Differential geometry MOC

Smooth geodesic

On a 𝐶𝛼-manifold, there are two ways to go about defining a geodesic:

1. As the straightest path between two points, i.e. tangent vectors are parallel by Parallel transport using the affine connexion;
2. 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 𝐶𝛼-manifold equipped with an affine connexion ∇. Consider a smooth path

𝛾:𝐼→𝑀:𝜆↦𝛾(𝜆)

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

˙𝛾𝑎∇𝑎˙𝛾𝑏=0.

all along 𝛾. In local coördinates 𝑥 :𝑈 →𝑀 we therefore have

¨𝑥𝛼+Γ𝛼𝜇𝜈˙𝑥𝜇˙𝑥𝜈=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,𝑔𝑎𝑏) be a semi-Riemannian manifold. Consider a definite smooth path

𝛾:[0,1]→𝑀:𝜆↦𝛾(𝜆)

and let

˙𝑓:=dd𝜆𝑓(𝛾(𝜆))

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

d𝑠2=𝑔𝜇𝜈˙𝑥𝜇˙𝑥𝜈d𝜆2

where everything is a function of 𝜆. Since 𝛾 is definite, without loss of generality we may assume that the factor in front of d𝜆2 is nonnegative.⁴ We may thus define the length functional

ℒ[𝛾]=∫𝜆∈[0,1]d𝑠=∫𝜆∈[0,1]√𝑔𝜇𝜈˙𝑥𝜇˙𝑥𝜈d𝜆=∫𝜆∈[0,1]𝐿𝑑𝜆

where we have introduced the Lagrangian function

𝐿=𝐿((𝑥𝜇),(d𝑥𝜇d𝜆)):=√𝑔𝜇𝜈˙𝑥𝜇˙𝑥𝜈.

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

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

˙𝑓=d𝑓d𝑠d𝑠d𝜆=d𝑓d𝑠𝐿.

It follows from the Euler-Lagrange equations that

𝜕𝐿𝜕𝑥𝛼−dd𝜆𝜕𝐿𝜕(d𝑥𝛼/d𝜆)=0.

For the partial derivatives with respect to 𝑥𝛼 we have

𝜕𝐿𝜕𝑥𝛼=−12𝐿𝜕𝑔𝜇𝜈𝜕𝑥𝛼˙𝑥𝜇˙𝑥𝜈=−𝐿2𝜕𝑔𝜇𝜈𝜕𝑥𝛼d𝑥𝜇d𝑠d𝑥𝜈d𝑠.

For the partial derivatives with respect to ˙𝑥𝛼 we note

dd˙𝑥𝛼[𝑔𝜇𝜈˙𝑥𝜇˙𝑥𝜈]=𝑔𝜇𝜈(˙𝑥𝜇𝛿𝜈𝛼+˙𝑥𝜈𝛿𝜇𝛼)=2𝑔𝛼𝜇˙𝑥𝜇

by the product rule and thus

𝜕𝐿𝜕˙𝑥𝛼=−1𝐿𝑔𝛼𝜇˙𝑥𝛼.

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

−dd𝜆(𝜕𝐿𝜕˙𝑥𝛼)=dd𝜆(1𝐿𝑔𝛼𝜇˙𝑥𝛼)=d𝑑𝑠(𝑔𝛼𝜇d𝑥𝜇d𝑠)𝐿=(𝑔𝛼𝜇d2𝑥𝜇d𝑠2+d𝑔𝛼𝜇d𝑠d𝑥𝜇d𝑠)𝐿=(𝑔𝛼𝜇d2𝑥𝜇d𝑠2+𝜕𝑔𝛼𝜇𝜕𝑥𝜈d𝑥𝜇d𝑠d𝑥𝜈d𝑠)𝐿=(𝑔𝛼𝜇𝜕2𝑥𝜇𝜕𝑠2+12(𝜕𝑔𝛼𝜇𝜕𝑥𝜈+𝜕𝑔𝛼𝜈𝜕𝑥𝜇)d𝑥𝜇d𝑠d𝑥𝜈d𝑠)𝐿.

Thus the Euler-Lagrange equations say

0=(𝑔𝛼𝜇d2𝑥𝜇d𝑠2+12(𝜕𝑔𝛼𝜇𝜕𝑥𝜈+𝜕𝑔𝛼𝜈𝜕𝑥𝜇)d𝑥𝜇d𝑠d𝑥𝜈d𝑠)𝐿−𝐿2𝜕𝑔𝜇𝜈𝜕𝑥𝛼d𝑥𝜇d𝑠d𝑥𝜈d𝑠.

We divide out by 𝐿 to get

𝑔𝛼𝜇d2𝑥𝜇d𝑠2=12(𝜕𝑔𝜇𝜈𝜕𝑥𝛼−𝜕𝑔𝛼𝜇𝜕𝑥𝜈−𝜕𝑔𝛼𝜈𝜕𝑥𝜇)d𝑥𝜇d𝑠d𝑥𝜈d𝑠

and raising indices gives

d2𝑥𝛼d𝑠2+Γ𝛼𝜇𝜈d𝑥𝜇d𝑠d𝑥𝜈d𝑠=0

where the Christoffel symbols are defined by

Γ𝛼𝜇𝜈=12(𝜕𝑔𝛼𝜇𝜕𝑥𝜈+𝜕𝑔𝛼𝜈𝜕𝑥𝜇−𝜕𝑔𝜇𝜈𝜕𝑥𝛼).

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Footnotes

1. 2024. General relativity workshop notes, §7, pp. 52–56. ↩

2. In the context of relativity, this corresponds to a timelike or spacelike worldline. ↩

3. This motivates the choice of connexion as the physical one. ↩

4. Otherwise we can just negate the metric. ↩