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Covariant differentiation
Covariant differentials DAa and DAa in a curved space contain Christoffel symbols Γabc
DAa = dAa + ΓabcAbdxc,
DAa = dAa - ΓbacAbdxc

Christoffel symbols and the metric tensor
The metric tensor gab defines the invariant length element in curved coordinates
ds2 = gabdxadxb .
The metric tensor also connects contra- and covariant vectors
Aa = gabAb .
Covariant derivative of the metric tensor is zero
Dgab=0 ,
from where one can then find
Γa,bc=1/2( dgab/dxc -dgbc/dxa +dgac/dxb)

Geodesic
A particle in a gravitation field moves in such a way that the covariant derivative of its velosity ua is vanishing
Dua/ds = 0 ,
d2xa/ds2 + Γabcdxb/dsdxc/ds = 0 .
This line is called geodesic.

Exercises
  1. For the Rindler space ds2=g2ρ22-dρ2
    1. Find gab, gab and calculate the Christoffel symbols.
    2. Using these Christoffel symbols write down the geodesic equations and compare with the equations for the motion of a free particle from exercise 2.
  2. Consider the polar coordinates x=r cosθ, y=r sinθ in a two-dimentional flat space.
    1. Assume that geodesics are the usual straight lines and find the geodesic equations (as in exercise 2).
    2. From the line element ds2=dr2+r22 find the metric tensor gab, gab and the Christoffel symbols and write down the geodesic equation.

Copyleft © 2003 D.V.Fedorov (fedorov@ifa.au.dk)