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. Using the metric one can define such notions as distance, volume, angle, future, past, and curvature. Two-dimensional visualisation of space-time distortion. ... The gravitational field is a field that causes bodies with mass to attract each other. ... Gravitation is the tendency of masses to move toward each other. ... World line of the orbit of the Earth depicted as a circle in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, Z, making the circle appear as a helix. ...

Notation and conventions: Throughout this article we work with a metric signature that is mostly positive (-+++); see sign convention. As is customary is relativity, we work in units where the speed of light c = 1. The gravitation constant G will be kept explicit. We also make use of the summation convention, where repeated indicies are automatically summed over.

The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ... In physics, Planck units are physical units of measurement originally proposed by Max Planck. ... Cherenkov effect in a swimming pool nuclear reactor. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

Definition

Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-rank, 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. In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In category theory, see covariant functor. ... A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the 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 and a point x in M, the metric can be evaluted on u and v to give a real number: In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

This can be thought of as a generalization of the dot product in ordinary Euclidean space. This analogy is not exact, however. Rather, than Euclidean space — where the dot product is positive definite — the metric gives each tangent space the structure of Minkowski space. In mathematics, the dot product (also known as the scalar product and the inner product) is a sesquilinear function (·) : V × V → F, where V is a vector space over the field F, having some further properties. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

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 Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

The quantities dx^μ are differential one-forms (given by the exterior derivative of the coordiante functions x^μ). The coefficients g_μν are a set of 16 real-valued functions (they are functions of the coordinates x^μ). In order for the metric to be symmetric we must have A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

giving 10 independent coefficients. If we denote the symmtric tensor product by juxtaposition (so that dx^μdx^ν = dx^νdx^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 means that this matix is non-singular (i.e. has non-vanishing 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). In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

With the the quantity dx^μ being an infitesimal coordiante displacement, the metric acts as an infitesimal invariant interval squared or line element. For this reason one often sees the notation ds² for the metric: The line element in mathematics can most generally be thought of as the square of the change in a position vector in an affine space equated to the square of the change of the arc length. ...

In general relativity, the terms metric and line element are often used interchangeably.

ds² imparts information about the causal structure of the spacetime. When ds² < 0, the interval is timelike and its absolute value is an incremental proper time. Only timelike intervals can be physically traveled by a massive object. When ds² = 0, the interval is lightlike, and can only be traveled by light. When ds² > 0, the interval is spacelike and acts as an incremental proper length. Timelike intervals cannot be traveled, 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. In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Proper time is time measured when the clock is at rest relative to the observer. ... In physics, the adjective light-like refers to a contour in spacetime in the context of special relativity whose proper length vanishes. ... In physics, proper length is the length of an object or a contour as measured in the reference frame of the object itself in the context of special relativity. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ... There are many kinds of events. ...

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 R⁴ with coordinates (t,x,y,z) 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. In the above coordinates, the matrix representation of η is

In spherical coordinates (t,r,θ,φ), the flat space metric takes the form This article describes some of the common coordinate systems that appear in elementary mathematics. ...

where

is the standard metric on the 2-sphere (i.e. the standard element of solid angle). For other uses, see sphere (disambiguation). ...

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 This article needs cleanup. ...

where, again, dΩ² is the standard metric on the 2-sphere. Here G is the gravitation constant and M is a constant with the dimensions of mass. For other uses, see sphere (disambiguation). ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ...

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 Lorentzian 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 local coordinates x^μ by the formula Curvature is the amount by which a geometric object deviates from being flat. ... In Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor. ... In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics, the term torsion has several meanings, mostly unrelated to each other. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

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: In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

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 famous field equations: Matter is the substance of which a physical object is composed. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or Einstein equation is a tensor equation in Einsteins theory of general relativity. ...

relate the metric (and the associated curvature tensors) to the energy-momentum tensor T_μν. This equation is a complicated set of nonlinear partial differential equations for the metric components. Exact solutions of this equation are very difficult to come by; see exact solutions of Einstein's field equations for more information. The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ... Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energy-momentum tensor (by the field equations). ...