title: Blandford-Znajek process: Why/how does the current flow along the magnetic field lines

text: Related: How would a black hole power plant work? _I have put a bit of commentary enumerating my confusions in parentheses_ I read in _Black Holes and Time Warps_ (Kip Thorne), that quasars can generate their jets from four different processes. These all involved the accretion disk, but there was one which doesn't make quite as much sense. It was called the Blandford-Znajek process, and it involved magnetic field lines carrying current. The process was visualized in two ways. A black hole, with magnetic field lines, is spinning. In the first visualisation (viewpoint actually), the magnetic field lines 'spin' along with the black hole, and nearby plasma is anchored onto the field lines by electrical forces (where did the electrical fields come from?). The plasma can slide along the field lines but not across them (why?). Since the field lines are spinning, centrifugal forces will fling them up and down the field lines, forming jets. The other viewpoint is this, and it makes even less sense (to me that is, I haven't had a formal education in GR): The magnetic fields and the swirl of space generate a voltage difference across the field lines (Why? How?). The voltage carries current across the magnetic field lines (why are the _field lines_ behaving like wires?). This current travels across plasma, which accelerates it, creating the jets. Now the main thing that doesn't make sense, is that magnetic field lines are behaving like wires. Why would they? I suspect the answer lies hidden somewhere in the equivalence of EM waves in different frames, but I can't think up any convincing argument from that side. If the answer involves GR equations, you don't need to solve it here (wouldn't make sense to me), but if you have to, just refer to the equation and what you did to it, along with the final result. Thanks!


title: Can there be black light? I mean is it possible to devise a machine that outputs darkness?

text: I understand there are various colours that light can have. But i was wondering why there is no 'black' light. What is the logical explanation for this? I mean I am expecting an answer that goes beyond mentioning the spectrum details. All I could think of was a machine as powerful as a blackhole; it could bend the light so hard that all we would see is darkness. But is there any other way? P.S. I am a programmer and didn't study much Physics beyond high school. This question is not a goof. I am not asking this question for fun. I seriously have this curiosity.


title: Timber floating in the river

text: There's an old puzzle (but the physical it is): Lumberjack chop wood and float it down the river. He noticed an interesting feature: in the spring, when the water is comming, timber is nailed to the shore. In the hot summer, when the water lowers, timber, by contrast, floats exactly in the middle of the river. Why is that?


title: How to show that basis of space of Dirac gamma-matrices is given by following matrices?

text: How to show that 16 matrices $$ \mathbf E , \quad \gamma^{\mu}, \quad \gamma^{5} = \frac{i}{4}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}, \quad \eta^{\mu \nu} = -\frac{1}{4}\left(\gamma^{\mu}\gamma^{\nu} - \gamma^{\nu}\gamma^{\mu}\right), \quad \gamma^{5}\gamma^{\mu} \qquad (.1) $$ create the basis in space of Dirac matrices, by having $$ [\gamma^{\mu}, \gamma^{\nu}]_{+} = 2g^{\mu \nu}\mathbf E ? $$ I don't want to calculate each of matrices $(.1)$.


title: Yield Strength versus Ultimate Strength

text: What is the qualitative difference between these two: As seen on the table Typical yield and ultimate strengths. I am trying to resolve the meaning of the phrase "contact yield stress" from C. Thorrton 1997 to real world values.


title: Were fermions ever massless?

text: In a discussion of the Standard model and Higgs mechanism it was claimed that accordingly: > "During an early phase of the cosmos all fermions were massless." I wonder whether this claim can be challenged. Therefore the following more detailed questions:   1. Is it fair to say that, according to the standard model, the mass of any fermion is an "effective mass", that it may vary depending on the Higgs field strength, and that it must necessarily vanish only if the Higgs field strength is exactly zero?   2. Is it fair to say that under conditions " _during an early phase of the cosmos_ " (or: at sufficiently large temperature, density, ...), and while fermions are present, the Higgs field strength is not necessarily exactly zero?


title: Zero Resistance in Quantum Hall Effect and Superconductivity

text: What is the difference between the zero resistance of $R_{xx}$ in integer quantum Hall effect and the zero resistance in superconductivity?


title: Classical electrodynamics formulated in terms of forces

text: The Newton's law of universal gravitation is described in terms of a force, which is produced by an action at a distance. It also can be described using the concept of a field, and that would be an equivalent formulation. Now, classical electrodynamics is formulated in terms of fields in a more complicated way that Newton's gravitation. My question is: Is there an equivalent formulation of classical electrodynamics in terms of action at a distance that is completely equivalent to the formulation in terms of fields (Maxwell's equations)? I would tempted to think that it is not possible, since in Maxwell's equations you may have electromagnetic waves without the presence of charges, and I don't know how to do it only in terms of forces. However, I'm not entirely sure, that's why I'm asking.


title: Rotation speed measurment

text: Could anyone please explain with simply how we can measure the rotation speed of an object using the Sagnac Interferometer ?


title: Is my grandmother alive for some alien who is located on some place and is moving with some speed?

text: _(I am science fiction writer. I had a goal in my novel to argue that time travel to past will not be available even in future. I asked here how to argue that. I collected answers and analyzed them and in conclusion I came to following question:)_ **Question: It is said that there can be found theoretical observer somewhere in universe (moving with specific speed to specific direction) for whom for in his current moment there will be for example 1900 on earth, while you are reading this question in 2013. Is this statement true ?** If yes: Then can this phenomena used to build time machine ? (For example if such kind of observer exists he theoretically can teleport on prehistoric or on future earth). If no: I d like to know why. This subject is described in documentary movie - "Time in nutshell" on the example of alien. You can see the scene directly on link below (youtube) http://youtu.be/MO_Q_f1WgQI?t=4m59s It is awkward because if that is true for some alien on some place my grandmother is still alive. Edit: Imagine the observer has no eyes to process light. It would be more interesting not to focus on the light or on the image of earth delivered to observer for the moment, but **to focus on observer's INDIVIDUAL "now" or what is happening on earth for observer in the moment when he is far on another planet and moves with specific speed.**


title: Point group symmetries and unit cell

text: I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have all point group symmetries. However, is it possible to define a unit cell that has all point group symmetries ?


title: What is the more fundamental quantity? The electromagnetic field F or the potential A?

text: Read somewhere that potential is a more fundamental quantity than EM field because if the latter is more fundamental then gauge transformation will reduce to nothing more than a mathematical trick. I am thinking it another way. Please point out whether my assumption is right or wrong. Since the action principle is the most fundamental of all, anything that is involved in it has to be more fundamental. 4-potential is involved in the action integral in the term describing the interaction of the particle and the field. So doesn't that make it in that sense more fundamental? After all EM field tensor itself is constructed from the potential term in the EOM.


title: Does the Chandrasekhar Limit scale for a Black Hole?

text: No physicist/astrophysicist I; All I know about the Chandrasekhar limit is that it apparently limits the mass a star may survive, beyond which it degenerates to a neutron star, or a black-hole. Does something similar apply to a black-hole? Is there an upper limit to black-hole mass? Can it acquire infinite mass? Is there a limit beyond which it may no longer acquire mass? Does a black-hole die? Is there a corresponding time-frame applicable?


title: What is the meaning of the word "particle" in particle physics?

text: I want to use Matt Strassler's definition of the word "particle" as a specific example: Matt Strassler writes: > (1) "...all the elementary “particles” (i.e. quanta) of nature are quanta of > waves in fields..." My first question is, in this sentence, is the word "quanta" short for "quanta of waves in fields?" Sentence (1) simply defines the word "particle" to be another name for "quanta". I looked up quanta and quanta means > a wave that for a given frequency, have minimum energy and amplitude. So, a quantum is a wave. Is this correct? From this analysis, I conclude that, in particle physics, a particle is a wave, because a particle is defined as a quantum. In mathematical terms: $quantum \equiv wave$ or $particle \equiv wave \equiv quantum$ So, in particle physics, the word particle, has the properties of a wave, and it has no other properties. A particle, in particle physics, for instance, is not a spherical object with finite radius. This is confirmed by Matt Strassler, who writes (2) "...the word “particle” in particle physics has the same meaning as the word “quantum”..." Sentence (2) is incomplete, because the word "particle" has several other meanings besides being another name for quanta. This is very confusing to non- physicists because the word "particle" has been defined many times and physicists pick and choose a meaning that saves the context. My questions are: 1) The word "quantum" is a perfectly good word meaning a wave with minimum amplitude; why are physicists insisting on calling a quantum a particle? 2) Is there a unique definition of the word particle in physics not dependent on context (not casuistic) and that all physicists can agree upon? If you cannot answer any of these, can you clarify if "particle" is another name for "quantum".


title: Is shear elasticity the same as shear modulus?

text: I've encountered both the terms "shear elasticity" and "shear modulus". Are these the same?


Fermi surface nesting and CDW/SDW/SC orders. What is the definition of a nesting vector? And why Fermi surface nesting gives rise to different orders at $T=0$? (CDW: charge density wave; SDW: spin density wave; SC: superconductivity)


title: Does zero change in magnetic flux always imply zero emf induced?

text: If you have a uniform B field, with a finite piece of wire inside it. Assuming the B field spans all space and the wire cannot leave the field. Are you able to create an emf by moving the wire ? I say there is only one possible way, and that is if you rotate it so from its starting perpendicular to the field position it does parallel to the field. That would create a change in flux. But are you able to create emf if the wire stays perpendicular to the wire?


title: Plotting a SHO in matlab

text: I have no prior experience of using matlab. My teacher want me to solve this question. I have been trying for a couple of hours now with no luck, please help! The mass of 100 g hanging in a spring with spring constant 40 N / m. The mass is set into vibration by the spring is stretched 14 mm. Neglecting any energy loss. Use MATLAB and plot a graph showing the distance from the equilibrium position as function of time with initial conditions x(0) = 14 mm and v (0) = 0 mm/s


title: Correlators at large N and large N factorization

text: I am having this very basic problem. In e.g Maldacena's AdS/CFT review (0309246) (page 5), he has defined operators as $\mathcal{O}=N\,{\rm tr}[f(M)]$ for some matrices $M$ and got the connected correlators as $\langle\mathcal{O}\mathcal{O}\rangle_c\sim N^2$ and $\langle\mathcal{O}\mathcal{O}\mathcal{O}\rangle_c\sim N^2$. Whereas in the MAGOO review (9905111) (page 14-15 e.g) we see some operators defined as $G$ which is added to the action as $S\rightarrow S+N\sum g_jG_j$ which gives the correlators as $\langle\prod_j G_j\rangle\sim N^{2-n}$. So that three points and higher vanish at large N. I thought this one is generic. If so I wanted to understand how is the first conventions/case are different and their difference (has it something to do with connected/disconnected diagram? etc.). Related to that I wanted to know about large N factorization. For a particular model if I calculate disconnected diagrams, I see that at large N I indeed get this factorization. I learned that its like WKB in $\hbar\sim 1/N$. But I wanted to feel it more generally in possibly a mathematical+insightful way if possible. Thanks for the help. Edit: If it has a long answer please at least let me know which way to think or any references where I can get some ideas. Thanks again.


