Metric tensor (general relativity)
Encyclopedia
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
, being used to define notions such as distance, volume, curvature, angle, future and past.
Notation and conventions: Throughout this article we work with a metric signature
that is mostly positive ; see sign convention
. As is customary in relativity, units
are used where the speed of light
c = 1. The gravitation constant G will be kept explicit. The summation convention, where repeated indices are automatically summed over, is employed.
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.
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 at a point x in M, the metric can be evaluated on u and v to give a real number:
This can be thought of as a generalization of the dot product
in ordinary Euclidean space
. This analogy is not exact, however. Unlike Euclidean space — where the dot product is positive definite — the metric gives each tangent space the structure of Minkowski space
.
(i.e. coordinates defined on some local patch
of M). In local coordinates
(where 
is an index which runs from 0 to 3) the metric can be written in the form
The factors
are one-form
gradient
s of the scalar coordinate fields
. The metric is thus a linear combination of tensor product
s of one-form gradients of coordinates. The coefficients
are a set of 16 real-valued functions (since the tensor g is actually a tensor field defined at all points of a spacetime
manifold). In order for the metric to be symmetric we must have
giving 10 independent coefficients. If we denote the symmetric tensor product
by juxtaposition (so that
) 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
. The nondegeneracy of 
means that this matrix 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 
themselves as the metric (see, however, abstract index notation
).
With the quantity
being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant interval squared or line element
. For this reason one often sees the notation
for the metric:
In general relativity, the terms metric and line element are often used interchangeably.
The line element
imparts information about the causal structure of the spacetime. When 
, the interval is timelike and the square root of the absolute value of ds2 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 light. When 
, the interval is spacelike and the square root of ds2 acts as an incremental proper length
. Spacelike intervals cannot be traversed, since they connect events that are out of each other's light cone
s. Events can be causally related only if they are within each other's light cones.
The metric components obviously depend on the chosen local coordinate system. Under a change of coordinates
the metric components transform as
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
In spherical coordinates
, the flat space metric takes the form
where
is the standard metric on the 2-sphere.
which can be given in one set of local coordinates by
where, again,
is the standard metric on the 2-sphere. Here G is the gravitation constant and M is a constant with the dimensions of mass
. Its derivation can be found here
. 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.
, Kerr metric
, Kerr–Newman metric, Kruskal–Szekeres coordinates, Lemaitre metric
, Reissner–Nordström metric, Rindler coordinates
. Some of them are without the event horizon
or can be without the gravitational singularity
.
, which can be used to integrate over spacetimes. In local coordinates
of a manifold, the volume form can be written
where det g is the determinant
of the matrix of components of the metric tensor for the given coordinate system.
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
by the formula
.
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
and its derivatives.
and energy
content of spacetime
. Einstein's field equations
:
where
relate the metric (and the associated curvature tensors) to the stress-energy tensor
. This tensor
equation is a complicated set of nonlinear partial differential equation
s for the metric components. Exact solutions of Einstein's field equations are very difficult to find.
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, 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
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
familiar from Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, being used to define notions such as distance, volume, curvature, angle, future and past.
Notation and conventions: Throughout this article we work with a metric signature
Metric signature
The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
that is mostly positive ; see sign convention
Sign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...
. As is customary in relativity, units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
are used where the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
c = 1. The gravitation constant G will be kept explicit. The summation convention, where repeated indices are automatically summed over, is employed.
Definition
Mathematically, spacetime is represented by a 4-dimensional differentiable manifoldDifferentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
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
Metric signature
The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
(-+++). A manifold M equipped with such a metric is called a Lorentzian manifold.
Explicitly, the metric is a symmetric bilinear form
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
on each tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of 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....
of M which 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:
This can be thought of as a generalization of the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
in ordinary Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. This analogy is not exact, however. Unlike Euclidean space — where the dot product is positive definite — the metric gives each tangent space the structure of Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
.
Local coordinates and matrix representations
Physicists usually work in local coordinatesLocal coordinates
Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...
(i.e. coordinates defined on some local patch
Atlas (topology)
In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
of M). In local coordinates
(where is an index which runs from 0 to 3) the metric can be written in the formThe factors
are one-formOne-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
s of the scalar coordinate fields
. The metric is thus a linear combination of tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
s of one-form gradients of coordinates. The coefficients
are a set of 16 real-valued functions (since the tensor g is actually a tensor field defined at all points of a spacetimeSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
manifold). In order for the metric to be symmetric we must have
giving 10 independent coefficients. If we denote the symmetric tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
by juxtaposition (so that
) we can write the metric in the formIf the local coordinates are specified, or understood from context, the metric can be written as a 4×4 symmetric matrix with entries
. The nondegeneracy of means that this matrix 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 themselves as the metric (see, however, abstract index notationAbstract index notation
Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical...
