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Introduction to General Relativity. Note
«
3
»
Covariant differentiation
Covariant differentials DA
a
and DA
a
in a curved space contain Christoffel symbols Γ
a
bc
DA
a
= dA
a
+ Γ
a
bc
A
b
dx
c
,
DA
a
= dA
a
- Γ
b
ac
A
b
dx
c
Christoffel symbols and the metric tensor
The metric tensor g
ab
defines the invariant length element in curved coordinates
ds
2
= g
ab
dx
a
dx
b
.
The metric tensor also connects contra- and covariant vectors
A
a
= g
ab
A
b
.
Covariant derivative of the metric tensor is zero
Dg
ab
=0 ,
from where one can then find
Γ
a,bc
=
1
/
2
(
dg
ab
/
dx
c
-
dg
bc
/
dx
a
+
dg
ac
/
dx
b
)
Geodesic
A particle in a gravitation field moves in such a way that the covariant derivative of its velosity u
a
is vanishing
Du
a
/
ds
= 0 ,
d
2
x
a
/
ds
2
+ Γ
a
bc
dx
b
/
ds
dx
c
/
ds
= 0 .
This line is called
geodesic
.
Exercises
For the Rindler space ds
2
=g
2
ρ
2
dη
2
-dρ
2
Find g
ab
, g
ab
and calculate the Christoffel symbols.
Using these Christoffel symbols write down the geodesic equations and compare with the equations for the motion of a free particle from exercise 2.
Consider the polar coordinates x=r cosθ, y=r sinθ in a two-dimentional flat space.
Assume that geodesics are the usual straight lines and find the geodesic equations (as in exercise 2).
From the line element ds
2
=dr
2
+r
2
dθ
2
find the metric tensor g
ab
, g
ab
and the Christoffel symbols and write down the geodesic equation.
Copyleft
©
2003
D.V.Fedorov
(
fedorov@ifa.au.dk
)