title: Is the temperature different and or salinity different can lead to different colour of the water bodies?

text: I had skimming on an article about Thermocline and Halocline. How can two seas not mix? However, I am still curious on:   1. Why the effect of Thermocline and Halocline are distinguishable based on colour different?   2. Is not the substance that is dissolved in the water will give a colour in the water bodies?   3. Do the temperature different and the salinity different will change the colour of the water bodies?   4. Do we measure the effect of Thermocline and Halocline by thermal and salinity parameter also instead of based on visual appearance? And explanation with figure would be nice. Thanks before.


title: Is a free proton a positive Hydrogen-1 ion?

text: The Hydrogen-1 isotope is a form of hydrogen with 1 electron, 1 proton, and 0 neutrons. If that isotope were to lose its electron (thus being a Hydrogen-1 ion with a positive charge) what would be the different between it and a free proton? The positive Hydrogen-1 would contain 0 electrons, 1 proton, and 0 neutrons and the free proton would obviously contain only 1 proton. Is there a defining factor that would distinguish between a free proton and positive Hydrogen-1 ion?


title: Does Mansuripur's Paradox involve fictitious charges?

text: Mansuripur's Paradox involves a magnet moving at relativistic speeds in an external electric field. _Additional:_ thanks to Retarded Potential, who found the original paper. If I understand correctly, the fictitious charges on either side of the magnet should cause a torque, which of course does not actually exist. Apparently this is conventionally resolved by adding an (equally fictitious) internal angular momentum. Mansuripur appears to be arguing that a better resolution would be to add a (fictitious) additional term to the Lorentz force law. Am I right in thinking fictitious charges are involved? If so, why are they introduced, i.e., what is the advantage in doing so? If not, where does the torque really come from?


title: How do laser rangefinders work when the object surface is not perpendicular to the laser beam?

text: I find the functioning of a laser rangefinder confusing. The explanation usually goes like this: "you shine a laser beam onto the object, the laser beam gets reflected and gets back to the device and time required for that is used to calculate the distance". Okay. But the object surface can be uneven and not perpendicular to the laser beam so only a tiny fraction of beam energy is reflected back to the device. And there's plenty of other radiation around, sunlight included. How does a rangefinder manage to "see" that very weak reflected signal in a reliable manner?


title: Discontinuities and nondifferentiability in thermodynamics

text: In physics and engineering sources, calculus-based formalisms - whether differential forms on a manifold, or "differentials" of functions of several variables - are presented as a way of modeling and reasoning about thermodynamic systems. However, I've found little to no mathematical background given to justify these formal manipulations when there are phase changes or other discontinuities. In such regions, one would expect the theory to break down since partial derivatives and differential forms are not defined. Nevertheless, if you "shut up and calculate", everything seems to work out fine. Why is this the case, and what is the proper mathematical state space and framework for systems with phase transitions or other non-smooth properties? (perhaps some sort of "weakly-differentiable" manifold?)


title: What's the difference between these two formulas and how are these called?

text: I just want to know the differences between these two formulas: $h = h_0 + v_0 t ± \frac{1}{2} g t^2$ and $y = y_0 + v_{0y} t + \frac{1}{2} g t^2$ Also, how are these called in English?


title: Sewing together flat spacetime pieces = flat spacetime?

text: I'm trying to imagine the geometry "operations" here:   Angular deficit   and   Conical spacetime of cosmic string **If we sew flat spacetime pieces together, what is the requirement for the sewing to not create curvature at that seam?** Clearly there has to be some condition, because in the examples above, sewing together the wedge "sides" doesn't produce cuvature, but somehow at the corner of the removed wedge, curvature appears. So, more concretely, if we remove this chunk of flat spacetime: $$(x>0) \mathrm{\ and \ } (0<y<x)$$ and now sew it together, somehow this creates flat spacetime everywhere _except_ at $x=y=0$. Is there some intuitive physical meaning to this? If the issue is the sharp angle there, then what if we remove instead a nice smooth parabola: $$(x>0) \mathrm{\ and \ } (-\sqrt{x}<y<\sqrt{x})$$ or maybe a wedge that is completely smooth to zero angle $$[x>0] \mathrm{\ and \ } [0<y<\exp(-1/x)]$$ Are these completely flat everywhere (no string defect) AND have angular deficit?


title: Physics behind 2 different wall kicks

text: I'm looking at a situation where a sprinter starts at some distance x from a wall. The goal is to sprint to the wall, hit it, and get back to the start as quickly as possible. There are two ways which this can be accomplished:   1. Run to the wall, take a slight leap and plant one foot on the wall. Push off the wall, and turn around as you do. Once you've returned to the ground, which should be approx no more half a meter from the wall, begin sprinting back.   2. Run to the wall, jump at it and turn around in the air to plant both feet on the wall (you are now somewhat parallel to the ground and "sticking" to the wall, in what might as well be a squat position). Jump forward from the wall (you should land farther than in the first case) and begin sprinting back. If clarification is needed, please say so. Assuming someone has the strength and capability to do this easily, from a Physics standpoint, which would be the quicker alternative? My gut is telling me the later because of the higher force you can get from using 2 feet (resulting from being able to more easily use all your weight), therefore you have a higher velocity when hitting the ground. However, a friend is going with the former due to leverage advantages.


title: Slinky reverb: the origin of the iconic Star Wars blaster sound

text: This is a fun problem that I came across recently, which I'm posting here for your delectation. We all love a good slinky: they can be used for all sorts of fun demos in physics. One example is the "spring reverb" effect, which is well known in the film sound effects industry. If you couple a microphone to a slinky, and then tap the slinky, it produces a wonderful chirping noise. This is actually how they made the sound of the blasters in the Star Wars film franchise. Watch and enjoy this video for a scaled-up example. The slinky in this case is a coil of 3.5mm galvanised steel, 100m in length. A piezo transducer is used to pick up the sound. What property of the coil allows it to produce such an interesting noise?


title: Does a magnetic field do work on an intrinsic magnetic dipole?

text: When you release a magnetic dipole in a nonuniform magnetic field, it will accelerate. I understand that for current loops (and other such macroscopic objects) the magnetic moment comes from moving charges, and since magnetic fields do no work on charges (F perpendicular to v) it follows that the work done on the dipole (that caused its gain in kinetic energy) must have come from somewhere other than the magnetic forces (like electric forces in the material). However, what about a pure magnetic moment? I'm thinking of a particle with intrinsic spin. Of course such a thing should be treated with quantum mechanics, but shouldn't classical electrodynamics be able to accommodate a pure magnetic dipole? If so, when I release the pure dipole in a nonuniform B-field and it speeds up, what force did the work? Is it correct to say that magnetic fields DO do work, but only on pure dipoles (not on charges)? Or should we stick with "magnetic forces never do work", and the work in this case is done by some other force (what?)? Thanks to anyone who can alleviate my confusion!


title: Statistical sum of physical quantities in a quantum system

text: Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true?   1. $\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$   2. $\mathbb{E}[C^3] = a^3 + 3a^2\mathbb{E}[B] + 3a\mathbb{E}[B^2] + \mathbb{E}[B^3]$


title: Radioactive Decay

text: > **Problem** :Nuclei of a radioactive element $\Bbb X$ having decay constant > $\lambda$ , **(** decays into another stable nuclei $\Bbb Y$ **)** is being > produced by some external process at a constant rate $\Lambda$.Calculate the > number of nuclei of $\Bbb X$ and $\Bbb Y$ at $t_{1/2}$ I tried to create an equation for rate of change of the number of nuclei a: $$\dfrac{dN_{X}}{dt}=\Lambda-N_X\lambda $$ I did that because in simple decay $\dfrac{dN}{dt}=-\lambda N$ holds and here it's also being produced by rate. But after integration should we write $$ln\Bigg(\dfrac{\lambda N_X-\Lambda}{\lambda N_0-\Lambda}\Bigg)=-\lambda t$$ or $$ln\Bigg(\dfrac{\lambda N_X-\Lambda}{N_0}\Bigg)=-\lambda t$$ First one because limit was on $N: (N_0\to N)$ And next what to substitute for $t$ (ie. what is $t_{1/2}$? $ln2/\lambda$ or something else?) Also how to do it for $\Bbb Y$? Just write $$\dfrac{dN_Y}{dt}=\lambda N_x $$?


title: On BE and FD Statistics

text: Lets consider the Bose-Einstein and Fermi-Dirac statistics:   1. Bose-Einstein statistics: $$\langle n_i\rangle = \frac{1}{\exp{[(\epsilon_i-\mu)/kT]} - 1}$$   2. Fermi-Dirac statistics: $$\langle n_i\rangle = \frac{1}{\exp{[(\epsilon_i-\mu)/kT]} + 1}$$ How would these laws respond to Lorentz boosts (Inertial frames being considered in this case)? It could provide some insight into the behavior of these statistical laws in fast moving bodies like galaxies moving at relativistic speed. Incidentally the quantity:$(\epsilon_i-\mu)/kT$ is a dimensionless quantity. But quantities like v1/v2 and E1/E2 are also dimensionless. But these ratios do change for Lorentz boosts. [v and E represent velocity and energy respectively] From the point of general relativity, is it necessary to give these laws a covariant form?


title: Does light escape when I open my blinds?

text: When I open my blinds, there is light enough to read a book outside while the lamp is on inside. So when I close my blinds, does the light that otherwise will escape, stay inside and thus being lighter inside?


title: whats the rate of energy increase required for constant acceleration between 0.0c and 0.99c?

text: I was wondering how much energy would be required to accelerate 1000kg to 0.99c at 1G. What I don't understand is what the rate of increase of energy is required as velocity increases. I was looking at the Lorentz factor curve but I'm not sure if this affects the energy required or just time dilation. eg. 0.99c = Lorentz factor of 7. So travel only takes 1/7th the time from the point of view of the traveler but does that mean you need 7 times as much energy to get to 0.99c than from say 0.0c to 0.1c. Does any body have a link to a graph that shows this rate of increase of energy required or is it the same as the Lorentz factor curve? I'm a programmer not a physics or math person so go easy on me. Thankyou.


title: If someone shoots a gun (45 caliber) at someone holding up a table or desk 10 feet away, does it slow or turn the bullet away?

text: Assume it's a standard .45 caliber handgun. Shooter is 10 feet away, and we hold a desk, mini-table, blackboard, etc on our chests(to protect vitals). How do you figure out the bullet's decrease in velocity, and likelihood of flying away from you. I got the idea thinking whether could we protect ourselves against a madman like the VT or Columbine guys?


title: How is North Korea planning to test a nuclear device underground?

