the equations of motions are more symmetric - both are just wave equations with the charge density and current as the sources respectively
so in Lorenz gauge, you can combine them into the 4-potential and have the equation for it just be the wave equation with the 4-current as the source, i.e. the Lorenz gauge is what you want to use if a relativistic/covariant formulation is important to you
@JonCuster @JohnRennie For finding the possible dislocation reaction (breaking into partials)is it at all necessary to check x y and z.Or are the square of Burgers vector sufficient?
Zone melting (or zone refining or floating-zone process or travelling melting zone) is a group of similar methods of purifying crystals, in which a narrow region of a crystal is melted, and this molten zone is moved along the crystal. The molten region melts impure solid at its forward edge and leaves a wake of purer material solidified behind it as it moves through the ingot. The impurities concentrate in the melt, and are moved to one end of the ingot. Zone refining was invented by John Desmond Bernal and further developed by William Gardner Pfann in Bell Labs as a method to prepare high purity...
The impurities are more soluble in the liquid than in the solid, so as the liquid region travels along the rod the liquid becomes increasingly enriched in impurities.
The impurities don't flow anywhere. They just concentrate in the liquid.
Then when the liquid zone reaches the end of the rod it is allowed to solidify, and the end of the rod with all the impurities is cut off.
@schn Choosing a gauge is like choosing a coordinate system. Always remember that V and A are not physical. The Coulomb gauge is designed to make electrostatics easy.
Hi @JohnRennie, can I ask about the conversion of the quadrupole moment tensor into TT gauge? From what exactly are we converting, and why does it involve the projection tensor?
I am having a tough time to understand as to why $g_{00}$ is taken to be 1 in the FLRW metric. The book says something like “ we can always choose our time coordinate to be the proper time”. Why then we didn’t do that in Schwarzschild metric? Would, anybody here like to explain that?
@JohnRennie This guy is a professor at TIFR!! (Tata institute of Funadmental Research which is quite reputable in India) I'm surprised by the behavior of someone so senior D:
One more question though, by projecting it, why can we simply ignore the non-transverse part of the tensor? Basically I have the same question as physics.stackexchange.com/questions/320889/…
:56854201@Slereah so does that mean we could have a “type” of Schwarzschild metric written in coordinates where $t$ is the proper time and $r$ is the proper distance, of course by having some other terms in the metric like, $g_{tr}$ and $g_{r\theta}? If that is correct why don’t we do so? The interpretation of $t$ and $r$ coordinates would be so nice. Also, can we then choose $t$ to be the proper time in any metric in GR ?
@Slereah @JohnRennie So is there a special reason to choose the $t$ and $r$ coordinates such that $g_{00} \neq 1$ $\g_{rr} \neq 1$ in the Schwarzschild case, when in fact we could and then also have a nice interpretation of the $t$ and $r$ coordinates? Is that more beneficial in some way?
@JohnRennie @Slereah Ahh I see. Thanks ! I was thinking that the only possible choice of coordinates in the Schwarzschild case was the way they are chosen. Now, at least theoretically it is clear we could have in fact chosen any other coordinates too which would respect spherical symmetry.
But is there any specific reason to choose $g_{00}=1$ in FLRW? There we would be having a diagonal metric nevertheless (once we have the metric for the space like hypersurface). So we could very well choose not to have $g_{00}=1$. And then plug the metric in the Einstein equations to see what all possible values could $g_{00}$ take. Is that right?
Sorry, but would anyone like to tell me the issue with Tejinder P Singh. I happened to met him a few weeks gaon in a QM foundations conference. He works with Bassi on the GRW theories and such. Was just curious to see the same person surface here
@Slereah where the time coordinate is not the conformal time, where $g_{00}\neq 1$
@MoreAnonymous No, I'm serious. If you look at my answers on GR you'll see that they are mostly explaining GR in a non-technical way, or at least with the bare minimum of maths.
I wanted to learn enough GR to understand the principles by which it worked, and I managed to do that. But my grasp of the intricacies of differential geometry is very limited.
