Metric tensor (general relativity)

TheInfoList

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 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

metric 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-dimensional
differentiable 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: :$g_x\left(u,v\right) = g_x\left(v,u\right) \in \mathbb.$ 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 in
local 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 $x^\mu$ (where $\mu$ is an index that runs from 0 to 3) the metric can be written in the form :$g = g_ dx^\mu \otimes dx^\nu .$ The factors $dx^\mu$ 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 $x^\mu$. 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 $dx^\mu$ 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 :$ds^2 = g_dx^\mu dx^\nu .$ 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 $x^\mu \to x^$, the metric components transform as :$g_ = \frac\frac g_ = \Lambda^\rho _ \, \Lambda^\sigma _ \, g_ .$

# Examples

## Flat spacetime

The simplest example of a Lorentzian manifold is
flat 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 R4 with coordinates $\left(t,x,y,z\right)$ and the metric :$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = \eta_ dx^ dx^. \,$ Note that these coordinates actually cover all of R4. The flat space metric (or
Minkowski 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 :$\eta = \begin-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end$ (An alternative convention replaces coordinate $t$ by $ct$, and defines $\eta$ 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 ...

$\left(t,r,\theta,\phi\right)$, the flat space metric takes the form :$ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2 \,$ where :$d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2$ 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

The
Schwarzschild 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 the
Schwarzschild 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 :$ds^ = -\left\left(1 - \frac \right\right) c^2 dt^2 + \left\left(1 - \frac \right\right)^ dr^2 + r^2 d\Omega^2$ where, again, $d\Omega^2$ 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 :$\left\left(x^0, x^1, x^2, x^3\right\right)=\left(ct, r, \theta, \varphi\right) \,,$ we can write the metric as :$g_ = \begin -\left\left(1-\frac\right\right) & 0 & 0 & 0\\ 0 & \left\left(1-\frac\right\right)^ & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2 \theta \end\,.$ 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 the
Kerr 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 $x^\mu$ for the manifold, the volume form can be written :$\mathrm_g = \pm\sqrt\,dx^0\wedge dx^1\wedge dx^2\wedge dx^3$ where $\det\left[g_\right]$ 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 $x^\mu$ by the formula :$\Gamma^\lambda _ = g^ \left\left( + - \right\right) = g^ \left\left( g_ + g_ - g_ \right\right)$ (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: :$_ = \partial_\mu\Gamma^\rho _ - \partial_\nu\Gamma^\rho _ + \Gamma^\rho _\Gamma^\lambda _ - \Gamma^\rho _\Gamma^\lambda _.$ 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 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 ...
. Einstein field equations, Einstein's field equations: :$R_ - R g_ = \frac \,T_$ where the Ricci curvature tensor :$R_ \ \stackrel\ _$ and the scalar curvature :$R \ \stackrel\ g^R_$ 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.