Non-existence of gluon singlets: Any recent theoretical progress?

An unanswered question from last year (2012) on gluon singlets asked whether there is any theoretical explanation for the experimental absence of the ninth or colorless (singlet) gluon. This is the gluon that, if it existed, would give allow the strong force to extend far beyond the range of atomic nuclei, with catastrophic results. (Non-experts can find an explanation of the singlet gluon in the Addendum below.) Qmechanic partially answered the question with a link to this 1996 paper by J.J. Lodder, which proposed that the singlet gluon exists but is so massive that its effects are negligible. However, Lubber's massive-singlet idea seems to have gone nowhere. On an cursory search I could not find any references to the 1996 draft, even though it was written seventeen years. I also do not find Ludder's approach very plausible. Theoretically he had to do a bit of mayhem to Standard Model symmetries to come up with his model. More importantly, though, its seems unlikely that the idea holds water experimentally. The mass of the singlet gluon would have to be astronomical indeed for it not to have shown up in high-energy experimental results, especially in the post-Higgs era. So, my version of the singlet question is this: Do there exist any plausible, Standard Model compatible theories for the observed absence of the singlet gluon, other than the apparently non-starter idea that the singlet exists but is very massive? Or alternatively, have searches for the Higgs boson produced any evidence that the singlet gluon state may in fact exist and have a very large mass? * * * **Addendum for non-experts: What is a "gluon singlet" and why is it important?** One way to understand the "colorless" gluon singlet is through an analogy with how photons work for the electric force. Photons interact with electrically charged electrons, but do not carry any electric charge themselves. Because they have no charge, even low-energy photons can easily escape an atom or electron and travel infinite distances. Imagine, however, what would happen if photons did carry electric charge. Such photons would have the same energetic difficulties leaving a neutral atom as an electron, dramatically altering and limiting how they behave. (I should probably mention that electrically-charged photons in one sense really do exist: They are more-or-less the $W^\pm$ particles of electroweak theory.) Curiously, the charged situation is reversed for the otherwise photon-like gluons. It is the gluons that convey the strong force, and thereby hold quarks together to form protons and neutrons, as well as secondarily binding protons and neutrons together within atomic nuclei. There are eight types of gluons instead of just one, due to there being more than two types of charge in the strong force. However, in sharp contrast to photons, all of the eight gluons _normally_ carry strong (color) charge. The color charges of gluons cause them to interact strongly with the quarks that emit them and with each other. As in the earlier hypothetical example of how electrical charges would dramatically limit the range of photons, the presence of color charges on gluons similarly limits the distances over gluons can convey the strong force. Consequently, the strong force has almost no impact beyond the scale of atomic nuclei. However, the same mathematical model that predicts the eight color-charged gluons also predicts a ninth neutral or strong-charge-free gluon, called the singlet gluon, that has never been seen experimentally. Its lack of color would make its impact far greater than that of any of the eight other gluons. In particular, just as charge-free photons can carry the electric force far beyond the range of atoms, a charge-free gluon, if it existed, would allow the strong force to extend far beyond the range of nuclei. The repercussions would be huge. In fact, the non-existence of the singlet gluon is best demonstrated by the fact that we exist at all. My best guess (only that, since I have not seen any papers on it) is that if the colorless singlet quark really did exist, every clump of two or more atoms in the universe would melt together into an amorphous sea of quarks. Even if that is not correct, I assure you it that the consequences of the existence of singlet gluons would be... very bad indeed! That is also why I find I think the non-existence of the singlet gluon is a truly interesting question, one that probably deserves more theoretical attention than it has received over the decades since strong force theory was first codified.


What's the difference between 'least count' and 'smallest division'?

I'm wondering the difference between the 'least count' and the 'smallest division' of an instrument. Can you explain that for me, plz?


a metal bar on a pair of conducting rails that carries a current

This is a homework question, and I solved it already, but something bugs me. So the problem is stated as following: A metal bar of mass M sits on a pair of long horizontal conducting rails separated by a distance $\mathcal{l}$ and connected to a device that supplies a constant I to the circuit, also the bar has length l. There is a constant magnetic field $\vec{B}$ that points into the screen where this whole setup is. The current is following in clockwise direction. Now if there is no friction on the rail, the bar will move to the right because of $$F = I\vec{L} \times \vec{B}$$ This is easy. And since there is no friction, the acceleration will be $$a = \dfrac{F}{M}$$ and hence the magnitude of the velocity at any later time $t$ will be $$v(t) = at = \dfrac{IlB}{M}t$$ The question is asking for the velocity at time $t$. However, what bugs me is that since the bar is moving to the right, thus the flux through the closed loop of the powersource, rails and the bar will be increasing, which means that there will be an induced electric potential trying to push back the current `I`. However, the problem does not give the resistance of the bar, and I assume that in this hypothetical situation, the whole thing has 0 resistance. So how does the electric potential make sense in this case? Since I am not given a resistance, I cannot calculate the "pushing back" current caused by the change in flux. Thank you!


Parallel Transport of a 4-vector

![enter image description here](http://i.stack.imgur.com/aEqXf.jpg) Why does the parallel transported $4$-vector change from $X^a(x)$ to $X^a(x) + \bar{\delta}X^a(x)$ ? This is also discussed in D'Inverno's relativity book [page - 72]; but the reason is not clear.


Modulus of action-reaction forces

I read somewhere that action-reaction forces satisfy the form $$ \mathbf{F}_{12}=-\mathbf{F}_{21}=f(|P_1-P_2|)(P_1-P_2) $$ meaning that the modulus can only depend on the distance between the two points $P_1$ and $P_2$. Is that right?   Cannot the modulus depend on, say, $|\mathbf{v}_1-\mathbf{v}_2|$?


Is the quantum state or information of a particle is all that differentiates it from another particle?

So I have learned in my QM classes that you can't tell one electron from another electron. They are indistinguishable. I also learned that the wavefunction of a particle includes the spacial part and the spin part. (There might be other parts I don't know! Please don't fault me for not knowing.) I have also learned that it is possible to use the quantum teleportation protocol to retrieve exactly, 100%, the spin state of one particle in another particle after particular measurements. http://en.wikipedia.org/wiki/Quantum_teleportation My question is, if you can teleport the spatial state and the spin state and whatever other state left there is that describes the particle, then wouldn't you be **effectively teleporting the original particle?** A sub question directly related to this is; Is matter/particles just a changeable glob of stuff that acts differently, thus being a different particle, depending upon what quantum state its in? (This kinda sounds like string theory to me... Oh no!!) Also, what's never answered is, **What state does the original particle assume when its state is teleported to the new particle?** Does the two particles, the original particle and the particle retrieving the quantum information, switch states?


Josephson effect - fractional or not

It is well known that a current proportional to $J\propto\sin(\phi_1-\phi_2)$ flows when two superconductor with phases $\phi_1$ and $\phi_2$ are connected. Also, a current which depends on the phases according to $\sin[(\phi_1-\phi_2)/2]$ is claimed to be fractional Josephson effect (which is believed to be a signature of Majorana fermions). But I see that even with two s-wave superconductors connected through a barrier, the current could be proportional to $\sin[(\phi_1-\phi_2)/2]$ (see for instance - http://arxiv.org/abs/cond-mat/9811026) when the barrier is zero. I do not understand this well. Does it mean that $J\propto\sin(\phi_1-\phi_2)$ is true only in the large barrier limit ? Then, what is holy about fractional Josephson current?


Electric potential of spherical water droplet

I'm trying to answer the following: > 1000 spherical water droplets, each of radius $r$ and each carrying a charge > $q$, coalesce to form a single bigger spherical drop. If $v$ is the electric > potential of each droplet and $V$ that of bigger drop, then find > $\frac{V}{v}$. It is not mentioned here whether $v$ is potential at a point outside or on the surface of sphere, where should I take the potential? As water is a good conductor of electricity so water droplet can be regarded as spherical shell.Potential due to spherical shell is *** $\frac{kq}{r}$.*** where **_k_** is constant and **_q_** is charge on the shell and **_r_** is the distance between the centre of spherical shell and point on which **_P_** is to be calculated and at a point inside or on the surface of shell **P** is constant and is equal to *** $\frac{kq}{r}$.*** here **_r_** is the radius of shell. So **_P_** depends on the distance between the centre of spherical shell and point on which **_P_** is to be calculated.So in the above problem **_P_** on which point should I need to consider. I am unable to understand what is meant by the potential of water droplet.I know potential at a point due to point charge,system of charge and due to continuous charge distribution.But they are calculated at point,I mean we calculate potential at point due to different charge distributions.How can we calculate potential of a body! I want to know what is meant by( " $v$ is the electric potential of each droplet" )this line of the question.


"+" and "-" sign in Maxwell Stress tensor

I have trouble in determining the "+" and "-" sign of momentum per unit time, per unit area of the following question. Why in the second part, $d\vec{a}$ is pointing in the $ -\vec{z} $ direction? I mean, the normal vector is pointing upwards, so the area vector should point upwards, so $d\vec{a}$ is pointing $ +\vec{z} $ direction, right? And for (b), why it is negative? Is it because $ \vec{T_{zz}}$ and $d\vec{a}$ point in the different directions? And for (c), this is the most confusing part for me. $ -{T_{zz}}$ should be minus, becasue in (a), we get the component of the tensor matrix to be "+". I just get really confused about the sign "+" or "-" of this problem; yet, the calculation is pretty easy for me. ![enter image description here](http://i.stack.imgur.com/2StCb.jpg) ![enter image description here](http://i.stack.imgur.com/w5zr0.jpg) ![enter image description here](http://i.stack.imgur.com/cwK23.jpg)


What would be the resulting effect of two ships passing near each other while using a Alcubierre drive

So, I don't have any formal background in physics or space. but I read about the Alcubierre drive, and now I'm curious. If two ships passed one another, what sort of interference would be generated? How dangerous would it be? I'm thinking it would actually neutralize the effect of warp for either ship since their fronts would be generating opposite forces. Also, am I correctly interpreting what a warp drive travel path would look like? If not, my question is pretty irrelevant. The arrows are supposed to indicate the ships travel path. ![Ship traveling in a warp. Arrows indicate direction of travel](http://i.stack.imgur.com/N7vtd.jpg)


Thermodynamic potentials and approach to equilibrium

I have seen some material which tries to show that when approaching equilibrium upon different constraints, the changes in some thermodynamic potential is always positive or negative (for example, that Helmoltz free energy decreases for constant temperature and volume). I cannot make sense of these arguments, however, because the quantities these thermodynamic potentials depend on are only defined at equilibrium. I think, that what one should show is that if there are several equilibria, the one which minimizes/maximizes the appropriate potential is the one which maximizes entropy. Is this the right way to think about thermodynamic potentials?


