Exact Solutions of the Field Equations for Empty Space in the Nash Gravitational Theory
John Nash has proposed a new theory of gravity. We deﬁne a Nash-tensor equal to the curvaturetensorappearingintheNashﬁeldequationsforemptyspace,andcalculateitscomponents for two cases: 1. A static, spherically symmetric space; and 2. The expanding, homogeneous and isotropic space of the Friedmann-Lemaitre-Robertson-Walker (FLRW) universe models. We ﬁnd the general, exact solution of Nash’s ﬁeld equations for empty space in the static case. The line element turns out to represent the Schwarzschild-de Sitter spacetime. Also we ﬁnd the simplest non-trivial solution of the ﬁeld equations in the cosmological case, which gives the scale factor corresponding to the de Sitter spacetime. Hence empty space in the Nash theory corresponds to a space with Lorentz Invariant Vacuum Energy (LIVE) in the Einstein theory. This suggests that dark energy may be superﬂuous according to the Nash theory. We also consider a radiation ﬁlled universe model in an effort to ﬁnd out how energy and matter may be incorporated into the Nash theory. A tentative interpretation of the Nash theory as a uniﬁed theory of gravity and electromagnetism leads to a very simple form of the ﬁeld equations in the presence of matter. It should be noted, however, that the Nash theory is still unﬁnished. A satisfying way of including energy momentum into the theory has yet to be found.
Aadne, Matthew Terje