Schwarzschild Solution of the Generally Covariant Quaternionic Field Equations of Sachs

TL;DR

This paper derives a quaternionic formulation of Einstein's field equations, constructs a spherically symmetric static solution, and shows it reproduces the Schwarzschild solution, highlighting differences due to quaternionic structure.

Contribution

It introduces a quaternionic approach to Einstein's equations and demonstrates that the Schwarzschild solution emerges within this framework.

Findings

Quaternionic field equations reproduce Schwarzschild metric.

Solutions differ in reflection covariance due to quaternionic structure.

Differential equations match classical Schwarzschild equations.

Abstract

Sachs has derived quaternion field equations that fully exploit the underlying symmetry of the principle of general relativity, one in which the fundamental 10 component metric field is replaced by a 16 component four-vector quaternion. Instead of the 10 field equations of Einstein's tensor formulation, these equations are 16 in number corresponding to the 16 analytic parametric functions {\partial}x^{{\mu}'}/{\partial}x^{{\nu}} of the Einstein Lie Group. The difference from the Einstein equations is that these equations are not covariant with respect to reflections in space-time, as a consequence of their underlying quaternionic structure. These equations can be combined into a part that is even and a part that is odd with respect to spatial or temporal reflections. This paper constructs a four-vector quaternion solution of the quaternionic field equation of Sachs that corresponds to a spherically symmetric static metric. We show that the equations for this four-vector quaternion corresponding to a vacuum solution lead to differential equations that are identical to the corresponding Schwarzschild equations for the metric tensor components.