Metric tensor (general relativity)

In general relativity, the metric tensor (or simply, the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future and past.

Notation and conventions: Throughout this article we work with a metric signature that is mostly positive (− + + +); see sign convention. As is customary in relativity, units are used where the speed of light c = 1. The gravitation constant G will be kept explicit. The summation convention, where repeated indices are automatically summed over, is employed.

1 Definition
2 Local coordinates and matrix representations
3 Examples
4 Volume
5 Curvature
6 Einstein's equations
7 See also
8 References

Definition

Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-rank, symmetric tensor on M, conventionally denoted by g. Moreover the metric is required to be nondegenerate with signature (-+++). A manifold M equipped with such a metric is called a Lorentzian manifold.

Explicitly, the metric is a symmetric bilinear form on each tangent space of M which varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors u and v at a point x in M, the metric can be evaluated on u and v to give a real number:

$g_x(u,v) = g_x(v,u) \in \mathbb{R}.$

This can be thought of as a generalization of the dot product in ordinary Euclidean space. This analogy is not exact, however. Unlike Euclidean space — where the dot product is positive definite — the metric gives each tangent space the structure of Minkowski space.

Local coordinates and matrix representations

Physicists usually work in local coordinates (i.e. coordinates defined on some local patch of M). In local coordinates $x^\mu$ (where $\mu$ is an index which runs from 0 to 3) the metric can be written in the form

$g = g_{\mu\nu} dx^\mu \otimes dx^\nu.$

The factors $dx^\mu$ are one-form gradients of the scalar coordinate fields $x^\mu$ . The metric is thus a linear combination of tensor products of one-form gradients of coordinates. The coefficients $g_{\mu\nu}$ are a set of 16 real-valued functions (since the tensor g is actually a tensor field defined at all points of a spacetime manifold). In order for the metric to be symmetric we must have

$g_{\mu\nu} = g_{\nu\mu}\,$

giving 10 independent coefficients. If we denote the symmetric tensor product by juxtaposition (so that $dx^\mu dx^\nu = dx^\nu dx^\mu$ ) we can write the metric in the form

$g = g_{\mu\nu}dx^\mu dx^\nu.\,$

If the local coordinates are specified, or understood from context, the metric can be written as a 4×4 symmetric matrix with entries $g_{\mu\nu}$ . The nondegeneracy of $g_{\mu \nu}$ means that this matrix is non-singular (i.e. has non-vanishing determinant), while the Lorentzian signature of g implies that the matrix has one negative and three positive eigenvalues. Note that physicists often refer to this matrix or the coordinates $g_{\mu\nu}$ themselves as the metric (see, however, abstract index notation).

With the quantity $dx^\mu$ being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant interval squared or line element. For this reason one often sees the notation $ds^2$ for the metric:

$ds^2 = g_{\mu\nu}dx^\mu dx^\nu.\,$

In general relativity, the terms metric and line element are often used interchangeably.

The line element $ds^2$ imparts information about the causal structure of the spacetime. When $ds^2 < 0$ , the interval is timelike and the square root of the absolute value of ds² is an incremental proper time. Only timelike intervals can be physically traversed by a massive object. When $ds^2=0$ , the interval is lightlike, and can only be traversed by light. When $ds^2 > 0$ , the interval is spacelike and the square root of ds² acts as an incremental proper length. Spacelike intervals cannot be traversed, since they connect events that are out of each other's light cones. Events can be causally related only if they are within each other's light cones.

The metric components obviously depend on the chosen local coordinate system. Under a change of coordinates $x^\mu \to x^{\bar \mu}$ the metric components transform as

$g_{\bar \mu \bar \nu} = \frac{\partial x^\rho}{\partial x^{\bar \mu}}\frac{\partial x^\sigma}{\partial x^{\bar \nu}} g_{\rho\sigma} = \Lambda^\rho {}_{\bar \mu} \, \Lambda^\sigma {}_{\bar \nu} \, g_{\rho \sigma} .$

Examples

Flat spacetime

The simplest example of a Lorentzian manifold is flat spacetime which can be given as R⁴ with coordinates $(t,x,y,z)$ and the metric

