When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized.
The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime.
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.
More precisely, the basic physical construct representing gravitation — a curved spacetime — is modelled by a four-dimensional, smooth, connected, Lorentzian manifold.
The rationale for choosing a manifold as the fundamental mathematical structure is to reflect desirable physical properties.
Any observer can make measurements and the precise numerical quantities obtained only depend on the coordinate system used.
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 equations).
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:
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 affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction.
This bilinear map can be described in terms of a set of connection coefficients (also known as Christoffel symbols) specifying what happens to components of basis vectors under infinitesimal parallel transport:
in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve
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.
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.
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.
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.
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,
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.
The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol.
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.
, 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.
A fairly universal way of performing these derivations is by using the techniques of variational calculus, the main objects used in this being Lagrangians.
Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising' solutions of the field equations.
For example, in classifying the Weyl tensor, determining the various Petrov types becomes much easier when compared with the tensorial counterpart.
This novel idea finds application in approximation methods in numerical relativity and quantum gravity, the latter using a generalisation of Regge calculus.
In general relativity, it was noted that, under fairly generic conditions, gravitational collapse will inevitably result in a so-called singularity.
A singularity is a point where the solutions to the equations become infinite, indicating that the theory has been probed at inappropriate ranges.
[1] The defining feature (central physical idea) of general relativity is that matter and energy cause the surrounding spacetime geometry to be curved.