Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of 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 separation of 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 will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over.Definition
Mathematically, spacetime is represented by a four-dimensional differentiable manifold and the metric tensor is given as a covariant, second-degree, symmetric tensor on, conventionally denoted by. Moreover, the metric is required to be nondegenerate with signature. A manifold equipped with such a metric is a type of Lorentzian manifold.Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of that varies in a smooth manner from point to point. Given two tangent vectors and at a point in, the metric can be evaluated on and to give a real number:
This is a generalization of the dot product of ordinary Euclidean space. Unlike Euclidean space - where the dot product is positive definite - the metric is indefinite and gives each tangent space the structure of Minkowski space.
Local coordinates and matrix representations
Physicists usually work in local coordinates. In local coordinates the metric can be written in the formThe factors are one-form gradients of the scalar coordinate fields. The metric is thus a linear combination of tensor products of one-form gradients of coordinates. The coefficients are a set of 16 real-valued functions. In order for the metric to be symmetric we must have
giving 10 independent coefficients.
If the local coordinates are specified, or understood from context, the metric can be written as a symmetric matrix with entries. The nondegeneracy of means that this matrix is non-singular, while the Lorentzian signature of implies that the matrix has one negative and three positive eigenvalues. Note that physicists often refer to this matrix or the coordinates themselves as the metric.
With the quantities 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
The interval imparts information about the causal structure of spacetime. When, the interval is timelike and the square root of the absolute value of is an incremental proper time. Only timelike intervals can be physically traversed by a massive object. When, the interval is lightlike, and can only be traversed by light. When, the interval is spacelike and the square root of 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, the metric components transform as
Examples
Flat spacetime
The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R4 with coordinates and the metricNote that these coordinates actually cover all of R4. The flat space 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, the flat space metric takes the form
where
is the standard metric on the 2-sphere.
Black hole metrics
The Schwarzschild metric describes an uncharged, non-rotating black hole. There are also metrics that describe rotating and charged metrics for black holes.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 bywhere, again, is the standard metric on the 2-sphere. Here, is the gravitation constant and is a constant with the dimensions of mass. Its derivation can be found here. The Schwarzschild metric approaches the Minkowski metric as approaches zero. Similarly, when goes to infinity, the Schwarzschild metric approaches the Minkowski metric.
With coordinates
we can write the metric as
Several other systems of coordinates have been devised for the Schwarzschild metric: Eddington–Finkelstein coordinates, Gullstrand–Painlevé coordinates, Kruskal–Szekeres coordinates, and Lemaître coordinates.
Rotating and charged black holes
The Schwarzschild solution supposes an object that is not rotating in space and is not charged. To account for charge, the metric must satisfy the Einstein Field equations like before, as well as Maxwell's equations in a curved spacetime. A charged, non-rotating mass is described by the Reissner–Nordström metric.Rotating black holes are described by the Kerr metric and the Kerr–Newman metric.
Other metrics
Other notable metrics are:- Alcubierre metric,
- de Sitter/anti-de Sitter metrics,
- Friedmann–Lemaître–Robertson–Walker metric,
- Isotropic coordinates,
- Lemaître–Tolman metric,
- Peres metric,
- Rindler coordinates,
- Weyl−Lewis−Papapetrou coordinates,
- Gödel metric.
Volume
The metric g induces a natural volume form, which can be used to integrate over a region of a manifold. Given local coordinates for the manifold, the volume form can be writtenwhere is the determinant of the matrix of components of the metric tensor for the given coordinate system.
Curvature
The metric 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 by the formula.
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:
The curvature is then expressible purely in terms of the metric and its derivatives.
Einstein's equations
One of the core ideas of general relativity is that the metric is determined by the matter and energy content of spacetime. Einstein's field equations:where the Ricci curvature tensor
and the scalar curvature
relate the metric to the stress–energy tensor. 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.