title: Show that the profile of the hill is a cycloid

text: I'm struggling with this problem: From George Simmons_ Differential Equations ![Question](http://i.stack.imgur.com/tqNV6.png) I've tried different things and I showed that the $x$ component and the $y$ component of the shadow should move with constant speed, but nothing really other than that. EDIT: I solved the problem, but I'm not sure if my argument is correct I start by defining $H$ as the height of the dome. By analyzing the motion of the rock, we know from conservation of energy that the initial velocity $v_o=\sqrt{2gH}$ in order to reach the top of the dome with $v_f=0$. We also can find the time it takes for the rock to reach the top. $$H=v_ot_T-\frac{1}{2}gt_T^2$$ $$0=v_f=v_o-gt_T \Longrightarrow t_T=\frac{v_o}{g}=\frac{\sqrt{2gH}}{g} $$ So we know that the time for the rock to reach the top is: $ t_T=\sqrt{\frac{2H}{g}}$ Now I define $X(t)$ and $Y(t)$ to be the parametric equations of the shadow. As stated in the problem and mentioned in the Math-stack-exchange counterpart, The Following Equation Holds $$\dot{S}^2={\dot{X}}^2+{\dot{Y}}^2=C^2 \tag{1}$$ Where $\dot{S}$ is the speed and $C$ is constant Now we must show that if the shadow moves as stated by parametric equation of the Cycloid a t constant speed the time it takes to reach the top is $t_T=\sqrt{\frac{2H}{g}}$ Now the parametric equation of a cycloid of Radius $H/2$ is : $$X(u)=H(u(t)-\sin(u(t)))$$ $$Y(u)=H(1-\cos(u(t)))$$ $$\dot{X}(u)=\dot{u}(t)H(1-\cos(u(t)))$$ $$\dot{Y}(u)=\dot{u}(t)H(\sin(u(t)))$$ We must note that $\dot{X}(t)=Y(t)$, which is a rather beautiful property of the Cycloid. As the we assume the shadow to be moving with a cycloid trajectory we can combine the previews equalities with (1). $$\dot{S}^2=\Big(\dot{u}(t)H(1-\cos(u(t))\Big)^2+\Big(\dot{u}(t)H\sin(u(t))\Big)^2$$ $$=(\dot{u}(t))^2 H^2\Big(1-2\cos(u(t))+\cos^2(u(t))+\sin^2(u(t))\Big) $$ $$=(\dot{u}(t))^2H^2\Big(2-2\cos(u(t))\Big)=(\dot{u}(t))^2 2H \cdot H\Big(1-1\cos(u(t))\Big)$$ $$=(\dot{u}(t))^2 2H(Y(t))=C^2 \tag{2}$$ As the sun is facing the rock we can assume that the motion of the $y$ component of the rock and the shadow is the same one, but whats clearly not the same is the $x$ component. Now we can say $$y(t)=Y(t)$$ So (2) now can be written as: $$C^2=(\dot{u}(t))^2 2H(y(t))=(\dot{u}(t))^2 2H(v_ot-\frac{1}{2}gt^2) $$ Now we try to find $\dot{u}$ $$(\dot{u}(t))^2=\frac{C^2}{2H}\frac{1}{v_ot-\frac{1}{2}gt^2}$$ $$\dot{u}(t)=C\sqrt{\frac{1}{2H}\frac{1}{v_ot-\frac{1}{2}gt^2}}$$ If I integrate $\dot{u(t)}$: $$u(t)=C \sqrt{\frac{1}{2H}} \int \frac{1}{v_ot-\frac{1}{2}gt^2} dt $$ by using Wolfram alpha -Integral Solution Asuming t is non negative I get: $$u(t)= C \sqrt{\frac{1}{2H}} \frac{2 \tan^{-1}(\frac{\sqrt{gt/2 }}{\sqrt{-gt/2 +v}})}{\sqrt{g/2}}+c$$ So I was able to find a parametrization for $u(t)$ such as the initial conditions were accomplished. Its easy to notice that $C=\sqrt{2gH}$, when t=0 in (1). Also now: $$u(t)= 2\sqrt{2} \tan^{-1}(\frac{\sqrt{gt/2 }}{\sqrt{-gt/2 +v}}) +c \tag{4}$$


title: Physical/geometrical interpretaion of the determinant of a matrix

text: Consider a matrix transformation $\mathbf{T}$ that acts on a vector $\mathbf{x}$: $$ \mathbf{x}' = \mathbf{T}\mathbf{x} $$. Now, I know that one-dimensional linear transformations expand the **length** by a factor $|det(\mathbf{T})|$, two-dimensional linear transformations expand the **area** by a factor $|det(\mathbf{T})|$, and three-dimensional linear transformations expand the **volume** by a factor $|det(\mathbf{T})|$. How can I see this mathematically though? I was thinking of calculating the norm $||\mathbf{x}'|| = ||\mathbf{T}||\cdot||\mathbf{x}||$ but don't know how to deal with $||\mathbf{T}||$ The physics of this question is: do all Lorentz transformations have determinant equal to 1? Because they preserve the space-time interval? EDIT: What I was asking was more about: how can I see that the determinant of a matrix carries information about the area/volume change of the system it is acting on? Mathematically, how can I show that the area spanned by two vectors is unchanged IFF the determinant of the transformation matrix is 1?


title: Regarding the free electrons on the conductor

text: 1. In a metal, why don’t the free electrons fall to the bottom of the metal due to gravity?    2. Also, charges in a conductor are supposed to reside on the surface so why don’t the free electrons all go to the surface? Regarding the first question, my hunch is that it is due to the electric force which is greater than the gravitational force and electric force act along the length of the conductor. I don't have any intuition for the second.


title: Why does earth have a minimum orbital period?

text: Using the equation for Kepler's constant for Earth $K$:   $$K = \frac{GM}{4\pi^2}$$ $$K = 1.01 \times 10^{13} \;\rm m^3/s^2$$ Finding time ($r = 6.4 \times 10^{6} \;\rm m$ (radius of Earth)): $$K = \frac{r^3}{T^2}$$ $$T = \sqrt{\frac{r^3}{K}}$$ $$T = 5095 \;\rm s = 85\; minutes$$ So 85 minutes is approximately the minimum time for a satellite to orbit the Earth. Why can't it just go faster to take less time? Would that mean it would fall out of orbit?


title: Why is there a controversy on whether mass increases with speed?

text: Some people say that mass increases with speed, some people say that the mass of an object is independent of its speed. I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be 'true' is any sense of the word. Either mass increases or it doesn't, right? Can't we just measure it, and find out which 'interpretation' is right? e.g.: (in some sophisticated way) heat up some particles in a box and measure their weight? **\------------------------------** **UPDATE** Right, so I've got two identical containers, each with identical amounts of water, each on identical weighing scales, and each in the same g field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and g is constant, does this mean that the m in w=mg has got bigger?


title: Phase coherence and interference effects in Anderson localization

text: Anderson localization results from wave interference of the between multiple- scattering paths from random impurities, yielding wavefunctions with exponentially decaying tails and absence of diffusion. What is the multiple- scattering paths and how to understand the interference between them?


title: Quantum field theory: field operators in terms of creation/annihilation operators

text: I am learning Quantum Field Theory and there is a step in my notes that I do not really understand. It starts with the classical definitions of position $q$ and momentum $p$: $$ q = \frac{1}{\sqrt{2\omega}}(a+a^{\dagger}) $$ and $$ p = -i\sqrt{\frac{\omega}{2}}(a-a^{\dagger}). $$ $a$ and $a^{\dagger}$ being the annihilation and creation operators. Then, it defines the field operators $\phi(\mathbf{x})$ and $\pi(\mathbf{x})$, equivalent to $q$ and $p$, in the following way: $$ \phi(\mathbf{x}) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{\mathbf{p}}}}[a_{\mathbf{p}}e^{i\mathbf{p}\cdot{\mathbf{x}}} + a_{\mathbf{p}}^{\dagger}e^{-i\mathbf{p}\cdot{\mathbf{x}}}]$$ and $$ \pi(\mathbf{x}) = \int \frac{d^3p}{(2\pi)^3}(-i)\sqrt{\frac{\omega_{\mathbf{p}}}{2}}[a_{\mathbf{p}}e^{i\mathbf{p}\cdot{\mathbf{x}}} - a_{\mathbf{p}}^{\dagger}e^{-i\mathbf{p}\cdot{\mathbf{x}}}].$$ Is there an obvious relation between the two expressions? What mathematical and physical assumptions have been made?


title: Two Loop ultraviolet divergence of $\phi^4$ theory

text: During the renormalization procedure of a massive $\phi^4$ theory at two loop level, one finds that the quantity $K_{\Lambda}(k^2,m^2)-K_{\Lambda}(0,m^2)$ that appears in the bare $\Gamma^{(2)}$ function is divergent, where $$ K(k^2,m^2)=\int_{\Lambda} \frac{d^4 q}{(2 \pi)^4}\frac{d^4 q'}{(2 \pi)^4} \frac{1}{(q^2+m^2)(q'^2+m^2) [(k-q-q')^2+m^2]}. $$ I computated explicitly $K_{\Lambda}(k^2,m^2)-K_{\Lambda}(0,m^2)$ and found $$ K_{\Lambda}(k^2,m^2)-K_{\Lambda}(0,m^2) = \int_{\Lambda} \frac{d^4 q}{(2 \pi)^4}\frac{d^4 q'}{(2 \pi)^4} \frac{-k^2 +2k(q+q')}{(q^2+m^2)(q'^2+m^2) [(k-(q+q'))^2+m^2][(q+q')^2+m^2]}, $$ but I can't see why this integral is UV divergent; is there an easy way to prove it?


title: What is the difference between complex field and two scalar fields?

text: Consider a complex scalar field $\phi$ with the Lagrangian:   $L = \partial_\mu\phi^\dagger\partial^\mu\phi - m^2 \phi^\dagger\phi$ Consider also two real scalar fields $\phi_1$ and $\phi_2$ with the Lagrangian:   $L = \frac12\partial_\mu\phi_1\partial^\mu\phi_1 - \frac12m^2 \phi_1^2 +\frac12\partial_\mu\phi_2\partial^\mu\phi_2 - \frac12m^2 \phi_2^2$ Are these two systems essentially the same? If not -- what is the difference?


title: Free electrons and energy states

text: Ok, background - studying for the physics part of my radiology exams, and came across a question that went something like this An electron fired through a tungsten target loses energy by: a) Bremsstrauhlung b) collisions with bound target electrons c) giving off characteristic radiation d) all of the above Now, by the process of elimination d must be right, because both a and b are My question regards c - characteristic radiation My understanding of this process is the excitation of the electron to a higher energy state, usually a more peripheral valence shell, which then releases that extra energy when it drops back (or another drops in to fill the hole) So, what I don't get is how a free electron (like one fired from the filament to the target) has energy states. By definition, it is unbound, and therefore there are no binding energies to create quantised energy states. Is there a property I am overlooking, is the question wrong, or is it a fudge? I apologise if this has been asked. I looked at the suggested answers, and did a google search of stack exchange, but my phone can't use the search bar here directly, so I could have missed something


title: What is the difference between translation and rotation?

