title: Interpretation of contribution of gravitational potential energy to the gravitational field

text: In terms of General relativity we have as a matter of principle that anything that has inertial mass contributes to gravity. All forms of potential energy have inertial mass, it follows that the potential gravitational energy of a star must contribute to the gravitational field of that star. Depending on the amount of matter that is present, (the remains of) a dying star may contract to a white dwarf, a neutron star, or a black hole. Let me compare contraction to a white dwarf to contraction to a neutron star. As a thought experiment, let's say a known amount of gas is distributed in a region of galactic space, and something triggers that amount of gas to contract to a star. From a thought experiment point of view, how much of that star's gravity must be attributed to arising from its internal gravitational potential energy? Here's why I ask:   In the case of ending up as a neutron star there is significantly more contraction, as gravity overcomes the electron degeneracy pressure. It seems to follow that in the case of contraction to a neutron star _more potential gravitational energy is released_. Let's say that in this thought experiment the number of atoms (and types of atoms) of the star is known, and the flow of atoms/particles being ejected from the star is known. So theoretically you can give an expression for the amount of potential energy per unit of rest mass. The critical case is the Chandrasekhar limit, of course. How much gravitational potential energy should be attributed to a star that is just at the Chandrasekhar limit? In the real world you never know the amount of atoms in a star, of course. All you can assess is the strength of the gravitational field of that star (the amount of spacetime curvature.) * * * [Later edit] As pointed out by Trimok, I had failed to appreciate that there is no explicit expression for potential energy in the equations, explained by Stan Liou. I have rewritten the title of the question accordingly. So the question only touches interpretation of the theory. It would seem a star that is just above the Chandrasekhar limit will collapse deeper than an star just just below it. In the interpretation, does that make a difference as to the amount of potential gravitational energy that is to be attributed to the system?


title: What is the probability that all the air ends up in the upper right corner of the room and we suffocate

text: Since someone commented this on this question(What is the probability of ice in boiling water?), I would like to ask what is the probability that all the air ends up in the upper right corner of the room and we suffocate, considering that brain death happens after 10 minutes, and an average room has 40 cubic meters of air.


title: Police laser speed detection guns

text: ### Situation Police shooting a laser device into his rearview mirror, bouncing off the oncoming car license plate. I want to know if this second surface mirror, and the symmetric system--laser gun to mirror to license plate of moving car to mirror to laser gun---has a built in error because of the wavelength shift due to the glass of the mirror. Velocity and wavelength change for the raypath inside the mirror's glass, but would revert to the original wavelength for the "in air" portion of the raypath. ### Question Is this symmetry perfect and creates no error in measuring the car's speed?


title: is action integral Lorentz invariant?

text: I need to find the Lagrangian for charged particles in EM fields considering relativistic effects. Is action integral Lorentz invariant. $$A = \int_{t_1}^{t_2} L (q_i, \dot q_i, t) dt $$ According to my note > According to the first postulate of special relativity, the action integral > $A$ must be invariant because the equation of motion is determined by > extreme condition $\delta A = 0$. I do not understand how does this make $A$ invariant.


title: Heat transfer by conduction is not possible for gases?

text: Today one of my instructor told me that gases cannot have heat transfer thru conduction because molecules are far apart and so it cannot transfer heat,infact the diffusion process of gases transfers the heat. my thot at this point was,even though molecules are far apart,they transfer momentum to next molecules and we can feel the pressure and temperature(Kinetic energy).But why not heat? if we compress the gas at very high pressure, when molecules are very closer to each other, it will start transfer of heat by conduction? can someone shed some light on this?


title: Same U(1) charge for the SU(2) doublet

text: Consider the symmetry $SU_L(2)\otimes U_Y(1)$. The entries of $SU_L(2)$ doublet will have same U(1)-charge. How can this be shown mathematically?


title: Lepton number conservation and global phase transformation

text: Why the lepton number conservation is connected with the invariance of the lagrangian under global phase (U(1)) transformation of the wave function? How to distinguish global gauge phase and global "leptonic" phase? And how to build the operator of the leptonic number of particles if there isn't "local leptonic invariance"?


title: "Velvet way" to Grassmann numbers

text: In my opinion, the Grassmann number "apparatus" is one of the least intuitive things in modern physics. I remember that it took a lot of effort when I was studying this. The problem was not in the algebraic manipulations themselves -- it was rather psychological: _"Why would one want to consider such a crazy stuff!?"_ Later one just gets used to it and have no more such strong feelings for the anticommuting numbers. But this psychological barrier still exists for newbies. Is there a way to explain or motivate one to learn the Grassmann numbers? Maybe there is a simple **yet practical** problem that demonstrates the utility of those? It would be great if this problem would provide the connection to some "not-so-advanced" areas of physics, like non-relativistic QM and/or statistical physics.


title: trying to understand Bose-Einstein Condensate (BEC)

text: I am a computer scientist interested in network theory. I have come across the Bose-Einstein Condensate (BEC) because of its connections to complex networks. What I know about condensation is the state changes in water; liquid-gas. Reading the wikipedia articles on the subject is difficult, not because of the maths, but the concept doesn't seem to be outlined simply. Other source share this approach going straight into the topic without a gentle introduction to set the scene for the reader. I would appreciate a few paragraphs describing the problem at hand with BEC (dealing with gas particles right? which kind, any kind? only one kind? mixed types of particles? studying what exactly, their state changes?), what effects can occur (the particles can form bonds between them? which kind of bonds? covalent? ionic?), what do we observe in the BEC systems (some particles form many bonds to particles containing few bonds? The spatial configurations are not symmetric? etc), and what degrees of freedom exist to experiment with (temperature? types of particles? number of particles?) in these systems. Best,


title: Do molecular bounded systems shield or reduce neutron cross-sections?

text: When talking about neutron cross-sections, literature is usually investigating isolated cases of Neutron + Atom. Here, the abundance of hydrogen is dominating neutron fluxes through material. I wonder whether the reflection or capture propability of neutron radiation changes when the flux is penetrating a system of (organic) molecules. I can imagine two effects potentially reducing the neutron cross-section for bounded hydrogen:   1. The dense grid of organic material (folded proteins etc, imagine thick tree or a croud of humans) could be able to shield their hydrogen atoms from beeing struck by neutrons. Assuming organic material to be much denser (nuclei/volume) then other material like soil, rocks.   2. Neutrons are not able to transfer their whole energy to the hydrogen nucleus when it is electrically bounded (p is bounded to its electron, which is bounded to the electrons of the molecular system, which introduces inertia to the protons ability to move). Concluding that the capability of slowing down neutrons is reduced compared to free hydrogen. Is this complete non-sense or could there be a measurable influence to neutron fluxes when using (a) non-organic or (b) organic material, containing the same amount of hydrogen?


title: Why is magnetic field vector perpendicular to magnetic force vector?

text: So recently in physics class, we learned about the magnetism right hand rules. One of them states that the index finger points in the direction of the velocity of a particle, the middle finger points in the direction of the magnetic field, and the thumb points in the direction of the magnetic force. I'm curious why the magnetic field vector is perpendicular to the magnetic force vector.


title: How the Bloch sphere of a Hahn echo in NMR looks like? 90-t-90-t-echo

text: I have tried to find in the literature a proper nice and beautiful Bloch sphere to describe the trajectory of a nuclear spin, starting in z-axis, using a pulse sequence of an initial 90º pulse with zero phase (which means that the pulse is launched in the x-axis), wait for a time $\tau$, and then launch another 90º pulse with zero phase, and wait the same amount of time tau to measure an echo. Please notice that I am talking about Nuclear Magnetic resonance, and most of the books don't offer a Bloch sphere to explain this echo, which is supposed to be the first! echo, and only use a 90º and 180º pulse to explain that. I have read the original article of Hahn echo, but I saw that his diagram is not clear.


title: How to explain the weak force to a layman?

text: I'm trying to explain in simple terms what the weak interaction does, but I'm having trouble since it doesn't resemble other forces he's familiar with and I haven't been able to come up (or find on the web) with a good, simple visualization for it.


What experiment would disprove Friedmann model of cosmology? As a layman, I have read a lot of articles and threads in specialized forums. I am probably wrong, but I developed an impression that that theory is circular, 99.9% self- referential. The only hard experimental datum seems to be the redshift, and then most other parameters, distance, luminosity, age, scale factor, expansion rate, etc. are model-dependent and therefore confirm the theory, a) - what are the other hard facts at the bottom of the theory, if any? If I am not wrong, the prerequisite of a scientific theory is that it can be disproved. b) - how do you disprove that redshift is determined by stretching of space, c) - how do you disprove that space can expand and that just a metric devised by a theorist can make it expand? > spacetime is not an object expanding like some rubber foam. It's a > combination of a manifold and metric. The expansion means the metric is time > dependant, and this is explained by the requirement that the metric be a > solution to Einstein's equation.


title: Unit of gradient/slope?

text: So I have a graph: ![enter image description here](http://i.stack.imgur.com/BL8Xn.png) The value of the gradient/slope is $1.6±0.4$ and the value of the intercept is $0.9±0.4$. But what are the units of the graph? Is the unit of the gradient $v^2M^{-1}$? What about the unit of the intercept?


title: How much work will a human do to lift a weight?

text: I saw a pretty simple homework question here that asked how much work it takes to lift a 200 kg weight, and while the math for a basic answer is simple the weightlifter in me instead wanted to actually approach this question with a critical and pragmatic eye. There are many factors that one could consider to easily get a value different from the answer in a conservative field (for example: 1960 N for 1 meter). I figure exact answers are obviously impossible. I want to instead see what methodologies can be used to start formalizing my upcoming reasoning (and also take this opportunity to correct any misconceptions I have) in hopes of closing in on a reasonable method to create an answer with more data. I also would love to see similar bio-mechanical studies and related material. I mainly broke this problem up into two levels: Mechanical: The human body is a collection of support structures (basically curved trusses), complex muscular tissue (non-linear springs with various attachment points), and connective tissue (constraints). Something I really could use clarification about here is the effect duration of the lift could have on the work exerted. Thermodynamic: Our body metabolizes chemicals to generate store energy in tissue, then later break down that tissue into work and heat. Let's ignore the energy lost due to digestion and other secondary factors, and instead rate the efficiency of the human "engine" as a function of how much work he creates from the energy he expends doing the exercise. The Problem: Let's say we have a (large) man who lifts a bar 1 meter up from the earth when he does a deadlift, squat, or bench press. How much energy does this body use to move 200kg doing each motion? How much energy does this body expend to move 200kg doing each motion?


