Smooth geodesic

On a $𝐶 𝛼$ -manifold, there are two ways to go about defining a geodesic:

As the straightest path between two points, i.e. tangent vectors are parallel by Parallel transport using the affine connexion;
As a shortest or extremizing path between two points, i.e. a path of maximal or minimal length as defined using the metric tensor.¹

Note that the latter only makes sense for a path which is definite, i.e. the line element is strictly positive or strictly negative.² When the connexion used is the Levi-Civita connexion, these notions coïncide.³

Straightest path

Let $𝑀$ be a $𝐶 𝛼$ -manifold equipped with an affine connexion $\nabla$ . Consider a smooth path

𝛾 : 𝐼 \to 𝑀 : 𝜆 \mapsto 𝛾 (𝜆)

with $˙ 𝛾 = d 𝛾 / d 𝜆$ . We say $𝛾$ is a geodesic iff its tangent vectors are related by parallel transport along $𝛾$ , i.e.

˙ 𝛾 𝑎 \nabla 𝑎 ˙ 𝛾 𝑏 = 0 .

all along $𝛾$ . In local coördinates $𝑥 : 𝑈 \to 𝑀$ we therefore have

¨ 𝑥 𝛼 + Γ 𝛼 𝜇 𝜈 ˙ 𝑥 𝜇 ˙ 𝑥 𝜈 = 0 .

Caveat emptor

This is parameterization dependent, since it cares about the speed at which we traverse the curve, and will only consider “affine parameterizations” as straight.

Extremizing path

Let $(𝑀, 𝑔 𝑎 𝑏)$ be a semi-Riemannian manifold. Consider a definite smooth path

𝛾 : [0, 1] \to 𝑀 : 𝜆 \mapsto 𝛾 (𝜆)

and let

˙ 𝑓 : = d d 𝜆 𝑓 (𝛾 (𝜆))

for any smooth function $𝑓 : 𝑀 \to 𝑁$ . The path $𝛾$ induces a 1-dimensional pullback metric on $[0, 1]$ so that in local coördinates $𝑥 : 𝑈 \to ℝ 𝑚$ we have

d 𝑠 2 = 𝑔 𝜇 𝜈 ˙ 𝑥 𝜇 ˙ 𝑥 𝜈 d 𝜆 2

where everything is a function of $𝜆$ . Since $𝛾$ is definite, without loss of generality we may assume that the factor in front of $d 𝜆 2$ is nonnegative.⁴ We may thus define the length functional

ℒ [𝛾] = \int 𝜆 \in [0, 1] d 𝑠 = \int 𝜆 \in [0, 1] \sqrt 𝑔 𝜇 𝜈 ˙ 𝑥 𝜇 ˙ 𝑥 𝜈 d 𝜆 = \int 𝜆 \in [0, 1] 𝐿 𝑑 𝜆

where we have introduced the Lagrangian function

𝐿 = 𝐿 ((𝑥 𝜇), (d 𝑥 𝜇 d 𝜆)) : = \sqrt 𝑔 𝜇 𝜈 ˙ 𝑥 𝜇 ˙ 𝑥 𝜈 .

We wish to find the extermizing path for the functional $ℒ$ .

By the Fundamental theorem of calculus, it is clear that $𝐿 = d 𝑠 / d 𝜆$ and thus for $𝑓 = 𝑓 (𝑠 (𝜆))$ we have

˙ 𝑓 = d 𝑓 d 𝑠 d 𝑠 d 𝜆 = d 𝑓 d 𝑠 𝐿 .

It follows from the Euler-Lagrange equations that

𝜕 𝐿 𝜕 𝑥 𝛼 - d d 𝜆 𝜕 𝐿 𝜕 (d 𝑥 𝛼 / d 𝜆) = 0 .

For the partial derivatives with respect to $𝑥 𝛼$ we have

𝜕 𝐿 𝜕 𝑥 𝛼 = - 1 2 𝐿 𝜕 𝑔 𝜇 𝜈 𝜕 𝑥 𝛼 ˙ 𝑥 𝜇 ˙ 𝑥 𝜈 = - 𝐿 2 𝜕 𝑔 𝜇 𝜈 𝜕 𝑥 𝛼 d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠 .

For the partial derivatives with respect to $˙ 𝑥 𝛼$ we note

d d ˙ 𝑥 𝛼 [𝑔 𝜇 𝜈 ˙ 𝑥 𝜇 ˙ 𝑥 𝜈] = 𝑔 𝜇 𝜈 (˙ 𝑥 𝜇 𝛿 𝜈 𝛼 + ˙ 𝑥 𝜈 𝛿 𝜇 𝛼) = 2 𝑔 𝛼 𝜇 ˙ 𝑥 𝜇

by the product rule and thus

𝜕 𝐿 𝜕 ˙ 𝑥 𝛼 = - 1 𝐿 𝑔 𝛼 𝜇 ˙ 𝑥 𝛼 .

Differentiating with respect to $𝜆$ and eliminating instances of $𝐿$ using derivatives with respect to $𝑠$ , we have

- d d 𝜆 (𝜕 𝐿 𝜕 ˙ 𝑥 𝛼) = d d 𝜆 (1 𝐿 𝑔 𝛼 𝜇 ˙ 𝑥 𝛼) = d 𝑑 𝑠 (𝑔 𝛼 𝜇 d 𝑥 𝜇 d 𝑠) 𝐿 = (𝑔 𝛼 𝜇 d 2 𝑥 𝜇 d 𝑠 2 + d 𝑔 𝛼 𝜇 d 𝑠 d 𝑥 𝜇 d 𝑠) 𝐿 = (𝑔 𝛼 𝜇 d 2 𝑥 𝜇 d 𝑠 2 + 𝜕 𝑔 𝛼 𝜇 𝜕 𝑥 𝜈 d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠) 𝐿 = (𝑔 𝛼 𝜇 𝜕 2 𝑥 𝜇 𝜕 𝑠 2 + 12 (𝜕 𝑔 𝛼 𝜇 𝜕 𝑥 𝜈 + 𝜕 𝑔 𝛼 𝜈 𝜕 𝑥 𝜇) d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠) 𝐿 .

Thus the Euler-Lagrange equations say

0 = (𝑔 𝛼 𝜇 d 2 𝑥 𝜇 d 𝑠 2 + 12 (𝜕 𝑔 𝛼 𝜇 𝜕 𝑥 𝜈 + 𝜕 𝑔 𝛼 𝜈 𝜕 𝑥 𝜇) d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠) 𝐿 - 𝐿 2 𝜕 𝑔 𝜇 𝜈 𝜕 𝑥 𝛼 d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠 .

We divide out by $𝐿$ to get

𝑔 𝛼 𝜇 d 2 𝑥 𝜇 d 𝑠 2 = 12 (𝜕 𝑔 𝜇 𝜈 𝜕 𝑥 𝛼 - 𝜕 𝑔 𝛼 𝜇 𝜕 𝑥 𝜈 - 𝜕 𝑔 𝛼 𝜈 𝜕 𝑥 𝜇) d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠

and raising indices gives

d 2 𝑥 𝛼 d 𝑠 2 + Γ 𝛼 𝜇 𝜈 d 𝑥 𝜇 d 𝑠 d 𝑥 𝜈 d 𝑠 = 0

where the Christoffel symbols are defined by

Γ 𝛼 𝜇 𝜈 = 12 (𝜕 𝑔 𝛼 𝜇 𝜕 𝑥 𝜈 + 𝜕 𝑔 𝛼 𝜈 𝜕 𝑥 𝜇 - 𝜕 𝑔 𝜇 𝜈 𝜕 𝑥 𝛼) .

#state/develop | #lang/en | #SemBr

2024. General relativity workshop notes, §7, pp. 52–56. ↩
In the context of relativity, this corresponds to a timelike or spacelike worldline. ↩
This motivates the choice of connexion as the physical one. ↩
Otherwise we can just negate the metric. ↩

Smooth geodesic

Straightest path

Extremizing path

Footnotes