title: Recommend good book(s) about the "scientific method" as it relates to astronomy/astrophysics?

text: I am interested in astronomy/astrophysics, but I am not science major (I am a computer science graduate). Facts and results of the field are presented to the public without showing how these facts/results got known or inferred. And I have that curiosity to know how we know what we know about the universe (either observationally or mathematically). So my question is, what book(s) do you recommend for someone who has knowledge of   * algebra, trigonometry and geometry   * college-level calculus   * classical mechanics and does not get intimidated by mathematical language? I expect the book(s) to answer questions like (not necessarily all of the questions, but questions of the same level and kind as these):   * How do we know how distant from the earth a celestial body (for example, a star) is?   * How do we know the volume/mass of celestial bodies?   * How do we know the materials that a planet is made of?   * How do we know that our solar system orbits around the center of the galaxy?   * How do we calculate the total mass of the galaxy?


title: Mixing and matching across eyepiece designs / manufacturers?

text: Right now, I am considering moving from 1.25" eyepieces to 2". While I'm convinced of the quality of the premium eyepieces, it would take me years to afford a complete set and, if I go that route, I will necessarily pick them up piecemeal. Would I be wise to follow that route, knowing that in a few years time I'll have not wasted any money? Or would I be wiser to buy a complete set of design _X_ , which have great price/performance ratios and slowly replacing them with premium eyepieces? Or would I be wiser to buy a few of type _Y_ for higher-power viewing, which are still pretty expensive, but not as expensive as the 100 degrees field-of- view eyepieces, but maybe just buy 1 of those for wide-angle viewing? Or another strategy? I like to look at everything, so _ultimately_ I do want a pretty wide range of eyepieces. But I kind of hate that with my current mish- mash, I have a "lumpy distribution" in quality.


title: SU(N) Yang-Mills $gg \to ggg$ scattering at tree level

text: When talking about the spinor-helicity formalism in his new textbook on quantum field theory, Matthew D. Schwartz claims as a highly nontrivial example, it is quite easy to use the Parke-Taylor formula to calculate the $gg\to ggg$ scattering cross section at tree level by hand, which is also one of the problems in the book. (Problem is found on page 560, problem 27.6) Can anyone tell me how to do this exactly? It looks like we cannot avoid summing $24^2$ terms, and in the paper, http://www.sciencedirect.com/science/article/pii/0550321386902300, the cross section of $gg\rightarrow gggg$ spans several pages! This makes me expect the cross section for $gg\rightarrow ggg$ is also complicated. Is there any smart way to do it?


title: How can stars make up 0.5% of whole universe?

text: I was reading Wikipedia article about universe and stumbled upon pie chart which represents what universe contains. It says that stars make up 0.5% of whole universe. I tried to find average distance between stars in the universe, but came to some different numbers, but either way distance is more than 1 light year. At the same time even largest stars in the universe which have 1400 sun radius are 0.00020584 light years in diameter (by my calculations), so it makes absolutely no sense that 0.5% of universe consists of stars, number should be many times smaller. What I am missing here?


title: Slight variation to method of images

text: Suppose a point charge $q$ is located at $(x=0,y=0,z=d)$, and that along the $x$-$y$ plane is a infinite plate of potential $V = 0$. Then the method of images solves Laplace's equation for the potential for $z>0$, $$V(x,y,z) = \frac{1}{4\pi\epsilon_0}\left(\frac{q}{\sqrt{x^2+y^2+\left(z-d\right)^2}} - \frac{q}{\sqrt{x^2 + y^2 + \left(z + d\right)^2}}\right).$$ This satisfies the boundary conditions $V(x,y,0) = 0$ and $V(r\rightarrow\infty) = 0$. Suppose instead the plate has potential $V(x,y,0) = V_0 \ne0$. Although very simple, it seems to me that the method of images cannot be used in this case. Is this correct?


title: Determining the location of magnetic moments

text: The magnetic moment of a closed current loop is $$\mathbf{m} = \frac{I}{2}\int\mathbf{r}\times d\mathbf{r}.$$ Note that $\mathbf{m}$ is independent of any coordinate frame. How, then, does one determine the _location_ of a magnetic moment? Certainly the vector potential due to a magnetic dipole $$\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{r^3}$$ _does_ depend on the location of the coordinate frame, since $r$ is by definition the distance between the magnetic dipole and the target point, therefore it is important to know the location of the dipole (or the current loop we are approximating as a dipole). How does one determine where to set the origin of the coordinate system so as to use the expression for $\mathbf{A}(\mathbf{r})$ above, given an arbitrarily-shaped current loop?


title: Interaction strength in BCS theory

text: I'm looking to plot the band gap $\Delta(T)$ as a function of temperature between $T = 0$ and $T = T_c$ by numerical evaluation of the band gap equation $$\frac{1}{\mathcal{N}(0)V} = \int_0^{\hbar\omega_D}{d\xi\frac{\tanh{\frac{\sqrt{\Delta^2+\xi^2}}{2k_BT}}}{\sqrt{\Delta^2+\xi^2}}}$$ where $\omega_D$ is the Debye frequency, $\mathcal{N}(0)$ is the density of states at the Fermi level and $V$ is the interaction strength in the BCS model. Now, in order to do this, I need values for both $\omega_D$ and $\mathcal{N}(0)V$. I think I have a pretty good idea about the order of magnitude for $\omega_D$ if I assume longitudinal optical phonons (I find $\hbar\omega_{LO} \propto 10\,\mathrm{meV}$, but please do correct me if that doesn't seem right). However, I still need a value for $\mathcal{N}(0)V$. Wikipedia mentions > The energy of the pairing interaction is quite weak, of the order of > $10^{−3}\,\mathrm{eV}$, and thermal energy can easily break the pairs. but it's not clear what quantity exactly is meant by 'the energy of the pairing interaction' (I could imagine it being one of two things). I'm guessing $\mathcal{N}(0) \propto 10^{28}$ for a metal (again, correct me if I'm wrong), so my question is really quite straightforward: what is the value of the interaction strength $V$ in the BCS model? (provide a reference please)


title: Gradient involved commutator in $\phi^4$ theory

text: In a phi fourth theory, the Hamiltonian density is: $$\mathcal{H}=\frac{1}{2}\pi^2+\frac{1}{2}(\nabla \phi)^2+\frac{1}{2}m^2\phi^2+\frac{\lambda}{4!}\phi^4$$ Now I impose the usual equal time canonical commutation relations for fields ($\hbar=1$) $$[\phi(\vec{x}),\pi(\vec{y})]=i \delta^3(\vec{x}-\vec{y})$$ where $$\pi=\frac{\partial \mathcal{L}}{\partial (\dot{\phi})} \equiv \dot{\phi}$$ Heisenberg equation of motion for the field is just the definition of the conjugate momentum $$\frac{d}{dt}\phi(\vec{x},t)=\pi(\vec{x},t) $$ and for $\pi(\vec{x})$ I have to calculate the commutator (not writing time dependence) $$[H,\pi(\vec{x},t)]=\int d^3x'\left[\frac{1}{2}\pi^2(\vec{x}')+\frac{1}{2}(\nabla \phi)^2(\vec{x}')+\frac{1}{2}m^2\phi^2(\vec{x}')+\frac{\lambda}{4!}\phi^4(\vec{x}'),\pi(\vec{x}) \right] $$ First term gives zero, third and fourth terms give $i\left(m^2\phi(\vec{x})+\frac{\lambda}{3!}\phi^3(\vec{x})\right)$ My question is, how can I calculate $$\frac{1}{2}\int d^3x' [(\nabla \phi)^2(\vec{x}'),\pi(\vec{x})] $$ As an analogy with the integral of the commutator is the commutator of the integral, may I write $\nabla \phi^2=\nabla \phi \cdot \nabla \phi$ and integrate by parts? How can I show that that is true?


title: Is a dynamical extension of non-commutative black holes feasible?

text: Non-commutative (sometimes called "fuzzy") black holes are solutions of Einstein's equations obtained with a previous basic assumption of non- commutativity of the coordinates $[x^{\mu},x^{\nu}]=i\, \theta^{\mu\nu}$. This assumption is considered helpful in the sense that removes the singularity in $r=0$. See this link for a review, even though there have been more recent developments. My question regards the possibility of extending this black hole solution to a time-dependent one, in order to describe an evaporating or accreting case.


title: 2-D Orientational Order Parameter

text: I am building a computational model of ellipsoidal cell network formation and I would like to use a particle order parameter to study my model's behavior. I have come across this article Steinhardt, Nelson, Ronchetti, PRB (1983), that was apparently the one that started everything, but the math is quite exotic for me. I have found that the formula: $$ S = 1/3<cos^2θ - 1> $$ is the order parameter for a 3-D system. This is convenient as S = 0 means no order, while S = 1 means complete alignment. A colleague at the univ has mentioned that for a 2-D particle system the previous equation for S becomes: $$ S = cos2θ $$ but I don't think this can be right, as in the following example: 2 particles have a 90 degree angle between them, so they are completely unaligned. `cos2θ = cos2*90 = -1` which would imply they are completely aligned, which is wrong. Also, the range is [-1:1] instead of 0 < S < 1. I would really appreciate it, if anyone could point me to the right direction.


title: What is the reason that relativistic corrections for hydrogen atom work?

text: Here I cite part from Sidney Coleman's lectures on Quantum Field Theory: > It is a phenomenal ﬂuke that relativistic kinematic corrections for the > Hydrogen atom work. If the Dirac equation is used, without considering > multi-particle intermediate states, corrections of $O \big(\frac{v}{c}\big)$ > can be obtained. This is a ﬂuke caused by some unusually low electrodynamic > matrix elements. What is the fluke about? Also, how can one justify the usage of Pauli- Schrodinger type equations that comes from first quantization of Dirac's equation? Schrodinger's equation is universal postulate valid for any quantum theory, and is equation for wave functionals in field theory. Could one go from non-relativistic QED field theory and then justify the usage of Pauli equation in which $\psi$ is interpreted as "wave function" in certain kinematical conditions (approximation)?


title: Collision of charged black holes

text: Suppose there are **twocharged black holes** which **collide to form a bigger black hole**. But when they combine, a lot of potential energy of the system is lost/gained depending on their charges (the opposite or the same). **Will it manifest as an increase/decrease in mass of the big black hole**? _If the mass were to come down (+ve ,+ve collision) will the resultant black hole shrink in size?_


title: How does heat energy travel in space?

text: How does the heat energy from the Sun reach us on the Earth? Since the kinetic energy of an atom is the amount of heat energy and there is no matter in space, how does heat from the Sun reach us?


title: How do astronomers measure the distance to a star or other celestial object?

text: How do scientists measure the distance between objects in space? For example, Alpha Centauri is 4.3 light years away.


