3.9 Geodesics in Lorentzian and Riemannian Manifolds: Two Simple Examples

vector t(0). So we can conclude that at every point p M there is a geodesic departing along any chosen direction in the tangent space Tp M .

We can define geodesics with respect to any affine connection Γ , yet nothing guarantees a priori that such straight lines should also be the shortest routes from one point to another of the considered manifold M . In the case we have a metric structure, lengths are defined and we can consider the variational problem of calculating extremal curves for which any variation makes them longer. It suffices to implement the standard variational calculus to the length functional (see (3.6.5)):

√

s = dτ ≡ 2L dλ

(3.8.5)

L≡ 1 gμν (x) d xμ d xν 2 dλ dλ

Performing a variational calculation we get that the length is extremal if

We are free to use any parameter λ to parameterize the curves. Let us use the affine parameter λ = τ defined by the condition:

In this case equation (3.8.6) reduces to δL = 0 which is the standard variational equation for a Lagrangian L where the affine parameter τ is the time and xμ are the Lagrangian coordinates qμ. It is a straightforward exercise to verify that the Euler-Lagrange equations of this system:

coincide with the geodesic equations (3.8.4) where for Γ we use the Christoffel symbols (3.2.7).

In this way we reach a very important conclusion. The Levi Civita connection is that unique affine connection for which also in curved space the curves of extremal length (typically the shortest ones) are straight just as it happens in flat space. This being true, the geodesics can be directly obtained from the variational principle which is the easiest and fastest way.

3.9Geodesics in Lorentzian and Riemannian Manifolds: Two Simple Examples

Let us now illustrate geodesics in some simple examples which will also be useful to emphasize the difference between Riemannian and Lorentzian manifolds. In a