text: I am reading the news about North Korea planning to test a nuclear device **underground**, and while fascinating, I have no idea what this phrase actually means. Could somebody explain, in absolute layman's terms, how an underground nuclear test actually works, what the dangers are, and how North Korea will use the results?


title: Resistance from voltage and charge

text: I'm sure this is pretty basic, but I'm not very good at electricity. > At both ends of a conductor we apply 12 volts. For one minute an electric > charge of 72 C passes through the conductor. What is the resistance of the > conductor?


title: Why are extra dimensions necessary?

text: Some theories have more than 4 dimensions of spacetime. But we only observe 4 spacetime dimensions in the real world, cf. e.g. this Phys.SE post.   1. Why are the theories (e.g. string theory) that require more dimensions taken seriously by scientists?    2. Is there any proof that these extra dimensions exist?   3. Is there a simple layman's explanation for the need [or strong hint] for extra dimensions?


title: Helmholtz decomposition in the plane

text: Prove or disprove the following proposition: For any smooth plane vector field $\mathbf{H}=\left(H_x,H_y\right)$, there exist scalar potentials $\phi$, $\psi$ such that $H_x=\frac{\partial \phi }{\partial x}+\frac{\partial \psi }{\partial y}$ $H_y=\frac{\partial \phi }{\partial y}-\frac{\partial \psi }{\partial x}$


title: Why is number of collisions equal to number of scattered particles times number of scatterers?

text: In Landau, Lifshitz "The Classical Theory of Fields" $\S12$ "Invariant cross- section" the following is said1: > ... according to the usual definition of the cross-section $\sigma$, the > number of collisions occuring in volume $dV$ in time $dt$ is $$d\nu=\sigma > v_{\text{rel}}n_1n_2dVdt,$$ where $v_{\text{rel}}$ is the velocity of > particle 1 in the rest system of particle 2 ... I'm trying to make sense of this formula, and here's what I got so far: $\sigma v_{\text{rel}}n_1dt$ is total number of scattered particles in time $dt$, and $n_2dV$ is total number of scattering centers. Why isn't total number of scattered particles equal to the number of collisions? Is it considered that there's more than one collision per scattered particle? If yes, then why should it appear that each scattered particle collides with _each_ of scattering centers (only in this case would we just multiply by number of scattering centers)? 1\. In English version which I cite the formula lacks a $dV$ factor. I put it here according to original Russian book.


title: Why doesn't a phone charge faster, rather than slower when it is in use

text: In Physics class, we were building parallel circuits, and as more lights were attached in parallel, they got brighter (as more power was being provided to the lights, and the resistance decreases). So, when I charge a phone, why doesn't the battery charge faster when it is in use (eg. more devices powered on within the phone), than when it is not in use?   Clarification: All lights at once became brighter


title: Uniqueness of Helmholtz decomposition?

text: Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ and $$\pmb{\nabla} \cdot \pmb{G}=0.$$ Is this decomposition unique? That is, given $\pmb{H}$, are the fields $\phi$, $\pmb{G}$ satisfying the above equations unique?


title: Do algorithms have an intrinsic time direction?

text: This article says > There is no intrinsic time direction in Newton's mechanics nor in the > differential equations of the new physics. My question is, do other types of mathematics, say a cellular automata as an example, have the intrinsic time direction that the equations lack? With a CA, all steps have to be completed sequentially. **UPDATE** : Here's what I mean. On this page, it describes projectile motion. It uses one approach with equations, and another approach with an algorithm. Very clearly both approaches yield the same answers. The biggest difference between them is how time works. Question: Is having "an intrinsic time direction" the same thing as "time having no choice but to march forward in the algorithm"?


title: Slowing of photons in a EB Condensate

text: Help me understand this. I have heard tell that they are now able to slow the forward progress of a photon traveling through an EBC to somewhere near 35mph. Is this actually the physical slowing down of the photon, or is it the slowing down of the absorption and emission process? If the photon is actually slowed down, does it gain mass? Or am I miss interpreting this all together. Thanks


title: Creating large aspheric lens profile for CNC cutting out of Acrylic plastic for huge Camera Obscura

text: I plan to use a CNC mill to cut a plano-convex aspheric lens from a large (300mm dia) thick(30mm) block of Acrylic (PMMA/Plexiglass). I need a focal length on the order of 800-2000mm The purpose of this large lens will be a lensed Camera Obscura that I hope will have a very bright image. I have built a lensed camera obscura before with a 150mm dia 2000mm FL glass lens. Documentation of project: https://vimeo.com/94692036 I have also used CNC routers to cut lenses from PMMA before, but they were spheric and made on low-tolerance machines. The lenses I made are 300mm dia 1000mm FL (Plano-Convex PMMA) and are in this video: http://vimeo.com/66783407 How does one go about generating a lens profile? Is there parametric software for the creation of an optimal projection lens as I describe? I am new to optical design, and would love some advice. I had considered buying such a lens, but they are prohibitively expensive or rare.


title: What would happen if Large Hadron Collider would collide electrons?

text: After some reading about the Large Hadron Collider and it's very impressive instruments to detect and investigate the collision results, there is a remaining question. What would happen if the scientists would use leptons instead of hadrons? Especially: What would happen if they would collide electrons? Isn't it intrinsic that all particles consist of smaller particles? With current technology, could we detect them?


title: What is the importance of the Fermi energy $E_F$ or the chem. potential $\mu$ for topological superconductors

text: A lot of effort is put into shifting the Fermi energy of a topological insulator to exactly zero which then provides some advantages when this TI is coupled with a superconductor. I don't understand why this is important or what advantages this should bring. Didn't Kane and Fu in 2008 (see: *) proposed the creation of a Majorana fermion explicitly for a TI that does not have the chemical potential at zero?   * http://www.physics.upenn.edu/~kane/pubs/p56.pdf


title: Heat reflection on distant planets

text: I was watching a documentary last night on the first planet discovered outside our solar system. The first one apparently is a giant gas planet that orbits very close to its sun over a very fast period (~1 week per orbit), and it's so hot all metal elements vaporise and form parts of it's atmosphere (so the current theory states anyway). Out of curiosity, when they estimate the surface heat of the planet (they give big numbers), did they take into account reflection of heat from it's metallic atmosphere, and internal reflection from heat bouncing off the surface and it's atmosphere (like global warming) when estimating surface temperature of the planet?


title: Do Sears Physics and Berkeley Physics Series provide intuition?

text: I'm freshman of Math major, because we don't have any physics course. But I'm very interested in Physics. But I'm not very sure which is better or say, more suitable for me.   1. Sears and Zemansky's University Physics (10th Edition)    2. Berkeley Physics Course Series (Vol 1, 2, 3, 4, 5)  Because I have talked to some of my friends who are in the major of physics. Some said that Sears physics is more oriented to High school level, with lots of pics, examples, hence to recommand me the Berkeley series which is more 'academic'. Some others said that the Berkeley Series is too beyond for me, because I'm in major of maths, with very limited foundation of physics. As I'm more into math, a book which focuses on the physical intuition would be nice. This leads me to my question _"Which of these books is better for gaining physical intuition? Which uses a more indtuitive approach, in other words."_


title: Is the speed of sound almost as high as the speed of light in neutron stars?

text: Have you ever wondered about the elastic properties of neutron stars? Such stars, being immensely dense, in which neutrons are bound together by the strong nuclear force on top of the strong gravity that “presses” them together, one would think they must have extremely large Young modulus, and the speed of sound could be on a par with the speed of light in the vacuum. If we let $c_s$ be the speed of sound, and also assume that the neutron star is isotropic, then using the well known equation for the speed for acoustic waves in solids, we can write the following equation for the crust of the neutron star $c_s=\sqrt{\frac{E}{\rho}}$ For a neutron star of density $\rho =5.9\times 10^{17}$ Kg m$^{-3}$ and Young modulus of about $E=5.3\times 10^{30}$Pa we get a value for $c_s=3.0\times 10^6$ ms$^{-1}$! **The Questions are:** 1) How can sound travel at such immense speeds inside a neutron star? 2) Should nuclear interactions, n-n and q-q , dictate the elastic properties of a neutron star, or is it just the gravitational force?


title: what is meant by "crossover phenomena"?

text: In many articles I read the term "crossover phenomena" and a lot of methodology discussed according to it, with little or no description about what is meant by it. Sometimes there is a connection to networks, matrices, and then to other structures of elements. Can someone describe the concept of the phenomena?


title: If matter creates space, shouldn't there be experimentally detectable consequences?

text: Ernst Mach, a man to who influenced Albert Einstein significantly in his approach to relativity, did not quite seem to believe in space as a self- existing entity. I'm pretty sure it would be correct to say he thought it was created by matter. So let's take that idea for a moment and run with it. Wouldn't the idea that matter creates space unavoidably create _testable_ predictions about the nature of how forces work, especially over astronomical distances? I say this because from my admittedly information-centric perspective, the only way matter can create something approximating what we call space is by creating _relationships_ between particles. Space would then become nothing more than a particular set of rules for how those relationships interact and change over time, capturing for example the idea that they have locality and interact based on rules such as the $\frac{1}{n^2}$ fall off of some forces with distance. With large enough sets of particles, the resulting interactions would become sufficiently smooth and fine-grained to create the abstraction we call space. But Euclidean space as we know it would necessarily be an illusion. (Incidentally, I have no idea if this line of thinking might be related to holographic universe ideas.) Now, the interesting thing about that argument is this: If one assumes as an experimental hypothesis that matter creates space, I don't easily see no way around the implication that the idea should be testable, at least in principle. That's because a simulation of smooth space created by the particle interactions will necessarily be incomplete and dependent on the distribution of those particles. So for example, if you only had a universe of two particles, only _one_ space- like relationship would exist. The resulting simulation of space could not possibly be as smooth or rich as the space we know while sitting at the Euclidean limit of a nearly number of particles and particle relationships. It would for example likely have some sort of predominantly one-dimensional field equations, e.g. a the universe with only one electron and one positron might maintain constant attraction between them regardless of their separation distance. (It's also interesting to note that constant 1D-like attraction is approximately how quarks behave when pushed far away from short-range asymptotic freedom envelopes.) Detecting deviations from the Euclidean limit would be hugely more difficult in our particle-rich universe, but I cannot easily see how it would be flatly impossible. Asymmetries of matter at the scale of the entire universe would for example have to affect the nature of space by creating asymmetries in the number and richness of the underlying particle relationships. If a precise model could be made for how such relationship asymmetries affected our local space abstraction, testable predictions would be possible. The first approximation in any such model would simply be to map out the density and orientation of the relationships based on our best guesses at particle distributions, then see if those relationship sets correlate strongly to any known effects. The unexpected and quirky distribution in dark matter could certainly be a candidate for such an effect. Again, as an information type I take high correlations pretty seriously, and some of curve fits between certain galactic phenomena and predictions made by the oddly simple MOND rule remain poorly explained at best. If such correlations somehow stemmed from asymmetries in space itself due to large-scale asymmetries in the galaxy-scale distribution of particles, a very different approach might be possible to explaining why the MOND rule sometimes produces such unexpectedly strong curve correlations. So again, the question is just this: Has the possibility of testing the Machian hypothesis every been explored theoretically? And if not, why not? What am I missing? (This question is a direct outgrowth of my earlier amusingly unproductive question about whether space exists independently of matter. For that one I received exactly one "yes" answer, exactly one "no" answer, and no up or down votes on either one. Flip a coin indeed!)


