Page 151 - Intro to Tensor Calculus

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then from the transformation equation (1.4.7) we can write, after changing symbols,

α δ 2 α
δ ∂u α ∂u ∂u ∂ u
= + . (1.5.85)
β
β
βγ ∂¯u δ δ ∂¯u ∂¯u γ ∂¯u ∂¯u γ
¯ a a

δ
This is a relationship between the Christoffel symbols in the two coordinate systems. If vanishes at
βγ
¯ a
apoint P, then for that particular point the equation (1.5.85) reduces to
δ
2 α
∂ u α ∂u ∂u
= − (1.5.86)
β
β
∂¯u ∂¯u γ δ ∂¯u ∂¯u γ
a
where all terms are evaluated at the point P. Conversely, if the equation (1.5.86) is satisfied at the point P,

δ
then the Christoffel symbol must be zero at this point. Consider the special coordinate transforma-
βγ
¯ a
tion
1 α
α α α β α
u = u +¯u − ¯ u ¯u (1.5.87)
0
2 βγ
a
where u α are the surface coordinates of the point P. The point P in the new coordinates is given by
0
¯ u α =0. We now differentiate the relation (1.5.87) to see if it satisfies the equation (1.5.86). We calculate
the derivatives
∂u α α 1 α β 1 α
γ
= δ − ¯ u − ¯ u (1.5.88)
τ
∂¯u τ 2 βτ 2 τγ u =0
α
a a
and
2 α
∂ u α
= − (1.5.89)
τ
∂¯u ∂¯u σ τσ u =0
α
a
u
where these derivative are evaluated at ¯ α =0. We find the derivative equations (1.5.88) and (1.5.89) do
satisfy the equation (1.5.86) locally at the point P. Hence, the Christoffel symbols will all be zero at this
particular point. The new coordinates can then be called geodesic coordinates.
Riemann Christoffel Tensor
Consider the Riemann Christoffel tensor defined by the equation (1.4.33). Various properties of this
tensor are derived in the exercises at the end of this section. We will be particularly interested in the
α
Riemann Christoffel tensor in a two dimensional space with metric a αβ and coordinates u . We find the
Riemann Christoffel tensor has the form

∂ δ ∂ δ τ δ τ δ
δ
R .αβγ = − + − (1.5.90)
∂u β αγ ∂u γ αβ αγ βτ αβ γτ
where the Christoffel symbols are evaluated with respect to the surface metric. The above tensor has the
associated tensor
R σαβγ = a σδ R δ (1.5.91)
.αβγ
which is skew-symmetric in the indices (σ, α)and (β, γ) such that
R σαβγ = −R ασβγ and R σαβγ = −R σαγβ . (1.5.92)
The two dimensional alternating tensor is used to define the constant
1 αβ γδ
K = R αβγδ (1.5.93)
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