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<p>I am assuming that zero (rest) mass particles don't interact gravitationally with each other and other particles. Does that mean they experience a "flat" spacetime instead of a curved one? I find it a little hard to imagine that. Also, does this mean massless particles can pass through an event horizon (not the singularity itself, I assume there would be some other way of interaction that would prohibit that).</p>
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<p>I am in an introductory physics class and we have been working with Gravitational Potential Energy, Elastic Potential Energy, Kinetic Energy and the Law of Conservation of Energy in relation to bungee cords. Recently, we delved into the topic of efficiency and with it I came upon the following stumbling block. </p>
<p>For a project, we need to design a bungee cord for an egg (Average mass: 57.3g +/- 3.0g) falling from a height of 10.0m, and try to get the egg's stopping point as close as possible to the ground. We also need to take efficiency into account.</p>
<p>Now I completely understand the basics of the calculation. Pick a desired stretch, use that to determine cord length, use that to determine the spring constant of the cord, etc. However, when you bring efficiency into account things get more complicated. Energy definitely escapes the system as the egg falls and the cord is stretched. Energy escapes in the cord as a result of internal friction, and during the fall as a result of air resistance. I think air resistance should be negligible given the mass of the egg and the shortness of the drop. I have scoured the internet for a model to determine the energy loss of the system due to internal friction to no avail. <strong>So...My question is, how can I quantify and predict the internal friction of the bungee cord, then use that to determine the % efficiency of my system?</strong></p>
<p>Sorry if my question's a little long and/or misplaced and/or I'm making some kind of mistake in my thinking here. This is my first post on the site.</p>
<p><strong>Edit:</strong> Here is all the given information for this problem:</p>
<ul>
<li>Average mass of the group of eggs being used: 57.3g</li>
<li>Standard Deviation: 3.0g</li>
<li>Height: 10.0m</li>
<li>Desired height at the egg's lowest point: 0.10m</li>
</ul>
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<p>I'm an engineering student and have a few ideas to bat at the scientific community. First premise is based upon common sense. I understand that if I am standing on a skateboard on a relatively smooth surface and a stone of significant mass is thrown to me and I catch it, the momentum of the projectile is transferred to me causing a forward acceleration. </p>
<p>My question: Suppose that a system is designed to provided constant acceleration in space utilizing the aforementioned intuitive concept. In the simplest form I can idealize such a system as a basketball constantly striking a flat surface. Opposed to a ball consider an arbitrary mass striking a greater mass. I suppose that if creative means are applied to control the striking action (e.g. magnetism, mechanical means) then this action can be repeated to provide constant acceleration without the use of carrying fuel loads. Would it not be possible to reset the system to an initial state without canceling the resulting acceleration? </p>
<p>Due to the length of this question I will post the next question later. Thx</p>
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<p>In a question titled <a href="http://physics.stackexchange.com/questions/2041/how-are-classical-optics-phenomena-explained-in-qed-snells-law">How are classical optics phenomena explained in QED (Snell's law)?</a> Marek talked about the probability amplitude for photons of a given path. He said that it was $\exp(iKL)$, and that "...this is very simplified picture but I don't want to get too technical so..." </p>
<p>I want to know how it arises, even if it is technical. I find it very strange. If we compare it to the case of a particle obeying the Schrodinger equation, we have $\exp(iS/h)$ where $S$ is the action of a given path. $S$ is what we want to minimize(in the classical limit). In this case the path is a space-time path. </p>
<p>But in the other case, of the photon, where $L$ is that we have to minimize(in the classical limit or if you want in the geometrical optics limit) the paths are only in space, and I can't find any temporal dependence. </p>
<p>when I check any book about QED, I can read about the photon propagator (about space-time paths) but I never have found out about the expression $\exp(iKL)$.</p>
<p>In general terms I find hard to relate what Feynman teaches in his book with what I have read in the "formal QED books" like Sokolov, Landau, Feynman or Greiner.</p>
| 7,782 |
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<p>I'm pretty certain that the answer to the question in the title is a no, but I don't understand why. I have some basic misunderstanding of quantum processes that I’d like clarified in the form of asking questions about a hypothetical scenario. </p>
<p>So, let’s say we have a pair of entangled photons…</p>
<p>1) Is it possible to prepare an entangled pair such that it can be known that, for example, the photon going down a certain path will be measured 25% of the time to be horizontally polarized and 75% likely to be vertically polarized? I guess this would be some sort of quantum decoherence? Partial collapse?</p>
<p>2) If that is possible, then would it also be possible to measure that photon’s diagonal polarization in order to "reset" the horizontal-vertical probability back to 50-50?
I've seen a trick done where you take some polarizing filters and stack them horizontal on top of vertical, so that no light gets through. But when you slide a diagonally oriented filter between them, some light does get through, because the horizontal/vertical polarization information gets "reset" when measured against the diagonal filter.</p>
<p>3) If these two things are possible, then wouldn’t it be possible to send information along the entangled pair? If you sent bursts of 1000 photons and were able to measure them all, and found that the first burst had ~750 vertically polarized photons but the second burst had ~500 vertically polarized photons, then wouldn’t you know that the second burst had their diagonal measurements taken? Or would that not work because the extra measurements "break" the entanglement..? </p>
<p>4) Along with that, how many times can the photon pairs be measured before there are no longer entangled? Just once? Or maybe just once per state/qubit/aspect (such as polarization angle)? I understand that measuring the photon would necessarily change it (like with the polarization filter trick), but then how do scientists measure the polarization angles of two photons like in experiments of Bell’s inequality?</p>
<p>I know that somewhere along the line, what I've described cannot be possible. If it were, then you could easily send information faster than light. E.g., say you had a photon-entanglement-generator-and-probability-fixer station in between two planets, 1 light year from the planet Foo and .99 light years from the planet Bar. After a year's time, after the photon stream has propagated to both planets, scientists on Bar could measure diagonally polarization of 1000 photons to encode a 1 or not measure to encode a 0. Then scientists on the planet Foo could measure vertically polarity and decode the 1s and 0s based on the statistical readings. The Bar scientists could encode that such a message and the Foo scientists would "instantly" receive the message. Except it would actually be from the past, since Foo is 1.99 years outside the Bar light cone. So since this is impossible, what am I misunderstanding?</p>
<p>Thanks for answering, and thanks for putting up with yet another clueless but curious layman.</p>
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<p>I was digging into <a href="http://latticeqcd.blogspot.com/2005/12/nielsen-ninomiya-theorem.html" rel="nofollow">Nielson-Ninomiya Theorem</a> and doubler fermions, as well as solutions to these problems using <a href="http://arxiv.org/abs/hep-lat/9410018" rel="nofollow">Domain Wall Fermions</a> and <a href="http://arxiv.org/abs/hep-lat/0405024" rel="nofollow">overlap lattice fermions</a>, both of which make effective use of a 5th dimension.</p>
<p>Does this imply that our universe is either a continuous 3+1 dimensional space, or a space that is 4+1 dimensions on a lattice?</p>
| 7,784 |
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<p>In organic light emitting diodes, only singlet excitons contribute to the light emission (this is flouresence). Everything I read, say that triplet excitons cannot recombine on the same timescale as the singlets because spin selection rules forbid it. </p>
<p>Can anyone expand on how the spin selection rules allow for singlet and not triplet emission? </p>
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<blockquote>
<p>I have given one-dimensional motion of the particle directed
horizontally. A problem says: "<em>...Show that for this given motion
Conservation of Energy Law holds.</em>".</p>
</blockquote>
<p>Since Energy can intuitively be thought of an operation that changes something over time, energy being conserved essentially means that the motion is not a function of time, but position only?</p>
<p>I.e. does energy conservation mean that a process is time-independent?</p>
| 7,786 |
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<p>May be this question might have already been asked but i couldn't find it, so let me know if its already there.</p>
<p>Consider a potential, $V(x) = -\frac{1}{|x|}$ and if we apply this to a one dimensional Schrodinger's equation, I'd like to know the solution for the wave function in 1-D. Does it have a simple analytical solution? Does it have any oscillatory behavior like $$\psi(x,t) = P(x) e^{ikx}e^{i\omega t}$$ I mean will there be a factor like $e^{ikx}$ ? From the internet search, looking at one-dimensional hydrogen atom, first of all i am not sure whether there is any analytical solution, but I guess it was suggested that an exponential decay, something like $$P(x) = e^{-\alpha x}$$ is present. But I am not sure about presence of oscillations like $e^{ikx}$. Hence I'd appreciate some suggestions and clarification.</p>
<p>PS : I am not interested in Hydrogen atom, but in this specific 1-D potential.</p>
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<p>I am currently designing a heat sink for OPA 549. Among the many considerations, for example, the choice of material, the geometry of heat sink, the design of fins and all the parameters that come into play during the modelling, I am most curious about the simple choice of material.
Copper has more conductivity than Aluminium while the heat capacity of Aluminium is more than twice of copper. So naturally there has to be a trade off between the two. And I have no constraint on weight. I cannot use the combination of the two as there is bonding problem. My question is simple- Which one to use and why?
Relevant details: Max Power 90W; Forced Convection with 21 cfm; Overall thermal resistance is around 0.5 SI units.