title: Why does everything spin?

text: The origin of spin is some what a puzzle to me, everything spin from galaxies to planets to weather to electrons. Where has all the angular momentum come from? Why is it so natural? I was also thinking do photons spin? we always think of the wave as a standard 2d sin wave but could this rotate in 3d? What implications would this have? And what about spacetime how does all the spinning effect? This has always been avoided in all lectures and classes I ever went to.


title: Why does each celestial object spin on its own axis?

text: AFAIK all the celestial objects have a spin motion around its axis. What is the reason for this? If it must rotate by some theory, what decides it's direction and speed of rotation? Is there any object that does not rotate about its axis?


title: Why planets are rotating only in one plane?

text: Since gravity is three dimensional why planets are rotating only in one plane around sun.


title: Why does earth spins?

text: I understand the governing force causing the earth's cycle around the sun is gravity. It can be described by Keplar's law. But what causes the earth's spin? what is its governing law?


title: Why does Venus rotate the opposite direction as other planets?

text: Given: Law of Conservation of Angular Momentum.   * Reverse spinning with dense atmosphere (92 times > Earth & CO2 dominant sulphur based).   * Surface same degree of aging all over.   * Hypothetical large impact is not a sufficient answer. Assuming any object large enough to alter a planets rotation or even orbit would likely destroy most of its shape, yet Venus has retained a spherical property with a seemingly flat, even terrain indicating no volcanoes,and few if any visible meteor impacts. It would be fragmented and dispersed for billions of years. Even the question of what meteor, comet, asteroid composition could survive traveling that close to the sun's temperature, radiation, electromagnetic energy, solar flares, or gravity to equal a mass reactionary change as to alter it's spin.


title: Why do we have a galactic and a solar plane

text: Why do all of the planets in our solar system move is (more or less) the same plane? It would seem that the planets should all have very different orbital planes, each at a different angel to each other. The fact that all of their orbits are so closely parallel suggests that a force is pulling and/or maintaining the orbits of the planets in a flat plane. What is that force? A similar force also seems to be working at a galactic level too.


title: Why are the orbits of the planets in our solar system along the same basic plane?

text: > **Possible Duplicate:**   >  Why are our planets in the solar system all on the same disc/plane/layer? After watching this video I realized that the orbits of the planets in our solar system basically reside along a similar 'flat' plane: ![enter image description here](http://i.stack.imgur.com/q4O5U.gif) Meaning that the angles of orbit are no greater than 20 degrees (if you still count Pluto as a planetary object). Why are there none of the planets at a 45 or even 90 degree orbit (in relation to the earth)? Is this due specifically to the gravitational pull from the Sun?


title: Single plane Ring system

text: > **Possible Duplicate:**   >  Why are our planets in the solar system all on the same disc/plane/layer? I've noticed this in many pictures, Planets are shown with a single ring around them (in some particular plane). Taking extreme case... As gravity should act in all the directions, such planets must be covered with asteroids all around them. Not just _single_ ring in some _single_ plane..! So, My question is: Why don't planets have many rings instead of just a single ring..?


title: Why the galaxies forms 2D plane (or spiral-like) instead of 3D ball (or spherical-like)?

text: > **Question** : As we know (1) the macroscopic spatial dimension of our > universe is 3 dimension, and (2) Gravity attracts massive objects together > and the gravitational force is isotropic without directional preferences. > **Why do we have the spiral 2D plane-like Galaxy(galaxies), (or more common > than), instead of spherical, 3D ball or elliptic like galaxies?** **Input** : Gravity is (at least, seems to be) **isotropic** from its equation of force law (Newtonian gravity). It should show **no directional preferences** from the form of force vector $\vec{F}=\frac{GM(r_1)m(r_2)}{(\vec{r_1}-\vec{r_2})^2} \hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically. If the gravity attracts massive objects together **isotropically** , and the macroscopic space dimension is **3 dimension** , it seems to be natural to have a spherical or ball like shape of massive objects gather together. Such as the Globular clusters, or **GC** , are roughly spherical groupings Star cluster, and its Wiki picture: ![Star cluster](http://i.stack.imgur.com/KKEUZ.jpg) However, my impression is that, even if we have observed some more-spherical or more-ball-like Elliptical galaxy, it is more common to find more-planer Spiral galaxy? ( **Is this statement correct? Let me know if I am wrong.** ) Such as our Milky Way Also, such as this NGC 4414 galaxy: ![Galaxy](http://i.stack.imgur.com/jrklh.jpg) **Is there some physics or math theory explains why theGalaxy turns out to be planer-like (or spiral-like) instead ball-like or spherical-like?**


title: Why does the earth rotate?

text: > **Possible Duplicate:**   >  Why does every thing spin? So why would the earth, or any planet for that matter, rotate along an axis? I know of no force which could come into play here, so i assume it just started off with an initial rotation. But where did this rotational energy come from? And are there any known planets which do not rotate? Assuming no force accounts for earth's rotation, would it be true that the angular velocity of the earth would decrease(by a very very small amount), when bombarded by meteors(Due to an increase in the moment of inertia by the increase in mass of the earth i.e. the additional mass of the meteors)?


title: On the origin of the rotation of celestial bodies

text: > **Possible Duplicate:**   >  Why does Venus rotate the opposite direction as other planets?   >  Why does every thing spin? As far as I can imagine, almost each celestial body, star, planet, solar- systems, galaxies do rotate on their center. Where this come from ? Is it the normal work of gravity? And why Venus has a retrograde rotation? (Or, why all the solar systems rotate in the same way - except Venus?)


title: why galaxies look like discs rather than spheres?

text: Black holes have much gravitation to hold all stars and nebulas but why they are aligned in disc type shapes rather than spheres because gravitation is everywhere around black holes. Even on upper and lower sides. Even if rotation is the cause why cant they move above black holes in smaller circles.


title: Why are our planets in the solar system all on the same disc/plane/layer?

text: I always see pictures of the solar system where our sun is in the middle and the planets surround the sun. All these planets move on orbits on the same layer. Why?


title: Rotating uniform rod

text: > A uniform rod of mass 1.2 kg and length 1.8 m is pivoted in the horizontal > position as shown (black point).![enter image description > here](http://i.stack.imgur.com/cGmYy.png) The rod is at rest and then > released. The acceleration due to gravity is $g = 9.8 m/s^2$. 1) Find the > rotational inertia of the rod relative to the axis perpendicular to the > screen and passing through the pivot point. 2) What is the angular speed (in > rad/s, but do not include units) of the rod as it passes through the > vertical position (when end marked B is at the bottom)? My attempt of solution: To find moment of inertia, I use the moment of inertia of a uniform rod respect to it's center of mass (middle in this case), which is $$I_{cm}=mL^{2}/12$$ where $m$ is mass of the rod and $L$ its length. Then I use the parallel axis theorem to find the moment of inertia in this case: $$mL^{2}/12+0.45^{2}m=0.567$$ For the 2º part, since the net torque on the rod is not constant I use potential energy to find first the speed of center of mass at the vertical position. The initial height of center of mass is 0.45 m taking as reference point position of the middle point of the rod at vertical position. So: $$mg0.45=1/2mv_{cm}^{2}+1/2I\omega ^{2}=1/2mv_{cm}^{2}+1/2I(v_{cm}^2/0.45^{2})$$ From here I get $v_{cm}$ and later the angular velocity. Where am I wrong?


title: Will an air-filled tire normally gravitate toward being nitrogen-filled just by refilling with air?

text: A big argument by the nitrogen-in-the-tire crowd is that: > _Nitrogen atoms are bigger and thus less likely to escape the tire, bringing > stability to your tire pressure._ If Earth's atmosphere is %78.084 percent nitrogen, then the non-Nitrogen composition is ~22%. If this 22% is more likely to seep out of the tire, and Nitrogen doesn't seep out, then simply filling the tire up with air will maintain the original 78% N, plus the added nitrogen which would be 78% of the remaining 22% (assuming total seepage). Simply, won't a tire that's been filled up 10 times in the course of its normal life already be disproportionately Nitrogen if this argument is true?


title: Why does the light intensity increase as I approach a distance light source?

text: Analogy: assume that I have constant rain fall and I have a water bucket to collect this rain. If I am rest relative to the earth, I will catch a certain amount of rain. However, if I now move towards the rain, I will increase the amount of water I collect. Now I want to apply this idea to a source emitting photons instead of rain drops. I image a distance light source (distance star) that is emitting a constant number of photons such that I can ignore the $1/r^2$ fall off of the intensity (that is, the intensity does not change very much over some appreciate distance). So instead of a water bucket, I now have a light collector that measures light intensity. Here is my question: **Does the light intensity measured by my light collector that is moving towards the distant star increase or stay the same when compared to a light collector at rest realtive to the distant star?** My first thought was that the light intensity would increase because the light is blue shifted and higher energy photons will therefore produce a higher intensity. However, I believe that there might be another contribution due to length contraction. Since an observer moving towards a light source has to account for length contraction, does this mean that there is an increase in photon density? If so, this higher photon density in the frame of the light collector will also contribute to increasing the light intensity. Can someone verify or correct my thinking.


title: Generalization of relation between charge and magnetic moment

text: In classical theory, for electrons or protons, charge and angular momentum combine to give the magnetic moment. Does a similar consequence hold for the generalization of charge to other forces, like the strong force or weak force?


title: Doubt in collision

text: > _If a ball under the influence of gravity falls straight down from a height > $h$, collides elastically with the floor, at the instant of collision, what > forces does it experience?_ Shouldn't the floor exert a force equal to the weight of the ball, on the ball itself? Then won't the forces cancel out?(since the only to forces on the ball are it's weight and the force of the floor, which are equal and Opposite) I know that the better way to find resulting velocity, etc. Would be to use work-energy or kinematics, but I would like to reconcile those with the force treatment


title: Accelerating expansion of the universe: so?

