Einstein's equations are supposedly a generalization of Poisson's equation for the gravitational potential, when the latter is replaced by the gravitational potentials in the indefinite metric. The connection, supposedly, arises in the weak field limit, where the gravitational potentials morf in the single Newtonian potential, and the static limit is taken.
Just because equations appear to be similar, doesn't make them so. We first have to distinguish between elastic and plastic deformations. This analogous to the Helmholtz-like decomposition for vectors, where any vector u can be decomposed into
u_i= U,i + e_(ij)V,j
where U is a scalar potential, V, a vector potential, e is the antisymmetric Levi-Civita (pseudo) tensor or simply "symbol", and the comma denotes the derivative.
Now, the generalized Beltrami representation of a smooth, and symmetric, tensor is
T_(ij) = curl(curl h)* + v_(i,j) + v_(j,i), (1)
where the star denotes the transpose, and h is a second order, symmetric tensor which is divergence free, div h=0. (1) is known as the Cesaro representation,
The proof resides in defining a symmetric tensor field, f_(ij) which is the solution to the inhomogeneous bi-harmonice equation
(div grad)^2 f_(ij)=f_(ij,kkll)=T_(ij).
The vector, v_i, is defined
v_i= (div grad)^2 f_(ij,j) -(1/2)f_(kl,kli).
The equilibrium stress, T_(ij) satisfies
div T_(ij)+F_i= T_(ij,j)+ F_i=0,
where F_i represent the body forces. From its definition, (1) it follows that
div grad v_i + v_(j,ji)+ F_i=0. (2)
If the transverse condition is satisfied, the second term in (2) vanishes, and v_i satisfies the Poisson equation. Note that h in (1) remains unrestricted by this condition, as it should be. In fact,
e_(imk)e_(jnl)h_(kl,mn)=G_(ij), (3)
which is the Einstein tensor in 3D. For 4D we have to increase the index on the Levi-Civita (pseudo) tensor. This shows that the generalized gravitational potentials, h, do not enter into the Poisson equation. Only the elastic part enters, v_i, not the plastic part, h_(ij).
Only if (3) vanishes, then
curl(curl T)* = 0, (4)
will be a necessary and sufficient condition for the validity of Stokes formula in a simply connected domain, e.g., Fosdick, "A Stokes theorem for second-order tensor fields...."
So the Poisson equation is completely independent of the solenoidal component of the tensor T_(ij), depending upon the irrotational component, v^S_(i,j), only, where S stands for symmetric part. Only when the Einstein tensor vanishes, can we expect that there will be a Poisson equation.
All what we have said is valid in 3D (or "3+1 general relativity). The generalization to 4D is to add another indice on the Levi-Civita (pseudo) tensor. Consequently,
G = k T
has no meaning where T is the so-called energy-stress tensor, and k is the Einstein constant. Moreover, when G=0, which is Einstein's condition of "emptiness", implies the Ricci tensor satisfies
Ric (g) =R/2 g, (5)
for a metric g, and R is the scalar curvature. In its linearized form g->h (5) becomes
Curl w= (R/2)^1/2 h (6)
where w=Curl h, is the vorticity. Apply the curl to the both sides of (6) gives (5) in its linearized form
Ric (h)= R/2 h.
This is none other than a Beltrami flow where the velocity and vorticity are parallel to one another. Long ago (1907), Silberstein has shown how a complex field, composed of E and H vectors, could be cast as a (complex) Beltrami flow. So is (1) a type of unification that Einstein sought when he wrote
“The present theory of relativity is based on a division of physical reality into a metric field (gravitation) on the one hand, and into an electromagnetic field and matter on the other hand. In reality space will probably be of a uniform character and the present theory be valid only as a limiting case. For large densities of field and of matter, the field equations and even the field variables which enter into them will have no real significance.” ?
However, the existence of one field invalidates the validity of the other! Electromagnetism is the solenoidal part of the 2-tensor T, while gravitation (at least in the form of elastic stresses) belongs to the irrotational part. To join the two in the form of (1) makes no sense since there is no connection between the two. The reciprocal effects of gravitation and electromagnetism are of a much sublter type than the mere addition of two mutually exclusive fields.