thanks in advance</p>
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<p>I am just reading at the beginnin of <a href="http://backreaction.blogspot.de/2013/06/passing-through-cosmic-walls.html" rel="nofollow">this</a> nice article, that axions could be responsible for spontaneously breaking of a symmetry in the early universe.</p>
<p>Does anybody know which symmetry is alluded to and how the mechanism of spontaneous symmetry breaking exactly works in this particular case?</p>
<p>I'd like to learn in some more detail about this.</p>
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<p>So far I have only actually calculated dimensional regularization and I just know about the idea of cutoff regularization. From what I understand, as the name suggests, you just ignore momentum as from some high value and integrate the virtual momenta until this arbitrary chosen value. Depending on which value you chose, and thereby to what order you consider the Feynman diagrams, you get a (little bit) different result. </p>
<p>Now what I was wondering is, why don't people just cutoff exactly at the value that corresponds to the momentum of the process you want to consider, if there is some specific experiment you want to calculate. E.g. if someone wants to calculate some cross section of an experiment that will be/has been done at CERN, why don't they just consider exactly the momentum that the virtual particle must (maximally) have (due to momentum conservation of the particles that are smashed together)? Intuitively it seems (to me) that this must give the best result. And let us ignore of course, that physicists that calculate cross sections for CERN probably do not use cutoff regularization.</p>
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<p>I was attending my philosophy class and in the middle of student presentations, I found myself mentally wondering off and thinking about light. After a few minutes of trying to piece together how the sun works I came to ask a question I could not answer myself. Why does each individual photon have such a low amount of energy? I am hit by photons all day and I find it amazing that I am not vaporized. Am I simply too physically big for the photons to harm me much, or perhaps the Earth's magnetic field filters out enough harmful causes such as gammy rays?</p>
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<p>I would like some <strong>super simple computational code for solving magnetohydrodynamics problems</strong>. High accuracy nor performance is not my concern. I wan't it just to visually explore qualitative behavior of plasma. Important for me is:</p>
<ul>
<li><strong>Generality</strong> - it should not be specialised on any particular subproblem
of MHD </li>
<li><strong>Easy to get it work</strong> - does not require any libraries difficult to instal. Input is strightforward and well documented by some examples</li>
<li><strong>Easy to read and modify</strong> - you can read the code and learn about MHD solution from it. You can use this code as starting point for your own MHD code development </li>
</ul>
<p>Something as simple as this <a href="http://www.openprocessing.org/sketch/10419" rel="nofollow">Navier stokes solver</a> would be best</p>
<p>best would be something in python-numpy or java, but C++ or Fortran is not a big problem for me.</p>
<p><strong>Up to now I found:</strong></p>
<p>some libraries ( <a href="http://certik.github.io/mhd-hermes/index.html" rel="nofollow">MHD-Hermes</a> and <a href="https://code.google.com/p/python-mhd/" rel="nofollow">python-mhd</a> ) which are in python, but doesn't seem to have any documantation or examples how to use it.</p>
<p>and this
<a href="http://www.cita.utoronto.ca/~pen/MHD/" rel="nofollow"><em>A Free, Fast, Simple and Efficient TVD MHD Code</em></a> which is in frotran but has just 400 lines</p>
<p>I was thinking about writing something by myself but I wan't to be sure there is nothing similar already avaible. </p>
| 7,792 |
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<p>I know that we can make water solid with high pressure, so I think water will be solid in the deep ocean?</p>
<p>If that is true, the depth of the ocean would be limited because water will become ice? Anyone know that maximum depth? </p>
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<p>In a gravitational field, the gravitational force acting on a body (of mass m) at a point x metres away from the attracting body (of mass M) is $\frac{GMm}{x^2}$. Integrating this force from a point at infinity to x gives $-\frac{GMm}{x}$, or the work done by the gravitational field on the mass in moving the mass from a point at infinity to x. Then, the gravitational potential at point x is $-\frac{GM}{x}$.</p>
<p>However, the definition that my school gives is : "Gravitational potential at a point in a gravitational field is the amount of work done by an <strong>external agent</strong> in moving one unit of mass from a point at infinity to the point in the gravitational field, without any acceleration". The lecturers explained that as a body moves closer to the attracting body, it accelerates and gains kinetic energy. In order to maintain a constant velocity, negative work must be done by an external agent to remove this extra kinetic energy.</p>
<p>Now I am very confused, because both seem valid to me. What does "gravitational potential" actually mean?</p>
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] |
<p>Would a projectile fired in space and assuming there were no stars or planets in the way, would the projectile continue on to infinity, since the x-component of the projectile's velocity never changes from it's original? </p>
| 7,795 |
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<p>One page 5 in Landau & Lifshitz <em>Fluid Mechanics</em> (2nd edition), the authors pose the following problem:</p>
<blockquote>
<p>Write down the equations for one-dimensional motion of an ideal fluid in terms > of the variables $a$, $t$, where $a$ (called a <em>Lagrangian variable</em>) is the
$x$ coordinate of a fluid particle at some instant $t=t_0$.</p>
</blockquote>
<p>The authors then go on to give their solutions and assumptions. Here are the important parts:</p>
<blockquote>
<p>The coordinate $x$ of a fluid particle at an instant $t$ is regarded as a
function of $t$ and its coordinate $a$ at the initial instant: $x=x(a,t)$.</p>
</blockquote>
<p>For the condition of mass conversation the authors arrive at (where $\rho_0 = \rho(a)$ the given initial density distribution):</p>
<blockquote>
<p>$$ \rho\,\mathrm{d}x = \rho_0 \mathrm{d}a $$</p>
</blockquote>
<p>or alternatively:</p>
<blockquote>
<p>$$ \rho\left(\frac{\partial x}{\partial a}\right)_t = \rho_0 $$</p>
</blockquote>
<p>Now the authors go on to write out Euler's equation, where I start to miss something. With the velocity of the fluid particle $v=\left(\frac{\partial x}{\partial t}\right)_a$ and $\left(\frac{\partial v}{\partial t}\right)_a$ the rate of change of the velocity of the particle during its motion, they write:</p>
<blockquote>
<p>$$ \left(\frac{\partial v}{\partial t}\right)_a = -\frac{1}{\rho_0} \left(\frac{\partial p}{\partial a}\right)_t
$$</p>
</blockquote>
<p><strong>How are the authors arriving at that equation?</strong></p>
<p>In particular, when looking at Euler's equation:
$$
\frac{\partial\mathbb{v}}{\partial t} + \left( \mathbf{v} \cdot \textbf{grad} \right) \mathbf{v} = - \frac{1}{\rho} \textbf{grad}\, p
$$
what happens with the second term on the LHS $\left( \mathbf{v} \cdot \textbf{grad} \right) \mathbf{v}$? Why does it not appear in the authors' solution?</p>
| 7,796 |
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<p>Got two accelerometers that are measuring accelerations oriented at +/- 45 deg from horizontal. I am looking to determine the conversion of the accelerometer measured data to accelerations in the vertical and horizontal direction. Can I use coordinate system conversion below ?</p>
<p>$$F_{Vertical} = (F_1 + F_2) \cos(\frac{\pi}{4})$$</p>
<p>$$F_{Horizontal} = (F_1 - F_2) \cos (\frac{\pi}{4})$$
<img src="http://i.stack.imgur.com/KKydJ.jpg" alt="enter image description here"></p>
<p>I don't know phase relationship, but I can assume they are out of phase since I know the vertical acceleration is high and the horizontal acceleration is small.</p>
<p>Are the conversions above correct ?</p>
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<p><a href="http://en.wikipedia.org/wiki/Work_%28physics%29">Work</a> is defined as</p>
<p>$$W = \vec{F}\cdot\vec{s}$$</p>
<p>But what what exactly is $\vec{s}$? Is it the displacement of the body on which the force is being applied? Or is it the displacement of the point of application of the force? </p>
<p>If you look at the derivation for the definition of work from kinetic energy (the way many textbooks do it), it would seem to be the former, but I'm not sure since I can think of many examples that contradict the fact.</p>
<p>Or is it something else entirely? </p>
| 7,798 |
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<p>Consider a glass of water and a glass of coffee. Their contents differ by no more than a few grams of particles coming from the roasted and ground coffee, yet the former lets almost all visible light pass through, and the latter blocks most of it.</p>
<p>I am wondering <strong>what are the most efficient ways to turn water opaque by pouring matter in it</strong>, under normal temperature and pressure. By efficient I mean minimising the mass of added matter. By opaque, let's say 99% of daylight is blocked by two inches of liquid.</p>
<p>There is no other reason for my asking than sheer curiosity from the many hours I spend gazing fixedly through the coffee pot.</p>
| 7,799 |
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<p>Possibly the most useful thing anyone could tell me about particle physics:</p>
<p>Naively, one could try and make an algebra by enumerating all the types of particles and defining equivalence relationships (some rules), for example</p>
<p>Objects:</p>
<p>object type 1: photon</p>
<p>object type 2: electron</p>
<p>object type 3 ...</p>
<p>object type n</p>
<p>Rules:</p>
<p>rule 1: a photon is equivalent to an electron and a positron</p>
<p>rule 2: a proton is equivalent to three quarks</p>
<p>rule ....</p>
<p>rule n</p>
<p>Then one could study the properties of these relationships as a mathematical structure containing objects that represent particles and rules that define equivalence - i.e. as an algebra. I assume such an algebra would be, if formulated correctly, represented by Feynman diagrams (for some particles at least). Does such an algebra have a name? Is there more than one? Is this at all connected to the idea behind stating that the standard model is U(1) x SU(2) x SU(3)?</p>
<p>By an algebra I understand a closed and consistent collection of mathematical objects and relationships.</p>
<hr>
<p>Thanks for trying to understand what I'm asking, with your input I can aim to make this more comprehensible. So I guess I'm looking for a formal structure which reflects particle reactions which is abstracted away from spatial, temporal and energetic considerations as far as is possible (is it at all possible?).</p>
| 7,800 |
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<p>How broad is the term "String theory"?</p>
<p>Typical <a href="http://en.wikipedia.org/wiki/String_theory" rel="nofollow">brief descriptions of string theory</a> focus on the fact that it describes fundamental particles as being string-like rather than point-like, or, more generally, being at least 1-dimensional.</p>
<p>So is any model in which the fundamental particles are at least 1-dimensional a form of string theory? If not, what are the other requirements?</p>
<p>To put it another way, given a specific model, what factors would you consider when deciding whether that model was a string theory model or not?</p>
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-0.06585178524255753,
0.031000565737485886,
-0.061886776238679886,
0.03738553449511528,
-0.026115847751498222,
-0.013261796906590462
] |
<p>Can we apply the <a href="http://en.wikipedia.org/wiki/Matter_wave#de_Broglie_relations" rel="nofollow">de Broglie relations</a> to a <a href="http://en.wikipedia.org/wiki/Sound_waves#Physics_of_sound" rel="nofollow">sound waves</a> ?
Is it possible?
if yes how do you do that?
what would be the mass(m) in the equation?</p>
| 7,802 |
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] |
<p>How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum?</p>
<p>I used the Gordon Decomposition to split the Probability Current of the Dirac Field into its orbital current and its spin current. I multiplied the currents by $mc$ to covert them into momentum and then crossed the momentum with position to obtain the orbital angular moment and the spin angular momentum. However the spin angular momentum was twice as large as the accepted value of spin angular momentum. I am stuck on how to get rid of the extra factor of two. </p>
<p>The Gordon Decomposition splits the probability current into two terms
\begin{align}
mc\langle \gamma^{0n} \rangle
=
\langle \gamma^{0} i\hbar \partial^{n} \rangle
+
\tfrac{\hbar}{2}\partial_{j}\langle i\gamma^{0nj} \rangle_{n\neq j}
,
\end{align}
where $\langle \gamma^{0} i\hbar \partial^{n} \rangle$ looks like a orbital momentum current and $\tfrac{\hbar}{2}\partial_{j}\langle i\gamma^{0nj} \rangle_{n\neq j}$ looks like a spin momentum current?</p>
<p>The angular moments of $mc\langle \gamma^{0n} \rangle$ can be computed by crossing then with position to obtain the following:
\begin{align}
mc
\left(
x^m\langle \gamma^{0n} \rangle
-
x^n\langle \gamma^{0m} \rangle
\right)
=
\langle \gamma^{0} i\hbar (x^m\partial^{n}-x^n\partial^{m}) \rangle
+
\tfrac{\hbar}{2}\partial_{j}
\left(
x^m\langle i\gamma^{0nj} \rangle_{n\neq j}
-
x^m\langle i\gamma^{0nj} \rangle_{n\neq j}
\right)
,
\end{align}</p>
| 7,803 |
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<p>I want to calculate the distance between two objects using an iPhone camera. </p>
<p>Suppose I'm standing with my iPhone 10 feet away from the objects. Now, I want to calculate the distance between those two objects. </p>
<p>Suppose these two objects are on a straight horizontal line. So, how can I calculate the distance between those two points?</p>
<p>I have only the one predefine value like 10 feet or 5 feet/meter which is the distance between camera and the object.</p>
<p>From the above information how can I calculate the distance between two point OR distance of maximum width of screen ? </p>
<p>Reference Android App: <a href="https://play.google.com/store/apps/details?id=kr.sira.speed&hl=en" rel="nofollow">Speed Gun</a></p>
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] |
<p>Much (All?) of quantum theory can be done in separable Hilbert spaces with a countable basis.<br>
How about quantum field theory? Is it “quite happy” (mathematically consistent) if everything is countable, or does it “need” to use an uncountable, continuous space (e.g. rigged Hilbert space) for mathematical consistency, or some other reason?</p>
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<blockquote>
<p>A sphere of radius $R$ is in contact with a wedge. The point of contact is $\frac{R}{5}$ from the ground as shown in figure. Wedge is moving with velocity $20~\text{m/s}$ towards left then the velocity of the sphere at this instant will be</p>
<p><img src="http://i.stack.imgur.com/8Qubm.png" alt="enter image description here"></p>
</blockquote>
<p><a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=513&t=530952" rel="nofollow">http://www.artofproblemsolving.com/Forum/viewtopic.php?f=513&t=530952</a></p>
<p>I have attempted a solution there , however , unable to make any progress . Kindly advice if that is the right path to the solution , if something is there that I am not considering.</p>
| 7,806 |
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<p>Supppose we are given a quantum state that isn't pure state, such that it is a linear combination of the eigenstates of a Hermitian operator $\hat O$.
$$|\psi\rangle=N\sum \alpha_i |i\rangle$$ where $N$ is just a normalization constant.