text: What is the difference between translation and rotation ? If this were a mathematics site, the question would be at best naive. But this is physics site, and the question must be interpreted as a question about physical theory, that is about hypotheses that can be tested, subjected to experiments and possibly disproved by experiments. **Restatement of the question**   _After about 3 days, 5 answers, 160 views and some more comments,   taking these contributions into account (hence the length)._ _First I thank all users who commented or tried to answer my ill stated question, and I apologize for not doing better. Hopefully, they did help me improve my understanding and the statement of my question. You can look at the awkwardprior formulations of the question which better explain the existing answers._ I am trying to understand whether and how translation differs from rotation, and whether or why it is a necessary physical concept or possibly only a mathematical convenience. There are two sides to the issue I am raising, one regarding space(time) symmetries, and one regarding motion. From what (very) little I understand of Noether's theorem, these cannot be unrelated as laws governing motion have to conserve charges that are derived from the space(time) symmetries. This may have been one source of my initial confusion. One of my point is that if there are situations when rotations is not distinguishable from translation: infinitesimal angles of rotation as suggested by user _namehere_. Then relevant phenomena can be analyzed either as rotations or as translation, with proper accomodation, particularly to account for the existence of a radius when rotation is concerned, which changes dimensionality and the mathematical apparatus. Of course this requires "care" when considering phenomena involving the center of rotation or phenomena indefinitely distant. One example is torque, moment of inertia, angular momentum, vs force, mass and momentum. The possible undistinguishability of translation and rotation would seem to indicate that they are really two guises for the same set of phenomena. They relate to two distinct symmetries, but is that enough to assert that they are fundamentally different ? This is precisely what is bothering me in the last comments of user _namehere_ attached to his answer and motivated my question initially. Actually, it was someone telling me that " _angular momentum is not linear momentum going round in a circle_ " that started me on this issue, as I was not convinced. It may have other manifestations, but it is also that. I am aware that the mathematical expressions, including dimensionality, are significantly different for translational and rotational concepts, and somewhat more complex for rotational concepts, as remarked by user _namehere_. But rotational concepts have to account for the existence of a center and a radius which may be directly involved in the phenomena being considered: this is typically the case for the moment of inertia which has to account for a body rotating on itself. If we consider a translational phenomenon about force, mass and momentum, occuring in a plane. We can analyse it indifferently as translational, or as rotational with respect to a center of rotation sufficiently distant on a line orthogonal to the plane, so that all radiuses may be considered vectorially equal up to whatever precision you wish. Since the radiuses may be considered equal, they can be factored out of the rotational formulae to get the translational ones. That is, the rotational mathematics can be approximated arbitrarily well by its translational counterpart. This should accredit the hypothesis that it is the same phenomena being accounted for in both cases. I am not trying to assert that rotating frames should be inertial frames. I am only asking to what extent physicists can see a difference, and, possibly (see below), whether inertial frames actually exist. **When do rotational phenomena differ in substance from translational ones ? Is there some essential phenomenon that is explained by one and not by the other ? And conversely ?** Mathematics is only a scaffolding for understanding problems. They are not understanding by themselves. Mathematical differences in expression do not necessarily imply a difference in physical essence. Then one could also question whether translation is (or has to be) a meaningful physical concept. This can be taken from the point of view of space(time) symmetries or from the point of view of motion. Why should it be needed as a physical concept, or can it (should it) be simply viewed as a mathematical convenience ? Does it have meaning independently of the shape (curvature) of space ? (I guess relativists have answers to that). Take the 2D example of the surface of a sphere. What is translation in that space ? The usual answer is "displacement along a great circle". This works for a point, but moderately well for a 2D solid, as only a line in that solid will be able to move on a great circle. I guess we can ignore that, as any solid will be "infinitesimal" with respect to the kind of curvature radius to be considered. However there is the other problem that, very simply, every translation is a rotation, and in two different ways, with a very large but finite radius. But it probably does not matter for scale reasons. Now, there is also the possibility that I completely missed or misunderstood an essential point. Which would it be ?


title: Commutator of operator and its derivative

text: Is it possible to calculate in a general way the commutator of an operator which depends on some variable and the derivative of this operator with respect to that variable? $$ \hat o = \hat o(\xi)\\\ \left[\partial_\xi \hat o(\xi), \hat o(\xi^{\prime})\right]=? $$ To be honest I don't quite understand, what the derivative of an operator is. When one plugs in the standard definition of the derivative of a function, it looks as though the operator commutes with its derivative, but I don't quite know how to feel about that "proof". In practice I need this to calculate the commutator of the field operator of a free scalar field and any one of its four-derivatives: $$ \left[\partial_\mu\hat\phi(\vec{x},t),\hat\phi(\vec{x}^{\prime},t)\right]. $$ (I'm looking at scalar field theory described by a Lagrangian density $\mathcal{L}=\partial_\mu \phi^*\partial^\mu\phi-m^2\phi\phi^*$.) One can of course simply compute this by plugging in the field operator, but I was wondering about the general situation. $$\\\$$ While we're at it: What about the more general case of two operators, whose commutator is known? $$ \left[\hat o(\xi),\hat u (\xi^{\prime})\right]=A(\xi,\xi^{\prime})\\\ \left[\partial_\xi\hat o(\xi),\hat u (\xi^{\prime})\right]=? $$


title: Is there a heat transfer equation that takes conduction, convection, radiation and dT/dt into account?

text: I see equations that take 2 or 3 of the listed parameters into account but I haven't been able to find one that is that complete. I'm seeking to solve this equation using matlab for a simulation project to study how the heat dissipates on a microheater given an electrical power or current as input and see how quickly the temperature rises, and so forth. But first, I need the proper equation.


title: How is the Saddle point approximation used in physics?

text: I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative description of the method and why it's used and for what it's used. So my question is, how is the saddle point approximation used in physics? What is it approximating? When can it be used?


title: Non-Integrable systems

text: Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities (n being the number of degrees of freedom), or n whose Poisson brackets with each other are zero. The way I understand it, these conditions correspond directly to us being able to do the Hamilton-Jacobi transformation, which is roughly equivalent to saying that the $2n-1$ conserved quantities are the initial conditions of the problem, which itself is a way of saying that the map from the phase space position at some time $t_0$ to that at time $t$ is invertible. But, if the last statement is right, why are there systems which are non-integrable at all? Shouldn't all systems' trajectories be uniquely determined by the equations of motion and initial conditions? Or is it that all non-integrable systems are those whose Lagrangians can't be written (non-holonomic constraints, friction etc)? I've heard that Poincare proved that the gravitational three-body problem in two dimensions was non-integrable, but he showed that there were too few _analytic_ conserved quantities. I don't know why exactly that means non- integrability so if someone could help me there that would be great too.


title: Cross sections and renormalization scheme

text: Can the result on cross section of some process be dependent on the renormalization scheme used?


title: Obtaining an expression for the Lorentz Force in the Dirac theory

text: We know that $P = p - \frac{e}{c} A$ How can we obtain a expression for the Lorentz force from the equation above using the Dirac Theory?? Could you please explain this to me step by step? The only idea I have now is: $i\hbar \frac{dP}{dt} = [P,H]$


title: What will the strength of a sintered steel piece be compared with a cast piece?

text: By sintering here, I mean specifically deposition laser sintering, i.e., put down a patch of steel powder, zap it with a laser so that it liquifies at least partially, lay down a new patch, laser, rinse, repeat :). I've not specified tensile, ductile, or a specific kind of strength to increase the quantity (and probability) of answers.


Is decoherence even possible in anti de Sitter space? The spatial conformal boundary acts as a repulsive wall, thus turning anti de Sitter space into an eternally closed quantum system. Superpositions remain superpositions and can never decohere. A closed system eventually thermalizes and undergoes Poincare recurrences. Whatever the dual is to an interacting conformal field theory, is it really a theory of quantum gravity, or does it only look like one superficially?


title: Distorted colors of Google StreetView photographs near electric power lines

text: This is a followup to my question: > Cyclist's electrical tingling under power lines Some users presented a convincing picture that the electric shocks under power lines are primarily from the electric fields, not the magnetic one, because the frequency is just too low, and the body must be considered as a capacitor to estimate the currents etc. Google Maps just included pretty much all of the Czech Republic to the Street View. That's why I could look at the place where I had felt the electric shocks. First, the mast over there clearly corresponds to 400 kV according to a list of masts (model 8a-3), a pretty high voltage. But it's even more interesting to see what the Google car was seeing in front of itself at the same point where I experienced the shocks. ![enter image description here](http://i.stack.imgur.com/0OYPo.jpg) A pretty nice colorful distortion. It is strongly correlated with the masts so I guess it has something to do with the electric fields, too. Am I right? If I am right, what are the electric fields that may cause similarly strong effects in the digital cameras used by Google or others? There are lots of capacitors in those digital cameras, aren't there? Why is the ordering of the anomalous colors yellow, cyan, violet (from the top towards the bottom where the treetops and roofs are located)? Is it linked to different capacities or voltages or other electric parameters of the three color-sensitive segments of the digital camera? A blog URL related to the Street View anomalies in Czechia.


title: How do voltage and voltage drops over a circuit relate to work done?

text: I have a few remaining uncertainties when it comes to voltage. I numbered the paragraphs in my thought sequence for easier reference.   1. In a series circuit, I know voltage is the electric potential, and represents the potential for work, but that doesnt mean that any electron travelling on any 10 volt circuit does the same amount of work from start to finish, does it?   2. What i mean is, let's say you have a 10 volt series circuit with a 1 ohm wire. Tracking an electron in the current from start to finish, as work is done on the electron by the voltage, its potential drops, until it reaches 0 at the end of the circuit.   3. But if we increase the resistance of the wire to 5 ohms, it would take 5x longer for the electron to get to the end of the circuit, because the current would be 1/5th, right?   4. But then that means that the electron in the 5 ohm circuit would have done 5x the amount of work (or work done on it) of the 1 ohm circuit over 5x the duration.   5. I know the work done per second is the same, but ultimately, if each electron in the 5 ohm circuit has more work done on it over the course of the circuit, then voltage cant be an expression of total work done from p1 (start of circuit) to p2 (end), right? I think i first thought that because voltage relates to potential for work, and the voltage always drops from full to 0 by the end of the circuit, so i assumed the work must be the same regardless of the circuit resistance.   6. So does this mean that when voltage is divided over the components of the circuit, it's really representing the ratio of where in the circuit the work is done?   7. i.e. in a 12v series circuit with a 3 ohm resistor and a 1 ohm resistor, there would be a voltage drop of 9v over the 3ohm resistor, and 3 volts over the 1 ohm, right? which would mean that 75% of total work done in the circuit was at the 3 ohm resistor?   8. But why is it that the more resistors you have in a series, the less the voltage drops over any one resistor? I know with more resistors the current would be lower. Does voltage drop over a resistor decrease as current decreases?