title: Why the vacuum polarization does not decrease the speed of light?

text: On one hand, in the classical electrodynamics polarization of transparent media yields in lowering the speed of light by the factor of $n=\sqrt{\epsilon_r \mu_r}$ (refractive index). On the other, in the quantum field theory the vacuum polarization does not decrease the speed of light. The thing it does is increase of strength of electromagnetic interaction. Why is it so? My two guesses are the following:   * In principle the vacuum polarization does decrease speed of light ($\rightarrow 0$) but we implicitly put on-shell renormalization to keep $c$ constant.   * For some reason the process is completely different from polarization of transparent media.


title: Pauli exclusion principle and Entangled pairs

text: It is true for fermions in the same potential that the total wavefunction of two particles must be antisymmetric with respect to exchange of electrons. Which means the spin wavefunction is given by $\chi=\frac{1}{\sqrt{2}}[\chi_+ (1)\chi_- (2)-\chi_+ (2)\chi_- (1)] $ which looks very much like the bell state, $\beta_{11}=\frac{1}{\sqrt{2}}[ |01\rangle - |10 \rangle]$. So, can we call those fermions, entangled states, as long as they are within the potential or there is something fundamentally special about entangled states (e.g. difference in measurement statistics) which makes them more unique? Apologies if the question is too simple for the level of this website. However, apparently it has made a lot of confusion for many people!


title: The status of $SU(3)_C$ symmetry in the Standard Model

text: In the Standard Model of Particle physics the $SU(2)_{EW}$ symmetry and the $SU(2)$ isospin symmetry are broken. What about $SU(3)_C$? Is it broken too? if YES, what breaks the symmetry? If NO, what are the consequences? Anything like "all baryons have almost the same mass"?


title: The Galactic Plane

text: I'm guessing this isn't a great physics question, but I just can't find an answer with Google. If the galactic plane is perfectly horizontal at what angle and rotation will the ecliptic plane of our Solar System be?


title: Photons, where do they come from?

text: 1. Photons, where do they come from?   2. What exactly is a photon?    3. I've certainly heard how they get produced, but it doesn't seem to make sense that some sort of particle should need to be produced just to mediate the interactions of electrons. I have yet to take quantum mechanics, can someone give a brief discussion of the conservation, creation and destruction of photons, and how we know they exist?    4. And furthermore why it makes sense to have a quantized bit of energy called a "photon"?


title: Have I calculated the air requirement for a pneumatic mail system correctly?

text: I googled How to calculate the air requirements of pneumatic conveying and found a clue to what might be a correct equation (http://uk.ask.com/question/pneumatic-conveying-calculation). Here is my attempt at running the calculation: Tube diameter: 200mm   Tube raduis: 100mm Tube length : 1675 meters   Weight of internal mail cannister: 1000kgs   Total cylinder volume: 1.76×10^6   Maximum velocity mail cannister velocity: 50m/s Steps followed in an attempt to find the answer:   square the radius of the cylinder   $100 \times 100 = 10000$ multiply the result by $\pi$ $10000 * \pi = 31415.9265359$ then divide the force require by the area of the cylinder face (I have just calculated the area of a circle for this part of the equation.) $31415.93/5102 = 6.15757154057$ I am Assuming this answer liter's of air? and it should be multiplied by the volume of the tube in order for me to arrive at the answer I require? forced required calculation   mass (kg)= 1000kgs/9.8 = 102.04 kgs   if $force$ = $mass$ $\times$ $acceleration$ ($F=ma$) then..   $102.04 \times 50m/s = 5102$ $Newtons$ required Am I googling in the right direction or should I be looking towards fluid dynamics equations like control volume analysis?


title: Do photons have acceleration?

text: Photons travels with the largest speed in our universe, the speed of light. Do photons have acceleration?


title: Does a photon instantaneously gain $c$ speed when emitted from an electron?

text: An excited electron looses energy in the form of radiations. The radiation constitutes photons which move at a speed $c$. But, is the process of conversion of the energy of the electron into the kinetic energy of the photon instantaneous. Is there a a simple way to visualize this process rather than math?


title: Is there any acceleration of light?

text: I find it hard to believe that photons always travel with 3 x 10^8 m/s just from the start. But there must be some acceleration of light. Maybe huge or taking place in picoseconds. So what maybe the acceleration of light or photons?


title: Does light initially accelerate?

text: Light travels with a speed of $3\times10^8{m\over s}$. My question is that was the light initially accelerating or it archived the speed in an instance? If it was accelerating then why it did not accelerate beyond that?


title: When photons are emitted, do they accelerate to reach the speed of light?

text: Photons are considered mass-less particle with a specific velocity but according to the electromagnetic theory, a photon is considered to have both energy and momentum. So what happen when they are emitted, do they accelerate to reach the speed of light or is it somehow instantaneous?


title: Does light photons have jerk?

text: While searching in web regarding whether rate of change of acceleration is possible or not; I came across the concept of jerk. I want to know whether photons which can be accelerated can also have jerk or not?


title: Deflection of magnetic compass sorrounded by alternating poles

text: What happens to the deflection of the magnetic compass if it is surrounded by south poles and north poles in alternating direction of a magnet around it in a circular pattern. Will it be deflected in any particular direction ? ![enter image description here](http://i.stack.imgur.com/MHqqH.jpg)


title: Can zinc nitride be used as light emitting material?

text: I cannot manage to find any journal papers about the applicability of zinc nitride as active layer of an light emitting diode (LED). But certain papers got mention that zinc nitride with a direct bandgap can be fabricated with potential applications in optoelectronics. Can anyone give me some opinions on this subject?


title: rectlinear motion with constant acceleration

text: Friends, this is a numerical homework problem. I tried my best to solve it but my answer is not matching with the one given at the back of the text book. Please help me out: A motor car moving at a speed of 72 km/h can come to a stop in 3 seconds, while a truck can come to a stop in 5 seconds. On a highway, the car is positioned behind the truck, both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it doesn't collide with the truck. The typical human response time is 0.5 sec. My logic and answer: since car can decelerate to a stop much faster than the truck, it only need to worry about human response time which is 0.5sec. car would cover 10m in 0.5seconds at a speed of 72 km/h. so it just need to be 10m behind the truck minimum. but the answer in the book is 1.25 m How is this possible?


title: Movement in outer space via Newton's law of every action has an equal and opposite reaction

text: What is more effective for travel in outer space ignoring all other factors like air radiation etc. I have a 10 kg bag of rice would I travel faster throwing the whole bag at once or throwing a grain at a time compounding my tiny acceleration or would they end up being equal?


title: Dangers of using liquid nitrogen

text: So, I remember in college physics the prof using liquid nitrogen in a demonstration. When he was done, he threw the container of LN at the front row, and of course it evaporated (or whatever) before it got to the students. I am watching a cooking show now and they are using LN -- if they touched that, what would happen? Would it be possible for the chefs to pour the LN onto their skin accidentally, or would it evaporate before it reached their skin?


title: Charges and Topology

text: I must apologize, I was a little bit excited when I began understanding some of this, I can not say I can compete with professionals, and words are still difficult concepts. In (1) S.H. et al. it is discussed that in extremal cases of charged black holes in a euclidean space, there is a topological change as compared to the non extremal case. This is viewed as a discontinuity by (2) who attempts a similar calculation in AdS. The notion of charge being associated with topology comes from some versions (3,4) of the calculation performed in (5). Where the "charges are Q1 and Q5 are generated by wrapping a number Q1 of D1-branes around the circle $x^5$ and a number Q5 of D5-branes around the five circles..." and N is the momentum around the circle $x^5$ (4). Since the entropy calculation relies on those charges, it should follow that the number of string states is also dependent on the topology. So in (3, 4, 5), the calculation is done on a supersymmetric extremal black hole. In (1) the calculation is not on a supersymmetric extremal black hole, but there is difference in topology at the extremal limit, which is implied in (2) to be discontinuous. In both case the change in topology is related to their being two event horizons that do connect to each other, implying a toroidal shape. In (1) the second horizon extends to infinity, causing the change in topology, in (3), the two horizons are described as meeting at the same radius. This suggests some sort of degeneration of the torus, implying a change in topology. My initial query is whether we should think of these changes in topology as discontinuous (assuming that what I described isn't horribly flawed), or possibly a smooth transitions due to quantum effects, such as uncertainty and tunneling, where the change only occur discontinuously at a microscopic level so that it appears smooth at a macroscopic level? The second part of the question is that charges are described as being dependent on topology, and they are also related to symmetries. So it seems natural to think that symmetries are dependent on topology. Although we live in a relativistic world, in our every day life, we still think in non-relativistic terms, and most of our theories have non- relativistic roots, so is it natural to think of there being some sort of low energy topological state (I don't know, something like Ball-3 x Real). That low energy state would apparently have certain symmetries that are dependent on that topology and would make it distinct from other possible topologies. Is that possible? 1\- Entropy, Area and Black Holes 2\- Extremal Limits and Black Hole Entropy 3\- String Theory Demystified 4\- A First Course in String Theory 5\- Microscopic Origin of Bekenstein-Hawking Entropy Further reference (pg 355 in Group Theory: An intuitive approach by R Mirman):


title: Back EMF & Current, how can we make I stable?

text: Lets assume we have a wire that has $10V$ across and $1$ $Amp$ flowing, now if this conductor is introduced to a changing magnetic field, $-EMF$ is induced, can we control our voltage to increase it and allow current to be stable at $1 Amp$? Thus more power is required correct? Generally, in any situation we can keep the current stable at the same value by increasing the voltage?


