In

^{4} with coordinates $(t,x,y,z)$ and the metric
:$ds^2\; =\; -c^2\; dt^2\; +\; dx^2\; +\; dy^2\; +\; dz^2\; =\; \backslash eta\_\; dx^\; dx^.\; \backslash ,$
Note that these coordinates actually cover all of R^{4}. The flat space metric (or

general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

, 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
In classical mechanics, the gravitational potential at a location is equal to the work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical w ...

of Newtonian gravitationNewtonian refers to the work of Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his time as a ...

. The metric captures all the geometric and causal structure
In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in ...

of spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

, 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 ametric signature
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

that is mostly positive (); see sign convention In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succes ...

. The gravitation constant $G$ will be kept explicit. This article employs the Einstein summation convention
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, where repeated indices are automatically summed over.
Definition

Mathematically, spacetime is represented by a four-dimensionaldifferentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

$M$ and the metric tensor is given as a covariant, second-degree
Degree may refer to:
As a unit of measurement
* Degree (angle)
Image:Degree diagram.svg, One degree (shown in red) andeighty nine degrees (shown in blue)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° ( ...

, symmetric tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
:T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_)
for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...

on $M$, conventionally denoted by $g$. Moreover, the metric is required to be nondegenerate
In mathematics, a degenerate case is a limiting case (mathematics), limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of ...

with signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...

. A manifold $M$ equipped with such a metric is a type of Lorentzian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a co ...

.
Explicitly, the metric tensor is a symmetric bilinear form In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

on each tangent space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of $M$ that varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors $u$ and $v$ at a point $x$ in $M$, the metric can be evaluated on $u$ and $v$ to give a real number:
:$g\_x(u,v)\; =\; g\_x(v,u)\; \backslash in\; \backslash mathbb.$
This is a generalization of the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

of ordinary Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

. Unlike Euclidean space – where the dot product is positive definite In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

– the metric is indefinite and gives each tangent space the structure of Minkowski space
In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...

.
Local coordinates and matrix representations

Physicists usually work inlocal coordinates
Local coordinates are the ones used in a ''local coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldes ...

(i.e. coordinates defined on some local patch of $M$). In local coordinates $x^\backslash mu$ (where $\backslash mu$ is an index that runs from 0 to 3) the metric can be written in the form
:$g\; =\; g\_\; dx^\backslash mu\; \backslash otimes\; dx^\backslash nu\; .$
The factors $dx^\backslash mu$ are one-form
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

s of the scalar coordinate fields $x^\backslash mu$. The metric is thus a linear combination of tensor product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of one-form gradients of coordinates. The coefficients $g\_$ are a set of 16 real-valued functions (since the tensor $g$ is a ''tensor field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

'', which is defined at all points of a spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

manifold). In order for the metric to be symmetric we must have
:$g\_\; =\; g\_\; ,$
giving 10 independent coefficients.
If the local coordinates are specified, or understood from context, the metric can be written as a symmetric matrix
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces ...

with entries $g\_$. The nondegeneracy of $g\_$ means that this matrix is non-singular
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

(i.e. has non-vanishing determinant), while the Lorentzian signature of $g$ implies that the matrix has one negative and three positive eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to it ...

. Note that physicists often refer to this matrix or the coordinates $g\_$ themselves as the metric (see, however, abstract index notation
Abstract index notation is a mathematical notation for tensor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and ...

).
With the quantities $dx^\backslash mu$ being regarded as the components of an infinitesimal coordinate displacement four-vector
In special relativity
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...

(not to be confused with the one-forms of the same notation above), the metric determines the invariant square of an infinitesimal line element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc le ...

, often referred to as an ''interval''. The interval is often denoted
:$ds^2\; =\; g\_dx^\backslash mu\; dx^\backslash nu\; .$
The interval $ds^2$ imparts information about the causal structure of spacetime. When $ds^2\; <\; 0$, the interval is timelike
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

and the square root of the absolute value of $ds^2$ is an incremental proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

. Only timelike intervals can be physically traversed by a massive object. When $ds^2=0$, the interval is lightlike, and can only be traversed by (massless) things that move at the speed of light. When $ds^2\; >\; 0$, the interval is spacelike and the square root of $ds^2$ acts as an incremental proper length
Proper length or rest length is the length of an object in the object's rest frameIn special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest.
The rest frame of compound obje ...

. Spacelike intervals cannot be traversed, since they connect events that are outside each other's light cone
In special and general relativity, a light cone is the path that a flash of light, emanating from a single event
Event may refer to:
Gatherings of people
* Ceremony
A ceremony (, ) is a unified ritual
A ritual is a sequence of activities ...

s. Event
Event may refer to:
Gatherings of people
* Ceremony
A ceremony (, ) is a unified ritual
A ritual is a sequence of activities involving gestures, words, actions, or objects, performed according to a set sequence. Rituals may be prescribed ...

s 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 $x^\backslash mu\; \backslash to\; x^$, the metric components transform as
:$g\_\; =\; \backslash frac\backslash frac\; g\_\; =\; \backslash Lambda^\backslash rho\; \_\; \backslash ,\; \backslash Lambda^\backslash sigma\; \_\; \backslash ,\; g\_\; .$
Examples

Flat spacetime

The simplest example of a Lorentzian manifold isflat spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rela ...

, which can be given as RMinkowski metric
In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...