title: Circuit analysis: Two nodes as one?

text: Can I treat these two nodes in the center as if there were only one node connecting these four components? ![enter image description here](http://i.stack.imgur.com/De05I.png)


title: Number of bits needed to express physical laws?

text: What is the minimum number of bits that would be needed to express a given physical law, like the law of universal gravitation? How many bits are needed to express each of the four fundamental forces? Is there a pattern here?


title: Can I detect/measure strength of a FM radio wave using a Electro magnetic field meter?

text: I understand that an alternating current produces an Electro magnetic field . The strength of this magnetic field can be measured by measuring devices. What I wanted to understand is   1. Are the FM waves generated by the radio transmitters a form of Electro magnetic field?   2. Given an EMF meter such as this one, can I detect the strength of such a wave/field that is coming my way? I have a basic understanding of how a magnetic field is generated by an alternating current and how it varies inversely with the distance from source. But I do not have an in depth understanding of this area. I am trying to understand the correspondence between the radio waves and the Electro magnetic field and then go forward on application from there.


title: Darcy law yields extreme speed for gas flow throgh packed spheres?

text: The darcy law is as follows: $u=-\frac{k}{\mu}\nabla p$. Assume we have a gas, then $\mu$ is about $10^{-5}$. $k$ for packed spheres a few mm in diameter is of order $10^{-8}$ $m^2$. Say the pressure difference across a 0.1 m bed is 10 Pa. Then we get $u= 0.1$ m/s which seems really high for such a small pressure difference. I mean, tens of Pascals is a variation which is comaparable to random pressure fluctations. Is this reasonable?


title: EM vector potential

text: We can write the electromagnetic field tensor as $$\begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\\ E_x/c & 0 & -B_z & B_y \\\ E_y/c & B_z & 0 & -B_x \\\ E_z/c & -B_y & B_x & 0 \end{bmatrix} = F^{\mu\nu}.$$ Erick J. Weinberg, _Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics_ (p. 43), states: > For a static solution with vanishing electric fields $F_{0j}$, and hence > $A_0 = 0$, [...] How can this be proven?


title: Liquid nitrogen and liquid helium

text: In cryogenics, liquid nitrogen and liquid helium are often used as coolants. Other than their low boiling point of 77K and 4.2K respectively, what properties make them suitable as coolants? Are there any other possible candidates around the same temperature range?


title: If a beam of light follows the path $y(x)=y_0\sin(x/y_0)$, what is the index of refraction?

text: An exam question that showed up while I was studying: > A plane wave refracts and follows a ray given by the equation > $y=y_0\sin(x/y_0)$ where $y_0$ is a constant. Find the refractive index > $n(y)$ and plot it. I believe I've made a mistake in my calculation. The wave propagates in the $x$-direction. So we are told the trajectory -- that's simple enough. Let us divide the material into thin sheets of height $dy$. If we zoom in on the path enough, we can use Snell's law to approximate the difference in the angle (between the sheets.) We get that $$ n(y)\sin(\theta)=n(y+dy)\sin(\theta + d\theta).$$ To be clear, $\theta$ is measured from the $y$-axis. (So $dx/dy=\tan(\theta)$.) We can expand to first order to get that $$ n(y)\sin(\theta)=n(y)\sin(\theta) + n(y)\cos(\theta)d\theta + n'(y)\sin(\theta)dy $$ whereupon we get (using the chain-rule backwards) $$-n(\theta)\cos(\theta)=\frac{dn}{d\theta}\sin(\theta)$$ where $n(y)$ has been written $n(\theta).$ (There is no difference.) Now we can separate variables. The equation $$\frac{dn}{n}=-\frac{d\theta}{\tan(\theta)}$$ yields $$n(\theta)=\frac{1}{\sin(\theta)}.$$ Now the issue is when I go through the trig identities, that is, replacing $\tan(\theta)$ with $dx/dy$, I get that $$\tan(\theta)=\frac{1}{\sqrt{1-(y/y_0)^2}}$$ which implies $n(y)=y/y_0.$ I have no idea if this is the right behavior. There is no divergence in $n(y)$ like I imagined there would be and strangely it seems to flip the sign with $y$. But $n(y)$ can never be negative to I am at a loss here. Edit: I believe the integral _should_ be $$n(\theta)=\frac{1}{|\sin(\theta)|}$$ which takes care of the sign issue.


title: One-photon-at-a-time effect on double slit experiment with detectors?

text: I do not have a access to a physics lab, nor have I come across this type of a double slit experiment, hence I am curious if someone has tried this. Given a double slit experiment setup with a detector. What happens if we have a detector at one of the slits, but we do not send one photon at a time (in other words, we send a bunch of photons without concerning ourselves about the count)? My assumption is that we will see a particle (bullet) pattern like we would if we sent one photon at a time, is this correct?


title: Apparent size of objects under water

text: A friend of mine has a homework question and we're having some trouble figuring out what physical mechanisms come into play for this. > An underwater swimmer sees a spherical air bubble that appears to have a > diameter $d=$ 1.5 cm. What is its actual diameter? We are having trouble, specifically, thinking of physical mechanisms that would change the apparent sizes of objects when seen underwater. Is it caused by refraction at the curved surface of the bubble?


title: Values of SM parameters at one certain scale

text: The general question is: What are the values of Standard Model parameters (in the $\bar{MS}$ renormalization scheme) at some scale e.g. $m_{Z}$? As its parametrization in Yukawa matrices is not unique - what are the values of gauge couplings, fermion masses and CKM matrix? The background: I want to solve renormalization group equations of MSSM and in order to have initial conditions for them I need to know SM parameters at one scale - not at few different, which one can find at Particle Data Group webpage.


title: Why do we have a TeV scale?

text: When model building we don't want to introduce any new scales into our theory. We usually try to have new particles at the Higgs (TeV) scale (to solve the hierarchy problem), at the GUT scale, or at the Planck scale. However, if the Higgs VEV already gives us a new scale, why would there not be new particles at some intermediate scale, for example say at $10^{5}$ TeV? In other words, what is unnatural about adding in new scales beyond the 3 that we are used to?


title: where i can find thermodynamics properties of octane

text: where can I find thermodynamics properties for octane at $T_s$? I need this properties or where can i find all ? $$liquid-density=\rho_l=?$$ $$vapor-density=\rho_v=?$$ $$latent-heat-of- evaporation=h_fg=?$$ $$liquid-kinematic-viscosity=\mu_l=? $$ $$liquid- conductivity=k_l=?$$


title: Charge neutrality of the Universe: evidences and theories

text: I've always wondered why the number of protons in the Universe exactly matches the number of electrons. They are such different particles with totally different cross sections. So, first of all, is this true? And, secondly, how could it be that they have been produced in the same amount after the Big- Bang?


title: How is light created?

text: I'm very curious about how light is created. I'm trying to get back to some type of real understanding about why everything on the sub-atomic level vibrates and the implications for the preservation of both the conservation of energy and conservation of momentum. How is light created at the atomic level?


title: How does light pass through rough glass?

text: Light incident on a rough surface will be diffuse after passing it. Angular intensity depends on the grinding of the glass surface. I'm trying to find information about the scattering indicatrix of light passing through the rough glass: how to calculate it numerically, which parameters are essential, etc., or at least where I should search for it. I would also be glad to find how to calculate it from the equations of electromagnetic fields.


title: How to calculate angular momentum per second for a photon beam?

text: Given a photon polarization state $$|\phi\rangle = \frac{3}{5}|x\rangle + \frac{4i}{5}|y\rangle,$$ a beam of photons transmit $N$ photons per second in such a state. An L-polarized photon has an angular momentum $\hbar$ along its direction of motion, and an R-polarized photon has an angular momentum of the same magnitude but opposite to direction of motion. If the beam described by the particular polarization state $|\phi\rangle$ above is totally absorbed by a surface, what angular momentum is added to the surface per second on the average?


title: How do I integrate the Poisson equation to determine the electric potential along a particular direction (e.g., $z$)?

text: This question is a sequel of sorts to my earlier (resolved) question about a recent paper. In the paper, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes. The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction. In my previous question, I asked about how particle number density is computed. In this question, I would like to ask about how the electric potential is computed, given the charge density distribution. Recall that in CGS (Gaussian) units, the Poisson equation is $$\nabla^2 \Phi = -4\pi \rho$$ where $\Phi$ is the electric potential and $\rho$ is the charge density. So the charge density $\rho$ is proportional to the Laplacian of the potential. Now suppose I want to find the potential $\Phi(z)$ along $z$, by integrating the Poisson equation. How can I do this? In the paper, on page 254, the authors write down the average charge density $\bar{\rho}_{\alpha}(z)$ at $z$: $$\bar{\rho}_{\alpha}(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^{\prime} \; dy^{\prime} \; \rho_{\alpha}(x^{\prime}, y^{\prime}, z)$$ where $\rho_{\alpha}(x, y, z)$ is the local charge density arising from the atomic charge distribution of ionic species $\alpha$, $\bar{\rho}_{\alpha}(z)$ is the average charge density at $z$ obtained by averaging $\rho_{\alpha}(x, y, z)$ over $x$ and $y$, and $\sum_{\alpha}$ denotes sum over ionic species. The authors then integrate the Poisson equation to obtain $\Phi(z)$: $$\Phi(z) = -4\pi \sum_{\alpha} \int_{-z_0}^z (z - z^{\prime}) \bar{\rho}_{\alpha}(z^{\prime}) \; dz^{\prime} \; \; \; \; \textbf{(eq. 2)}$$ My question is, how do I "integrate the Poisson equation" to obtain **equation (2)**? How do I go from $\nabla^2 \Phi = -4\pi \rho$ to **equation (2)**? In paricular, where does the $(z - z^{\prime})$ factor come from? Thanks for your time.


title: Proof of charge existence on a grounded conductor

text: A question regarding the existence of charge on grounded conductors is confusing me. Could there be charge on a grounded conductor? How does this not contradict Gauss's Law? Since every conductor has some capacitance attributed to it, doesn't the existence of charge on a grounded conductor contradicts the equation: $Q = C*V$ ?


title: How can one know if one has a Majorana fermion?

text: If the Majorana fermion is a fermion that is it's own antiparticle and exactly the same as its fermion counterpart, then how do they know that it's not just a fermion?


title: Can the equation $v=\lambda f$ be made true even for non sinusoidal waves?

text: The known relation between the speed of a propagating wave, the wave length of the wave, and its frequency is   $$v=\lambda f$$ which is always true for any periodic sinusoidal waves. Now consider: 1-a case were the wave is not sinusoidal but still periodic, like a sawtooth pattern say, that propagates with speed $v$, and of frequency $f$. It is obvious in this case we will not have a single $\lambda$ but infinite number of them, $\lambda_i$ (from Fourier analyzing that pattern). Can one define/calculate a new wave length $\lambda$ using $\lambda_i$'s such that $\lambda=v/f$? 2-another case where the propagating wave is not periodic, just a gaussian pulse say that propagates with speed $v$. Can one define something like $v=\lambda f$ in this case? how to calculate $f$ and $\lambda$ such that $v=f\lambda$


title: Why are objects at rest in motion through spacetime at the speed of light?

