ http://www.pne.ca/playland/rides-games-food/rides.html#sthash.Rej9u0z3.dpuf _The "propeller" (51.5M long) rotates at 10rpm. How fast are the riders suspended at ends travelling? (km/h)??_


title: Energy of quarks and the mass of the proton

text: We know that energy of quarks inside the proton can not be exactly fixed because if it,the 'proton decay' must not be exist. My question is if the energy of quarks inside the proton is not exactly fixed than the mass of the proton must be fluctuate because 99% of the proton mass is due to the kinetic energy of the quarks and to the energy of the gluon fields that bind the quarks together. Is this fluctuation in mass really occur or I am missing something. please explain.


title: Will a liquid boil quicker if it is stirred versus if it is not stirred

text: Will the whole liquid boil quicker on a stove-top if it stirred versus if not stirred? What should be the frequency of stirring to get to boiling point quickly? please explain your hypotheses with experiments(or links to) preferrably


title: concept of density in gravitational lensing

text: I may just be being very dense (no pun intended) but i'm reading up on gravitational lensing and it seems to require a notion of density (e.g. see here) I'm working on a question involving light bending around a point mass, and yet it still asks about the mean surface density of the lens (which produces an Einstein ring as an image). It being a point mass, i.e. delta function, i would surely expect an average density of 0, so I must have misunderstood the exact definition of density in this case. Could someone please lay out exactly what surface density relates to in gravitational lensing?


title: Spatial and polarizing beam splitters in a graphical calculus

text: Suppose I have four wires, and I tensor product them together $A \otimes B \otimes C \otimes D$ I pass $A \otimes B$ through a spatial beam splitter $Spl: A \otimes B \rightarrow A^\prime \otimes B^\prime$ and I pass $C \otimes D$ through a polarizing beam splitter $Pspl : C \otimes D \rightarrow C^\prime \otimes D^\prime $. What kind of product do I use to combine $Pspl$ and $Spl$? For instance, can I just tensor them and get $Spl \otimes Pspl : A \otimes B \otimes C \otimes D \rightarrow A^\prime \otimes B^\prime \otimes C^\prime \otimes D^\prime $? I guess this doesn't make perfect sense yet as there is no notion of a "wire". In my calculations so far, I am seeing 4 port devices as taking a state on two wires "1,2" and sending it to a state on two other wires "3,4". I recall someone (Phill Scott, Abramsky?) doing something with tensors where the tensor indices were labelled wire inputs/outputs. Upper indices were input and lower indices were outputs. Has anyone else seen that? I want to do everything in the string diagrams, so I want rules for rewriting diagrams with polarization beam splitters (call it a "P" box) and also regular beam splitters (call it an "S" box). Can anyone help?


title: ATLAS Higgs Interpretation

text: I came across this abstract, and I am curious as to what the ATLAS Team has actually discovered: > Abstract Motivated by the result of the Higgs boson candidates at LEP with a > mass of about 115~GeV/c2, the observation given in ATLAS note ATL-COM- > PHYS-2010-935 (November 18, 2010) and the publication “Production of > isolated Higgs particle at the Large Hadron Collider Physics” (Letters B 683 > 2010 354-357), we studied the γγ invariant mass distribution over the range > of 80 to 150 GeV/c2. With 37.5~pb−1 data from 2010 and 26.0~pb−1 from 2011, > we observe a γγ resonance around 115~GeV/c2 with a significance of 4σ. The > event rate for this resonance is about thirty times larger than the > expectation from Higgs to γγ in the standard model. This channel H→γγ is of > great importance because the presence of new heavy particles can enhance > strongly both the Higgs production cross section and the decay branching > ratio. This large enhancement over the standard model rate implies that the > present result is the first definitive observation of physics beyond the > standard model. Exciting new physics, including new particles, may be > expected to be found in the very near future. The abstract seems to be from a restricted web site (CERN Log-in required), however I have been able to track down a PDF that seems to be discussing the same phenomenon, however the abstracts are not the same. Currently, as I understand it, there is a lot of skepticism about the initial Higgs candidate. If this isn't the Higgs, then what part of the standard model is actually being represented? Or is this an entirely new phenomenon?


title: Sampling typical clusters between distant points in subcritical percolation

text: I have on several occasions wondered how one might proceed in order to sample large subcritical Bernoulli bond-percolation clusters, say on the square lattice. More precisely, let's consider the lattice $\mathbb{Z}^2$, and open each edge independently with probability $p<p_c=0.5$. I am interested in the event that the vertices $(0,0)$ and $(N,0)$ are connected by a path of open edges, when $N$ some large number ("large" meaning here "many times the correlation length"). It is well-known (rigorously) that this probability decays exponentially fast with $N$, and one has a lot of information about the geometry of the corresponding cluster (e.g., the corresponding cluster converges to a brownian bridge as $N\to\infty$ under diffusive scaling, it has a maximal "width" of order $\log N$, etc.). **Question:** Are there (not too inefficient) algorithms that would sample typical clusters contributing to this event? **Edit:** googling a bit, I have stumbled upon this paper. I have yet to read it (and to decide whether it is relevant to my question). What the author seems to be doing is growing his objects in a biased way, which helps create the very unlikely configurations he's after, but would result in the wrong final distribution. So he's at the same time correcting the bias (in a way I haven't yet understood by just browsing through the paper). As an example, he's sampling 18 disjoint crossing clusters for critical percolation on $\mathbb{Z}^3$ in a box of size $128\times128\times2000$, an event of probability of order $10^{-300}$. So: **Alternative question:** does anybody know how this works? (I can also simply read the paper, but it might be useful for me and others if a nice self- contained description was given here by someone knowledgeable). Apparently, looking more thoroughly, there is a lot of material that could be useful. It seems that relevant keywords here are: rare events simulation, importance sampling, splitting, etc.


title: Why does unbroken supersymmetry imply the absence of tachyons?

text: Just a quick question, same as in the title. I'm trying to understand stable D-branes.


title: Renormalization group evolution equations and ill-posed problems

text: There is a class of observables in QFT (event shapes, parton density functions, light-cone distribution amplitudes) whose the renormalization-group (RG) evolution takes the form of an integro-differential equation: $$ \mu\partial_{\mu}f\left( x,\mu\right) =\int\mathrm{d}x^{\prime}\gamma\left( x,x^{\prime},\mu\right) f\left( x^{\prime},\mu\right) . $$ It is well known for such equations that one should distinguish carefully between well-posed and ill-posed problems. A classical example of an ill-posed problem is the backward heat equation: \begin{align*} \partial_{t}u & =\kappa\partial_{x}^{2}u,\qquad x\in\left[ 0,1\right] ,\qquad t\in\left[ 0,T\right] ,\\\ u\left( x,T\right) & =f\left( x\right) ,\qquad u\left( 0,t\right) =u\left( l,t\right) =0, \end{align*} while the forward evolution (i.e., the initial-boundary value problem $u\left( x,0\right) =f\left( x\right) $) is well-posed. The fact that the backward evolution is ill-posed (the solution either doesn't exist or doesn't depend continuously on the initial data) models the time irreversibility in the sense of the laws of thermodynamics. Since the renormalization transformation corresponds to integrating out short- wavelength field modes, the RG transformations are lossy and thus form a semigroup only. My question is — if there is an explicit example (or a demonstration) of an ill-posed problem for RG evolution? I mean, RG evolution equation the solutions (of initial-boundary value problem) of which have some pathological properties like instability under a small perturbation of initial data, thus making a numerical solution either not sensible or requiring to incorporate prior information (like Tikhonov regularization). **Update.** Actually, I have two reasons to worry about such ill-posed problems. **The first one:** the standard procedure of utilizing the parton density functions at colliders is to parameterize these function for some soft normalization scale $\mu\sim\Lambda_{QCD}$ and then use DGLAP equations to evolve the distributions to the hard scale of the process $Q\gg\mu$. The direction of such evolution is opposite to «normal» RG procedure (from the small resolution scale $Q^{-1}$ to the large one $\mu^{-1}$). Thus I suspect that such procedure is (strictly speaking) ill-posed. **The second:** the observables/distributions mentioned above are matrix elements of some nonlocal operators. Using the operator product expansion (OPE), one can reduce the corresponding integro-differential equation to a set of ordinary differential equation for the renormalization constants of local operators. My intuition says that in this case the RG evolution for the distribution will be well-posed at least in one RG direction (thus I think the DGLAP equations are well-posed for the evolution direction $Q\rightarrow\mu$). Therefore, a complete ill-posed RG evolution appears when the OPE fails.


