Geodesic and Jacobi Equations

The geodesic equation selects curves whose tangent is parallel along itself; the Jacobi equation describes the linearized separation of a smooth family of such curves. Their coordinate forms are useful only after the parameter, connection, curvature convention, and comparison field are fixed. This record assumes a Levi–Civita connection and the site’s $(-,+,+,+)$ convention.

Geodesic motion

For a curve $x^\mu(\lambda)$ with tangent $u^\mu=dx^\mu/d\lambda$, affine geodesic motion is

$$\nabla_u u=0, \qquad \frac{d^2x^\rho}{d\lambda^2} +\Gamma^\rho{}_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}=0.$$

An affine change $\lambda\mapsto a\lambda+b$ preserves this form. A general monotone parameter $s$ preserves the same unparametrized path but gives

$$\nabla_vv=f(s)v, \qquad v^\mu=\frac{dx^\mu}{ds}.$$

For a Levi–Civita connection, varying the energy functional

$$E[x]=\frac12\int g_{\mu\nu}\dot x^\mu\dot x^\nu\,d\lambda$$

with fixed endpoints produces the affine equation. The square-root length or proper-time functional determines the same non-null path up to reparametrization; it cannot derive a null geodesic because the null length vanishes. The affine, variational, and exponential-map distinctions are developed in Carroll 1997, § 3 and Wendl 2021–2022, §§ 22–24, printed pp. 184–211, PDF.

If $K^\mu$ is a Killing field, an affinely parametrized geodesic has the conserved quantity

$$\frac{d}{d\lambda}(K_\mu u^\mu)=0.$$

This follows directly from $u^\mu u^\nu\nabla_{(\mu}K_{\nu)}=0$ and $\nabla_u u=0$.

For a timelike geodesic parametrized by proper time, $g(u,u)=-1$; for a null geodesic, $g(u,u)=0$ and the scale of an affine parameter remains conventional.

Jacobi fields and deviation

Let $F(s,\lambda)$ be a smooth family of affinely parametrized geodesics, with tangent $u=F_*\partial_\lambda$ and separation field $J=F_*\partial_s$. Since $[u,J]=0$, linearizing $\nabla_u u=0$ gives

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

or, equivalently,

$$\frac{D^2J^\mu}{d\lambda^2} =R^\mu{}_{\nu\rho\sigma} u^\nu u^\rho J^\sigma.$$

The sign follows from $[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma$; a source with the opposite curvature convention reverses the displayed curvature term. The family derivation and tidal interpretation are given in Carroll 1997, § 3 and Reall 2013, Chapters 5–6, printed pp. 49–66, PDF.

Case	Consequence
Flat spacetime	$J=A+\lambda B$ in a parallel frame
Tangential mode	$J=(a+b\lambda)u$ changes affine origin or scale rather than transverse geometry
Timelike transverse field	Frame components give the linear tidal relative acceleration measured along the reference geodesic
Null reference geodesic	Physical transverse separation lives in the screen-space quotient because $u$ is orthogonal to itself
$J(0)=J(\lambda_*)=0$, $J\ne0$	$\lambda_*$ is conjugate to the initial point along that geodesic

Initial data $J(0)$ and $DJ/d\lambda(0)$ determine one Jacobi field. A zero of $J$ diagnoses a singular differential of the exponential map, not by itself a curvature singularity or geodesic incompleteness. Positive-definite extremum statements about Riemannian geodesics do not transfer unchanged to null or timelike Lorentzian curves.

For the unified treatment connecting these equations to exponential maps and conjugate points, see Geodesics, Exponential Maps, and Jacobi Fields. Full derivations are given in Affine and Variational Geodesics and Jacobi Fields and Geodesic Deviation. For ideal free fall rather than abstract connection geometry, continue to Timelike Geodesics as Ideal Free Fall.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997), § 3. PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), Chapters 5–6, printed pp. 49–66. PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes (2021–2022), §§ 22–24, printed pp. 184–211. PDF.