text: I read that an object at rest has such a stupendous amount of energy, $E=mc^2$ because it's effectively in motion through space-time at the speed of light and it's traveling through the time dimension of space-time at 1 second per second as time goes forward. What troubles me here, is the fact that it is traveling through space-time at the speed of light. Why is it at the speed of light?


title: Why is the shadow of earth red?

text: Reading this article about the forthcoming eclipse on December 10th, I noticed mention that the moon passes through the red shadow of Earth; I'd always held the impression that a shadow is universally dark/gray. I'm guessing there is some kind of prism effect coming into play over there particularly with this configuration of the celestial bodies involved. Why is the shadow of earth red?


title: Why is scattering vector $\vec{q}$ called vector of 'momentum transfer'?

text: In the world of scattering the angle at which you detect the scattered radiation is known as $q$, where $$ \vec{q} = \frac{4\pi\eta}{\lambda}\sin(\theta/2) $$ I read in a lot of books that this is known as the scattering vector or "momentum transfer". What I am trying to understand is why it is called "momentum transfer"? In the physical sense, say a beam of laser light hits a particle and creates dipole oscillations which re-radiate the scattered wave. What does the particular scattering vector have to do with the oscillations? Doesn't the particle just act as new source of light scattering uniformly in all directions? In this example ignore form-factor.


Is it possible to make use of paper/bill permeate by chemical compound to became a paper-made bullet-proof vest? This is inspired by Greece and Italy tend to have more riot than rest of the Europe and if they use their own currency, they could make a cheaper bullet- vest with their money


title: How to calculate required energy to displace a pendulum?

text: How can one calculate the amount of energy needed to displace pendulum with given mass **m** and string length **L** to $\alpha$ degrees from resting position when acceleration due to gravity is known?


title: Acceleration of 2 bodies tied with a string

text: Find the acceleration of the block of mass M shown in the figure . The co- efficient of friction between the 2 blocks is μ1 and that between the bigger block and ground is μ2. ![enter image description here](http://i.stack.imgur.com/yzma2.png) Could someone help solve this problem using Lagrangian mechanics. I know the solution to this problem using Newtonian mechanics.However I am not finding the same answer when I am trying to do it using Lagrangian mechanics. Answer - $a=\frac{[2m-\mu_2(M+m)]g}{M+m(5-2(\mu_1-\mu_2))}$


title: Franck-Hertz Experiment explanation

text: Can anyone explain to me how the Franck-Hertz experiment works? (in term of electric current and voltage changes) I am getting all confused. More specifically, why is an accelerating grid (or mesh grid) needed in the experiment, and what is the accelerating grid? And why would the accelerating grid exhibit positive charge at first?


title: What happens to matter in extremely high gravity?

text: Though I am a software engineer, I have bit interest in sciences as well. I was reading about black holes and I thought if there is any existing research results on _How matter gets affected because of extremely high gravity._ I tried searching but these long equations really didn't help me. Can someone please put it in layman terms? More specifically if I ask _Why the infinite gravity doesn't tear down the atoms in to quarks and does it have enough potential that it can affect something a Planck scale?_


title: Does air resistance ever slow a particle down to zero velocity?

text: If a particle moves in a place with air resistance (but no other forces), will it ever reach a zero velocity in finite time? The air resistance is proportional to some power of velocity - $v^\alpha$, and I have to try it with different $\alpha$. I've solved for the function of position for several alphas and all functions I've gotten decay to $v=0$ as $t \to\infty$, but none ever reach exactly $v=0$ for a finite value of $t$. This should be the case, right?


title: What condition is fulfilled by the path of a mass sliding down a lumpy hill?

text: Suppose I have a hill that goes up and down unevenly. It's frictionless, and I want to slide a point mass down the hill. I am interested in the path it takes. (By "path" I mean the trail it leaves behind, not the time-dependent trajectory.) I can find the equations of motion of the point mass and, from there, find the path by finding the entire time-dependent motion. Is there a way to find the path directly, given the initial velocity of the particle? For example, does the correct path extremize something?


title: Help fill in my understanding of the Polywell fusion reactor

text: Polywell is a proposed new type of fusion reactor, which is designed to use magnetic fields to overcome the problems with the Elmore-Tuck-Watson fusor. I'm trying to understand exactly how it works. > An Elmore-Tuck-Watson fusor ... consists of a vacuum chamber containing a > negatively charged outer grid (which may be the chamber) and a positively > charged inner grid. Electrons are injected into the system and accelerated > toward the inner grid. Most of the time, the electrons pass through the > grid, through the core, and through the inner grid again, which then > decelerates them and reaccelerates them inward wherein they return through > the core. As they pass repeatedly through the core, they generate a > negatively charged zone, a potential well, which is called a virtual > cathode. **Question 1:** So this device uses a constant high voltage applied to the electrodes to produce a region of negative electric charge in the center. How is this possible? Electrons repel each other, so creating a region in empty space in which electrons are more dense than their surroundings would seem to require a constant energy input. Without any energy input, they'll just fly apart to a minimum energy equilibrium where the charge density is equal everywhere. Applying a voltage to electrodes with no current flowing between them does not use up any energy, however. So where is the energy to hold the clump of electrons together coming from? > Like the Elmore-Tuck-Watson (ETW) fusor, the polywell confines positive ions > through their attraction to the negative potential well which is created by > the electrons that are held inside a positively charged grid. However, to > avoid the losses related to the electrons striking the grid, the Polywell > uses magnetic fields to shield the grid. The magnetic fields are configured > in a way that adds to the confinement of the electrons so that there are > many more electrons inside the core than outside. **Question 2:** So the magnetic fields generated by the grid would supposedly prevent the electrons from hitting the grid. The positively charged inner grid is attractive to electrons, but it's also magnetized, and somehow the magnetic fields repel electrons away from the grid? But how can unchanging/steady-state magnets confine (or repel) electrons? A bar magnet doesn't create any kind of movement or displacement of charge.


title: Why is amplitude of a wavefunction to propagate from $q$ to $q'$ governed by $e^{-\frac{i}{\hbar}HT}$ unitary operator?

text: In the textbook _Quantum Field Theory_ by A. Zee, it says: > _In quantum mechanics, the amplitude to propagate from a point $q_i$ to a > point $q_f$ in time $T$ is governed by the unitary operator > $e^{−\frac{i}{\hbar}HT}$, where $H$ is the Hamiltonian._ I am having a hard time understanding this. Can anyone explain this in context of Dirac's formulation and relate this to Schrodinger equation?


title: If the size of universe doubled

text: My question is silly formulated, but I want to know if there is some sensible physical question buried in it: Suppose an exact copy of our universe is made, but where spatial distances and sizes are twice as large relative to ours. Would this universe evolve and function just as ours? Since mass is proportional to volume which is distance cubed, but strength of a rope is only distance squared so it would not work in proportion. Does this mean that a universe with our physical laws that evloves like ours can only have 1 certain size ? Or would the physical laws scale in proportion so it evolves the same ? How can it only have 1 size, what is it relative to ? Also, suppose another exact copy of our universe is made, but where everything happends twice as fast relative to ours, would it evolve in the same way as ours? (We can suppose if needed that our universe is deterministic)


title: Time dependent Lagrangian

text: Suppose I have a mechanical system with $\ell + m$ degrees of freedom and an associated Lagrangian: $$L(\alpha, \beta, \dot{\alpha}, \dot{\beta}, t),$$ where $\alpha \in \mathbb{R}^{\ell}$ and $\beta \in \mathbb{R}^{m}$. Now suppose I have a known $\mathbb{R}^{\ell}$-valued function $f(t)$ and define a new Lagrangian: $$M(\beta, \dot{\beta}, t)~:=~L(f(t), \beta, \dot{f}(t), \dot{\beta}, t)$$ Do the equations that derive from $M$ correctly describe the motion of the initial mechanical system, where the first $\ell$ degrees of freedom are constrained to the motion $f(t)$ (by means of an external force)?


title: Is it possible to use quantum mechanics for an effective time based encryption?

text: This is for an application in cryptography. There is a concept called "time based cryptography", where a message can be decrypted only after a certain time, Say "12/12/2060, 12:30 GMT". There are some ideas based on "proof-of- work" but they have many problems. Is it possible to use quantum mechanics to do this in a nicer way?


title: Will a ball slide down a lumpy hill over the same path it rolls down the hill?

text: Suppose I have a lumpy hill. In a first experiment, the hill is frictionless and I let a ball slide down, starting from rest. I watch the path it takes (the time-independent trail it follows). In the next experiment, the hill stays the same shape, but the ball now rolls without slipping down the hill. Assume there is no deformation of the ball or of the hill, no micro-sliding of the contact surface, and no other forms of rolling resistance. Energy is conserved. I release the ball from rest at same point on the hill. I track the path the rolling ball takes. Does the rolling ball follow the same path as the sliding ball, but more slowly, or does it sometimes follow a different path? Note: There may be some "trick" answers. For example, if the hill curves significantly on scales similar to the radius of the ball, the ball could get "wedged in" somewhere, or two parts of the ball could contact the hill at the same time. Let's assume that geometrically, the shape of the hill is such that it is only possible to have the ball contact the hill at one point, and that the ball always contacts the hill at a single point (i.e. it never flies off).


title: Making an Incomplete Set of Observables Complete

text: In quantum mechanics, it seems a standard procedure that if you have an incomplete set of observables, then one can make this set complete by adding more commuting observables until the set becomes complete. Can anyone describe briefly a procedure for doing this, in symbols? I think this would be more instructive than a word-based definition.


title: Momentum conservation problem

text: Lets a plastic ball of mass m which is collided with steel. After collision the ball is coming back with the half initial speed. If the steel doesn't move then how can I interpretate this ? Let the initial speed of the ball is $u_1$ and mass $m_1$ and mass of steel $m_2$ speed of steel before and after collision $0$. Therefore we can write according to the conservation of momentum, $$m_1 u_1 +m_2 u_2 = m_1v_1 +m_2 v_2$$ $$m_1 u_1 = m_1v_1 $$ $$ u_1 = v_1 $$ I have surmised $u_2 = v_2 = 0$. > Therefore speed is same then how can the speed of the ball can be halved > after the collision. Is this because of inelastic collision?


title: Change of variables in 3 dimensions

text: I'm confused about a basic mathematical step. Say we have $$\omega=\frac{2\pi|\vec n|}{L}.$$ Then why is it that $$\int^\omega d^3\vec n= 4\pi L^3\int_0^\omega\frac{d\omega}{(2\pi)^3}\omega^2$$ ? Thank you.