title: Word for the star around which an exoplanet orbits:

text: Is there accepted nomenclature for the star around which a particular exoplanet orbits? Meaning, if I were to say "The exoplanet blah blah blah's (noun)" what noun would I put there? Sun? Star? Designated Gravity Buddy? Something else? As I understand it, this would not come up often currently in the real world (because Exoplanets are named by their star), and it more in the realm of science fiction and fantasy at the moment. However, just because something seems currently useless (to me) from a scientific standpoint doesn't mean that someone hasn't thought up a name. Would this name change if you're actually on the planet or in the system? (ie, you hear about Tatooine having "two suns" not "two stars").


title: Is it feasible to detect light emitted from planet earth 2,000 years ago by reflection from an object situated 1000 light years away?

text: The possibility though remote, is intriguing as we may be able in the future to actually "see" our own planet's history. Though sounding science fiction, if we are able to detect bodies in space that are able of reflecting light emitted from our planet earth, using amplification systems and filters,this may - if possible, give us a tool of utmost importance.


title: Some questions on a version of the O'Raifeartaigh model

text: This form is taken from a talk by Seiberg to which I was listening to, Take the Kahler potential ($K$) and the supersymmetric potential ($W$) as, $K = \vert X\vert ^2 + \vert \phi _1 \vert ^2 + \vert \phi_2\vert ^2 $ $W = fX + m\phi_1 \phi_2 + \frac{h}{2}X\phi_1 ^2 $   * This notation looks a bit confusing to me. Are the fields $X$, $\phi_1$ and $\phi_2$ real or complex? The form of $K$ seems to suggest that they are complex - since I would be inclined to read $\vert \psi \vert ^2$ as $\psi ^* \psi$ - but then the form of $W$ looks misleading - it seems that $W$ could be complex. Is that okay?  Now he looks at the potential $V$ defined as $V = \frac{\partial ^2 K}{\partial \psi_m \partial \psi_n} \left ( \frac {\partial W}{\partial \psi_m} \right )^* \frac {\partial W}{\partial \psi_n}$ (..where $\psi_n$ and $\psi_m$ sums over all fields in the theory..) For this case this will give, $V = \vert \frac{h}{2}\phi_1^2 + f\vert ^2 + \vert m\phi_1 \vert ^2 + \vert hX\phi_1 + m\phi_2 \vert ^2 $   * Though for the last term Seiberg seemed to have a "-" sign as $\vert hX\phi_1 - m\phi_2 \vert ^2 $ - which I could not understand.  I think the first point he was making is that it is clear by looking at the above expression for $V$ that it can't go to $0$ anywhere and hence supersymmetry is not broken at any value of the fields.   * I would like to hear of some discussion as to why this particular function $V$ is important for the analysis - after all this is one among several terms that will appear in the Lagrangian with this Kahler potential and the supersymmetry potential.    * He seemed to say that if *``$\phi_1$ and $\phi_2$ are integrated out then in terms of the massless field $X$ the potential is just $f^2$"* - I would be glad if someone can elaborate the calculation that he is referring to - I would naively think that in the limit of $h$ and $m$ going to $0$ the potential is looking like just $f^2$.    * With reference to the above case when the potential is just $f^2$ he seemed to be referring to the case when $\phi_2 = -\frac{hX\phi_1}{m}$. I could not get the significance of this. The equations of motion from this $V$ are clearly much more complicated.    * He said that one can work out the spectrum of the field theory by _"diagonalizing the small fluctuations"_ \- what did he mean? Was he meaning to drop all terms cubic or higher in the fields $\phi_1, \phi_2, X$ ? In this what would the "mass matrix" be defined as?  The confusion arises because of the initial doubt about whether the fields are real or complex. It seems that $V$ will have terms like $\phi^*\phi^*$ and $\phi \phi$ and also a constant term $f^2$ - these features are confusing me as to what diagonalizing will mean. Normally with complex fields say $\psi_i$ the "mass-matrix" would be defined the $M$ in the terms $\psi_i ^* M_{ij}\psi_j$ But here I can't see that structure!   * The point he wanted to make is that once the mass-matrix is diagonalized it will have the same number of bosonic and fermionic masses and also the super-trace of its square will be $0$ - I can't see from where will fermionic masses come here!    * If the mass-matrix is $M$ then he seemed to claim - almost magically out of the top of his hat! - that the 1-loop effective action is $\frac{1}{64\pi^2} STr \left ( M^4 log \frac{M^2}{M_{cut_off}^2} \right ) $ - he seemed to be saying that it follows from something else and he didn't need to do any loop calculation for that!  I would be glad if someone can help with these.


title: Does decoherence explain all instances of wave function collapse?

text: Specifically, how can decoherence explain the appearance of flecks of metallic silver on a photographic plate when exposed to the very weak light of a distant star? EDIT: Perhaps the advocates of decoherence need some context for this question. There is a certain definite quantity of energy on the order of one or two eV to drive the chemical reaction 2AgBr -> 2Ag + Br2 This is the reaction responsible for the fleck of silver on the photographic plate. The amount of energy is far greater than can be accounted for in any realistic time frame by the classical e-m wave energy of the light of a distant star. Any explanation must explain where this energy comes from. How does "decoherence" claim to do this? I have heard over and over again that there is a matrix which is diagonalized, but no one has so much as volunteered to say just what matrix they are talking about. Is it, for example, the matrix of position states of the photon? Or perhaps it is the oxidation states of the silver atom? And I would really like a better explanation of how the matrix is "diagonalized" than to simply repeat that it is in "thermal contact with the environment." EDIT: I have reviewed the comments again and I find that no one has come close to dealing with the question. I cannot find anything wrong in the way I have asked it so far, so I am posting this edit as my only means to prompt people to attempt an answer.


title: What is energy in string theory?

text: Facts agreed on by most Physicists - GR: One can't apply Noether's theorem to argue there is a conserved energy. QFT: One can apply Noether's theorem to argue there is a conserved energy. String Theory: A mathematically consistent quantum theory of gravity. Conclusion - If one can apply Noether's theorem in String Theory to argue there is a conserved energy, String Theory is not compatible with GR. If one can't, it is not compatible with QFT. Questions - Is the conclusion wrong? What is wrong with it? Is there a definition of energy in String Theory? If yes, what is the definition?


title: Flux Over a Surface

text: I am teaching a multivariable calculus course and we are starting to go over surface integrals. I am a math professor with little knowledge of physics. At one point the book discusses fluid flow. It is trying to convince me that the surface integral of a velocity field is the flow of the fluid across the surface. If $\vec{v}$ is my vector field, then we integrate $\vec{v}\cdot \vec{n}$, where $\vec{n}$ is the outward normal. So, we are integrating the work of $\vec{v}$ in the normal direction. Doesn't this tell you how much the fluid wants to leave the surface in the normal direction? Why does it give you flow across the surface?


title: Imaginary time in quantum and thermodynamics

text: The following question is about chapter 2 of Sakurai's _Modern Quantum Mechanics_. I wish I could link to the Google book, but it doesn't seem to have a satisfactory preview to be able to read the section I'm talking about, so I'll do my best to write out the part I'm talking about... In the section about propagators and Feynman path integrals (p. 113 in my edition) he gives the following example: $G(t) \equiv \int d^3 x' K(x', t; x',0) $ $=\int d^3 x' \sum_{a'} |\langle x'|a'\rangle|^2 \textrm{exp} \left(\frac{-iE_{a'}t}{\hbar}\right)$ $=\sum_{a'}\textrm{exp} \left( \frac{-iE_{a'}t}{\hbar} \right)$ He goes on to say that this is equivalent to taking the trace of the time evolution operator in the $\\{|a'\rangle\\}$ basis, or a "sum over states", reminiscent of the partition function in statistical mechanics. He then writes $\beta$ defined by $\beta=\frac{it}{\hbar}$ real and positive, but with with $t$ purely imaginary, rewriting the last line of the previous example as the partition function itself: $Z=\sum_{a'} \textrm{exp} \left( -\beta E_{a'} \right)$. So my question is this: What is the physical significance (if any) of representing time as purely imaginary? What does this say about the connection between thermodynamics and quantum? The fact that you get the partition function exactly, save for the imaginary time, here seems too perfect to be just a trick. Can someone explain this to me?


