title: Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity

text: If I have the following metric: $$ds^2=(1-2\phi)c^2 dt^2 - (1-2 \phi)(dx^2+dy^2+dz^2)$$ $\phi$ being the gravitational potential with $|\phi| << 1$ everywhere. How do I find a coordinate transformation to a locally inertial coordinate system to first order in $\phi$? One method that I know of is writing $\phi$ as an equivalent acceleration $g\sqrt{x^2+y^2+z^2}$, and then make a coordinate transformation such that the particle is accelerating with this acceleration in the new frame. Is this correct? But, more importantly, I am looking for an **alternative method**. E.g. Is it possible to make a coordinate transformation such that the christoffel symbols are zero? How can I write any general metric in a locally inertial frame? Note: I am actually studying this in an SR course which introduces GR very very briefly. But, I know the basic definitions of manifolds, such as differential forms, and tangent spaces. So, I won't mind a technical answer, but it would be great if you could talk about any terms that I am not likely to know in a couple of lines. (or I can find it out for myself, if it is too much hardwork for you). I have studied GR uptil the geodesic equation.


title: Metronome synchronisation applied to swings

text: The movement of several metronomes can be synchronised when a movable floor is utilised which couples the movement of the different metronomes. Is it possible to apply this sort of synchronisation to several swings occupied by people (the ones you have on children playgrounds)? I visualise this as several swings connected to a rigid platform that is free to move parallel to the ground. I assume that the length of each swing line ($L$) is the same and that the length of the swing line is the only factor that influences the swing period ($T$) according to: $$T = 2 \cdot \pi \cdot \sqrt\frac{L}{g}$$ I understand that the metronome is driven by a wound spring and that this part is not present in a swing. People can move their c.o.g. back and forth during the swing, emulating the effect of the spring. Can the principles present in the metronome sync be carried over to swing sync such that swings start swaying together after a certain amount of time?   Are my assumptions correct? Which additional assumptions do I need to make this work?


title: How are these balls reflected after they hit each other?

text: > **Possible Duplicate:**   >  Physics of simple collisions I have 2 photos of the balls, one before the collision and one after the collision. They do a elastic collision. I want to know how is the reflected value calculated. I know it uses dot product and such stuff. But it all is very vague to me. Can anybody show me a working calculation and explain me based on this figures? I have been trying to understand this for past 2 days but I can't get how it all works. This is the image before collision: ![enter image description here](http://i.stack.imgur.com/nNpCv.jpg) This is the image after collision: ![enter image description here](http://i.stack.imgur.com/vlq45.jpg) Thanks in advance :)


title: Physics of simple collisions

text: I'm building a physics simulator for a graphics course, and so far I have it implementing gravitational and Coulomb forces. I want to add collisions next, but I'm not exactly sure how to go about doing it. A quick summary of how this is working so far is: All objects are spheres of a set radius, mass and charge. The mass's and charges of the spheres are treated as point charge/mass for the calculation. Every step in time (about 1/50th of a second) the forces acting on each object are calculated in a nice big nested for loop that figures out the coloumb and gravitational force between 2 objects, for every set of 2 objects, and then they are summed together. I use this net force to determine the acceleration, and the rest is fairly obvious from there. What I want to add in is collisions. I can deteremine pretty easily if a collision is happening (if the distance between them <= radius of one + radius of other), what I am not so certain of is how I should add in the collision force (and would need to do it component wise). I want elastic collisions, though for now I'd just be happy with getting conservation of momentum The information I have easily available are: the velocity of each object (and I mean velocity, speed and direction), the position of each object, the mass/charge (charge obviously not needed here) and the Net force calculated so far for each object for the next step in time. I dont need exact formulas (would prefer if they arent exact), just need a nudge in the right direction.


title: Collision of two balls of finite radius

text: This is your typical elastic collision problem except the balls have finite radius. To be clear:   * two billiard balls in the plane each with radius $\sigma$   * move at constant velocities $v_1, v_2$   * starting at positions $x_1, x_2$ What is the criterion for there to be any collision at all? When does it happen? And at what angle to they collide? I need it to write a computer simulation of the hard sphere model. ![enter image description here](http://i.stack.imgur.com/Lkndc.png)


title: The physics of two circles colliding

text: > **Possible Duplicate:**   >  Physics of simple collisions I have looked around the internet and gradually built up a better understanding. However I'm still not 100% clear on the physics. If I have 2 identical circles (in a 2D world, so they are on the same plane) and they collide, from any 2 directions, how do I calculate which direction they will travel in after the collision. My initial though was to get the tangent between the 2 circles, and each circle would bounce off the tangent in the same a ball bounces off a flat object (angle into the collision equals the angle out, but reversed around the normal): (where the red lines are parallel, and the line between the center of the circles is perpendicular to the red lines. And the angles x1 and x2 are equal in value, as are y1 and y2, but x1 and y1 are not [necessarily] equal) ![](https://docs.google.com/drawings/pub?id=13Kw1gxNjd_7Pd181QdVFzoOAtWXw4LO1mJerdnFrVj4&w=521&h=497) But after reading articles on the internet, it doesn't appear this is right? (NOTE: This is not homework, but rather learning by example)


title: Bouncing back of a ball

text: > **Possible Duplicate:**   >  Physics of simple collisions Let the unit vector along the positive x- axis be i and that along the Y-axis be j. Let us consider a rigid wall with the normal to the wall being along Y-axis. Let a ball with velocity V1 i - v2j move towards the wall. After hitting the wall will it bounce back with a velocity v1i +v2j or v2j-v1i?


title: 2 electromagnetic loops, with light-speed constraints: Is Newton's Third Law violated?

text: I'm a retired Physics Professor, but I admit I'm puzzled by this... Consider two loops of wire, 2 small dipoles B and C , with a common axis z (facing each other) and (say) 30 cm apart B to C. At the speed of light, information (including a change in magnetic field) will require about 1 nanosecond to travel from C to B.   1. Have the current on in coil B for some period of time at the start, so the B-fields at C is established in the +z-direction.   2. Turn loop B off rapidly (fall time < 0.3 ns, say) at the same time that a current in loop C is turned ON (rapidly, rise time <0.3ns, and current in opposite sense with respect to the previous current in loop B).   3. In this way, as the current is turned on in loop C, it is immersed in the field from loop B and therefore it receives an impulse to the right, in the +z-direction.  However, loop B will be "off" (and open so no effective eddy currents) when the "return" field from loop C arrives. **Thus, loop C (which is free to move) will experience a force giving it momentum in the +z direction (to the right), whereas loop B will not experience a force to the left. It seems that Newton's Third Law is violated.**   (The question can also be posed in momentum terms, but I chose to consider forces and the Third Law. )


title: Is accelerating particles through a chain of accelerators a continuous or batch process?

text: At the Advanced Photon Source, they use two accelerators before injecting the electrons into the large storage ring. Is the addition of particles to the storage ring done in "batches" (however small/short they may be), or continuously? By a "batch", I'm thinking where the LINAC would throw however many electrons into the booster ring, which would then accelerate them all together, then dump them all into the storage ring when they're at the appropriate energy.


title: How Earth communicates with Voyager I?

text: After taking a basic signals & systems class and learning about the frequency domain, I started wondering: ### How the heck do scientists still communicate with Voyagers I and II??    Do they send/receive signals in a frequency that is otherwise unusually silent? If not, then how do they send and receive faint signals from something _10 billion_ miles away, when the spacecraft has so little energy and where it seems practically impossible to aim everything 100% correctly (since things are so distant and moving at the same time)? (I wasn't sure what to tag this, so please retag as appropriate, thanks!)


title: Calculation of number density from material density

text: Material density is given by $ \rho =m/V$, where $m$ is mass and $V$ is volume. Again number density given by $n=N/V$, where $N$ is the total number of particle. How can I calculate number density $n$ from material density $ \rho $.


title: How to calculate pressure field in potential flow

text: I haven't ever studied fluid dynamics before and may mix something here, so please, be patient :). Given flow potential of the form (homogeneus flow over a dipole): $$ \phi = u_\infty x -\frac{M}{2\pi} \frac{x}{x^2 + y^2} $$ how do I calculate pressure field? I know the answer is like below, and it's derived using Bernoulli's principle. I just don't know how to get there analytically. ![enter image description here](http://i.stack.imgur.com/NI2t9.jpg) In my materials the answer is: $$ C_p = \frac{p-p_\infty}{\frac{1}{2} \rho u_\infty^2} = 1 - \left( \frac{u}{u_\infty}\right)^2$$ but I don't understand where it comes from.


title: When does voltage drop occur?

text: Why or when does it occur in a circuit? What does it imply when you speak of a voltage drop across a resistor? (Obviously, it probably means that the current's voltage before the resistor is higher than the voltage after the resistor, but why does this drop occur?)


title: Electric field lines

text: If electric field lines cannot terminate in empty space, in the simple example of two equal charges what happens to the line starting from one of the charges toward the other in the middle?


title: How much power can be drawn from stray electromagnetism in the atmosphere?

text: I know this probably varies quite a bit from place to place on earth. But just some rough estimates: if I were to pull power via multiple antennae tuned to a variety of different frequencies, how much "ambient" power could I draw from the atmosphere? Perhaps I'm not asking the right question, but I'm just trying to understand how much unused energy is floating around me right now and if I could use it to say, recharge batteries slowly without an active power source.


title: Oscillations of a fluid particle in a stably stratified fluid

text: Could anyone help me work through the following set of equations: ![enter image description here](http://i.stack.imgur.com/LVAjt.png) I found these online at http://marian.fsik.cvut.cz/~bodnar/PragueSum_2012/Staquet_1-2.pdf and I'm trying to work through them. I get the basics from this i.e. that if a water particle moves vertically it will be subjected to a buoyancy force due to it being surrounded by water of different density. However, I don't understand how the solution has worked i.e. how they went from one stage to the next. Could anyone provide more information about how this solution works? My main aim here is to figure out how the Brunt vaisala buoyancy frequency is derived.


title: What to do when you are living in very cold environment?

text: I am on my way to a shimla, and as it is one of the coldest places of India. Previously I lived in a warm environment of southern india. What would be the correct way to live in a cold environment? I mean to say that would keeping our body warm every time will prevent us from loosing temperature. Or no expenditure of energy would be better?