title: Single particle trajectory in a quadrupole potential

text: I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$


title: Lie Groups and group extensions?

text: Is $U(1)$ x $SU(2)$ x $SU(3)$ a vector space over a field? I saw an article here http://en.wikipedia.org/wiki/Group_extension that seemed to me that a similar concept to a field extension was being used... In QFT, is each particle considered to be its own vector space? And are the individual vector spaces then bridged together by these group extensions? If I have asked too many questions, feel free to respond to just one.


title: Calculate the polarization vector on reflection or refraction from a dielectric interface

text: I am interested in ray-tracing polarized photons. I have code that works very well for unpolarized light. When a ray hits a dielectric interface the photon is either reflected or refracted by comparing the Fresnel coefficients with a random number. The calculation of the reflected or refracted direction vector is relatively simple. Say we have a direction vector $\boldsymbol{v}$ and surface normal to the dielectric interface $\boldsymbol{n}$. **Reflected direction vector** $$ \boldsymbol{v} - 2d\boldsymbol{n} $$ **Refracted direction vector** $$ N\boldsymbol{v} + (Nd - c) \boldsymbol{n} $$ In the above $d=\left(\boldsymbol{n}\cdot\boldsymbol{v}\right)$, $N=\frac{n_1}{n_2}$ which is the ratio of the refraction index at the interface, and $c =\left(1 - N^2\left(1 - d^2\right)\right)^{1/2}$. These transformation are easily applied within the ray tracing loop because call values are known. **Question** My question is what transformation needs to be applied to calculate the correct polarisation vector of a reflected or refracted ray? Before hitting the interface the polaristaion vector of the ray $\boldsymbol{k}$ is known (I assume linear polarised states only). Clearly $\boldsymbol{k}\cdot\boldsymbol{v}=0$, but where in the plane perpendicular to the direction vector $\boldsymbol{v}$ does the polaristaion vector $\boldsymbol{k}$ lie? Can I apply a simple transformation, similar to the above, to find the new polaristaion vector?


title: Decibel level addition of multiple noise sources

text: If I have four computer fans of which each is said to run at 46 dB, and they run close to each other, how loud is the whole system? I somehow recall from my physics course that 10 decibel more means twice the noise, is that right? So would it be 66 dB for four such fans then?


title: Layman explanation for superconductivity?

text: I'm interested in reading an explanation of superconductivity, but though I have an undergraduate degree in engineering, I'm not sure I'm up to reading a detailed mathematical treatment of the theory. Still, I'm willing to give it a shot. Can anyone suggest the clearest overview of how superconductivity works, both with as a technical paper and a lay explanation if available?


title: Force field for Calcium ions, potassium ions and water in molecular dynamics

text: I want to adapt my existing MD simulation so that it can handle these three species. For the interactions of the ions I thought that I would use the coulomb potential, but I don't know what sort of potential I should use between water and the ions. It doesn't have to be super accurate, I'm not doing ab initio stuff. Also, am I correct in choosing coulomb between the force field between the ions? I am representing all 3 species at a molecular level of granularity. Thanks


title: Are Carnot theorem and Carnot Cycle related?

text: This has been in my head for a while, and I think I finally have the answer. I was wondering what was so special about the Carnot cycle so it was used to demonstrate the Carnot theorem; the demonstrations I had read didn't used any property specific to the Carnot cycle, but used a reversible engine. Why couldn't it use the Rankine cycle, for example? And now I've realized that the Carnot theorem hasn't anything to do with the Carnot cycle, does it? It just states that the efficiency of ALL reversible cycles will be the same, but the demonstration talks about an hypothetical reversible cycle, not necessarily the Carnot. Am I right?


title: At what velocity does sea water leave a pipe run into a fresh water lake (for equal surface levels)?

text: If I took a fresh water lake* whose surface is exactly as sea level, and connect it to the sea with a pipe filled with sea water, with both ends of the pipe at exactly the same depth from the surface, $h$, what is the correct way to calculate the velocity with which the salt water leaves the pipe from the depth $h$, and the densities of fresh and sea water, $\rho_\text{fw}$ and $\rho_\text{sw}$? I can't seem to apply Torricelli's law here, since that is not influenced by the fluid's density, yet that would be the only driving force here. Can I treat this as a hydraulic head of a fluid with a density of $\rho = \rho_\text{sw} - \rho_\text{fw}$? Should I treat this the same as a pipe of water on land that's pressurized with $(\rho_\text{sw} / \rho_\text{fw}) g \rho_\text{fw} h$ above ambient pressure ($g$ being gravitational acceleration)? For the purpose of this question, let's assume both water reservoirs are very large in comparison to the volume exchanged through the pipe, and the salt dissipates very rapidly in the fresh water (or the water sinks to a far away bottom immediately), so we neither alter the surface level (which would happen in communicating vessels containing liquids with different density) of either body nor do the salinities level out. * _(no actual lakes will be ruined, promise)_


title: Taking photos without photons?

text: I was looking up some science news and I came across this! **Blind quantum camera snaps photos of Schrödinger’s cat** https://www.sciencenews.org/article/blind-quantum-camera-snaps-photos- schr%C3%B6dinger%E2%80%99s-cat Can somebody please explain how is this possible?


title: A loop quantum gravity toy inspired by an Aharonov-Bohm ring

text: Comparing my question to Give a description of Loop Quantum Gravity your grandmother could understand what I'm looking for here is a toy for a toddler ($\approx$ a pre-QFT graduate student). I seek to generalize to gravity the following minimalistic model of magnetism (which is a textbook example for the phenomena of Aharonov-Bohm oscillations and persistent currents). Take a three-state Hilbert space with a Hamiltonian matrix $h_{nm}$ with $n,m=1,2,3$. Interpreting these states as localization sites for, say, an electron in a mesoscopic ring, one may ask how to include a magnetic field piercing the ring. The answer is given by the gauge-invariant phase $\varphi =\arg (h_{12} h_{23} h_{13}^{\ast})$ which is the Aharonov-Bohm phase for a closed trajectory (1-2-3) along the ring; $\varphi$ is equal to the magnetic flux through the ring expressed in units of $\phi_0=h/e$. If $\varphi \not = 0 (\rm{mod} 2 \pi) $, then the lowest energy state carries persistent current, and the model serves as a good starting point to discuss this piece of mesoscopic physics. I thinks it would be great to have such a toy example for gravity (classical and quantum). The can probably found in be found in a pedagogical book, so please give a reference if you know one. More specifically, my question would be: > What is the minimal number of points to define a meaningful finite- > difference analogue of Hilbert-Einstein action and how to do that? Can basic > ideas of LQG quantization be demonstrated on such a finite state model? I am looking for a simple, possibly less than $3+1$-dimensional example to demonstrate the possibility to eliminate gravity by local but not global Lorentz transformations the same way the above toy model demonstrates the relation between local and global $U(1)$ symmetry.


title: Question about the parity of the ghost number operator in BRST quantization

text: Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\\{c^i,b_j\\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are summed). The question raised in proving the BRST operator raising the ghost number by 1, given in the Example 6.1 on the page 116-118 of the book, String theory demystified. http://books.google.com/books?id=S4JyPgw4ZlAC&lpg=PP1&pg=PA117#v=onepage&q&f=false On the second and the third lines of the formula derivation of $UC^iK_i$ $UC^iK_i=...=c^iK_i-c^i\displaystyle\sum_rc^rb_rK_i=c^iK_i+c^iK_i\displaystyle\sum_rc^rb_r=...$ where we need $c^r U K_r =-c^r K_r U$ The change of sign above is not manifestly obvious to me. \---add--- Never mind. I find the second line cheated me. It should be $UC^iK_i=...=c^iK_i+c^i\displaystyle\sum_rc^rb_rK_i=c^iK_i+c^iK_i\displaystyle\sum_rc^rb_r=...$ Furthermore, in the proof followed in this example, there a few typos need to be corrected. But the final result is right. The BRST operator raises the ghost number by 1!


title: Does the Strong Nuclear Force follow Superposition?

text: I have just started the study of nuclear physics in my high school, and while reading about nuclear forces and binding energy per nucleon, I found out that the nuclear forces are highly short ranged and that is the reason for constancy of binding energy per nucleon $E_b / A $. The proof mentioned that for marginally large nuclei if the maximum number of neighbors that a nucleon can have is $ p $, then its binding energy would be proportional to p. $$ E_b = pK$$ where K is a constant having dimensions of energy. Does this imply that the strong nuclear force follows the principle of superpositon, as the resultant binding energy is the sum of the energy due to individual nucleons? Is it true that although the nuclear forces do not follow the inverse square law, they still follow superposition? Is superposition a more "fundamental" property of the universe than the inverse square rule? I need guidance along this line of reasoning of mine. Thanks in advance.


title: Conservation of energy and Doppler effect?

text: From what I understand, the frequency of light coming from the source moving towards an observer increases. From $ E=hv $ , this implies increase in energy of photon. What really is confusing, is where does that extra energy come from? And also where is the energy is lost during opposite Doppler effect (red shift)? Why doesn't this violate the conservation of energy?


title: Determining source of neutrinos

text: An article in Science, Evidence for High-Energy Extraterrestrial Neutrinos at the IceCube Detector, claims to have detected 28 neutrinos from an unidentified extra-solar location that has yet to be identified. From what I can gather, the only evidence for determining the origin of a neutrino is based only on the energy of said neutrino. Based on the large energy content of those observed, the hypothesis is that the neutrinos are from outside the solar system, making them the first extra-solar neutrinos ever detected. How accurate is using energy to identify the neutrinos? Presumably it is statistically possible, although perhaps very unlikely, that the energies observed were created inside the solar system. Is there a probability density function describing expected energy values for atmospheric and intra-solar- system generated neutrinos? Where do the newly observed neutrinos fit into those probabilities?


title: Can sound be propagated without initial mechanical interference?

text: I have researched up a little on sound, and it seems that sound is a mechanical wave that propagates through the air as energy, and that is how we hear it through our ears. Depending on the medium's density, the type of wave, and the amplitude, among other factors, we determine how loud it will sound, etc. So my question is how does a speaker do this with just a diaphragm, two magnets, etc.?


title: Is there a theoretical maximum for refractive index?

text: May there be materials yet to be discovered which may have a higher refractive index than today's known materials (for wavelengths within the visible range)? Is there a theoretical limit for the refractive index of a material?


title: Does a radio receiver add a load to the transmitter?

text: If I have a dynamo connected to a switch and a load (such as a light bulb), the load on the dynamo is increased when the switch is closed and the light comes on. The dynamo gets harder to turn. Is the same true for a radio transmitter and receiver? If the receiving aerial is initially not connected to a circuit and then is connected, can you detect at the transmitting end that the load was added at the receiving end? If not, what is the difference between the two examples?


title: Newton's 3rd Law: How can I break things?

text: If I punch a wooden board hard enough and it breaks in two, has the board still exerted a force of equal magnitude on my fist? When the board breaks in two due to my force, the halves have a component of acceleration in the direction of my striking fist...that implies the board did not exert an equal and opposite reaction, no?


