Jacobi Fields and Geodesic Deviation

A Jacobi field is the first-order separation vector generated by a smooth family of affinely parametrized geodesics. Linearizing the geodesic equation gives a covariant second-order equation in which the operator $R(u,J)u$ controls transverse separation. This page derives the sign from the site commutator convention, identifies the initial data, and separates genuine transverse deviation from a tangential change of parametrization.

Required background. Affine Connections and Covariant Derivatives supplies the general connection and covariant-derivative framework; Parallel Transport Along Curves supplies differentiation and parallel fields along a curve. The Riemann Curvature Tensor fixes $R(X,Y)Z=[\nabla_X,\nabla_Y]Z-\nabla_{[X,Y]}Z$.

Helpful background. Calculus and Linear Algebra for Relativity supplies linear maps and eigenstructure; the linear ODE facts used below are stated here. Presymplectic Potentials and Currents gives an advanced field-space interpretation of linearized solutions, but that interpretation is not assumed.

Geodesic variations and their fields

Let $F(s,\lambda)$ be a smooth two-parameter map such that, for each fixed $s$, the curve $\lambda\mapsto F(s,\lambda)$ is an affinely parametrized geodesic. Along the central geodesic $\gamma(\lambda)=F(0,\lambda)$ define

$$u=F_*\partial_\lambda, \qquad J=F_*\partial_s.$$

These are vector fields along the map $F$, not necessarily vector fields on an open subset of spacetime. Write $D_\lambda$ and $D_s$ for covariant differentiation along its two parameter directions. This remains meaningful when geodesics cross or the variation loses rank. Since the parameter directions commute and the connection is torsion free,

$$D_su=D_\lambda J.$$

The geodesic condition is $D_\lambda u=0$. Differentiate it with respect to $s$ and use the curvature commutator for the connection pulled back to the parameter domain:

$$0=D_sD_\lambda u =D_\lambda D_su+R(J,u)u =D_\lambda^2J+R(J,u)u.$$

Where the variation is an embedded surface, this is equivalently the familiar calculation with $[u,J]=0$ and $\nabla_Ju=\nabla_uJ$. Restricting to the central geodesic gives

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

Equivalently, using antisymmetry in the first two curvature arguments, $D^2J/d\lambda^2=R(u,J)u$. In components,

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

with the displayed index order understood through the invariant equation. Deriving rather than memorizing this form prevents convention errors. The variation argument is developed in Calegari 2022, §§7.5–7.6, pp. 59–62.

Initial data and the exponential map

The Jacobi equation is a linear second-order ODE along $\gamma$. Values

$$J(0)=A, \qquad \frac{DJ}{d\lambda}(0)=B$$

determine a unique Jacobi field. They have direct meanings: $A$ changes the initial point, while $B=D_su(s,0)|_{s=0}$ changes the initial velocity after connection-dependent comparison. Conversely, one can vary initial point and velocity with these derivatives and solve the geodesic initial-value problem; smooth dependence produces the corresponding Jacobi field locally. When $A=0$, the differential of the exponential map is encoded by

$$(d\exp_p)_{\lambda v}(\lambda B)=J(\lambda),$$

where $\gamma(\lambda)=\exp_p(\lambda v)$ and $J'(0)=B$. A nonzero Jacobi field with $J(0)=J(\lambda_*)=0$ therefore signals a kernel of the exponential-map differential and a conjugate point.

Flat space, sphere, and tangential modes

In flat space, $R=0$, and parallel components obey

$$J(\lambda)=A+\lambda B.$$

Initially parallel geodesics remain at constant separation when $B=0$; an initial relative velocity produces linear separation. This is the required flat limit.

On a round sphere of radius $a$, choose a unit-speed geodesic and a parallel transverse unit vector $E$. Constant curvature gives $R(E,u)u=E/a^2$, so $J=jE$ obeys

$$j''+\frac{1}{a^2}j=0.$$

With $J(0)=0$ and $J'(0)=E$, one obtains $J=a\sin(\lambda/a)E$. It vanishes first at $\lambda=\pi a$, matching the antipode and independently checking the curvature sign.

Not every Jacobi field represents transverse geometry. Fields

$$J=(a+b\lambda)u$$

solve the equation because $R(u,u)u=0$. They arise from shifting the affine origin or rescaling the affine parameter within the same geodesic image. For non-null curves, orthogonal projection cleanly separates these tangential modes. For null curves, the screen-space quotient is needed because $u$ lies in its own orthogonal complement.

Interpretation and limitations

The curvature combination depends on the tangent direction and the separation direction; it is not determined by the scalar curvature alone. Jacobi fields describe first-order behavior of nearby geodesics. Large separations can leave the linear regime, and a zero of a Jacobi field is a statement about the exponential map, not automatically a curvature singularity.

In relativity the same equation underlies tidal separation, but translating it into measured relative acceleration requires an observer frame and physical assumptions. Focusing theorems further require causal hypotheses, energy conditions, and often field equations; none follows from the Jacobi identity alone. The local and global geometric roles of Jacobi fields are also treated in Lee 2018, Chapter 10, pp. 283–313, whose positive-definite conclusions must not be transferred wholesale to Lorentzian geometry.

Two preservation laws are useful checks. For any Jacobi fields $J_1,J_2$, the Wronskian-like quantity

$$W(J_1,J_2) =g\!\left(J_1,\frac{DJ_2}{d\lambda}\right) -g\!\left(\frac{DJ_1}{d\lambda},J_2\right)$$

is constant. Differentiating it and inserting the Jacobi equation makes the curvature terms cancel by pair exchange symmetry. This conserved bilinear form is the linearized shadow of the symplectic structure of geodesic flow.

Also, $g(J,u)$ is affine in $\lambda$: differentiating twice and using $\nabla_u u=0$ gives zero because $g(R(J,u)u,u)=0$. Consequently, if both $g(J,u)$ and its first derivative vanish initially, $J$ remains transverse along a non-null geodesic. This provides a clean consistency check for numerical integration and shows precisely which initial conditions remove tangential reparametrization modes.

Exercises

Find all transverse Jacobi fields along a unit-speed great circle on $S^2_a$ that vanish at $\lambda=0$, and identify the first second zero.

Solution

There is one parallel transverse direction $E$ along the great circle. The scalar coefficient satisfies $j''+j/a^2=0$ and $j(0)=0$, so $j(\lambda)=C\sin(\lambda/a)$. Thus $J=C\sin(\lambda/a)E$. For $C\ne0$, its first positive second zero is at $\lambda=\pi a$, the antipodal point.

References

• Danny Calegari, Introduction to Differential Geometry, preliminary version, May 26, 2022, University of Chicago. PDF.
• John M. Lee, Introduction to Riemannian Manifolds, second edition, Graduate Texts in Mathematics 176, Springer, 2018. DOI.