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. 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.


Metric tensor of spacetime in general relativity written as a matrix

In general relativity, the metric tensor plays the role of the gravitational potential in the classical theory of gravitation, although the physical content of the associated equations is entirely different.[1] Gutfreund and Renn say "that in general relativity the gravitational potential is represented by the metric tensor."[2]

Notation and conventions edit

This article works with a metric signature that is mostly positive (− + + +); see sign convention. The gravitation constant   will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over.

Definition edit

Mathematically, spacetime is represented by a four-dimensional differentiable manifold   and the metric tensor is given as a covariant, second-degree, symmetric tensor on  , conventionally denoted by  . Moreover, the metric is required to be nondegenerate with signature (− + + +). A manifold   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   that varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors   and   at a point   in  , the metric can be evaluated on   and   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 edit

Physicists usually work in local coordinates (i.e. coordinates defined on some local patch of  ). In local coordinates   (where   is an index that runs from 0 to 3) the metric can be written in the form

 
The factors   are one-form gradients of the scalar coordinate fields  . The metric is thus a linear combination of tensor products of one-form gradients of coordinates. The coefficients   are a set of 16 real-valued functions (since the tensor   is a tensor field, which is defined at all points of a spacetime manifold). In order for the metric to be symmetric
 
giving 10 independent coefficients.

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   implies that the matrix has one negative and three positive eigenvalues. Physicists often refer to this matrix or the coordinates   themselves as the metric (see, however, abstract index notation).

With the quantities   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   imparts information about the causal structure of spacetime. When  , the interval is timelike and the square root of the absolute value of   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 (massless) things that move at the speed of light. When  , the interval is spacelike and the square root of   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  , the metric components transform as

 

Properties edit

The metric tensor plays a key role in index manipulation. In index notation, the coefficients   of the metric tensor   provide a link between covariant and contravariant components of other tensors. Contracting the contravariant index of a tensor with one of a covariant metric tensor coefficient has the effect of lowering the index

 
and similarly a contravariant metric coefficient raises the index
 
Applying this property of raising and lowering indices to the metric tensor components themselves leads to the property
 
For a diagonal metric (one for which coefficients  ; i.e. the basis vectors are orthogonal to each other), this implies that a given covariant coefficient of the metric tensor is the inverse of the corresponding contravariant coefficient  , etc.

Examples edit

Flat spacetime edit

The simplest example of a Lorentzian manifold is flat spacetime, which can be given as R4 with coordinates   and the metric

 
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
 
(An alternative convention replaces coordinate   by  , and defines   as in Minkowski space § Standard basis.)

In spherical coordinates  , the flat space metric takes the form

 
where
 
is the standard metric on the 2-sphere.

Black hole metrics edit

The Schwarzschild metric describes an uncharged, non-rotating black hole. There are also metrics that describe rotating and charged black holes.

Schwarzschild metric edit

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,   is the standard metric on the 2-sphere. Here,   is the gravitation constant and   is a constant with the dimensions of mass. Its derivation can be found here. The Schwarzschild metric approaches the Minkowski metric as   approaches zero (except at the origin where it is undefined). Similarly, when   goes to infinity, the Schwarzschild metric approaches the Minkowski metric.

With coordinates

 
the metric can be written 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 edit

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.[further explanation needed]

Other metrics edit

Other notable metrics are:

Some of them are without the event horizon or can be without the gravitational singularity.

Volume edit

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   for the manifold, the volume form can be written

 
where   is the determinant of the matrix of components of the metric tensor for the given coordinate system.

Curvature edit

The metric   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   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   and its derivatives.

Einstein's equations edit

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  . 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 edit

References edit

  1. ^ For the details, see Section 2.11, The Metric Tensor and the Classical Gravitational Potential, in Chow, Tai L. (2008). Gravity, Black Holes, and the Very Early Universe: An Introduction to General Relativity and Cosmology. Springer. ISBN 9780387736310.
  2. ^ Gutfreund, Hanoch; Renn, Jürgen (2015). The Road to Relativity: The History and Meaning of Einstein's "The Foundation of General Relativity", Featuring the Original Manuscript of Einstein's Masterpiece. Princeton University Press. p. 75. ISBN 9780691175812.