  John Nash has proposed a new theory of gravity. We define a Nash-tensor equal
to the curvature tensor appearing in the Nash field equations for empty space,
and calculate its components 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 the Nash 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 satifying way of including energy-momentum into the theory has
yet to be found.
