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

The connection between Noether's Theorems and symmetries in physics is well established. However, the role they play in gravitation, especially with the conservation of energy, is often overlooked. In this article, we will review Noether's Theorems and their application in General Relativity. This will include what is usually referred to as the Third Theorem. We will present Noether's Theorems in their original form. Then, they will be restated as they are usually applied to physics. Some basic equations of Special Relativity will be reviewed. Several applications in flat spaces are examined. This is so they can be contrasted with the equations in General Relativity. The equations of General Relativity are then developed, and applications are contrasted with those of Special Relativity. We will be most interested with the role of Noether's Theorems in conservation laws. A formulation of Noether's Theorems in curved space is presented. Of particular interest is the conservation of energy in General Relativity. This involves Noether's First Theorem. Here, there is no general form for a conserved tensor, as there is in the flat-space case. Pseudotensor formulations can be found. The Landau and Lifshitz and the Dirac pseudotensors are given as examples. In special cases, one can formulate a tensor equation for the conservation of energy. The standard cosmology is an example. The formulation of Noether's Second and Third Theorems are also presented. These formulations are related to things like the Bianchi identity and the response equation.