title: Magnetic multipoles in spherical harmonic

text: Does someone know explain me how to identify the multipoles magnetic terms of the multipolar expansion (Dipole, quadrupole, etc) in spherical harmonics?


title: About time measurements

text: What we regard as TIME is just a way of measuring duration for various phenomena. Like a ruler is a measuring device for measuring length ( or breadth or width). Saying Time is an illusion is like saying the measuring 'ability' of a ruler is an illusion. Is a measure of length, width, or breadth just an illusion? Why do some scientists or philosophers say Time is an illusion?


title: Is CO2 really a potent greenhouse gas for planets around M dwarfs?

text: The thing with global warming is that it absorbs infrared (IR) radiation from the planet and reradiates much of it back to the planet (whereas the Sun's peak flux is in the visible region, that is unaffected by CO2). But with red dwarfs \- it's different. The CO2 is going to reradiate back much of that incoming IR radiation from the Sun (in fact, this is why the upper atmosphere of Venus is so cold).


title: How big is the Solar System?

text: How big is the solar system? By "big", I guess I mean "wide", i.e. how far away from the Sun is the farthest object that is considered part of the Solar System? I've checked Wikipedia's pages on the Solar System, as well as Pluto, the Kuiper belt, and Trans-Neptunian Objects, but couldn't see the answer.


title: What is the fate of the asteroid belt?

text: From what I understand the asteroid belt never formed into a planet because Jupiter threw off the gravity of all the small objects and they never could gather to form a planet. As described here: Why did the asteroid belt between Mars and Jupiter form as it did? Are the asteroids in the asteroid belt really from 4.6 billion years ago or have they gotten material from somewhere? Also, what is the fate of the asteroid belt, in some long enough horizon are they all going to colide and the belt going to disappear? If it hasn't happened in 4.6 billions years, how long is that going to take? UPDATE: Are all the objects really still there from the LHB?


title: Extracting heat energy without a heat engine

text: Is it possible to extract the molecular kinetic energy from a system directly (without the use of a heat engine / temperature gradient) and convert that to another form of energy, such as electricity, or perform useful work? Similar questions have been asked before:   * Extracting heat energy from a material   * Can you extract energy from "hot" things without a temperature differential? However, I find the answers slightly lacking. They rest on one of two assumptions:   * that the asker wants to transfer the heat away through a heat engine; or   * that the Second Law of Thermodynamics applies to all systems in all situations — my understanding of thermodynamics (albeit rather limited) says that these laws are _statistical_ properties that are true _en masse_ , but not absolute laws of nature in the sense that they apply to all systems for arbitrarily short amounts of time. Even if my understanding of the second law is incorrect, I don't understand why we can't extract heat energy from an object without a temperature gradient by placing it under certain conditions. For example, heat transfer via infrared radiation could be extracted from a gas of any temperature, placed in a glass sphere and isolated from the environment via a vacuum chamber: ![diagram](http://i.stack.imgur.com/nPPPc.png) The gas would slowly radiate its heat through the glass to the ambient container housing the vacuum, and solar panels lining this surface could feasibly collect this energy. Note that this isn't a question about efficiency; I'm not concerned with how efficient this particular setup would be. It seems that if it works at all, one would be extracting thermal energy from an object without a heat engine.


title: Erosion effect from Coriolis force in a South-to-North water channel?

text: Imagine we could build a channel (like those cool ones in some countries) but very, very long, parallel to a meridian (ie South-North direction).. The channel would contain water, which would permanently flow from the beginning of the channel in the southern tip of South America towards the end of the channel in Alaska. Let's imagine the walls and bead of the channel are made from stone and the water is always running. Let's say there is a constant force which makes the water flow (and let's disregard the source and sink of the water in the channel). Flow is constant, that's what matters for now. I wonder if the walls of the channel will eventually wear out (thousands, maybe millions of years) differently? Eg. given that the water flows from South to North, would the West wall wear out in the Southern Hemisphere while the East wall wears out in the Northern Hemisphere? How important is flow speed for this answer?


title: Why are magnetic fields so much weaker than electric?

text: In EM radiation, the magnetic field is $ 3*10^8$ times smaller than the electric field, but is it valid to say it's "weaker". These fields have different units, so I don't think you can compare them, but even so it seems like we only interact with the electric field of EM radiation, not the magnetic field. Why is this?


title: Would incense burn at a different rate if oriented differently?

text: This is motivated by this question in the Puzzling.SE beta about measuring 90 minutes of time using two candles that burn for one hour. (Feel free to read up on that before I spoil the puzzle!) The solution given was to light one of the candles on both ends by holding it horizontally, but as one user mentioned, the candle's burning rate would be significantly affected by its orientation. This reminded me that I have heard of this puzzle before but using sticks of incense instead of candles. Does that actually improve the rigour of the puzzle though? How would the burn rate of incense be affected by the incense's orientation?


title: Writing Schrodinger equation with central potential in Atomic unit

text: I'm struggling to write Schrodinger equation with a central potential in Atomic unit. $$ \frac{\hbar^2}{2\mu}\Big(\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}-\frac{l(l+1)}{r^2}\Big)R(r)+\Big(E-V-\frac{e^2}{4\pi \epsilon_0 \epsilon r}\Big)R(r)=0 $$ Taking Effective Bohr radius $a_0$ as length scale and effective Rydberg as energy scale $$Ry=\frac{\mu}{m_e \epsilon^2}\big(\frac{hc}{e}\big)\frac{m_e e^4}{8\epsilon_0^2 \hbar^3c}$$ in MKSA or $Ry=\frac{\mu e^4}{2\epsilon^2 \hbar^2}$ in CGS. What I tried is: By substitution of $r'=r/a_0$ : $$ \Big(\frac{d^2}{dr'^2}+\frac{2}{r'}\frac{d}{dr'}-\frac{l(l+1)}{r'^2}\Big)R(r')+\frac{2\mu a_0^2}{\hbar^2}\Big(E-V-\frac{e^2}{4\pi \epsilon_0 \epsilon a_0 r'}\Big)R(r)=0 $$ where $\mu$ is effective mass $\mu=.06m_e$. Then we can write $m_e=\hbar=e=1$. The potential is for example $V=4 \ Ry$. Know I don't know what can I write instead of   $$ \frac{2\mu a_0^2}{\hbar^2}\Big(-\frac{e^2}{4\pi \epsilon_0 \epsilon a_0 r'}\Big) $$ Any help appreciated.


title: Lifting an Atwoods machine from the ground?

text: If we have an Atwood machine where masses $m_A$ and $m_B$ rest on the ground, then we apply an upwards force $F$ to the Atwoods machine. What is the acceleration of the blocks when $F=124N$, $m_A = 20$kg, $m_B = 10kg$? I am confused to about this problem,because I am not sure how this upward force $F$ on the Atwoods machine effects the two masses. For example, for each mass I would normally do this: $$F_{netA} = m_A*g - T = m_A*a$$ $$F_{netB} = T - m_B*g = m_B*a$$ But now I have another force directed upwards, of $124N$. Where does this come into play? I have thought about adding it to each of the equations, but I don't think this is right, because the force is acting on the Atwood's machine, not the mass. I believe this effects the Tension, but I'm not exactly sure how.


title: Derivation of the effective potential between a quark and an anti-quark

text: Typically in particle physics books (not in QFT books!) I have often seen this statement that the potential between a heavy quark and its anti-quark can be "empirically" represented as $V(r) = -\frac{\alpha_s}{r} + br$ where $\alpha_s \sim \frac{\hbar c}{2}$ and $b \sim \frac{0.18}{\hbar c} GeV^2$.   * Is there a way to get the above form or the expression from a QCD calculation? I have seen some approximate evaluations of the Wilson loops whereby one shows that the $2$-point gauge field correlator decays as $\frac{1}{r}$ to quadratic order of the coupling but thats about it. And I am not sure if somehow the proof that the beta-function of $QCD$ is decreasing to first order somehow implies the above form for $V(r)$. I would like to know how to derive the above expression for $V(r)$.


