General Relativity vs Ricci Flow: Propagation or Diffusion?
Whereas General Relativity operates in 4D, Ricci flows occur in 3D. But we can always contract the Riemann tensor, not with the metric tensor but, with its projection onto 3D space, which is a subspace normal to the unit time vector.
The vector identity
curl(curl A) =grad(div A) - div(grad A) (1)
also holds for second order symmetric tensors, g_(ij). If h_(ij) is its linearized version with respect to a flat metric then in 3D we have
G_(ij)=R_(ij)-(1/2)R =(1/2)(curl curl h)_(ij) (2)
where the linear Ricci tensor is
R_(ij) = (1/2){h_(ik,kj) + h_(kj,ik)-h_(kk,ij)-h_(ij,kk)} (3a)
and the scalar curvature is
R = h_(jk,jk)-h_(jj,kk). (3b)
Equations (3a,b) hold in any dimension. In 4D, specializing to the time component (ij)=(00),
and considering the time-independent case, only the last term subsists in (3a) showing that the Ricci tensor is -1/2 the laplacian, and the scalar curvature vanishes.
In 3D, the interpretation is that the Ricci tensor is the "laplacian of the metric", and by an analogy with heat, or diffusion, it is set equal to the time-rate-of-change of the metric. Whereas in 4D, with the application of the Lorentz gauge, or traceless condition:
h_(ij,i)-(1/2)h_(ii,j) =0 (4)
and the condition that the fields be time-independent, the linearized Einstein equations become
(d_t^2 - div grad)h_(ij)=-16 pi G T_(ij), (5)
where T_(ij) is the symmetric energy-stress tensor of Einstein. In the absence of sources, (5) becomes the free wave equation. So, it would appear on going from from 3D to 4D we go from diffusion to wave propagation!
In general, if we take the trace of the Einstein equations
G_(ij)= R_(ij)- (1/2)R g_(ij)= k T_(ij), (6)
where k is the Einstein gravitational constant, then we get
R=-trace T,
and when this vanishes in empty space,
R_(ij) =0.
This is none other than Einstein's condition of emptiness. Yet, Einstein introduced his cosmological constant, C, so that his equations would read
R_(ij)- (1/2)R g_(ij) + C g_(ij) = k T_(ij).
Now in empty space, the trace condition reads R=-2C so that the condition of emptiness is
R_(ij) -R g_(ij) =0, (7)
The Riemann metric is proportional to the metric itself in all cases where there is constant curvature. Equation (7) is said to define an Einstein manifold [e.g., Besse, Einstein Manifolds].