This article is a general description of the mathematics of general relativity. For a discussion of the minimal mathematics necessary to understand general relativity see Basic introduction to the mathematics of curved spacetime.
Most modern approaches to mathematical general relativity begin with the concept of a manifold. More precisely, the basic physical construct representing gravitation - a curved spacetime - is modelled by a four-dimensional, smooth, connected, Lorentzian manifold. Other physical descriptors are represented by various tensors, discussed below.
The rationale for choosing a manifold as the fundamental mathematical structure is to reflect desirable physical properties. For example, in the theory of manifolds, each point is contained in a (by no means unique) coordinate chart and can be thought of as representing the 'local spacetime' around the observer (represented by the point). The principle of local Lorentz covariance, which states that the laws of special relativity hold locally about each point of spacetime, lends further support to the choice of a manifold structure for representing spacetime, as locally around a point on a general manifold, the region 'looks like', or approximates very closely Minkowski space (flat spacetime).
The idea of coordinate charts as 'local observers who can perform measurements in their vicinity' also makes good physical sense, as this is how one actually collects physical data - locally. For cosmological problems, a coordinate chart may be quite large.
An important problem in general relativity is to tell when two spacetimes are 'the same', at least locally. This problem has its roots in manifold theory where determining if two Riemannian manifolds of the same dimension are locally isometric ('locally the same'). This latter problem has been solved and its adaptation for general relativity is called the Cartan-Karlhede algorithm.
One of the profound consequences of relativity theory was the abolishment of preferred reference frames. The description of physical phenomena should not depend upon who does the measuring - one reference frame should be as good as any other. Special relativity banished the singling out of inertial frames whereas general relativity eliminated any privileged reference frames for describing nature.
Any observer can make measurements and the precise numerical quantities obtained only depend on the coordinate system used. This suggested a way of formulating relativity using 'invariant structures', those that are independent of the coordinate system (represented by the observer) used, yet still have an independent existence. The most suitable mathematical structure seemed to be a tensor. For example, when measuring the electric and magnetic fields produced by an accelerating charge, the values of the fields will depend on the coordinate system used, but the fields are regarded as having an independent existence, this independence represented by the electromagnetic field tensor .
At each point of a manifold, the tangent and cotangent spaces to the manifold at that point may be constructed. Vectors (sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly dual vectors or one-forms) are elements of the cotangent space.
At , these two vector spaces may be used to construct type tensors, which are real-valued multilinear maps acting on the direct sum of copies of the cotangent space with copies of the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type at and denoted by . If the tangent space is n-dimensional, it can be shown that .
In the general relativity literature, it is conventional to use the component syntax for tensors.
A type (r,s) tensor may be written as
where is a basis for the i-th tangent space and a basis for the j-th cotangent space.
As spacetime is assumed to be four-dimensional, each index on a tensor can be one of four values. Hence, the total number of elements a tensor possesses equals 4R, where R is the sum of the numbers of covariant and contravariant indices on the tensor (a number called the rank of the tensor).
Although a generic rank R tensor in 4 dimensions has 4R components, constraints on the tensor such as symmetry or antisymmetry serve to reduce the number of distinct components. For example, a symmetric rank two tensor satisfies Tab = Tba and possesses 10 independent components, whereas an antisymmetric (skew-symmetric) rank two tensor satisfies Pab = -Pba and has 6 independent components. For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified.
Antisymmetric tensors of rank 2 play important roles in relativity theory. The set of all such tensors - often called bivectors - forms a vector space of dimension 6, sometimes called bivector space.
The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equation). Using the weak-field approximation, the metric can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'.
The metric is a symmetric tensor and is an important mathematical tool. As well as being used to raise and lower tensor indices, it also generates the connections which are used to construct the geodesic equations of motion and the Riemann curvature tensor.
A convenient means of expressing the metric tensor in combination with the incremental intervals of coordinate distance that it relates to is through the line element:
This way of expressing the metric was used by the pioneers of differential geometry. While some relativists consider the notation to be somewhat old-fashioned, many readily switch between this and the alternative notation:
The metric tensor is commonly written as a 4 by 4 matrix. Due to the symmetry of the metric, this matrix is symmetric and has 10 independent components.
A more explicit description can be given using tensors. The crucial feature of tensors used in this approach is the fact that (once a metric is given) the operation of contracting a tensor of rank R over all R indices gives a number - an invariant - that is independent of the coordinate chart one uses to perform the contraction. Physically, this means that if the invariant is calculated by any two observers, they will get the same number, thus suggesting that the invariant has some independent significance. Some important invariants in relativity include:
Other examples of invariants in relativity include the electromagnetic invariants, and various other curvature invariants, some of the latter finding application in the study of gravitational entropy and the Weyl curvature hypothesis.