Why does 'proper length' exist as a notion?

$$\text{proper time}= \tau= \sqrt{dt^2-d\mathbf{s}^2}$$ $$\text{proper length}= L= \sqrt{-dt^2+d\mathbf{s}^2}$$ What tangible benefit is brought about by calling $i \tau$ 'proper length' (applying when $\Im(L)=0$ (the spacetime intervals are spacelike))? Could one extend the notion of the interval to simulatneusly cover both proper time and distance? Call this $\text{Interval}=dt^2-d\mathbf{s}^2$, _which can be imaginary_. Is it impossible to merge proper time and length because they are fundamentally different things, as proper time is a explicitly measurable quantity, but proper distance must be determined implicitly? My problem is that I don't understand the rationale behind having proper length, rather than a cheap mathematical trick to make the interval real when $ dt^2-d\mathbf{s}^2<0$.


How to measure the diameter of a star?

I am thinking about something I read somewhere (if only I could find it again) in a textbook. It is about the size of a star and its ER peaks. It has to do with the waves coming off the edge (maybe) and arriving later than those from "head on" and therefore you can know something about its diameter. It has been puzzling me but I can't quite remember it. Just yesterday I was reading about a black hole that pulses at a minimum of 10 minutes and so it is at least 10 light-minutes across. (I probably am not getting it just right, but help me out!) Is this the same principle? Anyone know what I'm talking about? I would love to have this explained and/or would like to know what it is called, so I can look it up.


How fast will the sun become a red giant?

I've read many accounts of our sun's distant fate, but what I've never heard is on what time scale these events occur. For instance, when the sun runs out of hydrogen, I presume it doesn't just WHAM! become a red giant suddenly, engulfing the inner planets in seconds. But how long would it take for the sun to expand? Would anyone left on Earth have days, weeks, months, years to escape? Centuries? What kind of time scales _are_ we talking about, here?


Why isn't sunset time in sync with solstice?

The winter solstice in the Northern Hemisphere in 2011 is on December 22. But if I look at the sunset times for a location such as Washington, DC on the USNO site, the sunset time starts reversing much earlier (around December 6th). Shouldn't the sunset time start reversing (from earlier to later) on the same date as the solstice? **Update:**   Omega Centauri's explanation is correct.   * The Royal Observatory gives a similar account of this oddity on this page under the section titled "The apparently odd behaviour of sunrise/set times near the winter solstice".   * Cornell University provides more details here.


Behavior of black holes in higher- and lower-dimensional space-times

The behavior of black holes in 3+1 dimensional space-time as our own is rather well known: formation, event-horizon size, mass, spin, radiation etc. However, my question is what would black holes behave like in universes with different number of space-time dimensions, such as 2+1 dimensional or 4+1 dimensional space-time? Would they be able to form? Would the event horizon radius be smaller or larger? Would their densities be higher or lower? Would they spin, and how quickly? Would they emit Hawking radiation? How would the space-time behave close to it compared to in 3+1 dimensional space-time?


Visible light through a single-mode optical fiber?

If I understand things correctly, the optical fibers used for (long-range) data transmissions are generally single-mode fibers, transmitting light in the 1300-1500 nm spectrum. Now, could such a fiber transmit visible light (~400-700 nm) a short distance, say a few meters? Or does the fiber have a property that prohibits transmission of shorter wavelengths?


Are small speakers inherently limited to higher frequencies?

I am hoping to build a subwoofer using multiple smaller speakers (165mm) instead of a single larger speaker (380mm). My theory is that the displaced air volume is what matters, not the individual speaker's diameter.   The surface area of a 380mm speaker is $\approx900cm^2$.   The surface area of a 165mm speaker is $\approx140cm^2$.   Presuming the speakers are capable of being driven to the same $X_\max$, six or seven 165mm speakers should be roughly equivalent to a 380mm speaker.   Taken to an extreme, an array of 100 $9cm^2$ speakers should be equivalent to a single $900cm^2$ speaker. Is this correct, or do smaller speakers have some other limiting factor that limits their low-frequency reproduction?


How is barometric light porduced?

A shaken sealed glass tube containing liquid mercury in a dark room will glow. This has been know since the 17th century and is called barometric light. How does the addition of potassium permanganate crystals to the mercury influence the production of light?


Does Lyapunov exponent equate to exponential inflation?

Physics can be modeled by dynamical systems $f^t(x)$ as well as by PDEs. The most common dynamical system has hyperbolic fixed point and can be an attractor or a repellor. The dynamics at repellors and attractors are simple exponential expansion or contraction and are represented by a Lyapunov exponent. My question is how could one prove or disprove that exponential inflation is just an example of a repellor with a Lyapunov exponent?


Will a wave packet undergo dispersion when traveling down a hanging rope?

Suppose I tie one end of a rope to my ceiling and the other end to a spot on my floor directly underneath it. Because the rope has some mass, the tension varies along the rope, from highest at the ceiling to lowest at the floor. If a wave packet begins propagating down the rope, will its shape change? If so, it is possible to calculate the shape of a Gaussian wave packet as it travels down the rope? Intuitively, it seems to me that the wave packet will change shape for two reasons. First, the portions higher up the rope have higher tension, and therefore higher speed. They will "catch up" to portions of the wave packet further down. Second, because the tension is changing, the wave equation now has a term related to the first derivative of the displacement of the rope. Specifically, I tried assuming that the displacement of the rope is only horizontal and that the slope of the rope is never far from vertical. I made $y$ a coordinate measuring up from the floor and $x$ a coordinate to the right. Letting the tension be $T(y) = T_0+\lambda g y$, with $\lambda$ the mass per unit length, I got the wave equation ( **edit:** important typo corrected) $$\frac{\partial^2{x}}{\partial t^2} = g \frac{\partial x}{\partial y} + \frac{T_0+\lambda g y}{\lambda} \frac{\partial^2 x}{\partial y^2}$$ but I don't know what to do with it.


A vector field $\vec{F}$ from cartesian to spherical

I have a vector field F that has the same magnitude and directions at all points in space. If I choose $z$-axis parallel to $\vec{F}$, so the cartesian coordinate of $\vec{F}$ is $\vec{F}=F\hat{z}$, How can I express $\vec{F}$ in spherical coordinates? My guess = $\vec{F}=\cos(\theta)\hat{r} - \sin(\theta)\hat{\theta}$ I think?


Frequency Modulation

If FM radios work by modulating the frequency, how is it that we can tune into a specific channel, and hear a song or station? Wouldn't the channel need to be modulated along with the varying frequency?


Limit on velocity in Minkowski Spacetime geometry

Let A be a rocket moving with velocity v. Then the slope of its worldline in a spacetime diagram is given by c/v. Since it is a slope, c/v = tan(theta) for some theta > 45 and theta < 90. Does this impose a mathematical limit on v? If so what is it? As in, we know tan(89.9999999999) = 572957795131. And c = 299792458. Using tan(89.9999999999) as our limit of precision, the smallest v we can use is: c/v = tan(89.9999999999) => 299792458 / v = 572957795131 Therefore, v = 1911.18 m/s What is the smallest non zero value of v? Is there a limit on this?


Electron Fermi gas

My question is about 2-dimensional Fermi gas of electrons.   1. What is magnetic susceptibility when $T<<T_F$ (where $T_F$ is Fermi Temperature) and,   2. What is the ratio between Pauli and Curie spin susceptibilities?


Why is it wrong when I try to use Newton's Law?

> A 25 kg block is connected to a 30 kg block by a light string that passes > over a frictionless pulley. The 30 kg block is connected to a light spring > of force constant 195 N/m, as in Figure P13.59. The spring is unstretched > when the system is as shown in the figure, and the incline is smooth. The 25 > kg block is pulled 15 cm down the incline (so that the 30 kg block is 35 cm > above the floor) and is released from rest. Find the speed of each block > when the 30 kg block is 20 cm above the floor (that is, when the spring is > unstretched). ![image of blocks on ramp and spring](http://www.webassign.net/sf/p13_59.gif) I wrote out all the equations For the 25kg mass $$\sum F_x = T- mg\sin\theta = ma$$ $$\sum F_y = n = mg\cos\theta$$ For the 30kg mass $$\sum F_y = kx + Mg - T = Ma$$ Using energy, I got v = 0.91m/s, but using Newton's Law I got 0.95m/s EDIT: **Energy** $E = Mgh + \frac{1}{2}kh^2$ $E' = mgh\sin\theta + \frac{(M + m)}{2}v^2$ $E = E'$ Solving, $v = \sqrt{\frac{2Mgh + kh^2 - mgh\sin\theta}{m + M}}$ **Forces** $T- mg\sin\theta = ma$ $kh + Mg - T = Ma$ Add the two equations $kx + g(M - m\sin\theta) = (M + m)a$ $\frac{kh + g(M - m\sin\theta)}{M + m} = a$ $v^2 = 2\frac{kh + g(M - m\sin\theta)}{M + m}h$ $v = \sqrt{2\frac{kh + g(M - m\sin\theta)}{M + m}h}$ The source of the mistake seem to lie with the 2 in front of the kh


storing energy (as mass)

When chemical energy is released mass is reduced, if only by a negligible amount. Presumably that's true for all energy. And presumably that works in reverse as well: storing energy involves an increase in mass. It seems to follow that moving some object against a gravitational gradient, increases the mass of the object -- potential energy is being stored. Somehow that's difficult to understand. Is it really true?


Why is copper a metal?

**Context** : Solid state physics **Question** : Can it be reasoned in the context of solid state physics (perhaps within the band theory of solids?) that copper is a metal ?


Does a coronal mass ejection change solar neutrino emission rates?

Does the CME and neutrinos have any relation? CME is measured by Corona graphs.. How do they measure neutrinos coming from the sun? Does any of these have effects on earth's magnetic field or atmosphere..?


Why do rolling disc (coin) move in circular path?

We have a coin that is rolled such that it's tilted at an small angle $ \theta $. **Question::** What turns around rolling disc so that it traces circular motion (spiral as it's speed decreses)? **Problem statement::** In the following figure, what turns the disc so that it follows the circular path? ![image 1](http://i.stack.imgur.com/0Mp7R.jpg)   **Solution::**   I interpreted the problem incorrectly. Only a sliding body would have a torque like that.