).
With the quantity
being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant interval squared or line elementLine element
A line element ds in mathematics can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet–Serret formulas...
. For this reason one often sees the notation
for the metric:In general relativity, the terms metric and line element are often used interchangeably.
The line element
imparts information about the causal structure of the spacetime. When , the interval is timelike and the square root of the absolute value of ds2 is an incremental proper timeProper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
. Only timelike intervals can be physically traversed by a massive object. When
, the interval is lightlike, and can only be traversed by light. When , the interval is spacelike and the square root of ds2 acts as an incremental proper lengthProper length
In relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....
. Spacelike intervals cannot be traversed, since they connect events that are out of each other's light cone
Light cone
A light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime...
s. Events can be causally related only if they are within each other's light cones.
The metric components obviously depend on the chosen local coordinate system. Under a change of coordinates
the metric components transform asFlat spacetime
The simplest example of a Lorentzian manifold is flat spacetime which can be given as R4 with coordinates and the metricNote 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
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
. In the above coordinates, the matrix representation of η is
In spherical coordinates
, the flat space metric takes the formwhere
is the standard metric on the 2-sphere.
Schwarzschild metric
Besides the flat space metric the most important metric in general relativity is the Schwarzschild metricSchwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
which can be given in one set of local coordinates by
where, again,
is the standard metric on the 2-sphere. Here G is the gravitation constant and M is a constant with the dimensions of massMass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
. Its derivation can be found here
Deriving the Schwarzschild solution
The Schwarzschild solution is one of the simplest and most useful solutions of theEinstein field equations . It describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object...
. 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.
Other metrics
Other notable metrics are Eddington–Finkelstein coordinates, Friedmann–Lemaître–Robertson–Walker metric, Gullstrand–Painlevé coordinates, Isotropic coordinatesIsotropic coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often...
, Kerr metric
Kerr metric
The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...
, Kerr–Newman metric, Kruskal–Szekeres coordinates, Lemaitre metric
Lemaitre metric
Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric—a spherically symmetric solution to vacuum Einstein equation— obtained by Georges Lemaître in 1938...
, Reissner–Nordström metric, Rindler coordinates
Rindler coordinates
In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum. The Rindler chart was introduced by Wolfgang Rindler. The Rindler coordinate system or frame describes a uniformly accelerating...
. Some of them are without the event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
or can be without the gravitational singularity
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
.
Volume
The metric g defines a natural volume formVolume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
, which can be used to integrate over spacetimes. In local coordinates
of a manifold, the volume form can be writtenwhere det g is the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the matrix of components of the metric tensor for the given coordinate system.
Curvature
The metric g completely determines the curvatureCurvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of spacetime. According to the fundamental theorem of Riemannian geometry
Fundamental theorem of Riemannian geometry
In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric...
, there is a unique connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
∇ on any semi-Riemannian manifold that is compatible with the metric and torsion
Torsion
The word torsion may refer to the following:*In geometry:** Torsion of a curve** Torsion tensor in differential geometry** The closely related concepts of Reidemeister torsion and analytic torsion ** Whitehead torsion*In algebra:** Torsion ** Tor functor* In medicine:** Ovarian...
-free. This connection is called the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
. The Christoffel symbols
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
of this connection are given in terms of partial derivatives of the metric in local coordinates
by the formula.The curvature of spacetime is then given by the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
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
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 matterMatter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
and energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
content of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. Einstein's field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
:
where
relate the metric (and the associated curvature tensors) to the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
. This tensorTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
equation is a complicated set of nonlinear partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s for the metric components. Exact solutions of Einstein's field equations are very difficult to find.
See also
- Alternatives to general relativityAlternatives to general relativityAlternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation in competition to Einstein's theory of general relativity.There have been many different attempts at constructing an ideal theory of gravity...
- Basic introduction to the mathematics of curved spacetimeBasic introduction to the mathematics of curved spacetimeThe mathematics of general relativity are very complex. In Newton's theories of motions, an object's mass and length remain constant as it changes speed, and the rate of passage of time also remains unchanged. As a result, many problems in Newtonian mechanics can be solved with algebra alone...
- Mathematics of general relativityMathematics of general relativityThe mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold...