# 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.

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1. ### I A question about an Identity

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## ?
2. ### B Creating Metric Describing Large Disk

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...
3. ### A Exploring a QM particle in Motion with GR

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...
4. ### I Action of metric tensor on Levi-Civita symbol

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...
5. ### B What is the metric for a bag-of-gold spacetime?

What is the metric for a bag-of-gold spacetime?
6. ### B Intra-Universe Wormhole Metrics

Are there any metrics for intra-universe wormholes?
7. ### B Find Geodesics in Dynamic Ellis Orbits Metric

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.
8. ### B Calculate Unit Normal Vector for Metric Tensor

How do I calculate the unit normal vector for any metric tensor?
9. ### B Calc. Christoffel Symbols of Hiscock Coordinates

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...
10. ### I Calculating Spacetime Around Multiple Objects

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...
11. ### B Sign of Expansion Scalar in Expanding FLRW Universe

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?

13. ### I Calculate Gaussian Curvature from 4D Metric Tensor

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...
14. ### A Proper Volume on Constant Hypersurface in Alcubierre Metric

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...
15. ### I Calculating Relative Change in Travel Time Due to Spacetime Perturbation

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...
16. ### SH2372 General Relativity - Lecture 4

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

20. ### I Raising/Lowering Indices w/ Metric Tensor

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...

32. ### Show that the metric tensor is independent of coordinate choice

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...
33. ### Solving Metric Tensor Problems: My Attempt at g_μν for (2)

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...
34. ### A Anisotropic Universe and Friedmann Equations

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...
35. ### Did I Get These Metric Tensors Right?

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...
36. ### I The vanishing of the covariant derivative of the metric tensor

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...
37. ### I Metric tensor derivatives

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...
38. ### I Help Understanding Metric Tensor

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...
39. ### I Invariant properties of metric tensor

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?
40. ### I Convert Metric Tensor to Gravity in GR

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!
41. ### I Gradient vector without a metric

Is it possible to introduce the concept of a gradient vector on a manifold without a metric?
42. ### I Coordinates for diagonal metric tensors

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 -...
43. ### I How to keep the components of a metric tensor constant?

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...
44. ### B Density of the early Universe contributing to the red-shift?

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...
45. ### A On metric and connection independence

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...
46. ### A Normal to coordinate curve

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...
47. ### B G11 Metric Tensor: What is it & How Does it Work?

What is g11? I am very curious, can someone briefly describe what the metric tensor is, please?
48. ### A Causal Structure of Metric Prop.: Matrix Size Differs

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...
49. ### Finding the inverse metric tensor from a given line element

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...
50. ### I Lie derivative of a metric determinant

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...