Is pressure distribution affected by shape

We have two iron (assume real-life stiffness) manhole covers resting on friction-less, perfectly smooth shims on flat ground. One is circular and the other square. If a force F is applied vertically downwards at the center of each manhole cover, is the pressure on the manhole shim distributed uniformly in each case? i.e. is the downward force at any given cover-contact-point on the shim the same for both manholes.


expectation vaules of commutator and anti commutator (momentum and position)

what is expectation vaules of commutator and anti commutator when the case is momentum and position. when the case is commutator: $$\langle i\hbar\rangle=?$$ when the case is anti commutator: $$\langle - i\hbar \rangle=?$$


Why Are Maxwell's Equations Preferred Before Those Proposed by H. Hertz?

In a recent exchange it was pointed out that in certain situations Maxwell's equations need Lorentz force as a patch (the latter not being part of those equations neither is it derivable from them). It appears that the equations proposed by H. Hertz lack that deficiency while retaining Lorentz invariance. What might be the reason for using Maxwell's equations instead?


What are typical values of the critical thickness of insulation?

As most people who've had any elementary heat transfer course are aware, when insulating a pipe, wire, etc, there is a critical thickness for the insulation below which it causes greater heat transfer than no insulation would. In actuality, since the critical radius is a minimum, one needs to go even farther before it catches up to the uninsulated case. This effect is due to the thermal resistance from the added insulation (a function of material and thickness) being less than the decrease in thermal resistance to convection brought about by the increased surface area due to the thicker insulation. A more detailed explanation is found here What I wonder is how large this critical thickness typically is for realistic cases such as an electrical wire, a steam pipe, or a water pipe (all cases in STP air)? I realize exact answers aren't possible given the fact that it will vary with material type, temperature diff., and size. The motivation for this is that it theoretically could prevent wasting insulation or provide a practical avenue for energy dissipation, but Fundamentals of Heat and Mass Transfer 6e by Incropera et al. indicates that the effect is usually too small to be of practical importance.


Principle of Caratheodory and The Second Law of Thermodynamics

**Background** _Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:_ > In every neighborhood of any arbitrary initial state $P_{0} $ of a physical > system, there exists neighboring states that are not accessible form $P_{0} > $ along quasi-static adiabatic paths. The above statement is taken from 'Heat and Thermodynamics' 8th Ed by Zemansky and Dittman and it provides a very concise discussion on the topic which I did not find very illuminating. Moreover, Wikipedia states it slightly differently as: > In every neighborhood of any state S of an adiabatically enclosed system > there are states inaccessible from S. _With this formulation he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics._ **My questions are:**   * What is the exactly meant by _Adiabatic Accessibility_ and how is this relevant to the formalism of The Second Law?   * How is this formalism equivalent to the Kelvin Planck and Clausius statements of the Second Law?   * Except for the satisfaction of having an axiomatic approach, does this provide any advantage over the Kelvin-Planck formalism using heat engines? P.S: The text in _italics_ has been copied from Wikipedia.


Covariant derivative of a vanishing tensor component

Is the covariant derivative of a vanishing tensor component necessarily zero?


How do I calculate DC motor speed for a given load?

Suppose I have a robot of a given mass, and I'm choosing between 2 different wheels and 2 different motors to put on it. For each wheel I have the diameter, and for each motor I know the stall torque and free speed. How would I figure out which motor and wheel combination will make the robot move the fastest? My calculations show to use the big motor with the big wheels, but the small motor with the small wheels goes faster than the big motor with the small wheels. I am not sure my calculations are correct, I need to know the correct way to work this problem.


Diffussivity of gases, molecular weight and Lewis number

I am trying to understand the following figure of laminar premixed flames: ![enter image description here](http://i.stack.imgur.com/hCKz4.png) The Lewis number is defined as $Le = \frac{\alpha_{mix}}{D_{fuel}}$ where $\alpha_{mix}$ is the thermal diffusivity of the mixture and $D_{fuel}$ is the mass diffusivity of the fuel. The propane fuel has larger molecular weight than air and methane has lower molecular weight compared to air. It is not clear to me why mass diffusivity increases as the flame gets richer in the case for propane (Lewis number drops) but the converse is true for methane. I know it has to do with the difference in molecular weight, but why does diffusivity increase for propane if you introduce more propane into the system but decrease in the case of methane? I am therefore not sure why Le decreases in case of propane but increases in the case of methane as the mixture gets richer. Maybe the thermal diffusivity is changing?


What is the correct Hamiltonian for a system of coupled quantum oscillators?

The Hamiltonian (see Eqn. 1 in Appendix 2 of this paper) for a system of coupled quantum oscillators is given as $$H=\frac{1}{2}∑_{i}p^{2}_{i}+\frac{1}{2}∑_{j,k}A_{jk}q_{i}q_{k}$$ Yet, in my QM course, the Hamiltonian for such a system was given as $$H=\frac{1}{2m}∑_{i}p^{2}_{i}+\frac{k}{2}∑_{i}x^{2}_{i}+\frac{K}{2}∑_{i}(x_{i}-x_{i+1})^{2}$$ where the third term represents the coupling between the oscillators. Why aren't these equations equivalent?


Entangled particles

So we have two particles (A and B) that are entangled. From what I understand, entanglement isn't destroyed, it is only obscured by subsequent interactions with the environment. Particle A goes zooming off into outer space. 10 years later, Particle B becomes incorporated into my brain. The day after that, an alien scientist measures the entangled property on Particle A. This will have some immediate non-local effect on Particle B won't it? And since B's state has been altered (in some sense), and it is part of my brain, then my brain state has been altered as well, hasn't it? Maybe only a tiny amount, obscured by the many environmental interactions that the two particles have been subjected to since the initial entanglement, but in a way that is real and at least conceivably significant. And if that is true, then to the extent that mental states supervene on brain states, my mental state would also have been altered by non-local effects. Or is that wrong? Thanks!


Other possible theories (other than string theory) which are generalizations of the standard model with incorporation of gravity

The only finite mathematical framework that incorporates both the standard model of particle physics and gravity under one umbrella that I am aware of is string theory. I would like to know whether there are any other mathematical possibilities exist which do not depend on supersymmetry and still consistent with the standard model and gravity and produce finite answers. In a nutshell my question is: can there be any alternative to string theory? (Remember, I am not talking about only gravity. I am talking about gravity as well as other phenomena).


$\mathrm{d} \Omega_{CM}$ for a $1\rightarrow 2$ particle decay?

The differential solid angle is described in e.g. Srednicki's QFT text but only for the case of scattering. Because in the case of scattering it's defined with respect to the incoming three-momentum ${\bf k_1}$ and it's scattered three-momentum ${\bf k_1'}$. This definition would not make much sense for a particle decay (I can't at least see how it would easily carry over). What about the case of **particle decay**? What is $$\mathrm{d} \Omega_{CM}$$ in the case of particle decay? Any good references where it is properly defined?


Electric field inside a metal nanoparticle?

In localized surface plasmon resonance, the conduction electrons of a metal nanoparticle oscillate in tune with the frequency of the incident electromagnetic wave . The effect is dominated by electrons present on the surface of the particle as it is a nanostructure and surface atoms constitute a major portion of total number of atoms. As a dipole mode is created , electric field strength nearby the particle on outside is increased but what happens inside the volume of the nanoparticle ? For a small metallic nanoparticle , let us say 20 nm , the field (considereing a 500nmEM wave)totally penetrates the particle and the electrons of the whole particle (volume + surface ) will resonate at plasma frequency, then there will be a field gradient generated inside the particle too but inside metals , a field cannot exist. I am a little confused in this aspect.


Questions on electric field formed in conductor and the magnetice field it moves through

When a conductor (e.g. a wire) is moving through a uniform magnetic field, it will create a electric field in the wire as the magnetic field exerts a force on the moving conduction electrons in the wire and create a potential difference. ![the wire is so thick!](http://i.stack.imgur.com/2kBWr.gif) And here is my questions:   1. (If we can get the magnetic field to exert a super strong force on the electrons) Is it ever possible for the electrons in the wire to get separated from the wire like what happens when high energy photon hits an atom and knock off its electrons?   2. Since $F_e$ and $F_b$ always act in the opposite direction, why don't they cancel off one another (so the net force is 0) when the charges are in the electromagnetic equilibrium ($qE = qvB$) ?   3. What's so significant about the this equilibrium ($qE = qvB$)? The formula for motional emf $emf = vBl$ is derived from the fact that $E = vb$ and that was because the charges are in the electromagnetic equilibrium. But wouldn't this formula $emf = vBl$ be inaccurate if we have a motional emf that are caused by charges not in the electromagnetic equilibrium ($qE = qvB$)? Why when there is a motional emf we assume that the chargers are in this equilibrium? I'm still learning secondary school physics and I'm very new to this topic and so please explain it in laymen terms if possible. Thank you so much.


Determine whether the light will reach certain points or not

I've got an interesting question which goes like that: suppose we have a long cylindrical rod made of a transparent material (see illustration below). A point source $O$ which sits on rod's axis emits light in all directions. The light is partially reflected and partially refracted (i.e. goes to the air). The question is - which points will the light rays reach? (1) A only, (2) A and B only (3) All the four points (4) Not enough information ![enter image description here](http://i.stack.imgur.com/ARRM0.jpg) According to the book the right answer is (3), but I'm not so sure that this is correct - one the one hand, the light is emitted in all possible directions but on the other hand, my friend thinks there could be a total internal reflection at some point. In both cases there's no strict proof for either answer. By the way, there's no information about positions, indices of refractions and so on.


Radiative cooling time for a black body

Okay, so this is confusing me a bit. How can I calculate the time it takes to cool a perfect black body from an initial temperature $T$ down to equilibrium temperature (say, 3 K for space)? I know that $$\frac{P}{A}=\sigma T^4,$$ but I'm stuck at getting $t$ out of there.


speed of light observation

I was on a ship at night travelling parallel to the shore watching a lighthouse beam hitting the concrete sea wall. As the beam hit the wall at ninety degrees and continued to rotate I saw what appeared to be the light splitting into its spectrum and streaking away at incredible speed. Was I seeing the actual speed of light as the beam's direction changed?


Is the Earth negatively or positively charged?

The Earth carries a negative electric charge of roughly 500 thousand Coulombs. Does that mean the Earth is negatively charged?


Twin paradox - observers counter orbiting Earth

Imagine three observers - one (A) stationary on the surface of Earth (latitude 0 deg) and two others orbiting the planet in the same circular equatorial orbit just in the opposite direction. When the orbiting observers B and C meet each other just above the observer A all theisr clocks are synchronized. When their meet again what will their clock show? They move relative to each other, but experience equal acceleration and gravitational potential. Will the orbiting observers have their clock synchronized all the time? If the time intervals differ who measures greater time and why? Suppose Earth in non- rotating itself and with perfect spherical symmetry.


Why is radiation at the relativistic limit characterized by 1/gamma angle?