title: Homework Question Transformation Energy

text: A $1400kg$ car is approaching the hill shown in the figure at $14.0m/s$ when it suddenly runs out of gas. ![enter image description here](http://i.stack.imgur.com/RDU61.png) What is the car's speed after coasting down the other side? I think I have to use this equation $mg(\Delta h)=\frac 12 m(\Delta v)^2 $ $(1400)(9.81)(\Delta h)= \frac12 (\Delta v)^2$ I am not sure if this is correct. If it is, what do I plug in for $\Delta h$?


title: Can a wave propagate in an elastic fluid in the absence of volume forces?

text: A motion (wave) $\mathbf{x}: \mathcal{B}_0 \times [t_0,t_1] \to \mathcal{E}:$ such that   $q-o = \mathbf{x}(p,t)=(p-o)+\mathbf{a}_0 cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$ can propagate in an elastic fluid with constitutive equation:   $\mathbf{T}=-\pi(\rho)\mathbf{I}$ not subject to any volume force? What restrictions on polarization vector $\mathbf{a}_0$ and the vector $\mathbf{k}_0$ are needed? $\mathbf{a}_0$ and $\mathbf{k}_0$ should be parallel? What restrictions on $\pi(\rho)$ are needed? I can write the balance equation of linear momentum, in an eulerian description I suppose: $\rho \mathbf{a} = \mathrm{div}_q \mathbf{T} = -\nabla_q \pi = - \pi' \nabla_q \rho$ But it is easier to compute the acceleration in a lagrangian description:   $\ddot{\mathbf{x}}(p,t) = -\omega_0^2 \mathbf{a}_0cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$ At this point computing $\nabla_q\rho$ is not easy I think. There is some mistake here? There is a better and simpler way to prove the results? _Thanks and Please excuse my bad english_


title: Could we prove that neutrinos have mass by measuring their gravitational signature?

text: It is now said that neutrinos have mass. If an object has mass then it also emits a gravitational field. I appreciate the neutrinos mass is predicted to be small, but as there are so many produced by our sun and gravity works collectivity should we not be able to detect neutrinos through gravitational differences? An extract from Wikipedia states:- > "Most neutrinos passing through the Earth emanate from the Sun. About 65 > billion ($6.5 \times 10^{10}$) solar neutrinos per second pass through every > square centimeter perpendicular to the direction of the Sun in the region of > the Earth." That is a great deal of neutrinos. Would that many not produce a noticeable gravitational signature that we could detect? If we can't detect it does that mean that neutrinos are massless?


title: According to wave function collapse you only have one outcome, so what happens to the other superpositions?

text: 1. If the superpositions of a wave function are not needed because only one of the superpositions is allowed, what happens to the eigenvalues of the "null" superpositions?    2. Is the energy transferred elsewhere such as alternate universes, supporting theories such as higher dimensions and inflation theory?    3. Could these collapses come from string harmonics, therefore determining our universe and the others plausibly around us?


title: What reason(s) exist to suppose that all degeneracy pressures can be overcome in Black-Hole formation?

text: In models of stellar collapse to a black hole, it is a given that density increases without bound towards a singularity. Electron degeneracy I get. Neutron degeneracy I get. I assume there's some (theoretical) support that quark degeneracy can be ultimately overcome through some process Wikipedia doesn't describe - what is it?


title: Lightning driven electrolysis

text: Could a lightning bolt be harnessed to perform electrolysis, would a useful amount of hydrogen be produced? I'm imagining some industrial scale device in the ocean with some hydrogen capture mechanism.   1. Ignoring fact that lightning is unpredictable (though on a large flat surface eg ocean should be able to direct lightning somewhat)   2. Yes lightning is destructive but obviously many things are hit everyday without being utterly vaporised. Yes hydrogen is explosive but could the 'device' not be placed at some depth in the ocean where either the heat of the lightning does not ignite the hydrogen, or the hydrogen is separated from the oxygen produced by the reaction/and surface air (can hydrogen burn in water?)


title: Does the mathematics of physics require impure set theory?

text: Suppose for the sake of this question that all mathematics is ultimately reducible to set theory in such a way that the only mathematical objects there _really_ are, are sets. Now, there is a common distinction between pure and impure set theory. Pure set theory is built up from the empty set and only involves "pure" sets--- sets that only contain other sets as members. Impure set theory, by contrast, involves _urelements_ \--- non-set individuals which can be members of sets. My question is whether the mathematics needed for physics (assuming it all gets cashed out in terms of set theory) is of the pure or impure variety? Are there primitive relations of physics (where a primitive relation of physics is something unique to physics--- not something like equality/identity) that hold between mathematical entities (like numbers) and physical entities? If so, could you provide examples?


title: Do photons lose energy while travelling through space? Or why are planets closer to the sun warmer?

text: My train of thought was the following: The Earth orbiting the Sun is at times 5 million kilometers closer to it than others, but this is almost irrelevant to the seasons. Instead, the temperature difference between seasons is due to the attack angle of the rays, ~~so basically the amount of atmosphere they have to pass through~~. Actually, it makes sense, heat comes from the photons that collide with the surface of the earth (and a bit with the atmosphere) and gets reflected, and there's nothing between the earth and the sun that would make a photon lose energy over a 5 million km travel on vacuum. Or is it? (Note I'm not wondering about the possible lose of energy related to the redshift of the expanding universe.) Which made me wonder… So why then are the planets closer to the sun warmer? It seems silly, the closer you are to a heat source, the warmer it feels, but that's because of the dispersion of the heat in the medium, right? If there's no medium, what dissipates the energy?


title: is Bohr-sommerfeld formula valid if the potential is non-smooth?

text: let be a non-smooth potential , for example a linear combination of step functions $$ \sum_{n=0}^{10}H(x-n) $$ my question is, for this potential would be Bohr-sommerfeld quantization formula valid ?? is there any resource where they apply bohr sommerfeld formula to non-smooth potentials $$ 2\pi (n+\frac{1}{2})\hbar = \oint _{C}p.dq $$


title: Solving differential Equation for the Two-Body Problem

text: So, I'm following the derivation in D. Morin, _Introduction to Classical Mechanics,_ of the equations for a two-body system. I understand all of it, aside from this one step. When he's talking about solving Lagrangian for $r(\theta)$, I don't follow the step. $$\tag{*}L = \frac {1}{2} m\dot{r}^{2}+ \frac{\ell^2}{2mr^2}+u(r)$$ and some how he ends with $$\tag{7.16}(\frac{1}{r^2} \frac{dr}{d \theta})^2 = \frac{2mE}{\ell^2}- \frac{1}{r^2} - \frac{2mu(r)}{\ell^2}.$$ This is probably a case where my lack of differential equation skills causes trouble.


title: A thought about Quasars

text: If Quasars are "beams" of energy exiting a super-massive black hole, in order for them to get through the black-hole's event horizon, they'd have to be traveling faster than the speed of light. My question is the following: My understanding of particle physics is still sketchy, nevertheless I think that in order for a particle to move faster than the speed of light, it must be mass-less. What particles are involved in the creation and furthermore, the mechanics of quasars?


title: Reflectance and phase shift vs incident angle and wavelength

text: Plot the refectance and phase shift on reflection $\varphi$ for p and s polarized light versus incident angle $\theta$ for various wavelength $\lambda_1=.., \lambda_2=..,..$ in the simple case of reflection from a bare substrate given the complex refractive index $N=..$ of the substrate. Since the reflectance is $$R=(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})^* $$ where $\eta_{s,p}$ is $Ncos\theta$ and$\frac{N}{cos\theta}$ for s and p polarized light. Also, the phase shift on reflection is:$$\varphi=\arg(\frac{1-\eta_{s,p}}{1+\eta_{s,p}})$$ I can see the dependence of $R$ and $\varphi$ on the incident angle, but how can i plot it at different wavelength ? i don't see the dependence of $R$ and $\varphi$ on wavelength at all.


title: Böotes Void and Dark Galxies

text: Wouldn't it be possible that Böotes Void, a space we generally consider to be "empty" be filled with a number of undetectable dark galaxies?


title: Hows lensmaker formula works? if Radius is zero?

text: ![enter image description here](http://i.stack.imgur.com/LFEbx.png) I am doing my revision and there's such a question. However, when I tried to plug the values into the lensmaker formula, I find that one of the R is equal to zero, as it is not curved. $(n_2 -n_1)(1/r_1 - 1/r_2)$


title: Are majorana masses technically natural?

text: I understand that Dirac masses for fermions are "technically natural" (in the 't Hooft sense of technical naturalness) because they break chiral symmetry. In limit where the Dirac mass goes to zero, chiral symmetry is restored, so one expects any loop corrections to the Dirac mass to be proportional to the Dirac mass itself. That is to say, there is no power law UV sensitivity in a Dirac mass parameter. Is the same true for Majorana mass terms? Because the Majorana fermions are real, is it still true that the mass term breaks chiral symmetry? It seems so, since one may write a 4-component Majorana fermion in terms of a single Weyl fermion: $\Psi_D = \begin{pmatrix} \psi_\alpha \\\ \bar\psi^{\dot{\alpha}}\end{pmatrix}$, where the $\alpha$ and $\dot{\alpha}$ are Weyl (SL(2,C)) indices. Then a Majorana mass term takes the form: $M\bar\Psi_M \Psi_M = M(\psi^2 + \bar\psi^2)$, where the Weyl spinor contractions use the $\epsilon$ tensor. These terms appear to break chiral symmetry as well. Is this the correct picture?


title: Expectation value of momentum

text: I'm having a problem with an expectation value that doesn't seem to add up for me. What I know is, that $\psi(\vec{r})$ is a wavefunction for a particle in three dimensions. The Hamiltonian is given by: $\hat{H} = \frac{\hat{p}^{2}}{2m}+\hat{V}(\vec{r})$ The expectation value of the momentum in the state $\psi$ is given by: $\langle \psi,\hat{\vec{p}}\psi\rangle = \vec{p}_{0}$ Another state is given by: $\phi(\vec{r})=\psi(\vec{r}) \cdot e^{i\vec{k} \cdot \vec{r}} $, where $\vec{k}$ is a constant vector. Now I have to calculate the expectation value of the momentum in the state $\phi$. Then I just thought I had to do this: $\langle\vec{p}\rangle=\langle\phi|\hat{\vec{p}}|\phi\rangle = e^{-i\vec{k} \cdot \vec{r}}e^{i\vec{k} \cdot \vec{r}} \langle\psi|\hat{\vec{p}}|\psi\rangle$, which would end up being $\vec{p}_{0}$. But according to the solution I have to get: $\langle\vec{p}\rangle = \vec{p}_{0}+\hbar\vec{k}$, which I can't see how is done ? I know that the momentum is given by $p = \hbar k$ in general, but I can't see how it comes out in the solution ? So anyone who might be able to give me a hint ? Thanks in advance.