Why the pressure of atmosphere doesn't crush you when you e.g. walk outside? I mean the density of air is $1.26 kg/m^3$, so with $100 km$ above us, it exerts much pressure on you when you walk outside.


title: German book on introductory quantum mechanics

text: I'm looking for an originally German introduction to quantum mechanics. Is there such a canonical book used in German QM undergraduate courses?


title: Singular wave function

text: Given a wavefunction, $\psi(x)$, is it possible for $\psi$ to be singular at a point? Are there any rules against a wavefunctions having any singularities? For instance if $$\psi(x) =\frac{\gamma(x)}{x},$$ where $\gamma$ is a continuous function of $x$. Is this a valid wave function?


title: Kicking a soccer ball

text: I wonder which type of kicking may cause the ball to go a larger distance? One way is kicking the ball when it is at rest and another is kicking a moving ball in the opposite direction. If the ball goes farther in the second way,then what is the reason despite the fact that a moving ball will exert extra force to the player's leg?


title: How to calculate Riemann and Ricci tensors for a sphere?

text: Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it. The Ricci curvature must be $$ R_{ij}=\frac{2}{R^{2}}g_{ij}. $$ But when I use definition of Ricci tensor, I can't turn the expression into the expression for the metric tensor Maybe, there are siome hints, which can help?


title: How to increase grain size in iron

text: Cut, polished and etched iron meteorites have grain size about 1000 times greater than air quenched iron. The reason for the large grain size is the very slow cooling of the core of a planet or protoplanet. Check web for cross section picture Widmanstätten pattern What methods can be used to grow large iron crystals on earth. What is the largest iron grain size achieved on earth? How about zone refining?


title: Which magnet will last longer?

text: Say you have a magnet that is used in a brush-less generator. If the brush- less motor was ran (by external force such as water or human interaction) for 400 years, would the magnet that was under use have a weaker magnetic field than a magnet that had not been used at all? In other words, does using a magnet decrease its strength over time?


title: How can a clock work if the uncertainty principle is true?

text: If the uncertainty principle and Copenhagen Interpretation are true, then how can a clock tick? Supposedly particles can do all sorts of things when not measured, then how can they be formed into gears that make the clock tick when not measured?


title: Uncertain if invoking uncertainty principle for wave function is handwaving

text: Why doesn't the electron collapse onto the proton in a hydrogen atom? One explanation seems to be given by the Heisenberg uncertainty principle, which follows from the purely physical assertion that the wave function in the position representation is the Fourier transform of the wave function in the momentum (or more precisely the wave number) representation. A different explanation is obtained by solving (or inspecting) the Schrӧdinger equation. To what extent are these explanations independent? To what extent is the latter more fundamental? Is the latter explanation more correct, or more complete? The hydrogen atom is just an example, a more general form of the question would be to what extent it is correct to use the Heisenberg uncertainty principle to answer this kind of questions, as is often done in popular science texts? Elaborating a bit on what I mean to ask, the fact that the Fourier transform of the position wave function is the momentum wave function seems to allow us to draw conclusions about the behaviour of a system from this principle alone. However, when loosely drawing these conclusions, we tend to invoke our classical idea of momentum, whereas the dynamics of the system are really governed by the Schrӧdinger equation.


title: About the dimension of the longitudinal component of vector field

text: According to this lecture note http://www.staff.science.uu.nl/~wit00103/qft05.pdf page 115. Consider a Lagrangian for a massive vector field > $$L = -\frac{1}{4} (\partial_{\mu} V_{\nu} - \partial_{\nu} V_{\mu} )^2 - > \frac{1}{2} M^2 V_{\mu}^2 + ie V_{\mu} \bar{\psi} \gamma^{\mu} \psi - > \bar{\psi} (\partial \\!\\!\\!/ +m)\psi $$ It is said > Note that the dimension of the vector ﬁeld is generically equal to 1, but > since the longitudinal component carries no derivatives in $L$, its > dimension is equal to 2. I could not get why the dimension of the longitudinal component is 2. How to see that?


title: Capacitor with different charges on each plate

text: I am confused as to whether/how capacitance changes when each plate has a different charge. For example, consider a coaxial cable and put $20Q$ on the outer cable, and $-Q$ on the inner. Or how about concentric spheres, grounding either the inner or the outer?


title: Friction problem having two blocks on separate inclined planes connected by a string

text: This is a question I found in my old high-school textbook(I'm revising the topics for a course).Two blocks A and B of masses $1kg$ and $2kg$ are placed on a fixed triangular wedge by a massless in-extensible string as shown. ![enter image description here](http://i.stack.imgur.com/vkP06.png) The pulley is massless and friction-less. The coefficient of friction between block A and wedge is $\mu _1 = 2/3$ and that between B and wedge is $\mu _2 = 1/3$ (let's neglect the difference between static and kinetic friction).The inclines are at $45 ^\circ $ with the horizontal.We need to find the frictional force exerted on each block,and the tension in the string. I worked out the problem,and I thoroughly checked my steps.Since the maximum resistive(friction) force is higher than the components of weights of the bodies along the inclines,there should be a state of equilibrium.I got the equations: $$T= \frac {10}{\sqrt2}+f_A \space .........(1)$$ $$T+f_B=\frac {20}{\sqrt 2} \space .........(2)$$ $$=> f_A +f_B = \frac {10}{\sqrt 2} \space ....(3); f_A-f_B=2T-\frac{30}{\sqrt 2} \space ....(4)$$ $$max(f_A)=max(f_B)=\frac{10\sqrt2}{3}\space .........(5)$$ (Considering $g=10 m/s^2$) Where $f_A$ and $f_B$ are frictional forces experienced by blocks $ A$ and $B$. There seems to be no way to get exact values of $f_A$ and $f_B$.Still,the book says(only numerical answers are given in the end,no complete solution) that $$f_A=\frac {10}{3\sqrt2} , f_B=\frac{10\sqrt2}{3} and \space T= \frac{40}{3\sqrt2} $$ All I can see by looking at the book's answers is that the author simply guessed that let there exist limiting(max) friction between $B$ and the wedge,and he might have put the value $f_B= max(f_B)$ and solved.In that case,why not the friction between $A$ and the wedge be limiting and the answers be $$f_A=\frac{10\sqrt2}{3} , f_B=\frac {10}{3\sqrt2} and \space T= \frac{50}{3\sqrt2} $$ But when we look at eq$(3)$,we see that neither of the frictions needs to be limiting,their sum needs to satisfy $\frac{10}{\sqrt2}$. Any way to justify the book's answers?


title: If gravitation causes constant acceleration why moon does not fall into earth?

text: If moon travels with constant speed in one direction and earth gravitation causes constant acceleration in perpendicular direction why moon does not eventually fall into earth? I mean if gravitation causes moon to fall faster each second (10m/s2) shouldn't after time velocity toward earth be big enough to cause it to fall ?


title: Why does the atmosphere rotate along with the earth?

text: ![Now im isolating Atmosphere](http://i.stack.imgur.com/FkZ4H.jpg) I was reading somewhere about a really cheap way of travelling: using balloons to get ourselves away from the surface of the earth. The idea held that because the earth rotates, we should be able to land in a different place after some time. As we all know, this doesn't happen. Someone said that the reason why this doesn't happen is because the atmosphere (air, clouds etc.) also revolves around the earth (with the same angular velocity as the earth's rotation). Since we are also part of the atmosphere, our position doesn't change relatively. Well, I'm not convinced with that answer. Why does the atmosphere rotate along with earth? Gravitational force is towards the centre of the earth, but I don't see how it's making the atmosphere rotate.


title: Does the rotation of the earth dramatically affect airplane flight time?

text: Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S). During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles turning/moving away from our position? Or, in the opposite direction, would the flight to Sydney be faster since the Earth turns underneath us and moves Sydney closer? ===   * Please ignore jet stream effects and all other variables; this is a control case in an ideal environment.   * By "dramatically" I suppose I mean a delay of 1 hour or more.


title: Flight time Toronto to Moscow the same

text: I have a question. How come that the flight from Toronto to Moscow takes the same time as the flight from Moscow to Toronto. Should it take much longer because of the earth rotation? One direction we travel in direction or an orbit and traveling back we are going against the Earth orbit... From Toronto 9hours From Moscow 9:45 hours. If flying from Moscow we are traveling against the Earth rotation so we should add half of the travel to our current travel. It means that the travel from Moscow should take about 13 hours... Thanks,


title: Can a hovering helicopter travel half the globe in 12 hours?

text: Suppose we have a helicopter that is able to stay stationary in flight for extended periods of time. If such a helicopter stayed at point A in the sky for 12 hours straight, would it reach the other side of the globe?


title: Relation between Earth's rotation-gravity and moving object on Earth

text: This might be silly question. It's about the rotation of the Earth and the objects moving on it. If I jump into the air, why did I also move with the Earth though I am in the air? I am supposed to be land in backward/forward location, as it keeps rotating. But this doesn't happen. What is this phenomenon called?


title: How come the atmosphere moves with Earth?

text: now, I have read a lot of explanations on that but still can't really understand why it would happen so if you can give some examples for a such a thing happening. I mean lets say gravity attracts the air and due to air viscosity all of the air gets carried with the Earth but why will the speed be same ?


title: If I jump will I land in the same spot?

text: If I were to jump one meter in the air and hang for one second, would I fall back down in the same spot or would the earth rotate ever so slightly under me, causing me to land a short distance away from my original point of departure? I am conflicted on this. If I look at the equation for angular velocity, I see that $w = v/r$ where $w$ is the angular velocity, $v$ is the linear velocity, and $r$ is the radius of the object I am on (in this case the Earth). There is another version of this stating that $w$ = $2\pi/t_{rev}$ where $t_{rev}$ is the time it takes to complete one revolution. Being on the Earth, I have a certain linear velocity, $v_{G}$. When I jump in the air 1 meter, I am not applying any force except for vertically so I do not believe my linear velocity would change. However, I am increasing the distance I am from the center of the earth so now my angular velocity would be $w_{A} = v_{G}/(r+1)$. Therefore, it seems that my angular velocity in the air ($w_{A})$ would be slightly less than my angular velocity on the ground ($w_{G}$). If I were to recalculate the linear velocity given this discrepancy in $w$, I would get $v_{G}=w_{G}r$ and $v_{A}=w_{A}r$. This shows I get a small difference unless the radius in $v_{A}$ must account for the jump ($r+1$), in which case I end up with $v_{A} = v_{G}$. I've gone around in circles trying to decide if one would actually move. This top answer seems to think you would: Earth moves how much under my feet when I jump? Any insight would be appreciated. Thank you!


title: Earth moves how much under my feet when I jump?

text: If I'm standing at the equator, jump, and land 1 second later, the Earth does NOT move 1000mph (or .28 miles per second) relative to me, since my velocity while jumping is also 1000mph. However, the Earth is moving in a circle (albeit a very large one), while I, while jumping, am moving in a straight line. How much do I move relative to my starting point because of this? I realize it will be a miniscule amount, and not noticeable in practise, but I'd be interested in the theoretical answer.


title: Earth is rotating

text: > **Possible Duplicate:**   >  Why does the atmosphere rotate along with the earth? If i take off from land on a helicopter straight above the earth surface to a certain height and stay there for few mins/hours and come down. Why am i coming down to the same place where i took off? If the earth is rotating i should land on a different place right because i have not moved i am just coming down straight.I moved only vertically. I got this thought because i was thinking why are we spending so many hours on flights to reach a country on west if i started from a eastern country. May be there are a lot of scientific reasons behind this which i am not aware excuse me if it sounds silly.I thought this would be the best place to ask.


