Parallel transport and geodesics

Abstract

The mathematics of parallel transport and of affine and metric geodesics is presented. The geodesic equation is obtained in several different ways, bringing out its role both as a geometric statement and as an equation of motion. The Euler-Lagrange method to find metric geodesics, and hence Christoffel symbols, is explained. The role of conserved quantities is discussed. Killing’s equation and Killing vectors are introduced. Fermi-Walker transport (the non-rotating freely falling cabin) is defined and discussed. Gravitational redshift is calculated, first in general and then in specific cases.