I just want to make something clear about measuring. If we measure the corresponding eigenvalue of $\hat O$ and find it to be in the $|1\rangle$ state then immediately measure again, would I be right in thinking that the probability of getting the $|2\rangle$ state is $0$?</p>
<p>I think that because I believe the wavefunction collapses to the $|1\rangle$ state upon the first measurement...</p>
| 7,807 |
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] |
<p>How would the interference pattern look if one source emits blue light and the other source emits red light in Young's double slit experiment?</p>
| 7,808 |
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<p>When are two vectors orthogonal in curved spacetime?</p>
<p>From wikipedia:
<em>"In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90°, or π/2 radians. Hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces."</em> [<a href="http://en.wikipedia.org/wiki/Orthogonality%5d" rel="nofollow">http://en.wikipedia.org/wiki/Orthogonality]</a></p>
<p>How does this generalize for non-Euclidean space?</p>
<p>Is the following true?</p>
<p>Two vectors, $V_1$ and $V_2$, are orthogonal if and only if:
$$G_{MN} V_1^M V_2^N = 0$$
Where $G_{MN}$ is the metric of the space.</p>
<p>P.S.: I used Einstein notation: Summation over $M$,$N$ implied and $M$,$N$ $\in$ $\left\{1, 2, ..., d\right\}$.</p>
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<p>All of my textbooks mention, that entropy-change of all spontaneous physical, and chemical processes is positive, and that such processes need another condition to fulfill- decrease in the net internal-energy of the system, taking part in the process. </p>
<p>I just want to know that what is the reason behind this <strong>tendency</strong> of a system to increase its entropy, and to lose its internal-energy.</p>
| 7,810 |
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<p>I'm studying some materials on semiconductors, and a section on photoconductivity has left me somewhat confused.</p>
<p>In an extrinsic material, is photoconductivity the result of electrons moving from the valence band to the conduction band, or is it the result of dopant atom ionization? If it is the latter, is the complete ionization assumption invalid for photoconducting devices?</p>
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<p>Energy can be built from Lagrangian when considering the symmetry of time $(\frac{\partial L}{\partial t}=0)$. <a href="http://en.wikipedia.org/wiki/Free_energy" rel="nofollow">Free energy</a> is a generalization of energy when the system exchanges heat with the environment. I can't find any sources that building free energy directly from Lagrangian. Is there any way to do that? If it is, what symmetry is it based on?</p>
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<p>In Synchrotrons I think they use microwaves to accelerate the electrons bundles that fly through-how does putting a microwave through a cavity accelerate an electron? I know that the Electric and Magnetic fields could and do have an effect on the electron, but if the radiation is moving parallel to the electron, wouldn't the force be perpendicular? Also, how exactly does the microwave radiation look in the vacuum chamber? Are the two ends constant nodes? </p>
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<p>We are taught that the universe began as a singularity - an infinitely small and infinitely dense point. At the beginning of time there was a 'Big Bang' or, more accurately, 'Inflation'.</p>
<p>The main evidence for this is the observation of the red shift of all of the galaxies. This shows us that as time increases, the universe becomes bigger. A logical outcome of this is that going back in time, the universe shrinks. This is then extrapolated back to the beginning of time, where the universe was infinitely small.</p>
<p>However, the only thing that I can see that we know for sure is that the universe used to be smaller. This does not necessarily imply that it used to be infinitely small. How do we know, for example, that the universe doesn't oscillate and that we are in a time where the universe is expanding (and accelerating) and will eventually contract again?</p>
<p>What other evidence is there that suggests we started from an singularity?</p>
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<p>What is the equation for the saddle-like 2d surface (embeded in 3d Euclidean space with cartesian coordinates x, y and z) with constant negative curvature frequently used to illustrate open universe (for example in the following image is taken from Wikipedia)?</p>
<p><img src="http://i.stack.imgur.com/WNUNI.jpg" alt="enter image description here"></p>
<p>Flat universe (zero curvature) is illustrated by a plane, with equation say</p>
<p>$z = 0$</p>
<p>Closed universe (constant positive curvature) is illustrated by a surface of sphere, say</p>
<p>$x^2 + y^2 + z^2 = r$</p>
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<p>In an astronomy forum that I frequent, I have been having a discussion where the state of quantum gravity research came up. I claimed that Loop Quantum Gravity theories couldn't prove GR in the continuum limit, nor could they compute Mercury's perihelion because they didn't have any matter, at this point. Another poster claimed that one of the first things that any gravity scientist checks for the theory is the theory's prediction for the precession of Mercury's perihelion.</p>
<p>Now for cosmological extensions to GR like, MOND, TeVeS or STVG, I can see where this might make sense. But I can't see how this poster's claim holds for quantum-gravity theories that haven't been able to establish that they reproduce GR in the continuum limit and that don't have matter at all.</p>
<p>So I feel like I'm either missing something important about quantum gravity theories, in general, or the poster's comment only applies to cosmological-scale gravity modifications designed as an alternative to dark matter and not quantum gravity theories. Or perhaps a little of each.</p>
<p>How, for instance, would one compute the precession of Mercury's perihelion using CDT, or Rovelli et al.'s spin-foam version of LQG? It may be that the dynamics of any object are implied in the spacetime microstructure, I don't understand the math well enough yet to follow the implications to know if this is true. But how can you compute the orbit of a planet when you require mass and fermions in order to even have the Sun and a planet like Mercury?</p>
<p>I note the very very recent paper of this week, Dec. 21, <a href="http://arxiv.org/abs/1012.4719">http://arxiv.org/abs/1012.4719</a>, where the Marseille LQG group run by Carlo Rovelli claims they can now incorporate Fermions, so my question applies to research prior to this event, not that going forward.</p>
<p>I'm not even close to an expert, so I'm looking for enough information that I'll be able to respond intelligently to the other poster. I'll point them to the answer I received here so perhaps this will help the marketing for this excellent resource.</p>
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<p>Why in the irradiament mulipoles of Lienard-Wiechert's potential we say that electric quadrupole give a contribute of the same order of the magnetic dipole? How can we see it from their equations? And is there a physical reason for this? Sorry for my trivial question.</p>
| 7,816 |
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<p>Say that the airplane is going in 1000 km/h. On the side of the airplane, there is a 10x10 cm window. How much friction would this window cause. For the sake of the calculation, imagine that the airplane is infinitely long. The window is also perfectly level with the rest of the wall, and there is no turbulence on the side of the plane. The outside air is also standing perfectly still. The airplane is travelling at sea level.</p>
<p>I know it is neglectable compared to the turbulence etc. caused by the form of the airplane, but I really need the answer to the question.</p>
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<p>This question is posed in the context of thermodynamics/statistical mechanics. Suppose we define the correlation length as the $\xi$ in the exponential factor $e^{-r/\xi}$ that appears in the correlation function $$C(r) = \langle m(0)m(r) \rangle - \langle m \rangle^2$$ (where $m$ is some relevant order parameter).</p>
<p>Then my question is: how can we, from this mathematical definition, argue the following two physical properties.</p>
<p><strong>Suppose we have our thermodynamic system and we take a chunk out of it with side-length L.</strong></p>
<ol>
<li>If $L > \xi$, the sample has the <strong>same</strong> qualitative behavior as the original sample;</li>
<li>If $L < \xi$, the qualitative behavior is <strong>different</strong>.</li>
</ol>
<p>If one can add extra conditions for these statements to be true, that would be nice/welcome.</p>
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] |
<p>Consider a parallel-plate capacitor in free space. A negatively charged point particle with initial velocity $v$ passes through the space between the pair of parallel plates (with an initial path perpendicular to the normal vector of the plates). </p>
<p>The point particle accelerates towards the positively charged plate but passes beyond the edge of the plate. </p>
<p>How is energy conserved, given that the capacitor does work on the particle by accelerating it in the direction towards the negatively charged plate?</p>
<p>EDIT: Was reminded by Art Brown that a negatively charged particle accelerates towards the positive plate.</p>
| 7,819 |
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<p>Two open-car trains approach each other at fixed velocities. Each has a radar to see how quickly the other train is approaching, but apart from that the trains have no <em>a priori</em> knowledge of each other.</p>
<p>Each train has engineers on its first and last cars. Each engineer has an atomic clock and a laser for communicating with their partner on the same train. Long before the trains meet, the engineers on each train use their lasers to adjust their relative separations to exactly one kilometer. Using the synchronization procedure first defined by Einstein, the engineers also exchange time data and use it to adjust their atomic clocks until they are precisely synchronized within their shared frame.</p>
<p>The trains meet. The clocks are positioned so that they almost touch as they pass. At the moment of nearest contact, they exchange and record each other's values (timestamps).</p>
<p>Some time later, the clocks on the trailing cars meet and perform the same procedure.</p>
<p>The delay between the leading and trailing data events can now be measured in two ways.</p>
<p>If the two timestamps taken from the first train are compared, the result is the time between the events as measured from the first train. This value is meaningful because the engineers on that train previously synchronized their clocks and stayed within the same frame of reference at all times. If the timestamps from the second train are used, a similar but distinct measurement of the time between the two events can be obtained.</p>
<p>Now, three questions:</p>
<ol>
<li><p>Is there anything wrong or impossible with this experimental setup? If so, what is it?</p></li>
<li><p>If you accept the experiment as realistic and meaningful, will the delays calculated from the perspective of each of the two trains be the same, or different?</p></li>
<li><p>If you answered "different," what is the correct procedure for predicting in advance what the ratio of the two delays will be?</p></li>
</ol>
<hr>
<p><strong>2013-01-10 - The Answer(s!)</strong></p>
<p>I awarded the bounty to @FrankH for pages of excellent and educational work worth of reading by anyone who wants to understand special relativity better.</p>
<p>However, I've also taken the unusual step of re-allocating the <em>answer</em> designation to @MatthewMcIrvin.</p>
<p>There are two reasons: (1) Matthew McIrvin was the first one to spot the importance of the symmetric-velocities frame, which others ended up gravitating (heh!) to; and (2) Matthew's answer is <em>short</em>, which is great for readers in a hurry. FrankH, sorry about the switch, but I didn't realize I could split them before.</p>
<p>So: If you are not overly familiar with special relativity and want to understand the full range of issues, I definitely recommend FrankH's answer. But if you already know SR pretty well and want the key insights quickly, please look at Matthew McIrvin's answer.</p>
<p>I will have more "proximate data exchange" SR questions sometime in the near future. Two frames turns out to be a very special case, but I had to start somewhere.</p>
<hr>
<p>For folks with strong minds, stout hearts, and plenty of time to kill, a much more precise version of the above question is provided below. It requires new notations, alas. </p>
<p>To reduce the growing size of this question, I have deleted all earlier versions of it. You can still find them in the change history.</p>
<hr>
<p><strong>Precise version of the 4-clock conundrum</strong></p>
<p>Please see the first three figures below. They define two new operators, the frame view and clock synchronization operators, that make the problem easier to state precisely:</p>
<hr>
<p><img src="http://i.stack.imgur.com/JTWkQ.png" alt="Setup of the four-clock conundrum using the frame operators"></p>
<hr>
<p><img src="http://i.stack.imgur.com/347QJ.png" alt="Event T1: Leading edge clocks A1 and B1 pass very closely"></p>
<hr>
<p><img src="http://i.stack.imgur.com/EVZtI.png" alt="Event T2: Trailing edge clocks A2 and B2 pass very closely"></p>
<hr>
<p>Note that the four timestamps $\{T_{A1}, T_{B1}, T_{A2}, T_{B2}\}:>\{A,B\}$. That is, the four timestamps are shared and agreed to (with an accuracy of about 1 nanosecond in this case) by observers from <em>both</em> of the interacting frames A or B. Since T1 and T2 are historically recorded local events, this assertion can be further generalized to $\{T1,T2\}:>*$, where $*$ represents all possible frames. (And yes, $:>$ is two eyes looking left, while $<:$ is two eyes looking to the right.)</p>
<p>Using the four shared time stamps, define:</p>
<blockquote>
<p>$T_{\Delta{A}} = T_{A2} - T_{A1}$</p>
<p>$T_{\Delta{B}} = T_{B2} - T_{B1}$</p>
</blockquote>
<p>My main question is this:</p>
<hr>
<blockquote>
<p>Assuming that $T_{\Delta{A}} = f(T_{\Delta{B}})$ exists, what is $f(x)$?</p>
</blockquote>
<hr>
<p><strong>Analysis</strong> (why this problem is difficult)</p>
<p>Based purely on symmetry in the setup, the most obvious answer is $f(x)=x$, that is:</p>
<blockquote>
<p>$T_{\Delta{A}} = T_{\Delta{B}}$</p>
</blockquote>
<p>There are some interesting reasons to be troubled by that seemingly straightforward and even obvious conclusion, not the least of which is that it violates the whole concept of special relativity (the Dingle heresy).</p>
<p>That is, if you assume $f(x)=x$ and follow that line of logic through to its logical conclusion, you quickly end up with time that flows the <em>same</em> for all frames -- that is, no relativity. Such a conclusion is in flat violation of over a century of very detailed experimental evidence, and so is just not supportable.</p>
