Tensor Formalism for General Relativity

Abstract

Abstract This chapter introduces the basic tensor formalism needed for a proper formulation of general relativity. In a curved space, one must work with the covariant derivative, which is a combination of the ordinary derivative and the first derivatives of the metric (Christoffel symbols). The covariant derivative of a tensor is itself a tensor. The related concept of parallel transport is introduced, and the Riemann curvature tensor is derived by parallel transport of a vector around a closed path. The equation of geodesic deviation is presented as another method to arrive at the curvature tensor. The symmetries and contraction properties of the Riemann curvature (including the Bianchi identity) are considered in order to find the desired tensors for the Einstein equation. The approach to this general relativity field equation via the principle of least action is also sketched. The relevant mathematics of Schwarzschild solution and the cosmological constant are outlined.