title: What is (or where can I discover) the Burke Potential?

text: I have very much enjoyed William L. Burke's _Applied Differential Geometry_. Reading around on the web it seems that he discovered something which is called the (retarded) Burke Potential, but I have not been able to find any references to it. Can anyone point me at a reference or explain succinctly what it is?


title: How to derive the divergence leading to Kohn anomalies?

text: I'm trying to understand the mathematical derivation given in the book "A Quantum Approach to Condensed Matter Physics" on page 215 (see 1), for explaining how the phonon-energy perturbed by phonon-electron interaction, namely $$\hbar\omega_q^{(p)} = \hbar\omega_q - \sum\limits_k 2 | M_{kk'}|^2 \langle n_k \rangle (\varepsilon_{k'} - \varepsilon_{k})^{-1} $$ with $k' = k - q$ gives rise to the _Kohn anomaly_ by making $\partial \hbar\omega_q^{(p)} / \partial q_x$ diverge when $q \sim 2k_F$. The book explains that > Let us suppose $q$ to be in the $x$-direction and of magnitude $2k_F$, and > evaluate $\partial \hbar\omega_q^{(p)} / \partial q_x$. If we neglect the > variation of $M_{kk'}$ with $q$ the electron-phonon interaction contributes > an amount $$ 2 \sum\limits_k | M_{kk'}|^2 \langle n_k \rangle > (\varepsilon_{k'} - \varepsilon_{k})^{-2} {\partial \varepsilon_{k-q} \over > \partial q_x} .$$ On substituting for $\varepsilon_{k-q}$ one finds the > summation to contain the factors $\langle n_k \rangle (k_x - k_F)^{-2}$. > These cause a logarithmic divergence when the summation is performed [...] What I don't grasp yet is the step of substituting for $\varepsilon_{k-q}$ (what exactly does one substitute with? $\varepsilon_k = \hbar^2 k^2 / 2m$?) and what the summation then looks like concretely to cause a logarithmic divergence.


title: Why do higher pitches appear to be louder?

text: It may just be in a few cases, but in the case of a flute, a higher pitch appears to come with a perceived higher volume. Is this simply because you need to put more energy into the flute to get a higher harmonic? Why is that? It seems that one shouldn't need to put in more energy, and get a consequently a higher volume.


title: Expectation value of an operator

text: Suppose we have: $$ \hat{Q}|\psi_1\rangle=q_1|\psi_1\rangle \\\ \hat{Q}|\psi_2\rangle=q_2|\psi_2\rangle $$ with $q_1 \neq q_2$. Then consider the state: $$ |\Psi\rangle=\frac{1}{\sqrt{2}}(|\psi_1\rangle-i|\psi_2\rangle) $$ I want to calculate the uncertainty in $\hat{Q}$. Then first we must compute $\langle\hat{Q^2}\rangle$ and $\langle\hat{Q}\rangle^2$. I started with $\langle\hat{Q^2}\rangle$. First, $$ \hat{Q}|\Psi\rangle=\frac{1}{\sqrt{2}}(q_1|\psi_1\rangle-iq_2|\psi_2\rangle) $$ Then: $$ \langle\Psi|\hat{Q}=\frac{1}{\sqrt{2}}(q_1^{*}\langle\psi_1|+iq_2^{*}\langle\psi_2|) $$ It follows that: $$ \langle\Psi|\hat{Q^2}|\Psi\rangle = (\langle\Psi|\hat{Q})(\hat{Q}|\Psi\rangle) =\frac{1}{2}(q_1^{*}\langle \psi_1|+iq_2^{*}\langle \psi_2|)(q_1|\psi_1\rangle -iq_2|\psi_2\rangle ) $$ Using the fact that: $$ \langle A|bB+cC\rangle = b\langle A|B\rangle + c\langle A|C\rangle \text{ and}\\\ \langle bB+cC|A\rangle = b^{*}\langle B|A\rangle + c^{*}\langle C|A\rangle $$ I found: $$ \langle\Psi|\hat{Q^2}|\Psi\rangle = \frac{1}{2}(q_1^2-q_2 q_2^{*}) $$ Note that $q_1^2 \neq |q_1|^2$. On the other hand, if I act on the ket vector twice to compute $\hat{Q^2}|\Psi\rangle$ I find: $$ \hat{Q^2}|\Psi\rangle = \frac{1}{\sqrt{2}}(q_1^2|\psi_1\rangle -iq_2^2|\psi_2\rangle) $$ Then using this expression to compute $\langle\Psi|\hat{Q^2}|\Psi\rangle$ yields: $$ \langle\Psi|\hat{Q^2}|\Psi\rangle = \frac{1}{2}(q_1^2 - q_2^2) $$ Now using the representation in the basis: $$ \hat{Q}|\Psi\rangle = \frac{1}{\sqrt{2}} \left[\begin{matrix}q_1 \\\ -iq_2\end{matrix}\right] \\\ \langle\Psi|\hat{Q} = \frac{1}{\sqrt{2}}[q_1^{*} \text{ }iq_2^{*}] $$ Which gives: $$ \langle\Psi|\hat{Q^2}|\Psi\rangle = q_1^{*}q_1 + q_2^{*}q_2 $$ Why am I getting different results?


title: Having trouble understanding the work energy principle intuitively

text: I'm having trouble understanding the work energy principle intuitively. This is what I'm solid on so far: If you have a ball rolling down a hill, it loses potential energy and gains kinetic energy. What's happening here is that the potential energy is being transferred to kinetic energy, right? This can be summarised as -∆PE=∆KE or at any point KE+PE=constant, correct? I'm fine when just kinetic energy and gravity are involved, but I get a bit confused when other forces come into play. Now say that there's a frictional force opposing the ball. This frictional force is acting against gravity so am I right in saying that the gravitational potential energy lost is equal to the kinetic energy gained minus the work done by the force i.e. ∆PE=∆KE-Fs? I've been given the formula: work done by other forces = ∆KE±∆PE but I don't have an intuitive understanding of it, I want to know why it works!


title: How to calculate the velocity of fluid at the outlet when density and the pressure drop are known?

text: I have a U- like pipe. Its inlet has atmospheric pressure $p_o=10^{5} \, Pa$. Vacuum is applied to the other end with a pressure gradient $\nabla p_v=-30 \cdot 10^{3} \, kPa/s$. The total time of the simulation is $t=0.25$ s. I assume that the highest velocity would be reached at the end of the simulation. After 0.25 s the vacuum pressure is $p_v=-7.5 \cdot 10^{3} \, Pa$. I am interested in knowing the maximum speed of air ($\rho=0.146$ $kg/m^{3}$) at the outlet (assuming this is where it is highest). One idea that comes to mind is to use the Bernoulli equation: \begin{equation} p_0 + \frac{1}{2} \rho v_{0}^{2} +\rho g h_0 = p_v + \frac{1}{2} \rho v_{v}^{2} +\rho g h_v \end{equation} The $h$ value in both cases is the same, hence the equation simplifies. Assuming the speed of air at the inlet $v_0=0$ m/s, further simplifying the equation. \begin{equation} p_0=p_v+\frac{1}{2} \rho v_{v}^{2} \end{equation} \begin{equation} v_v=\sqrt{\frac{2}{\rho}(p_0-p_v)} \approx 1213.5 \, m/s \end{equation} However that seems unrealistically high. My rough computational simulation gives a value of $v_v=64 \, m/s$. How can I calculate $v_v$? Should I not be taking into account properties of the geometry? I am not really sure I can consider $v_0=0$, because surely the speed will be almost as high at the inlet as at the outlet due to the long term effects of the pressure gradient. The purpose of this is to give me a rough value of the maximum speed of air in such a scenario, so I could use $v_v$ value in a computational simulation. ![A U-shaped pipe.](http://i.stack.imgur.com/8o7nb.png)


title: Is there a way to define a function of a meniscus curvate?

text: If you cut a thin slit in thin opaque material and then put it into water and pull it out, the meniscus will be formed in the slit. For my research I need to know if it is analytically possible to define the curvature of meniscus( is it hyperbola, parabola, circle or something else). I need the function of curvature so I will be able to make calculations of how the light passes through the meniscus. I know there are numerical methods to make an approximate function, but I want to know if there is any other way.


What causes matter to initially rotate/spin/orbit? All I can find is the statement that in space particles of dust/gas/matter contract into a spinning disk due to gravity (to form stars, solar systems, galaxies etc.), with no explanation as to why the spin began. I see a lot about the conservation of angular momentum, but these discussions all presume that the 'spin' already exists. What caused the spin in the first place? Shouldn’t gravity simply attract particles of dust, gas or matter together along a straight path till they collide, as a magnet does to a paper clip? The magnet does not make the paper clip revolve around it, and if I fall off of a building, I don’t spin around the earth. I fall in a straight path till I collide with the earth. What am I missing?


title: What is Quantum jump of an electron?

text: Can any one define quantum jump 9 quantities jump of an electron ? I know it it is a silly question but can anyone please explain me in detail.I am a learning about the structure of atom and I want some help. Can anyone help.


title: Time taken for collision

text: We have three particles at the vertices of equilateral triangle of side $d$. At $t=0$ they start moving in such away that at all instant of time each of them has a speed $v$ towards adjacent one. We have to find the time after which they will collide. I know that you can solve this by calculating from reference frame of any one particle, but I want a way to solve this by calculating the total distance traveled first in ground frame and then dividing by speed.


title: tritium beta decay - probability of being in 1s state

text: Hydrogen-like wavefunctions have the form: $$R_{10} = \left(\frac{Z}{a_0}\right)^{\frac{3}{2}} 2\space e^{-\frac{Zr}{a_0}}$$ $$Y_{00} = \frac{1}{\sqrt {4\pi}} $$ where $a_0 = \frac{4\pi \epsilon_0 \hbar^2}{\mu e^2}$ Suppose Tritium $^3$H that's in its ground state beta decays to $^3$He suddenly. I want to find the probability the helium atom is in the 1s state. Firstly, is the reduced mass $\mu$ the same for both helium and $^3$He? I assumed it is $m_e$. Wavefunction of helium is $R_{10}' = \left(\frac{2}{a_0}\right)^{\frac{3}{2}}2\space e^{-\frac{2r}{a_0}} $. Wavefunction of tritium is $R_{10} = \left(\frac{1}{a_0}\right)^{\frac{3}{2}}2\space e^{-\frac{r}{a_0}} $. To find the probability, do I simply overlap the two and square them? $$\langle (1,0,0)_H| (1,0,0)_T \rangle = \left(\frac{1}{a_0}\right)^3 (2)^{\frac{3}{2}} (2)^2\left(\frac{1}{4\pi}\right) \int e^{-\frac{3r}{a_0}} r^2 sin\theta \space d\theta d\phi $$ $$ = \frac{2^{\frac{9}{2}}}{27}$$ Thus probabiliy is $\frac{2^9}{27^2} = \frac{512}{729} $?


