Newton's proposition 71 states that the force of gravity exerted by a spherical shell on any point in the interior is zero, and hence the potential due to that shell is constant in the interior. This, by the way, sounds vaguely similar to the maximum principle of partial differential equations whereby if the laplacian of the gravitational potential is non-negative, a constant solution is admitted at every point on the interval. That constant solution can be zero by redefining the zero of potential.
Now, the laplacian of Newton's potential satisfies the Laplace equation, which just so happens coincides with the vanishing of the Ricci tensor in every direction. However, converting the mass into density converts the Laplace equation into the Poisson equation. The Poisson equation is the Einstein equation for the density; there is no other Einstein equation for the pressure or linear combinations of the pressure and density. All such combinations lead to grave physical contradictions, like the existence of negative pressures that drive the inflationary scenario.
All relevant field equations can be derived from the eikonal equation
d^2x/d l^2=(1/2) dn^2/dx
where l can be time or the radial coordinate, and n is the index of refraction. If l is the radial coordinate, and G=x, then n^2=G'^2=1-2m/G, and
and the prime denotes differentiation with respect to the radial coordinate, and m is mass in units where Newton's constant and the speed of light is unity. This is the Schwarzschild solution.
If l is time, n^2=a*2=2m/a+E, x=a, the time dependent "radius of the universe", E the total energy, and the asterisk denotes differentiation with respect to time, then the eikonal equation reads
If the latter happens to be equal to a constant, K, constant density, then we have the Friedmann solution that is usually considered to be a model of the early universe. The corresponding pressure must be obtained through an equation of state, and not through the Einstein equations.
In the former case, the index of refraction,
is that of the Eaton lens (cf. Seeing Gravity p.29). The total energy E is proportional to the space curvature in the Friedmann model. E, however, does not determine the sectional curvature,
This was pointed out by X. Mei.
Replacing time by the radial coordinate, the index of refraction,
where G is usually interpreted as the distance from the center of the state, although, according to Besse "neither travel time measurements of reflected light signal nor trigonometric distance measurements would confirm that increments in G measure radial distances."
Finally what happens if we consider the Schwarzschild problem with a constant density so the index of refraction will be
where rho is the constant density? This is the antithesis of the Friedmann model. And it is also Schwarzschild's inner solution. The Poisson equation is applicable, and there is absolutely no black hole. It is what one gets from switching from Newton's potential, which satisfies the Laplace equation to a constant density which satisfies Poisson's equation.
But, what happens at infinity? Is rho really a constant there, or is their a tapering off of the potential at large distances, as we would expect. Einstein introduced a cosmological constant into Poisson's equation. When the density becomes "small", the equation becomes the Helmholtz reduced wave equation so that the cosmological constant, being proportional to the square of the angular velocity is sectional curvature. In the opposite regime, considered by Einstein, the gravitation potential becomes constant, inversely proportional to the cosmological constant. Einstein's universe is closed spherically being finite but unbounded.
Thus, the expression for the index of refraction for the Schwarzschild problem can be considered the antithesis of that of the Eaton's lens. The tidal forces, or the variation of the force, are what actually determine the evolution of cosmological systems, and not the gravitational acceleration. Moreover, Einstein's equations, apart from being inaccurate, are entirely superfluous.