Metric tensor (general relativity)

$\begin{pmatrix} g_{00} & g_{01} & g_{02} & g_{03} \\ g_{10} & g_{11} & g_{12} & g_{13} \\ g_{20} & g_{21} & g_{22} & g_{23} \\ g_{30} & g_{31} & g_{32} & g_{33} \\ \end{pmatrix}$ Metric tensor of spacetime in general relativity written as a matrix

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 potential familiar from Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separating the future and the past.

Notation and conventions

Throughout this article we work with a metric signature that is mostly positive (− + + +); see sign convention. The gravitation constant G will be kept explicit. The summation convention, where repeated indices are automatically summed over, is employed.

Definition

Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-order, 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 a type of 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, which is 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 quantities $dx^\mu$ being regarded as the components of an infinitesimal coordinate displacement four-vector, the metric determines the invariant square of an infinitesimal line element, often referred to as an interval. The interval is often denoted

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

The interval $ds^2$ imparts information about the causal structure of 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 outside each other's light cones. Events can be causally related only if they are within each other's light cones.

The components of the metric depend on the choice of 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

$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = \eta_{\mu\nu} dx^{\mu} dx^{\nu}. \,$

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

$\eta = \begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

(An alternative convention replaces coordinate t by ct, and defines η as in Minkowski space#Standard basis.)

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

$ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2 \,$

where

$d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2$

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

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

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:

• Alcubierre metric,
• Bondi metric,
• de Sitter/anti-de Sitter metrics,
• Eddington–Finkelstein coordinates,
• Friedmann–Lemaître–Robertson–Walker metric,
• Gullstrand–Painlevé coordinates,
• Isotropic coordinates,
• Kerr metric,
• Kerr–Newman metric,
• Kruskal–Szekeres coordinates,
• Lemaître coordinates,
• Lemaître–Tolman metric,
• Peres metric,
• Reissner–Nordström metric,
• Rindler coordinates,
• Weyl−Lewis−Papapetrou coordinates.

Some of them are without the event horizon or can be without the gravitational singularity.

Volume

The metric g induces a natural volume form (up to a sign), which can be used to integrate over a region of a manifold. Given local coordinates $x^\mu$ for the manifold, the volume form can be written

$\mathrm{vol}_g = \pm\sqrt{|\det [g_{\mu\nu}]|}\,dx^0\wedge dx^1\wedge dx^2\wedge dx^3$

where det[g_μν] 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

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

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:

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

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:

$R_{\mu\nu} - {1\over 2}R g_{\mu\nu} = \frac{8\pi G}{c^4} \,T_{\mu\nu}$

where the Ricci curvature tensor

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

and the scalar curvature

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

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.

See also

• Alternatives to general relativity
• Basic introduction to the mathematics of curved spacetime
• Mathematics of general relativity
• Ricci calculus

References

See general relativity resources for a list of references.

External links

• Caltech Tutorial on Relativity — A simple introduction to the basics of metrics in the context of relativity.