Abstract
Differential equations of electromagnetic and similar physical fields are generally solved via antiderivative Green’s functions involving integration over a region and its boundary. Research on the Kasner metric reveals a variable boundary deemed inappropriate for standard anti-derivatives, suggesting the need for an alternative solution technique. In this work I derive such a solution and prove its existence, based on circulation equations in which the curl of the field is induced by source current density and possibly changes in associated fields. We present an anti-curl operator that is believed novel and we prove that it solves for the field without integration required.
Highlights
Nonlinear field equations are generally impossible of solution—the few existing (Schwarzschild, Kerr) being well-known
Research on the Kasner metric reveals a variable boundary deemed inappropriate for standard anti-derivatives, suggesting the need for an alternative solution technique
We present an anti-curl operator that is believed novel and we prove that it solves for the field without integration required
Summary
Nonlinear field equations are generally impossible of solution—the few existing (Schwarzschild, Kerr) being well-known. These known solutions are static and expressed in spherical coordinates with boundaries at infinity. The Kasner solution, treated by Vishwakarma [1], is dynamic with dynamic boundaries and does not fit well into the static approach. Even physical interpretation of the exact solution has been largely nonexistent. For this reason an alternative approach was sought. The goals of this approach were: 1) A linear formulation that avoids the nonlinearity of Einstein’s field equations.
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