Why 21cm line observations is more helpful compared to other radiation? Is it only because of its abundance in whole universe or anything else?


title: Why does adding milk to my coffee produce a fractal?

text: Whenever I add milk to my morning coffee I often enjoy watching the patterns which are created. These patterns have a striking resemblance to certain fractals and my question is, "Why?" Oh dear, that is never a good question, so let's try "Why shouldn't they?" The obvious observations are that the water is very hot when the milk is added, so we expect the "coffee particles" (non-milk part) to have a higher velocity relative to the "milk particles". Secondly, the coffee part, having been recently stirred, is often still rotating (clockwise in my case). Finally, the image we observe is a projection of some distinct portion of a "top layer" of the milk and coffee mixture. Add diffusion into the mix and it really seems like some crazy stuff should go down! But I'm afraid I'm not satisfied. Why physics.SE... why?


title: If an electron can't go faster than light, how is energy conserved here?

text: I came across a physics question recently about accelerating electrons with potential differences and it got me wondering. Here is the question that brought the concept to mind. ![Physics Question](http://i.stack.imgur.com/w0coO.jpg) **First off, imagine if the potential difference is actually 260kV and not 350V.** Imagine that we put the electron there with our finger, so we are giving it energy which it stores as potential energy. The amount of energy we give it is equal to $$W=\Delta V\cdot q_e$$ In theory, when the electron accelerates to the entrance of the deflecting plates, it should have gained all of the energy we put into it as kinetic energy. $$1/2 \cdot m \cdot \rm v^2 = \Delta V \cdot q_e$$ $$1/2 \cdot 9.11\, \rm x \, 10^{-31} \cdot v^2 = 2.6 \, \rm x \, 10^5 V \cdot 1.6\, \rm x \, 10^{-19}\, \rm C $$ $$ \rm v \approx 3.02 \, \rm x \, 10^8 \, m/s$$ Therefore, if all of the potential energy stored in the electron was converted to kinetic energy, it should go faster than the speed of light. What would prevent this? And if this is prevented, how is the Law of Conservation of Energy Maintained?


title: Whenever we fill a bottle with cold water and close it, why is it that after a few seconds we notice cold water droplets outside the bottle?

text: Whenever we fill a bottle with cold water and place back the cap tightly closing it, why is it that after a few seconds we notice cold water droplets outside the bottle as if the water is coming out of the non porous plastic bottle? Also the same thing happens when the bottle contains hot water, so we can associate this with the thermodynamics' law of equilibrium, but then how and from where does this water form droplets outside the bottle? Because there can't be so much of moisture around in the air which is going to help in the droplet formation.


title: Warped AdS${}_3$ and symmetry breaking

text: In this article it is explained how on can (in suitable coordinate basis) get a so called warped AdS${}_3$ black hole, by introducing a warping factor. The original metric in 'Euler coordinates' for AdS${}_3$ is $$ds^2=\frac{\ell^2}{4}(-d\tau^2+d\omega^2+d\sigma^2+2\sinh \omega d\tau d\sigma)$$ which can be written as $$ds^2=\frac{\ell^2}{4}\left(-\cosh^2\omega d\tau^2+d\omega^2+(d\sigma+\sinh \omega d\tau)^2\right)$$ This metric has $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$ isometry with 3 Killing vectors for left and 3 for right sector. When we introduce a warping factor $\lambda$: $$ds^2=\frac{\ell^2}{4}\left(-\cosh^2\omega d\tau^2+d\omega^2+\lambda(d\sigma+\sinh \omega d\tau)^2\right)$$ the original symmetry breaks to $U(1)_L\times SL(2,\mathbb{R})_R$, which manifest itself by the fact that now we only have one Killing vector for $U(1)$, and that vector should be $J_0=2\partial_\sigma$. I have the original Killing vectors, but how do I show that warping factor will break the symmetry? Solve the Killing equation with warped metric, and show that the old Killing vectors won't solve it or? Is there an apparent way of seeing this?


title: How scalar curvature of following spacetime can be equal to zero?

text: For an interval of this spacetime, $$ ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})), $$ scalar curvature is equal to zero. Also, Ricci tensor is equal to zero. How can it be for curved space-time (the metric tensor isn't represented as diag (1, -1, -1, -1))? Addition. ![For an explanation odf reason of question creation](http://i.stack.imgur.com/Jf3kr.jpg)


title: Lorentz transformation of classical Klein–Gordon field

text: I'm trying to see that the invariance of the Klein–Gordon field implies that the Fourier coefficients $a(\mathbf{k})$ transform like scalars: $a'(\Lambda\mathbf{k})=a(\mathbf{k}).$ Starting from the mode expansion of the field $$\phi'(x)=\phi(\Lambda^{-1}x)= \int \frac{d^{3}k}{(2\pi)^{3}2E_{k}} \left( e^{-ik\cdot \Lambda^{-1}x}a(\mathbf{k}) +e^{ik\cdot \Lambda^{-1}x}b^{*}(\mathbf{k}) \right),$$ it's easy to see that it equals $$\int \frac{d^{3}k}{(2\pi)^{3}2E_{k}} \left( e^{-i(\Lambda k)\cdot x}a(\mathbf{k}) +e^{i(\Lambda k)\cdot x}b^{*}(\mathbf{k}) \right).$$ when using the property $\Lambda^{-1}=\eta\Lambda^{T}\eta$. Then doing a change of variable $\tilde{k}=\Lambda k$ and considering orthochronous transformations so that the Jacobian is 1, then I get the wanted result ($a'(\Lambda\mathbf{k})=a(\mathbf{k})$) when comparing with the original mode expansion. However, this is not quite right as I would have to justify that $E_\tilde{k}=E_k$ but I can't see how.


title: How does heat transfer between two atoms in solid material?

text: Been looking at heat equation and it's derivation, according to Wikipedia it uses 2 mathematical assumptions. My problem is that although it all seems OK, what is the physics of heat transfer in solids? so far haven't seen answer to the following questions regarding heat equation:   1. Given 2 adjacent atoms, how does one measure the hot and the colds one? ( theoretically at least, no need for a physical thermometer)   2. How does heat transfers from hot to the cold one? ( most likely by photons, but do electrons from one atom get knocked out and knockout the electrons of the other? is that considered a temperature increase or just change of charges?)   3. If this transfer occurs with discreet amount of energy being moved from the hot one to the cold one, what are the intervals (time wise) between each transfer? (In other words what's the time interval between photons being emitted within a solid?)


title: How Is It Possible To Measure Extreme Temperatures? (>2M Deg)

text: Linked Article: Fusion: X-ray laser zaps solid to 2 million degrees > The quest to create nuclear fusion may have come a step closer when > scientists heated solid matter to two million degrees with the world's most > powerful X-ray laser, a study reported on Wednesday. I don't see how is it possible that these scientists manage to measure two million degrees Celcius. Are there such measuring instruments that can measure extremely hot stuff without melting and still get a proper reading, or is it a theoretical deduction?


title: How to measure power used to sustain something in a (gravitational) field?

text: I've read this question and answers, but neither of the posts does seem to address my concerns. The problem is as follows: > Assuming a device that keeps something in a gravitational field on a > constant height, like an electromagnet, how does one measure the required > power? I'm stumbling on that because the equation $P = Fv$ would mean that, assuming the object stays in place, the power is zero. That might be true for, say, a rigid construction holding something up. What if I am _actively_ trying to keep something up, like holding it in my hand, or powering an electromagnet to create an electromagnetic field to keep something in the air? I am applying a constant force, but **the object doesn't move**! I've tried to imagine it from the point of _theoretical_ loss of potential energy I'm preventing, but it only got me to something like $P = g^2 m t$, which depends on time. In fact, every attempt to tackle the energy was in vain, because the velocity of the object in a gravitational field, if left without the support, wouldn't be constant (thus making the power depend on time), and is apparently not fit to describe the problem formally.


title: If I lift a box vertically, why is the work I do equal to the distance I lift it times the force of gravity on the box?

text: I have problems fully understanding the concept of work, so please forgive me if this is simple. If I take a box of mass $ m $, and lift it a distance $ d $ vertically, **why is the work I have done equal to $ gmd $, where $ g $ is the force gravity exerts on the box?** I understand that work is equal to force times distance--so I'm not asking about the definition of work--but if I exert an upward force equal in magnitude to gravity's, won't the box remain motionless, i.e., net zero force, in which case the velocity is constant, and displacement and work done will be equal to zero? Edit: To be clear, **what I'm asking _is not a duplicate_ of "Why does holding something up cost energy while no work is being done?", because I'm not asking about work done on an object with zero displacement, nor is it a duplicate of "What exactly is F in W=∫baFdx?", because I'm not asking about the distinction between the work done by an individual force and net force.**


title: How is work done by a system in static equilibrium?

text: For a little colour: this came from my wondering why fork-lifts are hydraulically driven, rather than electromechanically. My question is this: why is it that some systems, while in static equilibrium and seeming to do no work, actually are work. I'm thinking, specifically, of these examples:   * If I hold a weight straight out, my muscles eventually get tired. Thus, they must have been doing some work.   * If I suspend a weight attached to an pulley driven by an electric motor, I need to apply a voltage in order to keep the weight in place. Since, in both cases, the weight is not moving, a simple (simplistic) analysis would be that no work is being done. However, the expenditure of chemical (in my arm) or electrical (in the motor) energy indicates that some work is definitely being done. What am I missing? Is it that the equilibrium is not truly static and that there are minor oscillations in both systems that result in large numbers of small impulses? Is this simply a system with negative feedback of a small magnitude... it seems static, but isn't really? Or is there some other mechanism (or, indeed, analysis) that provides a more accurate insight into what's going on?


title: Is energy applied by the muscle when it holds a body in the air?

text: > **Possible Duplicate:**   >  Why does holding something up cost energy while no work is being done?    We all know the following Newtonian formulas:   PotentialGravitationEnergy=m*g*h   Kinetic Energy = 0.5m * v^2   W = F * distance   Therefore, if I put an apple on a table, it will not gain nor lose [potential] energy - because F = 0, and distance = 0.      But when I'm not using a 'mechanic lock' like that - but using my body muscle [without locking the bone] to hold the apple in the air - I'm damn right applying energy - my muscle is getting warmer [because glucose burning is very inefficient] - and I feel it stretching.      I DO apply a force towards up - but it's not a 'free force' like the Normal force - but I give no energy to the apple.      Can anyone please explain to me in detail what is actually happening? Thank you very much!


title: How do you calculate the power needed to keep something stationary?

