These calculations utilize the same equations that have been developed for the study of the acoustic wave in elastic media. The equations are developed in many textbooks, such as . The kinetic energy term is not used. In the first calculation the indices run from 1 to 4 instead of 1 to 3. There are no other changes to the acoustic wave equations, except that a cosmological potential energy term has been added. The tensor is shown in Section 2.
The solution is
div grad Ψ + 1/c ∂/∂t (div A) = 1/c (2 + b/s) ∂/∂t (div A + 1/c ∂Ψ/∂t) + Ψc1/Λ2
curl curl A + 1/c grad(∂Ψ/∂t) + 1/c2 ∂2A/∂t2 = (2 + b/s) grad(div A + 1/c ∂Ψ/∂t) + Ac2/Λ2
b is the bulk modulus, s is the shear modulus. c1 and c2 are small undetermined pure-number constants. It will be necessary to obtain a coupled (time-varying) solution in order to determine their values. If a system of units is utilized in which the scalar and vector potentials both have the units of distance then Λ also has the units of distance, and represents the range of the fields. Λ is the cosmological constant, with a value that is presumably near the Hubble distance.
At distances very small in relation to the range of the fields the cosmological term can be dropped. The terms on the left are the Maxwell potential equations. The remaining terms on the right are the gradient and the time derivative of the Lorentz condition. The Lorentz condition is zero for solutions to the Maxwell equations in the Lorentz gauge, so the surviving terms equivalent to the Maxwell equations in that gauge.
The vector equation can be restructured by the identity ∇2A = grad(div A) - curl(curl A). The Maxwell equations proper are expressed in terms of the fields, with the connection being B = curl A, E = -grad Ψ - 1/c ∂A/∂t.
It is interesting that the 4-vector calculation does not utilize
the antisymmetric part of the tensor, yet the solution is of the
same form as the 3+1 space solution in Section 5, which does utilize
the antisymmetric part. The correspondence seems to imply that a symmetric
4x4 complex-domain tensor can adequately represent a nonsymmetric 3+1 space.