title: Proper time of circular motion under Schwarzschild metric

text: I'm trying to calculate the proper time of a massive particle circulating Schwarzschild black hole, using EL equation of the following Lagrangian: $$L=-\frac{m}{2}\left(1-\frac{2M}{r}\right)\dot{t}^{2}+\frac{m}{2}\left(1-\frac{2M}{r}\right)^{-1}\dot{r}^{2}+\frac{m}{2}r^{2}\dot{\theta}^{2}+\frac{m}{2}r^{2}\sin^{2}\theta\dot{\varphi}^{2} .$$ At first I get from the Euler-Lagrange equation of $r$: $$\pm\sqrt{\frac{M}{R^{3}}}\dot{t}=\dot{\phi}.$$ Then, using energy conservation: $$\dot{t}\equiv\frac{E}{m\left(1-\frac{2M}{r}\right)}.$$ Now, I thought to integrate over $\tau$ (the proper time), as LHS does not depend on it, while at RHS it turns to integration over $\phi $ form $0$ to $2\pi$ . Eventually, I end up with: $$2\pi\left(\sqrt{\frac{R^{3}}{M}}\frac{m}{E}\left(1-\frac{2M}{R}\right)\right)$$ for the proper time for one circulation. This result doesn't seem to make a lot of sense. Where do I have wrong?


title: How to find the compression of a spring attached to an object

text: I am having some trouble figuring out the equation needed to solve this problem. > A 3.0-kg block slides along a frictionless tabletop at 8.0 m/s toward a > second block (at rest) of mass 4.5 kg. A coil spring, which obeys Hooke's > law and has spring constant k = 830N/m , is attached to the second block in > such a way that it will be compressed when struck by the moving block. ![Figure 1](http://i.stack.imgur.com/sc776.jpg) **Part A**               What will be the maximum compression of the spring?      **Part B**               What will be the final velocities of the blocks after the collision?      From what I understand, in order to answer Part A I need to find $x$ (the compression of the coil) when the velocity of object 1 is the same as the velocity of object 2. However, what confuses me is trying to figure out how the force of the smaller mass will accelerate the larger.


title: Why does wet skin offer less resistance?

text: I know that water with salts is a good conductor-However wetting a hand only adds water to a not very conductive surface. How does water actually reduce the resistance of skin? The way I am seeing it is that as opposed to just a wire touching skin, the wire is touching water then touching skin, which still does not seem to imply we would have less resistance.


title: What does $E$ field look like around an inductor? What does $B$ field look like outside of a capacitor?

text: For whatever reason I could not find a picture of $E$ field around inductor, and also couldn't find a picture of $B$ field outside of a capacitor. The reason I think is because these quantities are zero, and due to some obvious geometric intuition, it is not addressed in most texts. It has been years since I touched upon this topic and what may have once been obvious is now confounding for me. Can someone provide an intuitive answer as to why these quantities are zero? Thanks


title: Does rubber insulate lightning more effectively than air?

text: Last week, an Ars Technica writer was struck by lightning. He says that the 911 operators were concerned about whether or not he was wearing shoes at the time, but he didn't think it would make much difference. Apparently, air has a thousand times more resistance than hard rubber. Does this mean that wearing shoes wouldn't do anything? What about with electrical wiring? I've seen electricity spark through the air before, but I don't think I've seen it go through rubber, especially a greater volume of it. Is there something I'm missing here, or have I just never considered it?


title: Electromagnetic Inductance: Different voltmeter readings from different positions

text: In the setup below, the voltmeter on the right would read differently than the voltmeter on the left even though they are both connected to point D and point A. ![enter image description here](http://i.stack.imgur.com/3yrq7.png) This picture is taken from Lecture 16 of MIT course 8.02. It shows a circuit with a solenoid in the middle. The magnetic field of the solenoid is pointing out of the blackboard and at the moment it is turned on, there is an increase in magnetic field inducing a clockwise current in the wire. The EMF is given to be 1V. $ Given: V = 1 $   $ V=I(R_{1} + R_{2}) = I(100+900) = 1 $   $ \Rightarrow I = 0.001 $ By Ohm's Law,   $ Clockwise: V_{D} - V_{A} = IR_{2} = 0.9 $   $ Counter clockwise: V_{D} - V_{A} = -IR_{1} = -0.1 $ So the Voltmeter on the right will read 0.9 while the left will read -0.1. This is shown in an experiment where two voltmeters connected to the same points A and D read different values. How is this possible? How does the voltmeter "know" whether it is on the left or right? It seems entirely symmetrical. How would the experiment need to be setup? If I were to just take a voltmeter and connect it to point A and D, how would I know what it would read?


title: Is there work being done if no displacement occurs?

text: So the definition of work is $W = \vec{F}\cdot\vec{s}$. Say I have a point mass which is being pushed on both sides by equal forces and therefore does not move. Does this mean that no work is being done by any force? It's apparent that there is no net force, but could I calculate the work done by each side to be the work that would have been done absent the other? For example, assuming our point mass has a mass of 1 kg and would have been moved 1 m in a direction by our 1 N force if an equal and opposite force did not counteract it. Would our force have exerted $1N \cdot 1m = 1J$ of work, or did it not perform any work since our object didn't actually move?


title: Finding current in a sphere while given a changing current density

text: im trying to calculate the current running through a sphere while given a current density $$J=Jcos(2\theta)e^{-t/r} \hat r $$ J is given in spherical coordinates. I know that: $$I=\int J\,da $$ But I can't seem to understand how to "build" my integral. I was thinking that i need to do something like $$ \int r\,dr\,sin\theta\,d\theta*J $$ but this gets very messy because of the exponent containing expression with "r". Any tips/help please? Thanks


title: Bloch's theorem

text: I am studying Bloch's theorem, which can be stated as follows: > _The eigenfunctions of the wave equation for a period potential are the > product of a plane wave $e^{ik \cdot r}$ times a modulation function > $u_{k}(r)$, which has the periodicity of the lattice. In total: $\psi_k (r) > = u_k(r)e^{ik\cdot r} $._ [Reference: Kittel - Introduction to solid sate > physics.] I have some problems understanding Bloch's theorem in full. Can I view the wavevector $k$ as the actual, physical momentum of the electron, which moves in a periodic potential, i.e., does it define the wavelength via $\lambda = 2\pi/k$? And how does this relate to the fact that all wavevectors can be translated back to the first Brouillon zone?


title: Kronig-Penney model

text: I am studying the Kronig-Penney model as treated in the book by Kittel: _Introduction to Solid State Physics._ In this model one considers a period potential which is zero in the region $[0,a]$ (define as region I), $U_0$ in the region $[a,a+b]$ (define as region II) and again zero in the region $[a+b, 2a+b]$ etc. etc., so the period of repetition is $a+b$. One can solve the Schrödinger equation seperately in region I and II, yielding complex exponentials in region I. In region II one can have both complex and real exponentials, depending on the sign of $U_0 - \epsilon$. Kittel takes real exponentials without saying a word, but I guess that one should demand explicitly that $\epsilon<U_0$? Why can we do that? Furthermore, the solutions in region I and II are tied together by demanding $\psi_I(0)=\psi_{II}(0)$ and $\psi_I'(0) = \psi_{II}'(0)$. But then the crucial step comes in: Kittel writes that Bloch's theorem states (in this specific case) that $$ \psi_{II}(a<x<a+b) = \psi_{I}(-b<x<0)e^{ik(a+b)} $$ and uses this to write $\psi_{II}(a) = \psi_I(b)e^{ik(a+b)}$ and likewise for the derivatives. These two conditions, together with the previous two yield two equations which give a consistency equation on $k$, which will have no solutions in certain regions, yielding band gaps etc. I have some trouble understanding this. I understand that the crystal periodicity is $a+b$, but why does that mean that this combination occurs in the complex exponential? Furthermore, why can't we say $\psi_{II}(a) = \psi_{II}(-b)e^{ik(a+b)}$ (besides the fact that it will not help us solve the problem)? I do not really understand why one should tie the two different wavefunctions together in this specific way.


title: BICEP2 and e-foldings during inflation

text: After the BICEP2 results, we now know that $n_s = 0.96$ and $r = 0.2$. From what I understand, this fits extremely well with the basic chaotic inflation model given by $V(\Phi) = \lambda \Phi^4$. We also know that amplitude of density fluctuations is $\approx 10^{-5}$ and Energy scale of inflation is around $10^{16}$ GeV. My question: Given also this information, can we now make an educated guess for how many e-foldings happened during inflation ? Or at least a theoretical lower (upper ?) bound ? PS: I am only referring to inflation of the patch of space that now contains our observable universe, not inflation of Universe as a whole.


title: Why is there a voltage drop across a superconducting inductor?

text: If there is no electric field inside a perfect conductor, then why is there a potential difference across an (superconducting) inductor?


title: Postulate of Quantum Mechanics on Wavefunction collapse

text: There is a postulate in quantum mechanics about wavefunction collapse. Can anyone provide me some intuitive perspective into it along with the concept of projection (I mean mathematical treatment).


title: Motivation for spinors

text: 1. After it was found that the gamma matrices couldn't be Pauli matrices and only had to be larger and even, why was their need to define a new algebraic object (i.e a Dirac spinor)?    2. Why couldn't a vector or a tensor do?    3. Plus what exactly is a spinor as compared to vectors?


title: Equilibrium of of water level in punctured bucket?

