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 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.


Introduction
During the last twelve years of his life John Nash tried to develop an alternative theory to Einstein's general theory of relativity. He formulated field equations for empty space, but did not manage to include a description of matter in a way which satisfied his requirements. John Nash presented his work on this theory in lectures [1], but he did not publish any article on it. Neither have we been able to find any analysis by others on the Nash theory.
Nash mentioned several interesting properties of his theory even if the theory was only worked out for empty space. He did among others search for a theory giving a Yukawa-like gravitational field for the static spherically symmetric case, but later dismissed that point of view. Also Nash mentioned that it is a general consequence of his field equations for empty space that the Ricci scalar obeys a wave equation which reduces to Laplace's equation in the static case. Finally Nash mentioned that his field equations for empty space are satisfied by any solution for empty space of Einstein's field equations with a cosmological constant. This means that phenomena such as accelerated cosmic expansion that comes out of Einstein's theory as an effect of LIVE, causing repulsive gravitation [2], [3], are due to a natural tendency of empty space to expand according to John Nash's theory. This is agreement with the spirit in which Einstein originally interpreted the cosmological constant, before it was reinterpreted by Georges Lemaitre as an expression of the constant energy density of LIVE [4]. Nash did not, however, work out any solution of his field equations.
In the present paper we write out the field equations and find the general solution of the equations for empty static, spherically symmetric space, showing that it is a Schwarzschild de Sitter space time. Furthermore we show that the de Sitter universe solves the Nash equations for empty space with a line element of the FLRW-type.

The field equations for empty space of the Nash theory
In an unpublished manuscript [1] John Nash has presented a theory of space, time and gravitation. He gave the equations of empty space in the form where is the d'Alembertian operator. Defining the Nash curvature tensor the Nash field equations for empty space takes the form We shall here present the general solution of these equations for the static spherically symmetric case and a simple, exact solution for expanding, homogeneous and isotropic universe models.

Static spherically symmetric space.
The line element is described by introducing curvature coordinates where the area of a spherical surface about the origin with a coordinate radius r is 2 4 r  . Also, to simplify the calculations we shall restrict ourselves to solutions of the field equations fulfilling the restriction 1 tt rr gg  . Then the line element takes the form (using units so that the velocity of light is 1) Using the DifferentialGeometry package in Maple we then find the following non-vanishing orthonormal basis components of the Nash tensor, Hence the Nash equations for empty space imply that which may be written  

Homogeneous and isotropic universe models
The standard form of the line element is in this case   We shall integrate equation (19) in two steps in order to discuss an interesting special case. A first integration gives 3 1 which is the scale factor of the Milne universe, i.e. of the Minkowski spacetime as described in an expanding reference frame.
We then consider the case 1 0 C  . A particular solution of eq.(23) is then This is the scale factor of the deSitter universe, which in Einstein's theory is a spacetime dominated by LIVE with a constant density that may be represented by a cosmological constant.
Eq.(23) is an Abel differential equation of the first kind. The general solution of this equation can only be given implicitly with t as a function of y. This solution does not appear in Einstein's theory.

How can energy-momentum be represented in the Nash field equations?
The energy-momentum density tensor of electromagnetic radiation is trace free. We shall therefore start by considering the equation   0 We shall not try to find the general solution of this equation, but restrict ourselves to investigate a class of solutions obeying   2 20 HH  .