top of page

What Is Maxwell's Electro-kinetic Momentum Doing in Einstein's Theory of Gravitation?

The answer is that the electro-kinetic momentum, aka vector potential, is what generates gravitational waves. Gravitational waves are described by a linear(ized) wave equation, which has no nonlinear generalization.

In analogy with with elasticity, the general displacement field is now the acceleration, a_j, and we ask under what circumstance does a second-order symmetric tensor g_(ij) happen to be a strain tensor for the displacement field, a_i; that is,

g_(ij) = d_i a_j +d_j a_i= g_(ji). (1)

The answer to this question was found by Saint-Venant in 1864 in which he found that (1) applies when

(curl curl g)_(ij) = e_(iab) e_(jcd) d_a d_c g_(bd) = 0, (2)

where e_(ijk) is the Levi-Civita completely asymmetric tensor.

Equation (2) is a generalization of the condition for that a vector field be a gradient field, i.e., the curl of the field must vanish. And that is what is precisely done in general relativity. For small perturbations from a flat metric

g_(ij) = delta_(ij) + e h_(ij),

where delta is the Kronecker delta and e is a small parameter such that square terms and higher can be neglected, the Einstein tensor in 3D can be written as

G_(ij) = R_(ij) - (1/2) R,

=(1/2)e e_(ikl) e_(jmn) d_k d_m h_(ln), (3)

which, upon comparison with (2), is the Saint-Venant operator operating on the linearized metric. It's vanishing is precisely the condition for h_(ij) to be a strain tensor. What changes in 4D?

As we mentioned, specialization to the case where h_(ij) is time independent. This isolates the terms h_(00) and h_(0j), and the derivatives of these terms are with respect to the space coordinates only. What is done [See Ohanian & Ruffini, Gravitation and Spacetime] is to construct the analog of the Faraday tensor,

f_(ij) = (k/2){d_j h_(0i) - d_i h_(0j)}, (4)

where k=[16 pi G]^(1/2), the Einstein gravitational coefficient in units where c=1. Now comparing (1) and (4), we have introduced an asymmetric (pseudo) tensor, and cannot, therefore expect, that the "electric" field will be a gradient field.

In terms of the Faraday tensor (4), the equation of motion reads

dv_k/dt =- f_(0k) +2 f_(kj)v^j, (5)

for a 3-velocity, v_k. We can write (5) in vector notation,

dv/dt = E + v X B, (6)

where the E and B are the gravitational, and gravitomagnetic fields, respectively,

E_k =(k/2) d_k h_(00), (7)


B_k = k { d_k h_(0i) - d_i h_(0k)}v^i. (8)

Now, it is clear from (7) that E is a gradient field so that Faraday's law

curl E = - (1/2) dB/dt (9)

must vanish. This is equation (3.103) of Ohanian and Ruffini. It is apparent that they did not connect this to the fact that their gravitational acceleration, -(k/2) grad h_(00), is defined following their equation (3.104).

Furthermore, they write the second circuital equation (their equation (3.103)),

curl B = -16 pi G P, (10)

as if it were the definition of the Poynting vector, which they associate with the 0k component of the energy-stress tensor. Even if (9) and (10) were valid, they would not lead to a wave equation when either E or B is eliminated, since the Maxwell displacement current, dE/dt, is absent in (10). So the Poynting vector, P, is the momentum-energy flux of "nothing".

Since (7) corresponds to (1), the game is over. The description in terms of a scalar Newtonian potential of the tidal forces implies that (8)---the so-called gravitomagnetic field---vanish. Hence---just as in 3D---the vanishing of the Einstein tensor is the condition for the existence of a symmetric strain tensor. It is symmetry and traceless, which upon diagonalization represents the tidal force components in the 3 orthogonal directions. Without the time derivative of the vector potential, A, appearing in (7), there can be no Faraday induction (7) where B= curl A.

1 view0 comments

Recent Posts

See All

Never has so much confusion, and conflicting results, have been made that a theory can be made compatible with just about anything. Einstein built into his general theory of relativity that particles

The Schwarzschild metric is unique in general relativity, and its shows. The contradictions that it provides casts doubts on the validity of the whole theory. Depending on whether we reason from the

The idea that plane waves propagating away from electromagnetic sources are gravitational waves because the coefficients of the indefinite metric are called "gravitational" potentials is nonsense. The

bottom of page