text: Would a bucket with a hole in the bottom being filled at different rates with water come to to different equilibrium levels with in the bucket? Or would at any fill rate above initial flow rate of the hole cause the bucket to overflow? An answer with more math is preferred. **My thoughts:** If flow rates of water into bucket less than initial flow rate of hole then water level stays at or below height of hole in bucket. If water flow rates begin to increase, so does the depth of the water above the hole then since the product of the hole's size and pressure is positively proportional to flow rate, the water leaving and coming reach a equilibrium point with in bucket x height above hole. If water entering bucket becomes to fast then a equilibrium point in the size of the current bucket can not be reached, so the bucket over flows.


title: Producing energy from magnetic flux loops

text: I've simplified this down by quite a lot but as far as I understand it, magnetic fields on the Sun's surface twist together and when it gets all a bit too much they release energy.   My question is why can't we twist magnetic fields here on Earth to produce energy?


title: What happens if light/particles exceeded the speed of light for a particular medium

text: While the speed of light in vacuum is a universal constant (c), the speed at which light propagates in other materials/mediums may be less than c. This is obviously suggested by the fact that different materials(especially in the case of transparent ones) have a particular Refractive Index. But surely, matter or even photons can be accelerated beyond this speed in a medium? If so, what are the effects?


title: Why can't CO2 mix back with the liquid after a soda bottle has been shaken?

text: If you shake a soda bottle before opening it, and then open it, you get the fizz. That is the compressed CO2 releasing to the atmosphere which is at comparatively low pressure value. Two questions (related)   * Why does shaking a bottle make the compresses gases _un_ -dissolve?   * Why can't the gases dissolve back into the liquid?


title: What defines the brightness of a bulb?

text: So I have a question. There are three identical bulbs, 2 of them are connected in parallel and the third is basically in series, on the same circuit. If the one of the lamps in parallel breaks, what happens to the brightness of the other two? I don't know how to work this out, and what affects the brightness because I know its power but in this case we have to consider voltage or current.


title: Is there any example of an inelastic collision where final kinetic energy of the system is greater than its initial kinetic energy?

text: Is it even possible to have such kind of a collision?


title: A dictionary of string - standard physics correspondences

text: Motivated by the (for me very useful) remark ''Standard model generations in string theory are the Euler number of the Calabi Yau, and it is actually reasonably doable to get 4,6,8, or 3 generations'' in http://physics.stackexchange.com/a/22759/7924 , I'd like to ask: Is there a source giving this sort of dictionary how to relate as much as possible from general relativity and the standard model to string theory, without being bogged down by formalism, speculation, or diluted explanations for laymen? Or, if there is no such dictionary, I'd like to get as answers contributions to such a concise dictionary. [Edit April 30, 2012:] I found some more pieces of the wanted dictionary in http://www.physicsforums.com/showthread.php?p=3263502#post3263502 : ''branes and extra dimensions have proved to be implicit in standard quantum field theory, where they emerge from the existence of a continuous degeneracy of ground states. That multidimensional moduli space of ground states is where the extra dimensions come from, in this case! Branes are domain walls separating regions in different ground states, strings are lines of flux connecting these domain walls. Furthermore, in gauge theories with a small number of colors, it looks like the extra dimensions will be a noncommutative geometry, it's only in the "large N" limit of many colors that you get ordinary space.''


title: Radius dependence of Energy levels of spherical Quantum dots

text: I have frequently read in articles that the energy level of spherical quantum dots depends on radius. We simply match the wave functions inside and outside the particle and find the intersection of the curves which give us the appropriate $k$: $$ V(r)=0 \hspace{2cm} for \hspace{2mm}r<R $$ $$ \hspace{1.cm}V_0 \hspace{1.8cm} for\hspace{2mm} r>R $$ The wave functions inside and outside the dot of radius $R$ are $\phi_1$ \begin{equation} =N \frac{\sin(kr)}{r} \hspace{1cm} r<=R \\\ =N\sin(kR)\exp(KR)\frac{\exp(-Kr)}{r}\hspace{1cm} r>=R \end{equation} The transcendental equation $$ kR+KRtankR=0 $$ was solved to obtain the lowest ‘subband’ energy $E_1$.In $ r=R$ by matching the wave function we earn a $k$. $ k^2=\frac{2mE}{\hbar^2}$ inside and $K^2=\frac{2m(V_0-E)}{\hbar^2}$ outside the particle. I don't know how can we insert the radius dependence. ![enter image description here](http://i.stack.imgur.com/ZZlmb.png) Any help is appreciated.


title: Prior binary star system as possible explanation of high-velocity Type II Supernovae remnants

text: From wikipedia(http://en.wikipedia.org/wiki/Supernova) :- "A long-standing puzzle surrounding Type II supernovae is why the compact object remaining after the explosion is given a large velocity away from the epicentre;[96] pulsars, and thus neutron stars, are observed to have high velocities, and black holes presumably do as well, although they are far harder to observe in isolation. The initial impetus can be substantial, propelling an object of more than a solar mass at a velocity of 500 km/s or greater. This indicates an asymmetry in the explosion, but the mechanism by which momentum is transferred to the compact object remains a puzzle." My Question:- Could the remnant compact high-velocity object simply be the surviving member of a binary pair, flying off at a tangent from the original locality after the companion star has blown away in a supernova?


title: About star groupings and there apparent positions in the sky

text: Imagine a group of 3 stars that appear to be equally luminous ,star A, B, and C in the shape of an isosolese triangle. Imagine the 3 stars are in different $time-zones$. A,B and C look like they form an isosolese triangle right now on the Earth but when the light from star B reaches the Earth the light from star C still has say, 10 years of travelling to go. So when B is observed in its present position, the apparent position of C is from 10 years in the past. Where C is now relative to the present position of B might be different in such a way so as to not to appear to form an isosolese triangle. The 3 stars 'right now ' could form a zig-zag pattern as they appear in the sky but we might not 'see' that from the Earth for a 'while'. So should differences in many stars positions relative to the Earth be taken into account when the simultaneity or cause- and- effect of some stellar events are being analysed?


title: $c^4$ in Einstein field equations

text: I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $c^4$ in the denominator. the $8{\pi}G$ term can be obtained from Poisson's equation, but how does $c^4$ pop up. Most of the books say that in units where $c$ is not equal to 1, you get $\frac{8{\pi}G}{c^4}$. There is no need or mention of an explicit assumption that $c=1$.


title: State of Matrix Product States

text: What is a good summary of the results about the correspondence between matrix product states (MPS) or projected entangled pair states (PEPS) and the ground states of local Hamiltonians? Specifically, what "if and only if" type of equality/approximation hold? I know about this review-like paper by Verstraete, Cirac, and Murg but I feel some of its results are superseded by new ones...


title: Linearizing Quantum Operators

text: I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$ The right hand side is a sum of the "mean" value and the fluctuations about the mean. While I understand that the physical picture is reasonable, is this mathematically correct? If so what are the constraints this imposes? In literature this is designated as a "linearization" process. My understanding of a linear operator is that it is simply a homomorphism. I have never seen anything done like this and I'm having a hard time finding references which justify this process. I would be grateful if somebody can point me in the right direction!


title: Interesting topics to research in mathematical physics for undergraduates

text: I'm planning on getting into research in mathematical physics and was wondering about interesting topics I can get into and possibly make some progress on. I'm particularity fond of abstract algebra and topology and if possible any topics that involve abstract algebra would/topology/calculus of variations would be especially appreciated My mentor is involved in work dealing with topological field theories in regards to developments in string theory.


title: Pump pressure required to fill water tower

text: I have a water pump at a distance of x meters from a water tower located h meters above the pump through a pipe with diameter d. I am not concerned with the exact value just want to make sure my pump is strong enough. As far as I can remember the pressure required is equivalent to the weight of the water in the pump in kg multiplied by the height in meters of the tower. What is the formula for pressure required in kg as a function of x, h and d? All units metric.


title: The derivation of the Belinfante-Rosenfeld tensor

text: * It seems me that there is a "difference" (at least apparently) in how the Belinfante-Rosenfeld tensor is thought of in section 7.4 of Volume 1 of Weinberg's QFT book and in section 2.5.1 of the conformal field theory book by di Francesco et. al.  I would be glad if someone can help reconcile the issue. Schematically the main issue as I see is this -   * If the action and the lagrangian density is writable as $ I = \int d^4x L$ and $\omega_{\mu \nu}$ be the small parameter of Lorentz transformation then Weinberg is thinking of $\omega_{\mu \nu}$ to be space-time independent and he is varying the action to write it in the form, $\delta I = \int d^4x (\delta L = (A^{\mu \nu})\omega_{\mu \nu}) $ Then some symmetrized form of whatever this $A^{\mu \nu}$ comes out to be is what is being called the Belinfante tensor. ( Its conservation needs the fields to satisfy the equations of motion)    * But following Francesco et.al's set-up I am inclined to think of $\omega_{\mu \nu}$ as being space-time dependent and then the variation of the action will also pick up terms from the Jacobian and the calculation roughly goes as saying, $\delta I = \int (\delta(d^4x)) L + \int d^4x (\delta L)$. Since under rigid Lorentz transformations the volume element is invariant the coefficient of $\omega_{\mu \nu}$ in the first variation will vanish but the second variation will produce a coefficient say $B^{\mu \nu}$. But both the variations will produce a coefficient for the derivative of $\omega_{\mu \nu}$ and let them be $C^{\mu \nu \lambda}$ and $D^{\mu \nu \lambda}$ respectively.   Now the argument will be that since the original action was to start off invariant under Lorentz transformations, when evaluated about them, the $B^{\mu \nu}$ should be $0$ and on shifting partial derivatives the _sum_ of $C^{\mu \nu \lambda}$ and $D^{\mu \nu \lambda}$ is the conserved current (..and its not clear whether their conservation needs the field to satisfy the equation of motion..) So by the first way $A^{\mu \nu}$ will be the conserved current and in the second the conserved current will be, $C^{\mu \nu \lambda} + D^{\mu \nu \lambda}$ (the C tensor will basically look like $-x^\nu \eta^{\lambda \mu}L$) Is the above argument correct? If yes then are the two arguments equivalent? How or is Weinberg's argument taking into account the contribution to the conserved current from the variation of the Jacobian of the transformation?