title: Uniqueness of eigenvector representation in a complete set of compatible observables

text: Sakurai states that if we have a complete, maximal set of compatible observables, say $A,B,C...$ Then, an eigenvector represented by $|a,b,c....>$, where $a,b,c...$ are respective eigenvalues, is unique. Why is it so? Why can't there be two eigenvectors with same eigenvalues for each observable? Does maximality of the set has some role to play in it?


title: What is the connection between extra dimensions in Kaluza-Klein type theories and those in string theories?

text: This follows to some extent from a question I asked previously about the flaws of Kaluza-Klein theories. It appears to me that Kaluza-Klein theories attach additional dimensions to spacetime that are related to the gauge freedoms of field theories. I believe the original model was to attach a $U(1)$ dimension to the usual 4-dimensional spacetime to reproduce electromagnetism. But, as explained in the answers to my previous question, these extra dimensions have all sorts of problems. String theories also (famously?) require extra dimensions. So, is there a connection between the higher-dimensional descriptions? What do they have in common and how do they differ? For example, the $U(1)\times SU(2)\times SU(3)$ group is 7-dimensional, which, when attached to the 4 dimensions of spacetime, gives 11 dimensions. I hear the same number is bandied about in string theory although there's no obvious reason they should be related at all.


title: Total angular momentum in multielectron atoms

text: I have some confusion about orbitals in multielectron atoms. Let's say we consider an atom (Lithium, for example, $1s^2\, 2p^1$) and that the state of the last electron is [n=2, l=1, ml=0, s=1/2, ms=1/2]. Its wave equation should then be $$\psi(r,\theta,\phi)=R_{nl}(r)Y_{lm_l}(\theta,\phi)\chi_s(m_s)$$ Now we want to apply "LS coupling". Since the Hamiltonian for the perturbation commutes with $J^2$ and $J_z$ we express the wave function in terms of the eigenstates of these operators. So, we search for the Clebsh-Gordan coefficients and we have that $j$ can be either $1/2$ or $3/2$, and $j_z$ can vary in integer steps between $-j$ and $j$. So, if I understood well, we have 6 Clebsh-Gordan coefficients to care about, let's call them $K_{[l=1,m_l=0,s=1/2,m_s=1/2]}(j,j_z)$ and our wave equation should become: $$\psi(r,\theta,\phi)=\sum_{j=1/2,3/2}\sum_{m_j=-j}^j K_{[l=1,m_l=0,s=1/2,m_s=1/2]}(j,j_z)R_{nl}(r)Y_{jm_j}(\theta,\phi)$$ Now I'd like some clarification:   1. The spherical harmonics with the same parameters should be the same function. Are the Clebsh-Gordan coefficients and the spherical harmonics tuned such that $\sum_{j=1/2,3/2}\sum_{m_j=-j}^j K_{[l=1,m_l=0,s=1/2,m_s=1/2]}(j,j_z)Y_{jm_j}=Y_{lm_l}\chi_s(m_s)$? This seems reasonable, but what happens to the spin? How is it inglobed inside the $Y_{jm_j}$?   2. When you have more then one electron in the outer shell, from what I understood, you treat the "collection of electrons" as a _system_ with a certain $s$, $l$, $j$ and $m_j$. Am I right?


title: Feynman rules with helicity states.

text: * Whenever Feynman rules are stated they are always without any mention of the helicities - this I find to be very confusing. How does one introduce and account for that?    * Is there an intuitive/simple argument for why massless particles should have "helicities" (and not polarizations) and they can only be of the form $\pm\text{ some positive integer}$? (..i have seen some very detailed arguments for that which hinge on the representation theory for the little group of massless particles and various other topological considerations - i am here looking for some "quick" explanation for that..)    * Is there some reason why polarized gluon scattering amplitudes at the tree-level can somehow "obviously" be written down? Like for example, consider a process where two positive helicity gluons of momenta $p_1$ and $p_2$ scatter into two negative helicity gluons of momenta $p_3$ and $p_4$ then at tree level the scattering amplitude is,  $A(p_1^+,p_2^+,p_3^-,p_4^-)= \frac{ig^2}{4p_1.p_2} \epsilon_2^+ \epsilon_3^-(-2p_3.\epsilon_4^-)(-2p_2.\epsilon_1^+)$ where $\epsilon^{\pm}_i$ is the polarization of the $i^{th}$ particle. I have at places seen this expression being almost directly written down. Is the above somehow obvious?


title: Spring launches a mass on an incline plane with friction, find the height

text: So I've calculated the answer to this problem, but my answer is different from my book's, so I'm trying to find the error. A spring with coefficient $k=600N/m$ launches a mass of $1.2kg$ from an initial displacement of $0.15m$. It slides along a frictionless surface and then goes up an inclined plane with coefficient of friction $\mu_{k}=0.2$ and angle $\theta = 30^{o}$. What is the maximum vertical height it accomplishes? So I thought, initially the total energy of the system was $\frac{1}{2}kx^{2}$ but when it reaches maximum height it has potential energy $mgh$ and zero kinetic energy, and it has lost the energy equal to the work done by friction, which is $Fd$. The force of friction is $F=\mu_{k}mg\cos\theta$ and the distance $d= \frac{h}{\sin\theta}$ where $h$ is the final vertical height. So I get the equation $$\frac{1}{2}kx^{2} = mgh - \mu_{k}mg\cos\theta \left(\frac{h}{\sin\theta}\right)\Longrightarrow $$ $$h = \frac{kx^{2}}{2mg\left(1-\frac{\mu_{k}\cos\theta}{\sin\theta}\right)}$$ When I plug in all the numbers and compute (done here: Wolfram calculation) I get about .878 which is apparently incorrect by about a factor of 2. Any idea where this went wrong?


title: What size aperature would I need to find and see the supernova in M51?

text: On June 2, 2011, a new supernova was identified by an amateur French astronomer in M51. What size telescope aperture would one need to have any chance of seeing this? Given the right sized scope, how would I best identify which point of light is the supernova?


title: Space Rocket Lift Efficiency and the Gyroscope?

text: I'm no physicist apart from basic 3d web animation, I'm just curious and please feel free to correct my misuse of terms or inadequate speculations. I've been reading a lot on gyroscopes and aerodynamics and the various concepts of lift, drag, etc. I understand how a rocket utilizes lift and drag and how much it depends on thrust. I also understand that there are several forces acting upon a gyroscope which determines its stability and why the tool is so useful in viscosity navigation, among other things. It is my speculation that gyroscopes, once spun at the right velocity, can generate equally opposing momentum along it's various directional paths in 2d cross section i.e. up, down, left & right or (+/-) y and (+/-) x, and ideally, these equal, but, opposite forces generate equilibrium by shifting the object's center of gravity (point 0) to a variable location in the object i.e. relative to the direction in which the gyroscope is mounted or positioned. I also believe that these forces are constant relative to the constancy of the object's velocity. e.g. an electric gyroscope peaking at a constant spin velocity. With that said and in not having any way to test, I ask: If I am correct in saying that all the gyroscope forces given a fixed resistance, are equal and constant up to the given moment of external force (e.g. throwing a spinning electric gyroscope [straight] up in the air), will the upward force of throwing stack up with and amplify the upward (lift) force of the gyroscope? Will that gyroscope rise higher and faster than if it weren't spinning and will it descend much faster in the same manner i.e. a much stronger interaction with gravity upon descent? Also, if this were so, could the design concept of a gyroscope be applied to Space Rocket Designs (i.e. long shaft and a proportionally larger, extruding spinning wheel), where less fuel could be used in launch by piggy backing and amplifying the lift generated by the wheel?


title: Does the Fermi surface make sense for "Fermi liquids" with non-uniform charge density?

text: For a Fermi liquid, the Fermi momentum is determined by the singularity of the Green's function at $\omega=0$, i.e., $G(\omega=0,{\bf k}={\bf k}_F)\to\infty$. Suppose due to an external field or disorder, the charge density (or the chemical potential) is not uniform, i.e., it depends on the position. Now the system is not translational invariant, so the momentum is not a good quantum number, and we only have the Green's function in position space $G(\omega,{\bf x})$. Does the Fermi surface still make sense? is there a local Fermi surface, and how to define it?


title: What is the maximal height for a water rocket's flight?

