Metric Definition and 1000 Threads

I understand from the wiki entry on the Einstein-Hilbert action that: $$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$ What is the following? $$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$ Is there a place I could look up such GR expressions on the internet? Thanks

Does anyone have a reference or solution for a parallel plate capacitor in the Rindler metric? I'm particularly interested in the case where the capacitor plates are in the xz or yz planes, z being the direction of the acceleration. The motivation is to get an idea how a transmission line...

Hello all, let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##. In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable. We know that the Lorentz boost at every point of the...

The ansatz for the 5D metric is \begin{equation} G_{\mu \nu}= g_{\mu \nu}+ \phi A_{\mu} A_{\nu}, \end{equation} \begin{equation} G_{5\nu} = \phi A_{\nu}, \end{equation} \begin{equation} G_{55} = \phi. \end{equation} This information was extremely enlightening for me, but what's the analogous...

Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e., ## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...

In linearized gravity we define the spatial traceless part of our perturbation ##h^{TT}_{ij}##. For some reason this part of the perturbation should be gauge invariant under the transformation $$h^{TT}_{ij} \rightarrow h^{TT}_{ij} - \partial_{i}\xi_{j} - \partial_{j}\xi_{i}$$ Which means that...

Hello, I've always heard that the choice of signature for the metric was just a matter of convention, i.e. taking (+---) or (-+++) had no physical impact. The groups O(1,3) and O(3,1) being isomorphic it made sense to me. However, I came across an article discussing the Pin(1,3) and Pin(3,1)...

My attempt: Realize we can work in whatever coordinate system we want, therefore we might as well work in the rest frame of the fluid. In this case ##u^a=(c,\vec{0})##. The conservation law reads ##\nabla^a T_{ab}=0##. Let us pick the Levi-Civita connection so that we don't have to worry about...

Apparently "centibillionaire" is a term to describe someone worth over $100 billiion ( 100\cdot 10^9 \rm\ dollars =10^{11}\rm\ dollars=$100,000,000,000). (from 2019)...

Is there a spherically symmetric metric that doesn't have a singularity in the middle of it(like the schwartzchild metric). Something like our planet.

On the way to Kruskal coordinates, Carroll introduces coordinates ##\left(v^\prime,u^\prime,\theta,\phi\right)## with metric equation$$ {ds}^2=-\frac{2{R_s}^3}{r}e^{-r / R_s}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2 $$ ##R_s=2GM## and we're using a ##-+++## signature...

Aside from being symmetric, are there any other mathematical constraints on the metric?

I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...

This question is probably silly, but suppose I have a contraction of the form ##g_{\mu \nu} C^{\mu \nu} = 0## where ##C^{\mu \nu}## is a tensor* and ##g_{\mu \nu}## is the metric tensor. Can I say that it must vanish for any ##g_{\mu \nu}##, and since in the most general case all ##g_{\mu \nu}##...

We always can define a metric with a basis field ##g_{\mu\nu}=e_\mu \cdot e_\nu##, For a basis field ##e_\mu##, it can belong to a coordinate basis, then there is a corresponding coordinate system##\{x^\mu\}##,then we can have ##e_\mu=\frac{\partial}{\partial x^\mu}##, and ##[e_\mu , e_\nu]=0##...

About 10 years ago I worked on a project where I took a mater distribution and numerically solved for spatial curvature. Can this be done in the opposite direction? Can anybody point me to a resource that would allow me to calculate matter distributions when the metric is specified? What are...

Below are equations/formulas/text from https://en.wikipedia.org/wiki/Schwarzschild_geodesics https://hepweb.ucsd.edu/ph110b/110b_notes/node75.html I apologize for not remembering the source for the "v=" equation, or for my inability to find it again. For a circular orbit, the distance r and...

I understand that K(∞) = 0, and K(rs) = ∞ where rs = 2GM/c2. What is an equation for K(r) when rs < r < ∞? I have tried the best I can to search the Internet to find the answer, but I came up empty. I would very much appreciate the answer, or a reference that discusses the desired answer. I...

In a discrete metric space open balls are either singleton sets or the whole space ... Is the situation the same for open sets or can there be sets of two, three ... elements ... ? If there can be two, three ... elements ... how would we prove that they exist ... ? Essentially, given the...

##ds^2=g_{μν}(x)dx^μdx^ν= -(r^6/l^6)[1-(Ml^2/r^2)]dt^2+{1/[(r^2/l^2)-M]}dr^2+r^2dΦ^2## Does anyone know how to solve this?

I use metric, which describes spacetime upto second order terms in rotation. It is solution of Einstein equations expanded upto second order. My query is, how to manipulate with such metric during calculations? Concetrly I make inverse metric, produce effective potential (ie, multiplying...

I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha## ##\frac{\partial x^\alpha}{\partial...

If a proper time measuring clock goes along for the ride between events, then is such a clock physically possible as the scale factor changes / increases in the Friedman metric? How could any clock have zero spatial changes for that situation?

Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is: $$ \delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial...

Since ##\nu## is contracted, we form the scalar product of the metric and inverse metric, ##g_{\mu\nu}g^{\nu\lambda} = (\vec{e_\mu} \cdot \vec{e_\nu}) \cdot (\vec{e^\nu} \cdot \vec{e^\lambda}) = \vec{e_\mu} \cdot (\vec{e_\nu} \cdot \vec{e^\nu}) \cdot \vec{e^\lambda} = \delta^\lambda_\mu## I...

