The Euler-Lagrange equations of the second rank tensor in 3+1 space

These calculations are a direct extension of the equations that have been developed for the study of the acoustic wave in elastic media. The equations are developed in many textbooks, such as [2]. The kinetic energy term is not used. A cosmological potential energy term has been added.

It is necessary to evaluate the strain tensor from Section 2 under time reversal in performing these calculations. The stress tensor must also be time reversed. The inversion is accomplished by changing the sign of all time-dependent terms. It is odd that this procedure should be necessary, but since the Maxwell equations exhibit time-reversal symmetry it seems to be justifiable. The problem does not occur in the 4-vector calculations of Section 4, so it may represent a limitation of the real-number system.

A solution under forward time progression also exists. The terms in the solution are the same, but they are multiplied by multiples of 3. I obtained these solutions several years ago, and at the time concluded that the alternative solution was equivalent, having a meaning similar to that of a gauge transform. The question seems to be worthy of further study, and I will dust off the computer discs and post the calculations here after a while.

The vacuum solution is

div grad Ψ + 1/c ∂/∂t (div A) = 3/(2 c) ∂/∂t (div A + 1/c ∂Ψ/∂t) + Ψc12

curl curl A + 1/c grad(∂Ψ/∂t) + 1/c22A/∂t2 = 3/2 grad(div A + 1/c ∂Ψ/∂t) + Ac22

The solutions are of the same form as the 4-vector solutions in Section 4. It is shown there that, when the cosmological term is dropped, the equations are the Maxwell potential equations with a built-in constraint to the Lorentz gauge. The cosmological terms are also discussed in Section 4.

The static vector solution appears to represent the weak gravitational field. The gravitational solutions are investigated in another section.


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