title: Measuring polarization of a neutral object near a charged sphere

text: I am trying to design an experiment where I can calculate the distance at which polarization will not have a measurable effect on a neutral object, from a sphere charged by a Van Der Graaf generator. I am measuring the weight change caused by the polarization of the neutral object with a scale and measuring this effect at different distances. I have a problem in verifying my results, as i do not know an equation to measure polarization or something else which would help me verify this distances. Any help would be appreciated. Thank you


title: Lagrangian Coordinates in Fluid Flow

text: I apologize if this is not the right place to ask this question: I am currently reading a paper by Y. Brenier, where for the fluid flow he introduces a Lagrangian label $a$ instead of the vertical coordinate $z$, and defines a "Lagrangian foliation to be a family of sheets $z = Z(t,x,a)$, labelled by $a \in [0,1]$, where $Z$ is a smooth function such that: $$ 0\leq Z(t,x,a)\leq 1, \,\,\,Z(t,x,0) = 0, \,\,\,Z(t,x,1) = 1\\\ \partial_a Z(t,x,a) > 0\\\ \partial_t Z(t,x,a) + u(x,t,Z(t,x,a))\cdot \nabla_x Z(t,x,a) = w(t,x,Z(t,x,a))" $$ where $x \in \mathbb{R}^2$, $u = (u_1,u_2)$ is the horizontal flow velocity and $w$ is the vertical velocity and $\nabla_x = (\partial_{x_1},\partial_{x_2})$. A possible initial choice is $Z(0,x,a) = a$. My question is: What is the physical interpretation for this? It seems like instead of labeling each individual fluid parcel in the classical Lagrangian coordinates, we are only labeling horizontal planes instead, and then seeing how these sheets move and deform with the flow?


title: Looking out into the universe means looking back in time - how does that work?

text: This is a question that has been gnawing on me for many years now. Back a long time ago, as I recall in reference to a scene in a popular science show on TV, I was asked the following. The claim is that when you look out into the universe, you see the universe as it appeared at some past time. (This follows from the speed of light being finite, and looking at objects by observing their emitted light or other EM radiation.) The amount of time that one looks backwards is equal to the distance to the observed object in light-distance (so looking at something ten lightyears away means you are observing it as it was ten years ago). Matter moves at a rate of speed less than the speed of light (it has to, since matter has mass, even if miniscule at the atomic level). So $v_{mass} \lt v_{EM}$ (probably significantly less, since $v_{EM} = c$). So let's say you're looking at an object $10^{10}$ lightyears away by observing something for which $v=c$. To the observer, that object appears as it was $10^{10}$ years ago. But the Earth is much younger than $10^{10}$ years and we established that matter moves at a slower speed than EM radiation, so _wouldn't the radiation that was emitted $10^{10}$ years ago long since have passed Earth's current position?_ If, say, $v_{mass} = 0.5 \times c$ (a big assumption, but bear with me for a second), it would _seem_ that the radiation emitted $10^{10}$ years ago would have passed Earth's current position $10^{10} - (0.5 \times 10^{10}) = 5 \times 10^{9}$ years ago, around the time when the solar system was still forming. Setting $v_{mass} = 0.25 \times c$ (which seems more realistic) means the radiation would have passed "us" around $7.5 \times 10^{9}$ years ago. So **how could we be observing it now?** I'm not sure I'm posing this question in the best possible way (I'll admit I do find the concept somewhat confusing), and I'm _sure_ that there's a simple explanation for it all. _Just what am I missing?_ I did find Is it possible to look into the beginning of the Universe? which seems peripherally related but not quite the same thing. Qmechanic brought up Seeing cosmic activity now, really means it happens millions/billions of years ago? but that question seems to be about _whether_ it is so, not _why_ it is so.


title: Calculation of error in focal length?

text: ![enter image description here](http://i.stack.imgur.com/P8Wsp.png) $u=-10cm$ $v=10cm$ ![enter image description here](http://i.stack.imgur.com/wab5k.gif) Using the formula the focal length is 5 cm. But how do I get the fractional error in focal length when neither $\Delta u$ nor $\Delta v$ are specified? The options given are   A. (0.05±0.05) cm   B. (0.05±0.10) cm   C. (5.00±0.05) cm   D. (5.00±0.10) cm


title: Perceived weight in dumbbell vs. barbell

text: I had an observation today at the gym when I was lifting weights. Lifting a dumbbell of 10 kg on each hand was as difficult for me as lifting a barbell with 5 kg on either end. It's true that the bar has some weight of its own, but still the relative ease notwithstanding the weight difference was striking and counter-intuitive for me. I would love to know the physics involved in lifting a dumbbell and a barbell in a bicep exercise. Is there a relationship between effort, weight and angle of lifting in each of the two cases?


title: Can a mechanical systems on hold be switched off, in another way than just letting it do it's thing?

text: Can the value of the potential energy, which is responsible for driving the system, diminish in time, while the system itself is stationary during that time? Can there be dissipation in a system, while no physical degrees of freedom (at all levels) are changing? Maybe this should be formulated as "Can I write down a time dependend energy, whos total value reduces in time, while this change has no impact on the equations of motion." Maybe this is just equivalent to a time dependend exteriour force in every case - then I guess the answer is no, in the sense that the exteriour force is also a physical degree of freedom. (The question is probably motivated by me wondering if one can replace all (spatial?) boundary condition in a system by other forces: A chemical potential equilibration holds particles back from moving in one direction and so replaces a wall, which would be a coordinate restriction in a more mechanical pov. Can one similarly get rid of all spatial aspects of a classical theory? Maybe d'Alamberts principle goes in this direction. Mathematical phase space reduction kind of obscures physical system, I don't know what's real anymore.)


title: How do the energy eigenvalues of rotational degrees of freedom in statistical mechanics come about?

text: I want to understand the hierarchy different degrees of freedom of a mechanical system. Specifically, I want to understand which subsystems equibrilate faster and why. This question comes up: > Why is the rotational temperature $T_\text{rot}\propto\frac{\hbar^2}{I}$ > invesely proportional to mass? Shouldn't that energy increase, like in the case of the Maxwell-Boltzmann distribution, where it is proportional to the kinetic energy (translational degrees of freedom)? The positive energy-powers are in the Planck's constant $\hbar$ and essentially originate from the angular momentum (operator), which makes the particle dependend quantity $[I]\propto mass$. Probably, the answer becomes clear, once one has a general picture of the coupling of the system to it's rotational degrees of freedom.


title: Reflection of Electromagnetic Waves

text: Visible light - Being an Electromagnetic wave is reflected by glass (take mirror). Would all other waves in the electromagnetic spectrum be reflected in the same way by our simple mirror... For highly intense X-Rays & gamma rays, take that the mirror is more **stronger** such that it could resist for at least 30 seconds. Well If the mirror would reflect, then would the EM waves be the same after _several_ (take 30) reflections..? (At each reflection, the photons scatter in different directions... Would the intensity and energy of EM waves remain same?)


title: Spacetime and uncertainty principle

text: I only have limited knowledge of relativity and quantumphysics but as far as I know, the uncertainty principle relates the uncertainty of space and momentum of a particle. Einstein however, explained that space and time are tied together and the real fabric of the universe is spacetime through which all objects navigate. It feels as if space uncertainty should therefore be spacetime uncertainty. Is this wrong? Can it be that you know the position of a particle but not the exact time when it was there, and that that gives rise to uncertainty in momentum? If so, wouldn't this be a more elegant way to express the uncertainty principle? Googling for "spacetime uncertainty" gives papers that go far over my head. While my math is good, we barely touched on quantum mechanics in physics.


title: Why the pressure of water at 298 Kelvin is 25 mm Hg (torr)?

text: Here is the problem, is about the law of Dalton. NH4NO2 decomposes giving N2 (gaseous). NH4NO2 decomposes and collects on water at 198 Kelvin and 748 atmosphere (atm). Vokume 411 ml N2. How many grams of NH4NO2 have been decomposed? I understood all the procedure to solve the problem eccept the first one in which to calculate the pressure of N2 it says that the pressure of the water is 25 torr (mm Hg). Why is the pressure 25 torr? Maybe there is a table to see the pressure of the water on a certain temperature? And if yes, where is this table? Sorry for bad english, i'm italian.


title: Batteries connected in Parallel

text: "If we connect two batteries in parallel, the battery with higher voltage will discharge into the other one." What is the intuition behind this?


title: What are the applications of delta function potentials?

text: Are there real applications for using delta function potentials in quantum mechanics (other than using it as an exactly solvable toy model in introductory undergraduate quantum mechanics textbooks) ?


Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?


title: Emitting gravitational radiation

text: Is the following true:   * Two massive bodies with variable distance between them do not emit GR in any direction   * Two bodies that revolve around common center will not emit in the plane of their orbits   * But they will emit in all other directions with the maximum along the axis of rotation perpendicular to the orbit's plane


title: Only one electron?

text: If it is true that an electron can be anywhere in the cosmos at any given time, then is it even theoretically possible that there is only one electron, instead of multiple electrons in the cosmos? If it isn't, I'd appreciate it if someone could point out where I may have misunderstood something as far as the fundamental properties of electrons are concerned.


title: How is formula for converting pressure from mmHg to Pa derived?

text: Today my younger brother asked me from where does the 1 Pa = 0.00750061683 mmHg formula for mercury barometer come. He needs a way to derive it, or an academic source which can be cited. After doing some calculations we got the formula for a standard U-tube manometer: $P=\frac{h_2}{h_1}P_0$ where $P_0$ is atmospheric pressure, $P$ is pressure being measured, $h_1$ is height is mercury column exposed to atmospheric pressure and $h_2$ is height of the column exposed to pressure being measured. The problem is that in the case of a barometer, the $h_2$ is exposed to vacuum and I don't know how to use that. I've searched the Internet, and got countless sites which explain how a mercury column barometer works, but I was unable to find a site which explains which forces are acting there and how the number was derived. To make things even worse, none of the physics books I have access to have a detailed explanation.


title: What kind of research will go in nuclear fusion power once the commercial power plant is designed and implemented?

text: once the commercial nuclear power plant is built for electricity production , after this what kind of research does scientists do in nuclear fusion power. Why I am asking this question is ITER is planning to build the fusion reactor in 2030s , So after what research does nuclear fusion scientists do ?


title: What makes a material a good lens for CO2 lasers?

text: I read on Wikipedia that zinc selenide and germanium make good lenses for $\mathrm{CO}_2$ lasers. My question is, Why?


title: A question on IR cancellation caculation in peskin schroeder

text: In Peskin and Schroeder _Introduction to Quantum Field Theory_ book, above equation 6.64 on pg. 200, it was said that "to gain better understanding, of the divergence, let us evaluate the coefficient of the log in the limit of $-q^2\rightarrow\infty$." Why does he want to take $-q^2$ to infinity rather than $0$ to extract the divergence? I thought the form factor $F_1(q^2)$ contains an infrared divergence at $q^2=0$.


title: Fermion propagator is not a Grassmann-odd object?

text: Is the following differentiation correct: $$ \frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$ where $\eta$ is a Grassmann-valued field and $S_F$ is the Fermion propagator, or is the result actually with a minus sign?


title: Incommensurability between different observers describing the same universe?

text: According to black hole complementarity, if there is a black hole and Alice falls into it carrying a qubit, but Bob stays out, then Alice can measure the qubit inside the black hole, and confirm it exists inside, but Bob can observe the Hawking radiation coming out, collect all the data, feed them into a giant quantum computer, and extract the qubit from the outgoing Hawking radiation. Either the no xerox theorem is violated, or Alice' and Bob's observations are incommensurable. In plain old quantum mechanics, it is possible to set up an experiment where we can choose between measuring the position, or the momentum of a particle, but once that choice is made, all observers will agree upon whether it's the position or the momentum which is being measured. But not for black holes? Is the qubit inside, or outside? Bob has a problem. The qubit is scrambled and deeply encoded within the Hawking radiation. After collecting all the relevant Hawking radiation, would Alice have hit the singularity by then? If so, Alice measured first. Things are actually much worse. According to complementarity, encoded within the Hawking radiation is also an extremely scrambled version of Alice as she would look like after drinking too many "Drink me"s, eating too many "Eat me"s, and nibbling way too many mushrooms. To make matters even worse, the scrambling scrambles the scrambled Alice with the scrambled qubit, and to extract the qubit, his quantum computer would also have to unscramble Alice from the qubit. **PS** : If Alice is encoded holographically within the Hawking radiation, does that mean Bob can reconstruct what Alice did in the hole by observing all the Hawking radiation and feeding them into his quantum computer? And if he can reconstruct what Alice did, in what sense can't she communicate with Bob? Is it possible to simultaneously measure observables which don't commute with each other? Does Alice commute with Bob? They can't compare notes, that's for sure. According to Niels Bohr, something is objective only if an observer can measure it and unambiguously communicate the results in public using common language. If Alice can't communicate anything with the outside world, is anything she observes objective? Is she even qualified to be considered an observer? Let's say Humpty-Dumpty had a great fall to the black hole. According to Bob, as he approaches the hot stretched horizon, he becomes heated and scrambled, until we get scrambled eggs. The scrambled eggs evaporate as Hawking radiation. What Bob wishes to do is put Humpty-Dumpty back together again. To do that, he has at his disposal the entire royal team consisting of all the king's horses, and all the king's men. They engineer a device which can detect and measure every single quanta of Hawking radiation from the beginning right until the very end. Not only that, the measuring devices can preserve the quantum coherence of all the Hawking radiation quanta. All this data is transmitted along a coherence preserving perfect quantum channel to a quantum computer far away. Is it possible to reconstruct Humpty-Dumpty inside the quantum computer? Would this violate the second law of thermodynamics? According to Szilard, Landauer and Bennett, no, because we can convert entropy into information. Bob's quantum computer has enough memory to store all the information of the black hole. Not only that, it's much larger than the black hole in size so it doesn't collapse into a black hole of its own. It preserves quantum coherence. In fact, it's so large, and we are so patient to wait for an astronomically long time that all the quantum computer processes are low energy, and never remotely approach Planck energies at all. All the redshifted Hawking radiation are also low energy. Surely the universe is Turing complete enough for Bob to construct and program such a quantum computer? Can Bob reconstruct Humpty-Dumpty in principle? What about in practice? If something can be done in principle, but not in practice, is it still possible? According to Alice inside the hole, Humpty-Dumpty said a word inside the hole. This word is holographically encrypted within the Hawking radiation, and is all there outside in encrypted form for Bob to read. The information is all out there, and if Bob takes the detected encrypted pattern as the word itself, he gets the word. But to find out what that word means, he needs a "common language", and he can use the quantum computer as the translator. If translated, did Humpty-Dumpty send a word faster-than-light? If untranslated, did he still send a word faster-than-light? Inside the hole, Alice measures the qubit (Tweedle) along a basis consisting of the states "DEE" and "DUM". Outside, Bob detects all the Hawking radiation and feeds them into his quantum computer. He runs a program to extract Tweedle. But Tweedle is scrambled with Alice and her measuring apparatus, and in fact, just about almost everything else inside the hole. That's the nature of holographic encryption. According to Bohr, we have to include the measuring apparatus in the description. To unscramble Tweedle requires the quantum computer to extract a simulated Alice and simulated apparatus as well, and store them in the memory needed according to Landauer and Bennett. We can't erase the information! But the holographic Alice has already measured the holographic Tweedle using the holographic apparatus, and now, they are all entangled with each other. From the perspective of these holographic entities, this entanglement has decohered. To extract the pure qubit, Bob has to run another problem implementing the unitary transformation reversing the measurement and run them on the simulated Alice-apparatus-qubit system. This recombines all the decohered many worlds, recohering them. After extracting Tweedle, he measures it along an incompatible basis. But the simulated Alice now no longer has any memory of the measurement after the time-reversed transformation. The simulated Alice can remember the future, but not the past. Think about it this way; the Hawking radiation is what we get after evolving time forward for the system consisting of the exterior and the stretched horizon. To reconstruct Alice requires the quantum computer to reverse time in the simulation right back to the moment when Alice and the qubit fall into the hole, then run the simulation forward in time again from the perspective of Alice. To simulate this time evolution, the quantum computer also has to include a significant part of the state outside the hole to take into account infalling matter, but this can be done in principle. There was a time when a simulated Alice knew whether the simulated Tweedle was DEE or DUM, but Bob had no clue whatsoever about Tweedle, and can't without decohering the simulation along the DEE/DUM basis. After reversing time in the simulation, the simulated Alice had her memory of the value of Tweedle wiped clean, but now, Bob can measure the simulated qubit along the incompatible basis. There was never any time when both the simulated Alice, and Bob knew what the value of Tweedle was along incompatible bases simultaneously. Was Alice teleported from the interior into the simulation faster than light? The simulated Alice would have no clue she is inside a simulation, and would think she exists in an objective black hole interior. The faster-than-light interpretation only happens if we mix up incompatible frames. From Bob's frame consistently, it's the holographic Alice which was teleported, and this never happens faster than light. Do Alice and Tweedledee/Tweedledum really exist, or are they all a simulated dream inside the quantum computer? Remember that for Bob, the interior doesn't exist; only the stretched horizon does. Are the holographic Alice and Tweedledee/Tweedledum distinct as objects from the simulated Alice and Tweedledee/Tweedledum? Which of them is the real Alice? If you say neither of them are the real Alice, by the same reasoning, Bob doesn't have the real qubit either. Actually, as Alice and Tweedle are quantum systems, they obey the no Xerox principle. So, the holographic Alice and the simulated Alice can' both exist at the same time, although they can exist at different times. What no Xerox is telling us is from a quantum information theoretic point of view, both the holographic and simulated Alice are one and the same. According to Bob, the disembodied information encrypted within the stretched horizon and the Hawking radiation hallucinated itself as Alice, and this illusion hallucinated a nonexistent black hole interior. This disembodied delocalized illusion is transfered and reconstructed as a simulation inside a quantum computer, all for the sake of getting the qubit, but only after the original disembodied illusion had dematerialized to satisfy the no Xerox principle. The simulated Alice had her memory erased, all for the sake of getting the qubit. Things get even more exciting. After measuring the value of the simulated Tweedle along an incompatible basis, Bob has to decide what to do with the simulation running on his quantum computer. He could decide his task is done, and end the simulation, effectively killing the simulated Alice. But what if he decided to keep on running the simulation, this time with the simulated time running forward? The simulated Alice, who still thinks she's inside an objective black hole, would notice a violation of the laws of physics leading to an unexpected change of the qubit, projecting it into one of Bob's basis states. * * * There is an entirely incommensurable point of view to this story. According to Alice, she is objectively real, and is inside an objectively real black hole. The part of spacetime outside the black hole which exists before the black hole finally evaporates away is also real. More precisely, those events in spacetime for which there exist future directed trajectories which can end inside the black hole. So, Bob and his instruments measuring the Hawking radiation are real before the final evaporation. Their reality can be confirmed by messengers (maybe photons, or something more elaborate) from Bob falling into the black hole to be picked up by Alice. But time would end at the singularity, and with it, the exterior of the black hole too, because messengers can't reach beyond the singularity. Actually, there is no beyond the singularity. But Alice can imagine an extrapolation of Bob and his quantum computer into the future beyond that, but alas, not simulate them because the computational resources inside the hole are too limited. Maybe she does this because she's a cosmologist, or a philosopher ;) She knows from the messengers that the real Bob before the final evaporation had built a quantum computer and detectors collecting all the Hawking radiation with the express purpose of reconstructing her and Tweedle. She also knows that if time didn't end at the singularity, this Bob would actually have the resources to reconstruct her, and will, but she also knows time will end, so that won't really happen in reality. But she's free to imagine what would happen if time didn't end for Bob. The imaginary future Bob will be under the delusion he's the one who is real, and so is his quantum computer, but Alice, who is very real, is but a simulation running on his quantum computer. Well, Alice has already measured Tweedle, and observes it's in a definite state, either DEE or DUM. So, in her imagination, if she were in a simulation, which she's not, this would be the time before the imaginary future Bob made the measurement, when the simulation is being run backward... Ah, what about earlier, before she measured Tweedle? If the imaginary Bob decides never to run the simulation forward in time all the way to the singularity, then she won't experience the singularity if her imaginary scenario were actually true, and that would be a prediction distinguishing whether she was real, or simulated, but that's in the future and she hasn't experienced the future yet...


