On the quaternion transformation and field equations in curved space-time

TL;DR

This paper develops a quaternionic algebra framework to describe space-time field equations in curved space-time, including metric tensors, geodesics, and Einstein-like equations, offering a novel mathematical approach.

Contribution

It introduces a quaternionic formulation of space-time geometry and field equations, establishing transformation relations and deriving quaternionic Einstein-like equations.

Findings

Quaternionic covariant derivative formulated.

Quaternionic metric tensor and geodesic equations derived.

Quaternionic Einstein-like field equations established.

Abstract

In this paper, we use four-dimensional quaternionic algebra to describing space-time field equations in curvature form. The transformation relations of a quaternionic variable are established with the help of basis transformations of quaternion algebra. We deduced the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields.The quaternionic metric tensor and geodesic equation are also discussed to describing the quaternionic line element in curved space-time. Moreover, we discussed an expression for the Riemannian Christoffel curvature tensor in terms of the quaternionic metric tensor. We have deduced the quaternionic form of Einstein-field-like equation which shows an equivalence between quaternionic matter and geometry.