text: 1) If I place an object on a table, it will sit there forever. There's no change in potential or kinetic energy. If I hold the object at the end of my outstretched arm, eventually I can't do it anymore. But the object is still not changing kinetic/potential energy. Where is the loss coming from? 2) Related question: An airplane is flying level in the air. How much power is needed to keep it level? It seems clear that any explanation will revolve around inefficiency in the force-providing mechanism, something to do with how your muscles work for the first question and something to do with drag in the airplane example.


title: Keep an object at a certain hight, does it require energy?

text: If i have an object with a defined mass, like a stone, and put it on the shelf it will sit there forever (lets forget all external influences). But if I have to hold up the stone I will eventually get tired, feel pain, and give up. Obviously I am using up energy to keep the stone where it is. But, if it takes energy to keep an object at a certain hight it could not sit on the shelf forever because it would need infinite energy. What am I missing?


title: What exactly is $F$ in $W = \int_{a}^{b} F dx$?

text: I am trying to teach myself some basic physics, here is something I do not really understand about the definition of work: When moving from $a$ to $b$ (in one dimension), the work done is defined to be $$ W = \int_{a}^{b} F dx $$ Now, say, I lift something up from the floor to a table covering a distance $h$. The work done is supposed to be $mgh$. This result is obtained by setting $F = mg$ in the equation above, which is the force exerted by gravity on the object. However, gravity is not the only force acting upon the object. What about the force used by me to lift the object, why does it not enter the equation?


title: Propeller modelling

text: I need a (very) approximate model of a propeller on an aircraft. My principal question is this: what would the relationship be between:   * Propeller rate of rotation   * Aircraft speed   * Force generated by propeller As an aside, what would the relationship be between the above and:   * Energy required to drive propeller?   * Rotational force between the propeller and aircraft body Any insight would be very helpful, including stating that I haven't given enough information! Thanks


title: So there are 6 quarks, what are anti-quarks considered then?

text: I just recently got into particle physics and the quantum world and I love it. So my first big question is. I watch all these videos and people explain the quarks (up, down, top, bottom, strange, charm). And they all say there are 6 quarks. But every so often someone speaks of an anti-quark. What is this anti- quark if there are only 6 quarks? Is it anti-matter? Is it still a quark? (if so that means there are 12 quarks?) Second question, more for clarification. So there are force carriers and particles. Force carriers are bosons, they carry the strong weak and electromagnetic force and the gravity force carrier is still a mystery as to what carries it (mystery as in we just have not observed the gravity force carrying particle; i.e. higgs/gravitron)? Non-force carriers are leptons and are comprised of only quarks? And quarks can then make more massive particles like protons (uud)?


title: S-Wave for minimally coupled scalar field

text: This question is in reference to the paper here (Equation 3).The extremal 3-brane metic in $D=10$ can be written as: \begin{equation*} ds^2 = A^{-1/2}(-dt^2 +dx_1^2 +dx^2+ dx^3) + A^{1/2}(dr^2 +r^2 d\Omega_5^2) \end{equation*} where \begin{equation*} A = 1+ \frac{R^4}{r^4} \end{equation*} In this background the $s$-wave of a minimally coupled massless scalar satisfies: \begin{equation*} \left[\rho^{-5}\frac{d}{d\rho}\rho^{5}\frac{d}{d\rho}+ \frac{(\omega R)^{4}}{\rho^{4}}+1\right]\phi(\rho) =0 \end{equation*} How do I derive this result?


title: Is a soundproofed wall really only as strong as its weakest area?

text: I've seen a few questions about sound waves and sound travel here on Physics SE, so I'm hoping this question is a good fit for this site. During my internet research on soundproofing, I've come across many acoustics gurus who say that, when soundproofing a wall, you have to account for every single little opening such as a wall or ceiling even if the opening is as tiny as the shaft of a screw. If you don't, you may as well not soundproof the wall at all. Since I have approximately zero understanding of the physics of sound waves, my incredulity brings me to doubt those words. I imagine the contact of sound waves upon a wall being much like taking a handful of pebbles and slinging them against the same wall. When the pebbles leave your hand, they spread out. If I were to sling a handful of pebbles at a group of people from a distance, the chances are high that everyone's going to get hit. If you put up a solid wall in front of them, however, no one will feel a thing as the wall will stop 100% of the pebbles. If you cut a hole 2' in diameter and sling the pebbles, perhaps only a few people will be hit since only the pebbles that make it through the hole will have a chance of striking anyone. If I were to apply the "weakest link" argument here, it would suggest that slinging a handful of pebbles at a wall with a small hole in it would be the same thing as slinging the same pebbles as if the wall weren't there. The only way this could be true is if all the pebbles spread out after leaving your hand, magically coalesced enough to fit through the hole, then spread out again to hit the group of people. Now back to sound travel. How could a small, screw-sized hole totally negate the _entire_ soundproofing effort? While I can understand that a little sound would still make it through that hole, I am not seeing how that little leak would make it as if the wall isn't there at all. Can someone explain how a small opening in a well-soundproofed wall affects sound travel?


title: Activation energy and entropy

text: First assertion If a system is already in a high temperature, adding energy, will increment the entropy in a low amount (compared with a system in a lower temperature). Question (if assertion is right) What if the heat is enough that let molecules breaks (activation energy), this would lead to new multiplicity (more freedom) so more entropy. It is _a higher entropy grow than if the temperature were lower!_ that seems to be in contradiction with first assertion. I see there is something wrong here, but I don't know where.


title: Snowmaking in the tropics - an estimate of water evaporation

text: If I set up a snowmaker in the tropics and sprayed water with it how much water would I evaporate? How would I calculate?


title: The particle mesh ewald method in two dimensions

text: I am attempting to implement the particle mesh ewald method (http://dx.doi.org/10.1063/1.470117) in two dimensions. I am wondering what needs to be changed in the method from three dimensions to two dimensions? One thing I have recently discovered is that the electrostatic potential in two dimensions has a different formula (something like $-q_iq_j\text{log}(r_{ij})$) rather than $\frac{q_1q_2}{r_{ij}}$ as is stated in the paper. If anyone could shed some light on the particle mesh ewald method in two dimensions, and my issue with the electrostatic potential formula, that would be helpful. Cheers


An answer I gave elsewhere. > Some cases to ponder over. > > A closed string splits into two closed strings, which then merge again into > a single closed string. The overall string worldsheet has the topology of a > torus. There is an SL(2,Z) group of large diffeomorphisms acting upon this > worldsheet. The contribution to the partition function comes from summing up > over all contributions with this topology. Suppose you insist upon a > canonical description of this process. In the loop part in the middle, there > is a different slicing associated with each of the SL(2,Z) elements. We have > to sum up over all such contributions. Each such choice gives the same > contribution, but there is no canonical choice of which slicing. If you > consider summing up over all possible slicings, there is the possibility of > "interference" between different choices of slicings because the > configurations you get from a different slicing might be continuously > deformable into that of a different slicing. So, you can't just insist that > we sum up over one possible slicing only. However, if you sum up over all > possible intermediate slicings, you pick up an infinite multiplicative > factor compared to the no string interaction case. > > The other case is a "tree level" worldsheet with two incoming closed > strings, and two outgoing ones. There are the s-, t- and u-channels. They > correspond to different possible slicings. Each by itself gives the same > contribution. Each tells a different splitting story. You don't sum up over > all channels. The problem comes when the two outgoing strings are identical. > Then, you can't even distinguish between the s- and t-channels. > > The moral of the story is, there is no canonical description of string > interactions. How to deal with crossing duality and modular invariance in string field theory?


title: What does it mean to apply an operator to a state?

text: Let's say I have an operator $\hat{A}$ and a state $|\psi\rangle$. What exactly _is_ the state $\hat{A}|\psi\rangle$? Is it just another different state that I am describing using my $\hat{A}$ and $|\psi\rangle$? For example, if $$\hat{A} \doteqdot \text{put chocolate syrup}$$ Then is $\hat{A}$ just a tool to describe a state, like: $$|\text{vanilla ice cream with chocolate syrup}\rangle = \hat{A}|\text{vanilla ice cream}\rangle$$ But on the other hand, we have something like $$\hat{H} |\psi\rangle = E|\psi\rangle$$ I interpret this equation as "If you apply the (time-independent) Hamiltonian to a state, the result is proportional to your original state". But $E|\psi\rangle$ itself cannot be a new state, because it's in general not normalized. So here the operator $\hat{H}$ is just used to establish a mathematical property of $|\psi\rangle$, not to describe another state. You can't say that $\hat{H}$ is a machine that takes a state and returns another state, not in the same way $\hat{A}$ takes an ice cream and puts syrup on it. Or can you? And what does taking a measure mean? If you measure an observable, it returns an eigenvalue and the state collapses into an eigenstate. Is this resulting eigenstate the one you obtain when you apply the operator to the state?


title: Are "broken" planets possible in the real world?

text: In the Star Wars universe, there are mentioned (or rather shown) two planets that had some kind of great explosion in their history. First is Peragus, with its exposed core, that partially exploded during one of early fuel mining operations on the planet; the second is Lola Sayu; the nature of the cataclysm that caused it to break is unknown. Both of the planets are (relatively) stable, even though they are "a bit" cracked. Are such planets possible in the real world? Or, rather, how long would such a planet be stable, assuming that the explosion that happened on the planet (as I assume nothing else could cause such state) wouldn't push it out of its orbit?


title: What is Motivic mathematics and how is it used in physics?

text: In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being relevant. (For example in the last few seconds of this presentation). I would be interested if someone could give even a superficial hint of what motivic mathematics is and how it is applied in physical problems.


title: Correspondence principle

text: Is there a precise mathematical derivation for the correspondence principle for which I can replace $E \rightarrow i \hbar \frac{\partial}{\partial t}$ and $p \rightarrow -i \hbar \nabla$?


title: Electric potential difference from electric field of isolated spherical conductor

text: I'm wondering if my train of thought is correct. Say you have Point $A$ which is $A$ distance away from the center of an isolated spherical conductor, and Point $B$ which is $B$ distance away from the center. If I'm to find the electric potential difference, $V_A-V_B$, I know I can use the relationship $V_A-V_B= -\int_B^A \vec{E} \cdot d\vec{s} = -\dfrac{kQ}{r^2} \int_B^A d\vec{s}$ What I'm wondering is if $d\vec{s}$ is equal to $dr$ in this case. I don't really see any other thing it could be. Using that idea, things simplify down to, $\rightarrow = -\dfrac{kQ}{r^2} \cdot r \bigg|_B^A \rightarrow -\dfrac{kQ}{r} \bigg|_B^A$ Is this correct? I would also appreciate an explanation as to why it is or isn't, since I'm a obviously a little unsure of myself.