<p>The following four figures show why it's so hard to assert that $T_{\Delta{A}} = T_{\Delta{B}}$ without violating special relativity.</p>
<hr>
<p><img src="http://i.stack.imgur.com/G59Bl.png" alt="View of Event T1 from Frame A"></p>
<hr>
<p><img src="http://i.stack.imgur.com/j2K87.png" alt="View of Event T2 from Frame A"></p>
<hr>
<p><img src="http://i.stack.imgur.com/IFj5Q.png" alt="View of Event T1 from Frame B"></p>
<hr>
<p><img src="http://i.stack.imgur.com/jmoYk.png" alt="View of Event T2 from Frame B"></p>
<hr>
<p>While the above figures accurately capture the reality of relativistic contraction of both distance and time, the problem in this case is simple: How do you decide which frame to select? The experiment as described can only give one outcome. Which one will it be?</p>
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<p>I am studying the history of Modern Physics and Yang-Lee earned their Nobel the next year after the <a href="http://en.wikipedia.org/wiki/Wu_experiment" rel="nofollow">Cobalt experiments</a>. I am familiar with the chronology, but am not clear what those findings meant to the physics community to warrant such recognition for two relatively' unknown physicists.</p>
<p>Other household names had to wait decades for more seemingly significant work.</p>
<p>So can someone clarify what their work meant to the eventual development of the Standard Model?</p>
| 7,821 |
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] |
<p>Suppose I set up the <a href="http://en.wikipedia.org/wiki/Double-slit_experiment" rel="nofollow">double-slit experiment</a> using photons as my particle. Behind the left slit I place a <a href="http://en.wikipedia.org/wiki/Beam_splitter" rel="nofollow">beam splitter</a> that points some of the light off in the direction of a camera (represented as <code>C</code> in the diagram below). We'll say that the beam splitter splits a light beam passing through it into two equal beams. The light that passes through the beam splitter unaffected and the light that passes through the right slit continue on to another camera (represented as <code>CCC</code> in the diagram below). We'll say that camera <code>C</code> is closer to the beam splitter than camera <code>CCC</code> is.</p>
<pre><code> CCC
| |
| |
C-\ |
| |
-----|-|-----
|
|
Laser
</code></pre>
<p>If I fire a single photon through this apparatus, there's about a $1/4$ chance that it will be detected by camera <code>C</code> and about a $3/4$ chance that it will be detected by camera <code>CCC</code>. When the wavefunction reaches camera <code>C</code>, the photon is forced to either be detected by camera <code>C</code> or have the part of the wavefunction leading to camera <code>C</code> disappear.</p>
<p><strong>Question:</strong> In the case that the photon is not detected by camera <code>C</code>, how is the $1/4$ probability from that branch of the wavefunction redistributed to the remaining two branches of the wavefunction, the branch went through the left slit and straight through the beam splitter and the branch that went through the right slit? The first starts with about $1/4$ probability and the second starts with about $1/2$ probability. Does the first get a new probability of $1/3$ and the second a new probability of $2/3$? Does the first get a new probability of $1/2$ while the second keeps its old probability of $1/2$?</p>
| 7,822 |
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<p>I have three column data that has time-displacement-force from 1D tensile/compression test. Now I would like to get the standard mechanical properties of the material, like Young modulus, yield stress/strain, strain hardening/softening exponent, Fracture toughness, work done...</p>
<p>Is there algorithms/libraries available that will do it for me or do I have to code them myself? </p>
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<p>I've read that all paths inside the EH lead to singularity. ALL paths. Even the ones pointing away from it, right? Because there's NO pointing away from singularity, since ALL paths point to it. </p>
<p>So how could it be at the exact center? Or am I misunderstanding "center" here?</p>
<p><a href="http://physics.stackexchange.com/questions/56906/is-singularity-at-the-exact-centre-of-a-black-hole"></a></p>
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<p>From what I've so far understood about light, a photon is emitted somewhere and after some time it's absorbed somewhere else.</p>
<p>Have we had experiments that confirm the path taken or something akin to verifying that photon really do "travel"? If so what are they..</p>
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<p>In <a href="http://youtu.be/QGxNyaXfJsA" rel="nofollow">this youtube video</a> (which is an except from a documentation I believe) a DARPA development for a high-resolution surveillance camera is presented. This question is regarding the optics that are likely to be required to make it work.</p>
<h2>Summary of the specifications given:</h2>
<ul>
<li>$1.8\times10^9$ pixel</li>
<li>covering $35\,km^2$</li>
<li>from an altitude of $5\,km$</li>
<li>using $368$ cellphone camera chips</li>
</ul>
<h2>Some back of the hand calculations:</h2>
<ul>
<li>pixel-equivalent area: $\frac{35\times10^6\,m^2}{1.8\times10^9\,px^2} = 1.94\times10^{-2}\,\frac{m^2}{px^2}$</li>
<li>edge length of pixel-equivalent area: $\sqrt{1.94\times10^{-2}\,\frac{m^2}{px^2}}=1.39\times10^{-1}\,\frac{m}{px}\approx14\,\frac{cm}{px}$</li>
<li>pixel count per chip: $\frac{1.8\times10^9\,px}{368\,chips}\approx5\times10^6\,\frac{px}{chip}$</li>
</ul>
<h2>Optics question</h2>
<p>Cellphone chips have pixel pitches of the order of a few microns. Say they use a typical, decent quality 5 megapixel chip with $2.25\,µm$ pitch.</p>
<p>How are the optics likely to look like to image 15 cm at 5 km distance onto 2.25 µm on the chip within a camera that is at best a meter in diameter?</p>
<p>I am looking for answers beyond "they use a telephoto lens"... Lens focal lengths, combinations and setup for producing the required image is what I am struggling to work out.</p>
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] |
<p>The potential at the surface of an insulating sphere (radius R) is given by<br>
$$V(R,\theta) = k \cos(3\theta)$$
where $k$ is a constant. Use separation of variables to find the potential inside the sphere (r $\leq$ R) and outside the sphere (r $\geq$ R), and then use your answer to determine the surface charge density $\sigma(\theta)$ on the sphere. Assume there is no charge inside or outside the sphere.</p>
<p>My approach: I will set my reference point at infinity, therefore the potential at infinity is 0, to set that as a boundary condition. Essentially, the approach in finding the surface charge density would be to find the potential function outside the sphere, then integrate that with the surface area. I am following Griffiths for E&M, but he doesn't include a specific example I can refer to for the separation of variables, and how to apply it. I can follow his examples for following potentials pretty easily, but cannot understand the first steps to set up this problem with SOV.</p>
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<p>Let's say I have a projectile being thrown by a player in my 2-D game. I want to work backwards and find the initial velocity to apply to the projectile such that it passes through a target point in space given that the launch point might not be at $y=0$.</p>
<p>Some applications of this would be a basketball shot simulator or a pub darts simulation.</p>
<p>Given:</p>
<ul>
<li>initial or launch angle: $\theta_o$</li>
<li>initial or launch height: $y_o$</li>
<li>target point: ($x_f$, $y_f$)</li>
<li>constant acceleration due to gravity ($g$)</li>
<li>air resistance ignored</li>
</ul>
<p>I need to find:</p>
<ul>
<li>initial launch velocity: $v_o$</li>
</ul>
<p>Here's an image of what I'm talking about.</p>
<p><img src="http://i.stack.imgur.com/uRRkX.png" alt="enter image description here"></p>
<p>I am interested in learning the derivation of the answer not just a formula that I plug-and-chug. Thanks!</p>
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<p>I once came across the strange, artificial unit "hertz per dioptre", which is dimensionally equivalent to "metres per second". Could this unit, by some stretch of the imagination, be used in some artificial situation where the usage implications of both "hertz" and "dioptre" (frequency of periodic events and refractive power, respectively) would make for the ratio to actually be useful?</p>
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<p>I just saw this "<a href="http://www.youtube.com/watch?v=7ctaA2mERzI&feature=player_embedded" rel="nofollow">trick</a>" where a cup of water is turned over onto a table without spilling (using a piece of cardboard. After removing the cardboard from underneath the cup, the person then removes the cup in a particular way (lifts straight up and twists) and lo and behold, the water stays in it's position as if the cup were still there!? (watch the video to fully understand) Is this really possible? If so, and the real question I'm looking to have answered is, how? </p>
<p>After <a href="http://www.b3ta.com/links/592518" rel="nofollow">further research</a> it appears that deionized water is needed as well.</p>
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<p>What is the current <strong>record</strong> number of entagled qubits and how has this number been increased? </p>
<p>The latest result on stack exchange, which is 3 years old, reports 14 via
this post: <a href="http://physics.stackexchange.com/q/4226/">How many stabilised qubits have been achieved in Quantum Computing?</a></p>
<p>in accord with this paper: <a href="http://arxiv.org/abs/1009.6126" rel="nofollow">http://arxiv.org/abs/1009.6126</a>?</p>
<p>For the purposes of this post, I'd prefer to leave the D-Wave computers out of the answers. </p>
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<p>Okay, here is my understanding. Please correct me if anything here is incorrect. Suppose I have a cube with a side length of ten inches. It has a surface area of 600 square inches, and at an atmospheric pressure of 14.7 pounds per square inch, we can approximate 8800 pounds of air are applied on the cube by the atmosphere. The cube has a volume of 1000 cubic inches of air. But since there is air on the inside of the cube, and the pressure is equal, there pressure is cancelled out.</p>
<p>Now suppose we were to have a bunch of cubes with ten inch sides. We remove 1/2 of the air from the inside of one, and 1/4 of the air from another, and 1/10 from another. How can we calculate the actual pressure of the atmosphere on the cube?</p>
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<p>Currently, as described in <a href="http://physics.stackexchange.com/questions/39679/why-cant-collisions-be-elastic">this question</a>, collisions cannot be elastic. However, is there a way to make a collision more elastic?</p>
<p><strong>Background:</strong></p>
<p>We're working on a project right now involving high speed energy transfer to an exercise ball. One of the issues right now is that the exercise ball is going far slower than what basic (e.g. AP Physics AB) calculations say they should be. I know that this is most likely due to the less-than-perfect elasticity of the collision between the ram and the ball. Is there any way to make the collision more elastic?</p>
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<p>The one-particle states as well as the fields in quantum field theory are regarded as representations of Poincare group, e.g. scalar, spinor, and vector representations. </p>
<p>Is there any systematical procedure that one starts from the Dynkin label for a given representation, to construct a Lagrangian of that field theory? If yes, where can I find that procedure? </p>
<p>I don't care adding interaction by gauge invariance from these Lagrangians will cause non-renormalizability or not. I can live with effective theories. </p>
| 7,834 |
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<p>What's so special about second order equations in classical mechanics? I have a basic understanding of the Lagrangian and Hamiltonian formulations of classical mechanics, so I'm not looking for answers like 'because Newton's second law is a second order ODE' or 'because Euler-Lagrange equations act on a first time derivative of position'. I'm looking for a deeper physical reason - in the same sense that Energy conservation is not fundamental, it results from time translation invariance. I realise two boundary conditions are required to solve for the dynamics of a given system, but I see that more as a result of the equations being second order than the cause of them being second order. Is there a more fundamental organising principle that I am not aware of?</p>
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<p>I think this is a question about the mathematical axioms of quantum mechanics...</p>
<p>Consider the following operators $\tilde{x}$ and $\tilde{p}$ on Hilbert space $L^2(R)$, defined for fixed $\delta>0$:
$$
(\tilde{x}\psi)(x) = (n+1/2) \delta\ \psi(x)
$$
when $n\delta \leq x <(n+1)\delta$ for some integer $n$; and
$$
(\tilde{p}\psi)(x) = -i \hbar \frac{d\psi}{dx}(x)
$$
when $n\delta < x < (n+1) \delta$, and $(\tilde{p}\psi)(x) = 0$ when $x=n\delta$.</p>
<p>Both these operators are self-adjoint and they commute. Further, $\tilde{p}$ is identical to the usual momentum operator $\hat{p} = -i\hbar \frac{d}{dx}$ on the subspace of $L^2(R)$ on which $\hat{p}$ is defined, and thus has identical spectrum. Finally, by taking $\delta$ small we can make $\tilde{x}$ as close as we like to the usual position operator, $\hat{x}$. Thus naively it looks like $\tilde{x}$ and $\tilde{p}$ represent compatible observables which can be made arbitrarily close to $\hat{x}$ and $\hat{p}$.</p>
<p>I'm pretty sure that I've cheated somewhere, but I can't find anything in the axioms of quantum mechanics that rules out defining observables $\tilde{x}$ and $\tilde{p}$ as above (or at least, the axioms as they are usually presented in under/graduate physics texts). Is there some requirement that observable operators should map $C^\infty$ functions to $C^\infty$ functions? In any case, I'd be grateful if someone could tell me why the above is ruled out.</p>
<p>(As background, I am teaching an undergraduate class on quantum mechanics next semester and I wanted to explain to students that while it's not possible to make exact simultaneous measurement of position and momentum; it is possible to make an approximate simultaneous measurement. In other words, I wanted to find operators which approximate $\hat{x}$ and $\hat{p}$ on 'macroscopic' scales and which commute. This was before I looked in the literature and realised how technical constructing such approximate operators is.)</p>
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<p>I am wondering if there exist closed form-expressions for the time dilation experienced by an observer in different orbits around a Schwarzschild black hole, outside the event horizon, relative to some distant observer sitting fixed relative to the black hole. </p>
<p>I am no expert in GR, but I know these quickly turn into some complicated integrals. Any point in the right direction, even toward an approximate answer, would be much appreciated.</p>
