 Metric tensor (general relativity)

This article is about metrics in general relativity. For a discussion of metrics in general, see metric tensor.
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.
Contents
Definition
Mathematically, spacetime is represented by a 4dimensional differentiable manifold M and the metric is given as a covariant, secondrank, 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:
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^{μ} (where μ is an index which runs from 0 to 3) the metric can be written in the form
The factors dx^{μ} are oneform gradients of the scalar coordinate fields x^{μ}. The metric is thus a linear combination of tensor products of oneform gradients of coordinates. The coefficients g_{μν} are a set of 16 realvalued 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
giving 10 independent coefficients. If we denote the symmetric tensor product by juxtaposition (so that dx^{μ}dx^{ν} = dx^{ν}dx^{μ}) we can write the metric in the form
If the local coordinates are specified, or understood from context, the metric can be written as a 4×4 symmetric matrix with entries g_{μν}. The nondegeneracy of g_{μν} means that this matrix is nonsingular (i.e. has nonvanishing 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_{μν} themselves as the metric (see, however, abstract index notation).
With the quantity dx^{μ} 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:
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 ds^{2} 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 ds^{2} 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 the metric components transform as
Examples
Flat spacetime
The simplest example of a Lorentzian manifold is flat spacetime which can be given as R^{4} with coordinates (t,x,y,z) and the metric
Note that these coordinates actually cover all of R^{4}. 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
In spherical coordinates (t,r,θ,ϕ), the flat space metric takes the form
where
is the standard metric on the 2sphere.
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
where, again, dΩ^{2} is the standard metric on the 2sphere. 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^{μ} of a manifold, the volume form can be written
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 semiRiemannian manifold that is compatible with the metric and torsionfree. This connection is called the LeviCivita connection. The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates x^{μ} by the formula
 .
The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the LeviCivita connection ∇. In local coordinates this tensor is given by:
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:
where
relate the metric (and the associated curvature tensors) to the stressenergy tensor T_{μν}. 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
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.
Categories: Tensors in general relativity
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