title: Could gravity be an emergent property of nature?

text: Sorry if this question is naive. It is just a curiosity that I have. Are there theoretical or experimental reasons why gravity should not be an emergent property of nature? Assume a _standard model_ view of the world in the very small. Is it possible that gravity only applies to systems with a scale large enough to encompass very large numbers of particles as an emergent property? After all: the standard model works very well without gravity; general relativity (and gravity in general) has only been measured at distances on the millimeter scale. How could gravity emerge? For example, it could be that space-time only gets curved by systems which have measurable properties, or only gets curved by average values. In other words that the stress-energy tensor has a minimum scale by which it varies. * * * Edit to explain a bit better what I'm thinking of.   1. We would not have a proper quantum gravity as such. I.e. no unified theory that contains QM and GR at the same time.    2. We could have a "small" (possibly semi-classical) glue theory that only needs to explain how the two theories cross over:      * the conditions and mechanism of wave packet reduction (or the other corresponding phenomena in other QM interpretations, like universe branching or decoherence or whatnot)     * how this is correlated to curvature - how GM phenomena arise at this transition point. > Are there theoretical or experimental reasons why such a reasoning is > fundamentally incorrect?


title: Inclusive and exclusive searches

text: Please, I would like to know what means inclusive and exclusive searches in High Energy Physics. Thanks in advance. Fábio.


title: Translation operator to higher order

text: IN QM, the space translation operator, or generator of Translations is set to be Ie-ie/h_bar*P up to order e. Now my question is what is the physical justification of only going up to order e and do the higher order terms have any meaning? Is it correct to say(if we are only including up to order e) that we are doing a constant linear translation(equal distances in equal times) and hence assuming there is no change in how fast we do it?


title: what interactions would take place between a free proton and a dipolariton?

text: What interactions can be expected to take place between a free proton and a dipolariton, (a) at high energies and (b) at lower energies? A dipolariton is a bosonic quasi-particle mentioned in a recent article in Science. It is a static dipole that consists of an electron bound together with a photon. I suspect the proton and dipolariton would be attracted to one another, but I'm unsure of the specifics.


title: How close would Earth have to be for us to detect it was habitable, and then inhabited?

text: Given our current technology (or technology that is near implementation), how close would a clone of our Solar System (and so also Earth) have to be to us in order to detect that the cloned Earth was in habitable, and also how close would we have to be to detect that there is life on the planet (excluding radio signals the cloned humans have broadcast into space)? Basically I'm asking if we assume the worst case scenario where life only exists on Earth-like planets, and that the life is the same as ours (ie is inteligent, builds cities, etc), at what range with our current technology and methods/techniques would we be unable to detect a planet and civilisation the same as our own (seeing as it's the only civilisation we know about)? EDIT: Another way to put this, assume every star system is identical to the Solar System, using our current technology/techniques what is the furthest planet we could "see" that is habitable, and what is the furthest planet we could "see" that is inhabited by a species identical to our own (so the identical Earth 500 lightyears away would actually be in the year 2513, so we'd "see" it in 2013).


title: How to combine the error of two independent measurements of the same quantity?

text: I have measured $k_1$ and $k_2$ in two measurements and then I calculated $\Delta k_1$ and $\Delta k_2$. Now I want to calculate $k$ and $\Delta k$. $k$ is just the mean of $k_1$ and $k_2$. I thought that I would need to square-sum the errors together, like so: $$ \Delta k = \sqrt{(\Delta k_1)^2 + (\Delta k_2)^2} $$ But if I measure $k_n$ $n$ times, $\Delta k$ would become greater and greater, not smaller. So I need to divide the whole root by some power of $n$, but I am not sure whether $1/n$ or $1/\sqrt n$. Which is it?


title: Is there a simple way to derive a T-S diagram from a p-V diagram for arbitrary processes?

text: Often, for thermodynamic processes only a p-V diagram is shown. Even without hard figures, the shape of the curve can be helpful to evaluate the process. However, it is hard to figure out for real processes. Are there good rules of the thumb to arrive at a T-S diagram from a p-V diagram?


title: What does Power really mean?

text: I've been trying to solve a problem for some time. I have been given conflicting information by both literature, colleagues and people on this very forum. It's a very simple question: **What is the definition ofpower?** _In particular, I would like to know how constant power would accelerate a constant mass over a fixed distance, assuming that no resistive forces are at work, i.e. no friction, no air resistance, etc._ I know that power is the rate at which work is done with respect to time, i.e. $P = \operatorname{d}\\!W/\operatorname{d}\\!t$. One formula that I have seen is that $P = Fv$, i.e. power is the product of force and velocity. According to Wikipedia, this requires $v$ to be constant. Someone on this site, and someone I work with have both said that I need $v$ to be constant for $P=Fv$ to hold. However, I found an article from 1930, which found equations of motion for constant power. Just like the standard equations of motion $v=u+at$, $v^2=u^2+2as$, $s=ut+\frac{1}{2}at^2$, and $s=\frac{1}{2}(u+v)t$ all _assume a constant force_ , i.e. a constant acceleration, L. W. Taylor (1930) found six analogous formulae for motion under constant power. He used $P= Fv$ as his _definition_ for power. **Can I work with $P=Fv$, assuming that $F$ and $v$ vary, provided their product is constant?**


title: Macroscopic laws which haven't been derived from microscopic laws

text: Can you think of examples where a macroscopic law coexists with a fully known microscopic law, but the former hasn't been derived from the latter (yet)? Or maybe a rule of thumb, which works but hasn't been understood yet. Or some emergence which is hard to see from the individual interacting parts. The microscopic arrangement should be simple. So plain complexity like in biology is something different. Most thermodynamic results and entropy-related results you can derive from microscopic assumptions. I heard apparently Fourier's law isn't that clear http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Amath-ph%2F0002052 Maybe there are more examples from other physical areas (quantum mechanics, chemistry etc.)? Basically something where there are fully understood microscopic laws and also a macroscopically observed law which theoretically should be derivable.


title: Is it possible to witness a circular rainbow?

text: What conditions would make it possible to see a naturally occurring fully 360° circular rainbow? Would it even be possible?


title: Studying Feynman articles nowadays

text: I'm curious to know if it's useful to study Feynman article "Quantum Mechanical Computer" nowadays. I'm a computer scientist, and i don't know any of the literature in quantum computers. Since long time has passed from 1985 it's a waste of time studying that article? If yes, why? The article can be found at: http://www.cs.princeton.edu/courses/archive/fall05/frs119/papers/feynman85_optics_letters.pdf Thanks.


title: Derivation of relativistic uniformly accelerated motion

text: I'm trying to understand solution of the following problem from Landau, Lifshitz, Classical Theory Of Fields: ![enter image description here](http://i.stack.imgur.com/1cLN9.png) (ending skipped). What I see when I "write out the expression for $w^iw_i$", using definition $w^i=\frac{du^i}{ds}=\frac{du^i}{cdt\sqrt{1-v^2/c^2}}$: $$\frac1{c^2-v^2}\left(\left(\frac{d}{dt}\frac1{\sqrt{1-v^2/c^2}}\right)^2-\left(\frac{d}{dt}\frac{v}{c\sqrt{1-v^2/c^2}}\right)^2\right)=-\frac{w^2}{c^4}$$ I then take the derivatives and arrive at the following result: $$-\frac{c^2\dot v^2}{(c^2-v^2)^3}=-\frac{w^2}{c^4}$$ I can get rid of the squares, use some guessing to determine correct signs, and then solve the resulting equation, getting correct results (according to the book). But the original solution in the book looks way simpler and must be better justified than mine. So, the question: how did LL get their bottom-left equation? Did they "write out the expression for $w^iw_i$" in some other way than I did? How should I do here instead?


title: Clarification regarding Special Relativity

text: Empty the universe of all matter. Place the Earth in that universe sitting in an inertial frame. Take a pair of twins. One twin gets into a space ship and takes off from Earth (t=a). He accelerates rapidly and obtains .9c within a day of Earth's time and then turns off his engines (t=b). He then glides for 4 Earth years. After 3 Earth years 364 earth days (t=c), he starts to decelerate the ship, turns it around and returns to Earth the exact same way he left (1 earth day to decelerate 1 earth day to accelerate). After 4 years, he decelerates for one Earth day as he approaches Earth and then lands the ship 8 years to the day after he took off. My understanding is that the stay-at-home twin has now aged 8 years. The spaceship twin has aged less than that (about 7 years). If we assume linear acceleration the 4 Earth days of acceleration account for 2 days equivalent of time discrepancy. The remaining 7 years 361 days account for the majority of the discrepancy. Question 1: Is that analysis correct? Now, I repeat the experiment, spaceship twin takes off in the spaceship as before, but now as soon as he hits Earth orbit he parks his ship. stay-at-home twin attaches a really big rocket engine to the Earth itself and blasts the planet away from his brother. The distances versus time between the ship and Earth is exactly as before. I assume now the spaceship twin will have aged 8 years and the Earth twin will have aged about 7 years. Question 2: Is that correct? I repeat the experiment one last time. As before the spaceship twin blasts off of Earth. This time however he accelerates to .45c; his brother again attaches the rockets to the Earth and accelerates it .45c. Again they maintain the exact same distance to time relationship. My guess now is that when the brothers meet they will be the exact same age. Question 3: Is this correct? Question 4: In each of these scenarios from t=b to c the systems appear to be exactly equal to one another. The ship and the Earth are receding from one another at .9c. Where is the difference in their situations stored? How would an outside observer differentiate the 3 scenarios? Question 5: It appears that time dilation, length contraction is really caused by acceleration not velocity. Why does acceleration not show up in the special relativity formulas? Is perhaps velocity in these equations not really velocity, but rather the integral of acceleration?


title: AGT conjecture and WZW model

text: In 2009 Alday, Gaiotto and Tachikawa conjectured an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of $N=2$ super-conformal field theory. > I would like to know if anything similar shows up for WZW conformal blocks. * * * A much more elementary (but related) question is the following: > Is there any elliptic recursion for 4-point conformal blocks in WZW theory, > like in Liouville theory (see e.g. these papers). From a mathematician view-point, this relation between 4-point spheric and 1-point toric conformal blocks is not so surprising. Indeed, the moduli space $\mathcal M_{1,1}$ of elliptic curves and the unordered moduli space $\mathcal M_{0,[4]}$ of 4-punctured rational curves are the same. I would be anyway very interested in learning about recent advances in the WZW side of the story (if it exists).