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<p>I am a beginner who is learning QFT. When I was going through the quantisation of Klein-Gordon real-field. I got confused about something:</p>
<p>The solution to Klein-Gordon equations are of the form $ \psi(x^\mu) \sim e^{ik_\mu x^\mu} $. Now the solutions in Peskin and Schroeder have no time dependence. It gives a Fourier transform of this kind:</p>
<p>$$ \psi(\vec x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} \Bigl[a(\vec p) e^{i\vec k. \vec x} + a^\dagger(\vec p) e^{-i\vec k. \vec x} \Bigr] $$</p>
<p>My problem here is, why is the solution is a superposition of $ e^{i\vec k. \vec x} $ and not $ e^{ik_\mu x^\mu} $ ?</p>
<p>[EDIT]</p>
<p>For example in these <a href="http://www.physics.gla.ac.uk/~cwhite/lectures/QFTnotes.pdf">notes</a> in equations 90 and 113 the solutions are superposition of $ e^{ik_\mu x^\mu} $ and I don't where these two things disagree.</p>
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<p>Basically how boiling takes place? </p>
<p><em><strong>Also like to know...</em></strong><br>
What makes boiling point alter at various altitudes?<br>
Why bubbles rise through boiling water?</p>
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<p>I am reading Jan Boer's <a href="http://www-library.desy.de/preparch/desy/proc/proc02-02/Proceedings/pl.6/deboer_pr.pdf" rel="nofollow">review</a> of the AdS/CFT correspondence and I quote from end of page 1, where he is talking about equivalence of (d+1)-dim gravity to d-dim field theory </p>
<p>“If true, it implies [...]. If the degrees of freedom in gravity would be local, one would imagine that one can have arbitrarily large volumes with fixed energy density.[...]”</p>
<p>I don't quite understand that. What does it mean for “degrees of freedom to be local”? And how does that lead to fixed energy-density?</p>
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<p>Find the <strong>minimum value of the initial velocity</strong> $u$ of the particle such that the particle crosses the wheel of radius $R$.<br>
<img src="http://i.stack.imgur.com/hN0F9.jpg" alt="enter image description here"></p>
<p><strong>Details and assumptions</strong><br>
$R=2m$<br>
$g=9.8m/s^2$<br>
Neglect air resistance.<br>
All surfaces are frictionless.<br>
The value of $\theta$ (angle the projectile makes either with vertical or horizontal), range and $u$ is not known.<br>
Consider the motion in 2-D space only.</p>
<blockquote>
<p>I tried setting the maximum height equal to $2R$ and then finding the corresponding minimum value of $u$, but my answer was incorrect.<br>
Then I tried to set the latus rectum of the parabola (equation of trajectory) equal to $2R$ but that too didn't work.<br>
Can anybody suggest a way to do this question?<br>
Thanks in advance! </p>
</blockquote>
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] |
<p>Say we have a balloon, negatively charged, the voltage on it is 500 V. Can I measure the charge on it or in other words, the number of excess electrons?</p>
| 7,841 |
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<p>Mixing of two different fluids is associated with an increase of entropy. Conversely, separation of two gases must be associated with a decrease of the entropy of the two fluids.</p>
<p>Is there a minimum requirement imposed by this thermodynamic constraint on the minimum energy needed to separate two fluids? </p>
| 7,842 |
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] |
<p>Do the laws of physics change anywhere in the universe?Or will they change from place to place in the universe?</p>
| 7,843 |
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<p>A vector field $f^\mu$ is said to be Fermi-Walker transported along a curve $\gamma$ parametrized with $\tau$ if the following holds $$\frac{\mathrm{D}}{\mathrm{d}\tau}f^\mu = -(a^\mu v^\nu - a^\nu v^\mu) f_\nu,$$ where $v^\mu$ is the tangent vector $\gamma$ and $a^\mu$ is its derivative. The uppercase D denotes the covariant derivative. This is the usual transport law for nonrotating tetrads. However, in "Introduction to General Relativity" Lewis Ryder mentions the Fermi transport for which $$\frac{\mathrm{D}}{\mathrm{d}\tau}f^\mu = v^\mu a^\nu f_\nu.$$ What is the significance of this transport? Are there any applications of it?</p>
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] |
<p>Why any fermion can be written as a combination of two Majorana fermions? Is there any physical meaning in it? Why Majorana fermion can be used for topological quantum computation?</p>
| 7,845 |
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<p>In the case of a complex shape filiform distribution of current, are we allowed to determine the magnetic field created by sections of the distribution and then summing them up, like we do with a discrete distribution of charge when calculating the electric field?</p>
<p>Thank you</p>
| 7,846 |
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<p>How can the universe become infinite in spatial extent if it started as a singularity, wouldn't it take infinite time to expand into an infinite universe?</p>
| 7,847 |
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<p>Are there any examples of fermionic particles or quasiparticles for which the interaction potential is a globally smooth function? i.e. no singularities or branch points. </p>
<p>As an example, in Flügge's Practical Quantum Mechanics, problem 148 has two repulsive particles on a circle. This is supposed to model the two helium electrons in the ground state. The equation he gives is</p>
<p>$$ -\frac{\hbar^2}{2mr^2}\left(\frac{\partial^2 \psi}{\partial x_1^2}+\frac{\partial^2 \psi}{\partial x_2^2}\right)+V_0\cos(x_1-x_2)\psi=E\psi$$</p>
<p>I don't quite follow why this potential does not have a singularity when $x_2\rightarrow x_1$. Are there other such examples?</p>
| 7,848 |
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<p>Is there a way of calculating the mass of a galaxy, or even a nebula from the luminosity? </p>
<p>EDIT </p>
<p>I'm deleting this, and moving the question to Astronomy Stack Exchange - thanks david</p>
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<p>My <a href="http://www.cs.berkeley.edu/~vazirani/s07quantum/notes/lecture2.pdf">text shows</a> (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with $k$ and $l$ "bits of information", respectively, requires $kl$ bits to fully describe it. A critical consequence of this is that the $k+l$ bits in each of the individual subsystems are not sufficient to "span" their joint space, so that there must be states that are "entangled" (formally, cannot be expressed by the tensor product of the states of the subsystems). That much I (<em>think</em>) I understand.</p>
<p>But the text then seems to argue that this is a property that is exclusive to quantum systems. Is that true? Certainly there are classical systems where the state of one subsystem depends on the state of another. Doesn't any joint system that requires conditional probabilities to describe need "extra bits" beyond those necessary to describe the individual subsystems?</p>
<p>Perhaps I'm missing some subtlety about what constitutes a "sub system" or some unstated assumption about how systems are broken into parts. Perhaps a classical system that is "entangled" by conditional probabilities isn't thought of as consisting of valid "subsystems" in the same way that quantum system is.</p>
<p>Is "entanglement" unique to quantum systems? Are conditional probabilities just a kind of "classical entanglement"?</p>
<hr>
<p>Please be pedagogical. This is not meant to be a deep or advanced mathematical question, just a simple conceptual and definitional one. Just imagine I'm having basic probability explained to me (without reference to quantum vs. classical, or even physics). If one struck out all occurrences of the word "quantum" from the linked <a href="http://www.cs.berkeley.edu/~vazirani/s07quantum/notes/lecture2.pdf">text</a> (sections 0.2. and 0.3), wouldn't that be a perfectly valid part of such an explanation?</p>
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<p>I would like to start off by saying this is not a philosophical question. I have a specific question pertaining to physics after the following explanation and background information, which I felt was necessary to properly formulate my question.</p>
<p>I have done some research on <a href="http://en.wikipedia.org/wiki/Zeno%27s_paradoxes" rel="nofollow">Zeno's Paradoxes</a> as well as some of their modern day mathematical and physical "solutions." I understand that Zeno's dichotomy paradox can be explained mathematically by constructing an argument that makes use of the following infinite series:
$$ \sum_{n=1}^\infty {1 \over 2^n} = 1$$ Because this series converges to $1$, then that implies that if physically traveling from some point A to another point B truly does require an infinite amount of actions in which an object travels ${1 \over 2}$ of the distance, then ${1 \over 4}$ then ${1 \over 8}$ then ${1 \over 16}$ and so on, the sum of those actions would eventually result in traveling the full distance. Mathematically, this makes perfect sense. However, knowing that the mathematics we use to describe the physical world is only a model, adopted when there is sufficient evidence to support it, could it be that the solution gained by the above argument comes to the correct conclusion using the wrong path? Using the infinite series provides a solution to the dichotomy paradox if it does indeed require an infinite amount of actions to move from point A to point B. So, then the question becomes, does it truly take an infinite amount of actions to physically move from one point to another (in which case, I believe the mathematical solution is a perfect model), or are there other phenomena at play that cannot be described in this fashion?</p>
<p>I have one idea pertaining to a quantum mechanical description of the situation that does not disprove, but provides an alternative to the reasoning that it requires an infinite number of actions to move from point A to point B. </p>
<ul>
<li>Suppose we have a box of some macroscopic size (say $1m^3$), and it is in the process of moving with a constant velocity from some point A to another point B. Obviously, the box is made up of atoms, and if we zoom in far enough on the leading face of the box we could see that the motion of said box is really the motion of a very large number of atoms moving in the same direction. According to the uncertainty principle, it is impossible to determine both a particle's momentum and position simultaneously. According to the physics professor who first taught me about this, the reason for this uncertainty does not lie within our measurement methods, but rather, it is an impossibility inherent in the fact that all particles have wavelike characteristics. Thus, their position and momentum are not even clearly defined at a specific moment in time. Could it not be possible then, that if we zoomed in far enough on the leading face of our box, that we would find the face of the box does not have a definite position? If this is true, then wouldn't there be a certain point very close to the destination point, B, where the idea of cutting the remainder of the distance to the destination in half makes no difference to the particles in question? We may say that to complete the trip, we must travel another ${1 \over 2^x}$ portion of the distance we started with, but if the uncertainty in position is larger than this remaining distance, could the particle (and by the same argument, all of the particles that make up the leading face of the box), traverse the final distance to point B without physically having to travel the "infinite" amount of points remaining?</li>
</ul>
<p>Also, I have a second idea that I want to present as an aside, something that I don't believe can be proven as of now, but I wonder if, according to current understanding, it is possible.</p>
<ul>
<li>What if, instead of analyzing the situation at hand through the distance traveled, it was analyzed by analyzing the passage of time. Specifically, if time is quantized, would it resolve the dichotomy paradox in a physical sense? I realize that whether or not time is quantized is up for debate (there is even <a href="http://physics.stackexchange.com/questions/35674/is-time-continuous">this question</a> here at stack exchange that explores the idea), so I am not going to ask if time is quantized. Based off of my understanding, however, there is nothing known as of now that says time cannot be quantized (please correct me if I am wrong). Therefore, if time <em>were</em> quantized then could we not say that for a given constant velocity there is a minimum distance that can be traveled? A distance that corresponds to $\Delta t_{min} * v$? Would this not imply that as our moving box reached a certain distance, $d$ away from point B, and if $d < \Delta t_{min} * v$, the box would physically not be able to travel such a small distance, and in the next moment that the box moved, it would effectively be across the destination point B?</li>
</ul>
<p>All of that being said, here is my specific question: In my above arguments, is any of my logic faulty? Is there any known law that would disprove either of these explanations for Zeno's dichotomy paradox? If so, is there a better way of physically (not mathematically) resolving the paradox?</p>
| 7,851 |
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<p>As an assignment for uni I need to figure out an algorithm that explodes a particle of mass $m$, velocity $v$, into $n$ pieces.</p>
<p>For the first part of the assignment, the particle has mass $m$, velocity of $0$, the particle explodes into 6 equal pieces, and is not affected by gravity.</p>
<p>The problem I'm having is, how do I determine 6 equidistant unit vectors in three dimensions?</p>
<p>Assuming I need an initial vector, a unit vector in the direction of $v$ (the initial particle velocity) will do. In the initial case, where $v$ is $(+0, +0, +0)$, let the initial unit vector be in an arbitrary direction.</p>
<p>Also, how do I determine what speed each sub-particle will have? I know that the sum of sub-particle momentum will be equal to the momentum of the initial particle, and because each particle has the same mass they will each have the same speed... Do I need an explosion Force amount or something?</p>
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<p>I need to calculate the initial velocity required to launch a projectile at a given angle from point A to point B. The only force acting on the projectile after launch will be gravity – zero air resistance. The projectile is launched within a simulated, virtual environment; however, I am asking for help with the physics rather than the simulation itself.</p>
<p>I have had success (with a different equation) when point A and point B are at the same height; however, <strong>once point B is at a different height</strong>, my calculations become less precise – well, wrong in fact.</p>
<p>I researched the following formula for finding the range of a projectile on uneven ground. The parameters are all available to me, except the initial velocity that I need to solve for.</p>
<blockquote>
<p>$d$ : range or distance<br />
$v_i$ : initial velocity<br />
$g$ : gravity<br />
