Metric tensor Definition and 206 Threads

Given a metric tensor gmn, how to calculate the inverse of it, gmn. For example, the metric g_{\mu \nu }= \left[ \begin{array}{cccc} f & 0 & 0 & -w \\ 0 & -e^m & 0 &0 \\0 & 0 & -e^m &0\\0 & 0 & 0 & -l \end{array} \right] From basic understanding, I would think of divided it, that is...

I have very little knowledge in general relativity, though I do have a decent understanding of the theory of special relativity. In special relativity, points in space-time can be represented in Minkowski space (or a hyperbolic space) so that the metric tensor (that is derived in order to...

Hi, i was thinking about the metric tensor transformation law: g_{cd}(x) = \frac{{dx'}^a}{{dx}^c} \frac{{dx'}^b}{{dx}^d} g'_{ab}(x') and, in view of this definition, the differences between Poincare transformations and reparametrization-like transformation (f.e. various conformal...

Consider a flat 2-dimensional plane. This can be described by standard Cartesian coordinates (x,y). We establish a oblique set of axes labelled p and q. p coincides with x but q is at an angle θ to the x-axis. At any point A has unambiguous co-ordinates (x,y) in the Cartesian system. In...

Homework Statement Show that \frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d}) Homework Equations The Attempt at a Solution It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming...

Hello, Can anyone give me the answer of the following derivative? \frac{\partial{g}}{\partial{g^{\mu \nu}}} Thank you in advance !

I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a...

When I was studying general relativity, I learned that to change a vector into a covector (or vice versa), one used the metric tensor. When I started quantum mechanics, I learned that the difference between a vector in Hilbert space and its dual is that each element of one is the complex...

Hi This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions: {T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu...

Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)? I know that it's coordinate dependent, so suppose you have two coordinates: with one being like "latitude", 0 at the bottom pole, and 2R at the northern pole, and the other being like...

Can someone please explain to me how exactly you write down a metric, say the FLRW metric in matrix form. Say we have the given metric here. ds^2 = dt^2 - R(t)^2 * [dw^2 + s^2 * (dθ^2 + sin^2(θ)dΦ^2)] Thank you.

Hi. This is example 7.2 from Hartle's "Gravity" if you happen to have it lying around. Metric of a sphere at the north pole The line element of a sphere (with radius a) is dS^{2}=a^{2}(d\theta^{2}+sin^{2}\theta d\phi ^{2}) (In (\theta , \phi ) coordinates). At the north pole \theta = 0 and at...

I wish I could calculate the contraction: gabgab I wish someone could show me how to get n! Unfortunately, I find it difficult, for I am not familiar with Tensor Algebra ... My wrong way to calculate it: gabgab= gabgba (since gab is symmetric) = δaa = 1Why is it wrong?

I am a bit perplexed by the consequences of the fact that all covariant first derivatives of the metric tensor are zero. I think I can follow some of the proofs, as presented for example in John Lee "Riemannian Manifolds - An Introduction to Curvature". But intuitively it "seems wrong" to me...

General four-dimensional (symmetric) metric tensor has 10 algebraic independent components. But transformation of coordinates allows choose four components of metric tensor almost arbitrarily. My question is how much freedom is in choose this components? Do exist for most general metric...

Hi, I am in the process of trying to teach myself GR Maths, at the A101 level, and have been working through the idea of tensors as scalars, vectors and matrices, i.e. rank-0, 1 and 2 tensors. Think I have also acquired some idea of the concept of contravariance and covariance, which then seems...

In the previous section he derived the components, with respect to the coordinate bases associated with a polar coordinate system, of the Riemannian metric tensor field on S2, the unit 2-sphere: g = \begin{pmatrix}1 & 0 \\ 0 & \sin^2(\theta) \end{pmatrix} where \theta is the zenith angle...

