Geometrization of Mass in General Relativity

TL;DR

This paper introduces a $ ext{Z}$-graded tangent bundle with a Lie algebroid structure to geometrize mass in general relativity, leading to a purely geometric form of Einstein's field equations.

Contribution

It extends tangent bundle concepts to a graded bundle incorporating mass as a geometric component, simplifying Einstein's equations.

Findings

Mass is incorporated into the geometry of space-time.

Einstein's field equations are reformulated in a purely geometric manner.

The graded tangent bundle structure provides new insights into gravity and mass.

Abstract

In this paper we will extend the notion of tangent bundle to a $\z$ graded tangent bundle. This graded bundle has a Lie algebroid structure and we can develop notions semi-riemannian metrics, Levi-civita connection, and curvature, on it. In case of space-times manifolds, even part of the tangent bundle is related to space and time structures(gravity) and odd part is related to mass distribution in space-time. In this structure, mass becomes part of the geometry, and Einstein field equation can be reconstructed in a new simpler form. The new field equation is purely geometric.