title: Heuristics for definitions of open and closed quantum dynamics

text: I've been reading some of the literature on "open quantum systems" and it looks like the following physical interpretations are made:   * **Reversible** dynamics of a **closed** quantum system are represented by a **group of automorphisms** (of a $C^∗$-algebra, von Neumann algebra, or $B(H)$, depending on whom you read)   * **Irreversible** (but still deterministic) dynamics of a closed system are represented by a **semigroup of endomorphisms**   * The dynamics of an **open** system (which are non-deterministic, hence also irreversible) are represented by a **semigroup of unital completely positive maps**. I'm trying to motivate these interpretations based on my limited familiarity with quantum mechanics. If you use the formalism of QM wherein states are represented by density operators, then you get something like this:   * In the Schrodinger picture, a unitary time evolution is represented by $\rho \mapsto U(t) \rho U(-t)$, where $U(t) = e^{-itH/\hbar}$, and a measurement corresponding to an operator $X = \sum_\lambda \lambda P_\lambda$ is represented by $\rho \mapsto \sum_\lambda P_\lambda \rho P_\lambda$.   * In the Heisenberg picture, a unitary time evolution is represented by $Y \mapsto U(-t) Y U(t)$, and a measurement by $Y \mapsto \sum_\lambda P_\lambda Y P_\lambda$. The unitary time evolution in the Heisenberg picture is an automorphism group on $B(H)$, and the measurement process is a unital completely positive map. So I can intuit that, in an open quantum system (i.e. one which is constantly "measured" by interacting/entangling with the environment) the dynamics should be represented by a semigroup of unital completely positive maps. And certainly the idea that reversible dynamics for a closed system corresponds to an automorphism group make sense. What I still don't quite see is why you would represent irreversible dynamics of a closed system by a semigroup of endomorphisms. How does the difference between closed and open systems correspond to maps on $B(H)$ which are multiplicative or not? Also, is there an easily understandable example (for someone with a lot more math than physics) of where a closed system with irreversible dynamics might come from?


title: Why is AC more "dangerous" than DC?

text: After going through several forums, I became more confused whether it is DC or AC that is more dangerous. In my text book, it is written that the peak value of AC is greater than that of DC, which is why it tends to be dangerous. Some people in other forums were saying that DC will hold you, since it doesn't have _zero crossing_ like that of AC. Many others also say that our heart tries to beat with the frequency of ac which the heart cannot support leading to people's death. What is the actual thing that matters most? After all, which is more dangerous? AC or DC?


title: Is there any time-dependent hydrogen atom Schrödinger equation, solvable analytically?

text: It's well-known that hydrogen atom described by time-independent Schrödinger equation (neglecting any relativistic effects) is completely solvable analytically. But are any initial value problems for time-dependent Schrödinger equation for hydrogen solvable analytically - maybe with infinite nuclear mass approximation, if it simplifies anything? For example, an evolution of some electron wave packet in nuclear electrostatic field.


title: Would it be possible to distinguish rap and singing from just seeing the soundwaves?

text: For any person even remotely familiar with music it is obvious that rap and singing are easily distinguishable (save for some interesting mix styles). This means that there must be a clear difference, but as with many things (e.g. image recognition) it might be that humans are better at picking up the subtleties. What I am interested in is whether a computer would be able to recognize the difference between rap and singing? In other words: could you distinguish between rap and singing by just looking at the soundwaves in terms of frequency and amplitude and the likes? If so, how would that work?


title: Where does the 3rd and the 4th Maxwell's equations lead us in the end?

text: Take the 3rd and the 4th equation from this table. The first tells us that an electric field can be generated by a magnetic field. The second, says that a magnetic field can be generated from an electric field (or more specifically, from its variation in time). Well, at least, I've been told so. My question are two and are rather simple.   1. If the two fields can generate reciprocally, won't a single tiny variation in time of the first eventually always finish in two infinite fields? Maybe they change in different directions? If this is the case consider question number 2.   2. How can we study the effects separately? I mean, let's take for example an experiment, where the magnetic field changes in time. Something will happen, and we conclude that the electric field generated by the variation of the magnetic field (we can calculate it using the second equation), caused this thing. Can we really say that? (in my Physics book there are some exercises that are told this way). But shouldn't this variation be the point of origin of other infinite variations? Why do we make conclusions here?


title: Do Maxwell's Equations overdetermine the electric and magnetic fields?

text: Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and $\vec{B}=(B_x,B_y,B_z)$, there are only 6 unknowns. So we have 8 equations for 6 unknowns. Why isn't this a problem? As far as I know, the answer is basically because the equations aren't actually independent but I've never found a clear explanation. Perhaps the right direction is in this article on _arXiv_. Apologies if this is a repost. I found some discussions on PhysicsForums but no similar question here.


title: Why does electric field intensity $E$ can be uniquely determined by its divergence and curl?

text: My question is, the number of following equations $$\nabla\cdot E=\frac{\rho}{\varepsilon}$$ $$\nabla\times E=-\frac{\partial B}{\partial t}$$ is 4 while the number of unknown variables $E=(E_1,E_2,E_3)$ is 3. Intuitively, the equation is overdetermined and the solution may not exist unless four equations are correlated. Is my intuition right?


title: Why did the Homestake experiment only detect solar neutrinos?

text: The Homestake experiment was designed to detect solar neutrinos. In order to shield it from the background, it was set up in a deep underground mine. It was found that the only one third of expected neutrinos were detected, leading to the solar neutrino problem. What I wonder is why neutrinos created in the upper atmosphere by cosmic rays were not detected?


title: How does a MRI use 20.1kW yet creates over 204K BTU/h?

text: I've been reading through a manual about MRI operating procedures for a large healthcare provider. The manual (written by Siemens) states that the MRI machines use 9kW in stand-by mode and 20.1kW while operating - yet the cooling system is required to move 204K+ BTU/h. I'm confused as to where all the extra heat is coming from as 20.1kW![enter image description here](http://i.stack.imgur.com/qYCbo.png) should be 71k+ BTU/h. Where is the extra ~140k BTU/h coming from?


title: How is Magnetic force on a current carrying conductor $Blb$

text: I was reading an answer about torque acting on a rectangular current carrying loop kept in a uniform magnitude field B. Force acting on each sides is $F_1$, $F_2$, $F_3$, $F_4$. It's written here : Force $F_2$, $F_4$ acting on these sides has same magnitude $F' = Blb \sin(90-\theta) = Blb \cos\theta$. And $F1$ and $F3$, have $F=IlB\sin90° = IlB$. (where $l$=length; $b$=breadth) I understand $F_1$ and $F_3$. But I don't get how the magnitude of forces of $F_2$, $F_4$ become $Blb \cos\theta$ . Which law says that? Am i missing some knowledge? If you need the diagram or the book text for more details , please ask for it.


title: Lowest temperature possible in the universe?

text: The third law of thermodynamics states that nothing can reach to absolute zero temperature. What is the lowest possible temperature that can be in the universe? Has any experiment reached to a billionth of a Kelvin? Is there any restriction on how low it can be? Is $10^{-1000}\;\rm K$ possible? Or is there a lowest quantum of temperature?


title: Propagator and expectation value

text: Given a propagator of some system, is there a way of finding the expectation value of some operator without knowing the eigenstates involved? For example, if I have the propagator of a particle in a harmonic potential, and I would like to calculate $\langle\psi|p^2|\psi\rangle$ where $|\psi\rangle$ is the ground state, without actually knowing the ground state.


title: Symmetry of Minkowksi Metric -> Conserved Current?

text: My understanding of the Minkowski Metric is that we have the freedom to choose whether to place the negative sign on the time-component or on the spatial- components. That is, either basis should yield the same physics when dealing with Lorentz invariant Terms. Thus, if we have a Lorentz invariant Lagrangian, we should be able to take $\eta_{\mu \nu} \rightarrow - \eta_{\mu \nu}$ without changing the action. What is the associated conserved current with this symmetry? N.B. This transformation looks similar to a TP transformation. Is it identical?


title: Time slowing down problem

text: When someone moves, time slows down for him. Let, a man standing still and another moving very very very fast, this happens for an hour (as measured by the standing man). Time has moved slower for the moving man than for the still man. The fast has got less aged than the still. I've heard that length shortens for a thing when it moves. The faster it moves, length shortens more. I'm eager to know that; when a man is moving very fast while reading a book, will he comprehend the book's items faster than when he stands still while reading this same book? (I've thought neurons get smaller and impulses travelling lesser.)


title: What happens when a bare 3d topological insulator is subject to a magnetic field?

text: Effective field theory of 3d topological insulators (TI) predict some novel electromagnetic effects. Unfortunately it require a gapped surface which is hard to achieve experimentally. Then I have two questions. 1. > Is $\nabla P_3=0$ for a bare TI (without magnetic coating), and hence it > behaves like a trivial insulator? 2. > Why not put the bare TI inside a uniform magnetic field which, of course, > opens a gap on the surface? Can I still use the topological field theory > without worrying about the bulk? I am afraid that the bulk is no longer "topological" since time-reversal symmetry is broken. But the bulk band gap is robust under perturbation. As long as the gap is not closed, it remains a TI. > Take a look at the article topological quantization in unit of $\alpha$, in > which a $B$ field is applied instead of magnetic coating. I got more > confused after reading this paper. * * * The $P_3$ appeared in my question comes from Qi's paper topological field theory of time-reversal insulators. ![enter image description here](http://i.stack.imgur.com/AfPoE.png)


title: How to choose a suitable topic for PhD in Physics?

text: After completion of graduate courses when a student is supposed to start real research in Physics, (to be more specific, suppose in high energy physics), how does one select the problem to work on? The area is vast, mature and lots of problems remain to be solved. This vastness of the field and various levels of difficulty of unsolved problems may give rise to confusion regarding choice of problem. The time one can spend at graduate school is limited (let us assume about four years after courses are over) Can anybody guide me or share one's views about this issue? What type of problems should be avoided at PhD level? I guess problems which even the best theorists had failed to completely solve should be left. There are less difficult problems which are solved in collaboration of say three/four/five or more highly experienced physicists which may not be possible for a beginner who will be working practically alone. So should one start with the simplest unsolved problems? Or is it enough for a problem to be interesting to work on, irrespective of its level of difficulty? In general, what type of work is expected from a graduate student to be eligible for a PhD degree?


title: Nuclear fusion : ion vs atom fusion cross section?

text: I've read critique on ITER project (in Russian), and one of points was that cross section of ion-ion fusion is much lower than ion-atom and atom-atom, and that's the reason why it likely to not work. Can anyone suggest where I can find how much lower cross sections are? I am mainly interested in D-D, D-T and B11-H reactions.


