Tensor that describes the 4D geometry of spacetime
Metric tensor of spacetime in general relativity written as a matrix
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. 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.  Gutfreund and Renn say "that in general relativity the gravitational potential is represented by the metric tensor."
Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of that varies in a smooth (or differentiable) 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:
Physicists usually work in local coordinates (i.e. coordinates defined on some local patch of ). In local coordinates (where is an index that runs from 0 to 3) the metric can be written in the form
The factors are one-formgradients 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 (since the tensor is a tensor field, which is defined at all points of a spacetime manifold). In order for the metric to be symmetric
giving 10 independent coefficients.
If the local coordinates are specified, or understood from context, the metric can be written as a 4 × 4symmetric matrix with entries . The nondegeneracy of means that this matrix is non-singular (i.e. has non-vanishing determinant), 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 (see, however, abstract index notation).
With the quantities 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. 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 (massless) things that move at the speed of 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
The metric tensor plays a key role in index manipulation. In index notation, the coefficients of the metric tensor provide a link between covariant and contravariant components of other tensors. Contracting the contravariant index of a tensor with one of a covariant metric tensor coefficient has the effect of lowering the index
and similarly a contravariant metric coefficient raises the index
For a diagonal metric (one for which coefficients ; i.e. the basis vectors are orthogonal to each other), this implies that a given covariant coefficient of the metric tensor is the inverse of the corresponding contravariant coefficient , etc.
The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R4 with coordinates and the metric
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
The Schwarzschild metric describes an uncharged, non-rotating black hole. There are also metrics that describe rotating and charged black holes.
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, 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 (except at the origin where it is undefined). Similarly, when goes to infinity, the Schwarzschild metric approaches the Minkowski metric.
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.