title: Sideways motion between a vertical launch from a planet and landing

text: I saw a video some days ago (Hello Kitty in Space) of a schoolgirl successfully launching a balloon into space which later popped and landed ~47 km from launch site. If I vertically launch an object (say via balloon or rotors) from the surface of a spinning planet with an atmosphere like ours, then as it rises, will the planet's surface move away below it, or will the object rotate with the earth? (Assume that there are no local strong winds relative to the atmosphere, and it is completely calm.) The latter would mean the atmosphere, being itself a collection of such objects (air molecules), also spins with the planet. Related SE question: Why does the atmosphere rotate along with the earth? It seems from common sense that since at launch the object shares the motion of the surface it would keep rotating with the planet, and hence would later land at the launch site. However, as it ascends, the lateral speed required to maintain the same angular velocity with the planet rises linearly. Where would it get this extra momentum/energy from? From the rotating atmosphere? How did the atmosphere get moving in the first place then? It might seem a lot of questions for a single SE post, but they're all connected, so a single explanatory answer should suffice.


title: Does the ground beneath a levitating object change due to the rotation of the earth?

text: I don't know whether this question pertains to physics. But anyways, here goes... If I were to hover 1m above the ground in a helicopter, would the ground below me change after 1 hr due to the rotation of the earth?


title: Reaction-at-a-distance: Do charged plates immediately repel each other?

text: ![Retarded](http://i.stack.imgur.com/YMVBx.jpg) Imagine that we have a pair of parallel plates, $A$ and $B$, separated by some distance as in Fig. $1$ above. At time $t_1$ we simultaneously charge both the plates. This could be done by previously sending a light signal to a charging apparatus at each plate from a source located at the mid-point between them. According to standard electromagnetic theory a retarded electric influence travels at the speed of light from $A$ to $B$ and vice-versa. At time $t_2$ the electric influence from $A$ produces a force at $B$ and vice-versa. There are two points that I would like to raise about this description:   1. There are no reaction forces. It is as if a pair of boxers each punched the other on the nose simultaneously but neither felt a reaction back on their boxing glove.   2. Once the electric influences have left the charged plates at time $t_1$, and before they have produced forces on the opposite plates at time $t_2$, they must exist "somewhere". That somewhere is the electromagnetic field.  Now consider the picture described in Fig $2$ below which includes both retarded and advanced interactions. ![Retarded and Advanced](http://i.stack.imgur.com/zBS6w.jpg) Again at time $t_1$ we simultaneously charge both the plates. Now as well as a retarded electric influence that travels at the speed of light from $A$ to $B$ we also have an advanced electric influence which travels backwards in time from $B$ to $A$. Thus the force at plate $B$ at time $t_2$ is balanced by an equal and opposite force on plate $A$ at time $t_1$ (and vice-versa). Now as soon as we charge the plates up we measure electric forces on them. At first glance it seems that we have "action at a distance" but in fact we only have "reaction at a distance". In terms of spacetime, each plate at time $t_1$ is linked with the opposite plate at time $t_2$ in a manner that is consistent with the principle of locality provided we include advanced interactions. As there is no delay between charging the plates and measuring forces then there is no time interval during which the influences could be said to be in transit in the form of an electromagnetic field. Thus in this picture we have:   1. Reaction forces   2. No electromagnetic field Could one perform such an experiment to see if charged plates immediately repel each other?


title: What is the agent that does work in case of friction due to eddy currents?

text: In light of the well known fact that _magnetic forces do no work_ , In every such case that nature presents us, it is possible to find some agent which does the work while magnetic forces just give direction to it. As an example, in the case of an electromagnet lifting weights, it is the potential source that powers the electromagnet which is responsible for the actual work done. In the case of friction due to eddy currents, specifically _the case of slowing down of a metal object passing through a magnetic field_ , **what is the agent which is responsible for the work done?**


title: How to think physically about basic "fields"

text: "Field" is a name for associating a value with each point in space. This value can be a scalar, vector or tensor etc. I read the wikipedia article and got that much, but then it goes it into more unfamiliar concepts. My question is how to interpret a basic field. Lets say there is a field with momentum and energy. Does that mean, that any "object" which can interact with that field, can borrow that momentum and energy from that field and give back to it as well. Much like an electron in some (not its own) electromagnetic field, right? Now how do I extend this understanding to concepts like: > $U_\lambda$ is the radiation energy density per unit wavelength of a > thermodynamic equilibrium radiation field trapped in some cavity." Does that mean, that the "numbers" that make up the field in each point in space stay constant with temperature or time (or both, I'm not sure.) And if there are object is that field which can interact with it, for example electrons, then they can take energy from that field or lose energy to that field. Then the author calls this field a "photon gas" without explanation. So does that mean a bag of photons in cavity is mathematically equivalent to specifying some numbers in space? Or something else.


title: What's so unphysical about $W^{\mu}_3$ and $B^{\mu}$?

text: Imposing SU(2) and U(1) local gauge invariance introduces 4 gauge bosons, two of which correspond to $W^{\pm}$ bosons. The other two gauge fields $W^{\mu}_3$ and $B^{\mu}$ however are said not to correspond to $Z$ and $\gamma$ bosons because of the incorrect chiral coupling of the $Z$ boson if this identification is made. The "physical" bosons are said to be Weinberg rotated by $\sin^2\theta_W=0.23$, where the angle is obtained from experiment. This gives the observed coupling of $Z$ to both left and right chiralities and also a different mass from the $W$ brothers, again observed experimentally. Why? Why do the physical $Z$ and $\gamma$ bosons have to be Weinberg rotated? (by Nature that is, not by phenomenologists) If there was no rotation, the three $W$ 's would have the same masses and would couple only to lefties. This doesn't immediately seem particularly disastrous to me. I don't see how Gell-Mann's Totalitarian Principle can save the day here, because what I'm essentially asking is why our particle detectors observe the Weinberg rotated bosons and not the "natural" ones that come out of the Lagrangian? What is it that makes particular linear combinations of bosons the physical ones? And crucially, why aren't the $W^{\pm}$ bosons rotated in a similar manner? (Or would we not notice if they were?)


title: Would Oscars made of pure gold bend?

text: Since gold is often considered "soft": If the Oscars were made of pure gold (instead of gold-plated britannium), would they bend (deform) over time?


title: Stacking gold Bars

text: I once heard that because gold is so malleable, while also being very heavy; that if one were to stack several layers of bars on top of on another (like is so familiarly depicted in movies & TV), the shape of the ones on the bottom would be smushed down because of the weight of the ones on top and gold's high malleability. Is there any truth to this?


title: A Spaceship travels from Star A to B, but food/fuel and relativity!

text: The question is very simple, but complicated to solve. I am looking for a _genera_ form solution to this problem. Thus there will be no numbers, just variables. What I mean by that is often in physics you have equations like $F = ma$. I don't have the numbers to the problem yet, because that will vary based on the spaceship, its engine, the distance, etc. Newtonian physics works as an approximation, however its allows the spaceship to exceed $c$ during its travels. I would like to add relativity, with the frame of reference being the spaceship. I can already solve this problem in discrete time steps programatically, but it would be nice to have a general form solution that would be more accurate. (Because discrete time stepping is expensive computationally, and a general form solution would be much much faster.) From my current algorithm the variables are:   * $M$ = base mass of spaceship in kkg (ie 1000kg)   * $C$ = cargo space of spaceship kkg    * $D$ = distance to be traveled in km   * $T$ = thrust of the engines in kN   * $F$ = fuel consumed per year in kkg/year    * $E$ = food consumed per year in kkg/year   * $A$ = maximum velocity you can possibly achieve {$A < c$} in m/s   * $B$ = maximum velocity you are willing to achieve {$B <= A$} in m/s Some notes.   * Cargo space can be used to be store fuel and/or food.      * The way I solved this problem before, was calculated fuel costs first, and then decided if there was enough food afterwards. Based on $C_{remaining} > t_{travel} * E$   * Loaded cargo adds to $M$ so $M_{total} = M + C_{fuel} + C_{food}$   * As you fly the mass of the fuel is consumed, reducing the weight of the spaceship, allowing you accelerate faster.      * Engine efficiency is not part of the equation. The engine produces $T$ thrust and consumes $F$ fuel per year. (unless I am wrong, I shouldn't have to worry about it, if it does matter, lets say 100%; its ideal.)   * The ship needs to spend fuel accelerating (to the halfway point or until B it achieved), and then spend fuel decelerating on the other end. the $t_{accelerating} \neq t_{decelerating}$ due to the change in mass during voyage. This also means less fuel will be consumed to decelerate as well.   * As length inversely dilates with velocity, the length of the trip shrinks as the spaceship nears $c$ and expands as you decelerate as you near the destination. What I am looking for is a function that determines if the one-way trip is possible, how long it will take, and how much fuel is consumed in the process, in the form of $$f(M, C, D, T, F, E, A, B) -> (isPossible, t_{total}, C_{fuel-consumed})$$ I realize that with B there will be conditionals. As far as I can tell, there are 3 rate of change problems (more?)   * Mass increases as velocity increases $M_{relative} = M_{current} * \gamma$   * Mass decreases as fuel is expended $M_{current} = M + C_{fuel} - C_{fuel-consumed}$      * $C_{fuel-consumed} = F * t_{running-engines}$   * Distance decreases as velocity increases $D_{relative} = D_{base} / \gamma$   * Am I missing something? Some references:   * http://en.wikipedia.org/wiki/Speed_of_light#Upper_limit_on_speeds   * $\gamma = \dfrac{1}{\sqrt{1-v^2/c^2}}$   * $a = \dfrac{T}{M_{relative}}$ Note: $a$ will change as $M_{relative}$ changes, but $T$ remains constant


title: Global SU(2) invariance of QED Lagrangian

text: I'm having problems seeing the global SU(2) invariance of the QED Lagrangian. My specific problem is seeing why \begin{equation} e^{-i a_i \sigma_i} \gamma_\mu e^{i a_i \sigma_i} = \gamma_\mu \end{equation} In every book i looked it up, it was written this is trivial and i couldn't find a computation, so i guess i'm missing something trivial. Any tip or reading recommendation where the global SU(2) invariance is shown explicitly would be much appreciated.


title: How to caculate the pressure of a fluid when there is no gravity

text: Well for example we put a ball filled with water, with a density of $1\cdot 10^3 kg/m^3$, in space. How to calculate the pressure the water is exposed to?


title: Calculating capacitance of arbitrary plate shape and arrangement.

text: Looking for suggestions on how to approach calculating the capacitance of a capacitor where the plates have an arbitrary shape. I've seen derivations of capacitance for a few highly symmetric arrangement (eg coaxial cylinders) but nothing like a general approach to predicting the measurable capacitance of arbitrary arrangements of plates. Thanks for any suggestions or references.