$\theta$ : launch angle<br />
$y_0$ : launch height<br /></p>
</blockquote>
<p>$$d = \frac{v_i \cos\theta}{g}(v{_i} \sin \theta + \sqrt{(v_i \sin\theta)^2 + 2gy_0})$$</p>
<p>I attempted to solve for initial velocity (eq: <strong>A</strong>):</p>
<p>$$v_i = \sqrt{\frac{d^2g}{2\cos\theta^2(y_0+d \tan\theta)}}$$ </p>
<p>Using this equation in my simulation I apply the velocity to a normalised displacement vector and launch the projectile. It gets close to its target but:</p>
<ul>
<li>$x_{final}$ is always correct</li>
<li>$z_{final}$ is always incorrect - it is close to the desired $z$ but always offset by a seemingly proportional amount. </li>
</ul>
<p>I have spent many hours trying to review the equation but I have been unsuccessful. Any advice would be most appreciated:</p>
<ul>
<li>is this the correct equation?</li>
<li>have I solved for $v_i$ correctly? I have looked for an example equation online but I have not found one structured how I need it, i.e. solving for $v_i$</li>
<li>I have spent hours researching online - perhaps I have missed some good (but entry-level) resources. Do you know of any reference material that may help me?</li>
</ul>
<p><strong>Update - I have now named the equation above for referencing below</strong></p>
<p>Following on from the answer by @Pygmalion, which I am still gratefully working to understand:</p>
<p>I agree that your derived equation is equivalent to mine (A). Using yours (@Pygmalion's) in my simulation, I therefore observe the same failures: the projectile always lands short of the target.</p>
<p>I have simplified the simulation keeping the <strong>launch height and target height the same</strong>. I still encounter the same problem with equation A; however, when passing the same parameters to the following equation the projectile always hits the target precisely (eq: <strong>B</strong>):</p>
<p>$$v_i = \sqrt{\frac{dg}{\sin2\theta}}$$</p>
<p>Given the success of equation B when the launch and target heights are the same and the failure of equation A given the same conditions, I question whether equation A is the correct one to solve the problem since I would expect it to work for any combination of launch|target height.</p>
<p>I think the help I need is around why equation B works. Does it cater for conditions that equation A overlooks? If equation A is unsuitable, are you able to recommend an alternative?</p>
<p>In short, the $v_i$ calculated by equation A (in my simulations) is always less than $v_i$ from equation B given the same inputs - the projectile therefore always falls short of its target.</p>
| 7,853 |
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<p>I've been reading 'How the Hippies saved Physics', which describes a design for a superluminal communication device, of which the crucial part was a laser which duplicated an incoming photon many times. The reason this won't work is what is now known as the no-cloning theorem - a quantum state can't be duplicated in this way. It may appear that a laser can do this, but it can't.</p>
<p>The thing is that I have vague memories of reading that when the laser was first talked about, it was claimed that quantum theory forbade such a device. What I'd like to know is whether there were such claims, and if so were they based on the idea that a laser would duplicate a quantum state.</p>
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<p>I have a question to ask about the operationalist view of space-time. I am a mathematician who happens to be interested in physics, so if anyone thinks that my question is a silly or vague one, please feel free to close it.</p>
<p>Many physics textbooks that discuss Special Relativity mention that one way of setting up a coordinate system in flat Minkowski space is to imagine that there is an infinite grid of meter rules that pervades the universe, with synchronized clocks located at each node of the grid. In order to obtain more precise measurements of space-time events, one simply divides the grid into more nodes. I feel that this presents problems, because what does it mean to say that ‘all the clocks are synchronized’? Any act of moving a clock from a starting point to its designated node is sure to involve some acceleration, and this is going to upset any synchronization attempt. Also, such a description is not satisfactory because it is impossible to deal with an infinite number of meter rules and clocks.</p>
<p>Things are made worse in curved space-time because of the further effect of gravitational time-dilation. Hence, I believe that the only logically consistent way to perform measurements of space-time events is to use a single atomic clock carried by a single observer and to use the world-lines of photons emitted by a light source carried by the observer to ‘carve out’ a local-coordinate system. The exact description of how this can be done is beyond me, so I would like to gather some useful information from anyone in the community. Thank you very much!</p>
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<p>I have a small RC-controlled toy helicopter with removable tail rotor. </p>
<p>Suppose I remove the tail rotor, hold the tail with my hand, start the rotor until it moves with constant angular velocity and then let it go. Suppose that the heli remains on the floor all the time.</p>
<p>Then the body of the helicopter begins to turn into the opposite direction of the rotor. </p>
<p>My problem is to explain this in terms of angular momentum. At first this seemed to contradict angular momentum conservation because I assumed that the helicopter is a closed system and there is no external torque, but the rotor keeps the same angular velocity while the body starts to turn into the other direction. </p>
<p>My second thought was that I have to take the air into account. If I do that it seems plausible to me to explain it as follows:</p>
<p>The rotor transfers angular momentum from to the surrounding air. However to keep it running at constant angular velocity, the motor has to transfer angular momentum to the rotor, which is only possible if the body of the helicopter gets the same amount of angular momentum as goes to the rotor, but into the opposite direction. </p>
<p>But then I thought, the body should turn faster and faster to infinity which obviously doesn't happen. So I guessed I should also take the friction with the floor into account where the angular momentum of the body is transferred to.</p>
<p>This leads me to the following three questions:</p>
<ol>
<li><p>Is my reasoning described above correct or I am completely wrong?</p></li>
<li><p>If I would do the experiment on an almost frictionless turntable (which I don't have available at the moment), would the body of the helicopter indeed become faster and faster almost to infinity?</p></li>
<li><p>If I would do the same thing in a vacuum, what would happen? I guess (see my first thought above), that the body wouldn't turn at all. Is that correct?</p></li>
</ol>
<p>So it would be great if someone could clarify this in terms of angular momentum.</p>
<p>Since the same problem should occur in real helicopters, I guess that there should be some detailed literature about it's physics, but I didn't find any resources about this. So if you have any references for more details or further reading on this problem, it would be also great if you could post it.</p>
| 7,856 |
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<p>A function $f(ax)$ that satisfies
$$
f(ax)=a^\Delta f(x)\,\,\, (\Delta \in R)
$$
is said to be <em>scale invariant</em>. The most general function $f(x)$ that satisfies the previous condition is of the form
$$
f(x)=C x^{\Delta}
$$
If we consider the set of distributions, namely the set of all $f(x)$ such that $\int f(x)dx=1$, is it possible to <a href="http://math.stackexchange.com/questions/140962/scale-invariance-of-power-law-distribution">prove</a> that the only distribution that is invariant under <em>scale transformation</em> is
$$
f(x)=C x^{-1}.
$$
This means that in general I can build a generic distribution as a power law that still satisfies the scale invariance properties $f(ax)=a^\Delta f(x)$, but this generic distribution will not be invariant under scale transformations.
So, what is the physical meaning of the scale invariance property in a generic power law distribution? </p>
<p><strong>First Edit:</strong> For a distribution $f$, if I perform a change of variables, I must preserve the probability, namely
$$
|f(x)dx|=|g(y)dy|,
$$
so that the new distribution $g$ will be given by $f(x)|\frac{dx}{dy}|=g(y)$.
If a distribution is invariant under change of variables, I must have
$$
g(y)=f(y).
$$
Replacing $x$ with $ax$ and using the invariance of the distribution $f(x)$ under this transformation, I have $f(x)=C x^{-1}$. </p>
| 7,857 |
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<p>I was looking through some conformal Ward identity related things when I noticed that this paper (<a href="http://arxiv.org/abs/1212.3788" rel="nofollow">arXiv:1212.3788</a>) writes in their equation (33), a 3-point function between a conserved current and two scalars, complex conjugate of each other and having the same conformal dimensions. The equation reads
\begin{equation}
\partial_\mu\langle j^\mu\mathcal{O}(y_1)\bar{\mathcal{O}}(y_2)\rangle=-iq\langle \mathcal{O}(y_1)\bar{\mathcal{O}}(y_2)\rangle(\delta(x-y_1)-\delta(x-y_2))
\end{equation}
They also comment on the next page that this equation shows, Ward identity guarantees a vanishing three point function for scalar operators of unequal conformal dimensions. How to see that? Also, is this a general outcome of Ward identity that the operators inside (in such cases; may be instead of current, some other conserved tensor) the operators have to have equal conformal dimensions to have a non-vanishing correlator? A reference will be excellent. Thanks in advance!</p>
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<p>I'm currently taking a high school physics class and we're doing friction problems now. I've been doing the ones that use MKS perfectly well, but when the problems start to use the FPS system, I'm always getting a relative error of around 12%. This mostly stems from my not knowing how to deal with FPS using the $ F = ma $ formula. I've tried converting FPS to MKS for it, but it still ends up with a large relative error.</p>
<p>How do I apply the FPS system to a formula like $ F = ma $ that is defined using MKS units and why? For the why, I'm just wondering if I'm to consider pounds when used in a physics problem at my level as a force unit and my mistakes are just the results of not using proper significant figures, or am I supposed to convert to MKS force units? If I'm to convert to MKS units, is using the rate of 4.44822162 (NIST) lbf to one Newton appropriate?</p>
<p>Thank you for reading my question, I hope I'm not skirting on the edge of the rules too much. I tried consulting Google and searching this site for related questions, but I guess the answer is obvious and I can't see it :(</p>
<p><strong>EDIT:</strong> <br /> Here are two problems that display my attempt to convert, then plug in the values, and my attempt to just use the values as-is then plug in.</p>
<p><strong>Problem 1</strong> <br /> "A horizontal force of 6 lbs is applied to a 10 lb block which moves along a horizontal surface. If the coefficient of kinetic friction is 0.4, determine the acceleration of the block. <strong>($a = 6.4 ft/s^2$)</strong>" <br /><br />
<strong>$F_N$</strong> = Normal Force, <strong>$F_k$</strong> = Kinetic Friction Force, <strong>$F_h$</strong> = Forward Force, <strong>$F_w$</strong> = Weight Force <br /><br /><br />
$F_h = 6 lb$ <br />
$F_h = 6 lb * 4.44822162$ (conversion factor to Newtons) <br />
$F_h = 26.69 N$ <br /> <br />
$F_w = 10 lb$ <br />
$F_w = 10 lb * 4.44822162$ (conversion factor to Newtons) <br />
$F_w = 44.48 N$<br /> <br />
$F_w = mg$ <br />
$44.48 N = m * 9.8 m/s^2$ <br />
$m = 4.54 kg$ <br /> <br />
$μ_k = \frac{F_k}{F_N}$ <br /><br />
$0.4 = \frac{F_k}{44.48 N}$ <br /> <br />
$F_k = 17.79N$<br /><br /><br />
$-F_k + F_h = ma$ <br />
$-17.79 N + 26.69 N = 4.54 kg * a$<br />
$a = 4.54 m/s^2$<br /><br />
Need to convert now...<br /><br />
<strong>Final:</strong> $a = 14.89 ft/s^2$
<br /><br /><br /></p>
<p>There's another problem where I just directly put in the FPS values without converting, but if this method is right, there's no point in posting it. If this method is wrong, I'll write it out here.</p>
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<p>I found a really cool sounding order-of-magnitude modeling question but am a bit at a loss on how to approach this:</p>
<blockquote>
<p>The Milky Way contains 100 billion stars and has a radius of 250,000 light years. Suppose that humans set a long term goal to colonize every inhabitable star system in the Milky Way and that 10% of star systems are suitable. This colonization might be accomplished by sending out a ship of settlers to a nearby solar system, establishing a colony, and then waiting long enough - 500 years? - for the colony to develop enough that it can send its own settler ships. Do an order of magnitude estimate of how long it would take to colonize the galaxy, explaining any assumptions you make.</p>
</blockquote>
<p>My first assumption is to put our solar system into the center of the galaxy with the argument that this makes calculations more symmetric, assuming that it will change the results by a factor of order 1. (I'm shorter away from the "ends" of the solar system but I benefit more from the hopefully exponential growth). </p>
<p>Now the question is how to model things further. A crude estimate of the average distance between inhabitable star systems tells me that they are about 5 light years apart from each other. I guess since 5 is much smaller than 250,000 I could move to some sort of continuum description of colonization progress, but then I'm not sure on how to proceed from there. </p>
<p>Another important question would be the speed of my colony ships. I guess it should be at least on the order of the escape velocity of the earth-sun-system, so again using crude estimates I'll get something like $5\cdot 10^4\, \mathrm{m}/\mathrm{s}$ which means that a ship needs about 5000 years for a journey.</p>
<p>If the time per journey was significantly larger than the time to prepare the next ship (500 years), I guess I could ignore the 500 year delay and assume instant readiness, but that doesn't seem to be appropriate here. And if I don't do that, then the process seems a bit tricky: During the 5000 year journey of the first ship, I can produce 10 additional ships and send them to a journey, i.e., growth of colonized space is not just proportional to the number of fringe systems but depends in a non-obvious complicated way on the number of colonized planets AND when they were colonized. (Similar to "broadcasting in networks" scenarious?)</p>
<p>Maybe someone immediately recognizes a commonly used underlying model that is appropriate here? </p>
<p>PS: I'm open for suggestions re tags.</p>
| 7,860 |
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] |
<p>I have been reading the paper <a href="http://arxiv.org/abs/0810.1545" rel="nofollow">The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity</a>. In it they state, </p>
<blockquote>
<p>"An instability, if it occurs, must necessarily break a symmetry ...