1) Why are we talking about space curvature as if we know what space is? Every question about gravity seems to evoke an answer involving "space curvature" which seems like an undefined placeholder concept no different than 'ether'. 2) Can we define what "space" is and how its elements (and which of its elements) curve before we use it to build other definitions? 3) Are we truly confident that gravity propagates at the speed of light? 4) and is that truly the limit of gravity itself or is it the limit of the methods of its observation (themselves involving light and devices based on electromagnetism)? 5) Or is it the introduction of the undefined concept of a precurved "space time" our indirect acknowledgement of instantaneous propagation? The use of math can be a conceptual trap since the practice tends to provide accurate extensions cantilevering from original presumptions. Much like with Zeno's paradox, math does not negate the presumption, but merely confirms Achelies' distance to be asymptotic to a limit (the limit artificially set by our original presumption). 6) How confident are we really today that the speed of gravity is a done deal?


title: degree of freedom of a rigid body 5 or 6?

text: I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation.   But we have only three constraints, i.e. three equations that reduce 9 coordinates by 3, 9 - 3 = 6, which gives 6 degrees of freedom?? Did I miss something above?


title: Information relationship to Special Relativity

text: How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity. Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity). Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?


title: Current through two inductors after a long time

text: I'm having trouble with visualizing the following problem, which is asking me for the final, steady current in both inductors $L_{1}$ and $L_{2}$. I was thinking that after a long time, essentially the current will be stable and thus there will be no induced current in the inductors. However, how would the current in the circuit split in this case? (I suppose that we would have two short wires as the resistance of the inductors is 0). I know that the current through the resistor is: $I_{R} = \displaystyle\frac{e}{R}$ However, I really can't visualize how short circuits work, and mostly in this case in which we have two paths with zero resistance. How would the currents split? Here is a picture of the problem. ![enter image description here](http://i.stack.imgur.com/nrppf.jpg)


title: The average value of the the square of Dirac velocity operator

text: Let's have Dirac velocity operator (the case of the free particle: $$ \hat {\mathbf v} = i [\hat {H}, \hat {\mathbf r}] = \hat {\alpha}, \quad \hat {H} = (\hat {\alpha} \cdot \hat {\mathbf p}) + \hat {\beta} m, $$ where $\hat {\alpha} $ is $\hat {\gamma}_{0}\hat {\gamma}$. I need to find $\sqrt{\langle (\Delta \hat {\mathbf v})^{2}\rangle} = \sqrt{\langle \hat {\mathbf v}^{2}\rangle - \langle \hat {\mathbf v}\rangle^{2}}$. The second summand is equal to $\frac{\mathbf p^{2}}{p_{0}^{2}}$, and the first (maybe) $$ \langle \hat {\mathbf v}^{2}\rangle = \langle \hat {\alpha}^{2} \rangle = 3 \langle \rangle = 3. $$ So $$ \sqrt{\langle (\Delta \hat {\mathbf v})^{2}\rangle} = \sqrt{3 - \frac{\mathbf p^{2}}{p_{0}^{2}}}. $$ But it seems that this result is incorrect, because it is unnatural. Can anyone help me? What is the correct answer?


title: Creating electricity from mains water pressure.

text: Could someone cleverer than me help me out? I had a crazy thought going through my head the other day and I can't lay my mind to rest until I get an answer. Q. How much energy could be produced by using mains water pressure to turn a generator? And would it be feasible to install a system to feed whatever is produced back to the grid? Assuming that the system would be installed in a building where a constant water supply is needed so the generator would be turning continuously, and a rough water pressure of around 3-4 bar. Thanks in advance for any help


title: relativity and aberration of light

text: I am dealing with math and physics only at amateur level and I am writing regarding my question on relativistic aberration of light. Reading "Realtivity and Common Sense" by Hermann Bondi and some other texts I discovered that the following aberration formula is being used $$\cos a' = \frac{\cos a + v/c}{1+ (\cos a) v/c}$$ However in other physics websites the following aberration formula is being used $$\cos a' = \frac{\cos a - v/c}{1 - (\cos a) v/c}$$ I am sure that both they are correct and something I have missed. Could you help me?


title: How do instantons cause vacuum decay?

text: From what I read about on instantons (Zee, QFT in a Nutshell, pg 309-310), an instanton is a vacuum solution that maps $S^3 \rightarrow S^3$ (the boundary of Euclideanized spacetime), which comes from minimizing the Euclidean action for some Lagrangian with a nontrivial vacuum structure. I've also read (for example in Muckanov, Physical Foundations of Cosmology, pg 180-199) about how instantons can mediate quantum tunneling from one vacuum state to another. My question is: how are these two ideas/definitions of instantons related? All of the simples examples that I have looked at of nontrivial vacuum solutions involve solitons, vortices, or hedgehogs, which as far as I know cannot mediate decay from a metastable vacuum. Solitons etc are defined on spacial infinity so I know (suspect?) that the fact that an instanton lives on the boundary of space$time$ is related to its connection to rate of vacuum decay. I would greatly appreciated some simple examples/links to references as well.


title: Geodesics in Kerr

text: I'm interested in plotting the trajectories of null geodesics near an uncharged rotating black hole (described by the Kerr solution) which involves a system of first order differential equations. Kerr spacetimes are stationary and axially symmetric, and so because of the existence of these two Killing fields I'm really only interested in the "projection" of these trajectories in the $(r, \theta)$-plane. By using conservation laws the equations describing these null geodesic are given by the following system of ODE's (which can be found, specifically, at the end of $\S$62 in _Chandrasekhar's_ book _'The Mathematical Theory of Black Holes'_ , but also in just about any other book describing the geometry of Kerr black holes) $$ \rho^4 \dot{r}^2 = E^2r^4 + (a^2E^2 - L_z^2 - \mathscr{L})r^2 + 2Mr(\mathscr{L} + (L_z - aE)^2) - a^2\mathscr{L} \tag{1}$$ $$\rho^4\dot{\theta}^2 = \mathscr{L} + (a^2 E^2 - L_z^2\csc^2\theta)\cos^2\theta \tag{2}$$ $$ \rho^2\dot{\phi} = \frac{2aMrE + (\rho^2 - 2Mr)L_z\csc^2\theta}{\Delta}$$ $$ \rho^2\dot{t} = \frac{\Sigma^2E - 2aMrL_z}{\Delta}$$ Where $\dot{x^\mu}$ denotes differentiation w.r.t. some affine parameter $\lambda$. We have that $E, L_z$ and $\mathscr{L}$ are constants of motion/conserved quantities, $a$ and $M$ are the Kerr parameters such that $0 < a^2 < M^2, M> 0$ ($M$ is usually taken to be 1) and $$\rho^2 = r^2 + a^2\cos^2\theta, \quad \Delta = r^2 + a^2 - 2Mr$$ (at the moment I can't quite recall what $\Sigma$ is, but I don't think it's particularly important since I feel as though I'm really only interested in equations (1) and (2)). Equations (1) and (2) can be rewritten slightly by dividing both of them by $E^2$, yielding $$ \frac{\rho^4}{E^2} \dot{r}^2 = r^4 + (a^2 - \xi^2 - \eta)r^2 + 2M(\eta + (\xi - a)^2))r - a^2\eta \tag{3}$$ $$\frac{\rho^4}{E^2}\dot{\theta}^2 = \eta + a^2\cos^2\theta - \xi^2\cot^2\theta \tag{4}$$ Explicitly, $$\eta = \frac{\mathscr{L}}{E^2} \quad and \quad \xi = \frac{L_z}{E} $$ In this form it seems as though (according to $\S$63 of _Chandrasekhar_ ) the only restriction placed on the constants $E, \eta$ and $\xi$ is that $\eta > 0$. I have a couple questions regarding how (theoretically) one would go about numerically approximating (and subsequently plotting) the $r$ and $\theta$ solutions of this system of differential equations. First, to me, it seems as though since I'm only interested in the $(r, \theta)$ motion of any given null geodesic, would I then only have worry about numerically solving (3) and (4), that is, even though there are 4 equations governing the total motion of the geodesic, since (3) and (4) are independent of $t$ and $\phi$ it seems like equations (3) and (4) are kind of standalone equations governing the motion in the $(r, \theta)$-plane. Is this assertion of mine correct? Need I only deal with equations (3) and (4)? Another concern of mine is that; I've read that the geodesics in and around a black hole (with $\eta > 0$) oscillate symmetrically about the equatorial plane ($\theta = \pi/2$) and the geodesics which do this 'take infinitely long' to fall into the black hole, that is, as they oscillate toward the black hole it takes infinite coordinate time to fall into the black hole. This fact concerns me because when plugging (3) and (4) into Mathematica, the equations need to be parameterized, so really the system looks like $$ \frac{[r(\lambda)^2 + a^2\cos^2\theta(\lambda)]^2}{E^2} \dot{r}(\lambda)^2 = r(\lambda)^4 + (a^2 - \xi^2 - \eta)r(\lambda)^2 + 2M(\eta + (\xi - a)^2))r(\lambda) - a^2\eta$$ $$\frac{[r(\lambda)^2 + a^2\cos^2\theta(\lambda)]^2}{E^2}\dot{\theta}(\lambda)^2 = \eta + a^2\cos^2\theta(\lambda) - \xi^2\cot^2\theta(\lambda)$$ In which case, does this 'infinite amount of coordinate time' it takes to fall into the black hole affect how big of a domain one would have to allow for $\lambda$? if it does; theoretically this isn't an issue but practically it's not really feasible. These last two questions have a little more to do with implementation and less to do with the theory. Is dealing with these ((3) and (4)) equations actually any more useful than simply working directly from the geodesic equations, from a numerical stand point, that is? And lastly, I've recently found out that one can decouple (3) and (4) and I was wondering (again from a numerical standpoint) if there's any benefit in working with decoupled equations as opposed to these coupled ones? Any input on any of my above questions/concerns would be greatly appreciated.


title: Color of cementite

text: I have a question about micrograph of steel. In eutectoid steel, the pearlite exists as grains, within each colony the layers are oriented in essentially the same direction, which varies from one colony to another. The thick light layers are the ferrite phase, and the cementite phase appears as thin lamellae most of which appear dark. However in photomicrographs of a hypereutectoid steel (e.g.1.4 wt% C) , we can observe the proeutectoid cementite appears light. Why there is such a difference?


title: Renormalization is a Tool for Removing Infinities or a Tool for Obtaining Physical Results?

text: Quoting Wikipedia: > renormalization is any of a collection of techniques used to treat > infinities arising in calculated quantities. Is that true? to me, it seems better to define renormalization as a collection of techniques for adjusting the theory to obtain physical results. I'll explain. According to Wilson's renormalization group, a quantum field theory always inherently has a cutoff parameter, so in any case integrals should be done only up to the cutoff, so there are no infinite quantities. Yet the results are still not consistent with observation if you don't renormalize the calculations (e.g. using counterterms). I am correct? Is it true that the usual presentation of renormalization as a tool for removing divergences is a misinterpretation of the true purpose of it?


