A Covariant Approach to 1+3 Formalism

TL;DR

This paper introduces a covariant, basis-free 1+3 formalism for spacetime analysis, providing a deeper understanding of the 3+1 approach by splitting tensors and curvature without coordinate dependence.

Contribution

It develops a covariant 1+3 formalism that avoids basis or coordinate systems, clarifying the geometric structure of spacetime and Einstein equations.

Findings

Covariant splitting of tensors into temporal and spatial parts.

Derivation of Gauss, Codazzi, and Ricci relations covariantly.

Reduction to 3+1 formalism with normal congruence.

Abstract

I present a covariant approach to developing 1+3 formalism without an introduction of any basis or coordinates. In the formalism, a spacetime which has a timelike congruence is assumed. Then, tensors are split into temporal and spatial parts according to the tangent direction to the congruence. I make use of the natural derivatives to define the kinematical quantities and to investigate their properties. They are utilized in the splitting of covariant derivatives. In this way, the Riemann curvature is split into the temporal and spatial part, i.e. Gauss, Codazzi, and Ricci relation. Finally, the splitting of the Einstein equation is achieved by contraction. Choosing congruence as normal to a spacelike hypersurface, the formalism reduces to 3+1 formalism. This approach deepens our understanding of 3+1 formalism. All these processes are performed in a covariant manner without the complexities caused by the introduction of a coordinate system or basis.