Conservation Laws on Riemann-Cartan, Lorentzian and Teleparallel Spacetimes

TL;DR

This paper investigates the conditions under which conservation laws for energy-momentum and angular momentum exist in Riemann-Cartan, Lorentzian, and teleparallel spacetimes using Clifford bundle formalism, clarifying misconceptions in the literature.

Contribution

It provides a detailed analysis of the conditions for conservation laws in various spacetime geometries and corrects common misunderstandings in the context of General Relativity and related theories.

Findings

Identifies strong conditions for conservation laws involving matter fields.

Clarifies the existence of conservation laws for matter and gravitational fields.

Corrects misconceptions about conservation laws in Riemann-Cartan and teleparallel theories.

Abstract

In this paper using a Clifford bundle formalism we examine (a): the strong conditions for existence of conservation laws involving only the energy-momentum and angular momentum of the matter fields on a general Riemann-Cartan spacetime and also in the particular cases of Lorentzian and teleparallel spacetimes and (b): the conditions for the existence of conservation laws of energy-momentum and angular momentum for the matter and gravitational fields when this latter concept can be rigorously defined. We examine in details some misleading and even erroneous and often quoted statements concerning the issues of the conservation laws in General Relativity and Riemann-Cartan (including the particular case of the teleparallel one) theories.