The notion of a tensor field is of major importance in GR. For example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles. Another example is the values of the electric and magnetic fields (given by the electromagnetic field tensor) and the metric at each point around a charged black hole to determine the motion of a charged particle in such a field.
Vector fields are contravariant rank one tensor fields. Important vector fields in relativity include the four-velocity, , which is the coordinate distance travelled per unit of proper time, the four-acceleration and the four-current describing the charge and current densities. Other physically important tensor fields in relativity include the following:
Although the word 'tensor' refers to an object at a point, it is common practice to refer to tensor fields on a spacetime (or a region of it) as just 'tensors'.
The problem in defining derivatives on manifolds that are not flat is that there is no natural way to compare vectors at different points. An extra structure on a general manifold is required to define derivatives. Below are described two important derivatives that can be defined by imposing an additional structure on the manifold in each case.
The curvature of a spacetime can be characterised by taking a vector at some point and parallel transporting it along a curve on the spacetime. An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction.
By definition, an affine connection is a bilinear map , where is a space of all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients specifying what happens to components of basis vectors under infinitesimal parallel transport:
Despite their tempting appearance, the connection coefficients are not the components of a tensor.
Generally speaking, there are D3 independent connection coefficients at each point of spacetime. The connection is called symmetric if . A symmetric connection has D2(D+1)/2 coefficients.
For any curve and two points and on this curve, an affine connection gives rise to a map of vectors in the tangent space at A into vectors in the tangent space at B:
and can be computed component-wise by solving the differential equation
being the vector tangent to the curve at the point .
An important affine connection in general relativity is the Levi-Civita connection, which is a symmetric connection obtained from parallel transporting a tangent vector along a curve whilst keeping the inner product of that vector constant along the curve. The resulting connection coefficients are called Christoffel symbols and can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection.
for a covariant derivative of along associated with connection turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative.
It can be expressed using connection coefficients:
The expression in brackets, called a covariant derivative of (with respect to the connection) and denoted by , is more often used in calculations:
A covariant derivative of X can thus be viewed as a differential operator acting on a vector field sending it to a type (1,1) tensor ('increasing the covariant index by 1') and can be generalised to act on type (r,s) tensor fields sending them to type (r, s+1) tensor fields. Notions of parallel transport can then be defined similarly as for the case of vector fields. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field.
In the literature, there are three common methods of denoting covariant differentiation:
Many standard properties of regular partial derivatives also apply to covariant derivatives:
, if c is a constant
In General Relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. By definition, Levi-Civita connection preserves the metric under parallel transport, therefore, the covariant derivative gives zero when acting on a metric tensor (as well as its inverse). It means that we can take the (inverse) metric tensor in and out of the derivative and use it to raise and lower indices:
The Lie derivative is usually denoted by , where is the vector field along whose congruence the Lie derivative is taken.
The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. (In fact, any derivative will work, but covariant derivative is convenient because it commutes with raising and lowering indices.) The Lie derivative of a scalar is just the directional derivative:
Higher rank objects pick up additional terms when the Lie derivative is taken. For example, the Lie derivative of a type (0,2) tensor is
One of the main uses of the Lie derivative in general relativity is in the study of spacetime symmetries where tensors or other geometrical objects are preserved. In particular, Killing symmetry (symmetry of the metric tensor under Lie dragging) occurs very often in the study of spacetimes. Using the formula above, we can write down the condition that must be satisfied for a vector field to generate a Killing symmetry:
A crucial feature of general relativity is the concept of a curved manifold. A useful way of measuring the curvature of a manifold is with an object called the Riemann (curvature) tensor.
This tensor measures curvature by use of an affine connection by considering the effect of parallel transporting a vector between two points along two curves. The discrepancy between the results of these two parallel transport routes is essentially quantified by the Riemann tensor.
This property of the Riemann tensor can be used to describe how initially parallel geodesics diverge. This is expressed by the equation of geodesic deviation and means that the tidal forces experienced in a gravitational field are a result of the curvature of spacetime.
Using the above procedure, the Riemann tensor is defined as a type (1,3) tensor and when fully written out explicitly contains the Christoffel symbols and its first partial derivatives. The Riemann tensor has 20 independent components. The vanishing of all these components over a region indicates that the spacetime is flat in that region. From the viewpoint of geodesic deviation, this means that initially parallel geodesics in that region of spacetime will stay parallel.
The Riemann tensor has a number of properties sometimes referred to as the symmetries of the Riemann tensor. Of particular relevance to general relativity are the algebraic and differential Bianchi identities.