) is often denoted by the symbol ''η'' and is the metric used in special relativity
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

. In the above coordinates, the matrix representation of ''η'' is
:$\backslash eta\; =\; \backslash begin-c^2\&0\&0\&0\backslash \backslash 0\&1\&0\&0\backslash \backslash 0\&0\&1\&0\backslash \backslash 0\&0\&0\&1\backslash end$
(An alternative convention replaces coordinate $t$ by $ct$, and defines $\backslash eta$ as in .)
In spherical coordinates
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

$(t,r,\backslash theta,\backslash phi)$, the flat space metric takes the form
:$ds^2\; =\; -c^2\; dt^2\; +\; dr^2\; +\; r^2\; d\backslash Omega^2\; \backslash ,$
where
:$d\backslash Omega^2\; =\; d\backslash theta^2\; +\; \backslash sin^2\backslash theta\backslash ,d\backslash phi^2$
is the standard metric on the 2-sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

.
Black hole metrics

TheSchwarzschild metric
In Albert Einstein, Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass ...

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 theSchwarzschild metric
In Albert Einstein, Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass ...

which can be given in one set of local coordinates by
:$ds^\; =\; -\backslash left(1\; -\; \backslash frac\; \backslash right)\; c^2\; dt^2\; +\; \backslash left(1\; -\; \backslash frac\; \backslash right)^\; dr^2\; +\; r^2\; d\backslash Omega^2$
where, again, $d\backslash Omega^2$ is the standard metric on the 2-sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

. Here, $G$ is the gravitation constant and $M$ is a constant with the dimensions of mass
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...

. Its derivation can be found here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies
Here Technologies (trading as
A trade name, trading name, or business name is a pseudonym
A pseudonym () or alias () (originally: ...

. The Schwarzschild metric approaches the Minkowski metric as $M$ approaches zero (except at the origin where it is undefined). Similarly, when $r$ goes to infinity, the Schwarzschild metric approaches the Minkowski metric.
With coordinates
:$\backslash left(x^0,\; x^1,\; x^2,\; x^3\backslash right)=(ct,\; r,\; \backslash theta,\; \backslash varphi)\; \backslash ,,$
we can write the metric as
:$g\_\; =\; \backslash begin\; -\backslash left(1-\backslash frac\backslash right)\; \&\; 0\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; \backslash left(1-\backslash frac\backslash right)^\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; r^2\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \&\; r^2\; \backslash sin^2\; \backslash theta\; \backslash end\backslash ,.$
Several other systems of coordinates have been devised for the Schwarzschild metric: Eddington–Finkelstein coordinates In general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the olde ...

, Gullstrand–Painlevé coordinates, Kruskal–Szekeres coordinates
In general relativity Kruskal–Szekeres coordinates, named after Martin David Kruskal, Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cove ...

, 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 theKerr metric
The Kerr metric or Kerr geometry describes the geometry of empty spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dime ...

and the Kerr–Newman metric
The Kerr–Newman metric is the most general asymptotically flat, stationary spacetime, stationary solution of the Einstein's field equation#Einstein–Maxwell equations, Einstein–Maxwell equations in general relativity that describes the spacet ...

.
Other metrics

Other notable metrics are: * Alcubierre metric, *de Sitter
Willem de Sitter (6 May 1872 – 20 November 1934) was a Dutch mathematician, physicist, and astronomer.
Life and work
Born in Sneek, de Sitter studied mathematics at the University of Groningen and then joined the Groningen (city), Groninge ...

/ anti-de Sitter metrics,
*Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
Mathematics
* Metric (mathematics), an abstraction of the notion of ''dist ...

,
*Isotropic coordinates In the theory of Lorentzian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), offi ...

,
* Lemaître–Tolman metric (aka Bondi metric),
*Peres metric In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in p ...

,
*Rindler coordinates In relativistic physics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanics, quantum mechanical description of a system of particles, or of ...

,
* Weyl−Lewis−Papapetrou coordinates,
*Gödel metric.
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 (mathematics), region of a manifold. Given local coordinates $x^\backslash mu$ for the manifold, the volume form can be written :$\backslash mathrm\_g\; =\; \backslash pm\backslash sqrt\backslash ,dx^0\backslash wedge\; dx^1\backslash wedge\; dx^2\backslash wedge\; dx^3$ where $\backslash det[g\_]$ is the determinant of the matrix of components of the metric tensor for the given coordinate system.Curvature

The metric $g$ completely determines the curvature of spacetime. According to the fundamental theorem of Riemannian geometry, there is a unique connection (mathematics), connection ∇ on any semi-Riemannian manifold that is compatible with the metric and Torsion tensor, 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 $x^\backslash mu$ by the formula :$\backslash Gamma^\backslash lambda\; \_\; =\; g^\; \backslash left(\; +\; -\; \backslash right)\; =\; g^\; \backslash left(\; g\_\; +\; g\_\; -\; g\_\; \backslash right)$ (where commas indicate Covariant derivative#Notation, 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: :$\_\; =\; \backslash partial\_\backslash mu\backslash Gamma^\backslash rho\; \_\; -\; \backslash partial\_\backslash nu\backslash Gamma^\backslash rho\; \_\; +\; \backslash Gamma^\backslash rho\; \_\backslash Gamma^\backslash lambda\; \_\; -\; \backslash Gamma^\backslash rho\; \_\backslash Gamma^\backslash lambda\; \_.$ 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 ofspacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

. Einstein field equations, Einstein's field equations:
:$R\_\; -\; R\; g\_\; =\; \backslash frac\; \backslash ,T\_$
where the Ricci curvature tensor
:$R\_\; \backslash \; \backslash stackrel\backslash \; \_$
and the scalar curvature
:$R\; \backslash \; \backslash stackrel\backslash \; g^R\_$
relate the metric (and the associated curvature tensors) to the stress–energy tensor $T\_$. 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.
See also

*Alternatives to general relativity *Basic introduction to the mathematics of curved spacetime *Mathematics of general relativity *Ricci calculusReferences

* See general relativity resources for a list of references. {{tensors Tensors in general relativity Time in physics