General Relativity. Note 3

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Covariant differentiation in curved coordinates. Christoffel symbols.

General Relativity. Note: « 3 »

Contravariant and covariant vectors
A set of four quantities A^a, where a=0,1,2,3, is called a contravariant vector if under a transformation of coordinates x→x'(x) it transforms as coordinate differentials dx^a,

A^a = (^{∂x^a}/_∂x'^b)A'^b.

A set of four quantities A_a is called a covariant vector if under a transformation of coordinates x→x'(x) it transforms as derivatives of a scalar ^dφ/_dx^a,

A_a = (^{∂x'^b}/_∂x^a)A'_b.

The contraction A^aB_a≡∑_aA^aB_a is invariant under coordinate transformation A^aB_a = A'^aB'_a
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_bcA^bdx^c , DA_a = dA^a - Γ^b_acA_bdx^c

Basically all physical laws in curved space are the same as in flat space, only one has to chande the small differential "d" into the covariant differential "D".
Christoffel symbols and the metric tensor
The metric tensor g_ab defines the invariant length element in curved coordinates

ds² = g_abdx^adx^b .

The metric tensor also connects contra- and covariant vectors

A^a = g^abA_b .

Considering DA_a=g_abDA^b=Dg_abA^b, the 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 as the constant velocity trajectory.
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 ,

^{d²x^a}/_ds² + Γ^a_bc^{dx^b}/_ds^{dx^c}/_ds = 0 .

This line is called geodesic.
Exercises

For the Rindler space with the metric ds²=g²ρ²dη²-dρ²
1. Find g_ab, g^ab 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 note 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 note 2).
2. From the line element ds²=dr²+r²dθ² find the metric tensor g_ab, g^ab and the Christoffel symbols and write down the geodesic equation.