What is Metric tensor: Definition and 198 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.
In dimensional regularization I have seen this relation ##k^{\mu}k^{\nu}=\frac{1}{D}g^{\mu\nu}k^2## but this seems to hold for same types of four vectors k. Is there any similar identity for different vectors like ##k^{\mu}p^{\nu}=\frac{1}{D}g^{\mu\nu}k.p## ?
How can I create a metric describing the space outside a large disk, like an elliptical galaxy? In cylindrical coordinates, ##\phi## would be the angle restricted the the plane, as ##\rho## would be the radius restricted to the plane. I think that if ##z## is suppressed to create an embedding...
I encountered a problem in reading Phys.Lett.B Vol.755, 367-370 (2016).
I cannot derive Eq.(7), the following snapshot is the paper and my oen derivation,
I cannot repeat Eq.(7) in the paper.
##g^{\mu\nu}## is diagonal metric tensor and##g^{\mu\mu}## is the function of ##\mu## only...
We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in ##4D## spacetime, then
\begin{align}
g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr}
\end{align}
where ##g_{mn}## is the metric and ##\epsilon^{npqr}## is the Levi-Civita tensor.
The...
Does anyone see a way I can find geodesics in the metric ##ds^2=-dt^2+dp^2+(5p^2+4t^2)d\phi^2## (ones with nonzero angular momentum)? I'm hoping it can be done analytically, but that may be wishful thinking. FYI, this is the metric listed at the bottom of the Wikipedia article about Ellis Wormholes.
The Hiscock coordinates read:
$$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$
##dr=dx-vdt##
Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and...
In describing the spacetime around a massive, spherical object, one would use the Schwarzschild Metric. What metric would instead be used to describe the spacetime around multiple massive bodies? Say, for example, you want to calculate the Gravitational Time Dilation experienced by a rocket ship...
Considering the FLWR metric in cartesian coordinates:
##ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2)##
With ##a(t)=t##, the trace of the extrinsic curvature tensor is ##-3t##. But why is it negative if it's describing an expanding universe, not a contracting one?
I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...
Suppose you have the following situation:
We have a spacetime that is asymptotically flat. At some position A which is in the region that is approximately flat, an observer sends out a photon (for simplicity, as I presume that any calculations involved here become easier if we consider a...
0:00 The metric tensor
12:55 Curve lengths
28:17 Metric compatibility of connections
35:47 The Levi-Civita connection
40:27 Induced metrics
50:12 Curvature and the metric
1:04:18 Killing fields and symmetries
I'm still confused about the notation used for operations involving tensors.
Consider the following simple example:
$$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$
Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get...
Hi everyone! I'm having some difficulty showing that the variation of the four-velocity,
Uμ=dxμ/dτ
with respect the metric tensor gαβ is
δUμ=1/2 UμδgαβUαUβ
Does anyone have any suggestion?
Cheers,
Rafael.
PD: Thanks in advances for your answers; this is my first post! I think ill be...
The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...
I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways:
##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
I am working on a computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl Tensor, Einstein Tensors, Ricci Tensor, Ricci scalar. What are the other essential/needed...
The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics.
I want to know the reason why I was wrong.
My professor says about transformation of components.
But I cannot close to answer by using this hint, because I don't have any idea about "x"...
Hello all,
let's suppose we have, in a flat spacetime, two observers O and O', the latter speeding away from O, with an uniform acceleration ##a##.
In the Minkowski spacetime chart of O, the world-line of O' can be drawn as a parable.
We know that the Lorentz boost at every point of the...
Hey there,
I've been recently been going back over the basics of GR, differential geometry in particular. I was watching one of Susskind's lectures and did not understand the argument made here (26:33 - 35:40).
In short, the argument goes as follows (I think): we have some generic metric ##{ g...
Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is:
$$
\delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial...
I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system.
I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...
My attempt at ##g_{\mu \nu}## for (2) was
\begin{pmatrix}
-(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta)
\end{pmatrix}
and the inverse is the reciprocal of the diagonal elements.
For (1) however, I can't even think of how to write the...
The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic.
But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how...
I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.For the first one I get all my directional derivatives
(0, 0): (0)i + (0)j
(0, 1): (0)i + 2j
(1, 0): 2i + (0)j
(1, 1): 2i + 2j
Then I square them (FOIL):
(0, 0): (0)i...
I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy.
In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...
Calculating the christoffel symbols requires taking the derivatives of the metric tensor. What are you taking derivatives of exactly? Are you taking the derivatives of the inner product of the basis vectors with respect to coordinates? In curvilinear coordinates, for instance curved spacetime in...
I am trying to get an intuition of what a metric is. I understand the metric tensor has many functions and is fundamental to Relativity. I can understand the meaning of the flat space Minkowski metric ημν, but gμν isn't clear to me. The Minkowski metric has a trace -1,1,1,1 with the rest being...
Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?
I am still learning general relativity (GR). I know we can find the path of a test particle by solving geodesic equations. I am wondering if it is possible to derive/convert metric tensor to gravity, under weak approximation, and vice versa.
Thanks!
In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...
Does the relative density of the early universe contribute to the red-shift of distant galaxies?
If so, by how much? How would this be calculated?
Asked another way :
Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric...
Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast.
I will be very thankful if...
Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface?
I know how to find equation of normal to a surface. It is given by:
$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
Proposition: Consider an ##n + 1##-dimensional metric with the following product structure:
$$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$
where ##h## is a Riemannian...
Defining dS2 as gijdxidxj and
given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...
I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is.
Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...