text: Please help me out, I’m missing something. We know that, right now, space is expanding at roughly 73km/s/Mpc. This means: two points in space 1Mpc away from each other “move” 73 km farther away every second. Of course they are not actually moving, it’s the space between them that’s expanding: that’s why two objects can drift apart faster that the speed of light if they are far away enough). Now: scientists tell us that the universe is not just expanding, but that it’s expansion is accelerating. Initially my understanding of the accelerating expansion was: if now it’s 73km/s/Mpc, at some point in the future it will be 74, 75, …. But it turns out it’s not like that. It turns out that that 73 is actually DECREASING (which is why the Hubble sphere is expanding) and that “accelerating expansion of space” means something different. Apparently it means “accelerating growth of the scale factor” $a(t)=\frac{d(t)}{d0}$ (where $d(t)$ and $d0$ are the proper distance between two points at time t and time 0). So I understand: “accelerating universe” just means that $a’’(t)>0$ (i.e. the proper distance between two points changes over time at in increasing rate), while the Hubble parameter $H(t)=a'(t)/a(t)$. decreases over time (so at some point in the future two points in space will “move” away from each other at, say, 65km/s/Mpc instead of 73). In other words: the proper distance between the two objects increase faster and faster as the distance between them increases, BUT every single Mpc between them actually increases (slightly) SLOWER. BUT: if everything I’ve said is correct (and please tell me if it’s not), I don’t get this: If “accelerating universe” does NOT mean “increasing Hubble parameter”, but it just means that the proper distance between two objects increase at an increasing rate WHAT’S THE BIG DEAL? I mean: the very basic fact that space is expanding everywhere implies that as time pass two objects will drift away faster and faster, since their recession velocity increase with their distance. What am I missing?


title: Is heat added to an air bubble rising in water?

text: An air bubble was initially released from the bottom of an ocean. Assume that the air in the air bubble is an ideal gas and its **temperature remains constant** at 25 degrees Celsius. Is heat added or removed from the air bubble as the bubble rises? My reasoning is that since total internal energy, $U \propto T$, the total internal energy must have remained constant. But from the laws of thermodynamics, $U = W + Q$. Since $pv = nRT$ remains constant, no work is done on the bubble and hence $W = 0$. It follows that $Q$ must be equals to $0$, i.e. heat is neither removed nor added to the air bubble. Is my reasoning sound?


title: Has a theory of a small Big Bang in a much larger universe been proposed?

text: I have a theory I think might explain why the universe has accelerated in a way that doesn't require dark energy. I'm wondering if someone has proposed this theory before (did some research and couldn't find anything). The theory is that the big bang was essentially a huge super-nova-like eruption inside an even larger universe. Just like the visible universe has super novae that happen inside much much larger galaxies and dust clouds, so could the Big Bang have been a huge explosion in a much huger universe. The acceleration we have seen evidence of could, then, have been caused if the big bang was off center in the huge amount of matter in the outer universe - ie if one side exerted more gravity than the other, mass closer to that side would accelerate away from mass closer to the center of the big bang. I imagine a huge black-hole sucking in a giant swath of matter from the greater universe, coalescing into a small area that leaves a large space around it rather empty. Then when it explodes (or perhaps it just emitted large amounts of matter out of it via something like hawking radiation), it fills this space again with what we now see as our universe. This theory doesn't seem to require any sort of bizarre unknown physics, like dark energy, singularities or anything else that causes divide by 0 errors in established physical equations. Has it been proposed before? Also, I'd be interested to know if anyone has any concrete reasons why this theory wouldn't work.


title: Reference request: Hollow concave Earth hypothesis

text: Reference request concerning the Hollow concave Earth hypothesis. I am searching for this paper: > _A Geocosmos: Mapping Outer Space Into a Hollow Earth_ authored by M. > Abdelkader and published in Speculations in Science & Technology (vol. 6 pg > 81–89) in 1983. I would be very grateful if anyone could provide me with a link to the paper. NOTE: The paper discusses the mapping of all of space into a spherical body of earth's volume via non-Euclidean geometry. I have spent quite a bit of time searching for the paper. I heard that one of the AMS journals published between 1982-1983 contained the abstract. The paper has been cited once or twice, but there are no links available; vol. 6 has not been placed online as far as I can see.


title: Stellar Viscosity in Galaxies

text: Is there such as thing as the viscosity of stars in a galaxy, along the lines of gravitational attraction between stars changing the dynamics. If so, how is that put in terms of the Virial Theorem?


title: Hamilton's characteristic and principle functions and separability

text: Just hoping for some clarity regarding Hamilton's characteristic function (W). When we take a time independent Hamiltonian we can separate the Principle function (S) up into the characteristic function minus $ht$, yes I know its the Legendre transform but, \begin{equation} W=S+ht \end{equation} Meirovitch in his _Methods of Analytical Dynamics_ p 356 gives $h$ as being the Jacobi energy function $h$ as defined in earlier chapters both his and Goldstien's texts. This is the only time I have seen it being called $h$ rather than the Hamiltonian. I was just wondering if anyone had read through it and perhaps noticed something different in Meirovitch's definition that escapes me. Most authors define these integration constants as the Hamiltonian instead. I know the difference is subtle but it is intriguing as to why he chose $h$ not $H$! Is it just down to how you express the conjugate momenta?


title: Classical Field Theory - Continuum limit in forming the Lagrangian density and the elasticity modulus

text: I have been looking at taking the continuum limit for a linear elastic rod of length $l$ modeled by a series of masses each of mass $m$ connected via massless springs of spring constant $k$. The distance between each mass is $\Delta x$ which we use to express the total length as $l=(n+1)\Delta x$. The displacement from the equilibrium position is given by $\phi(x,t)$. The discrete Lagrangian in terms of the $i$th particle $\mathscr L$ is composed as follows, \begin{equation} \mathscr L=\frac{1}{2}\sum _{i=1}^{n}m\dot \phi _i^2-\frac{1}{2}\sum ^n _{i=0}k(\phi_{i+1}-\phi _i)^2 \end{equation} At this point we take the continuum limit such that the number of masses in the fixed length of rod tends to infinity and correspondingly the inter- particle distance tends to zero. It is fruitful to multiply top and bottom by $\Delta x$ such that we can define two quantities that remain constant during this limit namely the linear density ($\mu=m/\Delta x$) and the elastic modulus ($\kappa=k\Delta x$). \begin{equation} \mathscr L=\frac {1}{2} \sum _{i=1}^{n}\Delta x\bigg(\frac{m}{\Delta x}\bigg)\dot {\phi} _i^2-\frac {1}{2} \sum _{i=0}^{n}\Delta x(k\Delta x)\bigg(\frac{\phi _{i+1}-\phi _i}{\Delta x}\bigg)^2 \end{equation} It is easy to see why the linear density remains constant since both the number of masses per unit length increases while simultaneously the unit length decreases. However my question is regarding the elastic modulus, I fail to see how it remains constant in this limit. The argument goes as follows; Since the extension of the rod per unit length is directly proportional to the force exerted on the rod the elastic modulus being the constant of proportionality. The force between two discreet particles is $F_i=k(\phi _{i+1}-\phi _i)$, the extension of the inter particle spacing per unit length is $(\phi _{i+1}-\phi _i)/\Delta x$. Therefore (HOW) $\kappa=k\Delta x$ is constant. Its easy to relate them but why is it constant!?!


title: Is there a chance an object isn't where we think it is?

text: At first, I know the question sounds ambiguous and maybe pseudo-scientific, but it's a thing I've been arguing about with my colleage for quite some time and while neither of us knows much about quantum mechanics besides the popular statements you see in internet, it has been bugging me to finally end the argument. Here's the point: let's say there's a coin lying on the ground. He says, that there is a very small chance, but still a chance, that the coin isn't really there, but, say, 5 meters away. That's because there's a probability of finding an electron 5 meters away and thus, a probability of finding all of the object's particles 5 meters away. I'm not really sure if there IS a probability of finding an electron that far away, since I've always imagined the electron cloud as something existing around the nucleus but certainly not reaching infinity. Bet even so, doesn't the fact that I SEE the coin determine that the coin is in fact there? I'll never know the exact position of all the particles, but the uncertainty is small, for sure not as big as 5 meters. I'm sorry if I offended anyone by this question that probably seems stupid to many, that wasn't my intent.


title: Formula for getting time of sunrise at a particular location?

text: Is there a formula that can get sunrise at a particular latitude and longitude (and elevation from sea level)? If so, what it is? (Why does it work?)


title: Do the standard cosmology models spontaneously break Lorentz symmetry?

text: In standard cosmology models (Friedmann equations which your favorite choice of DM and DE), there exists a frame in which the total momenta of any sufficiently large sphere, centered at any point in space, will sum to 0 [1] (this is the reference frame in which the CMB anisotropies are minimal). Is this not a form of spontaneous Lorentz symmetry breaking ? While the underlying laws of nature remain Lorentz invariant, the actual physical system in study (in this case the whole universe) seems to have given special status to a certain frame. I can understand this sort of symmetry breaking for something like say the Higgs field. In that situation, the field rolls down to one specific position and "settles" in a minima of the Mexican hat potential. While the overall potential $V(\phi)$ remains invariant under a $\phi \rightarrow \phi e^{i \theta}$ rotation, none of its solutions exhibit this invariance. Depending on the Higgs model of choice, one can write down this process of symmetry breaking quite rigorously. Does there exist such a formalism that would help elucidate how the universe can "settle" into one frame ? I have trouble imagining this, because in the case of the Higgs the minima exist along a finite path in $\phi$ space, so the spontaneous symmetry breaking can be intuitively understood as $\phi$ settling **randomly** into any value of $\phi$ where $V(\phi)$ is minimal. On the other hand, there seems to me to be no clear way of defining a formalism where the underlying physical system will randomly settle into some frame, as opposed to just some value of $\phi$ in a rotationally symmetric potential. [1] The rigorous way of saying this is : There exists a reference frame S, such that for all points P that are immobile in S (i.e. $\vec{r_P}(t_1) = \vec{r_P}(t_2) \forall (t_1, t_2)$ where $\vec{r_P}(t)$ is the spatial position of P in S at a given time $t$), and any arbitrarily small $\epsilon$, there will exist a sufficiently large radius R such that the sphere of radius R centered on P will have total momenta less than $\epsilon c / E_k$ (where E is the total kinetic energy contained in the sphere). * * * Ben Crowell gave an interesting response that goes somewhat like this : Simply put then : Causally disconnected regions of space did not have this same "momentumless frame" (let's call it that unless you have a better idea), inflation brings them into contact, the boost differences result in violent collisions, the whole system eventually thermalises, and so today we have vast swaths of causally connected regions that share this momentumless frame. Now for my interpretation of what this means. In this view, this seems to indeed be a case of spontaneous symmetry breaking, but only locally speaking, because there should be no reason to expect that a distant causally disconnected volume have this same momentumless frame. In other words the symmetry is spontaneously broken by the random outcome of asking "in what frame is the total momentum of these soon to be causally connected volumes 0?". If I'm understanding you correctly, this answer will be unique to each causally connected volume, which certainly helps explain how volumes can arbitrarily "settle" into one such frame. I'm not sure what the global distribution of boosts would be in this scenario though, and if it would require some sort of fractal distributions to avoid running into the problem again at larger scales (otherwise there would still be some big enough V to satisfy some arbitrarily small total momentum).


title: Time Energy symmetry in General Relativity (not asking about energy conservation)

text: In General Relativity is there a TE symmetry similar to CPT symmetry in the Standard Model ? It's pretty easy to understand that by flipping charge and parity you merely get a time reversed equivalent of your system, so flipping time as well would lead to an equivalent description. Similarly, since metric perturbations in GR are sourced by energy density, it seems to me that GR _is_ invariant if we operate the transformation $(t,\rho) \rightarrow (-t,-\rho)$. Is this correct or am I missing something here ? Do the other $T_{\mu \nu}$ terms come into play in ways which I haven't considered here ? Is the symmetry actually $(g_{\mu \nu},T_{\mu \nu}) \rightarrow (-g_{\mu \nu},-T_{\mu \nu})$ ? The reason I think this might work is if you make the energy density negative in the Friedmann equations expansion turns into contraction, then you time flip it and it turns back into expansion. Is this a ubiquitous behavior ? If this is indeed generally true can we say there is indeed an ET symmetry ?


