Classical And Quantum Gravity. Riemannian Manifolds And Tensors

Abstract

This chapter has two purposes; to present the few elements of differential geometry which are required in different places in this volume and to provide, for completeness, a short introduction to the problem of quantization of gravity. It first briefly recalls a few concepts related to reparametrization (more accurately diffeomorphism) of Riemannian manifolds. It introduces the notions of parallel transport, affine connection, and curvature, in analogy with gauge theories as discussed in Chapters 19-21. To define fermions on Riemannian manifolds additional mathematical objects are required — the vielbein and the spin connection. The chapter constructs Einstein's action for classical gravity (General Relativity) and derive the equation of motion. In the last section, it studies the formal aspects of the quantization of the theory of gravity, following the lines of the quantization of non-abelian gauge theories of Chapter 19.