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Generalized Beltrami Representation for Gravitation

Beginning with the equation for geodesic equation, the symmetry of the Riemann tensor in the first and third indices was related to the electric tensor component of the Weyl tensor. It's Hodge product was associated with the magnetic tensor component of the Weyl tensor. Raising the vector potential to a tensor, the decomposition can be associated with the complete Beltrami representation.

The symmetry of the electric field tensor E_{ij}, which forms the irrotational component of the field, excludes the possibility of a dependency on time because the only way to introduce time is to add the time derivative of the vector potential to the gradient term in the expression for the electric field, and that would destroy the symmetric of E_ij. Then instead of the Hodge product to represent the magnetic field tensor, we use Beltrami's representation of the solenoidal component. For symmetric tensors A and E, the generalized Beltrami representation for a symmetric tensor T is

T= curl(curl A) + E,

with the condition

div A = 0,

since the gravitational field is static. With curl A=B, "Maxwell's" equations are

div(div E)=rho,

the density,

curl E=0,

div(div B) =0,

(curl(curl A))_ij=(curl B)_ij=J_i U_j,

the current density measured by an observer with a 4-velocity, U_j.

The first term in T is known as the "incompatible" part of E, inc E. It is related to the Riemann tensor by

e^{ijm} e^{klm}R_{ijkl}=(inc E)_{mn},

where e is the permutation tensor and R is the Riemann tensor. I found this representation more physical than the tidal stress formation of Costa and Herdeiro who define the magnetic field through the Hodge product

H_{ij}=*R_{ijkl}U^kU^l =(1/2)e^{cd}{im}R_{cdjn}U^mU^n,

involving 4-velocities, U^m.

The physics is contained in the last equation. Reverting to vectors, the cross product of curl B with B is the Lorentz force. In place of the magnetic field, Beltrami used the velocity field, and notwithing the fact that the equations of fluid motion are nonlinear, Beltrami showed that two velocity fields, v_1 and v_2 are superposable if

v_1 X curl v_2 + v_2 X curl v_1 = grad V_12,

where V_12 is some scalar potential. The same would apply to the superposition of two magnetic fields, B_1 and B_2. In the simplest case where the scalar potential is constant,

curl v =K(r,t) v,

where K is some scalar function. Beltrami analyzed flows satisfying this equation, and any flow which satisfies this equation has become known as Beltrami flow. Trkal show that it K is independent of space it is also independent of time.

For the magnetic field tensor,

(curl B)_ij=K B_ij,

showing that the field is determined by its curl. This describes magnetic field at cosmic length which are permanent. It has been shown that when the displacement current can be neglected, as in the case of high electric conductivity, the magnetostatic field is parallel to its own curl so that Beltrami's equation applies. This implies a proportionality between the current, J_iu_i and the magnetic field tensor, viz.,

J_i u_j=a B_ij,

where the constant of proportionality, a, can depend, at maximum, upon space.

Beltrami's equation can describe Maxwell's equations in free space, as Silberstein showed. Defining, what has become known as the Riemann-Silberstein vector, N=E+iB. the circuit equations become

i dN/dt = curl N.

Yet, such a time dependency is completely foreign to gravitation. Replacing i times d/dt by the frequency, the equation represents stationary, periodic plane waves tracing out circular vibration ellipses. Stationarity is crucial is E is not to lose its path independency. Or, translated into tidal forces, the geodesic deviations are symmetry in any given pair of directions.

Thus, if there is anything like a magnetic field tensor in gravitation, it would necessarily reduce to magnetostatics. Moreover, on time is permitted, the symmetric tensor becomes

T=(1/c^2) (d^2/dt^2)A + E,

which can be looked upon as a decomposition of the electric field into solenoidal and irrotational components in the sense of Helmholtz. The combining it with the previous expression for the tensor T, we find

curl(curl A) =-(1/c^2))d^2/dt^2) A,

under the condition that div A=0, and c is the propagation velocity. So once the symmetry of the Riemann tensor R_{ijkl) is broken in the first and third indices, tidal forces are no longer a valid description, and wave propagation can take place. However, if Einstein was correct in using the Riemann tensor to describe gravity, with a symmetric connection, and T reducing to E, meaning path independency as it must, then gravitational waves are non-existent.

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