title: Timelike Shell Collapsing into a Black Hole

text: Does anyone know where I can find the solution for a spherically symmetric thin shell of timelike matter falling into a Schwarzschild black hole? The matter should be pressureless, so that each particle on the shell follows a radial geodesic. I am interested in the case where the shell is released from rest at a given Schwarzschild radius and time. I think this is a special case of something called the Bondi-Tolman metric, but I'd rather not try to unpack that solution if there is a simpler way to find what I'm looking for. For null matter the answer is known (and simple), it's the Vaidya metric. Edit: After thinking about this a little more I think it might be trivial; is it just Schwarzschild inside the shell with the black hole mass $M$, and Schwarzschild outside the shell with mass $M+E$, where $E$ is the energy of the wave? Edit to the edit: The first edit looks wrong, I don't think it's a solution to Einstein's equations.


title: What could we observe if we see a 4 dimensional object and how could it change our physics view about our universe?

text: My question is little bit philosophical. I would like to explain my ideas with a 2 dimensional universe model. If we had lived in 2 dimensional universe like a plane, What could we observe when seeing a 3d object? For example: If a square pyramid that is inside full of material comes to the plane universe in right angle, what could the people who live in 2d universe observe? Firstly, we could see small square and slowly it would enlarge and then it would dissappear suddenly. If the scientists who live in the 2d universe observed such event , All scientists could get shocked and they could check their physics formulas because energy is not conversed in that event. They had believed that Energy is conserved and nobody could create energy from nothing and finally they could understand that total observed universe energy can be much more what they calculated with their formulas. My questions:   1. What could we see a 4d pyramid comes to our 3d universe? is it correct that Firstly we would see small cube and it would enlarge and then suddenly disappears?   2. Dark matter and Dark energy is related with such ideas? http://map.gsfc.nasa.gov/universe/uni_matter.html   3. I know we do not observe such strange events till now but maybe scientists can get some results during micro experiments. Is there any such experimental results at least in micro universe (atomic level experiments in quantum mechanics)? Sorry for your time if it was asked before.


title: How can I estimate the power between the guide and the sky diver?

text: I would like to estimate the Maximum power/ tension that can be, between the sky diving guider and the "tourist". In this picture what the max tenstion will be, I think it is maximum when the parachute is opened.


title: How to determine directions of vectors of an electromagnetic wave

text: I did an exercise which probably is quite popular,   in which you draw an electromagnetic wave and prove that it should propagate at the speed of light $1 \over \sqrt {\mu_0\epsilon_0}$ using Farday's law and Ampere's law. Basically if this is the wave:    ![Electromagnetic wave](http://i.stack.imgur.com/SqAzU.png) Let's say the E-field (red) is in the X direction, the B-Field (blue) is in the Y direction, and the velocity of the wave is in the Z direction. You take for example for ampere's law a surface in the ZY plane with a length L equal to the amplitude of the wave, and a width equal to $\lambda\over 4$ You do a similar thing with Faraday's law and you get the speed of light, assuming you know that the E-field and B-field propagate in this manner. I got the right answer but I wondered about this: Let's say I only had the E-field and I know the wave propagates at the speed of light, I assume this is enough information to draw the B-field at each point. But how will I know the direction? Both Faraday's law and Ampere's law say you need a closed loop integral and the rules I've been taught say you go over the loop in a clockwise direction for example and take the normal to the surface according to the right hand rule etc. But clockwise and counter-clockwise direction don't really give me much information in this case, so how can I determine the direction of the B-field if I only have the E-field?


title: What are characteristic ranges of time, distance, energy regions for physics and chemistry?

text: I've constructed such table from many different sources for my thesis introduction: ![enter image description here](http://i.stack.imgur.com/zixNi.jpg) I'm trying to make a bit of philosophy it the thesis first chapter. Are they realistic or not? Could you make any comments on the ranges presented here. I'm not trying to say that physics overcomes chemistry or something like this. These are two different and independent sciences. They study different objects. The question here: are ranges correct? For example for time range. It is easy to find this range in internet, but for chemistry it is not. So, my thought were this. What is minimum time in chemistry? It should be time when the fast chemical reaction can occur/or start, it is ~ tens of femtosecond. So, I've hoped that people in physics.StackExchange will give me some comments on what I've done by so far. I've discussed this topic with some other scientists and corrected my table. But it is not enough, I need more information. Please do not hesitate to swear:)


title: Energy of a cylinder rolling down a path

text: **Problem statement:** A cylinder rolls without slipping down a hill. It is released from height h. What is its speed when it come down? The cylinder mass may be completely concentrated on the radius R, which is the radius of the cylinder. **My thoughts:** At the top(hight h) the potential energy of the cylinder is E=mgh and at the bottom(h=0) all energy has become kinetic energy since friction and air drag is neglected in this context.(I assumed this). Thus: $mgh=\frac{1}{2}mv^2 \Leftrightarrow v=\sqrt{2gh}$ **Correct answer** is however $v=\sqrt{gh}$ which means that energy must have been lost,correct? What have i missed? ![picture](http://i.imgur.com/Ge3x1nu.png)


title: Tension in parallel springs

text: > _A block of mass $200$kg is connected to a horizontal ceiling by four > identical light elastic ropes, each having natural length $7$m and stiffness > $180$N/m. It is also connected to the floor by a single light elastic rope > having stiffness $80$N/m. All five ropes are stretched and vertical, and air > resistance is neglected._ > > _Find the tension in one of the upper ropes._ I know it's just $kx$, but I don't understand why it is not $\frac{kx}{4}$ because tension is shared between four ropes.


title: How to predict Iridium Flare?

text: I want to find out how to predict Iridium flares, but I don't know where to start. If I know Solar position, Iridium satelites' TLE and it's shape, I need a magnitude(time) function for some place on Earth. As I see now, main question is in satellite orientation (it's panels). Any ideas?


title: Principle of Least Action via Finite-Difference Method

text: I am reading Gelfand's Calculus of Variations & mathematically everything makes sense to me, it makes perfect sense to me to set up the mathematics of extremization of functionals & show that in extremizing a certain functional you can end up with Newton's laws (i.e. you could extremize some arbitrary functional $L$ & in examining the form of Newton's laws you see that one should define $L$ as $T - V$) & this way of looking at things requires no magic, to me it seems as though you've just found a clever way of doing mathematics that results in Newton's laws. However in books like Landau one must assume this magical principle of least action using the kind of thinking akin to Maupertuis, & I remember that every time I read some form of justification of this there's always a crux point where they'd say 'because it works'. I'm thinking that there may be a way to explain the principle of least action if you think of extremizing functionals along the lines Euler first did & use the method of finite differences (as is done in the chapter on the Variational Derivative in Gelfand, can't post a link unfortunately), i.e. because you're thinking of a functional as a function of n variables you can somehow incorporate the $T - V$ naturally into what you're extremizing, but I don't really know... I'm really hoping someone in here can give me a definitive answer, to kind of go through what the thought process is in this principle once and for all! As far as I know, one can either assume this as a principle as in Landau & use all this beautiful theory of homogeneity etc... to get some results, or else once can assume Newton's laws & then use the principle of virtual work to derive the Euler-Lagrange equations, or start from Hamilton's equations & end up with Newton's laws, or one can assume Newton's laws & set up the calculus of variations & show how extremizing one kind of functional leads to Newton's laws - but I'm not clear on exactly what's going on & would really really love some help with this.


title: Magnitude of force to keep stick in equilibrium

text: **Problem statement** A straight and homogenous stick with mass m is pressed against a wall with the force F. The stick is horizontal perpendicular against the wall. Given that the friction between the wall and the stick is μ, determine the horizontal component of F in order for the stick to not fall down. **My thoughts** Forces involved: We have: The gravitational force **mg** in the negative y-direction. The normal force from the wall, **N** (negative x-dreiction). The friction force in the positive y-direction which is **f=μN** and the force **F** which acts in the positive x-direction. Thus: $$\sum F_{x}: F-N=0 \Rightarrow F=N \\\ \sum F_{y}: mg-f=mg-\mu N=0 \Rightarrow N=F=\frac{mg}{\mu }$$ $$\boxed{F=\frac{mg}{\mu}}$$ ![picture](http://i.imgur.com/QFJq7wT.png) * _Correct answer is $\boxed{F=\frac{mg}{2\mu}}$ *_


Is it possible to see Fraunhofer lines with amateur equipment? Would it be possible (with reasonable effort) to identify elements or is this hard?


title: Why does positronium decay into 2 photons more often than into 3 photons?

text: I cannot find the answer to the above question. I know that para-positronium is created with a probability of $25\%$ and decays into 2 photons, while ortho-positronium is created with a probability of $75\%$ and decays into 3 photons. I also know that ortho-positronium has a way longer life time than para-positronium. This, in my understanding, should not affect the number of decays per time, but just means that the ortho-positronium will decay LATER into three photons. But in the end there should be $75\%$ 3-photon-decays and $25\%$ 2-photon-decays. But in reality 2-photon-decay happens about 300 times more often than 3-photon-decay. What information am I missing? Thank you!


title: Acceleration of electric charges and radiation

text: According to classical electromagnetic theory, accelerated charges should emit radiation and lose energy. The reason given in my book why atoms don't emit radiation (say, when the atom moves along a circle) is because the atom is neutral. I can understand how this works for a neutral particle like a neutron but the atom has constituent charges within in. How can the "presence" of an opposite charge nearby stop what seems to be an intrinsic process independent of the surroundings? Do the electrons and protons emit radiation that destructively interferes or something of that sort?


title: Why is adiabatic process isentropic?

text: I have read that adiabatic process is isentropic because there is no heat exchange in an adiabatic process and thus no change in entropy. But my question is - Even in adiabatic process, work can be done. Let's take an example of an adiabatic vessel with a piston attached. That vessel does not exchange heat but work can be done by pulling in or out the piston. If the volume of the system changes, isn't entropy also changed, even in a reversible adiabatic process? EDIT : I know that change in entropy is defined as change in heat divided by temperature. Since there is no change in heat in an adiabatic process, the entropy is zero. My question is different - What I see entropy as - It is a measure of the different microstates in which a system can be. So, even if there is no change in heat energy, can't number of thermally achievable microstates increase if we increase the volume by doing work? Why is only heat considered as a measure of entropy?


title: Can a force stop a Photon since Photons have momentum and What does momentum mean when talking about massless particles?

text: Momentum measures how hard it is to stop an object. While Photons are massless they still have relativistic mass and energy. My question is can something stop photons other than being absorbed by something? What does momentum mean when talking about massless particles?