title: Commutation relation with Hamiltonian

text: How do we get $[\beta , L] = 0$ , where $L$= orbital angular momentum and $\beta$= matrix from Dirac equation?


title: Pressure derivatives of the bulk modulus of materials? (Solved)

text: Isotherm bulk modulus is defined by $$K=\rho\left(\frac{\partial p}{\partial\rho}\right)_T=-v\left(\frac{\partial p}{\partial v}\right)_T$$ wherein $v=\tfrac{1}{\rho}$ is the specific volume. Studying around the Birch- Murnaghan equation of state and some other related topics I saw everyone talk about the pressure derivative of the isotherm bulk modulus very happily as if it is a function of e.g. pressure, like any other thermodynamic quantity usually is, so that one can calculate its nontrivial pressure derivative for it, even a nontrivial isotherm pressure derivative $\left(\tfrac{\partial}{\partial p}\right)_T$ of it. See for example this link that introduces the quantity $$K'=\left(\frac{\partial K}{\partial p}\right)_T$$ (the reference cited uses $B$ instead of $K$.) But to me its very odd. Let me compute the isotherm pressure derivative of $K$ so you understand why I'm confused: \begin{align} K'&=\left(\frac{\partial K}{\partial p}\right)_T\\\ &=\left(\frac{\partial}{\partial p}\right)_T \biggl[\rho\left(\frac{\partial p}{\partial\rho}\right)_T\biggr]\\\ &=\left(\frac{\partial\rho}{\partial p}\right)_T\left(\frac{\partial p}{\partial\rho}\right)_T+\rho\left(\frac{\partial^2 p}{\partial\rho\partial p}\right)_T\\\ &=1+0\\\ &=1 \end{align} A constant, not a function of pressure at all. Oh I don't mean that the Bulk modulus is not an ordinary thermodynamic quantity, but its isotherm pressure derivative can not be function of pressure as the above calculations suggest, yet isentropic pressure dependence could be non-trivial and etc. No reference could I have found to have computed this derivative, everyone who I encountered have only reported to have measured the quantity experimentally. That everyone say it is a function of pressure and experiments validate their idea makes me think I have missed a point in my computations. But there is a greater point that suggests i am wrong, that using the specific-volume-based form of the bulk modulus when is differentiated will result in $-1$ instead of $1$: \begin{align} K'&=\left(\frac{\partial K}{\partial p}\right)_T\\\ &=\left(\frac{\partial}{\partial p}\right)_T \biggl[-v\left(\frac{\partial p}{\partial v}\right)_T\biggr]\\\ &=-\left(\frac{\partial v}{\partial p}\right)_T\left(\frac{\partial p}{\partial v}\right)_T-v\left(\frac{\partial^2 p}{\partial v\partial p}\right)_T\\\ &=-1-0\\\ &=-1 \end{align} So that the contradiction $1=-1$ suggests that I must be wrong, but where is the point I am missing? Aren't the derivatives interchangeable in computing $\left(\frac{\partial}{\partial p}\right)_T \left(\frac{\partial p}{\partial\rho}\right)_T$ ( _I understand the Thermodynamic's underlying geometry can be non-euclidean in different cases_ ) or else? So the simple question is this: where have I make a mistake in the above calculations? Regards. _**UPDATE_. Wrong calculation led into an apparent contradiction, but as could be guessed it was wrong so no contradiction at all. It was a math question which is now resolved, so maybe better to be deleted from ere as it had no physics questioned behind it.**


I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also contains the Boltzmann weight. So, this is an Ising model. I am confused as there are more than one interaction between a pair of sites. How will I keep that consideration when I calculate the Edwards-Anderson Hamiltonian. Thanks in advance for your answers. The attachments cannot be posted as I am a new user. They are available at http://www.physicsforums.com/showthread.php?t=649548 . A quick supplementary question: should I model it as a frustrated Ising system?


title: How to determine velocity and mass of asteroid 2011 CQ1

text: As you may have seen, a small asteroid had a near miss with the earth a few days ago. As a physics teacher teaching momentum, I think this could be an excellent problem for my students, however, I don't know how to determine the mass or velocity of this object. I could get a decent estimate of the mass simply by assuming its a spherical rock, but I still don't see how to determine the velocity. I can get the orbital elements: ![Orbital Elements of 2011 CQ1](http://i.stack.imgur.com/m2jch.png). Is the 'V' term the magnitude of the velocity, and if so, what are those units? km/s? How could I get a heading for the asteroid?


title: Bosonic M-theory and moonshine gravity

text: Last month I asked about a 27-dimensional origin of the heterotic string. Now I'm looking at Witten's "Three-dimensional gravity revisited", where he proposes that pure gravity on AdS3 is dual to the "monstrous moonshine" CFT, which is bosonic string theory on a "Leech lattice" orbifold. Wouldn't that mean we were also talking about bosonic M-theory (the 27-dimensional theory) on AdS3 x R24/Λ24? But that would mean that on this occasion, the emergence of the "M" dimension was the same thing as the emergence of the radial AdS dimension. Does that even make sense?


title: Is electronvolt a mass or an energy unit?

text: I always thought electronvolt was an unit of energy and I knew its definition, but in these days I got some doubts because I saw 2 times it was used as an unit of mass: in my school textbook, in an article about the 4 fundamental forces, it was used as a mass unit; and in a famous magazine I read the same thing (same arguments) > In physics, the electron volt (symbol eV; also written electronvolt[1][2]) > is a unit of energy equal to... particularly there was a table about sub-atomic particles with .... spin | charge | **mass(GeV)** | ...


title: An approachable example of a field with a "mass gap"

text: **Preamble** : I have come to believe that alot of difficulties in explaining physics to people of all levels comes from the relatively mundane idea of a wave equation with a mass gap $$\left(-\partial^2_t +\nabla^2 -m^2\right)\phi = 0$$ or more generally a field that does not have propagating modes in some frequency band. Being able to demonstrate this behavior would be useful in explaining the differences between conductors and insulators, the difference between the Higgs field and the Higgs boson and other things - it would even be useful explaining gapless fields like EM to be able to refer to a gapped field. However despite being an extremely common and tame phenomenon I can't think of any _approachable_ examples of gapped wave equations. I can't think of anything where I could say, even to a junior undergraduate "its a gapped field, just like X". Nor can I think of a system I could show a video of, or a demonstration that would give intuitive understanding of gapped fields. **So my question is:** does anyone know of a "gapped system" that would be useful for pedagogical purposes? That is, a system that has a continuum of propogating frequencies and a continuum of non-propogating frequencies. The best I can think of is the following experiment: by scattering debris on the bottom of a pan filled with shallow water, you can localize the surface waves, but I assume the high frequency waves should still propagate. I don't like this for a variety of reasons, least of which is you are using Anderson localization to explain a mass gap. To reiterate I am looking for something which can give understanding - so something intuitive or something that can be played with until it becomes intuitive. I know there are lots of common things that are gapped (metals and EM radiation for example), but I can't think of anything that is pedagogically useful.


title: Propagation of uncertainty when integrating or differentiating

text: Lets say I have a polynomial $ax^4 +bx^3 +cx^2 +dx +e$ and the uncertainties on each coefficient. Now I need to calculate the tangent at some points as well as some areas under this curve. How would I find the uncertainties on those results? My initial idea was to simply differentiate or integrate it and then use the partial derivatives method on the obtained expression(ignoring the error on "x" itself). ex. $slope = 4ax^3 + 3bx^2 +2cx +d$ $$\sigma_{slope}^2 = (4x^3)^2*\sigma_a^2 + (3x^2)^2*\sigma_b^2 + (2x)^2*\sigma_c^2 + \sigma_d^2$$ But higher x will make this error much higher which doesn't seem logical to me. I couldn't find the answer online and my book doesn't cover it. Whats the right way to propagate uncertainties in such cases?


title: Energy-momentum tensor for dust

text: We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's four-velocity. I'm trying to derive the dust energy momentum tensor from the equation $T_{αβ}=-\frac{2}{\sqrt{-g}}\frac{δS_M}{δg_{αβ}}$ but I'm getting the wrong answer. The action for dust is $$S=\int -\rho_0\sqrt{-g}d^4x.$$ Thus $$\frac{\delta S}{\delta g^{\alpha\beta}}=\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}\sqrt{-g}-\rho_0\frac{\delta \sqrt{-g}}{\delta g^{\alpha\beta}}.$$ To evaluate $\frac{\delta (-\rho_0)}{\delta g^{\alpha\beta}}$, I define $K_\alpha=\rho_0 v_\alpha$. Then $\rho_0=\sqrt{g^{\alpha\beta}K_\alpha K_\beta}$ and thus $\delta \rho_0=\frac{1}{2\rho_0}K_\alpha K_\beta \delta g^{\alpha \beta}.$ It follows that $$T_{\alpha\beta}=\rho_0 v_\alpha v_\beta-\rho_0 g_{\alpha\beta}.$$ There's an extra term that I can't get rid of. Any idea where I went wrong?


title: How to avoid getting shocked by static electricity?

text: sometimes I get "charged" and the next thing I touch something that conducts electricity such as a person, a car, a motal door, etc I get shocked by static electricity. I'm trying to avoid this so if I suspect being "charged" I try to touch something that does not conduct electricity (such as a wooden table) as soon as possible, in the belief that this will "uncharge me".   * Is it true that touching wood will uncharge you?   * How and when do I get charged? I noticed that it happens only in parts of the years, and after I get out of the car...


title: why is mechanical waves faster in denser medium while EM waves slower?

text: Why is it that mechanical waves/longitudinal waves/sound travel faster in a denser/stiffer medium as in steel compared to say air, while EM waves/trasverse waves/light travels slower in a (optically) denser medium as in say glass? I know that for mechanical waves,$$v \varpropto \sqrt{T}$$, where $T$ stands for Tension and $$c \varpropto \sqrt{\frac{1}{\epsilon_0\mu_0}}$$ Any physical interpretation might be helpful. EDIT: What I understand from this is that, since higher tension implies stronger intermolecular interaction, which leads to better conveyance of energy across the material which causes the waves to move faster. But this analogy doesn't fit at all for light waves. Why?


title: Why don't the actinides start at neptunium?

text: In their physical and chemical properties, actinium, thorium, protactinium, and uranium are respectively similar to lanthanum, hafnium, tantalum, and tungsten, and thus would seem to belong respectively to the third, fourth, fifth, and sixth columns of the periodic table. Only with neptunium does this horizontal progression (arguably) break down. So why aren't elements 89 to 92 distributed across columns 3 to 6 of the table?