How do you prove the second law of thermodynamics from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide.


title: Mathematical proof of the Second Law of Thermodynamics

text: Is there some book or paper that formalizes statistical mechanics, like some people have done with relativity, and proves the second law of thermodynamics from more foundational axioms?


1. What is the relation between entropy and mass of black hole?   2. And what is the relation between symmetry of physics operation and entropy?For instance,measuring or doing measure on state of quantum system especially pure state of a system has to lead to increase in entropy of the system,and the measuring is not symmetric operation,but it seems that the statement is tautologic,because that operations make processes symmetric,means nothing has been changed.


title: metal - superconductor surface effect

text: What happens when You make a contact junction between normal metal with some fermi energy $E_{F}$ and superconductor with fermi level $E_{SC}$, but this levels are not necessarily the same ? There is some analogy between p-n junction or contact potential in metals?


title: What makes a Feynman diagram real or virtual?

text: Simple question: as the title says, what makes a real Feynman diagram real, and what makes a virtual diagram virtual? Or in other words, how do I tell whether any given diagram is real or virtual? I've never gotten a really satisfying explanation of this. I would imagine it has something to do with virtual particles, but all internal propagators are virtual particles and I know for a fact that having internal lines doesn't make a diagram virtual.


title: Rationale for writing wave function as product of independent wave functions

text: When solving Schrödinger's equation for a 3D quantum well with infinite barriers, my reference states that: $$\psi(x,y,z) = \psi(x)\psi(y)\psi(z) \quad\text{when}\quad V(x,y,z) = V(x) + V(y) + V(z) = V(z).$$ However, I cannot find any rationale for this statement. It may be obvious, but I would appreciate any elucidation.


title: What does really attracts a water stream to a charged object?

text: I saw this video by Varitasium and I was not 100% sure about the third phenomena, a statically charged object attracting a stream of water, especially because he explicitly mentioned that it is not due to the permittivity of the water. A week later he uploaded an video explaining all five phenomena and he says it is because of repelled equally charged ions, back up the water flow, such that the water actually falling along the charged object would be oppositely charged and therefore attracted. Here is a still image from the video illustrating this: ![enter image description here](http://i.stack.imgur.com/EbW0h.jpg) However this explanation seems strange to me, since this would mean that this way you could build up a charge in the water which has exited the tap (as well as in the water inside the tap as well). Therefore it should get harder to push those ions back over time, and therefore there also would be a decrease in the attraction of the water (assuming there is no other way to discharge it). I have tried a simple similar experiment, where I catched the water leaving the tap in an insulated bowl, but I did not notice a decrease in attraction. There a couple of reasons why my experiment does not rule out his theory:   * I was holding the charged (plastic) object by hand, which caused that its distance to the water stream varied, making it difficult to observe a decrease in attraction.   * I only conducted my experiment for roughly 30 seconds, which might be enough time for a opposite charge to build up. The reason that he mentioned why permittivity can't cause this phenomena is because the electric field would be homogeneous and therefore the attractive and repulsive force of both dipoles of a water molecule would cancel out. However the electric field would only be homogeneous if the charged object would be an infinitely large plate, which it isn't. Thus water molecules would be attracted to the charged object. In order to make the permittivity theory more plausible I thought of maybe doing a order of magnitude calculation to see if it could generate a large enough force would be able the water stream by the observed amount. However I do not know what a good guess would be for variables such as the amount charge on the charged object and how the electric field would fall off over distance. So, my question is whether his theory is really correct, or if the permittivity of the water can cause it.


title: Xray compression of secondary in hydrogen bomb

text: Accounts of the "secret" of the hydrogen bomb describe Xrays from a primary fission explosion reflecting off of the bomb case (occasionally passing through polystyrene foam) and compressing and heating the fusion fuel. No mention is made of gamma rays doing the same thing. Is there a physical explanation for why X-rays are able to compress the secondary, but gamma rays are not? Does the primary fission reaction produce gamma rays as well as x-rays?


title: Small oscillations of heavy string

text: I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points. First of all I wrote down this Lagrangian: $$S=\int dt ds \left[\frac{\rho}{2}(\dot{x}^2+\dot{y}^2)-\rho g y(s,t)+\frac{\lambda(s,t)}{2}\left(\left(\frac{\partial x}{\partial s}\right)^2+\left(\frac{\partial y}{\partial s}\right)^2-1\right)\right]$$ This Lagrangian describes heavy string with fixed ends in gravitational field. Where $\rho$ is density, $g$ is gravitational acceleration, $s$ is natural parameter. So I have 3 equations from Euler-Lagrange equations. $$\rho\ddot{x}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial x}{\partial s }\right)=0$$ $$\rho\ddot{y}+\frac{d}{ds}\left(\lambda(s,t)\frac{\partial y}{\partial s }\right)+\rho g=0$$ $$\left(\frac{\partial x}{\partial s }\right)^2+\left(\frac{\partial y}{\partial s }\right)^2=1$$ After that I've found stationary solution ($\frac{\partial x}{\partial t}=\frac{\partial y}{\partial t}=\frac{\partial \lambda}{\partial t}=0$). (I just put $\ddot{x}=\ddot{y}=0$) $$y_0(x)=-\frac{C_1}{\rho g}\cosh\left(\frac{\rho g x}{C_1}+C_2\right)$$ Where $C_1,C_2$ is integration constants (depends on positions of ends of string). And $\cosh(x)$ is hyperbolic cosine. To study small oscillations I've tried to use pertrubation theory. So, I put $$y(s,t)=y_0(s)+\bar{y}(s,t)$$ $$x(s,t)=x_0(s)+\bar{x}(s,t)$$ $$\lambda(s,t)=\lambda_0(s)+\bar{\lambda}(s,t)$$ But after that I get difficult differential equations, which I can't solve. **Maybe someone know the more simplier aproach to solve this problem or know how to solve it in this way?**


title: Does the color of a quark matter in a meson?

text: QCD and confinement specify that hadrons must be color-neutral. My understanding is that this means you can have mesons (quark + antiquark) or baryons with 3 quarks, one of each color: Red+green+blue=neutral. In a meson, does the color of the quark and antiquark matter? Must a red quark be paired with a red antiquark; or does the color change constantly due to the exchange of gluons which also carry color charg?


title: What is physics behind of explosion under Atmospheric pressure?

text: An explosion is a rapid increase in volume and release of energy in an extreme manner. A blast wave in fluid dynamics is the pressure and flow resulting from the deposition of a large amount of energy in a small very localised volume. The equation for a Friedlander waveform describes the pressure of the blast wave as a function of time: $$P(t)=P_oe^{-\frac {t}{t^*}}(1-\frac {t}{t^*})$$ where P$_o$ is the peak pressure and t$^*$ is the time at which the pressure first crosses the horizontal axis (before the negative phase). My **Question:** Why does the pressure temporarily drop below ambient after the wave passes?


title: Saturation of the Cauchy-Schwarz Inequality

text: Going to as little details as possible, here is a statement from Wald's text on QFT in curved spacetimes(I am not quoting the book) He considers two vector spaces ${\cal S}$ and ${\cal H}$. > Note - For more details about ${\cal S}$ and ${\cal H}$, read this box. I > believe for most part of the question, the following details are irrelevant, > but I will provide them nonetheless. Otherwise, skip below. > > He starts by considering the solution space of a classical system ${\cal S}$ > with symplectic structure $\Omega$. This has a natural vector space > structure. He complexifies it to ${\cal S}^{\mathbb C}$ and extends $\Omega$ > to ${\cal S}^{\mathbb C}$ by complex linearity on each variable. He then > defines the map $(\cdot, \cdot): {\cal S}^{\mathbb C} \times {\cal > S}^{\mathbb C} \to {\mathbb C}$ on ${\cal S}^{\mathbb C}$ as $$ (y_1, y_2) = > - i \Omega( \overline{y_1}, y_2) $$ This satisfies all the properties of an > inner product except positive-definiteness. He then considers the subspace > ${\cal H}$ of ${\cal S}^{\mathbb C}$ on which the inner product above is > positive-definite. (There are of course many such choices of ${\cal H}$. Any > one of them will do.) He then shows that there is a one-one onto map $K: {\cal S} \to {\cal H}$. He shows that one can define a real inner product $\mu: {\cal S} \times {\cal S} \to {\mathbb R}$ on ${\cal S}$. He then goes on to show that one can use this to define a complex inner product on $\cal H$ as $$ \left( K y_1, K y_2 \right)_{\cal H} = \mu(y_1, y_2) - \frac{i}{2} \Omega(y_1, y_2)~\forall~y_1, y_2 \in {\cal S} $$ where $\Omega: {\cal S} \times {\cal S} \to {\mathbb R}$ is an antisymmetric function on ${\cal S}$, i.e. $\Omega(y_1, y_2) = - \Omega(y_2, y_1)$. He then uses the Cauchy-Schwarz Inequality for ${\cal H}$. This reads $$ \left( K y_1, K y_1 \right)_{\cal H} \left( K y_2, K y_2 \right)_{\cal H} \geq \left| \left( K y_1, K y_2 \right)_{\cal H} \right|^2 \geq \left| \text{Im} \left( K y_1, K y_2 \right)_{\cal H} \right|^2 $$ Expanding it out, he writes $$ \mu(y_1, y_1) \mu(y_2, y_2) \geq \mu(y_1, y_2)^2 + \frac{1}{4} \Omega(y_1, y_2)^2 \geq \frac{1}{4} \Omega(y_1, y_2)^2 $$ More specifically $$ \boxed{ \mu(y_1, y_1) \mu(y_2, y_2)\geq \frac{1}{4} \Omega(y_1, y_2)^2 } $$ Now, here is the statement that confuses me > **Indeed, since $K$ is one-to-one and onto and since the Schwarz inequality > on ${\cal H}$ always can be ``saturated", we obtain the following stronger > version of the last inequality: For each $y_1 \in {\cal S}$ we have** $$ > \mu(y_1, y_1) = \frac{1}{4} \max_{y_2 \neq 0} \frac{ \Omega(y_1, > y_2)^2}{\mu(y_2, y_2)} $$ Here's my question **Q. Where did he get the above > expression from?** He seems to be claiming that the boxed inequality is always saturated for some vector $y_2 \in {\cal S}$. Is that true? Why? PS - I will understand if some people think that this question is more of a math question than a physics one. But, I thought that it might be possible that the answer relies on some of the assumptions we make in physics, so I asked it here. Any comments will be helpful


title: Rotating neutral for 3-phase system

text: What does it mean by oscillating neutral for 3-phase electrical system? Is it any effect of harmonic production due to magnetic core hysteresis nature? I have been searched a lot about this through internet, but I failed. So, please tell me about it & also mention if this question is wrong. Cause this question makes me confused.


