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.
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.
In SRT, the line element is ##c^2ds^2 = c^2dt^2 - dx^2 -dy^2-dz^2## and ##g_{00} = 1## (or ##-1## depending on sign conventions). In the Schwarzschild metric we have
g_{00}=(c^2-\frac{2 GM}{r}) .
So in the first example, ##g_{00}## is constant, in the second it depends on another coordinate...
Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R,
gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}##
,and I just know the metric tensor, but don't know that it is of a sphere.
Now I want to draw a 2D space(surface)...
Hello! I'd appreciate any help or pokes in the right direction.
Homework Statement
Show that a co-tensor of rank 2, ##T_{\mu\nu}##, is obtained from the tensor of rank 2 ##T^{\mu\nu}## by using a metric to lower the indices:
$$T_{\mu\nu} = g_{\mu\alpha}g_{\nu\beta}T^{\alpha\beta}$$
Homework...
After my recent studies of the curvature of the 2- sphere, I would like to move on to Minkowski space. However, I can not seem to find the metric tensor of the 4 sphere on line, nor can I seem to think of the vector of transformation properties that I would use to derive the metric tensor of the...
I have one question, which I don't know if I should post here again, but I found it in GR...
When you have a metric tensor with components:
g_{\mu \nu} = \eta _{\mu \nu} + h_{\mu \nu}, ~~ |h|<<1 (weak field approximation).
Then the inverse is:
g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}...
Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's:
g(\vec{A}\,,\vec{B})=A^aB^bg_{ab}
And, if...
Definition/Summary
The metric tensor g_{\mu\nu} is a 4x4 matrix that is determined by the curvature and coordinate system of the spacetime
Equations
The proper time is given by the equation
d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu}
using the Einstein summation convention
It is a symmetric...
I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards:
g11 = sin2(ø) + cos2(θ)
g12 = -rsin(θ)cos(θ)
g13 = rsin(ø)cos(ø)
g21 = -rsin(θ)cos(θ)...
Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...
From what I've understood,
1) the metric is a bilinear form on a space
2) the metric tensor is basically the same thing
Is this correct?
If so, how is the metric related to/different from the distance function in that space?
Some other sources state that the metric is defined as the...
Hello All,
Sorry if my question seems to be elementary. I am trying to find out a little bit details of the Riemann metric tensor but not too much in details. I found out the metric (g11, g12, g13, g14...). It provides information on the manifold and those parameters have the information...
I am trying to write the Einstein field equations
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$
in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$...
I've been watching the Stanford lectures on GR by Leonard Susskind and according to him the metric tensor is not constant in polar coordinates. To me this seems wrong as I thought the metric tensor would be given by:
g^{\mu \nu} =
\begin{pmatrix}
1 & 0\\
0 & 0\\
\end{pmatrix}
Since...
How do you find the inverse of metric tensor when there are off-diagonals?
More specifivally, given the (Kerr) metric,
$$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$
we have the metric tensor;
$$ g_{\mu \nu} =...
Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...
I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...
I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that:
"It is straightforward to show that the coordinate and dual basis vectors
themselves are related...
"ea = gabeb ..."
I have...
Our galaxy is rotating and is charged therefore the choice for the metric is the Kerr-Newman Metric.
I want to solve for the Kerr-Newman Metric Tensor.
There are a few questions.
1-What is the value for Q in the equation:
##r_Q^2=\frac{Q^2*G}{4*\pi*\epsilon_0*c^4}##
where
##G=6.674E-20...
I am looking for the Metric Tensor of the Reissner–Nordström Metric.g_{μv}
I have searched the web: Wiki and Bing but I can not find the metric tensor derivations.
Thanks in advance!
I want to find the ricci tensor and ricci scalar for the space-time curvature at the Earth surface. Ignoring the moon and the sun. I have used the scwharzschilds metric, but then the ricci tensor and the scalar where equal to zero.
If I have a 4x4 Covarient Metric Tensor g_{ik}.
I can find the determinant:
G = det(g_{ik})
How do I find the 4x4 Cofactor of g_ik?
G^{ik}
then g^{ik}=G^{ik}/G
I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :-p
Atrocious comedy aside, Spivak provides a parametric...