$ds^2 = -dt^2 %2B dx^2 %2B dy^2 %2B dz^2.\,$

Note that these coordinates actually cover all of R⁴. The flat space metric (or Minkowski metric) is often denoted by the symbol η and is the metric used in special relativity. In the above coordinates, the matrix representation of η is

$\eta = \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

In spherical coordinates $(t,r,\theta,\phi)$ , the flat space metric takes the form

$ds^2 = -dt^2 %2B dr^2 %2B r^2d\Omega^2\,$

where

$d\Omega^2 = d\theta^2 %2B \sin^2\theta\,d\phi^2$

is the standard metric on the 2-sphere.

Schwarzschild metric

Besides the flat space metric the most important metric in general relativity is the Schwarzschild metric which can be given in one set of local coordinates by

$ds^{2} = -\left(1-\frac{2GM}{r} \right)dt^2 %2B \left(1-\frac{2GM}{r}\right)^{-1}dr^2%2B r^2 d\Omega^2$

where, again, $d\Omega^2$ is the standard metric on the 2-sphere. Here G is the gravitation constant and M is a constant with the dimensions of mass. Its derivation can be found here. The Schwarzschild metric approaches the Minkowski metric as M approaches zero (except at the origin where it is undefined). Similarly, when r goes to infinity, the Schwarzschild metric approaches the Minkowski metric.

Other metrics

Other notable metrics are Eddington–Finkelstein coordinates, Friedmann–Lemaître–Robertson–Walker metric, Gullstrand–Painlevé coordinates, Isotropic coordinates, Kerr metric, Kerr–Newman metric, Kruskal–Szekeres coordinates, Lemaitre metric, Reissner–Nordström metric, Rindler coordinates. Some of them are without the event horizon ^[1] or can be without the gravitational singularity.

Volume

The metric g defines a natural volume form, which can be used to integrate over spacetimes. In local coordinates $x^\mu$ of a manifold, the volume form can be written

$\mathrm{vol}_g = \sqrt{|\det g|}\,dx^0\wedge dx^1\wedge dx^2\wedge dx^3$

where det g is the determinant of the matrix of components of the metric tensor for the given coordinate system.

Curvature

The metric g completely determines the curvature of spacetime. According to the fundamental theorem of Riemannian geometry, there is a unique connection ∇ on any semi-Riemannian manifold that is compatible with the metric and torsion-free. This connection is called the Levi-Civita connection. The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates $x^\mu$ by the formula

$\Gamma^\lambda {}_{\mu\nu} = {1 \over 2} g^{\lambda\rho} \left( {\partial g_{\rho\mu} \over \partial x^\nu} %2B {\partial g_{\rho\nu} \over \partial x^\mu} - {\partial g_{\mu\nu} \over \partial x^\rho} \right)$ .

The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇. In local coordinates this tensor is given by:

${R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho {}_{\nu\sigma} - \partial_\nu\Gamma^\rho {}_{\mu\sigma} %2B \Gamma^\rho {}_{\mu\lambda}\Gamma^\lambda {}_{\nu\sigma} - \Gamma^\rho {}_{\nu\lambda}\Gamma^\lambda {}_{\mu\sigma}.$

The curvature is then expressible purely in terms of the metric $g$ and its derivatives.

Einstein's equations

One of the core ideas of general relativity is that the metric (and the associated geometry of spacetime) is determined by the matter and energy content of spacetime. Einstein's field equations:

$R_{\mu\nu} - {1\over 2}R g_{\mu\nu} = 8\pi G\,T_{\mu\nu}$

where

$R_{\nu \rho} \ \stackrel{\mathrm{def}}{=}\ {R^{\mu}}_{\nu\mu \rho}$

relate the metric (and the associated curvature tensors) to the stress-energy tensor $T_{\mu\nu}$ . This tensor equation is a complicated set of nonlinear partial differential equations for the metric components. Exact solutions of Einstein's field equations are very difficult to find.

References

^ D. J. Leiter and S. L. Robertson, "Einstein revisited: Gravitation in curved spacetime without event horizons". Submitted to General Relativity and Gravitation on January 2001.

See general relativity resources for a list of references.