Metric Definition and 1000 Threads

In one of the lectures I was watching it was stated without proof that the Schwarzschild metric is spherically symmetric. I thought it would be a good exercise in getting acquainted with the machinery of GR to show this for at least one of the vector fields in the algebra. The Schwarzschild...

Homework Statement This is Exercise 19.1 in MTW - See attachment Homework Equations See attachment The Attempt at a Solution [/B] I've worked through 19.3a, 19.3b and 19.3c ( see post by zn5252 back in March 2013 replied to by PeterDonis) and proved them for my my own satisfaction and I've...

Hello there, Suppose $f$ smoothly maps a domain ##U## of ##\mathbb{R}^2## into ##\mathbb{R}^3## by the formula ##f(x,y) = (x,y,F(x,y))##. We know that ##M = f(U)## is a smooth manifold if ##U## is open in ##\mathbb{R}^2##. Now I want to find the determinant of the metric in order to compute the...

Two masses, m and M, are a fixed distance R apart. One of the masses is much larger then the other. At time t the masses start to fall towards each other. Using Newton's Law of Gravitation we can determine the acceleration of the small mass. Can one use the Schwarzschild metric in the...

Suppose we have a curved spacetime with metric g, how can we find out the path of light throughout that space?

Hello, I was enjoying Zee's book on GR when I noticed the location of this "a(t)" thing in the metric sound quite disturbing to me. BTW: I experience the same annoyance and went down to the same conclusions, when I watched a related Theoretical Minimum lesson...Here's the setup, the flat...

This is probably a really stupid question , but, Does it matter whether the metric is after or before the tensor? My guess is it doesn't because tensors can be positioned in any order, the equation is unchanged. E.g ##M_{ab}B^{c}T^{m}_{nl}=T^{m}_{nl}M_{ab}B^{c}## right? However the covariant...

It seems that, in general, mathematicians prefer the (-,+,+,+) signature for the Minkowski spacetime metric while most physicists prefer the (+,-,-,-) signature. Is there any evidence that Nature actually prefers one over the other? As usual, thanks in advance.

I'm stuck on an apparently obvious statement in special relativity, so I hope you can help me. Can I define Lorenz transformations as transformations that don't change the spacetime interval in M4 and from this deduct that the metric tensor is invariant under LT? I've always read that the...

Homework Statement Let ##E## be a metric subspace to ##M##. Show that ##E## is closed in ##M## if ##E## is complete. Show the converse if ##M## is complete. Homework Equations A set ##E## is closed if every limit point is part of ##E##. We denote the set of all limit points ##E'##. A point...

Homework Statement Let ##E \subseteq M##, where ##M## is a metric space. Show that ##p\in \overline E = E\cup E' \Longleftrightarrow## there exists a sequence ##(p_n)## in ##E## that converges to ##p##. ##E'## is the set of limit points to ##E## and hence ##\overline E## is the closure of...

I am trying to learn GR, primarily from Wald. I understand that, given a metric, a unique covariant derivative is picked out which preserves inner products of vectors which are parallel transported. What I don't understand is the interpretation of the fact that, using this definition of the...

Hello, I am looking for some nontrivial metric on ℝ^2 invariant under the coordinate transformations defined by the 2x2 matrix [1 a12(θ)] [a21(θ) 1], where aik is some real function of θ. In the same way that the Minkowski metric on ℝ^2 is invariant under Lorentz transformations. Does...

Why is it needed to consider a 4D Euclidean space to introduce the FLRW metric? Is it because with a fourth parameter, we can set the radius of the 4D sphere formed with the four parametres as constant?

I have read that Albert Einstein was quite (pleasantly) surprised to read Schwarzschild's solution to his field equation because he did not think that any complete analytic solution existed. However, of all the possible scenarios to consider, a point mass in a spherically symmetric field (ie, a...

Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...

The component of a one-form W can be represented using the metric as Wβ = gαβUα, where Uα is the component of a vector U, and one can always multiply Wβ by some Vβ to get the inner product between the two vectors U and V. My question is: since the inner product is defined with the two...