Suppose, I have the next metric:
g = du^1 \otimes du^1 - du^2 \otimes du^2
And I want to calculate g(W,W), where for example W=\partial_1 + \partial_2
How would I calculate it?
Thanks.
How to calculate something relating to the determinant of metric tensor? for example, its derivative ∂_{λ}g.
and how to calculate1/g* ∂_{λ}g, which is from (3.33) in the book Spacetime and Geometry, in which the author says that it can be related to the Christoffel connection.
We are stating with equivalence principle that passing locally to non inertial frame would be analogous to the presence of gravitational field at that point, so g^'_{ij}=A g_{nm} A^{-1} where g' is the galilean metric and g is the metric in curved space, and A is the transformation which...
my exploration of relativity followed by first reading various books which failed to explain to me how relativity worked but built a strong feel of how one can think about it. after which i decided to take the mathematical way of understanding it for which i am going on with the prof susskind's...
Homework Statement
This is not homework but more like self-study - thought I'd post it here anyway.
I'm taking the variation of the determinant of the metric tensor:
\delta(det[g\mu\nu]).
Homework Equations
The answer is
\delta(det[g\mu\nu]) =det[g\mu\nu] g\mu\nu...
I've read that the metric tensor is defined as
{{g}^{ab}}={{e}^{a}}\cdot {{e}^{b}}
so does that imply that?
{{g}^{ab}}{{g}_{cd}}={{e}^{a}}{{e}^{b}}{{e}_{c}}{{e}_{d}}={{e}^{a}}{{e}_{c}}{{e}^{b}}{{e}_{d}}=g_{c}^{a}g_{d}^{b}
Hello,
can anyone suggest a geometric interpretation of the metric tensor?
I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.
Barbour writes:
the metric tensor g. Being symmetric (g_uv = g_vu) it has ten independent components,
corresponding to the four values the indices u and v can each take: 0 (for the
time direction) and 1; 2; 3 for the three spatial directions. Of the ten components,
four merely...
H is a contravariant transformation matrix, M is a covariant transformation matrix, G is the metric tensor and G-1 is its inverse. Consider an oblique coordinates system with angle between the axes = α
I have G = 1/sin2α{(1 -cosα),(-cosα 1)} <- 2 x 2 matrix
I compute H = G*M where M =...
Hello,
So, given two points, x and x', in a Lorentzian manifold (although I think it's the same for a Riemannian one). If in x the determinant of the metric is g and in the point x' is g'. How are g and g' related?By any means can g=g'? In what conditions?
I'm sorry if this is a dumb...
I have a very limited knowledge of tensor calculus, and I've never had proper exposure to general relativity, but I hope that the people reading this forum are able to help out.
So I'm doing some stat. mech. and a part of a system's free energy is
\mathcal{F} = \int V(\rho)\nabla^2\rho dx
I'd...
Now let's say I have the metric for some curved two surface
ds^2=G(u,v)du^2+P(u,v)dv^2 ( the G and P functions being the 00 and 11 components, assuming the metric is diagonal)
Now my question is, since the metric defines the scalar product of two vectors, let's say
(1,0) and (0,1), for...
Hi all,
In flat space-time the metric is
ds^2=-dt^2+dr^2+r^2\Omega^2
The Schwarzschild metric is
ds^2=-(1-\frac{2MG}{r})dt^2+\frac{dr^2}{(1-\frac{2MG}{r})}+r^2d\Omega^2
Very far from the planet, assuming it is symmetrical and non-spinning, the Schwarzschild metric reduces to the...
Hi all,
I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution?
Thanks!
Joe W.
I am working on the weak gravitational field by using linearized einstein field equation. What if the metric tensor, hαβ turn out to be a complex numbers? What is the physical meaning of the complex metric tensor? Can I just take it's real part?
Or there is no such thing as complex metric...
I have looked at the definition of the metric tensor, and my sources state that to calculate it, one must first calculate the components of the position vector and compute it's Jacobian. The metric tensor is then the transpose of the Jacobian multiplied by the Jacobian.
My problem with this...
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...