title: Is the "dimension" in dimensional analysis the same as the "dimension" in "three spatial dimensions"?

text: When we talk about the dimension of a quantity (e.g. the dimension of acceleration is$[ L \ T ^ {-2}]$) are we talking about the same "dimension" as when we talk about three dimensional space? Are these two separate definitions sharing a word? or are they two uses of the same meaning?


title: Forces on an aircraft - thrust, lift, drag, weight

text: I'm extremely sceptical about the wikipedia page on aircraft flight mechanics. When describing 'straight climbing flight', it says: > lift is unable to alter the aircraft's potential energy or kinetic energy This seems like nonsense to me, given that 'lift acts perpendicular to the vector representing the velocity of the aircraft relative to the atmosphere'. In other words, I think there is always a vertical component of lift (except for the conditions where angle of attack for non-cambered aerofoil is 0 or 180 degrees). The reason I want to know this, is it also claims that > in straight climbing flight, lift is less than weight I also can't believe that this is always the case, since an aerofoil can generate lift when it has a zero angle of attack if it is cambered. So if the aeroplane is going fast enough, not only can the lift can exceed the weight, but just the vertical component of lift can exceed the weight. I'd appreciate criticisms of my understanding as well as the wikipedia page. (I wonder if it's an incorrect interpretation of the source, 'Aerodynamics' by Clancy). Thanks


title: Is a low-current electrical arc harmful to humans?

text: I've heard that electrical flux non-destructive particle testing machines are considered safe because they use less than 2 amps. I have seen an arc created between two objects do considerable damage, i.e. electrical burns. This is a low amperage, so would an arc with less than 2 amps and roughly 10,000 volts still be harmful to humans? If not because of the electricity itself, what about from the heat?


title: Usage of Schrödinger equation vs Madelung equations

text: It is well known that Madelung formulation is alternative to the Schrödinger Formulation, cf. this previous Madelung transformation Phys.SE post. I wanted to know what makes Schrödinger's formulation superior to that of Madelung, and in particular, when does the Madelung formulation fail?


title: How does gravity force get transmitted?

text: How does gravity force get transmitted? It is not transmitted by particles I guess. Because if it was, then its propagation speed would be limited by the speed of light. If it is not transmitted by particles how is it transmitted then?


title: How is Big Bang related to theory of relativity?

text: I'm not someone with good scientific knowledge, so if my question are weird, correct me. I was reading about big bang and I came by the theory of relativity. Can someone explain the relation between these two?


title: Could ions emitted by an ion thruster represent any realistic danger?

text: As ion thurster designs improve, the ions emitted could have a velocity (relative to the spacecraft) of well above 10^5 or even 10^6 m/s. It the likelihood of any such ions ever hitting a human completely negligible? (I'd love to see some rough calculation considering divergence angles, number of ions, and the solar system's dimensions). Also, what might be the effect of a single ion with such energy that hits a spacecraft or a human body, either on Earth (less likely I presume due to the atmospheric collisions) or in orbit?


title: What techniques can be used to analyze a rod rotating about the edge of a table?

text: > A uniform rod of length $4x$ is rotating about the edge $O$ of the table. > (The rod does not fall off the table.) The centre of mass $G$ of the rod is > distance $x$ away from $O$. The rod is making an angle $\theta$ with the > horizontal. > > The only forces present are the weight $W$ of the rod, the normal reaction > $N$ of the table on the rod and the frictional force $S$ that prevents the > rod from slipping off the table as it rotates. Let the Radial direction > point from O to G, and the Transverse direction be anticlockwise. I apologise for not including a diagram but it should be very quick to sketch. I would like to set up equation(s) of motion for the scenario above.   1. Would it be appropriate to approach this problem using Newton's 2nd Law and then resolving the equation into radial and transverse components? If so, am I suppposed to be considering the motion of a point on the rod, or the motion of the rod as a whole body?   2. Would it be appropriate to approach this problem by taking torques (say, about $O$), i.e. using $\tau=I \alpha$, where $\tau$ is the total external torque on the rigid rod?


title: Deriving the Electromagnetic Tensor

text: The electromagnetic tensor is given as: ![enter image description here](http://i.stack.imgur.com/b7xa5.png) How do you derive this? And how come there is a partial derivative in front of $A_\mu$? Do you multiply the derivatives or what?


title: Will a telescope show a more magnified image if the object is brighter?

text: I have a small telescope which shows me the moon's craters a little bit magnified. Consider a situation where the moon is very very brighter than its current brightness, so that the telescope's objective lens gathers much more light. Will the same telescope be able to show me the moon's craters in a more magnified image? If not, under what condition can this happen if the telescope is the same?


title: Fundamental equation(s) of string theory?

text: I often hear about string theory and its complicated mathematical structure as a physical theory, but I can't say that I've ever actually seen any of the related math. In general, I'm curious as to what the mathematics of string theory look like, can anyone point me to some references? In specific, I want to know if there is a fundamental equation in string theory that is assumed as a starting point for most problems, something comparable to Newton's second law in mechanics or the Schrodinger equation in QM?


title: Experimental evidence of Pauli's exclusion principle

text: A fermion is described by a set of quantum numbers, this set of numbers lead us to a unique wave function. If two fermions are described by the same wave function (violating the Pauli's exclusion principle), how can we differentiate this fermions if experimentally it produces the same result?


title: Block Sliding Down a Plane (Perpendicular Velocity)

text: I'm having trouble understand the concepts for this problem. Here's the problem: > _A block is placed on a plane inclined at angle $\theta$. The coefficient of > friction between the block and the plane is $\mu = \tan \theta$. The block > is given a kick so that it initially moves with speed $V $horizontally along > the plane (that is, in the direction perpendicular to the direction pointing > straight down the plane). What is the speed of the block after a very long > time?_ We can see that $F_x = mg\sin \theta - mg\cos (\theta )\mu = 0$, if we gave the block an initial velocity _along_ the plane, it should be subjected to a friction force opposite its velocity, where $f = mg \cos \theta \tan \theta = mg\sin \theta$ This however, is the answer my book gave: > The normal force from the plane is $N = mg \cos \mu$. Therefore, the > friction force on the block is $N = (\tan \mu)N = mg \sin \mu$. This force > acts in the direction opposite to the motion. The block also feels the > gravitational force of $mg \sin \theta$ pointing down the plane. Because the > magnitudes of the friction force and the gravitational force along the plane > are equal, **the acceleration along the direction of motion equals the > negative of the acceleration in the direction down the plane. Therefore, in > a small increment of time, the speed that the block loses along its > direction of motion exactly equals the speed that it gains in the direction > down the plane**. Letting v be the speed of the block, and letting vy be the > component of the velocity in the direction down the plane, we therefore have > $$v + vy = C$$ where $C$ is a constant. $C$ is given by its initial value, > which is $V + 0 = V$ . The final value of $C$ is $V_f + V_f = 2V_f$ (where > $V_f$ is the final speed of the block), because the block is essentially > moving straight down the plane after a very long time. Therefore, $2V_f = V > \rightarrow V_f = \frac{V}{2}$ I don't really understand what this is trying to say. The bolded part is what's giving me problems. I think it just may be poorly written, I think I need a clearer explanation.


title: Dimensional analysis to estimate order of magnitude of quantities

text: In the Coursera course From the Big Bang to Dark Energy on several occasions dimensional analysis was used to estimate the scale of quantities. This almost seems like a contradiction in terms to me, since you cannot strictly infer anything about dimensionless factors. Nevertheless it seems to work if you know some rules of thumb. The general procedure seemed to be to take the speed of light, Planck's constant and Boltzmann's constant, and then use dimensional analysis to relate these to the known quantities to get a quantity of the desired dimension. Examples included   * wavelength of cosmic microwave background from temperature   * kinetic energy of quarks from proton radius   * energy of W and Z bosons from length scale of weak force   * radius of hydrogen atom from electron mass via electron wavelength  In the last case the dimensionless fine structure constant has to be thrown in. I understand that when a force is involved, you will have to include some proportionality constant that cannot be obtained from dimensional analysis alone (like the fine structure constant above), but maybe that often is all that is needed. Is there just a handful of rules of thumb, using which more or less reliable estimates can be made? Apart from that, I would also be very interested in general comments on this theme. **Edit** As pointed out by QMechanic, a related (and very interesting) thread in this forum is found here In dimensional analysis, why the dimensionless constant is usually of order 1?. There certainly are similarities between the questions, but I am more specifically asking how to use dimensional analysis to obtain such estimates, and didn't find concrete answers in that thread. I find it fascinating that it can be done, but wouldn't dare to apply it myself. Rather than asking _why_ the constant is usually of order 1, maybe my question is _when_ the constant is of order 1, or more generally, how to estimate its value.


title: How would a physicist move?

text: Let's assume that Johnny Physicist has decided to move from his poor dingy second story apartment, into his much deserved home. Without making modifications to the existing structures? What would be the most work efficient way of moving all of his possessions? [Lets assume that the apartment is outside facing, and has stairs leading up to it] [Also lets assume that his possessions are a mix of carryable boxes, and large items]


title: What is the leading order Feynman diagram for nucleon-anti-nucleon annihilation into two mesons ($\psi^{\dagger} \psi \to \phi\phi$)?

text: I am working with a standard basic scalar Yukawa theory. I.e. the only interaction term is $-g\psi^\dagger\psi\phi$, where the $\phi$ field quanta are the mesons, the $\psi$ field quanta are the nucleons and the $\psi^\dagger$ field quanta are the anti-nucleons. I have tried fiddling around with various combinations and have looked at some relevant papers, but I have not found a solution which does not invoke other interaction terms. Is this process possible?


title: Review and mentions about time discretization

text: I would like to get basic knowledge in this topic of time discretization, at least basics. Because of I'm new in this field I don't know what is better way to start. Could anybody suggest any reviews and books about such a topic, or at least where it was mentioned and discussed?