that the background solution preserves."</p>
</blockquote>
<p>There is also discussion about the non-linear terms contributing to instabilities. Although I have some thoughts about how this should be interpreted. I wanted to see if there is some standard discussion relating to non linear instabilities in this context.</p>
<p>Addendum:</p>
<p>In order to provide more specificity, in standard discussion for understanding string theory, weak field approximations for gravity involve linearized fluctuations around a Minkowski metric: $$g_{\mu\nu}(x) = \eta_{\mu\nu}+h_{\mu\nu}(x)$$</p>
<p>In stronger fields, the fluctuations become highly non-linear. The non-linearity is thus understood as being associated with strong gravitational fields.</p>
<p>The Navier-Stokes equations can be understood as a composition of two types of equations, the <a href="http://en.wikipedia.org/wiki/Heat_equation" rel="nofollow">Heat equations</a> and the <a href="http://en.wikipedia.org/wiki/Euler_equations_%28fluid_dynamics%29" rel="nofollow">Euler equations</a>. In a crude sense, the unforced heat equations generally govern the diffusion of energy throughout an object, and the Euler equations govern the conservation of mass. </p>
<p>The incompressible Navier-Stokes equations in cartesian coordinates are:</p>
<p>$\partial_tu+u\partial_xu+v\partial_yu+w\partial_zu = -\partial_xp + \nu(\partial_{xx}u+\partial_{yy}u+\partial_{zz}u)$</p>
<p>$\partial_tv+u\partial_xv+v\partial_yv+w\partial_zv = -\partial_yp + \nu(\partial_{xx}v+\partial_{yy}v+\partial_{zz}v)$</p>
<p>$\partial_tw+u\partial_xw+v\partial_yw+w\partial_zw = -\partial_zp + \nu(\partial_{xx}w+\partial_{yy}w+\partial_{zz}w)$</p>
<p>$\partial_xu+\partial_yv+\partial_zw=0$ </p>
<p>Where one can identify the "heat equation terms" as (noting that in complex form these are the Schrodinger equations):</p>
<p>$\partial_tu = \nu(\partial_{xx}u+\partial_{yy}u+\partial_{zz}u)$</p>
<p>$\partial_tv = \nu(\partial_{xx}v+\partial_{yy}v+\partial_{zz}v)$</p>
<p>$\partial_tw = \nu(\partial_{xx}w+\partial_{yy}w+\partial_{zz}w)$</p>
<p>And one can identify the "Eulerian terms" as:</p>
<p>$u\partial_xu+v\partial_yu+w\partial_zu = -\partial_xp$</p>
<p>$u\partial_xv+v\partial_yv+w\partial_zv = -\partial_yp$</p>
<p>$u\partial_xw+v\partial_yw+w\partial_zw = -\partial_zp$</p>
<p>In the solution <a href="http://physics.stackexchange.com/q/52523/2451">I provided in a previous question</a>, the interesting characteristics is that solution satisfies both sets of equations (sans the pressure term for the heat equation). The pressure identified is also negative, and the solution is unbounded. This doesn't seem to be the type of space identified with the AdS in the above referenced paper or the <a href="http://arxiv.org/abs/0712.2456" rel="nofollow">related paper</a> discussing the gravity-fluid dynamics relationship. There is one other thing that is of interest, and that is the vanishing of the non-linear and pressure terms with the given solution (related to the apparent separability of terms).</p>
<p>If I relate the Eulerian terms with gravity, then the solution in my mind suggest that there are possibly some set of solutions that can be shared in common with both the Heat equation terms and the Eulerian terms (and the number of these are unknown). </p>
<p>It seems that those solutions would have some sort of privileged status, or perhaps they are completely trivial and not interesting, but I am thinking that answer is context dependent. However, I am somewhat encouraged that those solutions are not trivial, since <a href="http://en.wikipedia.org/wiki/Euler_equations_%28fluid_dynamics%29#Shock_waves" rel="nofollow">bifurcation is not generally considered a good thing in equations describing physical phenomenon</a>.</p>
<p>So getting back to the original question, in the above referenced paper it is implied that instability at high Reynolds numbers ($\frac{1}{\nu} >> 1$) is associated with symmetry breaking, which is a necessary physical phenomenon that is not well understood. So I am trying to tie these thoughts together and was wondering what misconceptions I might have on this or how these are tied together in academic settings.</p>
| 7,861 |
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<p>Iron is the most stable single element (I don't know about combinations of elements, but oh well, not applicable to question). Stars go through cycles where they change the materials of their cores, culminating in iron for the longest lasting and largest stars. The cores of stars naturally have tremedous forces. My question: Has humanity managed to combine to iron nuclei together? and if yes why can we do it but not stars? Alternatively, where would the progression past iron nuclei go, since no element is more stable?</p>
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<p>I'm having trouble with the following problem.</p>
<p><img src="http://i.stack.imgur.com/ed8nE.png" alt="Problem"></p>
<p><em>What I've done so far:</em></p>
<p>x-y is the usual coordinate system.</p>
<p>$a=\frac{F}{m}=\frac{800}{60}$ and the y component of this is $a_y=a\sin{60^\circ}$.</p>
<p>To figure out the force acting on the rod and contributing to the momentum, I take the sum of all forces in the y direction. Note that from now on, I am thinking about the rod as being separated from the rest of the body. The weld is represented by forces ($O_y, O_x$), and the rod is moving upwards with the acceleration calculated above.</p>
<p>$\Sigma F_y= O_y-20g = 20a_y\implies O_y=20a_y+20g$</p>
<p>Now, the momentum equation about the center of mass of the rod should be</p>
<p>$\Sigma M_G=0.7O_y+\bar{I}\omega_G=\bar{I}\alpha=0$</p>
<p>Because the rod is not rotating. Is this equation right? (Why not?)</p>
<p>If it is right, it implies that $\bar{I}\omega_G=-0.7O_y$. Where do I go from here? The things I've tried from here on don't yield the right answer, and I do not have much confidence in those strategies.</p>
<p>The correct answer, if anyone cares, is 196 Nm.</p>
| 7,863 |
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<p>Let's consider a spring. I am a strong man(well, lets assume) and I am pulling the spring. the work I do is being stored in the spring in the form of its elastic potential energy. Then suddenly, elastic limit is crossed and the spring reaches to the yielding region. then, at that very moment, i become exhausted and release the spring. then I realise that the spring doesn't budge. the energy i stored is lost. its gone. Nowhere to be found. Mind helping me find it?</p>
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] |
<p>In Mudelung's book, Introduction to Solid-State Theory, I am confused by the following statement.</p>
<blockquote>
<p>For many applications a further simplification is helpful. The concept of the hole presents us with the confusing situationthat a hole in the state $\mathbf{k},\sigma$ has a momentum $-ħ\mathbf{k}$, while an electron has a momentum $ħ\mathbf{k}$. This asymmetry can be avoided by defining new <em>quasi-particles</em> which have momentum $ħ\mathbf{k}$ inside and outside the Fermi sphere. When we note that the creation of a particle with $+ħ\mathbf{k}$ is achieved by the operator $c_{\mathbf{k}\sigma}^+$ when outside the Fermi sphere and by $c_{-\mathbf{k},-\sigma}$ when inside the Fermi sphere, it is only a small step to define the following operators:
$$
\begin{align}
\alpha_{\mathbf{k}\sigma}^+=u_\mathbf{k}c_{\mathbf{k}\sigma}^++v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma}
\\
\alpha_{\mathbf{k}\sigma}=u_\mathbf{k}c_{\mathbf{k}\sigma}+v_{-\mathbf{k}}c_{-\mathbf{k},-\sigma}^+
\end{align}
$$
with
$$
\begin{align}
u_\mathbf{k}=1,\; v_\mathbf{k}=0\quad\text{ for }\quad k>k_\text{F},
\\
u_\mathbf{k}=0,\; v_\mathbf{k}=1\quad\text{ for }\quad k<k_\text{F}.
