Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion

TL;DR

This paper introduces a new method to express curvature spinors in tensor form within a locally inertial frame, develops a novel algebra called sedon similar to sedenion, and explores their applications in gravitational theory.

Contribution

It presents a new tensor representation of curvature spinors and introduces the sedon algebra, expanding tools for analyzing gravity in the 2-spinor formalism.

Findings

Curvature spinors can be expressed in a rank-2 tensor form in a locally inertial frame.

The sedon algebra is structurally similar to sedenion but with a different basis multiplication rule.

Curvature spinors exhibit a chiral structure in the sedon representation.

Abstract

In the 2-spinor formalism, the gravity can be dealt with curvature spinors with four spinor indices. Here we show a new effective method to express the components of curvature spinors in the rank-2 $4 \times 4$ tensor representation for the gravity in a locally inertial frame. In the process we have developed a few manipulating techniques, through which the roles of each component of Riemann curvature tensor are revealed. We define a new algebra `sedon', whose structure is almost the same as sedenion except the basis multiplication rule. Finally we also show that curvature spinors can be represented in the sedon form and observe the chiral structure in curvature spinors. A few applications of the sedon representation, which includes the quaternion form of differential Binanchi indentity, are also presented.