title: Heuristic argument for the temeprature dependence of specific heat in the "low" temperature regimes

text: Here by "low temperature" I meant it in the scale of the characteristic $\hbar \omega$ of the system. One can calculate and show that in the low temperature regime $C_V$ of phonons goes like $T^3$ for free fermions it goes as $T$ and for free bosons its like $T^{\frac{3}{2}}$ In most of the above cases the power comes from some complicated calculation.   * I was wondering if there is a heuristic argument for "understanding" these powers.    * Further is the power of $T$ characteristic of the system? Like if in a substance one sees the low temperature behaviour of $C_V$ to be going as $T$ then does it says something like that the effective degree of freedom of the system is that of free fermions?   * Are there other examples of such calculation? Like typical systems where one can calculate the low temperature dependance of $C_V$?


title: Why do Bell tests give perfect correlations?

text: Suppose some decay process emits 2 electrons in opposite directions, and their spin is measured by a Stern-Gerlach type device in a particular direction, say Sz. The books say that if 2 detectors have the same orientation, then the spin measurements are 100% correlated (or anti-correlated). If one has Sz=+1/2, then the other has Sz=-1/2. But what is the quantum state of the electrons before the measurements? It seems to me that if it is anything but an eigenstate of Sz, then the perfect correlations are impossible. Is there any quantum mechanical explanation of the perfect correlations, other than with an action-at-a-distance collapse of the wave function?


title: Theoretical treatment of Hydrogen bond?

text: I would like to understand how the Hydrogen bond can be described through the Schroedinger equation. I don't need numerical methods that one uses them to simulate it, rather I need its treatment from theoretical point of view that can show also the probability of that electron will go around first and second atom, I searched the internet but couldn't find any treatment that shows what I mentioned. Can anyone offer an explanation?


title: Time for a wind-battred door to slam

text: Foolishly leaving the window too widely open, earlier today my bathroom door slammed shut with the wind. How could I theoretically work out the time it takes to close? Would it be impossible to reach an answer based on the conservation of energy, or does the nature of the wind's force ruin that method? That was my question, what follows are just hunches. I'm not sure how to approach the problem- the angular velocity is $w= \frac{d \theta}{dt}$, so the force on a point on the door is $\propto (u-wrsin\theta)^2$ (if you measure the angle so that it is 0 when the door is parallel to the wind's motion, u is the wind's speed and r the distance from the door's pivot). I assume a stationary thing hit b the wind is analogous to a moving thing hitting the stationary wind. Am I right in thinking you would then integrate to get the torque on the entire door, and from that, knowing the moment of inertia of the door, find the time for $\theta=90$? I am unfamiliar to most of angular mechanics, if that helps formulate a good answer.


title: Speed of light and virtual particles

text: After becoming extremely bored while studying for an Afrikaans exam, I started thinking about virtual particles. So, can light (photons) interact with virtual particles (even though they only exist for a very short time)? And can these virtual particles "slow down" light in the same way as atoms delay light when travelling through a medium?


title: Too small error on the calculus of wavelenght

text: I have this function: $$\lambda=d \sin(\arctan(\frac{x}{z}))$$ and I want to find its absolute error. $d$ is a constant ($10^{-6}$), $x =(0.716 \pm 0.001)$ m, and $z=(1.000 \pm 0.001) $ m. For the error of $\lambda$ I have used $$\Delta\lambda=\sqrt{\left(\frac{\partial\lambda}{\partial x}\Delta x\right)^2 + \left(\frac{\partial\lambda}{\partial z}\Delta z\right)^2}\\\ \Delta\lambda=d\sqrt{\left(\frac{z\Delta x}{(z^2+x^2)(\sqrt{\frac{x^2}{z^2}+1})}\right)^2 + \left(\frac{x\Delta z}{(z^2+x^2)(\sqrt{\frac{x^2}{z^2}+1})}\right)^2}$$ And I have obtained $6.6 \cdot 10^{-10}$ m. I expect a much larger error. Is this wrong?


title: Logarithmic intensity formula factor?

text: In formulae for sound intensity and sound pressure, there is a logarithm part for the logarithmic increase of the quantity we're studying, and also a numerical factor 10 or 20 depending on the formula: $L=10(or\ 20)\log\frac{I}{I_0}$ If the crux/intuition of the formula is already summed in the logarithm part, what can we use the 10 or 20 for? I thought of it as to "normalize" the obtained number (for instance 10log(2)) to human understanding (like 3dB instead of 0.3db without the 10), but it seemed too anthropic?


title: Does velocity or acceleration cause time dilation?

text: What causes time dilation? Acceleration or velocity? I've seen multiple comments on this forum that assert velocity is the cause, but that doesn't seem right to me. You can't have velocity without acceleration. It's the inertial force with acceleration that breaks the symmetry. My understanding is that the asymmetry is where the inertial frame changes. Measuring time between two objects with different inertial frames is where you have time dilation. When the acceleration ends, the object is effectively at rest in a new inertial frame and has velocity relative to another object in the original inertial frame. In other words, acceleration (changing reference frames) is the cause...velocity and time dilation is the effect. Am I right about this? If there are flaws in my logic I'd like to find and correct them.


title: Which surface to use in Ampere's law?

text: In calculating the current enclosed by an Amperian loop, one must, in general evaluate an integral of the form $$I_\text{encl} = \int \mathbf{J}\cdot\mathrm{d}\mathbf{a}$$ The trouble is, there are infinitely many surfaces that share the same boundary line. Which one are we supposed to use?


title: How do you define a reversible path for general processes?

text: The equation $dS = \frac{\delta Q}{T}$ is only defined for a reversible path. Given a irreversible path we typically calculate the entropy by choosing a reversible path from the same initial to final state. All examples I have seen involves some sort of reservoir that loses entropy such that $\Delta S_{system} =\Delta S_{original}+\Delta S_{res}= 0 $ for the total system in the reversible case, where by `original' I mean the system that was originally in consideration that I had to attached a reservoir to in order to find a reversible path. My question is this: (1) In the scheme of attaching a reservoir to our system, how would we generally find a reversible path to calculate the change in entropy for a system? (2) Depending on the answer to the above: Specifically, if I mix a bunch of liquids together (clearly irreversible) what is the reversible path I could use to find the change in entropy for the irreversible case, and does it involve a reservoir?


title: Noether's theorem in the realm of superfluids

text: In 1969 Keith Moffat showed helicity conservation for ideal fluids such as liquid Helium. This work is proving seminal in our understanding of turbulent flows and viscous fluids. In the case where helicity is conserved, namely for superfluids is there an underlying symmetry related to the conservation law?


title: conformal, Weyl transformations, apparent discrepancies and confusions

text: Because of the apparent discrepancy of how some CFT and GR books define conformal transformation unlike in string theory area, I wanted to get rid of all the confusion from McGreevy's lecture notes: > http://ocw.mit.edu/courses/physics/8-821-string-theory-fall-2008/lecture- > notes/lecture10.pdf I have understood and happy with the definition of conformal group and conformal transformation now, but I have the following two questions:   1. In page 2, where he defined conformal invariance, did he mean invariance under the diffeomorphism part only (which gives the conformal group)? The reason I think so is that he changed the metric which according to his definition of conformal transformation don't change. So, the reason I am asking is that I thought conformal invariance will be a suitable name for a theory, invariant under conformal transformation.   2. Just to make sure: the transformation in page 2: $\delta g_{\mu\nu}=2\lambda g_{\mu\nu}$ and $\delta g_{\mu\nu}=\delta\Omega(x)\delta g_{\mu\nu}$, these are results of diffeomorphism ..right? and not Weyl in any way. In other words, what are these transformations? All my questions are for general dimension and general space time (classically of course).


title: Pouring water in an aircraft while upside down?

text: What is the physics behind the following photo? Someone had told me that this is because the aircraft might be moving towards a lower altitude, but I am still not sure. ![Enter image description here](http://i.stack.imgur.com/1f2HZ.jpg)


title: What makes zinc reactive?

text: What makes zinc so desperate to give up its electrons? Silver has one extra electron, and zinc has two, but why is zinc more reactive?


title: Does higher voltage and plate area cause more separation between leaves in electroscope?

text: Does higher voltage cause more separation between leaves in electroscope? Will higher voltage and plate area attract more charge since Q = CV.![enter image description here](http://i.stack.imgur.com/hYXV5.png)


title: Force on a line

text: Say you have a rigid line of mass $m$ and length $\ell$ along the $x$ axis and you apply a constant force $f$ at one end in a direction that is always perpendicular to the line, starting in the $y$ direction. Assume there are no other external forces other than the applied force. How would you find the position and rotation of the center of the line at time $t$ (or $dt$)? Would any parts of this line remain motionless? How does the answer change as you move the force towards the center of mass?


title: Effects on a tree exposed to the vacuum of Space

text: My wife's kindergarten class asked, "What would happen to a tree planted on the moon?" Aside from the obvious that it would die from lack of water/air, what physical effects would happen to a tree exposed to the vacuum of Space?


title: Help in understanding the derivation for Fresnel Distance?

text: An Aperture of size $a$ illuminated by a parallel beam sends diffracted beam (the central maxima) in angular width approximately $λ/a$. Travelling a distance $z$, it aquires the width $zλ/a$ due to diffraction. The distance at which this width equals the size of the aperture is called the **fresnel distance**. But as far as I know, the angular width of central maxima is given by $2λ/a$ and not $λ/a$. Help?


title: How come neutrons in a nucleus don't decay?

text: I know outside a nucleus, neutrons are unstable and they have half life of about 15 minutes. But when they are together with protons inside the nucleus, they are stable. How does that happen? I got this from wikipedia: > When bound inside of a nucleus, the instability of a single neutron to beta > decay is balanced against the instability that would be acquired by the > nucleus as a whole if an additional proton were to participate in repulsive > interactions with the other protons that are already present in the nucleus. > As such, although free neutrons are unstable, bound neutrons are not > necessarily so. The same reasoning explains why protons, which are stable in > empty space, may transform into neutrons when bound inside of a nucleus. But I don't think I get what that really means. What happens inside the nucleus that makes neutrons stable? Is it the same thing that happens inside a neutron star's core? Because, neutrons seem to be stable in there too.


title: Why electrons are relativistic in Graphene and non relativistic in vacuum?

text: If a free region in space has a potential difference of one volt, an electron in this region will acquire kinetic energy of 1 eV. Its speed will be much smaller than the speed of light hence it will be a non relativistic electron. On the other hand conduction electrons in graphene are relativistic for the same potential difference. Question is how come that when the electrons are in vacuum they are non relativistic, and when they are inside Graphene they are relativistic (for the same potential difference)?


title: QFT basics for Klein-Gordon fields

text: I am teaching myself QFT from Peskin for next years maths course and I have two questions:   1. What is a c-number? Is it a complex number, and if so why does it mean, $[\hat{\phi}(x),\hat{\phi}(y)]~=~<0|[\hat{\phi}(x),\hat{\phi}(y)]|0> {\bf 1}$.   2. Is $[\hat{\phi}(x),\hat{\phi}(y)]|$ still equal to zero if $(x^{2}-y^{2})<0 $ ?