Updated: Jun 27, 2020
If the gravitational potential P=-GM/r, then it is easy to show that it satisfies the Laplace equation. Newton's potential for any point can be expanded in a power series of the radius of the mass exerting the gravitational attraction to any given position. If the radius is small compared to r, the actual position of the secondary mass, then the first term which is just P is a constant, the second term involves the square of the ratio, gravitational acceleration, and can be nullified by choosing the origin as the center of mass. This leaves the third order terms which are the tidal forces acting on the primary mass.
In a sense, Poisson's equation takes the opposite view. The second derivative of P, say in the x-direction is
where for lack of the partial derivative symbol, I am using Clerk-Maxwell's symbol, which is the opposite of that used today. Now, if we take the second derivatives with respect to the y and z directions and add,
which is Laplace's equation. Now, the curious thing is that if we look at the first equation above and transform to spherical coordinates, we have in the plane phi=0,
d^2P/dx^2=-GM P_2(cos theta)/r^3,
where P_2(cos theta) is a Legendre polynomial of degree 2 in the co-latitude theta. Thus, the summation is like an averaging out of the tidal contributions, like that of the above equation--inverse cubic in the radial distance between the two masses--to give Laplace's equation.
Also, from Newton's proposition 71, we can conclude that the force of gravity exerted by a spherical shell at any point in its interior is zero from which we can also conclude that the potential is constant or zero in the interior. Instead now we convert the M of the primary into a density of a shell of thickness dr equal to 4 pi (density) r^2 dr and integrate from r=0 to r, we get P_0=-2pi G(density) r^2. On the strength of prop 71, the potential on the rim is that same as if all the mass were concentrated at the origin so
P_r=-GM/r=-(4pi G/r)* density*r^3/3=-(4/3)pi G(density)r^2.
Differentiating this twice, for a constant density, we get
D^2 P=-8pi G*density,
where D^2 denotes the laplacian operator. Miraculously, the right-hand side different from zero! We started from a laplacian for a mass M and came up with a laplacian of P different from zero for a constant density! What is going on?
First note the similarity of Poisson's equation with Einstein's field equation
G=-8pi G T
where G and T are the Einstein tensor and energy-stress tensor, respectively. But this is neither the Poisson equation we want nor its generalization. Note that the sign on the right-hand side of Einstein's equation is conventional, and depends on whether + or - is taken in the contraction of the Riemann curvature tensor to obtain the Ricci tensor. Not so in the Poisson equation, for it determines whether the potential is a maximum or a minimum, and we want the latter. So we consider the difference in the potentials at the center of the sphere, -2pi G*density*r^2 and that at a point slightly displaced from it -(4/3)pi G*density*r^2. The laplacian of this difference is
D^2 P'= 4pi G(density).
Although this is the laplacian we want, it still causes difficulties.
First Newtonian's law for a mass M, has a zero laplacian whereas one for constant density has a source term. It doesn't seem logical that the transition from constant mass to one of constant density should produce a source term---or does it?
Second, if there is a source at the origin, the mass density should go to zero as we move further from it, since otherwise it wouldn't be a "source". Newton rejected such a possibility since it would mean that all matter would tend to collapse into a single point in the universe! However, we are not talking about mass, but rather its density.
Einstein tried a modification of Poisson's equation:
D^2 P +K*P=4pi G*density
with K as a "universal" constant. So at great distances from the source the mass density should be constant and tend to
P=4pi G *density_0/K,
where density_0 is some constant density. For, according to Einstein:
"If, without making any change in the mean density, we imagine matter to be non-uniformly distributed locally, there will be, over and above the P with constant value, [noted above], an additional P, which in the neighborhood of denser masses will be so much the more resemble the Newtonian field as K*P is smaller in comparison with 4 pi G*density*."
This is a verbal description that is not reflected in the equation!
In 1917 Einstein championed a model of the finite universe that is closed and spherical---and for good reason. We will now show that it corresponds to the inner solution of Schwarzschild's metric. Since we are so far from the the source we can set the density term in Einstein's modified Poisson equation equal to zero. We then obtain the Jacobi equation
for the Jacobi field G, which is a sole function of the radial coordinate, r. The Jacobi equation tells us how geodesic curves tend to spread out or bunch up. Think of it as a harmonic oscillator equation. For K>0 the geodesics will tend to bunch up like on a surface of a sphere, while for K<0, they will spread out like the paths of an unstable harmonic oscillator.
Integrating once gives G'=F=(1-KG^2)^(1/2), where the prime stands for differentiation with respect to r. This implies that G<1/K^(1/2), and the universe is finite. Like Einstein we find his "cosmological" constant equal to a constant density and G is proportional to the radial distance in the warped metric
where S^1 and S^2 are 1- and 2- spheres, respectively.
The Jacobi equation shows that the radial sectional curvature -G"/G is positive. Integrating leads to
which the sectional tangential curvature. It is equal to both translational sectional curvatures, F"/F=G"/G=K so that the eigenvalues of the Ricci tensor in each of these directions cannot vanish. Oddly enough, Poisson's equation is an affirmation of the presence of a source. Is this a mere coincidence?
Then it is also a mere coincidence that written in terms of the mass, M, the eigenvalues of the Ricci tensor all vanish, and Laplace's equation is satisfied! In this case the tangential curvature is
where K=M/G^3. Both translational curvatures are equal in magnitude to K but of opposite sign. I conjecture that the differences in the sign of K's in the Jacobi equation are what lead to compensation and the vanishing of the eigenvalues of the Ricci tensor. In one direction the geodesics spread out and in the other direction they bunch up.
Consequently, the Einstein's condition of emptiness is void of any meaning would be equivalent to the fact that the mass satisfies the Laplace equation, while its density satisfies the Poisson equation. Remember that the sectional curvatures determine the tidal stresses: in the case of constant density, they are all equal and the same sign, while in the case of a chunk of mass, they are different and so turn a sphere into an oblate ellipsoid.
So what is the significance of the energy-stress tensor, T? In the case of Schwarzschild's outer solution, its vanishing certainly does not mean "emptiness" since there is a mass M present. In the inner solution, it is proportional to three times the density since, each of the sectional curvatures contributes a density to the equation. The remaining components of T are nonsensical.
This is clearly seen in the companion model of Einstein's, known as the de Sitter model, after its author. There the vanishing of the sum of the mass density and hydrostatic pressure means that if the density is large and positive, the pressure will be large and negative, which in the words of Moller, is "quite incompatible with the properties of any known material." This led to the necessary conclusion that the de Sitter model corresponds to an "empty" universe.
It is due to the fact that what is the Ricci component for the density is its negative for the pressure. All that can be determined is the relation of the sectional components to the mass density and/or mass. These components are indicative of non-vanishing tidal stresses.
This is clearly confirmed in the Friedmann solution to Einstein's equations where a positive density implies a negative pressure, and vice-versa. The independent variable is time instead of the radial coordinate, but everything else remains the same. If pressure is to be introduced, it must be introduced as the gradient of the pressure proportional to the gradient of the potential times the density. It has no role in determining sectional curvatures.
Einstein considered his tensor G to be a marble edifice while T a humble wooden shack. The above analysis shows that it would be better if the wooden shack be demolished.
What is astounding (at least to me) is that how so many confirmations have been made to a lame theory. If the fundamentals have not been resolved, what hope is there to build castles in the sky that have absolutely no foundation?