title: How does adding electrons break the angular momentum degeneracy?

text: In the hydrogen atom, the energy does not depend on l. This degeneracy is sometimes called "accidental" (because it does not correspond to some symmetry?). However, there is l dependence in the energy multi-electron atoms. Since the original degeneracy was "accidental", is it no longer correct to talk about what breaks this degeneracy? Physically, I understand this is a result of the lower l-value states being concentrated closer to the nucleus, but is there a way to see that this would be the case in advance, without finding the eigenvalues of the Hamiltonian?


title: Relationship between current through a motor and it's load

text: When a motor, connected to a battery that has a constant voltage, spins without a load it's speed is higher than with load. I'm told that because of back emfs the current is very small when there's no load because of the higher speed. And so when there is a load the back emf is less as the motor spins slower, and so the current is higher. What is a back emf and what is the relationship between that and the speed of the motor (well the coils inside the motor)? Is the output energy of the motor constant, whether it has a load or not? (ignoring friction and electrical resistance) because I don't understand how the current can be higher when the motor is (or seems to be) doing more work due to the load?


title: What determines the form of the intensity curves in Laser-Induced Fluorescence (LIF) measurements?

text: What determines the form of the intensity spectra of different particle species in Laser-Induced Fluorescence (LIF) measurements? See e.g. I figure that bigger particles have more ways to get excited and so the intensities accumulate and make the curve wider? But how exactly do I derive an expected curve for a given molecule type? Why the steep rise and the slower fall? ![enter image description here](http://i.stack.imgur.com/5WEEM.gif)


title: Since radioactive material decays how is it possible that there is any left after 4.5 billion years?

text: I was not able to find an answer for this question... Some radioactive elements have half-life measured in thousands of years and some others even in millions, but over 4.5 billion years all the radioactive material that was part of the initial material that formed the planet earth should have decayed by now? However, there is still radioactive material with short half-life to be found in nature. How is this possible and if the answer is that the new radioactive material is constantly being generated somehow, can you explain the mechanism of how this happens? Thanks.


title: Conceptual question about field transformation

text: (c.f Conformal Field Theory by Di Francesco et al, p39) From another source, I understand the mathematical derivation that leads to eqn (2.126) in Di Francesco et al, however conceptually I do not understand why the equation is the way it is. The equation is $$\Phi'(x) - \Phi(x) = -i\omega_a G_a \Phi(x)$$ G is defined to be the generator that transforms both the coordinates and the field as far as I understand (so it is the full generator of the symmetry transformation) and yet it seems there are no instances of the transformed coordinate system (the primed system) present on the LHS of the equation. Another example is the equation $\Phi'(x') = D(g)\Phi(x)$, where the field transforms under a representation $D$ of some Lie group. $D$ is then decomposed infinitesimally as $1 + \omega \cdot S$ where $S$ is the spin matrix for the field $\Phi$. So, when $S$ acts on the field, should it not only transform the spin indices on the field? But it appears we are in the primed system of the coordinates on the LHS meaning that the coordinates have changed too? Can someone provide some thoughts? I was also wondering how Di Francesco obtained eqn (2.127). Here is what I was thinking: Expand $\Phi(x')$, keeping the 'shape' of the field (as Grenier puts it) the same, so $\Phi(x') \approx \Phi(x) + \omega_a \frac{\delta \Phi(x')}{\delta \omega_a}$. Now insert into (2.125) gives $$\Phi'(x') = \Phi(x') - \omega_a \frac{\delta x^{\mu}}{\delta \omega_a}\frac{\partial \Phi(x')}{\delta x^{\mu}} + \omega_a \frac{\delta F}{\delta \omega_a}(x)$$ which is nearly the right result except Di Francesco has a prime on the x at the final term. My thinking was he defined an equivalent function F in the primed system, i.e $F(\Phi(x)) \equiv F'(\Phi(x'))$. Are these arguments correct?


title: Gauge Invariance of the Non-abelian Chern-Simons Term

text: I'm trying to prove that, under the gauge transformation $$A_{\mu} \rightarrow A_{\mu}^{\prime} = g^{-1} A_{\mu} g + g^{-1} \partial_{\mu} g,$$ the non- abelian Chern-Simons Lagrangian density: $$\mathcal{L}_{CS} = \kappa \epsilon^{\mu \nu \rho} tr \left( A_{\mu} \partial_{\nu}A_{\rho} + \dfrac{2}{3} A_{\mu}A_{\nu}A_{\rho} \right)$$ becomes: $$\mathcal{L}_{CS} ~\longrightarrow~ \mathcal{L}_{CS} - \kappa \epsilon^{\mu \nu \rho} \partial_{\mu} tr \left( \partial_{\nu} g g^{-1} A_{\rho} \right) - \dfrac{\kappa}{3} \epsilon^{\mu \nu \rho} tr \left( g^{-1} \partial_{\mu} g g^{-1} \partial_{\nu} g g^{-1} \partial_{\rho} g \right)$$ as stated in Gerald V. Dunne's lecture notes 'Aspects of Chern-Simons Theory' pages 15-16. The second term in the last equation can be disregarded as it's a total derivative and the third term can be shown to be some integer multiple of $2\pi$ provided $\kappa$ is an integer. Now I understand that gauge invariance of the CS-term can be proven using some clever reasoning (see: Gauge invariant Chern-Simons Lagrangian). However I want to show how we can arrive at the second equation above using the `brute force' method of plugging in the gauge transformed vector field into the Lagrangian. Unfortunately I get stuck with a large number of terms that I'm not sure how to combine or cancel. Does anyone know of a source that goes through the above calculation in more detail, or does anyone have any tips for how to proceed. I've done a rather extensive search and can't find any sources that show some of the steps. I already tried using the cyclic properties of the trace and the cancelation of any symmetric term with the anti-symmetric $\epsilon^{\mu \nu \rho}$. Thank you in advance for any suggestions.


title: Gravitation as the source of redshift of light beams

text: According to Hubble's law, light and other kinds of electromagnetic radiation emitted from distant objects are redshifted. The more distant the source, the more intense is the redshift. Now, the expansion of the universe is expected to explain the redshift and its nearly linear dependence on distance between source and observer. But isn't there an other source influencing the redshift? We know that a light beam passing the Sun is deflected by the Sun's gravity in accordance with predictions made by Einstein's general theory of relativity. This deflection is dependant on a gravitational interaction between the Sun and the light beam. Thus, the position of the Sun is affected by the light beam, though by such a tiny amount that it is impossible to detect the disturbance of the Sun's position. Now, during its journey to the Earth a light beam, originating from a distant source in the Universe, is passing a certain amount of elementary particles and atoms. If the light beam interacts gravitationally with those elementary particles and atoms, affecting the microscopic mechanical properties of the individual elementary particles and atoms at issue, can this interaction be detected as a redshift of the light beam? If so, could we use this gravity redshift to measure the mean density of matter and energy in space? The beginning of the sentence "If the light beam interacts gravitationally with those elementary particles and atoms..." should be interpreted to say "If the light beam _interacts gravitationally_ with those elementary particles and atoms by way of leaving them in a state of acceleration different from their initial state of acceleration...." This clarification seems to necessitate the additional question: "why would a gravitationally interacting object (a cluster of photons) passing another gravitationally interacting object (a mass) leave that mass in the same state as before the passage?"


title: Why should energy have a lower bound but not an upper bound?

text: This question seems to be naive, but I really want some intuitive way of understanding the reason. Neglecting some trivial explanations, I have only one idea about this: the vacua state. Consider the harmonic oscillator problem in quantum mechanics, there are the creator (operator) $a^\dagger$ and the annihilator (operator) $a$. The lower bound of energy exists due to the fact that there is a vacua state which is invariant under annihilator $\left|0\right>$: $$ a \left|0\right> = 0$$ (while there is no such state for the creator) therefore the lowest energy state would be $\left|0\right>$ - a lower bound. Is there a more general explanation or principle that demands the existence of lower bound of energy? Is there any case there is a upper-vacua state (or should I say: saturated state)?


title: How to measure speed of a ceiling fan?

text: I thought for fun I might measure how my ceiling fan slows down when turned off, and maybe find out what forces act on it when it slows down. My problem is that for that I need to measure its rotational speed when it's turned on. When it's in the lowest speeds it's easy because you can see it rotate, but for higher speeds that's not an option. I thought one way would be to use a strobe light and try to match up the frequency, but I don't have one of those. Can you think of any DIY way to measure the speed of a very fast ceiling fan?


title: Tungsten Wire Heat discipation

text: Background Information: I'm doing an experiment in which I place a bare tungsten wire in to various liqids, to measure a coefficient $\alpha $ in the equation $$ Power Dissipated = \alpha * \Delta T $$ I was also given the equation: $$R=R_0*(1+0.0045 \Delta T)$$ I decided to measure $R_0$ at a low voltage and current to stop the wire from heating up. Then I placed it in the liquid and measured the current going through the wire for a variety of voltages. Originally I planed the substitute $Power Dissipated = V*I$ and obtain $R$ from the tangent of a V-I graph (can't use Ohms Law because the situation isn't ohmic) Unfortunately once I have substituted the values into the formulae I got a different value for every data point (all in the same range 10^-7). It made me wonder is there an equivalent formula for non-ohmic power I'm missing or is it something else entirely?