title: Why does a particle in an accelerating vehicle does not 'forget' its initial velocity after being dropped out of the vehicle?

text: A vehicle is moving with some constant acceleration 'a'. A particle is dropped out of the vehicle with no initial velocity. The net acceleration of the particle as soon as it loses contact with the vehicle is 'g', but what about its velocity at that very instant? Would the particle's velocity be a resultant of its horizontal component(due to vehicle) and vertical component (due to free fall)?


title: Hydrostatic pressure - doesn't density vary with depth?

text: Our class is learning about hydrostatic water pressure and we have been told that we can calculate the force of the liquid on an object at any depth using "the density x 9.8 x the depth". However, as the depth increases, wouldn't the density of the liquid increase because of the weight of the liquid above it compressing it? So should't there be something in the equation to account for the varying density? To me, "density x 9.8 x depth" seems like it is saying that the density will be constant...


title: Choosing sign for KVL (mutual inductance)

text: If I am writing KVL for one loop of the circuit which contains a coil coupled with another coil from another loop, the sign is chosen considering whether both currents enter or leave the stars AND if the original current has the same sense or not as the one chosen for the loop? EDIT: ![enter image description here](http://i.stack.imgur.com/ucP79.png) So, if I apply KVL to the left loop, in the shown direction, the voltage drop caused by the coupling should have the sign (-)*(-)?


title: More points vs. precision

text: I have a dilemma. In my lab exercise, I was measuring spectra with HPGe detector of several sources (gamma spectroscopy). To determine the energy of the unknown spectrum I first needed to calibrate the detector by making measurements of known elements: Co 60, Cs 137, and Ba 133. The spectra of Co and Cs have nice discernible peaks, and from table I know their energies. For Co ~1332 and 1173 keV peaks, and Cs ~661 keV peak. But I also had Ba 133, and there were 4 peaks, and even tho I have the energies of those peaks (I don't know with 100% certainty that they are correct), when I draw the calibration graph (energy/channel plot), I have some scattering, and my calibration fit is E=a*n+b, where a=($0.78\pm 0.01$) keV, and b=($80\pm 10$) keV. But if I only take the three distinct peaks (Co and Cs), the calibration parameters are a=($0.8200\pm 0.002$) keV and b=(26.1\pm 0.3)keV. The relative error in this case is minute. The picture is: ![enter image description here](http://i.stack.imgur.com/WSOQI.png) the dark cyan one is fit with all 7 points, and the red dashed is only 3 points. Now, my dilemma is: do I take more points (I remember from my stat class, that the more the better, since you never know how your data will behave in different regimes), or do I take these three which are totally distinct and I know with 100% the energy of those peaks. And then use that as my calibration? Is the ~40keV (around 200th channel) difference between these two, a big difference in determining the unknown spectra? Even tho my relative uncertainty is higher with Barium spectra, maybe it's good to include them, especially if my unknown spectra have peaks on lower energies (channel). What would you do? EDIT: Barium spectra: ![enter image description here](http://i.stack.imgur.com/LuEQz.png) And my detector has 2048 channels.


title: Determine viscosity using falling sphere (Stokes Law, Ladenburg correction)

text: ## Introduction I am trying to determine the viscosity of a fluid. Therefore, I let a sphere of known mass `m` and radius `r` fall (sink) into the fluid. I then measure the time it takes to sink/travel a given height. Then there are the following forces having a effect on the sphere: $F_g = mg$ and $F_a = \frac{4 \rho \pi r^3 g}{3} $ (buoyant force) with $g = 9.81 \ ms^{-2}$ and $\rho$ being the density of the fluid. The result is a motion downwards (sinking), hence I get friction $F_r$. Assuming this friction gets that strong to have no resulting force on the sphere (i.e. its velocity remains constant), I can use the following equation: $F_g - F_a - F_r = 0$ According to Stokes Law, the friction of a moving sphere in a fluid with viscosity `n` is: $F_r = 6 \pi r n v$ where `v` is the constant velocity. Using these equations, I can determine the viscosity: $n = \frac{g}{6 \pi r} (m - \rho \frac{4}{3} \pi r^3)\frac{1}{v}$ with $v = \frac{s}{t}$. ## Question about Ladenburg correction The fluid is inside of a cylinder, so I cannot use Stokes Law "as it is" because the walls of the container add more friction. Hence I have to apply the so called Ladenburg correction with a correction factor $f = (1 + \frac{(2.1 r)}{R}) (1 + \frac{(3.3 r)}{H})$ When / where do I have to apply this factor? ## Question about extrapolation As I am doing this with spheres of different radiuses, I get multiple values for the viscosity:               radius/mm | viscosity/(kg/(m*s))     --------------------------------     1.0       | 0.594     2.0       | 0.608     2.5       | 0.631      Extrapolation to $r = 0 \ mm$ gives a value $n_0 = 0.575$. I do not understand: Why do I need this extrapolation?


title: Does the curvature of space-time cause objects to look smaller than they really are?

text: What's the difference between looking at a star from a black hole and looking at it from empty space? My guess is that the curvature of space-time distorts the wavelength of light thus changing the color observed by observer 1. Is the size of the observed object distorted? ![enter image description here](http://i.stack.imgur.com/zksHv.jpg) Adopted from Being optically fooled by gravity [closed]


title: Does any lower frequency electromagnetic radiation naturally reach us from the Sun?

text: I am not sure whether the sun _originally_ emits _everything_ in the electromagnetic spectrum, (whatever the relative strengths of each portion might be), but I do read that many waves, including gamma / x-ray, and UVC, are filtered or 'converted' to heat after travelling through solar plasma and atmospheric layers like ozone. But what I am interested in is the lower end of the spectrum (frequency-wise) - like those we ourselves harness for communication. Does the sun emit low frequency radiation (below infrared), in the first place, and if so, how much of it - if any - actually reaches us on Earth within that same range? I am guessing it is either none, or only a very small amount as otherwise I assume it would interrupt severely with our own radio communication here on Earth.


title: Is there an intuitive description of vacuum entanglement?

text: People often refer to the fact that the vacuum is an entangled state (It's even described as a **maximally** entangled state). I was trying to get a feeling for what that really means. The problem is that most descriptions of this are done in the formalism of AQFT, which I'm not very familiar with. The entanglement definitions which I have some feeling for are those of the form > System S Hilbert space $\mathcal{H}$ factorizes as > $\mathcal{H}=\mathcal{H}_A \otimes \mathcal{H}_B$ where A and B are two > subsystems of S. An entangled state can't be written in the form $\phi_A > \otimes \phi_B$ There are then various measures of this, such as entanglement entropy. So my question is - is it possible to describe the entanglement of the QFT vacuum in these more familiar terms? Can such a description be given for a simple QFT example, say a Klein Gordon field on Minkowski space?


title: What Physical Quantity has SI Unit $kg \cdot m$,?

text: What is the physical quantity that has SI (System International) unit of $kg \cdot m$ (kilogram meters)?


title: What are some mechanics examples with a globally non-generic symplecic structure?

text: In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be $\mathbb R^{2n}$, where maybe the $q^i$-coordinates are cut off to get a finite volume. In the books about Hamiltonian mechanics, especially mathematical books, one needs a symplectic space $(\mathcal{M},\omega)$ and of course the Hamiltonian. Now necessarily, locally $\omega$ looks like the canonical form $\Theta=\text dq^i\wedge\text dp_i$. > Are there some relevant classical mechanics problems where one can state a > less trivial $\omega$, and that _globally_? I would like to see a global expression which is different from $\Theta$ (and also not just $\Theta$ in different global coordinates). That would be a nontrivial form, which might maybe arise over a more topologically complicated space than $\mathbb R^{2n}$, maybe due to restrictions of a mechanical system. And maybe you get such a form after a phase space reduction, but I don't actually know any explicit mechanical problem you need it for.