text: A water rocket works like this: there is a circular slot of area $A_1$ at the bottom centre of a cylinder of cross-sectional area $A_0$ and height $L$ that is filled with water to an initial height $h_0$. This slot will fall away during launch. The water has pushed all of the air that was originally in the cylinder to the top $L-h_0$ of the cylinder (I believe this is an isothermal compression: the compression was fast), so is at a higher pressure of $\frac{L-h_0}{L}P_0$, where $P_0$ is atmospheric pressure. ![enter image description here](http://i.stack.imgur.com/wuqUs.png) To launch, the slot is instantaneously removed (leaving a hole of area $A_1$ in the bottom of the cylinder), and water is pushed downwards, as the air pressure is higher inside the cylinder than outside, at a speed $u(t)$. There is no sloshing of the water in the cylinder: the body of water remains cylindrical. Thus the air in the cylinder now takes up more volume (has expanded in an adiabatic expansion), but because of the upwards impulse imparted to the cylinder by the leaving water, the cylinder is now moving upwards with speed $v(t)$. The rocket will reach a maximum height $H_{max}(h_0)$, where $h_0$ is the original height of the water. What $h_0$ will give the maximum value of $H_{max}(h_0)$ for fixed $A_0,A_1,l$? **Partial solution.** In the adiabatic expansion, let $V(t)$ be the volume of the air in the rocket and $P(t)$ the pressure. As the air is mostly diatomic (Nitrogen and Oxygen are), $$\displaystyle P(t)V(T)^{\frac{1+\frac{5}{2}}{\frac{5}{2}}}=k$$ $$P(t)V(t)^{\frac{7}{5}}=A_0P_0 \frac{(L-h_0)^2}{L}$$ $$ \frac{dV}{dt}= A_1 u(t)$$ The change of momentum per unit time of the water being spewed out the bottom is $$\rho \delta V(v(t)+u(t))$$


title: About electrostatic induction

text: when we approach a charged rod (+) to a neutral metal rod ( not touching) a number of electrons to that side ( lets call it side B) negating the effect of the introduced electric field, reaching equilibrium. But what about the electrons that where already on that side , will there presence affect the equilibrium , if some electrons ( those where already at side B before the electric field is introduced) disappeared what would happen to the equilibrium? Forgive me if some of my ideas are faulty , i am new to the world of physics


title: Mechanical system modeling

text: I have a mechanical system and I need to model in differential equations ![The Problem](http://i.stack.imgur.com/BHbsD.png) I tried to model the mass $M_1$ and got this $m_1\displaystyle\frac{d^2\,x}{dt^2}=F-K_1(x_2 - x_1)-Ba_1\displaystyle\frac{dx}{dt}.$ $F$ is the force. $K_1(x_2 - x_1)$ spring is multiplied by subtracting the distance of the spring and damper. $Ba_1\displaystyle\frac{dx}{dt}$ is the force of the damper. I don't understand how to model the mass on the right (D2). I tried, but I need help $m_2\displaystyle\frac{d^2\,x}{dt^2}=-K_1(x_2 - x_1)-Ba_2\displaystyle\frac{dx}{dt} $.


title: Is the second derivative of the effective potential always the mass square of the particle?

text: In quantum field theory, we can calculate the effective potential of a scalar field. My question is whether the second derivative of the effective potential always represents the mass square of the particle? Why or why not? $$ m_{eff}^{2}=\frac{\partial^{2}V_{eff}(\phi)}{\partial\phi^{2}} $$ For example, consider a scalar field which has an one-loop effective potential like $$ V_{eff}(\phi)=\frac{1}{4!}\lambda_{eff}(\phi)\phi^{4}. $$ If the field is sitting at some scale $\phi\neq0$, would the second derivative still be the physical mass square of the particle? If this is true, then the particle will have different masses as it rolls down the potential to its minimum. Would this affect the decay channel of the particle?


title: Intuitive explanation of difference in $r$-dependence between dipole and monopole

text: For an electric monopole, its potential scales with $\frac{1}{r}$, where $r$ is the distance from the point of interest to the charge. However, for a dipole, its potential scales with $\frac{1}{r^2}$. I understand how the latter is derived. But is there an intuitive explanation on why the dipole potential ends up having a second order dependence or $r$?


title: Why do some (older) wind generators have more than three blades?

text: Based on my personal observations, newer windmills seem to have three blades while older ones tend to have four or even more. This question has excellent discussion on my three is an optimal number. But what changed? For example, did people at some point not understand the relevant tradeoffs? Or does the availability of some new material shift the economics to favor fewer, longer blades? ![Old-fashioned and new-fashioned windmills](http://i.stack.imgur.com/XpwXx.jpg)


title: A problem with a basic thought experiment in Special Relativity

text: This is for a class on special relativity I am to give to some school children. > Moe is moving at 0.9c. At the instant he passed Joe he emits a flash of > light. One microsecond later, Joe(at rest) sees that Moe has traveled 270m > and the light flash has traveled 300m. That is, the distance between Moe and > the light flash is 30m. All perfectly good so far. > > But Moe, who does not know he is moving, waits one microsecond and observes > that the light has traveled 300m in front of him. Also perfectly good. > > The reason that Moe sees that the light has traveled 300m and not 30m is > that his time has been dilated, so that Moe's microsecond is longer than > Joe's microsecond. Also, when he measures how far ahead the light flash is, > he is using a shortened ruler. All very good. Except that if you use the Lorenz transformation equations, you find that his time has dilated to 2.29 microseconds and his ruler is shorter by 1/2.29. Those are not enough to explain a 300m measurement for Moe: It only provides for the light to be 157m ahead, not 300m ahead! What am I missing here?


title: Resonances in high energy physics

text: I still do not understand what a resonance precisely is. Is it exactly the same as a particle? Or only an excited state? And why does it make a peak in some diagrams? And which diagrams?


title: Stress in a thick-walled pressure vessel

text: I can find many references that give the stress in the walls of a pressure vessel for spheres and tubes, but they all seem to be limited to a thin-wall approximation. I'll limit my writing here to spherical vessels (I don't think it should be very different to find the answer for any structure with symmetry). My main trouble is in setting up the integral. $$ \sigma = \frac{P R }{2 t}$$ So my question is: how do you find material stress for a pressure vessel with a wall that goes from $R_i$ to $R_o$, and you want a solution that is still perfectly valid when $R_i \ll R_o$? My guess is that you would take the equation for the thin wall case, and recast it into an integral. $$ P = \frac{2 \sigma t }{ R }$$ In the more general case, would we write something like this? $$ P = \int_{R_i}^{R_o} \frac{2 \sigma }{r} dr$$ This would only be valid if the stress was constant for all differential shells. I'm wondering if that's a bad assumption, but if the material was highly elastic I think it would be decent assumption. My intuition is that the material would have to be much _more_ elastic for it to hold with the inner radius many times smaller than the outer radius. But regardless of that detail, is the above integral correct? Is there some good intuitive logic to justify it? And is there any other reference that gives the expression for stress in this case? It should be a logarithm if the above is correct, but I haven't seen this before.


title: Why does a remote car key work when held to your head/body?

text: I was trying to unlock my car, but I was out of range. A friend of mine said that I have to hold the transmitter next to my head. It worked, so I tried the following later that day:   * Walked away from the car until I was out of range   * Put key next to my head (it worked)   * Put key on my chest (it worked)   * Put key on my leg (didn't work) So first I thought it has to do with height. But I am out of range if I use the key at the same height as my head but not next to it, and normally my key is at the same height as my chest. So it has nothing to do with height. Or my whole body is acting like an antenna, but how is that possible if I am holding the key? Why would it be an antenna if I hold it against my head and not in my hand? Here's a vid of Top Gear demonstrating it.


title: The best way to cool the teapot

text: My professor today in the class made us a question: "Lets say we have a teapot with water in it.The water is hot.Now we want to cool the water. Will it cool faster if we put an ice cube above the teapot or under the teapot." My answer was the it will cool faster if we put the ice cube above it because the warm air stays up and the ice cube will melt faster. He didn't tell me if I was right or not. Can anyone help me ?


