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Einstein Metrics Are Beltrami Flows

Updated: Dec 20, 2022

An Einstein metric is defined as

Ric(g)=a g, (1)

where Ric is the Ricci tensor for the metric g, and a is a constant, 0,+/-1. It so happens that the linearized Ricci tensor for g = 1 + h, where higher powers in h are neglected is

Ric(h)= Curl x Curl h = -div grad h + grad div h.

Since the condition div h =0 is usually imposed, the linearized Ricci tensor becomes the negative of the Laplacian. So the condition that h is a harmonic function ensures that there exists a symmetric strain tensor. It also guarantees the existence of a metrizable space.

If the coordinates are harmonic with respect to a given metric, the Einstein condition (1) can be written as

-(1/2) grad div g + Q(g,g') =a g, (2)

where Q is quadratic in the metric g and its first derivatives. In fact, setting g=G^2, where G=G(r), only, (2) becomes

-G"G - G'^2+ Q = a G^2. (3)

For the uniformly rotating disc at an angular velocity w,

G'^2=1-w^2G^2, (4)

which can be look as the conservation of energy, (3) reduces to Q=1 if a=0.

Likewise, the conservation of angular momentum, h,

G'^2=1-h^2/G^2 (5)

and (2) is satisfied again with Q=1 and a=0. Equations (4) and (5) can be considered to be

equivalent for a 2D disc rotating at constant angular velocity with the angular momentum pointing in the direction normal to the disc.

We now claim that (4) is the inner solution to the Schwarzschild metric, while

G'^2 = 1- 2m/G, (6)

is that of the outer solution. Introducing (6) into (3) shows that it is satisfied for

2Q= 1+ G'^2,

and a=0. However, it is claimed that the exterior solution to the Schwarzschild problem is (cf. Moller, "The Theory of Relativity", Sec. 123) is

G'^2=1- 2m/G - b G^2/3, (7)

where b is a constant. If, as Moller claims, the last term can be neglected because of its "importance only for very large values of [G]", then the usual form of the exterior solution is obtained. That appears odd insofar as the interior solution is m=0 and b/3=w^2. The latter is related to the density of matter via Newton's free-fall time, and should be pertinent for "small" values of G.

(7) can satisfy (3) with an Einstein coefficient a= b/3. In the linearized case, (1) with a not equal to zero becomes a Beltrami flow. Beltrami (1889) [of the fame of the Beltrami metric and a model of non-euclidean geometry of constant curvature] is also accredited with the identification of the Ricci tensor in the solenoidal form.

Ric(h) = Curl(Curl h)= e_(imn) e_(nkl) h_(nl,mk) (8)

where e represent the Levi-Civita tensor, and the comma denotes the derivative. The incompatibility tensor, (8), generalizes to 4D by adding an additional index to both asymmetric tensors, i.e.,

Ric(h)_(ij)= e_(imnr) e_(jklr) h_(nl,mk).

And because of the nature of Levi-Civita tensor, in the time component,

R_(00) = e_(0mnr) e_(0klr) h_(nl,mk),

the indices (mn) and (kl) must take the spatial components only; the dummy index, r, vanishes after the summation is performed.

Only in the case a =0, does it reduce to the irrotational form of the symmetric stress tensor of linearized elasticity theory.

Now, if the Einstein constant, a, is finite then

Curl h = c h = V, (9)

where c is constant, and

Curl(Curl h)= c Curl h= c^2 h =a h, (10)

leading to the conclusion that the Einstein constant is positive. In this case, one can show that the space is compact, and that the curvature operator is nonnegative [cf. Petersen, "Riemann Geometry", p. 254.] Constant curvature is determined by a. The exterior Scharzschild is said to have Einstein constant 0, [ibid, p. 38]. As we have seen above the interiori solution also has an Einstein constant 0, in contradiction with Schwarzschild's solution. Even apart from all this, it has always been puzzling why a central mass (exterior solution) should be Einstein "empty", while the case of constant density (interior solution) should be "nonempty."

The constant, c, in (9) is known as the "abnormality" of the Belrami field. The condition that the flow be isochoric, or incompressible, is div h =0, The vorticity equation,

dV/dt= D grad div V, (11)

where D represents the viscosity, is analogous to the Hamilton equation for Ricci flows. A Trkalian field is one for which c, or equivalently the Einstein coefficient, a, is constant. The curl of a Trkalian field is again a Trkalian field, being the negative of the linearlized Ricci tensor in this case. This gives the Helmholtz equation for the vorticity

grad div V + a V =0 (12)

and introducing this into (11) gives

dV/dt= -Da V. (13)

(13) shows how the voriticity, and also the Ricci flow decays monotonically in time, since upon taking the curl of both sides of (13) and using (1)

dg/dt = - D Ric(g), (14)

which is Hamilton's flow equation for the Ricci tensor.

As a final point, only two sectional curvatures are necessary to specify the uniformly rotating disc, or the internal Schwarzschild solution, whereas three sectional curvatures are required in the exterior solution. Hence, the interior solution is a lower dimensionality than the exterior one, being confined to a rotating disc.

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