Metric Definition and 1000 Threads

Homework Statement (a) Find the FRW metric, equations and density parameter. Express the density parameter in terms of a and H. (b) Express density parameter as a function of a where density dominates and find values of w. (c) If curvature is negligible, what values must w be to prevent a...

I'm looking at Carroll's lecture notes 1997, intro to GR. Equation 7.27 which is that he's argued the S metric up to the form ##ds^{2}=-(1+\frac{\mu}{r})dt^{2}+(1+\frac{\mu}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}## And argues that we expect to recover the weak limit as ##r \to \infty##. So he then...

Can someone please type out the line element for the Godel metric (including any and all c terms and any other terms that one might omit if they were using natural units to set terms like c = 1)? I ask this because different sources on line have it written out in different ways which look...

Homework Statement [/B] (a) Find the christoffel symbols (b) Find the einstein equations (c) Find A and B (d) Comment on this metric Homework Equations \Gamma_{\alpha\beta}^\mu \frac{1}{2} g^{\mu v} \left( \partial_\alpha g_{\beta v} + \partial_\beta g_{\alpha v} - \partial_\mu g_{\alpha...

Homework Statement (a) Find ##\dot \phi##. (b) Find the geodesic equation in ##r##. (c) Find functions g,f,h. (d) Comment on the significance of the results. Homework Equations The metric components are: ##g_{00} = -c^2## ##g_{11} = \frac{r^2 + \alpha^2 cos^2 \theta}{r^2 +\alpha^2}##...

Hello, Let ##g_{jk}## be a metric tensor; is it possible for some ##i## that ##g_{ii}=0##, i.e. one or more diagonal elements are equal to zero? What would be the geometrical/ topological meaning of this?

Is an exact solution to Einstein's Field Equations known for the interior of a sphere of uniform density (to approximate a star or planet, for example?)

Hi. I've been struggling with a formulation of the twin paradox in the Kerr metric. Imagine there are two twins at some radius in a Kerr metric. One performs equatorial circular motion whilst the other performs polar circular motion. They separate from one another and the parameters of the...

Space in quantum mechanics seems to be modeled as a triplet of real numbers, i.e. a continuum. Same happens in special relativity. General relativity I do not know (nor field theories). And then we apply the Pythagorean theorem and triangle inequality and so forth... I have a few general...

Taken from Hobson's book: Metric is given by ds^2 = c^2 dt^2 - R^2(t) \left[ d\chi^2 + S^2(\chi) (d\theta^2 + sin^2\theta d\phi^2) \right] Thus, ##g_{00} = c^2, g_{11} = -R^2(t), g_{22} = -R^2(t) S^2(\chi), g_{33} = -R^2(t) S^2(\chi) sin^2 \theta##. Geodesic equation is given by: \dot...

What does the metric look like for constant uniform acceleration, say in the x-direction? ds^2 = g_tt (cdt)^2 + g_xx (dx)^2 g_tt = ? g_xx = ?

Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...

I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms; [1]##ds^{2}=dt^{2}-R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}))## [2] ##ds^{2}=dt^{2}-R^{2}(t)g_{ij}dx^{i}dx^{j}## where...

The principle of least action states that the evolution of a physical system - how a system progresses from one state to another- is given by a stationary point of the action. So I think this is varying the path and keeping two points fixed- the points of the initial and final state I know...

I have the expression ##g_{ab}=\eta_{ab}+\epsilon h_{ab}##, The indices on ##h^{ab}## are raised with ##\eta^{ab}## to give ##g^{ab}=\eta^{ab}-\epsilon h^{ab}## I am not seeing where the minus sign comes from. So I know ##\eta^{ab}\eta_{bc}=\delta^{a}_{c}## and...

I should calculate the variation of the Ricci scalar to the metric ##\delta R/\delta g^{\mu\nu}##. According to ##\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}##, ##\delta R_{\mu\nu}## should be calculated. I have referred to the wiki page...

I would appreciate any help with the following question: I know that for relativistic field theories, the stress tensor can be obtained from the classical action by differentiating with respect to the metric, as is explained on the wikipedia page...

What are the sufficient / necessary conditions for a metric to be stationary / static? - If the metric components are independent of time in some coordinate system , is this sufficient for stationary? - I've read for static if a time-like killing vector is orthogonal to a family of...

Hi... The ordinary plain vanilla conformal metric in spherical coordinates is: ds2 = a(t)2[dt2 - dr2/(1 - kr2) - r2 (dΘ2 + sin2(Θ) d(φ)2)] where a(t) is a function of time only. I am trying to find out what Rieman, Ricci and the Scalar Curvature are for this common metric when k=1 and...

Hi everybody! Can you explain me how I can obtain wave equation given a metric? For example, if I have this metric $$g_{μν}=diag(−e^{2a(t)},e^{2b(t)},e^{2b(t)},e^{2b(t)})$$, how can derive the relation $$\frac{1}{\sqrt{g}}\partial _t(g^{00}\sqrt{g}\partial _t...

When I study about the transformation of coordinates, especially while defining gradient, curl, divergence and other vector integral theorem in different co-ordinate system, a concept called metric is defined and it is said to used for transform these operators in different co-ordinates, it is...

Show that two metrics p and T on the same set X are equivalent if and only if there is a c > 0 such that for all u,v belong to X, (1/c)T(u,v)=<p(u,v)=<cT(u,v) Please help me , I'm so confused about Real Analysis.

Hello Say, the metric tensor is diagonal, ##g=\mbox{diag}(g_{11}, g_{22},...,g_{NN})##. The (null) geodesic equations are ##\frac{d}{ds}(2g_{ri} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{r}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0## These are ##N## equations containing ##N## partial...