title: Speed of gravity in cosmological codes and ephemeris generation

text: There are few questions in Phys.SE concerning the speed of gravity, and the answers are traditionally that the speed of gravity equals to the speed of light. But in that case I have three more specific questions which I didn’t find in those discussions, so let me ask them here separately.   1. All cosmological and astrophysical N-body simulation codes (e.g., the popular Gadget-2 code) calculate gravitational force acting on a given particle from all other particles taking their positions at the same moment of time. Well, that’s perhaps adequate if one simulates a small stellar cluster, but how can that be adequate when one simulates large-scale structure of the Universe? But most weird thing is that I failed to find any justification for that either in manuals or in Internet discussions – as if it is just obvious that gravitational interaction acts instantaneously irrelevant of the separation.   2. Astronomers calculate orbits and predict positions of celestial objects without taking into account the finite speed of gravity, and yet they get accurate results. They do make correction for the time that will take light to travel from the object to the Earth to observe it in its predicted position, but if similar correction is introduced for gravity itself, the result becomes in fact incorrect.   3. If the speed of gravity does not exceed the speed of light, how can black holes produce gravitational fields? The mass of the black hole is under the event horizon, and yet it manages to update its external gravitational field, whereas in fact the black hole should not reveal itself even gravitationally in this case.  The last two questions are actually spurred by the paper of Tom Van Flandern _The speed of gravity – what the experiments say_ (Phys. Lett. A 250, 1998) that I have come across recently. In fact, there are more interesting points raised in the paper, e.g. concerned with aberration - gravity has no observed aberration though it should have it if it propagates at the speed of light, and the ultimate conclusion is that the speed of gravity is at least ten orders of magnitude higher than the speed of light. There is some reaction to the paper. For example, Steven Carlip’s _Aberration and the speed of gravity,_ Phys. Lett. A 267, 2000. But Carlip focuses on aberration and concludes that in general relativity aberration might in fact be cancelled by velocity-dependent interactions. But what about the three questions outlined above? For me the most striking fact is that in predicting positions of celestial objects the speed of gravity is taken to be infinite and predictions turn out to be correct. If that is the case, I don’t understand at all why we are still taught that nothing can travel faster than light. Can anyone clear up the situation in more or less simple terms?


title: How to calculate strength of wind speed in a von Karman vortex?

text: I am working on a project involving Von Karman vortices coming off of a mountain. I was able to calculate the shedding frequency (thanks to tpg2114 in a prior question), but now find it necessary to calculate the wind speed of these Von Karman vortices. In about three days of searching I only came up with one article which mentioned how to obtain expected wind speeds from von Karman vortices (here if you would like to know, mostly with the sixth section of this being pertinent), through the use of the formula $V = \frac{0.72k}{2\pi r}$ where $V$ is the highest tangential velocity in the vortex and $r$ is the radius length at which the high speed occurs. Based upon the information which I have and the formulas from this article, I can find the radius if I know what the speed at which the vortices are traveling. The $k$ in this formula however is the tricky part as it is called the "strength of line vortex in an infinite medium", but never numerically defined. So here are my three questions in order from simplest to most difficult:   1. Is there an easy way to calculate the vortex speed? This article defines it as by the identity $$4\pi\frac{a}{h}\frac{v_v}{v_a}\left(1- \frac{v_v}{v_a}\right) = 1$$ with $v_v$ being vortex velocity and $v_a$ being air velocity, but when this is solved yield a quadratic with two solutions, thus being not very helpful.   2. What is a mathematical definition of $k$ so that I can get a value for it knowing the air speed (averaged at ~10 m/sec), shedding frequency, and mountain dimensions (height ~1000m)?   3. Is this even the right direction to go into if I am attempting to approximate the influence that Von Karman vortices will have on wind speed measurements? If not, what should I do in order to figure this out?


title: Dual Resonance Model: Fermions

text: I am going through Ramond's 1971 paper Dual Theory for Free Fermions Phys Rev D **3** 10, 2415 where he first attempts to introduce fermions into the conventional dual resonance model. I get the 'gist' of what he's doing: he draws an analogy of the bosonic oscillators satisfying the Klein-Gordon equation, and extends it to incorporate some version of the Dirac equation. Great. Now ~~without resorting to string theory (since I know nothing about it) and perhaps minimally resorting to field theory (after all, this is still S-matrix theory, right?)~~ , how can I understand his correspondence principle (eqn 3)? $$p^2-m^2=\langle P\rangle\\!\cdot\\!\langle P\rangle-m^2\rightarrow\langle P\\!\cdot\\! P\rangle-m^2$$ (the same correspondence principle appears in Frampton's 1986 book "Dual Resonance Models" equation 5.63). Is this a special property of harmonic oscillators?


title: What would happen if a hydrogen bomb were to explode in Saturn's atmosphere?

text: Purely hypothetical since any kind of testing in atmosphere/space is banned by international legislation/agreement. The humans have already bombed Luna so ... what could be expected to happen on Saturn if a hydrogen bomb were to explode in it's atmosphere? Would the explosion set the planet's atmosphere ablaze?


title: Potential collapse during an earthquake?

text: I was asked to determine the shear forces on 12" diameter table legs supporting a few hundred pounds of static weight. How does one calculate or determine the potential force on the legs and/or potential collapse of the legs given an earthquake? The legs are not secured to the flow. Where can I find information that will help me answer the question of potential collapse during an earthquake?


title: Light takes too long to get here

text: When looking at the night sky, we see lots of stars. Several places tell you that the light of those stars has traveled to many light years to reach Earth and there may be others where light has not made it here yet. How can this be? Assume we accept the Big Bang Theory that suggests the universe began in a single point (or at least very small space). When the "bubble" that is the universe reaches 1 light year in diameter, the light from one end should have been shinning the whole time, so the other side should still be able to see it. Continue on until the Universe is at present size, and the light should still be visible by the same logic. So how can there be systems from which light has not reached us yet?


title: Gauss's law problem. ( Need someone to point me in the right direction)

text: > _What force per square meter pushes 2 infinite planes charged positively > when their charge density is $0.3 \, \mu C/\mathrm{m}^2$_ * * * > _Second part is - using Gauss's theorem derive the equation for electric > field current between infinite electrified planes._ * * * Hi everyone, just need some general direction to how to start solving problems like these. I want to understand the task more then get the right answer.


title: Kinematics with non constant acceleration II

text: I'm getting crazy with this problem and I think that it's pretty simple. > An helicopter's helix is spinning at initial speed $w_0=200\ rpm$, all of a > sudden the motor stops and it decreases its velocity with a not constant > acceleration of $\alpha=-0,01\cdot w \frac{rad}{s^2} $. > > The question is: how many revolutions will it make until it stops? I know that it can't be solved using regular kinematics formulas because the acceleration is not constant so I tried with the next chain rule: $$a(x) = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = \frac{dv}{dx}v$$ that I've seen in another question where the acceleration depended on the $x$, and you can solve the problem with $a(x)dx = vdv$ but in my case it depends on the velocity $w$ and I don't know what to do! I'm trying to get something like $a(w)dw= ... $ but I don't know how to arrive. This is my closest attempt: $$\alpha=\frac{dw}{dt}\cdot\frac{dw}{dw}\rightarrow \alpha dw=\phi dw$$ and here I'm stucked. I will appreciate any help! Thank you!


title: How can the microstates be measured with zero energy expenditure?

text: _James P. Sethna. Statistical Mechanics. Exercise 5.2:_ > What prevents a Maxwellian demon from using an atom in an unknown state to > extract work? The demon must first measure which side of the box the atom is > on. Early workers suggested that there must be a minimum energy cost to take > this measurement, equal to the energy gain extractable from the bit. Bennett > showed that no energy need be expended in the measurement process. Why does > this not violate the second law of thermodynamics? Reference to Bennett's paper didn't help me much. Here the relevant model is a tape consisting of single atoms in pistons, where knowing which side an atom is in a piston counts 1 bit of information, which can be used to extract useful work by expanding the piston, as shown below: ![enter image description here](http://i.stack.imgur.com/aKzX4.jpg) ![enter image description here](http://i.stack.imgur.com/2MbfM.jpg) **My understanding is that after the measurement uncertainty of position is reduced by half and entropy decreases. But without energy expenditure, it seems that this decrease comes free. How can the second law hold if there's no corresponding increase of entropy elsewhere (which I can't identify)?** Something like an explanation is given at the end of the exercise: > The demon can extract an unlimited amount of useful work from a tape with an > unknown bit sequence if it has enough internal states to store the > sequence—basically it can copy the information onto a second, internal tape. > But the same work must be expended to re-zero this internal tape, preparing > it to be used again. **Does this mean after measurement, the reduced entropy in the first tape goes to the second "internal tape" which stores the information? How can such measurement take place?**


title: SMBC ball bouncing problem

text: This comes from a Saturday Morning Breakfast Cereal (SMBC) comic with a joke answer. The problem states: > _A 5 kilogram ball is shot directly right at 20 meters per second from a > height of 10 meters. The ball loses 1 joule whenever it touches Earth. > Assume no air resistance. When does the ball stop bouncing?_ How would one solve this problem? The best I could do was to assume the total energy of the ball, given by the sum of potential and kinetic energy when it's initially shot is completely lost when it stops bouncing. This would give us approximately 1490 bounces, with each bounce slowing the ball down and making it bounce ever so slightly lower. This still requires a ton of calculation (a huge series), even with the added assumption that there is no friction between the ball and the ground. Am I missing something?