title: Velocity of 2 balls with different masses on a moving train after the train brakes

text: # Question This is from my textbook: > There are 2 balls of the same size made of rubber and iron respectively kept > on the smooth surface of a moving train. Which ball will move faster when > the train brakes suddenly? # My Answer In that start it seems simple. Let the iron ball be called $A$ and the rubber one $B$. The mass of the iron ball is greater as: $$ (1)\ M = V \cdot D \\\ (2)\ D_A > D_B \\\ (3)\ V_A = V_B \ \ [From\ Q] \\\ (4) M_A > M_B \ \ [From (1), (2), (3)] \\\ $$ Then we have to consider what forces are acting on it. The only ones I could think of were friction with the floor and air resistance. If we consider friction, then we can easily conclude that the rubber ball moved slower as rubber creates a _lot_ of friction. But since they mentioned the floor was 'smooth', I thought we should ignore that. So all that's left is air resistance (denoted by $F$). $$ (5) F = m \cdot a \ \ \rightarrow \ \ a = \frac{F}{M} \\\ (6) Surface\ Area_A = Surface\ Area_B \ \ [From (3)] \\\ (7) F_{iron} = F_{rubber} \ \ [From (6)] \\\ (8) a_A < a_B \ \ [From (4),(5),(7)] $$ That means the rubber ball will move faster than the iron one (and therefore that the iron ball will stop before the rubber one does). # Another answer But my teacher (who'd given me this problem) disagrees. Her argument is: > Consider everything with reference to the train. Before braking, the balls > are stationary (w.r.t. the train). When the train brakes, negative > acceleration is provided to the train. That means the balls will move > forward w.r.t. the train. But since the iron ball has a greater mass and so > inertia, it will move slower. which give the same answer. ### Doubts So my questions are:   1. Was my teacher's approach correct? Seems a bit unsound to me, but that just may be b'coz I don't have enough intuition for switching inertial frames of reference.   2. Are there any other forces in play over here? I could only think of 2: air resistance and friction.   3. Should air resistance be considered?   4. Should friction be considered? In that case, the rubber ball will be significantly slowed.


title: Deriving the critical dimension of bosonic string theory

text: I am going through the lecture note by Gleb Arutyunov on the derivation of critical dimension for bosonic string theory. I was able to reproduce all the results till the last step given on page 62. For $[S^{i-},S^{j-}]$, the term that should vanish and give us the critical dimension is \begin{equation} \sum_{n=1}^{\infty}\left[\left(\frac{4\pi T}{(p^{+})^{2}} 2(n-1)-\frac{f(n)}{n^{2}}\right)\alpha_{n}^{[i}\alpha_{n}^{j]}\right] \end{equation} where \begin{equation} f(n)=\frac{4\pi T}{(p^{+})^{2}}\left[\frac{n(n^{2}-1)}{12}(D-2)+2an\right] \end{equation} I realize that I might be missing something obvious but at this moment I don't see how the above expression can vanish for all $n$ when $D=26$ and $a=1$. Thanks!


title: Interpretation of red shift

text: We see light as having travelled in a straight line from stars or galaxies light years away from us. However it's path is more likely of multiple curves as a result of gravity along the journey (gravitational lensing) . How can we be sure that the accelerations thus experienced have not produced braking radiation (bremsstrahlung) thereby reducing the energy and therefore frequency of the light spectrum leading to an apparent red shift?


title: Is there an equation that tells you more about the amplitude of an object which is in resonance?

text: I'm a high school senior and I have to write a paper about resonance and differential equations. I've been searching the Internet for a long time, but I haven't found an equation that is properly explained. Could anyone explain whether there is an equation to describe the amplitude and if so, whether you could explain to me how you can obtain this formula?


title: Understanding the Seebeck effect

text: Thermoelectricity is, as I understand it, the difference in voltage between the hot and cold ends of two dissimilar materials. If two materials are connected at two different junctions, the hot junction will effectively liberate electrons which will flow to the cold junction, along with the heat. All the diagrams of this I could find showed exactly two materials, connected at both the hot and cold ends. In order to have the hot side remain hot, and the cold remain cold, while transmitting the electricity, ideally a substance is desired that conducts electricity well, but heat poorly. This is the measure of quality. A good overview paper is: New Directions for Low- Dimensional Thermoelectric Materials by Dresselhaus et al. "ZT= S2rT/j, where S, r, T, and j are, respectively, the Seebeck coefficient, electrical conductivity, temperature, and thermal conductivity" Maximizing efficiency of a thermocouple is, in fact, a matter of finding materials that conduct heat poorly, while conducting electricity well. In practice, that's difficult, so in constructing nanomaterials to do so, they are trying to scatter phonons while not scattering electrons. Questions:   1. How do you make a series circuit with lots of thermocouples? Surely this must be done, yet when I draw a diagram, I can't come up with a way that doesn't look like two identical thermocouples are simply operating in opposing directions. This is answered below (and thanks), but note the answerer is incorrect about the need for thermal conductivity. Ideally, you want to insulate these things, and have as little heat as possible produce as much electricity as possible.   2. In the diagram drawn by the answer, it shows a thermopile as a zig-zag connection of two materials between hot and cold. This makes perfect sense now that I see it, but then, does material A transport electrons from hot to cold, while material B transports them from cold to hot? I thought I intuitively understood the Seebeck effect as a liberation of electrons from the hot end, effectively by "shaking them loose" but it appears one material must be doing the reverse? I'm guessing this is similar to galvanic action, where one material wants an electron to complete a valence shell, and the other wants to give one away?   3. I have now drawn a diagram showing a single thermocouple, a thermopile per the answer below, and my question, which is whether you can introduce a third material connecting the other two. I'm guessing you can, and that whatever its electropotential, since it's symmetric it should have no effect on the voltage of the thermopile other than its resistance. ![diagrams of thermopiles](http://i.stack.imgur.com/W714r.png)


title: Why is the anode (+) in a device that consumes power & (-) in one that provides power?

text: I was trying to figure out the flow of electrons in a battery connected to a circuit. Conventionally, current is from the (+) terminal to the (-) terminal of the battery. Realistically it flows the other way round; from the (-) terminal to the (+) terminal. My question is, assuming electron flow is from the (-) terminal, would the battery's cathode be located at the (+) terminal and it's anode at the (-) terminal or would it be vice versa? Another question: Why would the anode be positive in a device that consumes power and negative in a device that provides power?


title: Conservation of Energy and Quantum Fluctuations

text: Regarding conservation of mass-energy Wikipedia says: **"this is an exact law, or more precisely, has never been shown to be violated."** However, regarding quantum fluctuations, Wikipedia says here: " **That means that conservation of energy can appear to be violated, but only for small times** ". I thought to resolve the two conflicting statements by proposing that the virtual particles from quantum fluctuations are created/destroyed from the "noise energy" of the uncertainty principle. Hence, the energy is already there. Is this correct? If this is incorrect, how can we resolve the two?


title: Why does asphalt crack along painted lines?

text: My elementary school playground was made of asphalt and had a track painted on with white lines. While walking on the track, I noticed that cracks in the asphalt often followed the paint. The conclusion I came to and that I still hold is that the white tape on the black asphalt causes a temperature difference due to the different amounts of sunlight absorbed and that this difference accounts for the cracking. Is this correct? Is there more to the story?


title: Induced metric on the boundary of a manifold

text: The Gibbons-Hawking-York term which supplements the Einstein-Hilbert action is, $$S_{GH} = \frac{1}{8\pi G} \int_{\partial M} d^3 x\sqrt{-h} \, K$$ where $\partial M$ is the boundary of the manifold $M$, $K$ is the trace of the extrinsic curvature, and $h_{\mu \nu}$ is the induced metric on the boundary of the manifold. My questions are:   1. Is there a general formula for the metric $h_{\mu \nu}$ in terms of the metric $g_{\mu \nu}$ of the manifold M?   2. Is there a general formula for the inward/outward normal?


title: Spatial bound on the internal electron structure

text: In 2006 the radius for a possible internal structure of the electron has been pinned down to $10^{-18} m$. This validates the approximation of electrons as point particles at long distances, e.g. in an atom. The upper bound on the internal electron radius has been derived from a very precise measurement of the $g$-factor, see New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron What I don't understand is how did they determine the relation between the $g$-facor and the radius of the internal structure? As far as I understand they compare the $g$ factor of a point particle to the $g$ factor of an extend particle. But how do you calculate the $g$ factor of a point particle or an extend particle?


title: Adiabatic Expansion with expanding control volume

text: I have a tube with a length "L" and diameter "d" that is open on 1 side . At a certain point ( say "x" ) from the closed end of the tube, I have a gas with a high pressure. At this point, "x", there is also a weight of mass "m". It is also known what the velocity of the weight is at point "L". Initially : ![initial](http://i.imgur.com/DPXFndD.png) Finally : ![final](http://i.imgur.com/xu5Qmmr.png) Assuming that the tube is held stationary, what force is imparted on the closed section of tubing ( left side )? Basically, think of it as a rifle barrel with a bullet inside. I wish to find a expression for the force on the closed end of the tube as it varies with time. What I propose : Determine the Work by the gas. And since work is defined as $W = Fx $ and then $ \frac{dW}{dt} = \frac{d(Fx)}{dt}$ perhaps this could lead to an expression for force?


title: Are electrons simple? Do they have any inner structure?

text: The plank length is far smaller than the classical electron radius. Could the electron have structure?


title: Electric dipole moment(EDM) underlying physics

text: I read in a physics today paper, The electron can have nonvanishing EDM only if nature violates symmetry under time reversal (T) and under the combined operations of charge conjugation (C), which replaces particle by antiparticle (as we all know), and parity inversion (P). I checked here. So it will be nice if you can show me qualitatively and quantitatively about the above mentioned point. Note: Simplicity in explanation is always nice. (I don't have much idea of these big words)


title: What is the mass density distribution of an electron?

text: I am wondering if the mass density profile $\rho(\vec{r})$ has been characterized for atomic particles such as quarks and electrons. I am currently taking an intro class in quantum mechanics, and I have run this question by several professors. It is my understanding from the viewpoint of quantum physics a particle's position is given by a probability density function $\Psi(\vec{r},t)$. I also understand that when books quote the "radius" of an electron they are typically referring to some approximate range into which an electron is "likely" to fall, say, one standard deviation from the expectation value of its position or maybe $10^{-15}$ meters. However it is my impression that, in this viewpoint, wherever the particle "is" or even whether or not the particle "had" any position to begin with (via the Bell Inequalities), it is assumed that if it were (somehow) found, it _would_ be a _point_ mass. This has been verified by my professors and GSIs. I am wondering if its really true. If the particle was truly a _point_ mass then _wherever_ it _is_ , it would presumably have an _infinite_ mass density. Wouldn't that make electrons and quarks indistinguishable from very tiny black holes? Is there any practical difference between saying that subatomic particles are black holes and that they are point masses? I am aware of such problems as Hawking Radiation although at the scales of the Schwarchild radius of an electron (back of the envelope calculation yields $\tilde{}10^{-57}$ meters), would it really make any more sense to use quantum mechanics as opposed to general relativity? If anyone knows of an upper bound on the volume over which an electron/quark/gluon/anything else is distributed I would be interested to know. A quick Google Search has yielded nothing but the "classical" electron radius, which is not what I am referring to. Thanks in advance; look forward to the responses.