title: Solution of dynamics of density matrix

text: Given the dynamics of the density matrix: $ \frac{d}{d t}\begin{pmatrix} \rho_{00} & \rho_{01} \\\ \rho_{10} & \rho_{11} \end{pmatrix} = \begin{pmatrix} \lambda i(\rho_{10}-\rho_{01})+\lambda^2\rho_{11} & \lambda i(\rho_{11} -\rho_{00})+\lambda^2 \rho_{01} \\\ \lambda i(\rho_{00}-\rho_{11}) +\lambda^2 \rho_{10} & \lambda i(\rho_{01}-\rho_{10}) +\lambda^2 \rho_{11} \end{pmatrix} $ How can this system of differential equations be solved, since they refer to each other. with initial condition $\rho_{ij}\in \mathbb{R}$


title: Two-loop regularization

text: Working out some quantum field theory computations, I have to find out the value of the two-loop Feynman integral $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\frac{d^4p_2}{(2\pi)^4}\frac{1}{(p_1^2+m_1^2)(p_2+m_2^2)[(p-p_1-p_2)^2+m_3^2]}. $$ This integral is rather common and so, its value at small $p$ should be already well-knwon. But I was not able to find out it in literature. I would also appreaciate to see all the procedure to get the right value with whatever regularization procedure one likes. Thanks beforehand.


title: Positive test charge

text: Protons have positive charge on them. Protons aren't mobile. So how can a positive test charge move from the negative terminal of a cell to the positive terminal and gain electric potential energy? Why does negative charge not move from the negative terminal to the positive terminal? Electrons flow in a circuit, and electrons have negative charge on them so why is it said that positive test charge moves from the negative terminal to the positive terminal? Please explain.


title: Is it safe to ignore derivatives of velocity w.r.t. position and vice versa?

text: In a certain textbook a function is given as: $$f=f(x(t))$$ And then this is differentiated w.r.t. $t$ to get: $$f_t=\dot{x}f_x$$ (Where the notation $f_u=df/du$, $f_{uu}=d^2f/du^2$, etc.) This is then taken as a functional $A=A(x,\dot{x},t)=\dot{x}f_x$ and differentiated w.r.t. $x$ and $\dot{x}$ and set to zero: $$A_{\dot{x}}=f_x=0$$ $$A_x=\dot{x}f_{xx}=0$$ My concern is that in doing this the textbook has not completely differentiated $A_{\dot{x}}$ and $A_x$ In particular it has ignored the derivatives $\frac{dx}{d\dot{x}}$ and $\frac{d\dot{x}}{dx}$ If I'm not mistaken the complete differentiation would be: $$A_{\dot{x}}=f_x+\frac{dx}{d\dot{x}}\dot{x}f_{xx}=0$$ $$A_x=\dot{x}f_{xx}+\frac{d\dot{x}}{dx}f_x=0$$ Multiplying the first of these equations through by $\frac{d\dot{x}}{dx}$ or the second equation by $\frac{dx}{d\dot{x}}$ you get the relation: $$\frac{d\ln{f_x}}{dt}=-\frac{d\ln{\dot{x}}}{dt}$$ and $$\frac{A_x}{A_{\dot{x}}}=\frac{d\ln{\dot{x}}}{dt}$$ Whereas doing it the book's way you get $$\frac{d\ln{f_x}}{dt}=1$$ and $$\frac{A_x}{A_{\dot{x}}}=\frac{d\ln{f_x}}{dt}$$ By ignoring the derivatives of $x$ and $\dot{x}$ w.r.t. each other, then, the book is tacitly assuming that $$-\frac{d\ln{\dot{x}}}{dt}=1$$ EDIT: (and hence that $\frac{A_x}{A_{\dot{x}}}=1$, which introduces a contradiction b/c we also know from the fully executed derivative that $\frac{A_x}{A_{\dot{x}}}=\frac{d\ln{\dot{x}}}{dt}$ which by the said tacit assumption $=-1$) My question is: is this a safe assumption? What is the physical meaning of this assumption?


title: What happens when you shake a can of soda?

text: Of course the CO2 comes out of the liquid, and it will even build up a lot of pressure if needed. The question is what happens at a molecular level to cause it to leave the liquid?


title: Particle Physics : Conservation laws

text: As a whole, is conservation of a Lepton number valid, or is it basically the generalization of conservation of electron and muon numbers?


title: What is the physical importance of current density?

text: Why we have used the term current density, Is there any physical significance of it or does it affect anything apart from the drift velocity in equations only. Does it have a major significance like charge density in gauss's theorem.


title: How is the direction of the instantaneous acceleration determined?

text: I know from the text book that the direction of velocity at any point on the 2D path of an object is tangential to the path at that point and is in the direction of motion. But how would one determine the direction of instantaneous acceleration? In case, the only force influencing is gravity, direction of instantaneous acceleration is downward towards the earth's center. but what in the other cases?


title: friction between 2 bodies

text: _**There is a plank on which a block is kept. the block is given some velocity. Now if there is friction between block and plank it will act opposite to it's velocity for sure. but the plank is free to move. so where will the friction for the plank act? also if plank and block are the system is friction an internal force?? in other words will their centre of mass move??_**


title: Mixed quantum states and "complete knowledge of the system"

text: I ran across this statement in a professor's notes and I think it's just a typo, but I wanted to take the opportunity to check my understanding. So in his notes it says: > even if we have complete knowledge of quantum systems, they still can be in > the[a] mixed state As far as I understand, a mixed state is simply a classical mixture of quantum states. If we inherit the definition of maximum information from the von neumann entropy, then we should define maximum information to correspond to an entropy of ln(1) = 0. But since the density matrix is not a rank 1 projector in a mixed state, this condition is not met. Is there some other notion of maximum information that I am not aware of, do I have some misunderstanding, or is this just a typo?


title: Parity of spin states

text: Since orbital angular momentum commutes with the parity operator and since both are hermitian it is possible to build a common basis. These are the spherical harmonics, whose parity is known. Now, we also know that since the spin operator commutes with the parity operator we can build common eigenkets, but I have never been told which is the parity of a spin state. Moreover, when calculating the parity of, for example, a particle I always multiply the contribution of the orbital angular momentum by the intrinsic parity of the particle, and the spin part is never mentioned. I guess that the spin contribution is in that intrinsic parity. So i would like someone to tell me about the parity of the spin states and how they are related to the intrinsic parity of particles(if in fact my guess is right)


title: Number of Independent "Cross-ratios or Anharmonic-ratios"

text: The Cross-ratios or the Anharmonic-ratios are defined as, $${r_{ij}r_{kl}}/{r_{ik}r_{jl}}, \text{ where } r_{ij}=\mod{r_i - r_j}.$$ Now the claim is: conformal symmetry implies that for computing $N$ point correlation function there will be $N(N-3)/2$ number of independent cross- ratios. I can't prove this claim. I have seen the Ginsparg's explanation on this claim but I can't understand that. I need the proof. Can anyone help me?


title: Origins of Moment of Inertia

text: Where exactly does the equation $MR^2$ for moment of inertia come from? The quantity itself seems fairly arbitrary.


title: What is anti-screening?

text: In his book "The Lightness of Being" Frank Wilczek (page 48) writes about screening and anti-screening. In screening, a bare charge attracts virtual particles of opposite charge which lessen the effect of the bare charge. He writes that anti-screening is the "opposite of screening" with no further physical explanation. This is an attempt to explain why two quarks attract weakly at close range and more strongly with distance. Is there any physical explanation of this anti-screening?


title: How can heat turn into light

text: I am confused about how hot surfaces can radiate light to their surroundings. When I shine a light on a surface the light turns to heat spontaneously, and when I leave that hot surface it radiates light spontaneously. To me this suggests that the process isn't driven by entropy increasing because it is two directional. Could anyone explain how this process complies with the second law of thermodynamics?


title: Conformal time or cosmological time?

text: Let us assume a flat FRW metric $$ds^2 = -dt^2 + a(t)^2(dx^2+dy^2+dz^2)$$ Imagine a lightbeam traveling in the x-direction. It travels on a null geodesic with $ds=0$ so that its path obeys the relation $$dt = a(t) dx$$ Let us assume that $a=1$ at the present cosmological time. Thus presently in an interval of cosmological time $dt$ light travels a proper distance of $dx$. Now imagine a future time at which $a=2$. Thus in the future in the same interval of cosmological time $dt$ a light beam will travel a proper distance of $2dx$. Is this correct? It seems that if an observer uses cosmological time as his time measure then the local speed of light will change with the age of the Universe. However if he uses conformal time $\tau$ such that an interval of conformal time $d\tau$ is given by $$d\tau = \frac{dt}{a}$$ then while the proper distance the light travels increases in proportion to $a$ the number of time intervals will also be proportional to $a$ (because their size $d\tau$ is inversely proportional to $a$). Thus the local speed of light will remain constant if one uses conformal time rather than cosmological time (as it should).


title: If I were standing on the surface of the sun, how far could I see?

text: Assuming it would be possible to handle the intense heat... Is the surface even? If not, would average surface elevation be an appropriate definition of "surface"?Does the sun have an atmosphere that would degrade visibility? How far apart are small solar flares that would block the horizon? What affect would the heat have on the light rays? Assume I am 2 meters tall.


title: Is there a way to compute (trivalent) Feynman integrals inductively from smaller diagrams?

text: Suppose that I would like to compute the Feynman integral associated to the trivalent graph ![enter image description here](http://i.stack.imgur.com/RTPgp.jpg) One can argue that this diagram comes from two copies of the smaller diagram ![enter image description here](http://i.stack.imgur.com/3UYWV.jpg) glued together at the external vertices. We could argue even further that it really comes from two copies of the following diagram ![enter image description here](http://i.stack.imgur.com/LzEB6.jpg) where we instead cut one of the edges on each diagram, yielding the second, and then gluing them together. Suppose that I know how to compute the integral associated to either this second (or preferably third) diagram. Is there some sort of algorithm for reducing the integral of the first diagram to some combination of integrals for the second and third diagrams? I should remark perhaps that I am being deliberately vague about what the context of this computation is. I'm mostly hoping that there is some formal way to combine the smaller integrals to produce the larger ones, something that would look like, say, a multiplicative structure on an $R$-algebra freely generated by diagrams, for some ring $R$. Or something like that. **Edit:** I feel that I should note that one hope that I have (which may prove ill-founded) is that this will in some sense be computable in a manner akin to the gluing or other type formulae in Gromov-Witten theory. It'd be really nice if we could write something like $$ I_{\Gamma_g} = \sum_{g_1+g_2 = g} C_{g_1,g_2} I_{\Gamma_{g_1}}I_{\Gamma_{g_2}} $$ or something.


title: The particle content of a given state

text: In Carroll's we read > ...The Unruh effect teaches us the most important lesson of Quantum Field > Theory (QFT) in curved spacetime, the idea that "vacuum" and "particles" are > observer-dependent notions rather than fundamental concepts. I wonder are we talking about the same observer or two different observers using two different frames? Put it another way, can the same observer who observed nothing using a frame detect a thermal spectrum if switched to another frame that is uniformly accelerating with respect to his previous frame? Does it apply both ways? I mean if an observer detects particles, does this observant-dependent notion of particle content of a given state, implies that he can switch to another frame in the blink of an eye, and detect nothing?


