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Einstein Metrics are Beltrami Flows: Ricci Flows Vary From Navier-Stokes to Maxwellian

Updated: Oct 17, 2022

The most general decomposition of a tensor field is given by

T = E + Curl(Curl g) (A)

where E is the symmetric gradient tensor, and g is a symmetric tensor. The solenoidal component represents the Ricci tensor up to higher order terms that are quadratic in the first-order derivatives of g. They are, in fact, Einstein's pseudo-tensor which can be comfortably neglected.

If we are dealing with tidal forces, E is the second derivatives of the gravitational potential potential and the scalar Ricci potential vanishes, i. e. the sum of the diagonal terms of the tidal force metric vanishes. If the metric would be Einsteinian, or, equivalently, Beltramian, it would also mean that the Ricci tensor vanishes, since Curl g = c g, where c is a constant. Einstein's equation,

Ric - (1/2) R g = T (B)

could be interpreted as a deviation from Beltrami flow due to the presence of an energy-stress tensor, T, where the Ricci scalar, R is proportional to the square of c. But, this does no make sense Beltrami flows are the solution to the Navier-Stokes equations. This shows that it is ridiculous to add the pseudo-tensor to the right-hand side of (B) because it is already contained in Ric. Also the presence of (1/2) has nothing to do with the dimensionality of the problem.

Another approach is to start with a metric g_(ij) and "try to improve on it by means of a heat equation." This leads to Hamilton's Ricci flows that are governed by (1/2) dg/dt = R g/n- Ric, (C)

where n is the dimension of the space. Since the Ricci curvature is positive, (C) leads to a negative diffusion equation that cannot be handled by a minimization of an energy integral. The procedure invoked by Hamilton was to replace R by an average over the space. Equation (C) would always have a solution at least for a short time on any compact manifold of any dimension for any initial value of the metric at t = 0. The reason for requiring a positive Ricci tensor was to prove Poincare's conjecture.

Relinquishing this requirement leads to the equation for the vorticity if the first term on the right-hand side is neglected. The vorticity is proportional to the flow, w=c v, where the scalar is referred to as the "abnormality" of the Beltrami field. It is known that Trkalian (a sub-class of Beltrami flows in hydrodynamics), cannot develop from initial states in which the vorticity, w, is absent. If the vorticity is initially present it must decay exponentially. Using the fact that the curl of a Beltrami flow is also a Beltrami flow gives the Helmholtz equation

Laplacian w + c^2 w=0. (D)

For steady flow of a Newtonian fluid, w is harmonic so that no steady Beltrami flow is possible. If the flow is unsteady then substituting (D) into the vortex diffusion equation

dw/dt= c^2 Ric w = - c^2 Laplacian w, (E)

where c^2 represents viscosity, one gets

dw/dt +c^2w=0,

which shows that the vorticity decays asymptotically in time. Obviously, the Ricci tensor cannot be positive definite, implying that the manifold is hyperbolic rather than being spherical.

Now, there is nothing sacred about Hamilton's negative diffusion equation (C). If we replace the first derivative by the second derivative,

d^2g/dt^2 = Ric - g R/n, (F)

then we get Maxwellian flows. That is, following Silberstein, we set g= E +i H, where E and H are the electric and magnetic fields, and split (F) into two first-order equations,

i dg/dt = Curl g + J, (G)

much like Dirac split the Klein-Gordon equation into two first-order equations, where J represents the current and magnetic (for symmetry) current densities. Separating real and imaginary parts in (G) give Maxwell's circuit equations; taking the Curl of both sides of (G) gives the wave equation without source terms if J=0. Apart from the imaginary unit, the difference between (B) and (G) is that the Ricci tensor is the Curl of the Curl of g.

All this implies that the pseudo-tensor of Einstein can be neglected in the expression for the Ricci tensor. When tidal effects are included, by representing E as the Hessian of g, as in (A) one gets what is known as the "gradient shrinking of solitons" in Ricci flows. Ricci solitons change size but not shape, just as a soliton wave does, and is given by the half-plane metric, (dx^2+dy^2)/(x^2+y^2+e^(4t)), with an exponential time factor added. The half-plane model has negative, constant curvature implying a hyperbolic space. Also writing the equation

Ric(g) + Hess(f) = g/2, (H)

where the Hessian is of a potential function, f, is meaningless [Cao & Zhou, "On the complete gradient shrinking of the Ricci solution"] f and g are no independent functions and why the sum of irrotational and solenoidal components of g should be equal to g is a real mystery.

Einstein's equations (B) would give the solenoidal component of the field a dominant role: Anytime we energy-stresses (B) says that we can expect swirls and vortices. It also implies time harmonic wave propagation with "sources", T_(ij). But such time stationary wave propagation also occurs in free-space, so the effect of the sources are analogous to those of electromagnetism. Propagation in time only occurs because of the indefinite metric. Yet the time components of the Riemann tensor, R^i_(0j0), are not related to tidal forces which occur in Euclidean space. So the whole notion of spacetime seems to fall apart: the role of the Beltrami field would be to provide small corrections to potential flow through the presence of singularities, dislocations and disclinations, that would require a Cartan connection in place of the Levi-Civita connection. Such effects would be due to the material media which would replace spacetime. So the main effects of tidal forces due to gravity occur in Euclidean space (plastoelasticity), which may require a small modification due to endogenic factors (defects and singularities).

It is essential to note that to avail ourselves of the time-harmonic fields associated with plane waves, the compatibility condition (Curl(Curl g))=0 must fail. The existence of a Newtonian potential no longer exists together with the action of tidal forces. We have said this many times before: there is no induction in gravity.

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