Lectures on String Theory

– 128 –
D. Riemann normal coordinates
Consider a Riemannian manifold M of dimension n with coordinates x i , 1 = 1, . . . , m.
The geodesic equation is
ẍ i + Γ i jk(x)ẋ j ẋ k = 0 .
Let us consider two points p and q with coordinates x i and x i + δx i respectively. We
assume that these points are closed so there is a unique geodesic connecting them.
A parameter t on the geodesic can be chosen proportional to the length of the
arc connecting these two points (the natural parameter). The solution x i (t) can be
chosen so that x i (0) ≡ x i and x i (1) = x i + δx i . The tangent vector to the geodesic
at t = 0 is defined by ξ i = ẋ i (0). Then equation for the geodesic can be solved
perturbatively by assuming the following expansion
x i (t) = x i + c i 1t + c i 2t 2 + · · · .
Plugging this into the geodesic equation one finds
where
x i (t) = x i + ξ i t − 1 2 Γi j 1 j 2
(x)ξ j 1
ξ j 2
t 2 − 1 3! Γi j 1 j 2 j 3
(x)ξ j 1
ξ j 2
ξ j 3
t 3 − · · · ,
Γ i j 1 j 2 j 3
= ∂ j1 Γ i j 2 j 3
− Γ l j 1 j 2
Γ i lj 3
− Γ l j 1 j 2
Γ i j 3 l .
Here all the quantities are evaluated at x i . At t = 1 we have x i (1) = x i + δx i so that
x i + δx i = x i + ξ i − 1 2 Γi j 1 j 2
(x)ξ j 1
ξ j 2
− 1 3! Γi j 1 j 2 j 3
(x)ξ j 1
ξ j 2
ξ j 3
− · · ·
Hence, the point x i + δx i is parameterized by the tangent vector ξ i . We can therefore
take ξ i as the new coordinate system on our manifold. The coordinates ξ i are
called Riemann normal coordinates. They are used to define a map of an open
neighbourhood U of the zero-vector 0 ∈ T x M to the manifold M:
exp x :
U ∈ T x M → M
which is well-defined diffeomorphism of U on its image. Now we are parameterizing
the points of the curved manifold with tangent vectors. The image of the geodesic
through a point x in the Riemann normal coordinates is just a straight line.
How metric will look in the new coordinate system? Upon changing the coordinates
form x i to ξ i the metric transforms in the standard way
We have
g ij(ξ) ′ = ∂(xk + δx k ) ∂(x l + δx l )
g
∂ξ i ∂ξ j kl (x + δx)
∂(x k + δx k )
∂ξ i = δ k i − Γ k ijξ j + · · ·