Mathematics of general relativity

The principle of general covariance was one of the central principles in the development of general relativity. It states that the laws of physics should take the same mathematical form in all reference frames. The term 'general covariance' was used in the early formulation of general relativity, but the principle is now often referred to as 'diffeomorphism covariance'.

Diffeomorphism covariance is not the defining feature of general relativity,^[1] and controversies remain regarding its present status in general relativity. However, the invariance property of physical laws implied in the principle, coupled with the fact that the theory is essentially geometrical in character (making use of non-Euclidean geometries), suggested that general relativity be formulated using the language of tensors. This will be discussed further below.

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 this chart 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.

One of the profound consequences of relativity theory was the abolition of privileged 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 demonstrated that no inertial reference frame was preferential to any other inertial reference frame, but preferred inertial reference frames over noninertial reference frames. General relativity eliminated preference for inertial reference frames by showing that there is no preferred reference frame (inertial or not) 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 tensor .

Mathematically, tensors are generalised linear operators - multilinear maps. As such, the ideas of linear algebra are employed to study tensors.

At each point ${\displaystyle p}$ 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 ${\displaystyle p}$, these two vector spaces may be used to construct type ${\displaystyle (r,s)}$ tensors, which are real-valued multilinear maps acting on the direct sum of ${\displaystyle r}$ copies of the cotangent space with ${\displaystyle s}$ copies of the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type ${\displaystyle (r,s)}$ at ${\displaystyle p}$ and denoted by ${\displaystyle (T_{p})_{s}^{r}M.}$ If the tangent space is n-dimensional, it can be shown that ${\displaystyle \dim(T_{p})_{s}^{r}M=n^{r+s}.}$

In the general relativity literature, it is conventional to use the component syntax for tensors.

A type ${\displaystyle (r,s)}$ tensor may be written as

${\displaystyle T={T^{a_{1}\ldots a_{r}}}_{{b_{1}}\ldots {b_{s}}}{\frac {\partial }{\partial x^{a_{1}}}}\otimes \ldots \otimes {\frac {\partial }{\partial x^{a_{r}}}}\otimes dx^{b_{1}}\otimes \ldots \otimes dx^{b_{s}}}$

where ${\displaystyle {\tfrac {\partial }{\partial x^{a_{i}}}}}$ is a basis for the i-th tangent space and ${\displaystyle dx^{b_{j}}}$ 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 4^R, where R is the count of the number of covariant ${\displaystyle (b_{i})}$ and contravariant ${\displaystyle (a_{i})}$ indices on the tensor, ${\displaystyle r+s}$ (a number called the rank of the tensor).

Tensor fields on a manifold are maps which attach a tensor to each point of the manifold. This notion can be made more precise by introducing the idea of a fibre bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. A tensor field is then defined as a map from the manifold to the tensor bundle, each point ${\displaystyle p}$ being associated with a tensor at ${\displaystyle p}$.

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, ${\displaystyle U^{a}={\dot {x}}^{a}}$, which is the coordinate distance travelled per unit of proper time, the four-acceleration ${\displaystyle A^{a}={\ddot {x}}^{a}}$ and the four-current ${\displaystyle J^{a}}$ describing the charge and current densities. Other physically important tensor fields in relativity include the following:

• The stress–energy tensor ${\displaystyle T^{ab}}$, a symmetric rank-two tensor.
• The electromagnetic field tensor ${\displaystyle F^{ab}}$, a rank-two antisymmetric tensor.

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'.

At each point of a spacetime on which a metric is defined, the metric can be reduced to the Minkowski form using Sylvester's law of inertia.

Before the advent of general relativity, changes in physical processes were generally described by partial derivatives, for example, in describing changes in electromagnetic fields (see Maxwell's equations). Even in special relativity, the partial derivative is still sufficient to describe such changes. However, in general relativity, it is found that derivatives which are also tensors must be used. The derivatives have some common features including that they are derivatives along integral curves of vector fields.

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.

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 their 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.