Hello, I have two questions regarding the FLRW metric, it is more about its interpretation. The metric reads: ##dl²=dt²-a²(\frac{dR²}{1-kR²}+R²d\Omega²)## where ##a## is the radius of the 3-sphere (universe), and ##R=r/a## a normalized radial coord. What I don't understand is this statement...

What does the metric matrix look like for a binary star system? Does each follow its usual geodesic about the other? It seems like the solution would have to be different somehow than that for a tiny planet circling a big sun.

Hello, I need to find the angular velocity using Schwarzschild metric. At first I wrote the metric in general form and omitted the co-latitude: ds2=T*dt2+R*dr2+Φ*dφ2 and wrote a Lagrangian over t variable: L = √(T+R*(dr/dt)2+Φ*(dφ/dt)2) now I can use the Euler–Lagrange equations for φ...

This is probably a stupid question but does k=1,0,-1 correspond to closed,flat,open refer to space or space-times? Looking at a derivation what each geometrically represents is only done when talking about the spatial part of the FRW metric. As space can be flat and space-time still curved...

I was just wondering how you would go about calculating the proper time for an observer following a freely falling elliptical orbit in a Schwarzschild metric. I am happy with how to calculate the proper time for a circular orbit and was wondering whether if you had two observers start and end...

What is the next easiest solution to einstein field equation after the schwarschild metric?

So in deriving the metric, the space-time can be foliated by homogenous and isotropic spacelike slices. And the metric must take the form: ##ds^{2}=-dt^{2}+a^{2}(t)\gamma_{ij}(u)du^{i}du^{j}##, where ## \gamma_{ij} ## is the metric of a spacelike slice at a constant t QUESTION: So I've read...

I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf, deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0## I see that any christoffel symbol with a ##0## vanish and so so do any...

I'm looking at deriving the Schwarzschild metric in 'Lecture Notes on General Relativity, Sean M. Carroll, 1997' and the comment under eq. 7.8, where he seeks a diagnoal form of the metric... - Is it always possible to put a metric in diagonal form or are certain symmetries required? - What...

I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997. I don't understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric. I know that 2-spheres are...

Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...

I was wondering if I can expand the FRW metric in d spatial dimensions, like: g_{\mu \nu}^{frw} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & - \frac{a^2(t)}{1-kr^2} & 0 & 0 \\ 0 & 0 & - a^2(t) r^2 & 0 \\ 0 & 0 & 0 & -a^2 (t) r^2 \sin^2 \theta \end{pmatrix} \rightarrow g_{MN} = \begin{pmatrix} g_{\mu...

This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time? Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and...

How do you obtain this equation M=Gm/c^2. What does M stand for? Is is Newton law at infinity? Again what is this Newton law at infinity?

I need help with the proof of the Fixed Point Theorem for a metric space (S,d) (Apostol Theorem 4.48) The Fixed Point Theorem and its proof read as follows: In the above proof Apostol writes: " ... ... Using the triangle inequality we find for $$m \gt n$$, $$d(p_m, p_n) \le \sum_{k=n}^{m-1}...

I have recently come across the notion that Kerr metric describes the spacetime outside a rotating black hole but not outside a rotating (electrically neutral) star. Unlike Schwarzschild metric, which works both for non-rotating spherically symetric black hole without charge as well as any other...

I need help with the proof of Theorem 4.28 in Tom Apostol's book: Mathematical Analysis (2nd Edition). Theorem 4.28 reads as follows:In the proof of the above theorem, Apostol writes: " ... ... Let $$m = \text{ inf } f(X)$$. Then $$m$$ is adherent to $$f(X)$$ ... ... " Can someone please...

Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...

This may seem an odd question but it will clear something up for me. Are "The spacetime interval is invariant." and the "The spacetime metric is a tensor." exactly equivalent statements? Does one imply more or less information than the other? Thanks!

There are .67 gallons of paint in a can. A. How many cubic meters of paint are in the can? B. How many liters of paint are in the can? C. Imagine that all of this paint is used to apply a coat of uniform thickness to a wall of area 13m^2. What is the thickness of the layer of wet paint in metric...

Homework Statement Suppose everything is moving slowly, How can we find the metric tensor in GR in terms of the mass contained. Homework Equations I understand in case of everything moving slowly only below equation is relevant - R00 - ½g00R = 8πGT00 = 8πGmc2 The Attempt at a Solution None.

In considering special relativity as a limiting case of the general theory (without matter or curvature) the question arose as to whether the pseudo-riemann nature of the SR metric is actually an independant (essentially experimentally determined) assumption/property or derivable from the...

Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea...

I understand the Kerr metric has an off-diagonal term between the rotation and the time degrees-of-freedom? That a test mass falling straight down toward a large rotating mass from infinity will begin to pick up angular momentum? Is that what’s called “frame dragging”? Did the Gravity Probe B...

Does anyone know a reference with a discussion on the experimental determination of the metric tensor of spacetime? I only know the one in "The theory of relativity" by Møller, pages 237-240. https://archive.org/details/theoryofrelativi029229mbp

Some subtleties of the metric tensor are just becoming clear to me now. If I take ##g_{\mu\nu}=diag(+1,-1,-1,-1)## and want to write ##\partial_\mu\phi^\mu##, it would be ##\partial_0\phi^0 -\partial_i\phi^i##, correct? ##\phi## is a 4-vector.

Mod note: OP warned about not using the homework template. Let ##g_{ac}## be a 3-d metric. So the trace of a metric is equal to its dimension so I get ##g_{ac}g^{ac}=3## But I'm a tad confused with the expression : ##g^{ac}g_{ad}##=##delta^{c}_{d}## I thought it would be ##3delta^{c}_{d}###...