When determining the stability conditions of the motions of astronomical objects, account has to be taken of the gravitational waves emitted over billions of years. If the gravitational waves reach some state of equilibrium, a definite temperature must be assigned to them, in the same way as for a neutrino gas.
We may mention at this point that quite recently efforts have been directed toward developing another generalization of Einsteinian theory, involving "twisted" rather than curved space-time. The geometry of a space twisted on itself was proposed as early as the twenties by Cartan, Einstein, and Weizenbeck, but it failed to yield any concrete physical results. Not long ago, however, V. Rodichev and R. Finkel'shtein showed that in twisted space a nonlinear correction automatically crops up in the Dirac equation used to describe electrons, protons, and all other fermions, the correction being exactly of the pseudovector type required by a unified nonlinear spinor theory of matter. The point has also been made by Moller that it may be worthwhile adopting a special instance of twisted space (one in which there is absolute parallelism) in order to derive an effective expression for the energy of the gravitational field; in terms of Riemannian-Einsteinian curved space this has not yet been satisfactorily realized.
It seems likely that the new treatment of the gravitational field, similarly to the electromagnetic and other Bose fields, as some kind of "compensating" field (in accordance with the ideas of Yang and Mills and of Sakurai, which have been supported by Gell-Mann, Schwinger, and Feynman and applied to the case of gravitation in the studies of Utiyama, Kibble, and Brodskii Ivanenko-Sokolik) should normally lead to the existence of a twist in space.
It is still too early to say that space-time has been definitely proved to be twisted, but a significant attempt has certainly been made to generalize Einstein's theory in this direction.
Mention should also be made of the various cosmological investigations which seek to generalize A. A. Fridman' s theory, proceeding from the Einsteinian equations for a nonstationary expanding universe. The investi gations cover the case of a rotating anisotropic universe (Zel'manov, Godel, and Schilking) and the theory of a universe in expansion but having constant density (Bondi, Hoyle); this already goes beyond the Einstein-Fridman models proper.
There is a particular generalization of Einsteinian theory that stands apart, involved with the Dirac hypothesis on the slow secular decrease in the gravitational constant. Jordan, Dicke, and some other scientists have shown that this hypothesis has many astronomical and geological consequences, such as the fact the Earth is slowly expanding. Significantly enough, there are some geologists who also claim that the Earth is expanding at the rate of about half a millimeter per year (Egged, Wilson, cf. also V. B. Neiman). It would be interesting to correlate this expansion with the secular increase in the length of the day (Ivanenko-Sagitov), and perhaps also with the forma tion of the network of gigantic cracks on the Earth (Peive, Heezen, and others). At any rate, studies in gravitation, space-time and cosmology are proceeding at a lively pace, and they are bound to yield findings of fundamental significance for an integral world picture.