I am looking at this document I do not understand how the author gets 5.12 and 5.13 on page 133. I think the matrix of partials should be the transpose of the one shown. Even so I still can't figure out how you get 5.13. Any help would be appreciated.

I have no idea if this is an “A” level question, but I will put that down. From the Schwarzschild metric, and with the help of the Maxima program, one of the geodesic equations is: (I will have to attach a pdf for the equations...) I believe this integrates to the following, with ...

I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system. I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...

Let me begin by stating that I'm aware of the fact that this is a metric of de Sitter spacetime, aka I know the solution, my problem is getting there. My idea/approach so far: in the coordinates ##(u,v)## the metric is given by $$g_{\mu\nu}= \begin{pmatrix}1 & 0\\ 0 & -u^2\end{pmatrix}.$$ The...

The components of the energy tensor are defined sometimes as the flux of the ith component of the momentum vector across some component jth of constant surface. But isn't the tensor a function of points of spacetime just as the metric? How can you evaluate a surface of j when the tensor is a...

In chapter 3 of Sean Carroll's Introduction to General Relativity he 'makes the demand' of metric compatibility of a connection that ##\nabla_\mu g_{\lambda\nu}=0##. Metric compatibility becomes a phrase that is used frequently. However metric compatibility seems to arise naturally. One only...

Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?

I am self-studying GR, using principally Carroll’s textbook and Alex Maloney’s online lectures, and nice book by a guy called Herbert Roseman. I am a bit confused by alternative ways of expressing the metric and it would be most helpful if someone could clarify J Basically, I am perplexed by...

To arrive at the Robertson-Walker metric for a spatially homogeneous and isotropic cosmology, one first writes down the the metric for spatial sections i.e. constant t surfaces, dσ2 = d2 +f2(r) (dθ2 + sin2θ dφ2), where f(r) can take only 3 special forms, and then one promptly writes the...

I am trying to get a few concepts straight in my mind. There is no homework question here. 1) If we lived in Minkowski space and had to work in a rotating frame of reference would the Minkowski metric still be the one to use? I assume yes as even if the frame is non inertial the geometry of...

Hey! :o Let $(X, d)$ be a metric space. For $A \subseteq X$ und $x \in X$ we define $d_A : X \rightarrow \mathbb{R}$ by \begin{equation*}d_A(x):=\inf\{d(x,y)\mid y\in A\}\end{equation*} I want to prove the below statements: $A$ is closed iff for all $x\in X$ with $d(x,A)=0$ it holds that...

If I have a metric of the form ##g_{\mu \nu} = f_{\mu \nu} + h_{\mu \nu}## where ##f_{\mu \nu}## is the background metric and ##h_{\mu \nu}## the perturbation, how do I raise and lower indices of tensors? For instance, I was told that ##G_{ \ \nu}^{\mu} = f^{\mu \nu '} G_{\nu ' \nu }##. But...

The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.

I am trying to understand active diffeomorphism by looking at Schwarzschild metric as an example. Consider the Schwarzschild metric given by the metric $$g(r,t) = (1-\frac{r_s}{r}) dt^2 - \frac{1}{(1-\frac{r_s}{r})} dr^2 - r^2 d\Omega^2 $$ We identify the metric new metric at r with the old...

The line element given corresponds to the metric: $$g = \begin{bmatrix}a^2t^2-c^2 & at & 0 & 0\\at & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$ Using the adjugate method: ##g^{-1}=\frac{1}{|g|}\tilde{g}## where ##\tilde{g}## is the adjugate of ##g##. This gives me...

Hello, First post here. I have some data I am trying to do some forecasting on and was hoping somebody who knows what they're actually doing can verify what I have done. A few years ago, the company I work for developed a mobile app for its customers and about 1 year ago they added some new...

Here’s the metric: $$ds^2 = -dt^2+dl^2+r^2(l)d\Omega^2$$where ##r(l)## is minimum at ##l=0## with ##r(0)=r_0## and ##r## approaching ##|l|## asymptotically as ##l## approaches ##\pm \infty## Part a of the problem seemed pretty straightforward and intuitive, but part b asks which energy...

Sean Carroll says that if we have metric compatibility then we may lower the index on a vector in a covariant derivative. As far as I know, metric compatibility means ##\nabla_\rho g_{\mu\nu}=\nabla_\rho g^{\mu\nu}=0##, so in that case ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##. I can't see...

I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...

Can someone express the Godel metric line element in cylindrical coordinates? I keep looking for this line element, but no source clearly gives it to me. Can you please express it using the (- + + +) signature and while retaining all c terms? Thanks. Here is the line element in Cartesian...

My attempt at ##g_{\mu \nu}## for (2) was \begin{pmatrix} -(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta) \end{pmatrix} and the inverse is the reciprocal of the diagonal elements. For (1) however, I can't even think of how to write the...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...

The Minkowski metric for inertial observers reads ##ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2##. Is there a way to show that if it had off diagonal terms, the inertial observers would not see light traveling with the same speed?

I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms ... ... I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...Now ... on page 47 Carothers writes the following: In the above text from Carothers we...

Is it meaningful to introduce a Newtonian metric ##ds^2 = dx^2 + dy^2 + dz^2## in analogy with special relavity and the Lorentz metric?