Metric tensor (general relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) 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 formula_1 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 formula_2 and the metric tensor is given as a covariant, second-degree, symmetric tensor on formula_2, conventionally denoted by formula_4. Moreover, the metric is required to be nondegenerate with signature . A manifold formula_2 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 formula_2 that varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors formula_7  and formula_8 at a point formula_9 in formula_2, the metric can be evaluated on formula_7 and formula_8 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 (i.e. coordinates defined on some local patch of formula_2). In local coordinates formula_15 (where formula_16 is an index that runs from 0 to 3) the metric can be written in the  form The factors formula_18 are one-form gradients of the scalar coordinate fields formula_15. The metric is thus a linear combination of tensor products of one-form gradients of coordinates. The coefficients formula_20 are a set of 16 real-valued functions (since the tensor formula_4 is a "tensor field", which is defined at all points of a spacetime manifold). 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 formula_20. The nondegeneracy of formula_24  means that this matrix is non-singular (i.e. has non-vanishing determinant), while the Lorentzian signature of formula_4 implies that the matrix has one negative and three positive eigenvalues. Note that physicists often refer to this matrix or the coordinates formula_20 themselves as the metric (see, however, abstract index notation). With the quantities formula_18 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 formula_29 imparts information about the causal structure of spacetime. When formula_30, the interval is timelike and the square root of the absolute value of formula_29 is an incremental proper time. Only timelike intervals can be physically traversed by a massive object. When formula_32, the interval is lightlike, and can only be traversed by (massless) things that move at the speed of light. When formula_33, the interval is spacelike and the square root of formula_29 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 formula_35, the metric components transform as Examples. Flat spacetime. The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R with coordinates formula_37 and the metric Note that these coordinates actually cover all of R. 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 formula_43, 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 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 by where, again, formula_47 is the standard metric on the 2-sphere. Here, formula_1 is the gravitation constant and formula_2 is a  constant with the dimensions of mass. Its derivation can be found here. The Schwarzschild metric approaches the Minkowski metric as formula_2 approaches zero (except at the origin where it is undefined). Similarly, when formula_51 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: 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 formula_15 for the manifold , the volume form can be written where formula_56 is the determinant of the matrix of components of the metric tensor for the given coordinate system. Curvature. The metric formula_4 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 formula_15 by the formula (where commas indicate partial derivatives). 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 formula_4 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 the Ricci curvature tensor and the scalar curvature relate the metric (and the associated curvature tensors) to the stress –energy tensor formula_65. 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.