Trying to think of a reason as to why? Also, the factor that radiation is zero on axis? I haven't been able to resolve these two fundamental principles in my head :(


How do I know an initial speed of a thrown object using the max height

> The simulation being referred to is in box2d An object is thrown to the max height of $h$ with gravity of $g$, what is it initial speed?   I tried the following:   $v = v_0 - g t$   $0 = v_0 - g t$   $t = \frac{v_0}{g}$ $h = v_0 * t - \frac{1}{2}g * t^2$   $v_0 = \sqrt{2 * g * h}$ But putting it into physical simulation gives different max height, is the equation wrong or its a simulation artifact?


Cylindrical charge distribution

I'll just list the problem first and then explain what I need help with. > Consider a cylindrical charge distribution extending from $r = 0$ to $r = > 8.2 $cm of charge density $\rho = r/a_0$, where $a_0 = 21 cm · m^3/µC$. This > cylindrical charge distribution is surrounded by a dielectric shell of > dielectric constant $6.72$ whose inner radius is $17.7 cm$ and outer radius > is $25.9 cm$. I cannot seem to find a formula for finding the electric field at a given distance. Any suggestions are where to even start?


Proper Time with a non-zero Initial Velocity

I have a question regarding how to find the proper time for a body with an initial velocity to slow down to 0. For example, the equation I have been working with looks like: $$\int^\tau_0 d\tau=\int^t_0\frac{1}{\sqrt{1+\frac{(\alpha t+v)^2}{c^2}}}dt$$ Where $t$ is equal to the coordinate time for the rocket to slow down, $\alpha$ is the proper acceleration and $v$ is the intial velocity. Integrating gives: $$\tau=\frac{c}{a}arcsinh(\frac{\alpha t+v}{c})$$ I believe, however, that this is incorrect as the values it returns are far too small to be a legitimate value for proper time. I do believe that this is on the right track to deriving the correct value.


Light Rays that are Perfectly Parallel

I just heard this simple reasoning in a documentary film: > Light rays from distant stars are perfectly parallel. This is pretty interesting thought. In nature, it is hard to find something really perfect. For example, there are no isolated systems -- only quasi- isolated. But here it looks like indeed the light rays are perfectly parallel. **Maybe this can be confirmed by calculations?** \-- e.g. that angle between two light rays cannot be infinitely small (Planck constant will be probably involved?).


Why is renormalization necessary in finite theories?

Michael Brown made the following comment here: > The modern understanding of renormalization (due to Kadanoff, Wilson and > others) is hardly controversial and has nothing really to do with > infinities. It is necessary even in completely finite theories, but the fact > that it fixes infinities in QFT is a bonus. Can anyone explain what this means?


Optimal slit width in Young's double slit experiment

I'm trying to do Young's double slit experiment at home. Note that I don't have a laser, only a torch. I could get a bulb or use a candle though, if it helps I built the slits by cutting into a black chart paper with a knife. I tried to build a setup by placing a single slit, double slits and a screen one after the other, and shine a torch through the single slit. I tried varying distance between single-slit/double-slit and double-slit/screen which did not help. **_I did not observe any interference pattern._** All I got was two parallel bright fringes instead, like the ones you would get by shining a torch through two very thick slits. > I think that is precisely the problem, that the slits are pretty thick. > **What is the optimal slit width I should have to observe an interference > pattern? How do I build that with a chart paper?** I have seen videos of this experiment online, where people use pencil leads and _hair_ (!!), but they both use a laser. _I don't want to use a laser_ (just because I don't have one, and I'm probably too lazy to go and get one!). Note that you may suggest using any other materials that might be easily available at home.


Rigorous distinction between quasiparticles and collective excitations

I would like to hear your opinion on the question whether there is an accepted distinction between both concepts in condensed matter physics. I would tend to use the word quasiparticle for dressed fermionic particles and the term collective excitation in the case of bosonic excitation. But considering for example magnons in ferromagnets makes it very arbitrary kind of definition since magnons can be viewed (if I am not wrong) both as quasiparticles and also as collective excitations.


force of hail falling straight

There is a hail storm and it's falling straight down (0 wind factor) will the hail meet a moving car with greater force than a car standing still?


Dirac's remark that inspired Feynman when formulating path integral

When Feynman was trying to formulate path integral of quantum mechanics, he was inspired by Dirac's remark which roughly states that $e^{i\frac{S}{\hbar}}$corresponds to the transition amplitude, where $S$ is the action. I'm wondering what was Dirac's original argument behind this statement.


Gauss' law and ions?

My text book says that with we have a singly ionized sodium atom net charge +e and if we choose a spherical surface centered on the ion and large enough to contain it all we do not need to know the distribution of charge inside the ion. And using Gauss' law can find the the electric field outside the sphere radius R as $E(R)=-\frac{e}{4\pi \varepsilon_0R^2}$. Here is my problem surly we need to know that the charge is distributed evenly around the center?? If we this was not the case then all the charge could, say, be all at one side of the sphere and thus the electric field at that side will be greater then the electric field at the other side of the sphere will it not??


Work and transfer of energy

A businesswoman is applying force of 12 N [or more] to carry her briefcase for a horizontal distance of 200 m. How much work is she doing on the briefcase?. I tried multiplying force by distance but the answer I got is different from the answer in the answers sheet. The correct answer is 0. I don't know why We have to use this formula: W = F * D


Formula for getting energy required to accelerate to a certain speed?

I am curious how many newtons it takes to accelerate an object with a given coefficient of drag, reference area, weight, and air pressure? I know this may sound like a simple equation but I have no idea where to find this formula. Thanks.


Eigenfunctions v.s. eigenstates

Is there a difference between "eigenfunction" and "eigenstate"? They seem to be used interchangeably in texts, which is confusing. My guess is that an "eigenfunction" has an explicit representation/argument, e.g. in terms of position, $x$, whereas an "eigenstate" is abstract. Thanks in advance for any clarification! :) But then again, some texts violate this rule, so I'm prolly wrong...


Was Planck's constant $h$ the same when the Big Bang happened as it is today? Planck's constant : $$h= 6.626068 × 10^{-34}\, m^2 kg / s,$$ $$E=n.h.\nu,$$ $$\epsilon=h.\nu$$


Totally antisymetric wavefunction: clarification about terminology

Pauli's Principle says: "The wavefunction of two identical fermions must be totally antisymmetric". I know that, for a antisymmetric wavefunction, $(-1)^L*(-1)^{S+1}*(-1)^{I+1}=-1$ "totally antisymmetric" means this relation or it means that these 3 relations: $(-1)^L=-1$ and $(-1)^{S+1}=-1$ and $(-1)^{I+1}=-1$ must be verified simultaneusly?


With these two elements how can the isotope half-life be determined using first order kinetics? nobelium-259 half-life recorded @ 58 minutes   fermium-253 half-life recorded @ 3 days Although I do not know, there is probably an average starting concentration that can probably be determined by the name of the element. Maybe in the comments someone can hook me up with some estimate averages so the expansion of the math could be given in the answer.


How to quote answer when error is smaller than significant figures of data?

We have these rules:   1. Can't quote answer to more significant figures than original data.   2. Error quoted to one significant figure unless first significant digit is a one. In which case quote error to 2 significant figure. But with original data to 3 significant figure, I get the answer **13.15(6)**. i.e. **13.15 +/- 0.06** How do I quote this to 3 significant figure without breaking the rules? If I round the error to 1, I still have to have a trailing 0: **13.20(10)**


Verlet timestep

I am using velocity Verlet in molecular dynamics. Is just a simple question, if a simulation using time-step femtosecond, in velocity Verlet is just necessary $dt = 10^{-15}$ to use femtosecond?


Tension in vertical circular motion

In vertical circular motion we conserve energy for calculating velocities at a point (if initial velocity given). But, energy can only be conserved when forces are conservative. Tension is not a conservative force. Does it not affect the particles velocity? Is the tension's role only to provide centripetal acceleration?


Extremal black hole with no angular momentum and no electric charge

A black hole will have a temperature that is a function of the mass, the angular momentum and the electric charge. For a fixed mass, Angular momentum and electric charge are bounded by the extremality condition $$M^2 - a^2 - Q^2 \gt 0$$ Exactly at the extremality boundary, both entropy and temperature are zero. Suppose i create a black hole with a spherically symmetric, incoming wavefront of electromagnetic radiation **in a pure quantum state** (that is, the density matrix satisfies the property $\rho^2 = \rho$). The wavefront is shaped in a way such that the whole energy of the packet will be inside the Schwarzschild radius, which will form an event horizon. Since the wavepacket is as nearly as pure as it is physically possible to create, the quantum (Von Neumann) entropy is zero or nearly zero. But the formation of the black hole does not create nor destroy entropy, so the black hole must contain zero or nearly zero entropy as well. So the black hole seems to be extremal (it has zero temperature) but it nonetheless does not have any angular momentum (it is formed from a wavefront with zero net polarization over the whole sphere) and it does not have any charge (electromagnetic radiation is neutral). > **Question:** what "hair" does have a black hole formed from such a pure > state, so that it can be extremal and still do not have angular momentum or > electric charge (which are the _classical_ hair that we come to expect from > classical general relativity) _This question is a mutation ofthis question, but while that specific question tries to look what input black hole states create specific output (Hawking) radiation states that are far from thermal from a statistical point of view, this question is specific about extremality that is unrelated to angular momentum and charge_


Sound - for purposes of vibration

What is the best way to distribute noise from more than one source (I'm envisioning a system with many), within a dome, with the ground as its primary target, at optimal frequencies and volumes to create maximum vibrations on the ground? Think, this picture - http://hyperphysics.phy- astr.gsu.edu/hbase/acoustic/imgaco/foc3.gif \- but with more than one source. ![enter image description here](http://i.stack.imgur.com/Zt4xK.gif) I'm assuming you would want the dome to be perfectly circular, and have the point at which the dome wall meets the ground at a slight incline (as opposed to vertical) so all the sound stays directed at the ground. Imagine a tennis ball which is cut in half, but was split just before the equator. You take the side that is smaller, and flip it over. Thats what I am imagining. I'm just judging from the above drawing that you would want the sound sources crammed as close as possible in the center of the dome, in as perfect as a circle as you could design it, and maybe directed so that they are all pointed to the center of their own, independent, equally sized areas (the dome being the total area of the surface that the sound is being pointed at). It seems like if you could cram all the sound into a point of singularity and put it at the center of the dome, and then release it somehow - that would be ideal (although that sounds pretty complicated). Not sure about frequencies or volumes.


Is there a good Modern English translation of Newton's "Philosophiæ Naturalis Principia Mathematica" available for free viewing or download?