title: Formalizing Quantum Field Theory

text: I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or conformal field theories, and of older approaches such as algebraic QFT, and the classic works of Wightman, Streater, etc. etc . I am more interested in more current approaches, in particular such approaches that incorporate the modern understanding of the subject, based on the renormalization group. I know such approaches exists and have had occasions to hear interesting things about them, I'd be interested in a brief overview of what's out there, and perhaps some references. Edit: Thanks for all the references and the answers, lots of food for thought! As followup: it seems to me that much of that is concerned with formalizing perturbative QFT, which inherits its structure from the free theory, and looking at various interesting patterns and structures which appear in perturbation theory. All of which is interesting, but in addition I am wondering about attempts to define QFT non-perturbatively, by formalizing the way physicists think about QFT (in which the RNG is the basic object, rather than a technical tool). I appreciate this is a vague question, thanks everyone for the help.


title: Rigor in quantum field theory

text: Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, which in general have no mathematical meaning (at least not yet) ect. My question is a little vague, but i am interested in hearing what is the status of rigor in QFT. What is known to be mathematically rigorous and consistent, what is known to be not rigorous? Any examples and references are welcome. **Added:** Just to clarify by rigorous I meant anything that a mathematician would find satisfactory. Also my question wasn't for books with rigorous (in some sense) approach, although that was welcomed. It was about specific examples of what is considered mathematically satisfactory and what not. For example the quantization of free fields satisfying the Klein-Gordon equation can be done rigorously. There is no mathematical definition in general of the Feynman path integral and so on.


title: Will a charged capacitor discharge if one lead is connected to ground?

text: If I charge a capacitor and connect one lead to ground keeping the other lead floating, will the capacitor discharge ?                  G-------||------ open/floating              +q -q      (G for ground)


title: Integers, Energy levels, and wavenumbers for a particle in a 2D box

text: _(This question is **not** about coding)_ I have built a little code in Python that allows the user to plot the energy vs the wave number of particle in a 2D box, depending on what values for the integers $p$ and $q$ they desire. The energy $E_{p q}$ is given by: $E_{p q} = \frac{\hbar^2}{2m}[\frac{p^2\pi^2}{a^2} + \frac{q^2\pi^2}{b^2}]$ The wave numvber $k_{p q}$ is given by: $k_{p q} = \frac{p^2\pi^2}{a^2} + \frac{q^2\pi^2}{b^2}$ **Should the user be allowed to enter any values for $p$ and $q$ or should they only be allowed to enter matching values?** (e.g.: $p=3$, $q=3$ and not $p=3$, $q=1$) **Which results are more interesting, or are they all equally significant (why?)** This is the sort of graph you get when you plot matching values: ![enter image description here](http://i.stack.imgur.com/QbuLF.png)


title: Describing the Higgs mechanism to non-particle physicists

text: I'm sure I'm not the only person with this problem at the moment. I have been asked to give a public (not quite public, scientists, just not physicists) about 'this Higgs boson thing'. I am trying to think of a good analogy because even my own rather limited knowledge of the Higgs mechanism stretches to realising how bad the swimming through treacle metaphor is. I was wondering about refraction although I'm not sure how well understood that will be by a general audience? Superconductivity might be ok for physicsts but again probably rather specific for a more general audience. Any other tips on how to pitch such a talk are appreciated but I'm aware I have to pose a specific question and describing the Higgs mechanism is the real conundrum.


Can fusion and fission happen at the same time, in the same place? I was talking with a friend, and he thinks that fission and fusion happen at the same time at the sun, is that true? I guess this would cause perpetual motion, then I speculated that they could happen, but in different proportions and I believed that fusion is happening in a higher proportion than fission. What do you think? I've searched on wikipidia, and I still found no mention to fission happening in the sun.


title: Divergence of cross product of transverse component

text: If I define the vector as $V_i=V^T_i+V^L_i$ and the transverse part is defined by $$V^T_i=\Big(\delta_{ij}-\frac{\partial_i\partial_j}{\partial^2}\Big)V_j$$ then is is obvious that $\nabla.V^T=0$ as well as $\nabla \times V^L=0$. What happened if I took the the divergence of the cross product of two different vectors with only transverse component? Are they zero too? For example is $\nabla.(A^T\times V^T)=0?$


title: Why is information indestructable?

text: I really can't understand what Leonard Susskind means when he says that information is indestructible. Is that information that is lost, through the increase of entropy really recoverable? He himself said that entropy is hidden information. Then, although the information hidden has measurable effects, I think information lost in an irreversible process cannot be retrieved. However Susskind's claim is quite the opposite. How does one understand the loss of information by an entropy increasing process, and its connection to the statement "information is indestructible". Black hole physics can be used in answers, but, as he proposes a general law of physics, I would prefer an answer not involving black holes.


title: When it is said that the Higgs field is a scalar, do they mean Lorentz scalar?

text: We often hear that Higgs boson is a scalar boson, and that Higgs field is a scalar field. I was always thinking that this means "4-scalar". In other words, it is space-time invariant, .i.e. it's properties are not changing if move relative to it. Unlike photon which changes it's frequency, polarization and other properties, if we look at it from moving frame, Higgs is always the same. Is above correct? If not then what is correct and what is not?


title: Two-Body with external force - energy confusion

text: **Setup** Imagine a two-body system of masses under a classical mechanics model. The separation and mass-ratio doesn't matter for this example. Presume they are initially stationary. Now suppose that we hold one of the masses, m1, permanently stationary with an external force opposing the gravitational force due to the second mass. This external force changes over time such that it always balances the gravitational force on m1. Take the frame-of-reference centred on m1 - the stationary mass. **Problem**   1. If we think of the two bodies as as system, then we would say that the centre-of-mass will accelerate from stationary, and will be displaced in the same direction as the external force. Therefore, positive work is being done on the system by the external force (energy is added to the system).   2. However, consider the bodies individually. For m1, it experiences no net force. Nor is it displaced. Therefore, we must conclude that no work is done on m1. m2 only experiences a gravitational force, and so is accelerated from stationary in the direction of the force. Therefore work is done on m2. But, if energy is transferred to the system from an external source (as suggested from 1), which of the two objects received this energy? It can't have been m1, since it was stationary. But how m2 gain this external energy when the external force acted on m1?


Does a complete theory of quantum gravity require anthropic post-selection? Certainly the black hole complimentarity and causal patch conjectures highlights the essential role of observers, at least in the asymptotic future of their future timelike trajectories. Does the measure problem in quantum gravity cosmology suggest that a "global god's eye view" of the universe might be an incoherent fiction? However, if observers are essential, don't we have to post-select to those states containing the observer of interest? It might be suggested the S-matrix of string theory in a superselection sector with only a finite energy difference from the vacuum over a BPS background provides a counterexample, but can an asymptotic future state of noninteracting Fock space particles really support the future trajectory of an observer?


title: Are physics and philosophy compatible?

text: Lately, there has been a war between physicists and philosophers. Some of the exchanges can be found at http://www.nytimes.com/2012/06/10/opinion/sunday/what-physics-learns-from- philosophy.html?bl, http://opinionator.blogs.nytimes.com/2012/05/10/can- physics-and-philosophy-get-along/?ref=sunday and www.3ammagazine.com/3am/time- lord/. are physics and philosophy compatible? Does physics need philosophy?


title: Constant Velocity 'Force'?

text: According to Newtons second law: F = ma, if acceleration is zero then the force must be zero, but assuming you have an object moving with a constant velocity of say 2 m/s, and that object strikes you, then obviously some sort of 'force' would be felt by you, so my question is what do you call that 'force' since it actually is not a 'force' and is there an equation to calculate it?


title: exponential potential solution

text: let be the Schroedinguer equation $$ - \frac{d^{2}}{dx^{2}}y(x)+ae^{cx}y(x)=E_{n} $$ (1) here a and c are constants. i know how to solve it from http://eqworld.ipmnet.ru/en/solutions/ode/ode0232.pdf but what is the condtion to get the energies ?? on the interval $ [0. \infty) $ or in other interval if you wish the solution to (1) i know that is given in terms of the Bessel function but i have problems to get the energy quatnizatio condition i have tried a get a nonsense like $$ J_{\sqrt{E_{n}}}(b)=0 $$ for a certain real number b but is this true ?? of course i know that semiclassically the energies satisfy $$ 2\pi n \sim \int_{0}^{\infty} \sqrt{E_{n}-ae^{cx}} $$


title: Idea of precursors of the electro-magnetic waves

text: The idea of the material Maxwell equation is almost clear. But I'm curious about the idea that except for material equation the pure Maxwell equation should work, but in harder sense: more currents and charges. There are plenty of vacuum in solid body, so, I think, that there should be some precursors of electro-magnetic waves that propagates through the body: $v = c/n$ the speed of wave in body, but part of wave propagates in vacuum with the speed of $c$. And for **any** frequency some part of amplitude would be observed much earlier. Is there some theory behind this and does this phenomena is observed?


title: What is the relativistic action of a massive particle?

text: all Lorentz observers watching a particle move will compute the same value for the quantity $$ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2,$$ $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ and ''ds/c'' is then an infinitesimal proper time. For a point particle not subject to external forces (''i.e.,'' one undergoing inertial motion), the relativistic action is: $$S = -m_oc \int ds.$$ **Remark:** An edit to the comment. we can write a Lagrangian for a relativistic particle, which will be valid even if the particle is traveling close to the speed of light. To preserve Lorentz invariance, the action should only depend upon quantities that are the same for all (Lorentz) observers.


title: Where does the smell of electrostatic charge come from?

text: Everybody knows you can produce electrostatic charge rubbing two different materials together. But have you ever smelt e.g. at the plastic after charging it? There actually is a distinct electrostatic charge smell :-) While normally smelling involves the transport of molecules, what you are smelling here, is the electric field itself, or do you? What is happening?


title: Particle physics plots

text: I'm having a hard time understanding what some of the plots that are presented by ATLAS/CMS actually show. See for example: http://resonaances.blogspot.com/2011/07/higgs-wont-come-out-of-closet.html What is the y-axis? How would the plot look like if a Higgs was in the given mass range? ![enter image description here](http://i.stack.imgur.com/IbqWV.png)


title: Quantum mechanics + General relativity =?

text: I have learned (the basics) of how one can incorporate the principles of special relativity to quantum mechanics to obtain quantum field theory. Can the same be done with GR and QM to obtain a new theory? If not, what happends if one tries, what exactly goes wrong? Is there a mathematical inconsistency in any theory that combines them, such that either current GR or QM must be modified, or is it just that we dont know how to yet?


