Exact Solutions in General Relativity - Constructing Solutions

Constructing Solutions

The Einstein field equation, when fully written out as a system of partial differential equations, takes the form of a rather complicated system of coupled, nonlinear partial differential equations. As such, in general, it is very hard to solve.

Nonetheless, several effective techniques for obtaining exact solutions are available.

The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum).

This naive approach usually works best if one uses a frame field rather than a coordinate basis.

A related idea involves imposing algebraic symmetry conditions on the Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor. As will be apparent from the discussion above, such Ansätze often do have some physical content, although this might not be apparent from their mathematical form.

This second kind of symmetry approach has often been used with the Newman-Penrose formalism, which uses spinorial quantities for more efficient bookkeeping.

Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS).

But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie's notion of the point symmetry of a differential equation (or system of equations), and as Lie showed, this can provide an avenue of attack upon any differential equation which has a nontrivial symmetry group. Indeed, both the Ernst equation and the NLS have nontrivial symmetry groups, and some solutions can be found by taking advantage of their symmetries. These symmetry groups are often infinite dimensional, but this is not always a useful feature.

Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an infinite sequence of conservation laws. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible.

There are also various transformations (see Belinski-Zakharov transform) which can transform (for example) a vacuum solution found by other means into a new vacuum solution, or into an electrovacuum solution, or a fluid solution. These are analogous to the Bäcklund transformations known from the theory of certain partial differential equations, including some famous examples of soliton equations. This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. Unfortunately, even when applied to a "well understood", globally admissible solution, these transformations often yield a solution which is poorly understood and their general interpretation is still unknown.