title: Will a hole cut into a metal disk expand or shrink when the disc is heated?

text: Suppose you take a metal disc and cut a small, circular hole in the center. When you heat the whole thing, will the hole's diameter increase or decrease? and why?


title: Is there an abstract notion of heat within a microscopical system?

text: The microstates of a system are said to be unobservable. I can introduce the entropy as a measure of the number of microstates, which lead to the same macroscopic variables. So in this detailed definition of entropy, there are two theories at work, where one can be viewed as approximation to the other. If I introduce an energy function of the macroscopic system, then there is heat too. Dissipation in the big system can microscopically understood in the small system. * * * I formulated the above intro rather general, without reference to any particular system. I guess I can sort a quantum mechanics description underneath the classical one, I can sort quantum field theory description underneath certain quantum mechanical systems, and I can even semi-consider deeper theories in renormalization procedures. However, all the models I know which consider problems involving heat are usually fluids. > Can I, in a similar way as macroscopic fluid dynamics and statistical > mechanics is related, introduce dissipative systems in the microscopic > system and explain it in terms of even lower level systems? And if it's > possible to introduce two levels of dissipation in that way (three theories > at work), how are the first and the third related? To formulate an aspect of the second part more practically: If I have a thermodynamical entropy, then I don't know the absolute value. If I construct a microscopic description I can get such a value. If I now construct another microscopic description, then I'll get another value for the same macroscopic entropy, right? * * * I never encountered the microscopic-macroscopic entropy and heat viewpoint as such developed as a general meta theory or mathematical theory, independend of a certain physical viewpoint of how the real world is made of. > What is left if I strip off explicit statistical systems (namely these we > considered to be of physical relevance so far, especially particles and > fields descibed by this and that physical differential equation) of these > considerations?


title: Detecting dim objects

text: What are the minimum conditions (switching times, sensitivity ...) for a camera to detect dim objects, e.g. Oort cloud or Kuiper belt objects, passing by in front of a star in the background? I've read the Hubble did this. Is it possible also from ground? How do these conditions change, if the camera would be in space? Or do I need Hubble?


title: Are there certain alignments in planetary orbits that create interesting effects in the moons or planets?

text: We have some interesting examples such as Mercury's perihelion > The planet Mercury is especially susceptible to Jupiter's influence because > of a small celestial coincidence: Mercury's perihelion, the point where it > gets closest to the sun, moves around at a rate of about 1.5 degrees every > 1000 years, and Jupiter's perihelion moves around only a little slower. One > day, the two may fall into sync, at which time Jupiter's constant > gravitational tugs could accumulate and pull Mercury off course. This could > eject it from the Solar System altogether1 or send it on a collision course > with Venus or Earth.[10] And also the alignment of Jupiter and Saturn > Yes. When Jupiter and Saturn align, Jupiter's plasma shadow may disrupt > magnetospheric activity at Saturn. Jupiter's 1-billion-km plasma shadow > (magnetotail) is the largest structure within the Solar System. When Jupiter > and Saturn align, Jupiter's plasma shadow can envelop Saturn. Based on > Voyager 1 observations, it is plausible - but not proven - that the > reduction in solar wind pressure reduces magnetospheric activity around > Saturn. This is like drafting an upwind cyclist. > http://www.solarviews.com/eng/vg... But are there any other interesting effects that could be created by other planetary orbit alignments? Especially in exoplantary systems?


title: Is it possible to see satellites with the naked eye?

text: Every now and then I notice some very bright "stars" in the sky. They tend to be very few (one or two, usually), and are quite much brighter than any other star out there. Often they're perfectly visible and bright even in dusk, when no other star is visible yet or will be for some time to come. Now, I'm writing the word "star" in quotes because I really don't think that they're stars. But what are they? Planets? Satellites? Latest Chuck Norris encounters leaving the atmosphere? My best guess would be that they are satellites, because planets should be about the same brightness as stars, but that sounds pretty implausible as well... And yes, I'm very certain that they are NOT airplanes.


title: Why don't any of the gas giants have moons of Earth's mass (or greater) that orbit them? Is this generalizable to exoplanet gas giants?

text: Now, we know that the composition of the outer planets tend to contain a much higher ratio of ice over rock. [1] So the lack of Earth-like moons around the outer planets that this could be an artifact of the fact that in our solar system, all the gas giants are beyond the "ice line". But could other factors be involved too? If Jupiter was at the distance of the Earth (probably by migrating in), then is it unlikely for it to have a moon of Earth's mass or greater? [1] This is probably because the amount of rocky material falls off with distance from the Sun, but this is not true for icy material because icy material will volatilize if it's too close to the Sun.


title: Confusing description of Orions belt, could someone clarify?

text: The Wikipedia article about Orion's belt contains this confusing sentence: > They also mark the northern night sky when the sun is at its lowest point, > and were a clear marker for ancient timekeeping. Is this a reference to the winter solstice? In what sense?


title: How do we know the masses of single stars?

text: I have recently read that we can only know the masses of stars in binary systems, because we use Kepler's third law to indirectly measure the mass. However, it is not hard to find measurements for the mass of stars not in binary systems. So how is the mass of these stars determined?


title: Distribution of each element and molecule as a function of distance from the Sun and as a function of time

text: Are there any graphs that show the distribution of each element and molecule as a function of distance from the Sun? And maybe even the time-evolution of each distribution over the solar system's lifetime?


title: Force and Area are proportional to each other, but

text: I have read on book the next following sentence: > _The force is proportional to the area, for if we increase the area but keep > the number of molecules per cubic centimeter the same, we increase the > number of collisions with the piston in the same proportion as the area was > increased. You should know that is happening on a cylinder with a piston in > it._ my question is: if we increase the area. The force should become less. Why is that? Because we make the collisions between molecules with the piston or the wall of cans or cylinder less. we drive: More area = less force (less pressure) as I thought .. Is that what is really happening ?!! more area = more space = more free for molecules = less pressure = less force . I hope someone explains to me what is author wants to tell me if I misunderstood.


title: Unitrary groups and infinitesimal transformations - Schwingers way of deriving Lie groups

text: In Schwinger's source theory book, he suggests if $G_a$ are the hermitian generators of the Unitary group, then we have an infinitesimal transformation is given by : $$ G = \sum_{a=1}^n \delta\lambda_a G_a $$ Now if we subject to the infinitesimal transformation operator to arbitrary unitary transformation of the group ($\\{\lambda\\}$ are the group parameters) , $$ U^{-1}G_aU = u(\lambda)G_b $$ Alternatively, $$ UG_aU^{-1} = G_bv(\lambda) $$ And then, these two quantities are related by $ v = (u^T)^{-1} $ Firstly, I am not able to see why the author is trying to do this kind of unitary transform on the group element, Is it possibly to get a representation? How further do we proceed with this.


title: Does the discreteness of spacetime in canonical approaches imply good bye to STR?

text: In all the canonical approaches to the problem of quantum gravity, (eg. loop variable) spacetime is thought to have a discrete structure. One question immediately comes naively to an outsider of this approach is whether it picks a privileged frame of reference and thereby violating the key principle of the special relativity in ultra small scale. But if this violation is tolerated doesn't that imply some amount of viscosity within spacetime itself? Or am I writing complete nonsense here? Can anybody with some background in these approaches clear these issues?


title: Does a charged particle accelerating in a gravitational field radiate?

text: A charged particle undergoing an acceleration radiates photons. Let's consider a charge in a freely falling frame of reference. In such a frame, the local gravitational field is necessarily zero, and the particle does not accelerate or experience any force. Thus, this charge is free in such a frame. But, a free charge does not emit any photons. There seems to be a paradox. Does a freely falling charge in a gravitational field radiate?


Could a people do all sort of gymnastics movement in vacuum space? I asked this because I am worry about that the astronaut leave the space shuttle during emergency could not go back to earth by himself if there are no fuel on the astronaut, could he swim in space to get back on earth even if there are no water?


title: why people choose condense matter physics the most in 2007 and 2008 for their phd? And what are they studying and what is application in this subject?

text: Page 11 of this report from the American Institute of Physics shows that condensed matter physics is by far the most common PhD dissertation topic in 2007 and 2008. Can anyone here explain why this would be so? Additionally, what are they studying. and what is application with this subject? On a related note, why are atomic and molecular physics so uncommon as a PhD dissertation topic? Personally, I think that atomic and molecular physics should be more popular, since atomic physics is closely associated with chemistry, making it more broadly applicable. Is this the case?


title: How does electron know when to change into a wave?

text: It is known that electron behaves as a wave also.   How does electron know that it has to change into a wave?   Are there any factors that influence the behavior of electron changing into wave?


title: Where does the CSHS inequality really come from

text: I'm reading about Bell's inequalities and hidden variables theory. And I wonder where does the CSHS inequality really come from ? I read the paper and the inequality we find is $$\left|P(a,b)-P(a,b')\right|+P(a',b)+P(a',b')\leq 2 \tag{1}$$ while the most common form I found elsewhere is $$P(a,b)-P(a,b')+P(a',b)+P(a',b')\leq 2 \tag{2}$$ Where does the absolute value disappear ? I don't understand how (2) could be contained in (1)


title: Infinite Atwood's machine

text: I tried doing the following problem just for fun (please see the following link): ![HaklssckcwuhbKlABDSKBTHBhbhOkookolxockcnn](http://i.stack.imgur.com/JTa4E.jpg) (I'm not allowed to post the entire problem here since I'm a new user.) Here are my equations:   $$T - mg = ma$$ (equation 1)   $$Mg - T = Ma$$ (equation 2)   (Upper case "M" used for infinite series of masses on the right arm of the first pulley. Lower case "m" for the small, single mass on the left arm.) algebra is as follows:   $$Mg - (ma + mg) = Ma$$   $$g (M - m) = a (M + m)$$   $$a = g (M - m) / (M + m)$$   Here is my reasoning to get final answer: The limit of $a$ for large $M$ goes to $g$. Therefore, acceleration of the top mass is $g$. However, the correct answer is $g/2$. Why does my reasoning get the wrong answer? Any ideas? (Please excuse the formatting, I'm a newbie here).