I wonder if it is possible to write the components of the metric tensor (or any other tensor) as a summ of functions of the coordinates? Like this: g^{\mu\nu} = \sum_{\mu}^{D}\sum_{\nu}^{D} g_{_1}(x^{\mu}) g_{_2}(x^{\nu}) where g1 and g2 are functions of one variable alone and D is the...

That the metric tensor is not uniquely determined by the EFE and what this might entail has been a source of debate for about a century. A way to view the problem is to decide what the manifold that has the property of diffeomorphism invariance and background independence exactly is in the...

Schwarzschild coordinates for the Schwarzschild black hole solution become very weird near the event horizon because the radial coordinate is based on the proper circumference of a sphere but that has a minimum at the event horizon. This is easy to see in isotropic coordinates, where the...

I am studying special relativity, and I found that you have to work with a four dimensional space, where time is a complex variable. If you do so, you end up with the Minkowski metric, were the time component is negative and space components are positive (or vice versa). My questions are, why do...

Hello everyone 1. Homework Statement I have a homework question where I need to find out if the geometry is flat or not. The metric is shown below. Homework Equations The Attempt at a Solution So far I have written the metric in the form guv but and I am trying to find coordinates in which...

In the presence of torsion, is it correct to say that the metric doesn't give rise to a unique connection? So, if we were using, say ECKS theory, which unlike GR includes torsion, in order to find the connection, we'd need not only the metric, but a specification of the torsion. My thinking is...

Hello! I have a question that has been bothering me since I first started learning about Special Relativity: Given only the Minskowskian metric and/OR the spacetime interval, how can one reach the conclusion that the speed of light is invariant for every observer and how can one conclude that it...

I was thinking about the metric tensor. Given a metric gμν we know that it is symmetric on its two indices. If we have gμν,α (the derivative of the metric with respect to xα), is it also valid to consider symmetry on μ and α? i.e. is the identity gμν,α = gαν,μ valid?

Homework Statement What is the Schwartzchild metric. Calculus the 4-velocity of a stationary observer in this spacetime (u). Show that u2 = c2. Homework Equations Schwartzchild Metric d{s^2} = {c^2}\left( {1 - \frac{{2\mu }}{r}} \right)d{t^2} - {\left( {1 - \frac{{2\mu }}{r}} \right)^{ -...

Hello everyone: I studied in differential geometry recently and have seen a statement with its proof: Suppose there is a Riemannian metric: ##dl^2=Edx^2+Fdxdy+Gdy^2,## with ##E, F, G## are real-valued analytic functions of the real variables ##x,y.## Then there exist new local coordinates...

I'm not sure on how to do this problem. Can someone please help and explain? Thank you! Recall (Exercise 3.2.8) that the square metric distance between two points (x1, y1) and (x2, y2) in R^2 is given by D((x1, y1), (x2, y2))= max{|x2 − x1|, |y2 − y1|}. Show by example that R^2 with the square...

Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...

Homework Statement (From Di Francesco et al, Conformal Field Theory, ex .2) Derive the scale factor Λ of a special conformal transformation. Homework Equations The special conformal transformation can be written as x'μ = (xμ-bμ x^2)/(1-2 b.x + b^2 x^2) and I need to show that the metric...

Hello I just got interested into quasars and I have a question:What metric do you use for quasars?

So the Schwarzschild metric is given by ds2= -(1-2M/r)dt2 + (1-2M/r)-1dr2+r2dθ2+r2sin2θ dφ2 and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0## with L = dτ/dσ. So for each α=0,1,2,3 we have ##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime also...

Hello so if we have geodesic equation lagrange approximation solution: d/ds(mgμνdxν/ds)=m∂gμν∂xλdxμ/ds dxν/ds. So if we have schwarzschild metric (wich could be used to describe example sun) which is:ds2=(1-rs/r)dt2-(1-rs/r)-1dr2-r2[/SUP]-sin22. But that means that ∂gμν/∂xλ=0. So that means that...

Homework Statement Let ##(X,d)## be some metric space, and let ##A## be some subset of the metric space. The interior of the set ##A##, denoted as ##int A##, is defined to be ##\bigcup_{\alpha \in I} G_\alpha##, where ##G_\alpha \subseteq A## is open in ##X## for all ##\alpha \in I##. The...

Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂tgμν=-2Rμν. And we have metric tensor for schwarzschild metric: Diag((1-rs/r),(1-rs]/r)-1,(r2),(sin2Θ) and ∂tgμν=0 so 0=-2Rμν and we get that Rμν=0.But Rμν should not equal to...

I finally found a result I believe for the the asymptotic metric (valid for large r) of a pair of bodies in a circular orbit emitting gravitational waves. I use spherical coordinates, ##[t, r, \theta, \phi]##. If we let the linearized metric ##g_{\mu\nu}## be equal to the sum of a flat metric...

Hi, I was wondering if anybody could help me understand the derivation of the Schwarzschild metric developed by the author of mathpages website. Rather than reproduce all the equations via latex, I have attached a 2-page pdf summary that also points to the mathpages article and explains my...

Hey all, in every theory that involves GR you see they give their space-time metric, but very few show any other math related to it, how does one know if a metric is valid?

In the paper Strong field limit of black hole gravitational lensing, the amplification of images in the Schwartzschild metric was given by $$ \frac{1}{\beta}\sqrt{\frac{2\,D_{LS}}{D_{OL}D_{OS}}} $$However the authors did not derive this expression or explain its origin. Does anyone know how to...

I'm looking for the linearized metric in the far field for a pair of mutual orbiting bodies that are emitting gravitational waves (GW's). I gather finding this (approximate) metric should be possible using the quadrupole moment of the source. From Landau & LIfshitz "Classical theory of...

Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...

As I understand it, the notion of a distance between points on a manifold ##M## requires that the manifold be endowed with a metric ##g##. In the case of ordinary Euclidean space this is simply the trivial identity matrix, i.e. ##g_{\mu\nu}=\delta_{\mu\nu}##. In Euclidean space we also have that...

Hello I have been reading about Schwarzschild metric and scources what I read said that Schwarzschild metric is used to describe a non-rotating black holes. And what I can not understand is what can you calculate with it? It would be good if you give some examples where you can use it.

I am trying to follow Nakahara's book about Holonomy. if parallel transporting a vector around a loop induces a linear map (an element of holonomy group) {P_c}:{T_p}M \to {T_p}M the holonomy group should be a subgroup of GL(m,R) then the book said for a metric connection, the property...

Here is the Godel solution: ds2 = -dt2/(2ω2) - (exdzdt)/ω2 - (e2xdz2)/(4ω2) + dx2/(2ω2) + dy2/(2ω2) Here is the metric tensor for it: g00 = -1/(2ω2) g03 & g30 = -ex/(2ω2) g11 & g22 = 1/(2ω2) g33 = -e2x/(4ω2) Every other element is 0. Now to my question: What shape is this metric? To clarify...

Here is the Morris-Thorne Wormhole line element: ds2 = - c2dt2 + dl2 + (b2 + l2)(dθ2 + sin2(θ)d∅2) Now my main question here (even though I've asked this before, but never quite understood) is: What exactly is l? I know that b is the radius of the throat of the wormhole. I know that the rest...

Dear PF Forum, While learning why the net energy of the universe is zero. I've been reading about the expansion of the universe, and of course in it, Hubble Flow. https://en.wikipedia.org/wiki/Hubble's_law = 73km/s per Mega parsec https://en.wikipedia.org/wiki/Parsec = 3.26 light year. In the...

hi, when I dug up something about metric tensors, I found a equation in my attached file. Could you provide me with how the derivation of this ensured? What is the logic of that expansion in terms of metric tensor? I really need your valuable responses. I really wonder it. Thanks in advance...

Original Question (Please ignore this): I knew this when I read about it the first time a while back but can't put the two and two together anymore. :) I understand ##\vec{a_i}\centerdot \vec{a^j}=\delta_i^j## and ##g_{ij}=\vec{a_i}\centerdot\vec{a_j}## How does one get from there to...

Goodmorning everyone, is there any implies to use in general relativity a metric whose coefficients are harmonic functions? For example in (1+1)-dimensions, is there any implies for using a metric ds2=E(du2+dv2) with E a harmonic function? In (1+1)-dimensions is well-know that the Einstein...