In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter.
Moreover, taking covariant derivatives of the field equations and applying the Bianchi identities, it is found that a suitably varying amount/motion of non-gravitational energy–momentum can cause ripples in curvature to propagate as gravitational radiation, even across vacuum regions, which contain no matter or non-gravitational fields.
Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator.
On the one hand, these conditions are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable.
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.
Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations.
This may give some idea of how just tiny a typical "large" family of exact solutions really is, in the grand scheme of things.)
However, this crude analysis falls far short of the much more difficult question of global existence of solutions.
We can imagine "disturbing" the gravitational field outside some isolated massive object by "sending in some radiation from infinity".
This approach is essentially the idea behind the post-Newtonian approximations used in constructing models of a gravitating system such as a binary pulsar.
However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations.
The desired result, sometimes expressed by the slogan that the Minkowski vacuum is nonlinearly stable, was finally proven by Demetrios Christodoulou and Sergiu Klainerman only in 1993.
This result, known as the positive energy theorem was finally proven by Richard Schoen and Shing-Tung Yau in 1979, who made an additional technical assumption about the nature of the stress–energy tensor.
Roger Penrose and others have also offered alternative arguments for variants of the original positive energy theorem.