What the Riemann Tensor allows us to do is tell, mathematically, whether a space is flat or, if curved, how much curvature takes place in any given region. In order to derive the Riemann curvature tensor we must first recall the definition of the covariant derivative of a tensor with one and two indices;

1. ${\displaystyle \nabla _{\mu }V_{\nu }=\partial _{\mu }V_{\nu }-\Gamma ^{\rho }{}_{\mu \nu }V_{\rho }}$
2. ${\displaystyle \nabla _{\sigma }[V_{\mu \nu }]=\partial _{\sigma }V_{\mu \nu }-\Gamma ^{\rho }{}_{\mu \sigma }V_{\rho \nu }-\Gamma ^{\rho }{}_{\nu \sigma }V_{\mu \rho }}$

For the formation of the Riemann tensor, the covariant derivative is taken twice with the respect to a tensor of rank one. The equation is set up as follows;

${\displaystyle {\begin{aligned}\nabla _{\sigma$ ;\mu }V_{\nu }&=\nabla _{\sigma }[\nabla _{\mu }V_{\nu }]\\&=\partial _{\sigma }[\nabla _{\mu }V_{\nu }]-\Gamma ^{\rho }{}_{\mu \sigma }[\nabla _{\rho }V_{\nu }]-\Gamma ^{\rho }{}_{\nu \sigma }[\nabla _{\mu }V_{\rho }]\\&=\partial _{\sigma }[\partial _{\mu }V_{\nu }-\Gamma ^{\alpha }{}_{\nu \mu }V_{\alpha }]-\Gamma ^{\rho }{}_{\mu \sigma }[\partial _{\rho }V_{\nu }-\Gamma ^{\alpha }{}_{\nu \rho }V_{\alpha }]-\Gamma ^{\rho }{}_{\nu \sigma }[\partial _{\mu }V_{\rho }-\Gamma ^{\alpha }{}_{\rho \mu }V_{\alpha }]\\&=\partial _{\sigma }\partial _{\mu }V_{\nu }-\partial _{\sigma }(\Gamma ^{\alpha }{}_{\nu \mu }V_{\alpha })-\Gamma ^{\rho }{}_{\mu \sigma }\partial _{\rho }V_{\nu }+\Gamma ^{\rho }{}_{\mu \sigma }\Gamma ^{\alpha }{}_{\nu \rho }V_{\alpha }-\Gamma ^{\rho }{}_{\nu \sigma }\partial _{\mu }V_{\rho }+\Gamma ^{\rho }{}_{\nu \sigma }\Gamma ^{\alpha }{}_{\rho \mu }V_{\alpha }\\&=\partial _{\sigma }\partial _{\mu }V_{\nu }-\partial _{\sigma }(\Gamma ^{\alpha }{}_{\nu \mu })V_{\alpha }-\Gamma ^{\alpha }{}_{\nu \mu }\partial _{\sigma }(V_{\alpha })-\Gamma ^{\rho }{}_{\mu \sigma }\partial _{\rho }V_{\nu }+\Gamma ^{\rho }{}_{\mu \sigma }\Gamma ^{\alpha }{}_{\nu \rho }V_{\alpha }-\Gamma ^{\rho }{}_{\nu \sigma }\partial _{\mu }V_{\rho }+\Gamma ^{\rho }{}_{\nu \sigma }\Gamma ^{\alpha }{}_{\rho \mu }V_{\alpha }\end{aligned}}}

Similarly we have:

${\displaystyle \nabla _{\mu$ ;\sigma }V_{\nu }=\partial _{\mu }\partial _{\sigma }V_{\nu }-\partial _{\mu }(\Gamma ^{\alpha }{}_{\nu \sigma })V_{\alpha }-\Gamma ^{\alpha }{}_{\nu \sigma }\partial _{\mu }(V_{\alpha })-\Gamma ^{\rho }{}_{\sigma \mu }\partial _{\rho }V_{\nu }+\Gamma ^{\rho }{}_{\sigma \mu }\Gamma ^{\alpha }{}_{\nu \rho }V_{\alpha }-\Gamma ^{\rho }{}_{\nu \mu }\partial _{\sigma }V_{\rho }+\Gamma ^{\rho }{}_{\nu \mu }\Gamma ^{\alpha }{}_{\rho \sigma }V_{\alpha }}

Subtracting the two equations, swapping dummy indices and using the symmetry of Christoffel symbols leaves:

${\displaystyle \nabla _{\sigma$ ;\mu }V_{\nu }-\nabla _{\mu ;\sigma }V_{\nu }=(\partial _{\mu }\Gamma ^{\alpha }{}_{\nu \sigma }-\partial _{\sigma }\Gamma ^{\alpha }{}_{\nu \mu }+\Gamma ^{\rho }{}_{\nu \sigma }\Gamma ^{\alpha }{}_{\rho \mu }-\Gamma ^{\rho }{}_{\nu \mu }\Gamma ^{\alpha }{}_{\rho \sigma })V_{\alpha }}

or

${\displaystyle R_{\nu \mu \sigma }^{\alpha }V_{\alpha }=(\partial _{\mu }\Gamma ^{\alpha }{}_{\nu \sigma }-\partial _{\sigma }\Gamma ^{\alpha }{}_{\nu \mu }+\Gamma ^{\rho }{}_{\nu \sigma }\Gamma ^{\alpha }{}_{\rho \mu }-\Gamma ^{\rho }{}_{\nu \mu }\Gamma ^{\alpha }{}_{\rho \sigma })V_{\alpha }}$

Finally the Riemann curvature tensor is written as;

${\displaystyle R_{\nu \mu \sigma }^{\alpha }=\partial _{\mu }\Gamma ^{\alpha }{}_{\nu \sigma }-\partial _{\sigma }\Gamma ^{\alpha }{}_{\nu \mu }+\Gamma ^{\rho }{}_{\nu \sigma }\Gamma ^{\alpha }{}_{\rho \mu }-\Gamma ^{\rho }{}_{\nu \mu }\Gamma ^{\alpha }{}_{\rho \sigma }}$

You can contract indices to make the tensor covariant simply by multiplying by the metric, which will be useful when working with Einstein's field equations,

${\displaystyle g_{\alpha \lambda }R_{\nu \mu \sigma }^{\lambda }=R_{\alpha \nu \mu \sigma }}$

and by further decomposition,

${\displaystyle g^{\alpha \mu }R_{\alpha \nu \mu \sigma }=R_{\nu \sigma }}$

This tensor is called the Ricci tensor which can also be derived by setting ${\displaystyle \alpha }$ and ${\displaystyle \mu }$ in the Riemann tensor to the same indice and summing over them. Then the curvature scalar can be found by going one step further,

${\displaystyle g^{\nu \sigma }R_{\nu \sigma }=R}$

So now we have 3 different objects,

1. the Riemann curvature tensor: ${\displaystyle R_{\nu \mu \sigma }^{\alpha }}$ or ${\displaystyle R_{\alpha \nu \mu \sigma }}$
2. the Ricci tensor: ${\displaystyle R_{\nu \sigma }}$
3. the scalar curvature: ${\displaystyle R}$

all of which are useful in calculating solutions to Einstein's field equations.

The sources of any gravitational field (matter and energy) is 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 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:

${\displaystyle G_{ab}+\Lambda g_{ab}={8\pi G \over c^{4}}T_{ab}}$

where ${\displaystyle G_{ab}}$ is the Einstein tensor, ${\displaystyle \Lambda }$ is the cosmological constant, ${\displaystyle g_{ab}}$ is the metric tensor, ${\displaystyle c}$ is the speed of light in a vacuum and ${\displaystyle G}$ is the gravitational constant, which comes from Newton's law of universal gravitation.

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 ${\displaystyle {\vec {U}}}$; i.e., ${\displaystyle \nabla _{\vec {U}}{\vec {U}}=0}$. This condition, the geodesic equation, can be written in terms of a coordinate system ${\displaystyle x^{a}}$ with the tangent vector ${\displaystyle U^{a}={\frac {dx^{a}}{d\tau }}}$:

${\displaystyle {\ddot {x}}^{a}+{\Gamma ^{a}}_{bc}\,{\dot {x}}^{b}\,{\dot {x}}^{c}=0}$

where ${\displaystyle {\dot {}}}$ denotes the derivative by proper time, ${\displaystyle d/d\tau }$, with τ parametrising proper time along the curve and making manifest the presence of the Christoffel symbols.

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).

Having outlined the basic mathematical structures used in formulating the theory, some important mathematical techniques that are employed in investigating spacetimes will now be discussed.

• Ricci calculus – Tensor index notation for tensor-based calculations

^[1] The defining feature (central physical idea) of general relativity is that matter and energy cause the surrounding spacetime geometry to be curved.

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• Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
• Landau, L. D. & Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.
• Petrov, A. N.; Kopeikin, S. M.; Tekin, B. & Lompay, R. (2017). Metric Theories of Gravity: perturbations and conservation laws. Berlin: De Gruyter. doi:10.1515/9783110351781. ISBN 978-3-11-035173-6.