Why Gravity attracts all objects with the same speed? Is this question was solved? What is the exact answer?


title: Correlation, Time Ordering, and Observables

text: In general, the product of two Hermitian operators $\phi$ will not be Hermitian, unless the two operators commute. Question: is $X = T \phi(t_1) \phi(t_2)$ Hermitian? It doesn't seem to be if $T \phi_1 \phi_2 = \phi_1\phi_2\theta(t_1-t_2) + \phi_2\phi_1\theta(t_2-t_1).$ (Weinberg) If not, how can we interpret $\langle 0 |X|0\rangle$ as the vacuum correlation function, if X is not an observable?


title: New to physics and need some help with understanding acceleration

text: 1. The free fall acceleration of an object under earth's gravity is directed toward?   2. When the same amount of force is applied to two different masses, the smaller mass will be accelerated **___ ___** than the larger?


title: Fitting of data to a model

text: Imagine that I have some observable value predicted with a theory for some process to be: $1+a x + b x^2$ and observed value is 1.3 with an error 0.2; a and b are some numerical constants. I also have another observable influenced by the same x to be $1 + cx+ dx^2$ and the measured value 1.2 with an error 0.6, c and d are some numerical constants. My question is: How to calculate value of x that explains both measurements the best? I guess this is called benchmark point, but how to get it and how to quantify it?


title: Are the laws of physics applied mathematics?

text: This questions started with a question I had about gravity. If two objects of different weights fall to the earth at the same rate of acceleration, then it seems to me that gravity is in some ways 'calculating' the weight of each item and applying the appropriate force to each item so as to have it fall at the same rate of acceleration. Is this true (or at least close to the truth)? This got me think that perhaps this is what all of the mathematical equations of physics are really saying - namely that there are mathematical equations that are getting applied to the real world in one way or another. Is this right? If not, why not?


title: What physical phenomena underly the shape of this graph of diurnal UV index?

text: When I dig up a graphs of UV index versus time-of-day, most graphs are characteristically shaped like one of the plots shown here: ![enter image description here](http://i.stack.imgur.com/IaKjX.png) That graph is from this site.   1. What physical phenomena underly that characteristic shape?   2. Is there a (perhaps simplified and approximate) function for UV index versus latitude and time of day if we assume constant atmospheric conditions and altitude?


title: When is use of the 'effective mass' concept appropriate?

text: In textbooks the characteristic length scale of an exciton, or an electron bound to dopant atom, in silicon is calculated by analogy to the vacuum case. Bohr radius in vacuum: $$a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_0 e^2} = 0.53\ {\rm Å}$$ where $\varepsilon_0$ is the vacuum permittivity, $m_0$ is the electron mass, and $e$ is the electron charge. Exciton bohr radius in material: $$a = \frac{4 \pi \varepsilon_{\mathrm{r}} \varepsilon_0 \hbar^2}{m_e e^2} = a_0 \varepsilon_{\mathrm{r}} \frac{m_0}{m_e}$$ where $\varepsilon_{\mathrm{r}}$ is the relative permittivity, and $m_e$ is the effective mass of the electron in the material. For silicon, these can be looked up in a properties table and are: $$\varepsilon_{\mathrm{r}} = 11.7$$ $$m_e = 0.26 m_0$$ Which gives: $$a = \frac{11.7}{0.26} 0.53\ {\rm Å} = 23.85\ {\rm Å}$$ However, experimental evidence suggests an exciton radius of $4.3\ \mathrm{nm}$ in silicon (according to Yoffe, AD. Advances in Physics, 42(2) pg 173, 1993). This seems quite a bit off to me. Is the idea of effective mass not entirely appropriate for this analogy?   And more specifically, what about in heavy fermion systems where the effective mass of the electron can be even larger than a muon!? This would give an incredibly small scale for the exciton or defect bound radius.


title: Gravitational time dilation at the earth's center

text: I would like to know what happens with time dilation (relative to surface) at earth's center . There is a way to calculate it? Is time going faster at center of earth? I've made other questions about this matter and the answers refers to: $\Delta\Phi$ (difference in Newtonian gravitational potential between the locations) as directly related, but I think those equation can't be applied to this because were derived for the vecinity of a mass but not inside it. Any clues? Thanks


title: the higher you go the slower is ageing

text: as per to einstein as we go far from the earth the TIME tends to slow down , so it means when I am one metre above the earth's surface , the time has slow down for me as per http://en.wikipedia.org/wiki/Gravitational_time_dilation what would happen if I am at the centre of earth


title: What affects the damping of a spring?

text: What variables affect the damping of a spring executing simple harmonic motion? What are the independent variables, and what variables would need to be controlled in an experiment? I'm attempting to complete an investigation where I measure the decrease in amplitude of a damped spring, and to prove the relationship between variables in the motion. Thanks!


title: How did Einstein come up with his postulates for Special Relativity using thought experiments on simultaneity?

text: In his popular book on relativity, in chapter IX, "The relativity of simultaneity", Einstein describes an experiment in which a flash happens simultaneously on A and B, as defined by the fact that an observer at the middle point M can see the light coming from A and B at the same moment: > >               --train--> >   > > \---embankment---A-------M-------B---- Then, he proceeds to say that an observer on the train "is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A". So Einstein concludes that simultaneity is not absolute. And an explicit calculation using Special Relativity principles confirms this. The conclusion is that: > So the answer is that the observer on the train sees the lightning strike > the front of the train at $t' = -\gamma\tfrac{vd}{2c^2}$ and the rear of the > train at $t' = \gamma\tfrac{vd}{2c^2}$. The time between the lightning > strikes is $\gamma\tfrac{vd}{c^2}$. By setting $\gamma=1$, we obtain the result in Galilean Relativity (ie: "The time between the lightning strikes is $\tfrac{vd}{c^2}$"), which is the theory of space time before Einstein came out with Special Relativity. The point is that regardless of whether we are calculating using Galilean Relativity or Special Relativity, the conclusion is that different people on different speed will always have different notion of simultaneity. Since this is a conclusion known to everyone before Einstein, how did Einstein use this insight to derive his Special Relativity? Truly, I fail to see how can this thought experiment on simultaneity can give Einstein insights into Special Relativity. Anything I miss?


title: Why electrons behave as a particle and also as a wave?

text: Why do electrons (and other very small particles) sometimes behave as particles (i.e. when we are not looking at them) where as other times they behave as waves?


title: Compton Wavelength

text: I have the formula for Compton wavelength: $$\lambda_{c}= \frac{h}{m_{0}c}$$ In this equation, is $m_0$ the mass of the electron that the photon hit? I got online that this might be the photon rest mass, but it is basically 0, and it is impossible to do the calculation to get $\lambda_c$ if it is equal to 0.


title: Wireless electricity through Wi-Fi power signal?

text: Could there be a method to obtain and send worldwide electricity through WiFi? I mean if we have internet everywhere and only need a WiFi device, can we apply the same to electricity technology? or electrons conducted by WiFi waves?


title: Logical positivism and black hole interiors

text: A black hole exists. Eventually, it will completely evaporate away. Alice falls into the hole. Imagine you are Alice. According to logical positivism, the interior of the black hole exists. But you will die at the singularity. You can't measure anything after you're dead, so any event outside after the black hole has vanished doesn't exist. Bob stays outside the hole. Imagine you are Bob. The interior doesn't exist. Enter Quentin. Right now, he holds a qubit in the state ${1\over\sqrt 2} (|0\rangle + |1\rangle)$. He will measure the value of the qubit shortly in the future. If 0, he will jump into the hole. If 1, he will stay out forever. Imagine you are Quentin. Right now, does the interior exist? Curiouser and curiouser.


title: Do events exist after our death if we can't measure them?

text: The great physicist Raphael Bousso predicted time will end in this article. We can't measure anything after our death in principle. So, does time end when we die?


title: Deriving the law of moments

text: Recall the Law of Moments for a one dimensional rod: > " _When an object is in equilibrium the sum of the clockwise moments is > equal to the sum of the anticlockwise moments._ " I understand that we may _define_ a quantity $F \times d$, and call it a moment. Then, experimentally, we can observe that $F_1\times d_1 = F_2\times d_2$ is a necessary condition for equilibrium. However, is it possible to _derive_ the Law of Moments from some more fundamental principle, **e.g.** the principle of least action?


title: Why can't my eye see itself in the mirror through polarizing 3D-glasses?

text: I found a pair of polarizing "3D glasses" lying around, and tried to look at myself in the mirror while wearing them. To my utter confusion, when closing the left eye and only looking through the right eye, I could not see the right eye in the mirror. The light could not pass through the same polarized lens twice. (I could, however, see the closed left eye clearly.) I would expect the opposite to be true, as light going out the right lens with polarization X and coming back in with the same polarization X should pass through unaffected.


title: Boundary layer theory in fluids learning resources

text: I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. Not rigorous as in keeping track of all epsilons and deltas, but more rigorous than an heuristic argument. Hope you understand what I mean. Some free resources available on the web would be preferred, but if you can suggest book titles that's also helpful. Thanks. Edit: The applications I have in mind are the calculation of damping in surface waves in basins of various shapes (circular, rectangular, etc).


title: Questions about Statistical Mechanics

text: 1. For grand partition ensemble, is it true that the introduction of chemical potential allows us to have the sum of number of the particles in each state to be the total number of particles ("On average")? and this fluctuation in the constraint of $N$ will shrink as the number of particles you got is large.   2. Next question is about Boson gas, why would we have to consider the chemical potential to be negative or zero? Is there anything to do with the reference to the ground state energy level?   3. What would it mean if we were to integrate the energy of those states below ground states? NB For last question, I mean when I were to find the total energy of this boson gas system, which I know below a critical temperature Tb, the Bose Einstein condensation becomes important, and we would do the separate sum thing as usual ![enter image description here](http://i.stack.imgur.com/ufZ9N.png)(see first Pic) , Now I wonder why when we do the sum of the energy we integrate the states above ground state , does it mean all the particles at ground state have zero energy?? and we only have to concern those above the ground state? ![enter image description here](http://i.stack.imgur.com/d3ALk.png)


title: Behaviour of duvet covers in washing machines

text: When you put a duvet cover in a washing machine without buttoning it up any other items of bedding (e.g. pillow cases) inevitably end up inside it. Instinctively, I would think that the opposite would happen (although I have no rigourous basis for believing this) due to the contents of the duvet cover being pushed out of it as it rotates. Why does this (to me at least) counter- intuitive behaviour occur? (Or are my findings not reproducible?)