title: Do electrons have shape?

text: According to the Wikipedia page on the electron: > _The electron has no known substructure. Hence, it is defined or assumed to > be a point particle with a point charge and no spatial extent._ Does point particle mean the particle should not have a shape, surface area or volume? But when I searched Google for the "electron shape" I got many results (like this and this) that says electrons are round in shape.


title: What is the volume of electron?

text: I know that electron has mass , and that is particle( a body which has only mass and whose size is negligible) but can we ever calculate the volume of the electron . if yes how much it is . if no why?


title: Size of an elementary particle

text: Do we have a well defined mathematical expression denoting the size of a fundamental particle with no internal structure (electron for example) ? If we do, how does it fit in with the uncertainty principle ? And if we don't then what exactly do the experiments reveal claiming radius of electron is of the order $10^{-18} m$ or something like that ? Also in this particular case, does this mean that the probability of the electron being within that radius is 1 and zero outside ? A proper clarification would be very helpful.


title: Question about Majorana fermion and Majorana representation

text: In Chiral representation, a Majorana spinor looks like: $$\psi=\begin{pmatrix} \psi_L\\\ -i\sigma^2\psi_L^*\end{pmatrix}$$ In this representation the Right handed field is the charge-conjugate of the left handed field. i.e., $(\psi_R)^c=\psi_L$, where $$\psi_R=\begin{pmatrix} 0\\\ -i\sigma^2\psi_L^*\end{pmatrix}$$ and also $\psi^c=e^{i\phi}\psi$   1. How does it look like in Majorana Representation, explicitly in the form of a column vector? What is the usefulness of Majorana representation?   2. Can I use the condition $\psi^c=e^{i\phi}\psi$ to be the definition of a Majorana fermion?


title: Boundary currents for Asymptotic Symmetry Group (ASG)

text: In the context of asymptotic symmetry groups, what is a boundary current? Why is it called a "current"? Context: I'm reading Strominger's recent paper on Asymptotic symmetry group of Yang-Mills (link here) and he has a section on the boundary current (section 2.3). I can follow the math completely fine, but some of the words are confusing to me.


title: Is the Higgs a quantum field or a particle?

text: The Higgs is not detected in the asymptotic data, so it is possible that there is no particle interpretation for the Higgs quantum field. Indeed, the Higgs potential is only positive definite if the quartic term is included --- the quadratic term corresponds to a negative mass term. It would seem, therefore, that the Higgs field does not have an on-shell mass spectrum, so that there is no straightforward particle interpretation. One can say that there is an effective field theory in which there is a resonance near a given mass that we will call the Higgs _resonance_ , but in the absence of a pure mass shell spectrum (that is, if there is a continuous mass spectrum), it is generally taken in QFT that there is no particle interpretation. The resonance is clearly not a $\delta$-function, so is there some other precise way in which we can call the Higgs a particle? Of course this doesn't call into question the empirical effectiveness of the Standard Model of Particle Physics, it only asks about its interpretation and about how we put the Mathematics into words. I was somewhat struck by Rolf Heuer's observation (this morning) that this is the first observation of a scalar particle. Indeed, according to the SM, there are no quantum fields that have non-zero mass terms in the absence of interactions. In the absence of interactions, the Higgs field is a massless scalar field. Should we say that it is the Higgs _interaction_ that gives mass to the standard model? (EDIT: Is it better to say that every term that is not quadratic in the fields _contributes_ towards the effective masses of each of the asymptotically observed fields? Or what alternative phrasing is closer to the Mathematics of the interacting fields?) EDIT(2, $\scriptstyle\mathsf{see\ below\ for\ the\ comment\ that\ prompted\ this\ possible\ rephrasing}$): Is there any part of the definition of "particle" that is not a matter of convention? Does the Higgs cross that bar?


What is the principle behind centrifugation? I understand the idea that you spin something around the centripetal force will cause an apparent force on the spinning system. However I don't quite grasp how particles (in the non subatomic sense) with different density should be affected differently. Quite coarsely, I would expect to write down Newton's second law, but then the mass would simplify and the acceleration of every particle would be the same, regardless of mass. Is friction the answer? Or am I missing something silly?


title: Could life survive a pole shift caused by an asteroid collision?

text: > Could life on earth survive a large pole shift caused by an asteroid > collision? I became aware that there are people who believe that the earth's pole suddenly shifts. That is, its rotational axis changes rather than its magnetic axis. This is listed in wikipedia as follows: **Cataclysmic pole shift hypothesis** The cataclysmic pole shift hypothesis is the conjecture that there have been rapid shifts in the relative positions of the modern-day geographic locations of the poles and the axis of rotation of a planet. For the Earth, such a dynamic change could create calamities such as floods and tectonic events.   ...   The field has attracted pseudoscientific authors offering a variety of evidence, including psychic readings.   ...   http://en.wikipedia.org/wiki/Cataclysmic_pole_shift_hypothesis While watching a chess game at the local club a fellow member told me that he'd found the solution for where oil came from. It was long and involved. One of its features required that the earth's axis be changed by 90 degrees due to the collision of an asteroid a few million years ago. The idea was that the earth's axis used to be in the plane of the earth's orbit, but then it was knocked into its current rotation axis tilt of around 23 degrees. This changed the climate and buried a lot of stuff. The axis shift seemed to be an obvious weak link with the theory in that it involves the simplest physics. I told him that any asteroid large enough to change the earth's axis an appreciable amount (i.e. 70 degrees!) would cause so much damage as to destroy all life on the surface of the planet including its oceans. I'd have forgotten about it, but the other night, while listening to the radio, a guest appeared who had an idea that was similar. So this seems like a common target for pseudoscience. And so my question is this: Would an asteroid collision sufficient to immediately change the earth's rotation axis wipe out all life on the planet?


title: What is the difference between pole and running mass?

text: For example, when we meassure Higgs boson mass to be 125 GeV, do we think about renormalized or pole mass? Should the mass of the Higgs change if it is produced at higher energies?


title: Is there an analytical solution for fluid flow in a square duct?

text: I couldn't find one but assumed it must exist. Tried to find it on the back of an envelope, but got to an ugly differential equation I can't solve. I'm assuming a square duct of infinite length, incompressible fluid, constant pressure gradient. The flow is steady. I'm also assuming there's only flow down the duct (z direction). I get to here (seemed trivial, might still be wrong), then I'm stuck. $$ \frac{\partial^2 v_z(x,y)}{\partial x^2} + \frac{\partial^2 v_z(x,y)}{\partial y^2} = \frac{\Delta P}{\mu \Delta X} $$


title: Is the S-Matrix the only quantum field observable?

