Updated: Oct 17, 2022
In the same way that the scalar gravitational potential is related to its source, the mass density, the vector potential, to the current density, we show that the generalized gravitational potentials are related to their source, the Einstein tensor, interpreted as a stress tensor. The non-vanishing of the latter prohibits a metrizable analysis, implying that tidal forces cannot be described by elastic forces.
The gravitational potential can be expressed as
V = - G Int (p/r) dV,
where p is the mass density, r denotes the distance from a point in the domain of integration to the point at which V is estimated, and the integral is taken throughout space. The quantity, 1/4 pi e_0, where e_0 is the dielectric current is replaced by -G, the negative of Newton's gravitational current. Consequently, we have the Poisson equation
div grad V= 4 pi G p. (1)
The right-hand side is the inverse square of Newton's free-fall time. It does not come out of a weak gravitational limit of Einstein's equation for the mass density. In fact, Einstein's equation lose their significance. In particular, there can be separation between gravitation, viewed solely in geometrical terms and an energy-stress tensor that excludes gravitational interactions. Mass cannot be separated into a component representing the mass density of everythin except graivity and a contribution that represents the mass equivalent of the gravitational energy density. In other words, gravitational enteractions may enter without having to consider the mass equivalent, E/c^2, where c is the speed of light and E is the energy (of what?).
The vector analogy with the vector potential in electrodynamics follows, although its physical implication is not clear since there is no theory of gravitation with masses in motion, that would be comparable to Weber's theory. Weber's theory is a particle theory, and certainly would not fit into a field interpretation of gravitational interactions.
So we'll do it for electromagnetism. The vector potential, A, is defined in terms of its source, the electric current, j, by
A=-(1/4 pi) Int j/r dV,
where A has a vanishing divergence, div A =0. The Poisson equation reads
- curl curl A= div grad A = j.
Now, the tensor case is quite clear if the Einstein tensor, G_(ij)=R_(ij)-R g/2, where R_(ij) is the Ricci tensor, and R is the scalar curvature, represents the relevant gravitational stresses. In the linearized regime where the gravitational potentials, g_(ij), are replaced by their linear counterparts,
h_(ij)= -(1/4pi) Int G_(ij)/r dV.
The condition div G =0 is taken as a conservation condition. But without its relation to the energy-stress tensor, one may ask, conservation of what?
Poisson's equation then follows
curl curl h_(ij) = - e_(ikm) e_(jln) h_(mn,kl) = G_(ij), (2)
where e is the Levi-Civita antisymmetric pseudo-tensor, and the comma denotes the derivative. The fourth-order Riemann tensor may easily be constructed from (2), i.e.
R_(ijkl)= e_(ijm) e_(kln) G_(mn). (3)
The St. Venant condition of compatibility is that there exists a symmetric tensor of the gradient of the displacement only when either (1) or (2) vanishes. This identifies the Einstein tensor as the incompatibility tensor.
These results apply strictly to 3D. The incompatibilty, or Einstein's tensor, can be extended to spacetime in 4D,
G_(ij)=e_(iklp) e_(jmnp) h_(ln,km), (4)
where the Levi-Civita tensor has now 4 indices. Setting p=4 (time), the repeated index vanishes leaving purely spatial components. This says that 44 component contains indices that range only over the spatial components. The conclusion remains: the Einstein tensor represents incompatibility, and its non-vanishing prohibits a metrizable approach to gravitation.
Setting the Einstein tensor equal to an energy-stress tensor has absolutely no meaning, as Eq. (2) shows. Poisson's equation, whose source is the density, Eq (1), follows from the existence of a scalar gravitational potential, and that depends on the vanishing of the incompatibility tensor, (4). That requires the vanishing of the Ricci tensor, which is the average of the sectional curvatures in the different directions. These can be looked upon as equilibrium stress conditions when referred to an oblate ellipsoid. And that requires 3D, not 4D!