\end{align}
$$</p>
</blockquote>
<p>He showed the motivation of introducing Bogoliubov tansformation. But I don't understand his statement. "Creation of a particle with $+\mathbf k$ is acvieved by the operator $c^{+}_{\mathbf k,\sigma}$ when outside the Fermi shere and by $c_{-\mathbf k,-\sigma}$ when inside the Fermi sphere".</p>
<p>$c^{+}_{\mathbf k,\sigma}$ means create a particle with $\mathbf k$. So, why must we distinguish ouside and inside the Fermi sphere? And why does annihilation operator $c_{-\mathbf k,-\sigma}$ inside the Fermi sphere have the same effect as creation operator outside the Fermi sphere?</p>
| 7,865 |
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<p>The <a href="http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect" rel="nofollow">Aharonov–Bohm effect</a> shows the vector potential $\mathbf{A}$ is more fundamental than magnetic flux density $\mathbf{B}$. However, vector potential is introduced by
$$ \nabla \times \mathbf{A} = \mathbf{B} $$ or $$ \mathrm{d} \mathbf{A} = \mathbf{B} $$
which requires the Poincaré's lemma to ensure the existence of $\mathbf{A}$. If the manifold is not contractible, there is no guarantee that $\mathbf{A}$ exists. On a non-contractible manifold, what would happen for the Aharonov–Bohm effect?</p>
| 7,866 |
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-0.008267965167760849,
-0.042952265590429306,
0.02175750769674778,
-0.08840577304363251,
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-0.05823284387588501,
0.016431866213679314,
-0.019319651648402214,
0.024478398263454437
] |
<p>How I determine the center of mass of a semicircle using the definition of center of mass? I only know solve this using the Pappus theorem. Consider that the semicircle is centered on the origin and a homogeneous mass distribution.</p>
| 7,867 |
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<p>I'm supposed to calculate the inversion pressure $p_i$ of a <a href="https://en.wikipedia.org/wiki/Van_der_Waals_gas" rel="nofollow">Van der Waals gas</a>. The state equation of the Van der Waals gas is: $$(p + \frac{a}{V^2})(V-b) = RT.$$ To get a hold of the inversion temperature, I differentiate this equation with respect to $T$, while keeping $p$ constant. This gives me:
$$\frac{(\frac{\partial V}{\partial T})_p \cdot (2ab - aV + pV^3)}{V^3} = R,$$ which with the condition for the inversion curve $T(\frac{\partial V}{\partial T})_p = V$ I get:
$$\frac{1}{T_i}(\frac{2ab}{V^2} - \frac{a}{V} + pV) = R,$$ which gives me an easy to solve equation for $T_i$. But to get $p_i$, I'd have to solve the equation for $V$ and put it back into the state equation. But solving this equation for $V$ gets really messy. That's why I'm thinking I went wrong somewhere.</p>
<p>Can you guys help me out here?</p>
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<p>This is sort of a silly question because I'm a total beginner, and I debated whether it was better to ask here or on Math.SE. I decided on here because it's about how physicists use terminology, even though the terminology is intrinsically mathematical and there's not actually any physical content in the question.</p>
<p>It's a well-known fact in theoretical physics that the conformal group of Minkowski space is $\operatorname{SO}(4,2)$ (or sometimes its double cover $\operatorname{SU}(2,2)$ is chosen instead). This is infinitesimally generated by the standard Poincare generators $M_{\mu \nu}$ and $P_\mu$ together with scale transformation $D$ and special conformal generators $K_\mu$. I've also seen conformal groups discussed for other manifolds (the case which lead me to this question is Witten's 2003 paper <a href="http://arxiv.org/abs/hep-th/0312171">Perturbative Gauge Theory As A String Theory In Twistor Space</a>, which includes the conformal group of $\mathbb{R}^4$ with all possible signatures.)</p>
<p>However, the above doesn't seem to agree with the definition of the term "conformal group" which I know from the minimal amount of conformal geometry I know. Specifically, the conformal group of a pseudo-Riemannian manifold $M$ is the group of conformal self-maps on $M$, that is, self-diffeomorphisms for which the pullback metric is conformally equivalent to the original metric. (Two metrics $g$, $h$ are conformally equivalent if there is a positive real-valued function $u$ such that at each point $x \in M$ we have $g_{x, \mu \nu}=u(x)h_{x, \mu \nu}$.)</p>
<p>The trouble is that the conformal group of Minkowski space isn't $\operatorname{SO}(4,2)$ with the above definition. The special conformal transformations, given by the expression (for some vector $b_\mu$) $$ x' ^\mu = \frac{x^\mu - x^2 b^\mu}{1-2b \cdot x +b^2 x^2}$$ aren't self-maps of Minkowski space at all. The denominator can become 0 (while the numerator remains finite), and at that particular $x^\mu$ the transformation given by $b^\mu$ is undefined. For an example, take $x=(1,0,0,0)$ and $b=(0,1,0,0)$. So in fact the special conformal transformations are only maps from a subset of Minkowski space back to the full Minkowski space, and shouldn't be included as part of the conformal group if we take the above definition. It's difficult to tell from this formula, but from the fact that $$ \frac{x'^\mu}{x'^2} = \frac{x^\mu}{x^2} - b^\mu$$
we can tell that the inverse transformation is just given by $-b^\mu$, another special conformal transformation, which means these maps aren't surjective either.</p>
<p>These problems stem from the fact that the conformal group is typically derived in physics by considering the algebra of vector fields (which we think of as infinitesimal coordinate transformations) which satisfy the conformal Killing equation. There's in general no reason to expect that the corresponding finite transformation to be well-behaved away from the origin.</p>
<p><strong>How, exactly, are conformal groups defined in physics for arbitrary (in particular noncompact) pseudo-Riemannnian manifolds so that we get the expected result for Minkowski space?</strong></p>
<p>To be perfectly clear, I don't want answers of the form "find the Lie algebra, then exponentiate everything". I know that this works, but I want a definition for the conformal group as a group, without referencing algebras. In my opinion, there ought to be a way to define it this way, and it seems a bit important to me since <em>a priori</em> I don't see any reason why the group of conformal self-maps of a manifold needs to be a (possibly infinite-dimensional) Lie group at all. There are probably some high-power theorems (perhaps in PDE) which would imply this trivially, but naturally we need to know an actual group structure to be able to apply them. In the case of the conformal group as I defined it above (so excluding special conformal transformations) I believe I know a way to prove this for the "mathematician" definition, but when we start throwing in maps that are not everywhere defined the approach I came up with fails. Hence, even if this is how most physicists would define it, I'm looking for an alternate global definition to get some better geometric intuition which produces the correct answers.</p>
<hr>
<p>Some thoughts on this (feel free to ignore, especially if I'm on the wrong track):</p>
<p>It seems to me possible to remedy this in a fairly trivial way. We can consider all injective conformal maps from dense open sets of Minkowski space to (possibly distinct) dense open sets of Minkowski space. We also need to define compositions to be restricted to the subset of Minkowski space on which they are defined. Furthermore, we impose an equivalence relation that two maps which restrict to the same map on the intersection of their domains are equal. Using these definitions, we can at least get a group which contains all of the conformal transformations it should. It also contains some other less desirable sorts of transformations (for instance, take two rotations and patch them together on the interior and exterior respectively of $\mathbb{R} \times S^2$; incidentally, this is essentially why the proof I have could potentially fail for the "physicist's" conformal groups), but by imposing some conditions on the behavior of the maps in a neighborhood of the boundary of the region on which they are defined it should be possible to get back the get back the ordinary conformal group. (I have not checked this, nor do I know exactly what conditions should be applied, though I could figure it out in the case of Minkowski space based on an alternate model below for the conformal group.)</p>
<p>That approach, while easy to justify based on our above assumptions, is not very intuitive. It requires a lot of technical work to get everything to work out properly. We're implicitly using the fact that finite intersections of dense open sets are dense and open. The conditions required as the functions approach the boundary of their domains to get everything to work are also not obvious. More importantly, the whole thing seems to be suggesting an alternate approach which is much more geometrical; that we should view Minkowski space as contained within some slightly larger space on which the conformal group acts more nicely, where we view the complement of Minkowski space as points at $\infty$.</p>
<p>Specifically (and at this point I'm just making conjectures because they seem helpful) for any psuedo-Riemannian manifold $M$, there ought to be some unique space $\bar M$ which has the properties that $M \subset \bar M$ is dense (this is based on the above observations, and might not be necessary or desirable in the general case; however, it seems reasonable), the restriction of the metric on $\bar M$ to $M$ produces a conformally equivalent metric to the standard metric on $M$, and such that all the conformal self-maps of $M$ which "should exist" can be legitimately interpreted as conformal self-maps of $\bar M$. I don't know that this actually implies $\bar M$ is compact, but certainly if $M$ is compact then the sorts of problems we run into for Minkowski space can't happen. We can then just define the conformal group of an arbitrary manifold $M$ to be the conformal group, in the "mathematics definition", of $\bar M$. A construction of $\bar M$ would thus be a perfectly reasonable answer to this question, though at this point I'm still not sure that any space $\bar M$ need exist in general. I briefly tried to do this myself, but I was not able to come up with anything general enough to be satisfactory, nor was I able to prove that the few cases I know of this construction are unique.</p>
<p>I do know that this approach works for Minkowski space in some sense. Specifically, we can complete Minkowski space in this sense by including it into the projective light cone (i.e. the space of all null lines through the origin) in 6-dimensional Euclidean space with $(4,2)$ signature. The particular embedding is well-known and not so important for my purposes. However, this doesn't generalize particularly well when presented in this way. The reason why it works, at least intuitively for me, is that $\operatorname{SO}(4,2)$ naturally acts on this space, which to me seems like a big coincidence that can't be expected to continue to work if we consider arbitrary spaces. I considered using Penrose diagrams to try to do the equivalent construction for all of the cases I care about, but these only work for asymptotically flat spaces as far as I know and that misses some cases of interest for me (e.g. AdS), plus a general construction would be far more satisfying.</p>
<p>I also considered the possibility that physicists may be using the term inconsistently outside the context of Minkowski space. That is, there may be a number of constructions of "conformal groups", all of which give the same $\operatorname{SO}(4,2)$ on Minkowski space. That would actually be more interesting for my purposes if it's true.</p>
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] |
<p>The <a href="http://en.wikipedia.org/wiki/Cross_section_%28physics%29" rel="nofollow">cross section</a> is a tiny area, measured in barns ($10^{-28} \mathrm{m^2}$), which relates to the probability of a reaction.</p>
<p>But I have trouble interpreting that number.</p>
<p>For example how can I calculate, when two deuterons collide what's the chance that they fuse? I read in forums that they much more likely scatter.</p>
| 7,870 |
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<p>Imagine some sort of textile (as in very light, not metallic) that is tightly rolled up, can be easily unrolled by applying some force (i.e pulling it) but when released will eventually return to it's rolled up state, over and over again.</p>
<p>Does something like this exist?</p>
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<p>I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving math, which would be all of them.</p>
<p>So lets say we drop an object from a height, $R+r$, it falls toward earth. This height, $R+r$, is far enough away that the $g$ it experiences is a fraction of $g$ at sea-level. Let's just say that air resistance is negligible, and it wouldn't be that complicated to just integrate from $0$ velocity to the terminal velocity piece wise and deal with the rest of it later.</p>
<p>So the key here is that acceleration is changing with time. I thought I could simplify this by saying it changes with distance, and it has nothing to do with time, but this didn't really help, my guess is that maybe time is important (doh).</p>
<p>I tried integrating acceleration with time, and ended up nowhere. I tried integrating $a=GM/R^2$ with respect to $R$ from $R+r$ to $R$ and ended up with a negative function.</p>
<p>I saw somewhere someone tried to expand with taylor series, they even have something similar on hyperphysics, but I can't figure out how to obtain the polynomials that precede the variables.</p>
<p><a href="http://hyperphysics.phy-astr.gsu.edu/hbase/images/avari.gif" rel="nofollow">http://hyperphysics.phy-astr.gsu.edu/hbase/images/avari.gif</a></p>
<p>This is the hyperphysics site where they use polynomials to find the distance.
<a href="http://hyperphysics.phy-astr.gsu.edu/hbase/avari.html#c1" rel="nofollow">http://hyperphysics.phy-astr.gsu.edu/hbase/avari.html#c1</a></p>
<p>Maybe I can't solve this because I haven't taken a course in differential equations yet.
What I want to know is how to calculate the distance at any time.</p>
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<p>In paramagnetic-to-ferromagnetic phase transitions, the symmetry spontaneously breaks down from SO(3) to the subgroup SO(2) below $T_\text{crit}$. This implies that there should be two Goldstone modes and not one because SO(3) has three generators and SO(2) has one. How do we distinguish between these two excitations physically?</p>
| 7,872 |
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<p>When a photon is emitted from a far away source and then measured by an observer, there is a loss of energy or redshift which takes place. Why does this happen?</p>
<p>I have read <a href="http://physics.stackexchange.com/questions/4821/redshifted-photon-energy">this</a> similar post, however the confusion I'm having is due to the invariance of photons, which travel at the speed of light, to non-inertial reference frames. If they are invariant then why would there be a perceived energy loss as it switches from one reference frame to the other?</p>
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<p>(Please bear with me.)
If the universe included multi-universes, and black holes transferred matter between them, AND yet there was a large, but FINITE amount of matter and space, How would that max-limit change our understanding/perception or theories of the universe? Would it make any difference or change current concepts? </p>
| 7,874 |
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<p>An electron in a cathode-ray tube is traveling horizontally at 2.10×10^9 cm/s when deflection plates give it an upward acceleration of 5.30×10^17 cm/s^2 .</p>
<p>B.) What is its vertical displacement during this time?</p>
<p>My work <a href="http://i.imgur.com/AqKlTuX.jpg" rel="nofollow">http://i.imgur.com/AqKlTuX.jpg</a></p>
<p>I am asking this because the answer my teacher gave is 2.3cm</p>
<p>Thanks in advance</p>
| 7,875 |
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