text: If the S-Matrix is the only observable, that rules out both generalized free fields and Wick-ordered polynomials of generalized free fields as interesting Physical models, because both result in a unit S-matrix. Neither possibility has been developed since the 1960s when these results were proved, and when the S-Matrix was ascendant as the only observable in Particle Physics. If the S-matrix is not the only quantum field observable, which it certainly seems not to be in Condensed Matter Physics and in Quantum Optics, to name just two fields in which Wightman or correlation functions play a large part in modeling, does that encourage us to construct Wick-ordered polynomials of generalized free fields as algebraic, non-dynamical deformations of free fields? In such a construction, the point is to avoid deforming a Hamiltonian or Lagrangian evolution, instead working with deformations of the observables, requiring for example that all observables and all states will be constructed as Wick-ordered polynomials in a Wick-ordered operator-valued distribution such as $\hat\Phi(x)=:\\!\\!\hat\phi(x)+\lambda\hat\phi^\dagger\hat\phi(x)^2\\!\\!:$. The Hamiltonian is taken to be a derived quantity in this approach. The S-Matrix seems remarkable, in that it requires us to set up a hyperplane at an initial time $t_I$ and another at a final time $t_F$, and another two hyperplanes between which the interaction will be nontrivial, at times $t_I<t_A<t_B<t_F$, all of which is not Lorentz-invariant, then take the limits $t_I\rightarrow-\infty,t_A\rightarrow-\infty,t_B\rightarrow+\infty,t_F\rightarrow+\infty$. It seems that in no other branch of Physics would we construct such an idealization and say it's the only possible way to report the experimental data. In other words, I wonder, going further, whether the S-matrix even is an observable? Particle Physics seems to be the only field in Physics that seems to think it is. **post-Answer addition.** It's Greenberg who proves in JMathPhys 3, 31(1962) that Wick-ordered finite polynomials in the field have a trivial S-matrix, and I think, without looking it up, that it's also Greenberg who introduced and ruled out generalized free fields. I've resisted doing something else because Wick-ordered polynomials in a generalized free field seem to give such a lot of freedom to create models that surely they can be useful, which I know is not a good reason, but I had to feel how bad it is before I could move on. I suppose I was almost ready to give up this way of doing things before I asked this question. While Tim van Beeks' response is definitely of interest, and Matt's Answer is clearly the "right" answer, but doesn't go into the periphery of the question in the right way to help me, it was Marcel's response that particularly pushed me. My comment to Marcel indicates the way I'm now going to take this, to functions of the field such as $\hat\Phi(x)=\tanh{(\hat\phi(x))}$. Insofar as we can say that the measured value of $\hat\phi(x)$ is almost always $\pm\infty$ in the vacuum state, because the expected value $\left<0\right|\hat\phi(x)\left|0\right>$ is zero and we can say, loosely, that the variance is $\infty$ in the vacuum state, presumably $\left<0\right|\delta(\hat\Phi(x)-\lambda)\left|0\right>$ is non-zero only for $\lambda=\pm 1$ (though with some worries about this construction). It's not clear that we even have to introduce normal-ordering. $\tanh{(\hat\phi(x))}$ is of course a bounded operator of the unbounded Wightman free field, whereas no nontrivial polynomial in the field $\hat\phi(x)$ can be a bounded operator. The particular choice of $\hat\Phi(x)=\tanh{(\hat\phi(x))}$ is clearly a particular coordinatization; if we change the coordinatization by taking a function of $\hat\Phi(x)$, we in general get a measurement operator that results in two discrete values of the field, mapping $\pm 1$ to $a,b$ respectively. If we take something different from the real Klein-Gordon field, my aim as I envisage it four hours after first imagining it is to map something like an $SU(3)$ invariant Wightman field, say, to a finite number of discrete values. If such an $SU(3)$ symmetry is unbroken, the relationships between each of the discrete values will all be the same, but if the $SU(3)$ symmetry is broken, there presumably has to be a coordinate-free way in which the relationships between the different discrete values are different. There will, I now suppose, have to be enough distinction between raising and lowering operators between different sectors of the theory and measurement operators to allow there to be a concept of an S-matrix. This may look crazy, but I'll also put on the table here why the structures of the Feynman graphs formalism encourage me. We introduce connected Feynman graphs at different orders to calculate $n$-point connected Wightman functions. Although we typically expand the series in terms of the number of loops, we can alternatively expand the series in terms of the number of points we have introduced between the $n$ points at which we measure; the extra points ensure that there are infinite numbers of different paths between the $n$ points. With something like $\hat\Phi(x)$, we introduce an infinite number of paths directly between the $n$ points, without introducing any extra points, so we need, in effect, a transformation of the superposition of an infinite number of Feynman path integrals into a superposition of an infinite number of weighted _direct_ paths between the $n$ points. Getting the weights on the direct paths right is of course rather important, and I also imagine the analytic structure has to be rather carefully done, particularly if I don't use normal-ordering. I'll do any amount of work to avoid renormalization, even in its modern gussied up form. It seems to me significant that the quantum field $\hat\Phi(x)=\tanh{(\hat\phi(x))}$ is not reducible. A number of proofs concerning Wightman fields rely on this property. If anyone else understands this (or reads this far) I'll be surprised. In any case I expect it will look very different a few years down the road if I ever manage to get it into a journal. Although I've worked in and around the Wightman axioms for the last few years, I find it interesting that I can now feel some pull towards something like the Haag-Kastler axioms. _Lots_ of work to do! Thank you all! Good luck with your own crazy schemes! **That's the question (and the to me unexpected state of play a day later). Completely separately, as an example** , _to show the way I'm going with this, hopefully_ (which, a day later, looks as if it will be only a background concern for the next little while, but I think not likely to be completely forgotten by me), the real-space commutator of the creation and annihilation operator-valued distributions of the free field of mass $m$ is, in terms of Bessel functions, $$C_m(x)=\frac{m\theta(x^2)}{8\pi\sqrt{x^2}}\left[Y_1(m\sqrt{x^2})+i\varepsilon(x_0)J_1(m\sqrt{x^2})\right]$$ $$\qquad\qquad+\frac{m\theta(-x^2)}{4\pi^2\sqrt{-x^2}}K_1(m\sqrt{-x^2})-\frac{i}{4\pi}\varepsilon(x_0)\delta(x^2).$$ If we take a weighted average of this object with the normalized weight function $w_{\alpha,R}(m)=\theta(m)\frac{R(Rm)^{\alpha-1} {\rm e}^{-Rm}}{\Gamma(\alpha)}, 0<\alpha\in \mathbb{R},\ 0<R$, $\int w_{\alpha,R}(m)C_m(x)dm$, we obtain the commutator of a particular generalized free field, which can be computed exactly in terms of Hypergeometric functions and which at space-like separation $\mathsf{r}=\sqrt{-x^\mu x_\mu}$ is asymptotically $\frac{\Gamma(\alpha+1)R^{\alpha}}{\pi\Gamma(\frac{\alpha+1}{2})^2\bigl(2\mathsf{r})^{\alpha+2}}$, and at time-like separation $\mathsf{t}=\sqrt{x^\mu x_\mu}$ is asymptotically $$\frac{\cos{(\frac{\pi\alpha}{2})}\Gamma(\frac{\alpha}{2}+1)R^\alpha} {4\sqrt{\pi^3}\Gamma(\frac{\alpha+1}{2})\mathsf{t}^{\alpha+2}} -i\frac{R^\alpha}{4\Gamma\left(-\frac{\alpha}{2}\right)\Gamma\left(\frac{\alpha+1}{2}\right)\mathsf{t}^{\alpha+2}},$$ except for $\alpha$ an even integer. At small space-like or time-like separation, the real part of this generalized free field is $-\frac{1}{4\pi^2x^\mu x_\mu}$, identical to that of the massless or massive free particle, independent of mass, but we can tune the 2-point function at large distances to be any power of the separation smaller than an inverse square law. On the light-cone itself, the delta-function component is again identical to that of the massless or massive free particle, independent of mass. There is of course an infinity of possible normalized weight functions, a half-dozen of which I have worked out exactly and asymptotically, and somewhat obsessively, by use of MAPLE and Gradshteyn & Ryzhik, though I've managed to stop myself at the moment. In a subsequent edit, I can't resist adding what we obtain if we use the weight function $w_{\mathsf{sm}[R]}(m)=\frac{\theta(m)R\exp{\left(-\frac{1}{mR}\right)}}{2(mR)^4},\ 0<R$ [using 6.591.1-3 from Gradshteyn & Ryzhik]. This function is smooth at $m=0$, and results at time-like separation in $$\left\\{Y_2\\!\left(\sqrt{\frac{2\mathsf{t}}{R}}\right)+iJ_2\\!\left(\sqrt{\frac{2\mathsf{t}}{R}}\right)\right\\} \frac{K_2\\!\left(\sqrt{\frac{2\mathsf{t}}{R}}\right)}{8\pi R^2} \asymp \frac{\exp{\left(-i\sqrt{\frac{2\mathsf{t}}{R}}+\frac{\pi}{4}\right)} \exp{\left(-\sqrt{\frac{2\mathsf{t}}{R}}\right)}} {16\pi\sqrt{R^3\mathsf{t}}}$$ and at space-like separation in $$\frac{K_2\\!\left((1+i)\sqrt{\frac{\mathsf{r}}{R}}\right) K_2\\!\left((1-i)\sqrt{\frac{\mathsf{r}}{R}}\right)}{4\pi^2R^2} \asymp \frac{\exp{\left(-2\sqrt{\frac{\mathsf{r}}{R}}\right)}} {16\pi\sqrt{R^3\mathsf{r}}}.$$ I wish I could do this integral for more general parameters, but hey! This weight function has observable effects only at space-like and time-like separation $\mathsf{r}<R$ and $\mathsf{t}<R$, so this is essentially unobservable if $R$ is small enough. The kicker is that this decreases faster than polynomially in both space-like _and_ time-like directions. The generalized free field construction always results in a trivial $n$-point function for $n>2$, however by introducing also Wick-ordered polynomials of these generalized free fields, we can also tune the $n$-point connected correlation functions, which are generally non-trivial and to my knowledge finite for all $n$. All this is far too constructive, of course, to prove much. I think I propose this more as a way to report the $n$-point correlation functions in a manner comparable to the Kallen-Lehmann representation of the 2-point correlation function than as something truly fundamental, because I think it does not generalize well to curved space-time.


title: How can I calculate the force that is applied on a tube by another tube?

text: Let's say there is two tubes(cylinders with no tops or bottoms) with charges $q_1$ and $q_2$, radii $b_1$ and $b_2$, lengths $l_1$ and $l_2$. These tubes are located along the axis of each other's surfaces like in this figure: ![two cylinders along the same axis, separated by a gap](http://i.stack.imgur.com/stO7E.png) If the electric field that the first tube creates on a point is; $$ E = \frac{q}{4\pi\varepsilon_0}\left(\frac{1}{\sqrt{b^2 + (c-a)^2}} - \frac{1}{\sqrt{b^2 + (c+a)^2}}\right) $$ where $b$ is the radius of the tube, $c-a$ is the distance between the centre of the furthest part of the tube and the point, $c+a$ is the distance between the centre of the closest part of the tube and the point, $q$ is the total cahrge on the tube and $\epsilon_0$ is the electric constant. Here is the figure of the tube and the point for those who didn't understand from my description: ![variable definitions for one tube](http://i.stack.imgur.com/ZqnNg.png) The question is how can I calculate the force between these two tubes? If I have to integrate something please show me how to do it since I have never taken a course on calculus. Update:The electric field formula I found is not true since it is valid for a point on axis of the cylinder. Thus I would be pleased if you could show me how to solve the problem from the beginning.


title: Is the force carrier of the magnetism in a common household magnet a photon?

text: As I have understood it, the Standard Model includes particles that carry the different forces, e.g. the electromagnetic (EM) force, the gravitational (G) force. When talking about EM fields such as visible light or microwaves, the associated particle is said to be the photon. But what about a static EM field without any electricity, like a common household magnet? How does that magnet communicate its force? Via photons?


title: The Effect of Tortoise Coordinates

text: Referring particularly to http://arxiv.org/abs/hep-th/9909056 in regard to the wave equation for Schwarzschild-AdS black holes (p.4), I'm trying to understand tortoise coordinates. So starting with the 4-dimensionalSchwarzschild-AdS metric in the general form $$ds^2=-f(r )dt^2+\frac{dr^2}{f(r )}+r^2(d\theta^2+sin^2 \theta d\phi^2),$$ if I want to find the wave equation $\Box \phi=0$ in the Schrodinger-like form. This is done by introducing the separation of variables $$\phi=\frac{\psi(r ) Y(\theta,\phi)e^{-i\omega t}}{r}$$ and then using the tortoise coordinate $dr_*=\frac{dr}{f(r )}$ to get $$(\partial_{r_*}^2+\omega^2-V(r_*))\psi=0.$$ But I don't fully understand what this tortoise coordinate really does. In fact when I go through these calculations myself, I use the transformation $$\psi'(r ) \to \frac{\psi'(r )}{f(r )}$$ and (fortunately) get the Schrodinger like form as in the paper above. However, they never explicitly state the potential and what I find is $$V(r_*)=\frac{-\ell(\ell+1) f(r )+rf'(r )}{r^2}.$$ where $\ell$ is the angular momentum mode. **But note** , in my transformation, I never mentioned $r_*$ and hence why my $V(r_*)$ doesn't actually mention an $r_*$. This is where my confusion lies. Is my potential right if I just replace the $r$ by $r_*$? i.e $$V(r_*)=\frac{-\ell(\ell+1) f(r_* )+r_*f'(r_* )}{r_*^2}?$$ (I highly doubt it.) And if not, how do I recover $V(r )$ from here? P.s. It would actually also be extremely helpful if someone knew $V(r )$, i.e. potential in original coordinates, for the Schwarzschild-AdS black hole.


title: How to differentiate between rotating frame and linearly accelerating frame?

text: Two friends, $A$ and $B$ are part of an experiment. $A$ is placed in a closed box and made to accelerate in free space at an acceleration $g$. $B$ is also placed in a closed box, but is made to rotate in a circle at uniform speed, such that the radial acceleration is also $g$. Can $A$ and $B$ perform some experiment from their boxes to tell who is moving radially and who is moving linearly?


title: Why does frozen water burst a pipe?

text: When water freezes in a pipe it can crack the pipe open. I assume this takes quite a lot of energy as when I try to crack a pipe it can be hard work! I think water freezing is a result of energy (heat) being lost from the water and out of the pipe into the freezing environment around it. So what energy is cracking the pipe and how? When warm and not frozen there is more energy in the pipe than when frozen? My secondary question might be - is this a particular phenomenon of water or would other matter crack open a pipe when it freezes solid from liquid? Andrew