Differential geometry MOC

Smooth geodesic

On a -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 be a -manifold equipped with an affine connexion . Consider a smooth path

with . We say is a geodesic iff its tangent vectors are related by parallel transport along , i.e.

all along . In local coördinates we therefore have

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 be a semi-Riemannian manifold. Consider a definite smooth path

and let

for any smooth function . The path induces a 1-dimensional pullback metric on so that in local coördinates we have

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

where we have introduced the Lagrangian function

We wish to find the extermizing path for the functional .

By the Fundamental theorem of calculus, it is clear that and thus for we have

It follows from the Euler-Lagrange equations that

For the partial derivatives with respect to we have

For the partial derivatives with respect to we note

by the product rule and thus

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

Thus the Euler-Lagrange equations say

We divide out by to get

and raising indices gives

where the Christoffel symbols are defined by

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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. ↩