title: Difference between theoretical physics and mathematical physics?

text: I'm a huge _fan_ of mathematical physics and I know what the _formal_ definitions of those two areas are, I've seen them. But I still get completely baffled when someone asks me to explain it simply. The difference is obvious to me, but I just can't seem to put it into words in a satisfying enough manner. So I'm asking you to help me... **What is the essential difference betweentheoretical physics and mathematical physics?** or if you prefer the rephrased version... **What was the motivation for introducing the name "mathematical physics" as a separate entity?** (It doesn't matter if you're one of those people who don't like labels, the thing is that we _do have_ these two separated very often in academia.)


title: Why does a photon colliding with an atomic nucleus cause pair production?

text: I understand that the photon needs to have enough energy to produce a lepton and it's antimatter partner, and that all of the properties are conserved, but why does the photon do this in the first place? What's going on "behind the scenes" to transform a single, neutral particle (the photon) into two charged particles (i.e. electrons)? Why is the nucleus necessary?


title: Spontaneous symmetry breaking in the quantum 1D XX model?

text: The ground states of the quantum 1D Ising and Heisenberg models exhibit spontaneous magnetization. Is this also true for the 1D XX model?


title: How does rust behave as a PN junction?

text: Reading about old-fangled radios the reference quoted below indicates it was/is possible to use a rusted razor (or perhaps any similar thin corroded metal strip) as a detector. **Some resourceful GIs found that a crude crystal set could be made from a coil made of salvaged wire, a rusty razor blade and a pencil lead for a diode. By lightly touching the pencil lead to spots of blue on the blade, or to spots of rust, they formed what is called a point contact diode and the rectified signal could be heard on headphones or crystal ear pieces.** How does a rusted razor perform the same task as a PN junction? Is it merely that rust (being minute oxidized metallic particles) migrates under the influence of the electrical field?


title: Defy gravity torques with gyroscopes?

text: **Context** On the following drawing, a platform is hung from the ceiling not exactly from its centre of gravity. Because of this it can't sustain an arbitrary orientation for long; I want to increase its stiffness with the use of 3 orthogonal fast spinning flywheels, but before building those flywheels I need to know how big they need to be by making a dynamic model... ![enter image description here](http://i.stack.imgur.com/eab5e.jpg) ... And I have been unable to model the gyroscopic effect [Edit: end removed for clarity]. **Question** How can I model the 3 flywheels reaction to (themselves and) the gravity torque, be it through a rate increment or a torque (to inject in the plant model)? * * * **Implementing Lelesquiz's answer** ## Top level equations: Euler-Lagrange The goal is to solve for each timestep for the angles $(\phi, \theta, \psi)$ using the equations, in inertial frame: $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right)-\frac{\partial L}{\partial \phi}= (\boldsymbol{M_g}-\sum_{k=1}^3 \boldsymbol{M_{wk}})\cdot \frac{\partial \boldsymbol{\omega_b}}{\partial \dot{\phi}}$$ $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right)-\frac{\partial L}{\partial \theta}= (\boldsymbol{M_g}-\sum_{k=1}^3 \boldsymbol{M_{wk}})\cdot \frac{\partial \boldsymbol{\omega_b}}{\partial \dot{\theta}}$$ $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\psi}}\right)-\frac{\partial L}{\partial \psi}= (\boldsymbol{M_g}-\sum_{k=1}^3 \boldsymbol{M_{wk}})\cdot \frac{\partial \boldsymbol{\omega_b}}{\partial \dot{\psi}}$$ Where $M_g$ is the gravity torque, $M_{wk}$ is the motor torque applied on the flywheel k (opposite of what's applied on the platform), and $\omega_b$ is the platform rate vector in inertial frame. 3 other equations based on $\alpha_k$, the angles of the flywheels, should have been added to completely characterize the system but the rates of the wheels are inputs, measured directly. The torques on the wheels are inputs as well (and the command of the closed loop system). Knowing that the lagrangian $L=T-U$ in the inertial frame. ## Kinetic energy T $$ T(\phi, \theta, \psi, \dot{\phi}, \dot{\theta}, \dot{\psi}) = \frac{1}{2} \sum_{k=1}^3 \left(\boldsymbol{N_{k,Ri}^T}\boldsymbol{[I_{wO,Ri}]}\boldsymbol{N_{k,Ri}} \right) + \frac{1}{2} \left( m_{b} \boldsymbol{v_{b,Ri}}^T\boldsymbol{v_{b,Ri}}+\boldsymbol{\omega_{b,Ri}^T} \boldsymbol{[I_{bG,Ri}]} \boldsymbol{\omega_{b,Ri}} \right)$$ Where $m_b$ is the mass of the system (with wheels on), $\boldsymbol{v_{b}}$ the velocity vector of the center of gravity of the system, $\boldsymbol{\omega_{b}}$ the system's rate vector (wheels not spinning) and $\boldsymbol{[I_{bG}]}$ the inertia matrix of the system about its center of gravity (which needs to be transported from the guimbal centre 0), $[I_{wO}]$ the inertia matrix of the flywheels about their axis, and $\boldsymbol{N_{k}}$ the rate vector of the flywheels in the inertial frame. $$\begin{array}{lc}\boldsymbol{[I_{wO,Ri}]}=\boldsymbol{[R_{ib}]}\left[\begin{matrix}J_w & 0 & 0\\\ 0 & J_w &0\\\0 & 0& J_w\end{matrix}\right] \boldsymbol{[R_{bi}]} \end{array}$$ $$\boldsymbol{v_{b}}=\boldsymbol{\omega_{b,Ri}}\times\boldsymbol{\Delta_{b,Ri}}$$ $$\begin{array}{lc} \boldsymbol{[I_{bG,Ri}]}= \boldsymbol{[R_{ib}]}\boldsymbol{[I_{bO,Rb}]}\boldsymbol{[R_{bi}]}+m_b& \left(\begin{matrix} \Delta_{by,Ri}^2+\Delta_{bz,Ri}^2 & -\Delta_{bx,Ri}\Delta_{by,Ri} & -\Delta_{bx,Ri}\Delta_{bz,Ri} \\\ -\Delta_{bx,Ri}\Delta_{by,Ri} & \Delta_{bx,Ri}^2+\Delta_{bz,Ri}^2 & -\Delta_{by,Ri}\Delta_{bz,Ri} \\\ -\Delta_{bx,Ri}\Delta_{bz,Ri} & -\Delta_{by,Ri}\Delta_{bz,Ri} & \Delta_{bx,Ri}^2+\Delta_{by,Ri}^2 \end{matrix}\right) \end{array}$$ $\boldsymbol{\Delta_{b,Ri}}$ being the guimbal centre - center of gravity vector of the system expressed in the inertial frame. See below "Euler angle expressions" for the transformation body-inertial $\boldsymbol{<vector>_i}=\boldsymbol{[R_{ib}]}\boldsymbol{<vector>_b}$. ## Potential energy U $$U(\phi, \theta, \psi) = -m\boldsymbol{g_{Ri}}.\boldsymbol{\Delta_{b,Ri}}$$ The minus sign comes from the fact U is minimum with the guimbal-CG vector pointing downward (dot product positive). ## Euler angles expressions The Euler angles 313 have been selected as a coordinate system, which gives the rotation matrix from inertial frame $(0 \boldsymbol{X_i Y_i Z_i})$ to body frame $(0 \boldsymbol{X_b Y_b Z_b})$: $$ \begin{array}{lc} \boldsymbol{R_{RbRi}}(\phi, \theta, \psi)= & \left[\begin{matrix} cos \psi & sin \psi & 0 \\\ -sin \psi & cos \psi & 0 \\\ 0 & 0 & 1 \end{matrix}\right] \left[\begin{matrix} 1 & 0 & 0 \\\ 0 & cos \theta & sin \theta \\\ 0 & -sin \theta & cos \theta \end{matrix}\right] \left[\begin{matrix} cos \phi & sin \phi & 0 \\\ -sin \phi & cos \phi & 0 \\\ 0 & 0 & 1 \end{matrix}\right] \end{array}$$ The rate vector is obtained by summing the projections on each axis of the inertial frame the $\boldsymbol{\dot{\phi},\dot{\theta},\dot{\psi}}$ vectors): $$\begin{array}{lc} \boldsymbol{w_{b,Ri}}= & \left[\begin{matrix} \dot{\theta}cos\phi+\dot{\psi} sin\theta sin\phi \\\ \dot{\theta}sin\phi+\dot{\psi} sin\theta cos\phi \\\ \dot{\phi}+\dot{\psi}cos\theta \end{matrix}\right]\end{array}$$ ## Problem The kinetic energy of the wheels is independent on the Euler angles, so the rates of the wheels have no effect, which is wrong! Since it's absolutely normal that no matter the rotation matrix the norm of the wheels rates vector does not change, the kinetic energy expression or even the top level equations must be wrong.