title: Decompression speed (rate)

text: I've heard that decompression speed or rate in which the gas will leave the greater pressured container is equal to the speed of sound. For example, if the spacecraft gets damaged the air cannot exit out to vacuum faster than about 340 m/s I don't understand how the flow rate of the fluid is determined by the density. Please explain...


title: When does a particle go through the Higgs Field?

text: This is a short and simple question... I have been reading my book on particle physics and quantum physics when I had thought of a question that it failed to answer: "Does a particle enter/interact with the Higgs Field when created, or at some other time? And if the answer is neither of these two, does it constantly go through/react with the Higgs Field?" I have been looking for this everywhere (here, Wikipedia and other sites) and haven't found an answer yet. Does anyone know that answer (or possible answers) to this?


title: Wavefuntion for Wigner Crystal

text: Quantum wavefunctions of infinite variables can be written that describe certain Fractional Quantum Halls states, such as the Laughlin family of wavefucntions $ \Pi_{i<j} (z_i-z_j)^k $ that describe the the $ \frac{1}{k} $ states. However, can wavefunctions also be written that describe the called "crystal" states, that form when the electron density becomes too small, and the electrons converge to a lattice. If so, are there any fundamental differences between those of Wigner crystals and those of "observable" Fractional Quantum Hall states? i.e. Would a hypothetical wavefunction that has maxima at the vertices of an equilateral triangle on the complex plane describe a triangular lattice Wigner crystal? What about a wavefuntion with zeroes at the vertices of an equilateral triangle?


title: Second law of thermodynamics

text: I think this is a simple question. If I have that $E(L)=\tau L$ and we are told that $\tau=BTL$ would this mean that $E=BTL^2$ implies $dE=(2BTL)dL$ or should I sub $\tau$ straight into the second law giving $dE=\tau dL=(BTL)dL$? $\tau=tension \ L=length \ T=temperature \ B= constant$


title: Lorentz force for electrically conducting fluid flow in homogeneous magnetic field

text: I am mathematician and have paper which models situation when homogeneous magnetic field is applied to moving electrically conducting fluid. There is such Lorentz force formula on which all the work is made: $$F= \sigma (E + V \times B) \times B $$ But I think that the last $\times B$ is not needed. Is this correct formula for this case or the work is not correct?


title: Trouble with calculating Christoffel symbols of FLRW metric using Lagrangian method

text: The FLRW metric which I am using is $$ds^2 = dt^2 - \frac{a(t)^2}{c^2} \left( dx^2 + dy^2 + dz^2 \right)$$ where $a(t)$ is the so-called 'scale factor'. I did not want to calculate the Christoffel symbols directly by using the metric and its derivatives. Instead, I took the Lagrangian, $$\mathcal{L} = g_{\mu \nu} \frac{\mathrm{d}x^{\mu}(s)}{\mathrm{d}s}\frac{\mathrm{d}x^{\nu}(s)}{\mathrm{d}s}$$ and then applied the Euler-Lagrange equations for each coordinate, i.e. $t,x,y,z$. Note that the scale factor $a(t)$ is a function of time, $t$, which is a function of $s$, that is, $t(s)$. I think I reach my problem when I compute the term: $$\frac{\partial}{\partial s}\frac{\partial \mathcal{L}}{\partial \left( \frac{\mathrm{d} x(s)}{\mathrm{d}s} \right)}$$ It leads to the equation: $$\frac{\mathrm{d}^2x(s)}{\mathrm{d}s^2} + \frac{2 \dot{x} a'(t)}{a(t)}=0$$ where $\dot{x}$ is short-hand for the derivative with respect to $s$, and the $a'$ is short-hand for the derivative of $a(t)$ with respect to $s$. I do not see the derivative of another one of the coordinates in the second term, so when attempting to discern the Christoffel symbols by comparing the equation to the geodesic equation, I do not know what the $\gamma$ index should be: $$\frac{\mathrm{d}^2 x^{\alpha}(s) }{\mathrm{d}s^2} + \Gamma^{\alpha}_{\beta \gamma} \frac{\mathrm{d}x^{\beta}(s)}{\mathrm{d}s}\frac{\mathrm{d}x^{\gamma}(s)}{\mathrm{d}s}$$ $\alpha=1$ because of the second derivative of $x$ in the equation, $\beta=1$ because of the first derivative of $x$ in the second term, but I do not know what $\gamma$ is. There is no other derivative.


title: Heisenberg formalism of QFT

text: Has anyone tried to develop relativistic quantum theory along the lines of the Heisenberg picture and what's so difficult about promoting time to an operator??


title: Extension to continuous in proofs of rigid body mechanics

text: I'm studying rigid body mechanics and I've seen several proofs of properties related to total angular momentum, kinetic energy, etc. that all regard discrete set of points. For example, to show that in an inertial frame $\frac{\text d \mathbf {L}}{\text d t}=\mathbf{\Gamma}^{\text {ext}}$, one writes down the sum $$\mathbf{L}=\sum \mathbf{r} \times \mathbf{p},$$ and does the required derivatives. How can such arguments be extended with rigour to continuous rigid bodies? To give an example, in "Rudin, Principle of Mathematical Analysis", in the chapter of Riemann Stieltjes integral, there's an example regarding the moment of inertia of a "straight line" body, that can be defined uniquely with the Riemann-Stieltjes integral $$\int _0 ^{\ell} x^2\, \text dm(x). $$ Altough it's restricted to a "straight-line" body, the result is very elegant. Back to the example of angular momentum, I think that one could define the total angular momentum as $$\mathbf{L} =\int _{V} \mathbf {r} \times \mathbf {v(\mathbf{r})} \,\text d m(\mathbf{r}),$$where the integral is extended over the volume of the body (actually I've never seen a definition of total angular momentum for continuous rigid bodies, since it always appears as $$\mathbf{L} = \mathbf{I}\mathbf{\omega},$$ and, of course, this is the useful identity). So, how can the discrete arguments be extended to continuous arguments?


title: Is it possible to calculate distance if non-constant acceleration is unknown?

text: I know that the distance travelled in non constant acceleration is $d=\int_a^b f(x)\mathrm{d}x$, but is it possible to calculate the distance without knowing the value of $f(x)$?


title: How do you estimate colour temperature based on the position of the sun in the sky?

text: I'd like to estimate the colour-temperature of sunlight (as applied in photography) based on the position of the sun in the sky for a mobile phone app I'm working on (app link from a more appropriate question -Are there any mobile applications that calculate sunrise/sunset based on location?). I'm already able to determine the position of the sun in the sky based on the date, time, latitude and longitude. From this position (sun elevation from the horizon) I'd like to estimate the colour-temperature assuming clear skys. I understand there are factors such as how overcast the sky is that I won't be able to take into account. I'll assume an unobstructed view of the horizon. E.g. Something like the following.               Light Source                    Colour Temperature in K     ============                    =======================     Sunrise and Sunset              2,000 to 3,000     Sunlight at 10 Degree elevation 3,500     Sunlight at 20 Degree elevation 4,000     Sunlight at 30 Degree elevation 4,500     Noon Sun and Clear Sky          4,900 to 5,800     Start of Blue Hour              ?      **Is there a formula I can use that directly relates the estimated colour temperature to the suns position in the sky?** Note, this question was originally asked on the photography stack exchange but was migrated here as more of a physics based problem. My current understanding of the problem is that I'll need to apply Rayleigh scattering and possibly Mie scattering. The following image from Cambridge in colour shows an (exaggerated?) colour temperature scale with dawn (1), sunrise (2), midday (3), sunset (4) and dusk (5) marked. ![Exaggerated sky colour temperature](http://i.stack.imgur.com/HdWS8.png)


title: Debye temperature for the steel?

text: I'm looking for data on Debye temperature of steel (ideally with a known carbon concentration, structure, and set of phases), but find only data on elements. Do You happen to meet the data in papers, or textbooks?


title: Computing the Christoffel symbols with the geodesic equation

text: I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$ g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm e^\lambda,\quad g_{22} = -r^2,\quad g_{33} = -r^2 \sin(\theta)^2, $$ where $\nu$ and $\lambda$ are functions of $r$. The Lagrangian is $$ L = \mathrm e^\nu \dot t^2 - \mathrm e^\lambda \dot r^2 - r^2 \dot \theta^2 - r^2 \sin(\theta)^2 \dot \phi^2.$$ From this, I have computed for Euler Lagrange equations: $$ 0 = 2 \mathrm e^\nu \ddot t $$ $$ \mathrm e^\nu \nu' \dot t^2 - \mathrm e^\lambda \lambda' \dot r^2 - 2 r \dot \theta^2 - 2 r \sin(\theta)^2 \dot\phi^2 = -2 \mathrm e^\lambda \ddot r $$ $$ - 2 r^2 \sin(\theta) \cos(\theta) \dot \phi^2 = - 2r^2 \ddot \theta $$ $$ 0 = - 2 r^2 \sin(\theta)^2 \ddot \phi $$ With the second I got: $$ \Gamma^1_{00} = \frac{\nu'}2 \mathrm e^{\nu-\lambda}, \quad \Gamma^1_{11} = - \frac{\lambda'}2, \quad \Gamma^1_{22} = - r \mathrm e^{-\lambda}, \quad \Gamma^1_{33} = - r\sin(\theta)^2 \mathrm e^{-\lambda}$$ And the third one: $$ \Gamma^2_{33} = - \sin(\theta)\cos(\theta) $$ From the first and fourth equation I would deduce that any $\Gamma^0_{\mu\nu} = 0$ as well as $\Gamma^3_{\mu\nu} = 0$. The solution says that this is not the case. How can I obtain the other nonzero Christoffel symbols?


