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In Search of the Induction Terms in Einstein's Field Equations

If there are no induction terms in Einstein's equations, there still remains the nagging question that, by brute force, the Ricci tensor can be linearized to give a second-order wave equation. Thus, it would appear that there are induction terms in Einstein's field equations after all!

If the gravitational potentials g_(ab) are linearized to h_(ab) about a flat Minkowski metric and the gauge conditions

d_a h^a_b - (1/2) d_b h^c_c =0, (1)

are imposed on the Einstein equations, there result

d^c d_c h_(ab) = (d_0^2 - nabla^2) h_(ab)=-16 pi G/c^4 (T_(ab)-h_(ab)T),

where "0" is time and T_(ab)= p u_au_b is the energy stress tensor, p is the density and u_a are the 4-vectors. So the 00 and 0i terms on the right hand side are pc^2/2 and pcu_i, respectively.

If g and H are the GEM analogs of the electric and magnetic fields, then they satisfy

2g =-d_0 A - grad V, (2)

where A and V are the vector and scalar potentials, respectively, and

curl A = H. (3)

The four gauge conditions (1) then can be expressed as

div A + 2 d_0 V = 0 (4)


d_0 A = 0. (5)

Condition (5) is disasterous since it effectively separates the two fields where

2g = - grad V, just like in electrstatics. And when (5) is introduced into (4), d_0^2V=0,

and plugging this back into the wave equation, shows that the second time derivative

of the h_(ab) vanish, since the h_(ab) consist of diagonal terms in V and bordered on both sides by -A.

You would rightly think that this would put the damper on the wave equation

(d_0^2- nabla^2)A=(16 pi G/c) pu, (6)

since the first term vanishes on account of (5). But just introduce

nabla^2 A= grad(div A)- curl(curl A),

and using 2g=- grad V, the stationary wave equation becomes

curl H = (4/c)d_0 g - 16 pi G/c p u. (7)

This "looks" like the Ampere-Maxwell equation with the Maxwell displacement current d_0 g!

So here is testimony to the fact that although there is an induction term in (7), there is no traveling wave equation in (6)! Moreover, since A is time-independent, so too will be H and consequently there is no Faraday induction since g is potential. Therefore, we should not even be able to get standing waves in (6).

So what appears to reduces the arbitrariness of the potentials (1) actually raises havoc with the linearization of the Ricci tensor.

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