The connection and curvature of any Riemannian manifold are closely related, the theory of holonomy groups, which are formed by taking linear maps defined by parallel transport around curves on the manifold, providing a description of this relationship.
The sources of any gravitational field (matter and energy) are represented in relativity by a type (0,2) symmetric tensor called the energy-momentum tensor. It is closely related to the Ricci tensor. Being a second rank tensor in four dimensions, the energy-momentum tensor may be viewed as a 4 by 4 matrix. The various admissible matrix types, called Jordan forms cannot all occur, as the energy conditions that the energy-momentum tensor is forced to satisfy rule out certain forms.
The corresponding statement of local energy conservation in special relativity is:
This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'.
The Einstein field equations (EFE) are the core of general relativity theory. The EFE describe how mass and energy (as represented in the stress-energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor). In abstract index notation, the EFE reads as follows:
The solutions of the EFE are metric tensors. The EFE, being non-linear differential equations for the metric, are often difficult to solve. There are a number of strategies used to solve them. For example, one strategy is to start with an ansatz (or an educated guess) of the final metric, and refine it until it is specific enough to support a coordinate system but still general enough to yield a set of simultaneous differential equations with unknowns that can be solved for. Metric tensors resulting from cases where the resultant differential equations can be solved exactly for a physically reasonable distribution of energy-momentum are called exact solutions. Examples of important exact solutions include the Schwarzschild solution and the Friedman-Lemaître-Robertson-Walker solution.
The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol. 16 Issue 1).
Once the EFE are solved to obtain a metric, it remains to determine the motion of inertial objects in the spacetime. In general relativity, it is assumed that inertial motion occurs along timelike and null geodesics of spacetime as parameterized by proper time. Geodesics are curves that parallel transport their own tangent vector , i.e. . This condition - the geodesic equation - can be written in terms of a coordinate system with the tangent vector :
A principal feature of general relativity is to determine the paths of particles and radiation in gravitational fields. This is accomplished by solving the geodesic equations.
The EFE relate the total matter (energy) distribution to the curvature of spacetime. Their nonlinearity leads to a problem in determining the precise motion of matter in the resultant spacetime. For example, in a system composed of one planet orbiting a star, the motion of the planet is determined by solving the field equations with the energy-momentum tensor the sum of that for the planet and the star. The gravitational field of the planet affects the total spacetime geometry and hence the motion of objects. It is therefore reasonable to suppose that the field equations can be used to derive the geodesic equations.
When the energy-momentum tensor for a system is that of dust, it may be shown by using the local conservation law for the energy-momentum tensor that the geodesic equations are satisfied exactly.
The issue of deriving the equations of motion or the field equations in any physical theory is considered by many researchers to be appealing. A fairly universal way of performing these derivations is by using the techniques of variational calculus, the main objects used in this being Lagrangians.
Many consider this approach to be an elegant way of constructing a theory, others as merely a formal way of expressing a theory (usually, the Lagrangian construction is performed after the theory has been developed).
A frame field is an orthonormal set of 4 vector fields (1 timelike, 3 spacelike) defined on a spacetime. Each frame field can be thought of as representing an observer in the spacetime moving along the integral curves of the timelike vector field. Every tensor quantity can be expressed in terms of a frame field, in particular, the metric tensor takes on a particularly convenient form. When allied with coframe fields, frame fields provide a powerful tool for analysing spacetimes and physically interpreting the mathematical results.
Some modern techniques in analysing spacetimes rely heavily on using spacetime symmetries, which are vector fields (usually defined locally) on a spacetime that preserve some feature of the spacetime. The most common type of such symmetry vector fields include Killing vector fields (which preserve the metric structure) and their generalisations called generalised Killing vector fields. Symmetry vector fields find extensive application in the study of exact solutions in general relativity and the set of all such vector fields usually forms a finite-dimensional Lie algebra.
The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation given initial conditions. In the context of general relativity, it means the problem of finding solutions to Einstein's field equations - a system of hyperbolic partial differential equations - given some initial data on a hypersurface. Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising' solutions of the field equations. Ideally, one desires global solutions, but usually local solutions are the best that can be hoped for.
Another appealing feature of spinors in general relativity is the condense way in which some tensor equations may be written using the spinor formalism. For example, in classifying the Weyl tensor, determining the various Petrov types becomes much easier when compared with the tensorial counterpart.
Regge calculus is a formalism which chops up a Lorentzian manifold into discrete 'chunks' (four-dimensional simplicial blocks) and the block edge lengths are taken as the basic variables. A discrete version of the Einstein-Hilbert action is obtained by considering so called 'deficit angles' of these blocks, a zero deficit angle corresponding to no curvature. This novel idea finds application in approximation methods in numerical relativity and quantum gravity, the latter using a generalisation of Regge calculus.