title: Relatvity of Promise

text: Suppose there are two observers moving at a speed c/2 w.r.t. each other. Now they won't agree on simultaneity of space separated events. Now they decide to host a race between two racers, with common starting point but different ending points. They bet on their racers, that whosoever wins will pay the other one. Say for an observer A, both racers are tied and no one is liable to pay the other. But for the second observer, one racer would have won, since the events won't be simultaneous w.r.t. him , so he will demand money. So both are correct in holding their grounds in their respective frames. Suppose on no payment being made by A, B shoots him and A dies. Who will win in the court of law if killing for money is acceptable ? So even keeping a promise is relative. Doesn't this put a limit on proper functioning of machines. Like in A's frame B has malfunctioned. So, isn't this kind of saying machines in general will give us the desired output only in a particular frame


title: How has the age of the Universe been derived from the observations made by the Planck mission?

text: The parameters of $\rm\Lambda CDM$ model have been determined to an amazing high precision from the measurements made by the Planck mission. In particular, the Hubble "constant" (the value of Hubble parameter at the present day) has been determined to be $H_0 = (67.3 \pm 1.2 )\, \text{km} \, \text{s}^{-1} \text{Mpc}^{-1}$. They have also given values for the density parameters at the present day (the density divided by the critical density) coming from cosmological constant, $\Omega_{\Lambda,0} = 0.6817$, and from matter, $\Omega_{m,0} = 0.3183$. The age of the Universe $t_0$ can be calculated from other cosmological parameters in this way $$t_0 = \frac{1}{H_0}\int_0^1 \frac{a\,\text{d}a} { \sqrt{\Omega_{\Lambda,0}a^{4} + \Omega_{k,0}a^{2} + \Omega_{m,0}a + \Omega_{r,0}} },$$ where $\Omega_k = 1 - \Omega$ is the space curvature parameter, $\Omega = \Omega_{m,0} + \Omega_{\Lambda,0} + \Omega_{r,0}$ is the total energy density parameter (the energy density divided by the critical density) and $\Omega_r$ is the energy density parameter coming from radiation. I have read that the age of the Universe has been established from the Planck mission measurements as $t_0 = 13.82 \times 10^9$ years. My question is: how is this value been calculated? I mean, it has been calculated assuming a flat space geometry, that is, assuming that $\Omega_k = 1 - \Omega = 0$? If not, what values for $\Omega_{r,0}$ and $\Omega_{k,0}$ have been used to perform the calculation?


title: Twins Paradox Paradox

text: I've recently has special-relativity explained to be in a rather elegant way. All objects travel at the speed of light in space time. Thus, when you travel faster through the three dimensions of space, the speed at which you travel through time decreases. Photons do not experience time because all their velocity is in the spatial directions and none of it is in time. Considering the 4-dimensional universe this invokes, how can I ever interact with anything that has traveled at a different velocity than me unless it changes direction to meet me? In the terms of the twin paradox, how is it possible for the twins to meet? Twin A, stays on earth traveling at a fixed speed in spacetime. Twin B leaves earth traveling faster in space and slower in time but still traveling at c in spacetime. Some basic geometry tells me that if two objects forced to travel at a fixed speed from the same origin in changeable directions (in any number of dimensions) that there is no way their future locations can be the same unless they **both** change direction to provide the possibility of intersection. If Earth is traveling at a fixed speed and Twin B is traveling at a fixed speed in a space (no matter the dimension) then there is no way they could ever be in the same location in spacetime again unless the earth changes direction and meets Twin B in the middle. Wouldn't they be forever doomed to be in different places along the axis of time unless Twin A goes on a relativistic voyage to allow Twin B to catch up? Should we not expect time to behave the same way as the spacial dimensions?


title: Why the peak of spectrum gets vague when the dimension is lower?

text: In a many-body system, we can know the spectrum function at a particular temperature from Green function. It means density of states. A peak of spectrum represents one mode. My question is that in the same model why the peak of the spectrum gets broader with lower dimension?


title: What strategies can a researcher use when confronted with a long and complicated symbolic expression?

text: When doing research in theoretical physics, a frequent task one encounters is trying to express some physical quantity as a function of other quantities. A lot of times this can't be done analytically, but even when it can - it sometimes results in very long and complicated symbolic expressions. Although technically such an expression is a "solution", it is not of much use for a researcher that wants to gain physical insight from the solution, and maybe rely on it to do more research. **What are some general strategies a researcher can use to gain insight when confronted with such expressions?** For starters - here are some ideas I use:   * Check certain limits of the expression, i.e. when one of the variables is very low or very high - these are often simpler and can shed light on the behavior in the general case.   * Look for a recurring pattern in the expression and give it a name. The newly defined variable usually has some physical significance in itself, and also when all the occurrences of the pattern are replaced with the new variable the whole expression becomes simpler.   * Substitute some of the variables with reasonable numerical estimates, and plot the expression as a function of the rest of the variables.


title: How can the speed of light be a dimensionless constant?

text: This is a quote from the book _A first course in general relativity_ by Schutz: > What we shall now do is adopt a new unit for time, the meter. One meter of > time is the time it takes light to travel one meter. The speed of light in > these units is $$\begin{align*}\end{align*}$$ > > $$\begin{align*} c &= \frac{ \text{distance light travels in any given time > interval}}{\text{the given time interval}}\\\ &= \frac{ \text{1m}}{\text{the > time it takes light to travel one meter}}\\\ &= \frac{1m}{1m} = 1\\\ > \end{align*}$$ > > So if we consistently measure time in meters, then c is not merely 1, it is > also dimensionless! Either Schutz was on crack when he wrote this, or I'm a dope (highly likely) 'cos I can't get my head around this: The space-time interval between different events at the same location measures time, and between different events at the same time measures space. So they're two completely different physial measurents: One is a time measurement using a clock, the other a space measurement using a ruler. In which case the units of $c$ should be $ms^{-1}$ Does Schutz correctly show how $c$ can be a dimensioness constant?


title: Carnot Engine Work from Heat Exchanger between Two Gas Streams

text: To be clear, this is not a homework question, but it is something I am studying for an exam. The question is about a hilsch tube which separates a stream of high pressure air to a high temperature and cold temperature stream. After this, a carnot engine is placed between the streams to convert some of the heat from the hot stream to work and the rest of the heat is given to the cold stream such that the two streams are exiting at the same temperature. I have given the full problem diagram at the end of the question. **The main question I have is this:** **Most definitions of Carnot engine work have isothermal temperature reservoirs, cold and hot, at temperatures $T_{hot}$ and $T_{cold}$ such that** $\eta=W/Q_{hot}=1-T_{hot}/T_{cold}$ **However since the temperatures here are constant, how does this relationship change if they continually change as energy is removed or added as in this heat exchanger between two streams?** * * * The manner in which I have approached this problem is: a) Apply an Energy Balance in which the Hilsch vortex tube is isenthalpic: $H_{in} = H_{out}$ $\implies$ $n_AC_p(T_A-T_{ref})=n_BC_p(T_B-T_{ref})+n_CC_p(T_C-T_{ref})$ Since $C_p$ drops out, $n_A$ is given, and $T_{ref}$ is arbitrary (can be chosen to be 0), $n_A$ and $n_B$ can be found along with using $n_A=n_B+n_C$. Which for this specific example ends up being $n_A=1mol/s$, $n_B=0.833mol/s$, $n_C=0.166mol/s$. b) **This is where the problem originates.** If we use the equation for efficiency above, we can end up using another energy balance where we take advantage of the equation $Q=mC_p\Delta T$, so we have: $Q_{hot}=n_CC_p(T_C-T_D)$ where $Q_{hot}$ is the heat being removed from the hot source, or stream $C$. So, similarly, $Q_{cold}=n_BC_p(T_B-T_D)$ and $W=Q_{hot}(1-T_{cold}/T_{hot})$ **This _would_ (if the temperatures were constant as reservoirs) be:** $W=Q_{hot}(1-T_B/T_C)=n_CC_p(T_C-T_D)(1-T_B/T_C)$ So, The energy balance would be: $W = Q_{cold} + Q_{hot}$ $\implies$ $n_CC_p(T_C-T_D)(1-T_B/T_C)=n_BC_p(T_B-T_D)+n_CC_p(T_C-T_D)$ $\implies$ $n_C(T_C-T_D)(1-T_B/T_C)=n_B(T_B-T_D)+n_C(T_C-T_D)$ All variables in this equation are known except $T_D$, which allows us to calculate it, but I don't know if the assumptions are right to get here. ![problem](http://i.stack.imgur.com/ForRJ.jpg)


title: Why is the charge transferred by electrons and not by protons?

text: charges are transferred by electrons which we all know but why it cant but it cant be transferred by protons.Well i searched on google where i found similar questions already being asked on many sites.But i ddnt found any answers of them convincing.they were saying that electrons are mobile and similarly protons are busy being nucleons.and similarly someone was saying"By the simple mechanism of separating two things, electrons are pulled out of their Fermi energy levels and into a conduction band to effect charge transference and the creation of a static charge" which i didnt understand at all.


title: How long does it take to travel 36 light years with tolerable acceleration and deceleration?

text: The recent discovery of HD85512b only 36 light years from Earth has promising attributes to harbor life. Assuming we want to travel there, we cannot instantaneously jump to light speed, (StarTrek euphemisms aside), we'll have to accelerate the conventional way by building momentum. Now, how long would it take to first reach the speed of light at a rate of acceleration that wouldn't kill the occupants of the spacecraft ? (We can't continuously accelerate at 4g's for prolonged periods because it would eventually kill you from physical stress.) Secondly, then given the time to accelerate to the speed of light, how long would it then take to travel the remaining distance and reach our new utopia ?


title: what's the physical significance of the off-diagonal element in the matrix of moment of inertia

text: In classical mechanics about rotation of rigid object, the general problem is to study the rotation on a given axis so we need to figure out the moment of inertia around some axes. In 3-dimensional case, we have a matrix (i.e. moment of inertia tensor) $$ I = \left( \begin{matrix} I_{xx} & I_{xy} & I_{xz}\\\ I_{xy} & I_{yy} & I_{yz}\\\ I_{zx} & I_{zy} & I_{zz} \end{matrix} \right) $$ I am curious what's the physical significance of the matrix element. I guess the moment of inertia in element $ij$ is the moment of inertia when the object is rotating about the axis $ij$. For example, $I_{xy}$ is the moment of inertia when the object is rotating about $xy$ axis and $I_{yy}$ is the moment of inertia when the object is rotating about $y$ axis, is that correct? When I read further into the text, it then introduce the a method to diagonalize the moment of inertia tensor such that the non-vanishing elements only appears on diagonal. In that case, the text call the diagonal elements as the principal moment of inertia, my question is what's the physical significance of the principal moment of inertia?