Exact Solutions in General Relativity - Types of Exact Solution

Types of Exact Solution

Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stress-energy tensor:

• Vacuum solutions: ; these describe regions in which no matter or nongravitational fields are present,

• Electrovacuum solutions: must arise entirely from an electromagnetic field which solves the source-free Maxwell equations on the given curved Lorentzian manifold; this means that the only source for the gravitational field is the field energy (and momentum) of the electromagnetic field,

• Null dust solutions: must correspond to a stress-energy tensor which can be interpreted as arising from incoherent electromagnetic radiation, without necessarily solving the Maxwell field equations on the given Lorentzian manifold,

• Fluid solutions: must arise entirely from the stress-energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.

In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields:

• Scalar field solutions: must arise entirely from a scalar field (often a massless scalar field); these can arise in classical field theory treatments of meson beams, or as quintessence,

• Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): arises entirely from a nonzero cosmological constant.

One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. Presently, it seems that no exact solutions for this specific type are known.

Below we have sketched a classification by physical interpretation. This is probably more useful for most readers than the Segre classification of the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts:

• nonnull electrovacuums have Segre type and isotropy group SO(1,1) x SO(2),
• null electrovacuums and null dusts have Segre type and isotropy group E(2),
• perfect fluids have Segre type and isotropy group SO(3),
• Lambdavacuums have Segre type and isotropy group SO(1,3).

The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stress-energy tensor.