A Brief Review Noether's Theorems and their Application to General Relativity

TL;DR

This paper reviews Noether's Theorems and their application to General Relativity, focusing on conservation laws and energy conservation issues in curved spacetime, contrasting with flat space cases.

Contribution

It presents a comprehensive review of Noether's Theorems in both flat and curved spaces, highlighting their role in conservation laws within General Relativity.

Findings

Noether's Theorems are fundamental in understanding conservation laws.

Energy conservation in General Relativity is complex and lacks a general conserved tensor.

Pseudotensor formulations can describe energy conservation in specific cases.

Abstract

In this article, we will review Noether's Theorems and their application in General Relativity. We will present Noether's Theorems in their original form and restate them as they are usually applied to physics. Some basic equations of Special Relativity will be reviewed and contrasted with the equations in General Relativity. We will be most interested in the role of Noether's Theorems in conservation laws. Several applications in flat spaces are examined. A formulation of Noether's Theorems in curved space is presented. Of particular interest is the conservation of energy in General Relativity. Here, there is no general form for a conserved tensor, as there is in a flat space case. Pseudotensor formulations can be found and two examples are given. In special cases, one can formulate an equation for the conservation of energy. The standard cosmology is an example.