I have found several nice versions available for purchase, but all of the free ones are either Latin, old English, or pixellated. I searched google and was only able to locate the free ones linked to from the Wikipedia page on Philosophiæ Naturalis Principia Mathematica.


Which one conducts heat better: water, or air?

If we were to assume room temperature of 20C, which of the fluids conducts heat better, and why?


Do all the electrons form Cooper pairs at absolute zero?

At $T=0K$, do all the electrons form Cooper pairs, or just the electrons near the Fermi Surface do?


Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?


Cross-Section Compendium?

Are there any books or references that exhaustively list cross-sections for various processes in QED or particle physics? I think such a book would make for a very useful reference, and in particular I would like to practice deriving cross-sections.


Is the fine-structure constant a parameter of the standard model?

According to the wikipedia entry on the fine-structure constant: > In fact, α is one of the about 20 empirical parameters in the Standard Model > of particle physics, whose value is not determined within the Standard > Model. but, the wikipedia list of parameters does not mention α:               me  Electron mass       511 keV     mμ  Muon mass       105.7 MeV     mτ  Tau mass        1.78 GeV     mu  Up quark mass   μMS = 2 GeV     1.9 MeV     md  Down quark mass     μMS = 2 GeV     4.4 MeV     ms  Strange quark mass  μMS = 2 GeV     87 MeV     mc  Charm quark mass    μMS = mc    1.32 GeV     mb  Bottom quark mass   μMS = mb    4.24 GeV     mt  Top quark mass  On-shell scheme     172.7 GeV     θ12     CKM 12-mixing angle         13.1°     θ23     CKM 23-mixing angle         2.4°     θ13     CKM 13-mixing angle         0.2°     δ   CKM CP-violating Phase      0.995     g1 or g'    U(1) gauge coupling     μMS = mZ    0.357     g2 or g     SU(2) gauge coupling    μMS = mZ    0.652     g3 or gs    SU(3) gauge coupling    μMS = mZ    1.221     θQCD    QCD vacuum angle        ~0     v   Higgs vacuum expectation value      246 GeV     mH  Higgs mass      ~ 125 GeV (tentative)      Is α one of the basic parameters of the Standard Model? If not, then is there a simple formula for α in terms of these other parameters? (My guess is that α can be derived from g1/g2/g3. However, I have been unable to find an explicit formula so far.)


Equivalence Theorem of the S-Matrix

as far as I know the equivalence theorem states, that the S-matrix is invariant under reparametrization of the field, so to say if I have an action $S(\phi)$ the canonical change of variable $\phi \to \phi+F(\phi)$ leaves the S-matrix invariant. In Itzykson's book there is now an exercise in which you have to show, that the generating functional $$Z^\prime(j)=\int \mathcal{D}[\phi] \exp{\\{iS(\phi)+i\int d^4x\hspace{0.2cm} j(x)(\phi +F(\phi)\\}}$$ gives the same S-matrix as the ordinary generating functional with only $\phi$ coupled to the current due to the vanishing contact terms. He then writes, that this proves the equivalence theorem, which I do not fully understand. Suppose I take this canonical change of variable, then I get a new action $S^\prime(\phi)=S(\phi+F(\phi))$ and a generating functional $$Z(j)=\int \mathcal{D}[\phi] \exp\\{iS(\phi+F(\phi))+i\int d^4x\hspace{0.2cm} j(x)\phi\\} $$ If I now "substitute" $\phi+F(\phi)=\chi$ I get $$Z(j)=\int \mathcal{D}[\chi] \det\left(\frac{\partial \phi}{\partial \chi}\right) \exp\\{iS(\chi)+i\int d^4x\hspace{0.2cm} j(x)\phi(\chi)\\} $$ with $\phi(\chi)=\chi + G(\chi)$ the inverse of $\chi(\phi)$. Therefore comparing $Z(j)$ and $Z^\prime(j)$ I get an extra jacobian determinant. Where is my fallacy, or why should the determinant be 1? Thank you in advance


Srednicki's book chapter 8

Reading first page in chapter 8 of Srednicki's it reads: > To employ the $\epsilon$ trick, we multiply $H_0$ with $1-i\epsilon$. The > results are equivalent to replacing $m^2$ with $m^2-i\epsilon$. Now, our Hamiltonian is $H_0=\frac{1}{2}\Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2$ so making the replacement he suggests ($m^2\to{}m^2-i\epsilon$) we would have $H_0(1-i\epsilon)=\frac{1}{2}\Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2-\frac{i\epsilon}{2}\phi^2$ while on the other hand, in reality $H_0(1-i\epsilon)=\frac{1}{2}\Pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2-\frac{i\epsilon}{2}\Pi^2-\frac{i\epsilon}{2}(\nabla\phi)^2-\frac{i\epsilon}{2}m^2\phi^2$ And we see that both terms don't agree. How can both expressions be equivalent?


Sackur-Tetrode equation - clarification required - problem with units

I'm a 2nd year physics undergraduate and recently I've volunteered to give a short presentation on the Sackur-Tetrode equation derivation and its use at removing the Gibbs paradox. I've looked on the Internet and in some books and found many methods for deriving this result. The one I'm going with is the derivation from the book _Thermodynamics and Statistical Mechanics_ by W. Greiner. However, there are some things I'm still not certain about (I dared to expand the argument a little) and in order to seem like I know what I'm talking about, I need some clarification. Here's more or less how I see it going: * * * In order to obtain an expression for the entropy of an ideal gas with $N$ (identical) particles of mass $m$ occupying a volume $V$ and with internal energy $E$, we use Boltzmann's entropy formula: $$S(N,V,E)=k\ln \Omega(N,V,E).$$ This reduces the problem to finding $\Omega(N,V,E)$, the number of microstates of our system which realise the macrostate $(N,V,E)$. A microstate of the system gives information about the entire system at a microscopic level, that is, it specifies the exact momentum and position of each constituent of the system. There are $N$ particles and each one has three directions of position and of momentum, therefore our system has $6N$ degrees of freedom. We imagine a $6N$ dimensional space, called the phase space, where each axis corresponds to a degree of freedom or the system. Each point in this space corresponds to a specific microstate: $(r_x^{(1)},r_y^{(1)},r_z^{(1)}, p_x^{(1)}, p_y^{(1)}, p_z^{(1)},...,r_x^{(N)},r_y^{(N)},r_z^{(N)}, p_x^{(N)}, p_y^{(N)}, p_z^{(N)})$. The microstates which correspond to our particular macrostate $(N,V,E)$ form some sort of high-dimensional hypersurface. The "volume" of this surface, $\phi(N,V,E)$, would give us a quantity related to the number of microstates. Of course in phase space, "volumes" of surfaces have dimension of action raised to some power, and since we're interested in a number, the idea is to introduce some sort of elementary bit of "volume", $\phi_0$ and then divide to produce a number $\frac{\phi(N,V,E)}{\phi_0}$ which we claim to be the number of relevant microstates. Fortunately, it physically makes sense for such an elementary "volume" to exist, since from the uncertainty principle we know that a point in phase space really cannot be specified with infinite precision, and that there is a quantum of action, which is nice because our "volumes" have dimension of action to some power. We already used the fact that we have $N$ particles. Now it's time to take our other givens into account. Let's start with $E$. Classicaly, for an isolated system of noninteracting particles, the total energy is purely kinetic and given by the sum of the kinetic energies of each particle: $$E=\sum_{k=1}^{N} \frac{ [p_{x}^{(k)} ]^2 +[p_{y}^{(k)} ]^2 +[p_{z}^{(k)} ]^2 }{2m}$$ That gives a bound for our momenta coordinates. The position coordinates can be likewise bounded using our last given: the volume $V$. We may assume that the system occupies a cube of length $l$, then for every $k=1,2,...N$ we have: $$0\le r_{x}^{(k)} \le l ; \quad 0\le r_{y}^{(k)} \le l ; \quad 0\le r_{z}^{(k)} \le l ; $$ This seems like cheating a bit (why assume a particular shape of the container), but the assumption can be later dropped because in further calculation we don't really care about the bounds on the position coordinates, thanks to the fact that the total energy is independent of them. So this works for arbitrary $V$. The two previous equations specify the desired "volume" in our phase space. The dimension of this object is $6N-1$ because we only have one constraint, namely the energy being constant. To calculate its "volume" we need to consider the wonderful looking integral: $$\phi(N,V,E)=|A|=\int_{A} dr_{x}^{(1) }dr_{y}^{(1)} dr_{z}^{(1)} dp_{x}^{(1) }dp_{y}^{(1)} dp_{z}^{(1)} ... dr_{x}^{(N) }dr_{y}^{(N)} dr_{z}^{(N)} dp_{x}^{(N) }dp_{y}^{(N)} dp_{z}^{(N)}$$ Where $A$ is the set given by the previous equations. We can simplify this notation by writing $$\phi(N,V,E)=|A|=\int_{A} d^{3N}r \ d^{3N}p$$ Notice that the position coordinates are independent of momentum coordinates, so really, the integral factors: $$\phi(N,V,E)=|A|=\int_{A_r} d^{3N}r \int_{A_p} d^{3N}p$$ Where $A_r$ is the set given by the position conditions and $A_p$ by the momentum conditions. The first factor is simply $V^N$, or $l^{3N}$. The second factor is slightly trickier to calculate, but it's just the surface of a 3N dimensional sphere with radius $\sqrt{2mE}$. This already implies that the units will be $\mbox{[momentum]}^{3N-1}$, and so the units of $|A|$ are $\mbox{[momentum]}^{3N-1} \mbox{[length]}^{3N}$. The actual derivation of the surface of an $n$-dimensional sphere of radius $r$ will be left out of this, and we just quote the result: $S_n = \frac{2 \pi ^{n/2}}{\Gamma (n/2)}r^{n-1}$ Anyway, we have obtained a solution for our "volume" $|A|$: $$\phi(N,V,E)=|A|=2 V^N \frac{\pi ^{3N/2}}{\Gamma(3N/2)}\sqrt{2mE}^{3N-1} \quad \mbox{[momentum]}^{3N-1} \mbox{[length]}^{3N}$$ * * * Here I have a big problem. As said before, the number of states is $\frac{\phi(N,V,E)}{\phi_0}$. I would _like_ for $\phi_0$ to be something like $h^{3N}$ or $h^{3N-1}$ but this gives wrong units! Ignoring this for the moment and just taking $\phi_0 = h^{3N-1}$ here's the rest of the derivation: * * * We thus obtain an expression for the number of microstates: $$\Omega(N,V,E) = 2 V^N \frac{\pi^{3N/2}}{\Gamma(3N/2)} \left( \frac{\sqrt{2mE}}{h} \right) ^{3N-1}$$ Taking the log of this and using appropriate approximations etc.(I skip this here but won't during the presentation ofc.) we arrive at: $$ S(N,V,E) = kN \left( \frac{3}{2} + \ln V \left( \frac{4 \pi m E}{h^2} \right) ^{\frac{3}{2}} \right)$$ This unfortunately leads to the Gibbs paradox. The reason for this is that we've assumed the particles to be disinguishable when in reality they are not. In order to account for this, we need to introduce the Gibbs correction factor into the total amount of states $\Omega$: $$\Omega_C(N,V,E) = \frac{1}{N!} \Omega(N,V,E) $$ Dividing by the amount of permutations accounts for indistinguishability. Going through the same derivation leads to the proper formula now. Thus the paradox is fixed. * * * I would love to hear some feedback about this presentation and particularly with the problem I had with $\phi_0$. Additionally, the introduction of the Gibbs correction factor seems kind of shady to me, I read somewhere that it's again just an approximation and that proper counting would produce a more complicated result than just diving by the factorial, but once again the error is negligible.