The Wikipedia article Metric tensor (general relativity) has the following equation for the metric tensor in an arbitrary chart, g = g_{\mu\nu} \, \mathrm{d}x^\mu \otimes \mathrm{d}x^\nu It then says, "If we define the symmetric tensor product by juxtaposition, we can write the metric in...

Homework Statement An interval in Minkovski space is given in spheric coordinates as; ds^{2}=c^{2}dt^{2}-dr^{2}-r^{2}d\theta^{2}-r^{2}sin^{2}\theta d\phi^{2} Now I have to find the covariant and contravariant components of the metric tensor. Homework Equations General expression...

Homework Statement \mbox{Prove that}\,g^{ij} \epsilon_{ipt}\epsilon_{jrs}\,=\, g_{pr}g_{ts}\,-\,g_{ps}g_{tr} Notation : e_{ijk}\,=\,e^{ijk}\,=\,\left\{\begin{array}{cc}1,&\mbox{ if ijk is even permutation of integers 123...n }\\-1, & \mbox{if ijk is odd permutation of...

This is a bit of a novice question I'm sure, but I was reading through Griffiths introductory particle physics text and in his section on QED, he states that the propagator of the photon is \frac{ig_{\mu \nu}}{q^2} but I can't see where he gets this from. Where does the metric come from? Can...

I've seen a Riemannian metric tensor defined in terms of a matrix of its components thus: g_{ij} = \left [ J^T J \right ]_{ij} and a pseudo-Riemannian metric tensor: g_{\alpha \beta} = \left [ J^T \eta J \right ]_{\alpha \beta} where J is a transformation matrix of some kind. I've...

I wonder how much information about metic tensor of Riemmanian manifold can be extracted if only the Levi-Civita connection is given. Conversly, if the metric on manifold is given there is formula for Christoffel symbols which define connection so there exists only one symetrical metric...

Hi, I'm having problem with understanding tensors and the Einsteins summation convention, so I decided to start doing explicit calculations, and I'm doing it in the wrong way. Hope someone could help me to clarify the concepts. In flat spacetime we have \eta with the signature (-+++). Under...

Okay, we have Einstein's field equation: R_ab + 1/2 R g_ab = 8pi T_ab Let's say we have T_ab defined for some region of space, and we want to calculate the spacetime from that. How would you calculate R_ab, R and g_ab? Supposedly you can write it as a system of PDEs but I cannot find them...

Hi all, I have just started learning about general relativity. Unfortunately my book is very math-oriented which makes it a bit challenging to understand the content from a physical point of view. I hope you can help. :smile: I am familiar with special relativity theory and Minkowsky Space...

Hi! I'm trying to learn some geometry for general relativity, and I am having a bit of trouble understanding how to tell flat and curved spaces apart. Specifically, I heard that a space is flat if you can "flatten" the metric by finding a coordinate system where ds^2 = dx^2 + dy^2 + ...

Hi All, let me preface I am an engineer, in classical terms, a trivial techician. Still I am putting some effort in improving my tensorial calculus. I am struggling with the definition of a metric tensor, as found for example in...

Homework Statement Not really homework, but thought this might be the best place to get a quick answer. Question Calculate g^{ca}g{ab} for the FLRW metric. I would have thought this would be g^{ca}g{ab}=\delta^c_b=4 I thought 4 because I assumed there should be "1" for each...

Homework Statement Given a N-dimensional manifold, let gab, be a metric tensor. Compute (i) gabgbc (ii)gabgab Also, just need a clarification on something similar. gcdTcd=gcdTdc=tr T? I'm pretty sure its yes. Probably even a stupid question but a clarification would be useful...

Homework Statement Prove that \Gamma^\mu_{\mu\lambda}=\frac{1}{\sqrt{-g}}\partial_\lambda(\sqrt{-g}) where g is the determinant of the metric, and \Gamma are the Christoffel connection coefficients. The Attempt at a Solution From the general definition of the coefficients I got...

I have started to teach myself General Relativity and I have been pointed to a book by Robert M. Wald called General Relativity. I really like it actually, I like how it doesn't skip the math behind the theory. It makes it appear more beautiful to me. However I think the book is quite vague...