title: Do object reflect radio waves?

text: Do object reflect radio waves just like light waves? I don't know much about their use but can anyone explain how we use this reflection?


title: Conformal Coupling for QFT in Curved Spacetime

text: I have seen it stated but not explained that consistency requires you to couple massless fields to gravity using the conformal coupling, so that $trT_{\mu \nu}=0$. What is the reason for this?


title: Dimensional analysis of magnetic energy: dimensions of µ0 and H

text: When calculating the energy difference between the normal and the superconducting state in a superconductor at zero magnetic field, the result is as follows: ![enter image description here](http://i.stack.imgur.com/tIq7w.png) Now I'm quite confident of this result, as it is the same as my textbook tells me it is. The problem is that when I do a dimensional analysis, I find ![enter image description here](http://i.stack.imgur.com/IfgUL.png) The dimension on the left is energy, the dimension on the right is vacuum permeability times H-field squared. The units of the first are found on Wikipedia, $N/A^2$, as are the units of the second, $A/m$ (Ctrl+F for _``The H-field is measured in''_ ). As you can see, the lengths don't cancel out, we have $[L]^2$ on the left and $[L]^{-1}$ on the right. What am I doing wrong? Is the first equation not in SI? * * * _Edit:_ In case it might be of any help, the derivation of the above equation is (using the Meissner effect in the final step, $M=-H$) as follows: ![enter image description here](http://i.stack.imgur.com/Bq0cB.png)


title: Physical Role of Batter in Baseball

text: Physically, what is the role of a batter in baseball? My question is inspired by How does the speed of an incoming pitch affect the speed of a baseball after it's hit? The answer to that question, that a faster pitch results in a farther hit, surprised me. I had thought that a batter was solely applying impulse to the ball (mass * velocity of bat = Force * time on ball). Clearly the batter is doing more. Specifically, I'm interested in what information we'd need to know and what math we'd use to calculate the final velocity of the ball.


title: Questions (intuitive) about circuits and the flow of electrons

text: Why don't electrons accelerate in a circuit? If there is a potential difference and the electron moves in the circuit a force is exerted on it. Why does that force produce no acceleration, keeping the current constant? If we add a resistor to a circuit why is the resistance 'spread' over the circuit? Is it because electrons moving faster in the areas of lower resistance would create charge imbalances at places in the conductor?


title: question about elasticity and tensile strength of wires

text: To the best of my knowledge, if you double the cross sectional area of a wire you double the maximum weight it can support before breaking. But what if you use two wires of the original cross sectional area and tie them together? Can it still support twice the weight? Are there any new forces involved in this scenario?


title: Fock matrix elements for RHF formalism

text: Here I refer to a particular book _Molecular Quantum Mechanics_ by Peter W. Atkins and Ronald S. Friedman, but similar derivation could be found in many other texts. So, when obtaining the explicit form of the Fock matrix elements for RHF formalism (p. 295 in 4th edition), authors go from ![enter image description here](http://i.stack.imgur.com/GtcVE.png) to ![enter image description here](http://i.stack.imgur.com/kGKFJ.png) just by mentioning that $\psi_u$ is expanded as a linear combination of basis functions $\theta$. The only way I could go from the first equation to the second one is by expanding the same spatial orbital $\psi_u$ on the left and on the right sides of integrand expression, i.e. before and after $1/r_{12}$, _differently_ , $$ \psi_u = \sum_l c_{lu} \theta_{l} \quad \text{"on the left side"} \, , \\\ \psi_u = \sum_m c_{mu} \theta_{m} \quad \text{"on the right side"} \, . $$ Is it true? And if yes, why on earth this should be done this way?


title: Why does a single mode fibre have a cutoff wavelength?

text: I earlier had a doubt based on ray theory that light must reflect and so all wavelengths should propagate through SMF. This Q/A Single-mode fibers and ray- theory of light does specify that we have to consider wave theory for propagation of wave in SMF, but i still don't get why a SMF has a cutoff wavelength.


title: Input parameters for the reconstruction algorithm in Digital Holography

text: I've been browsing some Digital Holography papers these days, and have come across this fundamental question. When you reconstruct the complex amplitude for the object image, you use e.g. Fresnel Transform to simulate diffraction. The thing is, one of the parameters in this process is the distance, d, between the holographic plate (or CCD) and the object. However, the whole point of Digital holography, I believe, is to find out the depth profile of the object, that is to say, the value of d. We wouldn't have to do DH in the first place if we had known the precise (down to nanometric realm) value of the distance! Could anyone clarify this for me?


title: What different approximations yield Gravitoelectromagnetism and Weak Field Einstein Equations?

text: This question is inspired by this answer, which cites Gravitoelectromagnetism (GEM) as a valid approximation to the Einstein Field Equations (EFE). The wonted presentation of gravitational waves is either through Weak Field Einstein equations presented in, say, §8.3 of B. Schutz “A first course in General Relativity”, or through the exact wave solutions presented in, say §9.2 of B. Crowell “General Relativity” or §35.9 of Misner, Thorne and Wheeler. In particular, the WFEE show their characteristic “quadrupolar polarization” which can be visualized as one way dilations in one transverse direction followed by one-way dilations in the orthogonal transverse direction. GEM on the other hand is wholly analogous to Maxwell’s equations, with the gravitational acceleration substituted for the $\mathbf{E}$ vector and with a $\mathbf{B}$ vector arising from propagation delays in the $\mathbf{E}$ field as the sources move. My Questions:   1. The freespace “eigenmodes” of GEM, therefore, are circularly polarized plane waves of the gravitational $\mathbf{E}$ and $\mathbf{B}$. This does not seem to square exactly with the WFEE solution. So clearly GEM and WFEE are different approximations, probably holding in different approximation assumptions, although I can see that a spinning polarization vector could be interpreted as a time-varying eigenvector for a $2\times 2$ dilation matrix. What are the different assumptions that validate the use of the two theories, respectively?   2. The Wikipedia page on GEM tells us that GEM is written in non-inertial frames, without saying more. How does one describe these non-inertial frames? Are they, for example, stationary with respect to the centre of mass in the problem, like for Newtonian gravity? There would seem to be very few GR-co-ordinate independent ways to describe, when thinking of GEM as an approximation to the full EFE, a departure from an inertial frame. It’s not like you can say “sit on the inertial frame, then blast off North from there at some acceleration”.   3. Are there any experimental results that full GR explains that GEM as yet does not? I’m guessing that these will be in large scale movements of astronomical bodies.   4. Here I apologise for being ignorant of physics history and also because I am at the moment just trying to rehabilitate my GR after twenty years, so this may be a naïve one: if GR can in certain cases be reduced to analogues of Maxwell’s equations, what about the other way around: are there any theories that try to reverse the approximation from GR to GEM, but beginning with Maxwell’s equations instead and coming up with a GR description for EM? I know that Hermann Weyl did something like this – I never understood exactly what he was doing but is this essentially what he did? I am currently researching this topic, through this paper and this one, so it is likely that I shall be able to answer my own questions 1. and 2. in the not too distant future. In the meanwhile, I thought it might be interesting if anyone who already knows this stuff can answer – this will help my own research, speed my own understanding and will also share around knowledge of an interesting topic.


title: Is general relativity holonomic?