Any known relationships among quark masses?

Here is some numerology: Gen 1 Mass(d)/Mass(u) = 2 = 2 * 1 Gen 2 Mass(c)/Mass(s) = 12 = 4 * 3 Gen 3 Mass(t)/Mass(b) = 40 = 8 * 5 Does standard model make any predictions of relations? Do any GUT or string theories make any predictions of relations?


How to find distance of closest approach for a Schwarzschild geodesic?

What is the distance of closest approach in this Wikipedia article? I can't seem to find its definition, and this other question doesn't have an answer I can understand.


Interpreting the equation $\vec F = R\vec v$

> The Aristotle's law of motion, which is incorrect, states that > > _The velocity of an object $\vec v$ is directly proportional to the force > $\vec F$ acting on it_ > > or $\vec F \propto \vec v$ > > or $\vec F = R\vec v$, > > where $R$ is the constant of proportionality which is the resistance to > motion. I want to know how can we interpret this equation $\vec F = R\vec v$ ? Does this equation imply that $\vec F$ and $\vec v$ have the same direction? If it does, then how do we know this through this equation? Does equality mean that they have the same direction? The book in which I actually saw this equation, wrote this equation in this way: $\vec v = \frac {\vec F}{R}$.


Do the same experiments yield different results due to the principle of uncertainty?

When thinking about small particles and their uncertainity, I've allways rather seen them being _all over the place_ rather than randomly changing location. I would think that, in the same time, you'd always have same result, have you tried to do something to them (eg. shoot another particle at them). However, in certain situations, like the evaporation of a black hole, the particle just either is here or it is not. This makes me wonder: if you try something with uncertain particles, does every experiment end up differently because, event though all conditions are the same? (The last requirement seems to be unfeasible so consider it just a thought experiment). Similarly, if there was a separated Big bang would the other universe look different than ours because particles ware randomized differently? Out of the question, I must say that I find it very confusing for particles to really have random locations. Wasn't there ever a suspicion that their location (as well as the probability of it) has a rule we can find out?


Air vs. Water attraction of Oxygen Molecules

The air we breathe is made up of nitrogen and oxygen gases. The water in a pond is made of a single hydrogen/oxygen molecule. If it wasn't for the surface tension on top of the water, oxygen molecules can enter the water and occupy space next to the hydrogen/oxygen molecules. My question is this: All things being equal, is oxygen more attracted to water or air?


Difference between momentum and kinetic energy

From a mathematical point of view it seems to be clear what's the difference between momentum and $mv$ and kinetic energy $\frac{1}{2} m v^2$. Now my problem is the following: Suppose you want to explain someone without mentioning the formulas what's momentum and what's kinetic energy. How to do that such that it becomes clear what's the difference between those two quantities? From physics I think one can list essentially the following differences:   1. momentum has a direction, kinetic energy not   2. momentum is conserved, kinetic energy not (but energy is)   3. momentum depends linear on velocity, kinetic energy depends quadratically on velocity I think is is relatively easy to explain the first two points using everyday language, without referring to formulas. However is there a good way to illustrate the 3. point?


How exactly, or whats the process, rather, of energy changing into matter?

$E=mc^2$ this is the equation by Einstein claiming energy can change from energy to mass. this would happened at the big bang I assume, when electrons and protons were made to create hydrogen and some helium.


Can lightning be used to solve NP-complete problems?

I'm a MS/BS computer science guy who is wondering about why lightning can't (or can?) be used to solve NP complete problems efficiently, but I don't understand the physics behind lightning, so I'm posting here. What seems peculiar to me about lightning is that it seems to follow the "easiest" path to a given destination, as opposed to just the shortest path. This seems analogous (identical?) to the weights in the traveling salesman problem. Since the lightning process ionizes the air and seems to pick the easiest route quickly, I have 2 questions: 1) Does lightning use a method similar to Dijkstra's approach (such that a "good enough" path is found in polynomial time, but not necessarily the "best of all options")? Is it basically an NP "solver"? If the answer to the above question is "no" and it is indeed the best pick: 2) Could the selection process of lightning be used in a computer algorithm to solve a traveling salesman-type problem in polynomial time, and 3) If not, could an arbitrary series of weighted nodes be physically set up somewhere and have electricity passed through it and have the results fed back into some kind of computer mechanism for every time a user wants to know a shortest path? If you could use this approach to solve NP-hard problems quickly, you would have your name on the news and maybe in the history books. So why hasn't anyone used lightning to solve this problem? UPDATE: Another aspect I find interesting: when lightning strikes it ionizes the air to find the shortest path. If the world were totally flat (and the clouds were also) this would ionize all the air equally. Which leads me to believe the route "picked" by the lightning incorporates all points, which seems to be a requisite for finding the best of all solutions.


Flux compactification

How flux compactification solves the moduli space problem in string theory? Please provide some details and, if posible, an example. Thanks in advance.


Why is string theory a two dimensional quantum (conformal) field theory on its worldsheet?

In string theory, we quantize the two dimensional field theory on the string's worldsheet. I have a question about this kind of quantization of string theory: did we have similar theory for point-like particle theory, i.e., we quantize the classical theory on worldline, then we get the usual quantum mechanics? If the string theory is the two dimensional quantum field theory on its worldsheet, then what does the momentum operator measure? Does it measure the expected momentum from initial time to final time represented by the worldsheet (because usually the moment operator measure the moment of particle at some specified time)?


Meaning of the phrase "dipole moment of the combination"

Here is a question I came across in a book: > Three point charges $-q$,$-q$ and $2q$ are placed on the vertices of an > equilateral triangle of side length $d$ units.What is the **dipole moment of > the combination**? I had learnt that the dipole moment is defined for 2 point charges only with equal magnitudes but opposite character.I am in a bit of a fix regarding what the author means by that phrase in bold.Thank you in advance! Edit:I just noticed theoretical physics has been merged here;I sincerely apologise if the question is unbearably stupid.


Supersymmetric Sigma Model

I was working through the Mirror Symmetry book by Clay Math Institute. It deals with supersymmetric sigma model in 10.4 section. It doesn't derive how the action is invariant under the variation. I am trying hard, but stuck at few places. Does any one have any references for such a calculations ? The Lagrangian and variation is given in the following image. ![I have given the lagrangian, if one doesn't have the soft copy of the book.](http://i.stack.imgur.com/mu6Ta.jpg)


Where to find probability density plots for all elements?

Does anyone know where I can find something similar to this, but for all elements? I would love to find something with the same image quality. Also, is there any software that can produce images like these?


Singularities in Schwarzchild space-time

Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element $$ ds^{2}~=~(1-\frac{2GM}{r})(dt)^{2}-(1-\frac{2GM}{r})^{-1}(dr)^{2}-r^{2}(d\theta)^{2}-(r^{2}\sin^{2}\theta)(d\theta)^{2}$$


Do electrons need specific energies to excite electrons

Photons need specific energy levels, equal to the difference between two energy levels to excite an electron in an atom. Is this the same case with electrons that collide with atoms?


What final states can proton-proton collision generate? Where to find listing of possible final states of a particle interaction, for example proton colliding with proton?


What is the dominant interaction in elastic collision pi- + neutron -> pi- + neutron? Is it strong nuclear interaction or electromagnetism? Edit: sorry I meant to say "elastic". There is no matter - energy transformation.


Small scale water power, how does water volume and hight convert into electric energy?

I was playing a little bit with the basic physics behind water power production but I can't get the numbers right. Let's say that I put a windmill that pumps water into a watertank on the top of my house, then I connect some kind of pipe with generator and starts to drain the watertank. How much electric power (kWh) can I get out from a watertank with size $X\text{ m}^3$ placed $Y\text{ m}$ above the ground? How does the formulas look like? * * * Let's put some numbers on this problem, and see where we end up: Let's say the tank is $1\text{ m}^3$, and it is $10\text{ m}$ off the ground so the water will fall $10\text{ m}$ to the generator. Let's connect the generator with a standard garden hose that has a 1 inch diameter, with an area of $2.54\text{ cm}/(2\pi) \sim 5.1\text{ cm}^2$. And then I guess we would get a $10\text{ m}$ column of water pressure, that could be transformed with the area into the force the hight is putting on the system. Something like the earths gravity (9.82)*density*height = 9.82*1*10 ~ 98 Newton (???). And then maybe use that we can find that pressure=Force/Area, but how to move from pressure to energy? * * * Thanks David Zaslavsky for the example, and in theory that would mean that to store 1kWh I need like 40m3 at 10m height. That more or less mean that if one would try to build something like this in real life things need to be quite big. Also thanks Fortunato for illustrate the practical problem in extracting the energy, and that even thou it is hard to get hight numbers it can be worth the effort anyway.


observable quantities are gauge invariant?

I have a simply question, that is whether spatial velocity is gauge invariant. It is seems that under a infinitesimal coordinate transformation the velocity is just transform as other vectors, and it is not invariant. on the other hand the velocity surely can be measured. Best regards @Chern it might be adding a sentence containing this issue to your question, so the essence of the question is "I've read that observable quantities are gauge invariant, but if GR is a gauge theory under Diff(M), then why aren't things like particle velocities counterexamples?" – twistor59 yesterday


Effect of spin on gravitational field?

If a large sphere of uniform density is spinning around one axis, is its gravitational field uniform? If yes, then what if it's not a sphere or not uniform? For example if   * we remove a chunk of the exterior?   * it's made of lead on one side of the equator and aluminum on the other?   * 50/50 lead/aluminum, symmetric around the axis?   * it's slightly pancake-shaped?