@Slereah Buy the supermarket own brand! 99.999% of the effectiveness of toothpaste comes from brushing well, and the stuff in the toothpaste makes little difference. Most of it is just abrasive (silica), surfactant and flavouring anyway.
If your teeth are still forming then the fluoride in toothpaste can help make teeth stronger, but for those of us long paste those years even this doesn't matter.
@Slereah The silica in toothpaste is a silica precipitate with a low hardness. Sand is much harder than teeth so if you use sand you will quickly have no teeth left to clean.
@MoreAnonymous that doesn't mean Feynman didn't also have a deep technical understanding of what he was doing - beware of the fallacy that because some people are good at explaining things non-technically in a way that sounds sensible, a non-technical understanding suffices to argue/reason about the topic
@Slereah I've found that phenomenologists and other physicists focused on applying QFT to specific scenarios mostly focus on clever methods to compute how the QFT amplitudes translate to stuff in colliders and as such usually care very little about anything in "theoretical" QFT beyond how to use Feynman diagrams - maybe they were like that?
@RewCie What about the Wiki article is unclear to you? This has been considered pseudoscience ever since we understand enough biochemistry to see that the same reactions can happen in a test tube that happen in a cell.
@MoreAnonymous many unis have small research groups under a single professor, and that professor is the only specialist in their field at that place - if you need to get a committee together, you have no choice but to pick non-specialists
@RewCie physicists are ultimately pragmatists. If a theory works it's a good theory. If vitalism improved on our current theories we'd adopt it. But it doesn't. And that's all there is to it really.
Can unit vectors in curvilinear coordinates also be defined as $ \vec{e_i}=h_i\nabla x_i$? I'm only familiar with $ \vec{e_i}=\frac{1}{h_i}\frac{\partial\vec r}{\partial x_i}$, and I'm unable to prove the former using the latter.
It's totally a different feeling when you miss class lectures and don't listen to teachers, solve questions on board and get appreciated in front of everyone.
@MoreAnonymous if you're international anyway there's no need to focus on the UK, e.g. there's several unis with good string theory research in Germany, too (e.g Munich, Heidelberg, Hamburg (DESY))
I don't know the situation in the UK but I suspect you're blinded by PR there, too, and there's more places where you might find the kind of research you're looking for
@JohnRennie There's a lot of stupid politics going on. For example, if a prof sets an exam which students are finding difficult to pass without memorising the students remove him via the means of negative feedback
These days Nottingham (the city not the university) has a major crime problem. I used to go there on business, and the local radio was full of stories of how many people had been stabbed the previous day.
@JohnRennie Also maybe I'm wrong but compared to my classmates they had far inferior math abilities to mine. Which suggested to me something went terribly wrong in UK schooling eduction system
The Definite Integral Problem (with a twist)
In the Riemann integral one essentially calculates the area by splitting the area into $N$ rectangular strips and then taking $N \to \infty$.
Here's something I asked myself related to the Riemann integral.
Let's say I split the area into say $3$ st...
@JohnRennie @JohnRennie Thanks for the article. The time coordinate in that link is not, the proper time, right? And also is $H$ there a constant or a variable? In one of the previous comments above you mention that in the Gullstrand Painleve metric the time coordinate for some observer is the proper time. But in that metric $g_{00}$ is not equal to 1. So why does the time coordinate there correspond to the proper time.
H is the Hubble parameter so it is time dependent not a constant.
In the GP metric the time coordinate is the elapsed time for an observer falling freely into the black hole i.e. it is the proper time for that observer.
@JohnRennie And in the PDF you linked, eqn 1.38, the time coordinate is not the proper time ( $g_{00}\ neq 0), but it’s just the Schwarzschild metric with M=0 and a cosmological constant. Does that eqn suggest an expansion of the universe ?
@JohnRennie and will the t coordinate in that linked answer correspond to the proper time for any observer (i.e one for whom H =0)?