text: Is it meaningful to ask whether general relativity is holonomic or nonholonomic, and if so, which is it? If not, then does the question become meaningful if, rather than the full dynamics of the spacetime itself, we consider only the dynamics of test particles in a fixed globally hyperbolic spacetime? For the full problem, my immediate conceptual obstacle is that it's not obvious what the phase space is. In ordinary mechanics, we think of the phase space as a graph-paper grid superimposed on a fixed, Galilean space. The definitions of holonomic and nonholonomic systems that I've seen seem to assume that time has some special role and is absolute. This isn't the case in GR. There is a Hamiltonian formulation of GR, which would seem to suggest that it's holonomic. In classical GR, information can be hidden behind a horizon, but not lost. This suggests that some form of Liouville's theorem might be valid. The motivation for the question is that Liouville's theorem is sort of the classical analog of unitarity, and before worrying about whether quantum gravity is unitary, it might make sense to understand whether the corresponding classical property holds for GR.


title: Total time taken for an accelerating frame in special relativity

text: How does one derive the equation for the total time traveled in a constantly accelerating frame? I found some help at this question: Special Relativity and Constant Acceleration   But the information pertains to the measured time at a specific time along the way. I have taken the scenario from the question above, but am asking different questions: _A rocket is constantly accelerating at 1g to reach Andromeda (2.4 x 10^22 m) . Assume travel is in only one direction and no external forces act on the rocket, like gravity. The frames are in standard configuration. Halfway there, it begins to constantly decelerate at 1 g and the change takes no time or energy. Find:   \- the maximum speed   \- the total time of the journey measured in the rocket's frame   \- the total time of the journey measured by someone on Earth_ I have tried to do some work on the matter and have derived the following equations so far: $$ v_i ' = c^2 - \sqrt{\frac{(c^2-v_i^2)}{\gamma^2(1-v_x\frac{v}{c^2})^2}} $$   Where $v_i'$ is the instantaneous velocity as measured by the rocket, $v_i$ is the instantaneous velocity as measured by Earth, and $v_x$ is the velocity of travel along the x-axis as measured from Earth (the rocket travels in a straight line along the x-axis). And I have:   $$ \frac{\gamma(v_i')}{\gamma(v_i)}= \gamma(v)\frac{c^2-v_xv}{c^2} $$ Where $\gamma$ is the Lorentz factor as a function. I am completely stuck here however! I do not know how to move forward with deriving the necessary equations to answer the questions above. I assume it has something to do with the rapidity functions, but I am not entirely well-versed in their use. I do have a working knowledge of calculus, which I assume is necessary.


title: Calculating molal depression constant from latent heat of fusion

text: > Calculate molal depression constant of water if latent heat of fusion of ice > at $0^\circ \text C$ is $80 \text{ cal/g}$. All I could find was the equation $K_f=\frac{0.002T^2}{L_f(\text{in cal/g})}$ at http://www.sciencehq.com/chemistry/depression-of-freezing-point.html **How was this equation derived?**


title: How long will my fridge magnet stay attached to my fridge?

text: I have a 500 gram neodymium magnet. It is stuck to my fridge. With the constant pull of gravity trying to pull it off how long would it stay attached to fridge? For arguments sake lets say my fridge is very sturdy and will never need to be replaced! If my fridge was very sturdy would it still be attached when the earth was swollowed up by the sun in its red giant phase?


title: Free falling objects

text: My teacher and I are in the middle of an argument because she says that if you were to drop two objects at the same time and the same height, but with different initial velocities, both of them would hit the ground at the same time. She also said this was proven by Galileo's unregistered experiment when he let two objects fall from the Leaning Tower of Pisa. So I told her that she was right, they would fall at the same time only if both initial velocities had been the same, and that Galileo's experiment was mainly to prove that mass has nothing to do with the timing registered on two objects falling from the same height. Could you please help me out, using 'pure' physics to prove my teacher wrong? I've already searched the internet, but I haven't found something that would convince her because it's what I already have explained to her, but she's in denial and I'm thinking she just doesn't want to accept that she's wrong. Assume no drag/air resistance.


title: Writing a tensor with respect to a particular basis

text: When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way: $$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu \otimes f^\rho$$ where $\\{e_\mu\\}$ is a basis for the vector space, and $\\{f^\mu\\}$ is its dual basis. I know how to expand the tensor when applied to some generic vectors and covectors. So in the example above: $$T(\alpha,\beta,X) = T(\alpha_\mu f^\mu,\beta_\nu f^\nu, X^\rho e_\rho) = \alpha_\mu \beta_\nu X^\rho T(f^\mu,f^\nu,e_\rho) = \alpha_\mu \beta_\nu X^\rho\, T^{\mu\nu}_{\rho}$$ How would I proceed from here? Can I just say $\alpha_\mu = \alpha_\nu f^\nu(e_\mu) = \alpha (e_\mu) $, and similarly for the others? Then I would get: $$T(\alpha,\beta,X) = T^{\mu\nu}_{\rho}\, \alpha(e_\mu)\, \beta(e_\nu)\, f^\rho(X)$$ I am not sure how to generalise from here. Apologies if this is really obvious.


title: Showing Hubble constant is time-independent

text: I have the following question for homework: > _Show that the Hubble constant $H$ is time-independent in a universe in > which the only contribution to energy density comes from vacuum energy._ So in this case we have $\rho = \rho_{\Lambda} = $ constant and $\rho_{\Lambda} = -p_{\Lambda}$. Then using the (simplified) Friedmann equation: $$\dot{a}^2 - \frac{8 \pi G \rho_{\Lambda}}{3}a^2 = -k$$ Dividing by $a^2$ gives $$H^2 - \frac{8 \pi G \rho_{\Lambda}}{3} = -\frac{k}{a^2}$$ But this says that $H$ depends on $a$ which isn't constant. Am I supposed to assume $k=0$? (In that case $H$ is clearly constant ...)


title: SPTs and systems with Topological Order

text: I am an undergrad interested in Condensed Matter Theory. Particularly topological phases and systems exhibiting topological order. A potential research advisor doing a lot of work in Symmetry Protected Topological (SPT) Phases pointed me to a review paper by Senthil (http://arxiv.org/abs/1405.4015) but I have found that despite having taken some of the CMT grad courses at my institution (many-body quantum field theory, cond-mat topics) there are quite a few tools I still lack practice in. Chern-Simons Theory being a prime example. Anyone know some good review papers or perhaps a good path through the literature for someone interested in doing this kind of work?


title: Notation for Sections of Vector Bundles

text: (Reformulation of part 1 of Electromagnetic Field as a Connection in a Vector Bundle) I am looking for a good notation for sections of vector bundles that is both invariant _and_ references bundle coordinates. Is there a standard notation for this? **Background:** In quantum mechanics, the wave function $\psi(x,t)$ of an electron is usually introduced as a function $\psi : M \to \mathbb{C}$ where $M$ is the space- time, usually $M=\mathbb{R}^3\times\mathbb{R}$. However, when modeling the electron in an electromagnetic field, it is best to think of $\psi(x,t)$ as a section in a $U(1)$-vector bundle $\pi : P \to M$. Actually, $\psi(x,t)$ *itself* is not a section, it's just the image of a section in one particular local trivialization $\pi^{-1}(U) \cong U\times\mathbb{C}$ of the vector bundle. In a different local trivialization (= a different gauge), the image will be $e^{i\chi(x,t)}\psi(x,t)$ with a different phase factor. Unfortunately, I feel uncomfortable with this notation. Namely, I would prefer an invariant notation, like for the tangent bundle. For a section $\vec v$ of the tangent bundle (= a vector field), I can write $\vec v = v^\mu \frac{\partial}{\partial x^\mu}$. This expression mentions the _coordinates_ $v^\mu$ in a particular coordinate system, but it is also _invariant_ , because I also write down the basis vector $\frac{\partial}{\partial x^\mu}$ of the coordinate system. The great benefit of the vector notation is that it automatically deals with coordinate changes: $\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial y^\nu}\frac{\partial y^\nu}{\partial x^\mu}$. **My question:** Is there a notation for sections of vector bundles that is similar to the notation $\vec v = v^\mu \frac{\partial}{\partial x^\mu}$ for the tangent bundle? What does it look like for our particular example $\psi$? If no, what are the usual/standard notations for this? How do they keep track of the bundle coordinates?