What is Metric tensor: Definition and 200 Discussions
In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.
A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.
While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.
The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.
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}\\}...
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...
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
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...
Q1 If given a 2D Riemannian space, ds^2 = dx^2 + x^2dy^2, do the componets of the metric tensor are these:
g_11 = 1, g_12 = 0
g_21 = o, g_22 = x^2 ?
In addition, I got a question from my lecturer:
Q2. 2 metrics, defined in a Riemannian space, are given by ds^2 = g_ijdx^idy^j
and ds'^2 =...
My cosmology textbook tells me that if I raise the indicies on the metric tensor (from subscript to superscript), then all I have to do is divide one by each element. But from what I know about inverting matricies, the process is quite a bit more involved. When I raise the indicies on the...
I was wonder if some can explain to me what exactly are the 10 parameters for the metric tensors. I know the reason for getting 10 parameters, 3^2=9 + 1, you get three for every spatial dimensions plus one for time. But why exactly three parameters for each spatial dimension? And what exactly...
Hi I'm new here and I hope that you will be able to give me a lot of help. My english is far to be perfect but sufficiant to asks you a lot of questions... (i hope so :wink:).
First question :
I'm looking for a complete proof (with all steps) of :
\partial _h g = gg_{ij} \partial _h...