title: Evaporative cooling in a tropical environment

text: I am an architect and am trying to understand the effects of evaporative cooling in a humid climate for buildings. I would like to use the example of a zeer pot for my question. I have read everywhere that evaporative cooling is not possible in a humid climate simply because humid ambient air is already approaching 100% saturation. Obviously when saturated air cannot take any more moisture - evaporation does not take place. I get that. However if humid air is heated then the saturation decreases quite a lot. For instance I remember looking at a chart that shows if air temperature is doubled then its ability to hold moisture increases about ten fold. In a tropical climate there is a lot of sun and heating air is easy. If air were heated to increase its moisture holding capacity and then passed over an extremely porous, moist surface will cooling take place or will the whole system gain heat? If a zeer pot for instance were placed inside a simple black metal chimney in the sun, open on the bottom and top, where humid air is heated and the heated air rises and passes over the pot will the pot cool or will it gain heat? Is there a formula (something I can understand) I can use to calculate heat gain and loss in this example? Roughly speaking is the concept of evaporative cooling possible in a humid climate if the humid air is heated?


title: Capturing Energy from a Positron-Electron Collision

text: According to the book Physics of the Impossible, the catastrophic collision between a electron and a positron yields an output of 1.02 MeV. Assuming you have an isotope like $^{22}\text{Na}$, which is known to emit photons during positron emission decay, will they annihilate upon any contact with electrons, or do they have to be accelerated at one another. Secondly, Is it possible to efficiently capture this energy yielded in the reaction. Also, in what form does this 1.02 MeV get emitted in? This question is excluding the fact that the generation of positions in very large amounts is not feasible at the moment.


title: Must length equal distance?

text: In his book The Elegant Universe, Brian Greene points out that when a foot ruler is laid out by an observer on a spinning “Tornado” ride (a disk) that a “A shorter ruler means that he will have to lay it out - head to tail - more times to traverse the whole circumference” and that the total physically measured distance will exceed what is required to satisfy $C=2πR$ See http://www.onereads.org/elegant-universe-brian-greene?page=0,31 My understanding of relativity is that the circumference will be longer by the gamma factor : $$\gamma=1/(1-(v/c)^2)^{1/2}$$ For example, the distance between muon and earth (in earth's rest frame) is related to time by$$t_{earth} = d/v$$ If d is the "at rest" distance of the muon to the Earth, then in the muon’s frame $$t_{muon} = (d/γ)/v$$ Therefore, the two times are related to each other by $$t_{muon}=(1/γ)t_{earth}$$ This agrees with time dilation; the muon’s time is less than the Earth’s time by a factor of $1/\gamma$. \--Let's repeat the rationale by regarding travel time instead-- **Using an analogous example, the travel time between muon and earth (in earth's rest frame) is related to distance by** $$d_{earth}=vt$$ If t is the "at rest" time travelled by a muon to the earth, then in the muon's frame $$d_{muon}=vtγ$$ Therefore, the two distances are related by $$d_{muon}=γd_{earth}$$ It appears from the mathematics that distance as measured in the muon's frame expands by the gamma factor. This can only mean that in the earth's frame the distance contracts by a reciprocal factor. We should regard the length contracted ruler, as was also assumed in the spinning disk example, to support this interpretation. (Because more foreshortened rulers are required to measure an expanded distance in the muon's frame).


title: Non-local structure of field theory

text: Can someone explain what is non-local structure of field theory? I know you cannot have $\phi(x) \phi(y)$ term in Lagrangian which indicates the non- locality. However, why I cannot have the non-local terms as long as I have causality maintained? In QFT, one should not write an operator like $\phi(x)^2$ which will yield singularities like $\delta (x-x)$ if one does OPE? How should I understand the locality in field theory and OPE sense consistently?


title: Will we see interference in a double slit experiment in these three cases?

text: First case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. A detector irreversibly erases the information about particles before they hit the film   3. The particles hit the film   4. A person observes the film. Second Case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. The particles hit the film   3. A detector irreversibly erases the information about particles before they hit the film   4. A person observes the film. Third case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. The particles hit the film   3. A person observes the film   4. A detector irreversibly erases the information about particles without anyone ever knowing what it was


title: How to calculate transmission of photonic band gap fiber?

text: How to calculate transmission of hollow core photonic fibers (multilayer fiber and 2D band gap fiber)? Which books/article you can recommend ? I see that comsol can do it, but is there a model that runs faster maybe semi- geometrical approximation ? I dont see chapter about photonic fibers in the Agrawal book.


title: Can relativistic kinetic energy be derived from Newtonian kinetic energy?

text: Relativistic kinetic energy is usually derived by assuming a scalar quantity is conserved in an elastic collision thought experiment, and deriving the expression for this quantity. To me, it looks bodged because it assumes this conserved quantitiy exists in the first place, whereas I'd like a derivation based upon using KE $= \frac12 mv^2$ in one frame, and then summing it in another frame say to get the total kinetic energy. Can this or a similar prodecure be done to get the relativistic kinetic energy?


title: Is the EmDrive, or "Relativity Drive" possible?

text: In 2006, _New Scientist_ magazine published an article titled _Relativity drive: The end of wings and wheels_ 1 [1] about the EmDrive [Wikipedia] which stirred up a fair degree of controversy and some claims that _New Scientist_ was engaging in pseudo-science. Since the original article the inventor claims that a "Technology Transfer contract with a major US aerospace company was successfully completed", and that papers have been published by Professor Yang Juan of The North Western Polytechnical University, Xi'an, China. 2 Furthermore, it was reported in Wired magazine that the Chinese were going to attempt to build the device. Assuming that the inventor is operating in good faith and that the device actually works, is there another explanation of the claimed resulting propulsion? Notes:   1\. Direct links to the article may not work as it seems to have been archived.   2\. The abstracts provided on the EmDrive website claim that they are Chinese language journals which makes them very difficult to chase down and verify.


title: Temperature change inside pressure chamber

text: Let's say there is a pressure chamber with some sort of sample / specimen (e.g. protein crystal) in it. Now I apply a certain amount of gas pressure, e.g. 10 or 20 atm. Let's say I use xenon as a gas. I'm wondering what will happen to the temperature inside my chamber - from the **ideal gas law** you get a reciprocal relationship between temperature and pressure, so given all other things stay the same (i.e. the chamber _volume_ ), what is the temperature change / final temperature as seen by my sample? I'd start at room temperature and go from there .. Obviously, the gas is compressed to start with (and has a certain temperature as well).


title: What is sound and how is it produced?

text: I've been using the term "sound" all my life, but I really have no clue as to what sound exactly is or how it is created. What is sound? How is it produced? Can it be measured?


title: What Hermitian operators can be observables?

text: We can construct a Hermitian operator $O$ in the following general way:   1. find a complete set of projectors $P_\lambda$ which commute,   2. assign to each projector a unique real number $\lambda\in\mathbb R$. By this, each projector defines an eigenspace of the operator $O$, and the corresponding eigenvalues are the real numbers $\lambda$. In the particular case in which the eigenvalues are non-degenerate, the operator $O$ has the form $$O=\sum_\lambda\lambda|\lambda\rangle\langle\lambda|$$ **Question:** what restrictions which prevent $O$ from being an observable are known? For example, we can't admit as observables the Hermitian operators having as eigenstates superpositions forbidden by the superselection rules. **a)** Where can I find an exhaustive list of the superselection rules? **b)** Are there other rules? **Update:** **c)** Is the particular case when the Hilbert space is the tensor product of two Hilbert spaces (representing two quantum systems), special from this viewpoint?


title: Deriving torque from Euler-Lagrange equation

text: How could you derive an equation for the torque on a rotating (but not translating) rigid body from the Euler-Lagrange equation? As far as I know from my first class in Classical Mechanics, there is no potential defined for a rotating body, so the only term I see in the Lagrangian is the $\frac{1}{2}I\omega^2$. Since there is no $\theta$, I'm just getting that torque always equals $0$.


title: Faraday's law in a ring

text: What role does the induced emf in Faraday's law play in generating current in a ring in which the magnetic flux is changing ?


title: Phase diagram of simplified QCD

text: Consider QCD with a single generation of massless quarks (u, d). This is probably the simplest variant of QCD which bears some relation to the real world. The theory has the following exact global symmetries:   * U(1) acting on u and d (baryon number)   * Chiral U(1) acting in opposite matter on left-hand and right-handed quarks. It is destroyed by an anomaly so we will not consider it further   * SU(2) isospin rotation mixing u and d   * Chiral SU(2). It is spontaneously broken   * Charge conjugation C   * Parity (spatial reflection) P   * Time reversal T A thermal equilibrium of the model is characterized by 3 parameters:   * Temperature   * Chemical potential associated to baryon number. Alternatively, baryon number density. We can use C to fix its sign   * Chemical potential associated to isospin. Alternatively, isospin density. It is a vector but using isospin rotation symmetry we can align it along a prescribed axis, so we are left with a positive scalar parameter Hence, the theory has a 3-dimensional phase diagram > How does the 3D phase diagram look like? Which phases do we have? Which > phase transitions? What is the type of each phase transition? Above a certain temperature T, chiral symmetry is restored. Btw, is this the same phase transition which breaks confinement? Above this T, we can introduce a 4th parameter, namely the chemical potential associated to chiral isospin. I suppose there a certain inequality governing the maximal value of this parameter as a function of the temperature? > How does the 4D phase diagram look like?


title: Time reversal symmetry and T^2 = -1

text: I'm a mathematician interested in abstract QFT. I'm trying to undersand why, under certain (all?) circumstances, we must have $T^2 = -1$ rather than $T^2 = +1$, where $T$ is the time reversal operator. I understand from the Wikipedia article that requiring that energy stay positive forces $T$ to be represented by an anti-unitary operator. But I don't see how this forces $T^2=-1$. (Or maybe it doesn't force it, it merely allows it?) Here's another version of my question. There are two distinct double covers of the Lie group $O(n)$ which restrict to the familiar $Spin(n)\to SO(n)$ cover on $SO(n)$; they are called $Pin_+(n)$ and $Pin_-(n)$. If $R\in O(n)$ is a reflection and $\tilde{R}\in Pin_\pm(n)$ covers $R$, then $\tilde{R}^2 = \pm 1$. So saying that $T^2=-1$ means we are in $Pin_-$ rather than $Pin_+$. (I'm assuming Euclidean signature here.) My question (version 2): Under what circumstances are we forced to use $Pin_-$ rather than $Pin_+$ here? (I posted a similar question on physics.stackexchange.com last week, but there were no replies.) * * * EDIT: Thanks to the half-integer spin hint in the comments below, I was able to do a more effective web search. If I understand correctly, Kramer's theorem says that for even-dimensional (half integer spin) representations of the Spin group, $T$ must satisfy $T^2=-1$, while for the odd-dimensional representations (integer spin), we have $T^2=1$. I guess at this point it becomes a straightforward question in representation theory: Given an irreducible representation of $Spin(n)$, we can ask whether it is possible to extend it to $Pin_-(n)$ (or $Pin_+(n)$) so that the lifted reflections $\tilde R$ (e.g. $T$) act as an anti-unitary operator.