Geodesics, Exponential Maps, and Jacobi Fields

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Geodesics connect a local connection to finite curves, the exponential map packages their initial-value problem, and Jacobi fields describe the first-order behavior of neighboring solutions. Together these objects separate local existence from global uniqueness, variational extremality from affine parametrization, and conjugate points from cut points or incompleteness. This chapter develops those distinctions before any physical free-fall interpretation is added.

Required background. Affine Connections and Covariant Derivatives supplies the connection and differentiation along curves; The Riemann Curvature Tensor supplies the curvature commutator and sign convention used in geodesic deviation.

Helpful background. Calculus and Linear Algebra for Relativity supplies multivariable differentiation, linear maps, bilinear forms, and integration by parts. Presymplectic Potentials and Currents offers an advanced connection between variational boundary terms and currents on solution space; the chapter develops its finite-dimensional variational arguments directly. Tensor and Geometry Bridge offers a shorter review of the notation.

The initial-value viewpoint

An affinely parametrized geodesic satisfies

\nabla_u u=0.

Its initial point and tangent determine a unique local solution. Affine and Variational Geodesics compares this definition with stationary energy and length. Linear affine changes preserve the equation, while a general reparametrization introduces acceleration parallel to the tangent. Null length is identically zero, so null geodesics require the energy action with a constraint or an auxiliary-field formulation rather than a naive length variation.

Exponential Maps and Normal Neighborhoods collects the initial-value solutions into $\exp_p(v)=\gamma_v(1)$ . Its differential at zero is the identity, so it is a local diffeomorphism and produces normal coordinates. That local fact does not imply a globally defined, injective, or surjective map. On a sphere every vector of length $\pi a$ reaches the antipode; on an incomplete manifold some initial vectors cannot be evolved to parameter one.

The local construction and parameter issues are developed in Tong 2019, §3.3, pp. 118–129.

Linearized geometry and variation

Jacobi Fields and Geodesic Deviation derives

\frac{D^2J}{d\lambda^2}+R(J,u)u=0

by differentiating a family of affine geodesics. The sign is fixed by the chapter’s curvature convention, not by analogy. In flat space $J=A+\lambda B$ . On a round sphere, transverse separation oscillates as $\sin(\lambda/a)$ and first vanishes again at $\pi a$ . Tangential solutions $(a+b\lambda)u$ represent shifts and rescalings of the affine parameter, so they must be separated from transverse geometry.

Conjugate Points and the Index Form turns this linearized equation into a variational criterion. A nontrivial Jacobi field vanishing at both endpoints makes the exponential-map differential singular and gives a kernel direction of the second variation. In Riemannian geometry the first conjugate endpoint gives degeneracy, while extending beyond it destroys local minimality. For timelike Lorentzian geodesics, the corresponding causal statement concerns local maximal proper time. Spacelike geodesics can be saddles even in flat spacetime; null variations require their own causal hypotheses.

Global behavior and Hamiltonian structure

Geodesic Completeness and Cut Loci separates three frequently conflated events. A conjugate point is a rank failure of $d\exp$ ; a cut point marks loss of global optimality or uniqueness; incompleteness means some maximal affine geodesic has a finite parameter endpoint. On a round sphere the antipode is both conjugate and cut, yet every geodesic continues forever. On a flat cylinder, cut points can arise without curvature focusing.

Hopf–Rinow equates metric completeness and geodesic completeness for connected finite-dimensional Riemannian manifolds and guarantees minimizing geodesics. It is not a Lorentzian theorem. A Lorentzian metric gives no positive-definite path distance of the needed kind, and compact Lorentzian manifolds can be geodesically incomplete.

Geodesic Flow and Hamilton–Jacobi Structure gives the cotangent-bundle route. With

H=\frac12g^{\mu\nu}p_\mu p_\nu,

Hamilton’s equations project to affine geodesics. The Hamilton–Jacobi equation is $g^{\mu\nu}\partial_\mu S\partial_\nu S/2=\kappa$ , with $\kappa=-m^2/2$ for the timelike mass shell and $\kappa=0$ for null characteristics in signature $(-,+,+,+)$ .

Reader routes and scope

For a first encounter, read affine geodesics, exponential maps, Jacobi fields, and then completeness. For a variational route, read affine geodesics, Jacobi fields, the index form, and Hamilton–Jacobi structure. For global geometry, focus on the exponential map, conjugate points, and the completeness comparison, carrying every Riemannian hypothesis explicitly.

The chapter does not assert that metric geodesics describe physical free fall; the General Relativity volume supplies the equivalence principle and matter coupling. It also stops short of causal hierarchy, singularity theorems, and extension theory, which belong to Global Structure and Mathematical Relativity. A caustic or conjugate point is not automatically a curvature singularity.

Chapter review

On a round two-sphere of radius $a$ , start at $p$ with a unit-speed great circle. Determine the first conjugate point, the first cut point, and whether the geodesic is complete. Then explain which conclusions would remain valid merely from knowing that the curvature is finite.

Solution

A transverse Jacobi field with $J(0)=0$ is $J=a\sin(\lambda/a)E$ , so its first second zero is at $\lambda=\pi a$ . The antipode is therefore the first conjugate point. Great-circle arcs stop being uniquely minimizing there, so it is also the first cut point. Nevertheless, the great circle continues for every $\lambda\in\mathbb R$ , and the sphere is geodesically complete. Finite curvature alone proves none of these global conclusions: it neither locates cut points nor guarantees completeness, and it does not determine topology.

The next conceptual step is Symmetries, Lie Derivatives, and Killing Fields, where invariant flows generate constants of geodesic motion.

References

David Tong, General Relativity, University of Cambridge Part III Mathematical Tripos lecture notes, 2019. PDF.