Hello, I am still having a hard time with tensors... The answer is probably obvious, but is it always the case (for an arbitrary metric tensor g_{\mu \nu} that g_{ab}g_{cd}=g_{cd}g_{ab} ? I was trying to find a formal proof for that, and was wondering if we could use the relations: (1)...

Hi, every one I'm newbie here. I have a few problem with my study about GR. Here's a problem \partial_a(g^{ad})g_{cd}-\partial_d(g^{ad})g_{ac}=\\0 Could I prove these relation by change index (in 1st term ) from a -> d and also d -> a? and let's defined {g}=det{\\g_{ab}\\}...

I would like to know the Metric Tensor of the Earth in the form of g = [g11,g21,g31;g12,g22,g32;g13,g23,g33].

Hi guys. I'm taking a GR course right now, my first one. I was reading the textbook and I was wondering if you guys could help me out just to make sure I'm getting things straight here. I'm reading about the metric tensor, and I'm pretty sure I am expected to know what the metric tensor for a...

I know what the equation for proper time is in basic Euclaiden space. But when space-time is concerned, I get a bit confused. The equation is: \Delta\tau=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}} I realize that g_{\mu\nu} is the Metric tensor. However i don't understand the dx's and their indices...

I'm just wondering if the metric tensor (in its matrix form of 1 and -1's along the diagonal) is the same even when the direction of velocity of the "moving" frame isn't along the x-axis of the "stationary" frame but is in some arbitrary direction. This would obviously alter the Lorentz...

Is there an identity for \frac{\partial g^{\mu\nu}}{\partial g_{\lambda\sigma}}? Note raised and lowered indices.

The covariant derivative of any metric tensor is zero. Is this an axiom or something that is derived from other axioms?

Could anybody help to spot the inconsistency in the following reasoning? When calculating the normal derivative of the metric tensor I get: \partial_\mu g^{\rho \sigma} = g^{\rho \lambda} g^{\sigma \gamma} \partial_\mu g_{\lambda \gamma} + 2 \partial_\mu g^{\rho \sigma}, (1) which...

Hi, I'm trying to verify that the covariant derivative of the metric tensor is D(g) = 0. But I have a few questions: 1) This is a scalar 0 or a tensorial 0? Because it is suposed that the covariant derivative of a (m,n) tensor is a (m,n+1) tensor, and g is a (0,2) tensor so I think this 0...

How can I calculate reimannian metric tensor for the vector. I know about matrix it is equivalent to W(tranpose)W but don't know what it will be for vector w

Hi, I don't think this belongs in the homework section since this is a graduate course. My question is regarding making a field theory generally covariant by including a metric tensor g_{\mu\nu}(x)in the Lagransian density, and it's transformation under infinitesimal coordinate change...

in my fields course we are using the metric tensor g=diagonal(1,-1,-1,-1), off diagonal(0) i'm looking for an explanation of why the time coordinate has to be orientated oppositely to the spatial coordinates. can anyone give me an explanation of this? i'm also lost with upper and lower...

Hi, This might sound a very basic question to most of you all. But could you kindly give me some information on what eactly is a Metric Tensor and what is its significance in the higher dimension study? (Wiki or Google info seemed too cryptic. Thus...I ask you) Thank You

How do we derive the metric tensor?

That's a crucial point of GR ! And I have always problems with that. Back to the basics, with your help. Thanks Michel

What is the metric tensor on a spherical surface?

I read some books and see that the definition of covariant tensor and contravariant tensor. Covariant tensor(rank 2) A'_ab=(&x_u/&x'_a)(&x_v/&x'_b)A_uv Where A_uv=(&x_u/&x_p)(&x_u/&x_p) Where p is a scalar Contravariant tensor(rank 2) A'^uv=(&x'^u/&x^a)(&x'^v/&x^b)A^ab Where A^ab=dx_a...