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.

In general relativity, the metric tensor plays the role of the gravitational potential in the classical theory of gravitation, although the physical content of the associated equations is entirely different.

[1] Gutfreund and Renn say "that in general relativity the gravitational potential is represented by the metric tensor.

"[2] This article works with a metric signature that is mostly positive (− + + +); see sign convention.

This article employs the Einstein summation convention, where repeated indices are automatically summed over.

Mathematically, spacetime is represented by a four-dimensional differentiable manifold

Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of

This is a generalization of the dot product of ordinary Euclidean space.

The metric is thus a linear combination of tensor products of one-form gradients of coordinates.

is a tensor field, which is defined at all points of a spacetime manifold).

If the local coordinates are specified, or understood from context, the metric can be written as a 4 × 4 symmetric matrix with entries

means that this matrix is non-singular (i.e. has non-vanishing determinant), while the Lorentzian signature of

being regarded as the components of an infinitesimal coordinate displacement four-vector (not to be confused with the one-forms of the same notation above), the metric determines the invariant square of an infinitesimal line element, often referred to as an interval.

Only timelike intervals can be physically traversed by a massive object.

, the interval is lightlike, and can only be traversed by (massless) things that move at the speed of light.

Spacelike intervals cannot be traversed, since they connect events that are outside each other's light cones.

The components of the metric depend on the choice of local coordinate system.

The metric tensor plays a key role in index manipulation.

provide a link between covariant and contravariant components of other tensors.

and similarly a contravariant metric coefficient raises the index

The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R4 with coordinates

The Schwarzschild metric describes an uncharged, non-rotating black hole.

There are also metrics that describe rotating and charged black holes.

The Schwarzschild solution supposes an object that is not rotating in space and is not charged.

[further explanation needed] Other notable metrics are: Some of them are without the event horizon or can be without the gravitational singularity.

The metric g induces a natural volume form (up to a sign), which can be used to integrate over a region of a manifold.

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

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.

The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates

Exact solutions of Einstein's field equations are very difficult to find.