Lorentz force law in Newtonian relativity

I know that in special relativity Electric and Magnetic fields mix together in different reference frames, but my question is about classical mechanics. It seems weird to me is that the Lorentz Force law has velocity in it, it doesn't make much sense in classical mechanics, and I assume the Lorentz force law was derived before Special relativity. In classical mechanics acceleration should be the same in all reference frames, so let's take two examples:   1. There's a constant magnetic field and a particle moving, if I'm moving with the particle it looks to me as if it is stationary so it should not have any magnetic field acting on it. so why is it accelerating?    I assume the problem here is that it is impossible to have a constant magnetic field in both frames, but I'm not sure why.   2. There are two charged particles moving in parallel to each other, meaning they have the same velocity, again if I'm moving together with the particles it looks to me as if they are stationary and should not have any magnetic field acting on them. I realize the real answer is to use special relativity, but my question is how did Lorentz think about this before special relativity was discovered and how is this problem "solved" in classical mechanics? I assume this also has some connection with what the definitions of Electric Field and Magnetic Field actually are.


Commutation of operators in quantum theory

I have always written the commutation rules of quantum theory as , $[q,p] = i\hbar\delta _{ij}$ But seems that some people write this as, $[q^i,v_j]= \frac{i\hbar}{M}\delta^i _{j}$ (..this is often done in the context of taking the Galilian group limit of the Poincare group…though I am not sure which aspect of it does it emphasize-- the non-relativstic aspect or the non-quantum aspect?..)   * But somehow dimensionally the second form doesn't look okay. Am I missing something? In the same strain, it seems that the operators for "finite boost by $v$ " and is done by the operator $exp(\frac{iK.v}{\hbar})$ and the "finite translation by $q$" is effected by the operator $exp(\frac{iMv.q}{\hbar})$. (..where $v$, $q$ and $K$ are all $3-vectors$..)   * I would like to know how the above is rationalized. To ask again - is the above taking just the non-relativistic limit or is it also a non-quantum limit ?


The physical laws of thermodynamics tell us that electrical impulses must produce heat?

> Health - Controversial New Idea: Nerves Transmit Sound, Not Electricity - By > Robert Roy Britt, LiveScience Managing Editor posted: 14 March, 2007 1:00 pm > ET Rendering shows a biological membrane at its melting point. The green > molecules are liquid, and the red are solid. Molecules of anesthetic reduce > the number of red areas so that the sound pulse can no longer transport its > signal. The nerve is anesthetized. Credit: Heiko Seeger, Ph.D. Niels Bohr > Institute Nerves transmit sound waves through your body, not electrical > pulses, according to a controversial new study that tries to explain the > longstanding mystery of how anesthetics work. Textbooks say nerves use > electrical impulses to transmit signals from the brain to the point of > action, be it to wag a finger or blink an eye. "But for us as physicists, > this cannot be the explanation," says Thomas Heimburg, a Copenhagen > University researcher whose expertise is in the intersection of biology and > physics. "The physical laws of thermodynamics tell us that electrical > impulses must produce heat as they travel along the nerve, but experiments > find that no such heat is produced." How should I read: > "The physical laws of thermodynamics tell us that electrical impulses **must > produce heat** as they travel along the nerve, but experiments find that no > such heat is produced." ? I learned: If only resistive [being coupled to an electrical resistance] electrical signal transmission takes place, there is dissipation of electric energy in the relevant resistance and hence development of heat. But also, I learned: If the signal is purely capacitive [a circuit of ideal capacitors] or inductively [ideal coils], no energy is dissipated. Under ideal capacitors and inductors current and voltage differ always 90 ° in phase. Therefore φ = 90 °. If the current _I_ and the voltage _V_ , then the energy dissipation is always given by: _I_ × _V_ × cosφ And because cos 90 ° = 0, is also the dissipation 0. Is "purely capacitive" or "inductively" not the explanation why no such heat is produced?


How are functional determinants of Laplace-type operators used in physics?

Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in quantum geometry/string theory. Can someone give further details (or may refer to good literature) how the determinant of the Laplacian is used in that kind of settings? What is the "meaning" of calculating the determinant? For example, what is the correct physical interpretation of the following example: Consider a surface $(M,g_0)$ (a Riemannian Manifold of dimension 2) with a conformal structure, so I have the conformal class related to $g_0$. Naively I think: The surface represents a world sheet of a propagating string. What role plays the determinant of the Laplacian? In another context I read about using such determinants to calculate path Integrals (?) Thanks for illuminating comments!


Equivalence principle and radiation from falling particle

I am currently having a hard time solving a problem of GR from Lasenby's book. I can't make it more clear than by quoting the exercise: 7.2 A charged object held stationary in a laboratory on the surface of the Earth does not emit electromagnetic radiation. If the object is then dropped so that it is in free fall, it will begin to radiate. Reconcile these observations with the principle of equivalence. Hint: Consider the spatial extent of the electric field of the charge. Could someone give me a second hint, currently I am stuck because I try to think about an energetic argument: from the laboratory the particle is losing energy from radiation and potential energy from falling, but in the particle none of them is lost. And I am stuck there.


Orbital speed for a circular orbit?

I wanted to look up the formula for orbital speed for a circular orbit on Wikipedia, and I found 2 formulas: > All bounded orbits where the gravity of a central body dominates are > elliptical in nature. A special case of this is the circular orbit, which is > an ellipse of zero eccentricity. The formula for the velocity of a body in a > circular orbit (orbital speed) at distance r from the centre of gravity of > mass M is $ v = \sqrt{\dfrac{GM}{r}}$. I found this weird, because this leaves out the mass of the body orbiting $M$. I would think that this would have some effect on the orbital speed. I figured that perhaps the radius of the orbit indicates the mass of the orbiting body through some physical relation, but I wasn't sure so I continued looked up stuff on Wikipedia and I found: > The relative velocity is constant: $v = \sqrt{\dfrac{G(M+m)}{r}}$. This was the equation I intially expected to see, but now I'm confused because these are 2 formulas for the same situation, right? Or does the word 'relative' indicate a difference?


Vacuum expectation value in QFT

In QFT, one writes the VEV of a field $\psi$ as $\langle0|\psi|0\rangle$. But as I understand it, the fields in QFT are not operators, but just some functions which we use to calculate cross-sections. What does one mean by bra- keting it in such a way as if $\psi$ is an operator? (Edit: a field in QFT is also a solution of the equation of motion, of course.)


Self Stirring Boiling Liquid

A couple of weeks ago, a friend and I were brewing a new batch of beer. We boil our wort in a turkey fryer. Pretty standard setup for beginners. Anyway, our wort was getting pretty close to boiling (not quite bubbling on the top yet.) I started stirring the wort and then stopped. However, the wort continued spinning and the spinning was accelerating. The " **pit** " of the whirlpool was continually getting deeper until it was a few inches deep. Then, while the whirlpool continued spinning at a seemingly constant rate, the wort started shooting up from the middle of the whirlpool in spurts like a tiny geyser. This went on for about 30 seconds or so as we gawked and thought it was cool. But it started making a mess, so we stuck our stirring spoon in there to stop the wort from stirring, which successfully stopped the whole process. A few minutes later when I stirred it again, the same thing happened. I tried using Google to find what this effect might be, but I was unsuccessful in turning up any info.


Why doesn't an electrometer's capacitance influence the measurement of voltage?

I've read on the subject in several books, and none of them mentioned whether we can neglect electrometer's influence on the measurement of the voltage or not. Maybe my question sounds a bit stupid, but I really can't understand why they do not address the fact that electrometer has a certain capacitance, and when we connect its parts to the desired points, the electrometer and the conductor become one equipotential conductor, and certainly its capacitance no longer depends only the conductor we're testing but also on the electrometer. And we know that $q=CV$ so it influences the voltage as well. So why the measurements are still fine? Can we really neglect the capacitance of an electrometer? Or am I completely wrong here? Thanks in advance.


Definition of pressure in Thermodynamics

In Mechanics we define _pressure_ as "the amount of force acting per unit area". It appears naturally in fluid mechanics where we consider a volume of fluid $W$ contained in a region filled with fluid and we ask about the force due to the rest of the fluid on $W$ accross its boundary $\partial W$. Now, in Thermodynamics, pressure is defined by $$P = -\dfrac{\partial U}{\partial V}$$ That is the rate of change of internal energy with respect to the volume. Now why is that? I mean, what is the motivation to define pressure in that way in Thermodynamics? If it's still force per unit area, what force per area are we talking about here? And where is this force applied? In fluid mechanics, for instance, pressure is defined _pointwise_ , that is, if $D\subset \mathbb{R}^3$ is the region of the fluid, then pressure is a field $p : D\times \mathbb{R}\to \mathbb{R}$ such that if $S$ is a surface in $D$ with normal $\mathbf {n}$ then the stress force accross $S$ at $a\in S$ is $p(a, t)\mathbf{n}(a)$. So, $p$ is a force per area which we know _where_ it acts and by _whom_ it is applied. In Thermodynamics, however, it seems the pressure is the same for the entire system. So if we pick a thermodynamic system, pressure is a characteristic of the state of the entire system. This confuses me, because I can't see what this thermodynamic pressure really is and why it is defined as the partial derivative of internal energy with respect to volume.


Why is more angular momentum required to stabilize the orientation of a heavier bullet?

It's often stated that "heavier" bullets require more "spin" to stabilize vs. "lighter" ones. My question is whether this is really a matter of _mass_. I'm assuming that the destabilizing forces/torques are primary aerodynamic and that those aerodynamics are almost entirely a function of bullet shape, not bullet mass. If two bullets are identical in shape and differ only in density, would the heavier one _really_ need more angular momentum to keep it stable during its flight?


Method of images tutorial?

I'm having an exam in Electrodynamics soon. I think I have most of it under control, but the method of images I'm not quite sure about. There is not much in my book about, so I was thinking some of you might knew a website, article or something like that, maybe even with examples, that could help be get that last piece of the puzzle in my head? I have tried Google, but I didn't find much to be honest. That's why I ask here.


Is time not a universal parameter, really, in special relativity?

I want to check that I get it right. The way that I have it in my mind is that, in my frame of reference and from my point of view, SR associates only one time parameter for the whole space, and not distinct one for each point in space. And that it's only when we want to translate the view of the world from a frame of reference to another, do we need to use Lorentz transformations which mix space with time, depending upon the usual parameters ($v$, $x$ and $t$). And that when we say moving clocks evolve more slowly, we don't mean that time depends on the velocity of that clock, because in that case we would be associating time with objects and not with the points of space itself. From our frame of reference, the point at which the clock exists has a time parameter equal to all other points in my frame. Am I wrong/correct?