If I want to prove that for any scalar field $f:\;\mathbb{R}^3\to\mathbb{R}:$
$$\int_V \boldsymbol{\nabla} f\;\mathrm{d}V=\int_{\partial V} f\;\mathrm{d}\mathbf{S}$$
Can I apply the divergence theorem to $\mathbf{a}_1=(f,0,0),\;\mathbf{a}_2=(0,f,0),\;\mathbf{a}_3=(0,0,f)$ and then stack the equal...
principal $\rm G$-bundles over $X$ are classified by homotopy classes of maps from $X$ to $B\rm G$
We love $\mathbb{R}^{3, 1}$ which we can shuffle around to $\mathbb{R}^4$ and then $S^4$ for kicks, and $\pi_4(B\text{SU}(2)) = \mathbb{Z}$, call this number A, which classifies isomorphism classes of principal $\text{SU}(2)$ bundles over $X$
Then there's also the second Chern class which sends $c_2 : B\text{SU}(2) \rightarrow K(\mathbb{Z}, 4)$. Since we're only working up to homotopy equivalence, principal $\text{SU}(2)$ bundles over $X$ can be classified by elements of $H^4(X, \mathbb{Z})$
Two questions here, is the Chern class map also preserve homotopy classes? I would assume it does, since the composition has to respect that.
Second, taking $S^4$ again, the "instanton number" is an element of $H^4(S^4, \mathbb{Z})$. Is this the same thing as "A" in the previous (less general) derivation?
This might be really obvious but I'm blanking out here
For the second question, in the case of $X=S^4$, you have $H^4(S^4,\mathbb{Z}) = \mathbb{Z}$ and so the Chern class is just a number we call the instanton number
@NiharKarve In this case I'd probably rather try it on math.SE since it's much more likely to be answered by a "mathematical" gauge theorist than a "physical" one
In mathematics, the method of steepest descent or stationary-phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.
The integral to be estimated is often of the form
∫
C...
what additional details would you like? The partition of unity trick reduces the integral into a sum of integrals of functions that each just have a single saddle point, uses the original asymptotic for each of these, then sums them together again
the path integral is just a way to calculate the amplitudes, the copenhagen interpretation is still possible, since it just states that that particle collapses to such a state with the given probability
@Slereah any idea if we can get back schrodinger’s eqn from dirac’s eqn. Dirac proceeded the other way round, I know. But I was trying to see if under proper non relativistic assumptions, the opposite is possible. I got into trouble because Dirac eqn is 1st order is the time as well as space coordinates
In non-QFT, the path integral is just a different way to compute amplitudes and expectation values. In QFT, the waters get a bit murkier since it's technically unclear whether or not it is equivalent to the operator formalism in general, but in neither case does it have anything to do with interpretation
although of course you might argue it lends itself a bit more naturally to interpretations like MWI or consistent histories than it does to Copenhagen or other collapse-y interpretations
@Charlie the path integral over arbitrary fields is not mathematically well-defined so you cannot ask whether it is equivalent to the operator formalism before you've figured out its definition :P
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
\begin{eqnarray}
P & = & \dfrac{i\hbar}{2mc^2}\left(\Phi^*\dfrac{\partial\Phi}{\partial t}-\Phi\dfrac{\partial\Phi^*}{\partial t}\right) ...
@Slereah looks interesting. Let me see. Looks interesting.
@bolbteppa but that doesn’t mean that the correct non relativistic should always be T-V. Further I can always (should always) be able to derive the correct non relativistic eqns from the relativistic ones (which I get by the relativistic action)
This is a problem that requires the extension to general relativity. It is not possible to solve this problem remaining in special relativity.
Make the relativistic ansatz (in 2d spacetime)
$L = - m c \sqrt{-g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}$
where the dot is with respect to the geodes...
If you can get a free particle Schrodinger equation from a free particle relativistic equation such as Klein-Gordon or Dirac that's enough before things go absolutely haywire
Sorry to bother u all, Could u please check the statement mentioned below and make the possible correction, {statement: MM experiment: Results and conclusion: (I) No deflection detected in the fringes pattern, So there is no 'constant relative speed of light' with respect to any medium hence, No Aether Medium exist. (II)The speed of light will be the same in every direction in a particular inertial frame of reference.}