title: Graphing measurements with uncertainties

text: I need to draw a graph of measurements of time for a ball to drop on an inclined plane, with error bars, along with lines of best and worst fit. The problem is that I need to do this on the computer and I am unsure about what graphing package I can use to do this and how to go about doing it. I have access to Wolfram Mathematica 7, but I'm unsure about graphing points with uncertainties.


title: Why is the covariant derivative of the metric tensor zero?

text: I've consulted several books for the explanation of why $$\nabla _{\mu}g_{\alpha \beta} = 0,$$ and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta} $ $$\Gamma ^{\lambda } _{\beta \mu} = \frac{1}{2} g^{\alpha \gamma}(\partial _{\mu}g_{\alpha \beta} + \partial _{\beta} g_{\alpha \mu} - \partial _{\alpha}g_{\beta \mu}).$$ But I'm getting nowhere. May be I've to go through the concepts of manifold much deeper.


title: What is the relation between surface area and radiation, if any?

text: Basically I wonder what happens to emitted radiation by douubling a light e.g. twice the surface area of the sun will emit how much more radiation? 4 times more? Is there a formula?


title: Why are neutrino and antineutrino cross sections different?

text: Particularly in the case of Majorana neutrinos, it seems a little odd that the particle and antiparticle would have differing cross sections. Perhaps the answer is in here, but I've missed it: http://pdg.lbl.gov/2013/reviews/rpp2013-rev-nu-cross-sections.pdf In the caption of figure 48.1 of the PDG excerpt linked above, it says "Neutrino cross sections are typically twice as large as their corresponding antineutrino counterparts, although this difference can be larger at lower energies." Is it common for particles to have different cross sections from their corresponding antiparticle? Is there a reason for this difference? Can we theoretically predict the magnitude of this difference?


title: Torque due to Gravity

text: I have learned that a torque on an object due to the gravity can be calculated as if the gravity acts on the center of mass of the object when the object is near the surface of the earth. Now I want to prove it... So, I have $$\tau=\int d\tau=\int r(dmg)sin\theta=g\int rsin\theta dm.$$ Then, how do you obtain $$\int rsin\theta dm= mr_{cm}sin(\theta_{cm}),$$ where $r_{cm}$ is the distance between an axis of rotation and the center of mass, and $\theta_{cm}$ is the angle between the two vectors $\vec{r_{cm}}$ and $m\vec{g}$ at the center of mass??? Would $$\int r dm=mr_{cm}$$ be helpful to prove this? Would the integral be dependent on the object I use, or would there be a general way to prove this?


title: Question about Rayleigh scattering

text: To quote from Wikipedia on elastic scattering, "In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed." How does this work? The impacted particle will also start moving to conserve momentum. So it'll also carry some energy. How can the photon not lose any energy during the collision? Another possibly unrelated question: how exactly can the photon and the particle interact? The photon essentially carries information about a changing electromagnetic field. How can it influence a particle that carries neither charge nor magnetism?


title: Can someone explain to me how to solve $6.6738410*10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}$, the formula to find the gravitational constant?

text: Can someone explain to me how to solve $6.6738410\times 10^{-11}\ \text{m}^3\ \text{kg}^{-1}\ \text{s}^{-2}$, the formula to find the gravitational constant? I'm writing a bunch of reports about the scientific perspective about the video game Minecraft, and I'm trying to find the gravitational constant of Minecraft. I'm having a little bit of trouble solving $6.6738410\times10^{-11}\ \text{m}^3\ \text{kg}^{-1}\ \text{s}^{-2}$, as I don't know what many of the variables are. Some I get, like $\text{m}$ is the mass of something, $\text{kg}$ is something's weight in kilograms, and $\text{s}$ [for seconds] is the time it takes an object to fall in seconds. It would be really helpful if someone could explain to me what some of these variables mean, and help guide me through the equation.


title: What does the Rayleigh Phase Function tell us

text: I am working on some radiative transfer equations, and struggling as I'm fairly new to this field. I have read about the Rayleigh Phase Function which is: $P(\theta) = \frac{3}{4}(1 + cos^2 \theta)$ I can plot this function, and generate values from this function in my computer program (which is what I need to do) - but what is it actually telling me? If I know I have a ray of incident light at a certain angle, will this function tell me what angle it will scatter at? Or is it more of a statistical generalisation where the areas where the function is higher are the angles at which the scattering is more likely to happen? Furthermore, when I see the function plotted it is always from 0-180, rather than 0-360 as I would expect for the scattering. This makes me think that the function may be giving the difference between the incoming angle and the outgoing angle rather than the absolute angle it is scattered at. Is that correct?


title: Was there some phenomenological motivation for Ramond and Neveu-Swarz models?

text: This remark from Lubos puzzles me > Ramond's string - and Neveu-Schwarz string - wasn't really an "origin of > string theory". String theory had "origin" as bosonic string theory which > has no fermions. All SUSY/fermioncs strings are "new". How to fit this view, which seems to be the common one, with the fact that the R and NS papers date from 1971, before the concept of supersymmetry, long before superstrings themselves, and only three years later than the Veneziano model, and almost simultaneous (less than two years) with the idea of a String interpretation of the model? Were the R-NS models just a hep-th proposal, without any link to, nor motivation from, phenomenology? It could be so. I can not find any clear reference to baryonic regge trajectories in the early literature, and I can not find any duality diagram involving fermions, before 1971. But on other hand, Susskind 1970 abstracts keep speaking of "model of hadrons", not just "mesons", when exposing the idea of strings. So perhaps I am just being sloppy in my spires searches.


title: How do I determine the amount of lift needed to launch an object?

text: So as part of a video I'm working on, I'm trying to launch a 10kg table about 8m into the air. A machinist friend of mine has provided a solution that involves pneumatic cylinders and a pump to generate the force needed to propel the table. What we're trying to calculate is the amount of PSI needed to lift the table. All these calculations don't need to be exact (if it only goes 16m up, then there's no issue) I don't know where to even start to figure it out. Once gravity gets involved, it throws me off. Any real help would be appreciated, thanks!


title: two pendulums, which pendulum has shorter period

text: > **Possible Duplicate:**   >  A pendulum clock problem There are two pendulums. First pendulum consists of a rod of length L and flat heavy disk of radius R (R < L), disk is connected rigidly to the rod such that the plane of the disk is vertical. Mass of the rod is negligible compared to mass of the disk. Center of the disk is at distance L from the upper point of the rod. Second pendulum has same rod of length L and same disk, but the disk is not connected rigidly to the rod. Disk can rotate freely around its center at which it is attached to the rod at distance L from the upper end of the rod. Disk remains in vertical plane, as if there is ball bearing atached to the rod. Which pendulum has smaller period at small deviations, if their periods are different.


title: Do extra-dimensional theories like ADD or Randall-Sundrum require string theory to be true?

text: What I mean is could it turn out that the world is not described by string theory / M-Theory, but that nevertheless some version of one of these extra- dimensional theories is true? I have no real background in this area. I just read Randall and Sundrum's 1999 paper "A Large Mass Hierarchy from a Small Extra Dimension" (http://arxiv.org/PS_cache/hep-ph/pdf/9905/9905221v1.pdf). Other than the use of the term "brane" and a couple of references to string excitations at TeV scale, I don't see much about string theory, and I notice their theory only requires 1 extra dimension, not 6 or 7.


title: Why does lambda decay violate parity?

text: When a lambda particle decays into proton and a pion, I am told it does not conserve parity. Why?


title: Current Density and the Dipole Moment $\int _v \vec{J} dV = d \vec{p}/dt$

text: Show that: $$\int _v \vec{J} dV = \frac{d \vec{p}}{dt}$$ Here's my attempt: $$\vec{p} = \int _v \rho r dV \rightarrow \frac{d}{dt} \int _v \rho r dV = \int _v \vec{J} dV = \int _v \frac{\partial p}{\partial t}r = \int _V \vec{J} dV$$ Because of the fact that $\nabla \cdot \vec{J} = - \frac{\partial p}{dt}$ we have $$ \int _v (\nabla \cdot \vec{J})rdV$$ I'm not exactly sure how to turn this divergence of the current density per unit volume into the original $\int _v \vec{J} dV$ The last expression is very close to equaling the current, maybe we could use that?


title: The Energy Eigenvalue of a Wavefunction

text: I've been reading an introduction to quantum mechanics online, and while constructing the Schrodinger equation for a free particle, the equation $i\hbar \frac{d \Psi}{dt}=\hbar\omega\Psi$ is obtained. It is then explained that if $i\hbar \frac{d}{dt}$ is considered as an operator, then the equation means "that when it operates on $\Psi$, the result we get back is the energy eigenvalue of the wave." It's clear to me that $\Psi$ is an eigenfunction of $i\hbar\frac{d}{dt}$, with the eigenvalue being $\hbar\omega$, but I don't understand why they say that $\hbar\omega\Psi$ is the "energy eigenvalue of the wave." As far as my understanding goes, $\hbar\omega$ gives the energy of a photon with angular frequency $\omega$, and so it's unclear to me why the energy of a photon multiplied by $\Psi$ would be of particular significance. Furthermore, I don't think I understand what is meant by the statement "the energy eigenvalue of the wave". If anyone could help explain these concepts (preferably without presupposing too much knowledge of QM) that would be great.


title: Interpretation of field operator

text: Consider a real scalar field operator $\varphi$. It can be written in terms of creation and anihilation operators as $$\varphi(\textbf{x})=\int \tilde{dk}[ a(k)e^{i\textbf{kx}}+a(k)^{\dagger}e^{-i\textbf{kx}}]$$ where $\tilde{dk}$ is a Lorenz-invariant measure. If $\varphi$ is interpreted as creating a particle at $\textbf{x}$ when acting on the vacuum, what is its action on a generic state? It seems to be creating a superposition of a state with one added quantum of energy through the creation operator, and a state with one less quanta of energy through the annihilation operator.


title: How is the view of charge and forces in quantum mechanics different from classical physics?

text: I am taking an introductory course on quantum mechanics and would like to understand this. Do forces act in quantum mechanics? In orbitals, is the charge of an electron dispersed, with a higher charge at places with a higher probability of finding an electron? Is there any centripetal force/acceleration at the quantum mechanical level? Thank you.