Speed of Light Question and Einstein

Einstein's clock experiment. What would happen if you were to travel at the speed of light or faster than the speed of light towards the clock? Can we translate this into traveling large distances in space? By that I mean, if light is traveling towards us while we are traveling towards it at the speed of light or faster, shouldn't we be able to arrive at that Galaxy (Sun/planet) in no time at all? Meaning that we would be able to reach a Star before their light has even hit our planet or hit us.


What happens when the string slackens during vertical circular motion?

Consider a particle attached to one end of a string of length $l$ moving anti- clockwise in a vertical circle whose centre is $O$. What exactly happens physically when the string becomes slack and leaves circular motion? I'm guessing that the particle falls downwards until its distance from $O$ is again $l$, whereupon it will re-enter circulation motion but now in a clockwise direction. But I think that this is either incorrect or incomplete.


What are zero modes?

What are zero modes in quantum field theory, and what are they used for? Or, where can I read about them? I was never able to find a good introduction on the subject. I am particularly interested in zero modes that appear in the context of Yang- Mills, QCD and QCD-like theories, at zero or finite temperature. My doubt consists on the nature of these zero modes in these theories: are they special field configurations, or quantum states, or something else? Basically, I'd like to know what's their physical significance.


How big an egg can be?

Please hold on to flag as off-topic. Few days ago I received a question - how big an egg can possibly big? Can it be as big as 1 meter, as a human? (Imagining that there could be really huge animals which would lay these eggs) My thoughts are that an egg should have an upper limit and basically you can find out the biggest possible egg for Earth. By big I assume the biggest volume. I started to think this way - the thing inside the egg should breath, therefore the egg should allow CO2 and O2 to go in and out. The thing inside the egg consumes constant amount of O2 which is related to the volume of the egg. On the other hand the surface of the egg is related to the amount of air the baby can breath. But with the increase of the size of the egg - the volume is growing much faster then the surface. So at some point of time the amount of air going inside of the egg would not be enough. I think that I am on the right track, but can not finish it to receive the answer. P.S. I hesitated about the Math.stackexchange and physics, but assumed that this one is more applicable but please feel free to move it to math if you think it is better


Conformal Compactification of spacetime

I have been reading Penrose's paper titled "Relativistic Symmetry Groups" where the concept of conformal compactification of a space-time is discussed. My other references have been this and this. In case you cannot see the paper above, let me describe what I understand so far: The idea of conformal compactification is to bring points at "infinity" on a non-compact pseudo-Riemannian manifold $M$ (equipped with metric $g$) to a finite distance (in a new metric) by a conformal rescaling of the metric ${\tilde g} = \Omega^2 g$. This is done so that $(M,{\tilde g})$ can be isometrically embedded into a compact domain ${\tilde M}$ of another (possibly non-compact) pseudo-Riemannian manifold $M'$. This allows us to discuss the asymptotic behaviour of the manifold under consideration. In the references above, they mention the following without explanation: > Then observe that any regular extension of $\phi$[$=\Omega^2$] to the > conformal boundary $\partial {\tilde M} \subset M'$ must vanish on said > boundary. This reflects the property of a conformal compactification that > “brings infinity to a finite distance”. This I do not understand. Firstly, what is meant by conformal boundary? Secondly, why should $\Omega = 0$ on the conformal boundary? Is there any good reference for this material?


Why is it easier to measure the specific charge of an electron over the charge?

The electron was discovered in 1897 and the $e/m$ ratio was measured at that time ,but the charge $e$ itself was measured in 1911. Why was it not possible to measure it earlier?


Can a combination of two or more x-ray type beams be made to burn tumours at their intersecting focal points inside a body?

A single x-ray or similar type beam can pass through a human body with little or no damage. Has anyone ever tried to use a second beam that is phased to have no harmful reactions with the tissue except at the exact spot where two or more beams collide. I know they use sound waves to smash gall stones but I have never heard of using two waves that interact on each other. Is it even possible to do this? They also use chemicals collected in a tumour that will react when hit by a beam.


Dimensions of obstacle needed to block light?

How many dimensions are required minimum for an object to block light? in 1d I doubt it's possible. in 2d I think it could do it if only the light was limited to a dimension perpendicular to the 'thickness' dimension in 3d I think it could do it no matter what dimension the object was looked at in. Clarification: 3d -> A block of wood can "block light" (make a shadow) no matter how you shine light at it. Edit: What about what are the minimum dimensions needed to block light assuming you are dealing with an $n$ dimensional space?


Conductors connected with a wire

Let's say I have two spherical conductors with different radii and different amount of positive charge on them. The spheres are far enough from each other. I connect them with a conducting wire. I'm told that the charge is going to flow until both of the conductors gain the same potential (on top/inside of them). Now, intuitively it seems fine but I do not quite understand a few things. 1) The intuition is that the electric field should be zero everywhere when they reach the same potential, because then the charge is not going to flow. But it is not always the case. 2) Shouldn't the wire also contain some charge after the rearrangement? Because if no, then its potential will be zero, and so there will be a difference in potentials between one of the conductors and the wire, so the charge should still flow into the wire. I'll try to rephrase the question, because it seems people keep getting my question wrong - in general, I can't understand why charges should always flow until both of the conductors reach the same potential. I mean, the fact that the conductors have same potential doesn't really tell us that the field which drives charges is zero. What's the **proof** that connected conductors reach the same potential?


Is hydrogen the same everywhere?

Silly thought. Feel free to shoot it down Does a hydrogen atom undergo any kind of change subject to it's environment? If one were to study a hydrogen atom on the surface of Mercury, another above Earth, and a third in interstellar space - would they exhibit any difference/s?


Moment of inertia about a tilted axis

I apologize if it seems foolish, I am a beginner. Suppose I am given an object, whose moment of inertia along the x,z axes are known. Suppose it rotates around a tilted axis, say at angle $\theta$ with the x axis. Can I find out the moment of inertia by the following method? I resolve the rotatory motion of the object into its x and z components- $\omega \cos \theta$ and $\omega \sin \theta$. I know its MOI along the x and z axis, so I find out the work done by the rotation along the x and z axis- $2W_x= I_x \omega^2 \cos^2 \theta$ and $2W_z= I_z \omega^2 \sin^2 \theta$. If the axis passes through the CM, I know the values of $I_x$ and $I_y$ readily. If not, I use the parallel axis theorem to find out $I_x$ and $I_z$. I think the total work done will be $W= W_x + W_y$, so from here I can find the total work done (I am not sure about this, but in analogy to translational motion, it makes sense. But the total work done is also $\frac12 I \omega^2$. So making them equal, I can find I (note- the $\omega^2$ term will cancel out. Will my method give the correct answer here, or do I just have to use the classic irritating method of integration? To be precise, I claim $$I^2= I_x \cos^2 \theta + I_z \sin^2 \theta$$


Physics behind the flow of gas coming out of a balloon

I'm working with stratospheric balloons (latex ones) and I want to put a valve on it so it can float for a longer time. I'm trying to define which valve I should use, which demands I estimate the flow os gas it can address. Also, it would be great to be able to model the problem mathematically, so I can design an altitude control system. Thinking about this, I got some doubts related to the physics of the latex balloon membrane and the gas inside of it. I'm trying to understand what exactly makes the gas inside a balloon come out when we open a valve placed on its "mouth". I know the pressure inside the ballon must be greater than the external in order to the gas flow. But when the system is in equilibrium, the internal pressure must be equal to the external, so the ballon doesn't expand/contract. Or ALMOST that. I think most people thinking about this just ignore or deem negligible the force the membrane of the balloon exerts in the gas inside of it. I came to the following conclusion: $$p_i = p_o + p_b$$ where: $p_o$ = external pressure $p_i$ = internal pressure $p_b$ = pressure exerted by the balloon membrane into the gas in its interior * * * This justifies why the internal pressure would be greater than the external and the system would, still, remain in equilibrium. If this is correct, I can conclude that what will make the gas come out is pb, at any altitude. So, if I have pb, I could use Bernoulli's principle to calculate the velocity with which the gas come out and, from this, find which is the flow rate I need to model the control problem and guide me on the valve specification. I have two concerns with this approach:   1. Is it correct? Physically speaking, this is what really happens? Can someone give me more insights on the dynamics of mechanical tension on the membrane and gas pressure inside the balloon?   2. I can't find a way to calculate the pressure the membrane would exert on the gas (pb). I know that I can calculate the tensions that appears in the membrane surface accordingly with the internal pressure (with pressure vessels theory). I have a feeling that I can derive the force with which the membrane compress the gas from this tensions, and it seems to me it would be only a geometrical/calculus problem, but I'm struggling getting it forward. I have idealized a simplified model in which I apply Hook's Law, but I'm stuck with infinitesimal areas. Any one can guide me here?


Calculation of the quantized Hall coefficient in the Integral Quantum Hall Effect

I have been reading about the QHE over the past couple of days. I am facing difficulty understanding a calculation in this review. www.nimt.or.th/nimt/upload/linkfile/sys-metrology-248-434.pdf In this paper, equation 16 gives the energy levels of a 2d electron gas as $$E_k =\frac{\hbar^2}{2m}[k_x^2+k_y^2]$$ which I have written as $$E_k=\alpha k^2$$ The author then says (eqn 17) that the number of states with energy less than $E$ is $$N(E)=\frac{S_k}{(2\pi)^2}=\frac{\pi k^2}{(2 \pi)^2}$$ $S_k$ is the area in k space. I don't know what the $k$ in this equation is, and if it is the same $k$ that I have taken. I am not being able to understand how they got this equation. I integrated the equation for $E_k$ from $-\sqrt{\frac{E}{\alpha}}$ to $\sqrt{\frac{E}{\alpha}}$. But I am getting it as $\frac{2}{3}\sqrt{\alpha}k^3$. How do I get that equation? Why is that equation true in the first place? What is the $(2\pi)^2$ in the denominator for?


Quantum Anomalies in Non-Gauge Theories?

I'm reading about quantum anomalies in QFT and all the examples seem to arise in gauge theories. Is it true that theories without a local gauge invariance don't have quantum anomalies? I can't think of examples of symmetry violation in any non-gauge theories, particularly the free theories. Then again, there might be a well-known example I'm not aware of!


energy efficiency, transformation from chemical to thermal energy, for district heating plants

If I have a system in which I introduce 100e.u.F (energy units) derived from chemical energy (i.e. fuel) and generate thermal energy to heat water what are e.u. absorbed by water? I.e. the efficiency of the transformation? Basically I would like to know the average/general efficiency of energy transformation for a district heating plant. Considered that the only energy input is fuel and the only energy output is heat (no electricity)? What are the energy loss? Thanks, Luther