title: Five-component field

text: Recently I was reading about 5-component field $(\varphi , \psi_{\mu})$, for which $$ \hat {p}^{\mu} \varphi = mc\psi^{\mu}, \quad \hat {p}_{\mu}\psi^{\mu} = mc\varphi . $$ This field refers to the spin-zero representation of the Lorentz group and obeys the Klein-Gordon massive equation (so it reproduces Poincare group irreducible representations). What physical system is described by it?


title: Is every quantum measurement reducible to measurements of position and time?

text: I am currently studying Path Integrals and was unable to resolve the following problem. In the famous book _Quantum Mechanics and Path Integrals,_ written by Feynman and Hibbs, it says (at the beginning of Chapter 5 _Measurements an Operators,_ on page 96): > _So far we have described quantum-mechanical systems as if we intended to > measure only the coordinates of position and time. Indeed, all measurements > of quantum mechanical systems could be made to reduce eventually to position > and time measurements (e.g., the position of a needle on a meter or time of > flight of a particle). Because of this possibility a theory formulated in > terms of position measurements is complete enough to describe all > phenomena._ To me this seems to be a highly non trivial statement (is it even true?) and I was unable to find any satisfying elaboration on this in the literature. I would be thankful for any answer to resolve this question and any reference to the literature!


title: What does non magnetic and nonconducting mean in reflection and transmission of waves?

text: So, we were ask to consider the Fresnel Equations for parallel and perpendicular waves (with index of refractions). Then, we are ask to prove some equations in which "... for nonmagnetic non- conductors" The Fresnel equations got reduced and the indexes of refraction disappeared. I do not really know where to start, but in our reference book we have: normal incidence and oblique incidence topics. Any help?


title: What is the meaning of electron's magnetic effect cancelling each other?

text: I'm trying to understand, why magnets attract certain metals but not objects made out of paper, plastic etc. And the answer I got is "in paper, electrons cancel each other". What does it mean?


title: How to interpret $t^2$?

text: I can't think of the meaning of squaring the Time (multiplying it by itself). It makes sense in Mathematics. But how can you figure it out in nature (or physics). As an example, the formula $$s=u*t+(1/2)*a*t^2.$$


title: Intuition behind the notion of reflection positivity

text: I came across Yuji's question. I'm finding it difficult to parse the meaning behind what's said on Wikipedia. Could someone give an explanation of the concept involved? I would also appreciate references.


title: Why isn't data lost when sent over large distances?

text: I was thinking about how information is sent, for example through the atmosphere. There are plenty of obstacles, as well diffraction, etc. Still, no information is lost. How is information sent to allow this?


title: What is the constant $K_1$ in these orbital equations?

text: I want to compute the derivatives of argument of periapsis and longitude of the ascending node of the orbit of a GPS satellite from the following formula. $$\frac{d\Omega}{dt} = -K \cos{i} \\\ \frac{d\omega}{dt} = K ( 2 - 2.5 \sin^2{i}) \\\ K = \frac{nK_1}{a^2(1-e^2)^2}$$ But what is $K_1$?


title: Explanation for the physics behind magnetic monopoles?

text: Ok so I am looking for a simple explanation of how the process is done from the information I could access and the knowledge I was allowed I have the idea that it is a matter of quantum physics and all i have understood is:   1. They freeze two magnets and split them not allowing the inner particle constellation to move freely and so it is stuck it its state as a monopole.   2. At room temperature it will go back to normal state (and doesn't this contradict with Curie I mean the Curie temperature isn't even close to room temperature and they say that magnetic materials maintain their magnetic properties before they reach that temperature).   3. When they touch they lose that state and they become regular magnets. so how does this work being explained for a high school student in simplistic means i have seen some equations about it but they contain some theoretical physics equations and PDEs some even started explaining the theory of everything and so on,and though i love physics i plan not to self-learn PDEs and theoretical physics. I would really appreciate all the help I can get even if you give me a book that i can read to understand it.


title: Universe Expansion and two tennis balls

text: Clear the universe of all matter except for two tennis balls. Place the two tennis balls in the same inertial frame 1 Mpc apart.   * Are the tennis balls getting further apart?   * Will the tennis balls remain in the same inertial frame? EDIT: Don't assume the balls are massless, but please ignore the gravitational attraction two tennis balls separated at 1 Mpc would exhibit on one another. (Previously, I had asked to assume the tennis balls were massless. I couldn't switch to a strike out font, so I just removed it entirely.)


title: "Cloud maps" of a nearby brown dwarf: what do they actually show?

text: This paper recently made headlines: > _A global cloud map of the nearest known brown dwarf._ I. J. M. Crossfield > et al. _Nature_ **505** , 654–656 (30 January 2014). In short, they claim they've been able to observe, using light-curves and Doppler shift measurements, the surface features on a brown dwarf, and they've produced some pretty impressive maps of the brown dwarf surface: ![enter image description here](http://i.stack.imgur.com/MpY2N.jpg) There's also a fair bit of information in the Max Planck Institute of Astronomy press release and in this Sky and Telescope article. I can't quite make out, though, what those images actually are. Are those visible, real-colour images? Or are they some sort of temperature map? Or simply brightness maps? If so I can't seem to find the scale, and I'm having trouble figuring out whether the variations have been amplified with respect to the mean (as is the usual practice in, say, cosmic microwave background maps). Does anyone know exactly what these maps show?


title: If light is linearly polarized, does it have some spatial extent?

text: If light (a photon) is linearly polarized, say vertically, does it have some vertical _spatial_ extent (perhaps in amplitude)?


title: How is celestial navigation done on a low-level?

text: When we send a probe off to Jupiter or Saturn, or even Earth orbit, how are the rocket firings timed and coordinated? For instance, when I want to drive to another city I pull onto the highway and use the lines on the road to adjust my steering. I also use intuition to adjust my velocity, even though that won't make a difference _where_ I wind up (only _when_ ). However, spacecraft don't have lines on the road to line up with. Nor do they have a direct measurement to take for speed, which is critical in orbits and in planetary transfers. I suppose that some bright stars near opposition or off the ecliptic (Capella?) and GPS could be used for interplanetary navigation or LEO, but how was it done before GPS? Inertial navigation? How did the Soviets get the Vostok to enter a clean orbit (I understand that the Sputnik orbit was very elliptical)? How did early interplanetary probes orient themselves?


title: Does our local time speed up as the Universe expands?

text: Starting from a simplified radial Freidman Walker metric we have $$ds^2 = -c^2 dt^2 + a(t)^2 dr^2 $$ How does one measure one's proper time operationally? One times a light beam along an element of proper distance $ds$. Thus an element of proper time $d\tau$ is given by $$d\tau = \frac{ds}{c}$$ From the metric above we have an element of proper distance $ds$ (where $dt=0$) given by $$ds = a(t) dr$$ Thus $$d\tau = \frac{a(t)dr}{c}\ \ \ \ \ \ \ \ \ \ \ \ (1)$$ Now light travels on a null geodesic where $ds=0$ so from the above metric we also have $$\frac{a(t)dr}{dt} = c\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ Substituting Equation (2) into Equation (1) we find $$d\tau = dt$$ Thus an interval of our proper time $d\tau$ is the same as an interval of cosmological time $dt$ provided we measure the time light takes to travel an **expanding** distance $ds$. But our clocks don't expand with the Universe. They are rigid like we are. Thus if we measure our proper time interval $d\tau$ we would use $$d\tau = \frac{dr}{c}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$$ where $dr$ is a distance interval that does not expand with the Universe. If we now substitute the light-path expression (2) into Equation (3) we find $$d\tau = \frac{dt}{a(t)}$$ This seems to imply our local proper time will speed up as the Universe expands. Is this right?? :)


title: Proof of minimum distance between object and image of a convex lens image being 4*focal length

text: I can prove that the minimum distance between an object and its image, through a convex lens is 4*focal length, if I assume that the distance between the object and the lens is the same as the distance between the image and the lens, i.e. if: $$1/u + 1/v = 1/f$$ that $u=v$, but is there a way of showing that the distance $u+v$ is minimised at $u=v$, because I can't work one out.


title: How can fiber optic image conduits consist of only fiber core, no cladding?

text: Looking at fiber optic image conduits, I found that they have a packing fraction (the ratio of the fiber core's area to the total area) of 100% (source here). In other words, they consist only of fiber core, no cladding. But without cladding, wouldn't the light travel between the individual fibers, messing up the output image? Or is the cladding so extremely thin that this is simply a result of rounding up to 100%?


title: What is the definition of physically meaningful?

text: I asked a question recently where I wanted to know whether it was physically meaningful to talk about the arrow of time in other universes. Although many people apparently have an intuitive notion of what is meant by "physically meaningful", the only real context I have found is in terms of "physically meaningful observables" where in a quantum mechanical context we talk about the possible eigenvalues associated with eigenstates of an operator: $H |\psi\rangle = \lambda|\psi\rangle$ The observables being possible outcomes of experiments or other operations on the state vector. So in the context of the arrow of time, where there are claims of the possibility of a vector time, one would like to think of some possible operation that would produce set of possible values for that time. So the first question one might ask is, "what operation can I perform to observe the vector values of the arrow of time?" It would seem that there is some difference between defining such an operation and the possibility of it actually being executable. So one might think of a thing being physically meaningful if one can define a set of executable operations (executabality being determined by some set of constraints). However, I suspect there is a better definition, or a better way to conceptualize what is meant by "physically meaningful", and I would like to read what those are.


title: can a system with nested building blocks (atoms,cells) NOT be "fine tuned"?

text: Never mind whether the universe is "fine tuned" for anything in particular,just the idea that there is a nested hierarchy seems incredibly constraining on the outcome, and anything but accidental.