title: Continuous vs. Discrete Spectra in various materials

text: I read that the reason solids emit continuous spectra is that they don't have time to let their electrons decay-they are too close together. Given that electrons decay on the order of 100 nanoseconds I find this difficult to believe. Also, do electromagnetic waves move the electrons, or the atom, or both? If it is simply exciting the electrons, I don't know why is should also give way to the vibration of the atoms. If it does give way to vibration, then shouldn't gases _also_ give way to continuous spectra?


title: 2D Ghost CFT and two-point functions

text: For some reason I am suddenly confused over something which should be quit elementary. In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb C)/\mathbb Z_2$ invariance to have the form $$\langle \phi_i(z)\phi_j(w)\rangle = \frac{d_{ij}}{(z-w)^{2h_i}}\delta_{h_i,h_j}.$$ So a necessary requirement for a non-vanishing two-point function is $h_i = h_j$. Now consider the Ghost System which contains the two primary fields $b(z)$ and $c(z)$ with the OPE's $$T(z)b(w)\sim \frac{\lambda}{(z-w)^2}b(w) + \frac 1{z-w}\partial b(w),$$ $$T(z)c(w)\sim \frac{1-\lambda}{(z-w)^2}c(w) + \frac 1{z-w}\partial c(w).$$ These primary fields clearly don't have the same conformal weight for generic $\lambda$, $h_b\neq h_c$. However their two-point function is $$\langle c(z)b(w)\rangle = \frac 1{z-w}.$$ Why isn't this forced to be zero? Am I missing something very trivial, or are there any subtleties here?


title: Equations for an object moving linearly but with air resistance taken into account?

text: I know (from Kinematics) that for an object moving linearly with an acceleration and without air resistance the following equations can be used to determine v(velocity) or x(position of the object) at any time: $v=v_0+at$ $x=x_0+v_0t+\frac{1}{2}at^2$ Where $x_0$ is the position of the object at the start of accelerated movement, $v_0$ is the velocity at the start of the motion and a is the acceleration. Now if I want to add air drag, say we take the formula for air drag as: $D=0.5CA\rho v^2$ $C$ is a coefficent, $A$ is the reference Area, $\rho$ is the density and $v$ the velocity. Since $D$ is a force the air drag results in an acceleration, so in other words: The resulting acceleration is $a-D/m$, however $D$ depends on $v$, which depends on the acceleration. And that's where my problem is, I simply can't wrap my mind around that. How would I go about getting the Equations for $x$ and $v$ including air drag (and don't forget there's acceleration too)?


title: Electrical force between two objects

text: I tried to solve the following problem: There are 2 objects . The object m1 with charge q and the object m2 with charge q.(same charge).The object m2 is connected with a rope to the ceiling. at the initial state , m1 and m2 were on the same height. When the object m2 is placed in distance d from the object m1, the following situation occurs: ![enter image description here](http://i.stack.imgur.com/KhKjc.png) Please ignore the force of gravity between the masses and ignore any kind of friction. just consider the tension , electrical force and mg.               m2 : ΣFx,y=0      You have to find x. You can use the parameters k(Coulomb's constant), q , L , d , mg. I was able to find only 2 equations that would help to solve the problem but one is still missing. I wasn't able to calculate the components of the electrical force. The equations that I found are : 1) r ^ 2 = (L-Lcos a) ^ 2 + (x+d) ^ 2 (r is the distance between the two objects) 2) sin a = x / L 3) as you can see ΣFx = 0 and ΣFy = 0 (The object m2), but without the components of the electrical force I wansen't able to calculate anything.


title: Does gravity affect radiation incident on the planet?

text: Just curious ... Known facts:   * When outside the atmosphere levels of radiation are higher   * Typically such radiation is CBR, Solar Radiation Question: If the matter of the atmosphere and magnetosphere as shields is kept aside for a moment, does gravity affect radiation incident on the planet?


title: How do you find conserved quantities for linear second order ODEs?

text: I have a differential equation of the form $ \frac{d^2 y}{dt^2} + f(t) \frac{dy}{dt} + g(t) y = 0 $ where $f$ and $g$ are known functions of time. Is there a systematic (or otherwise) way of finding the conserved quantities, if there are any? I've been trying to google this topic, but haven't had much success yet. Perhaps someone with more mathematical knowledge than myself (i.e. almost everyone) can point me in the right direction. Even phrases to search for would help.


title: Left and Right-handed fermions

text: 1. Is there a simple intuitive way to understand the difference between left-handed and right-handed fermions (electrons say)?    2. How to experimentally distinguish between them?


title: Photon absorption

text: [sorry, this way below the level of this forum -- flames are most welcome] When a photon is absorbed by a piece of matter that does not reflect it -- where does the photon "go"? Eg, one shines light at a black object -- clearly the photon arrives at the object and then, well, vanishes. Or does it? Thanks very much.


title: What does a positive gradient on a graph of V plotted against I mean in terms of EMF and internal resistance?

text: According to the equation $V = E-Ir$, the gradient of a graph of $V$ against $I$ should be $-r$ (internal resistance) and the Y intercept should be the EMF. Am I right? In an experiment I used a constant wind speed to drive a small motor, keeping everything constant and changing the resistance in the circuit using a variable resistor, measuring the output in volts. How is it that the graph shows a positive gradient (and therefore a negative internal resistance ?). The equation of the line is $0.0732934 + 0.0396359x$. http://cl.ly/image/0b3i393E0E0G


title: What could cause a planet to have apparently random season lengths?

text: The world _A Song of Fire and Ice_ series is set on is characterized by seasons that last several years (how years are measured on this planet is not specified to my knowledge). However, they are of unpredictable, variable lengths, though seemingly in balance (long summers are followed by long winters). Rather than some sort of massive El Niño, it seems to be a result of something astrophysical, as winter can be forecast by the shortening of days. Is there some possible mechanic for this or is it outside the realm of possibility?


title: Non-relativistic Kepler orbits

text: Consider the Newtonian gravitational potential at a distance of Sun: $$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$ I write the classical Lagrangian in spherical coordinates for a planet with mass $m$: $$L ~=~ \frac{1}{2}m (\dot{r}^{2} + r^{2}\dot{\theta ^{2}} + r^{2}\dot{\phi ^{2}}\sin^{2}\theta ) + \frac{GM}{r},$$ and find that the canonical momentum $p_{\phi }$ is a constant of motion, because: $$\dot{p_{\phi }}~=~ \frac{\partial L}{\partial \phi} ~=~ 0.$$   1. What is the physical interpretation of the canonical momentum?   2. How can we from the Lagrangian see that it is a constant of motion?


title: Introducing emf of a chemical cell as a hint towards quantum mechanics

text: Today I had a discussion with a colleague who teaches electricity and magnetism to 2nd year undergraduate physics students. He is seeking the best way to explain how is the emf generated inside a battery with a minimal appeal to physics beyond classical. We have lamented that some textbooks refer to "non-electromagnetic chemical forces" since all of chemistry is essentially electrostatics+quantum mechanics. Our proposal is to draw the students' attention to the existence of atoms which can not be explained by classical mechanics + classical electromagnetism. In the same vein the forces on charge carriers in galvanic cells are electromagnetic but the response is not classical. Thus a battery "amplifies" non-classicality to the level macroscopic electricity. It is not that forces are non electrostatic but the systems response is not classical. Can you recommend textbooks or online sources that use/expand this idea? (the main tag for this question should be "teaching" but I'm too much of a rookie here to create one)


title: How can I determine the feasibility of Pd(d,f) fission chains?

text: The conclusion of this paper (p. 6) discusses some hypothetical Pd(d,f) yields as a possible explanation for anomalous results that the author observed. Suspending disbelief in the data reported in the paper for a moment, what are some good ways to go about exploring the feasibility of a set of fission chains along these lines, given an arbitrary amount of energy? Here is one that was mentioned as a possibility: > ${{}^{104}Pd} + 2D \rightarrow [Pd;2D]* \rightarrow {{}^{44}Ca} + > {{}^{64}Ni} + 45.44 MeV$ The proposed [Pd;2D]* intermediate step is an "intermediate short-lived neutron excessive compound-nucleus," according to the paper. I was able to find an interface to the EXFOR database, and using a query with the the parameters [Target="Pd-104", Reaction="d,*"], the results seem to indicate that there are only nine known or hypothesized Pd-104 + D reactions, and none of them involve fission. My questions:   1. Am I missing a basic detail here? Have I done something wrong?   2. Are there better ways to go about exploring the feasibility of the proposed chains? For example, would a simple back-of-the-envelope calculation have gotten me this far (aside from an objection about Coulomb repulsion)?   3. How likely is it that an actual Pd+D chain would not be available in EXFOR? What conclusions can I draw about the presence or absence of a chain in EXFOR? (I understand that some of the entries are hypothetical.) These questions are methodological ones, about how to look into the merits of the paper's proposal about palladium fission chains, rather than ones about the actual merits of the proposal (which I assume takes the author into left field). **EDITS:** Changed example reaction from one of deuteron absorption to an actual fission chain (also from the paper), modified details to relate to the new reaction. Mentioned an important assumption regarding input energy.


title: Is there a difference in binding energy between a regular material and a doped one?

text: Say Silicon and boron doped silicon. Would the doping affect the binding energy? Could I see this in an XPS spectra?


title: Do human bodies give off a consistent but unique radiation/electromagnetic/energy signature?

text: Is there any facet of the energy emitted by a human body that is consistent and unique - like a fingerprint, but a signal that could be detected by a remote device?


title: Are energies non-transferable in the Born-Oppenheimer approximation, and when does it apply?

text: Adiabatic approximation or the Born-Oppenheimer approximation is used whenever the electronic motion is too fast that the electrons effectively see static nuclei and the nuclei, in turn, see an averaged electronic cloud. My question is:   1. As the word 'adiabatic' suggests no heat/energy transfer, does this mean that the energies associated with nuclear and electronic motions are non-transferable?   2. If my understanding is right, the limiting criterion of the approximation is when transfer can no longer be neglected. My other question is how to judge when the transfer is significant?


title: Killing vector fields

text: I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.


title: QM : Perturbation theory with multiple operators

text: When doing perturbation theory in quantum mechanics, if the perturbation hamiltonian is made of three terms : $$W = W_1 +W_2 + W_3,$$ can I treat each term separately and performing perturbation theory for each one of them, then adding the perturbed energies together to get the total energy ? $$ E \stackrel{?}{=} E^0 + E_1^1 +E_2^1 +E_3^1 $$ where $E^0$ is the energy of the unperturbed hamiltonian, and $E^1_j$'s are energies at first order of $W_j$'s. Edit : What about degenerate perturbation theory ?