The Geodesic-Deviation Equation

Curvature becomes directly observable when two freely falling probes are compared. For an infinitesimal family of affinely parametrized geodesics, the separation vector obeys the Jacobi equation: its covariant relative acceleration is the Riemann tensor acting on the separation. The statement is local, first order in separation, and independent of coordinates; it does not describe finite bodies, forced trajectories, or self-interaction.

Required background. Affine and Variational Geodesics supplies affine geodesics and Jacobi fields; The Riemann Curvature Tensor supplies the curvature commutator and algebraic symmetries; Observers, Rest Spaces, and Measured Components supplies observer projections.

Helpful background. Events, Intervals, and Minkowski Geometry reviews timelike normalization; Tensor Fields and Natural Operations reviews vector fields and commutators; The Newtonian Gravitational Field and Potential provides the limiting comparison.

A two-parameter family of free-fall curves

Let $x^\mu(\tau,s)$ be a smooth surface in spacetime. Holding $s$ fixed gives a geodesic, while changing $s$ selects a neighboring geodesic. Define

$$u=\frac{\partial x}{\partial\tau}, \qquad \xi=\frac{\partial x}{\partial s}.$$

The assumptions are consequential: the connection is Levi-Civita, $\tau$ is an affine parameter shared across the family, every $s={\rm constant}$ curve satisfies $\nabla_u u=0$, and the coordinate vector fields commute, $[u,\xi]=0$. The vector $\delta s\,\xi$ is then the first-order displacement between the curves at equal $\tau$. This construction and its parametrization conditions are given explicitly in Reall 2013, §6.5, pp. 60–61, PDF and MIT 2017, §13.1, pp. 54–56, PDF.

The quantity $D\xi/D\tau=\nabla_u\xi$ is a covariant relative velocity. Its second derivative is a relative-acceleration vector, not automatically the second derivative of a finite radar distance. That distinction matters whenever the separation changes direction or the probes are no longer infinitesimally close.

Deriving the Jacobi equation

Torsion freedom and $[u,\xi]=0$ give $\nabla_u\xi=\nabla_\xi u$. Therefore

$$\begin{aligned} \frac{D^2\xi}{D\tau^2} &=\nabla_u\nabla_u\xi =\nabla_u\nabla_\xi u \\ &=\nabla_\xi\nabla_u u+R(u,\xi)u =R(u,\xi)u. \end{aligned}$$

The last equality uses the geodesic equation. With the site’s curvature convention, the equivalent component form is

$$\frac{D^2\xi^\mu}{D\tau^2} =-R^\mu{}_{\alpha\nu\beta} u^\alpha\xi^\nu u^\beta.$$

The minus sign in this index ordering follows from antisymmetry in the final pair; it is not an additional convention. The same derivation appears, with the commutator step visible, in Reall 2013, §6.5, pp. 61–62, PDF, Carroll 1997, §3, pp. 86–88, PDF, and MIT 2017, §13.1, pp. 55–57, PDF. Source formulas must be translated by their stated index order before signs are compared.

If $u^2=-1$ throughout the family, then

$$\nabla_u(u\mathbin{\cdot}\xi) =\frac12\nabla_\xi(u^2)=0.$$

Thus a separation initially orthogonal to $u$ remains orthogonal. A nonzero constant longitudinal component instead reflects how neighboring worldlines were synchronized and labeled; it is not a new tidal degree of freedom.

What the equation measures

Along one reference geodesic choose a parallel-propagated orthonormal tetrad with $e_{\hat0}=u$. For a spatial separation,

$$\frac{d^2\xi^{\hat i}}{d\tau^2} =-R^{\hat i}{}_{\hat0\hat j\hat0}\xi^{\hat j}.$$

This is the ideal gradiometer reading: the acceleration of either probe vanishes, yet their relative acceleration need not. A coordinate system can remove the connection on the reference worldline, but it cannot remove this curvature coefficient from every nearby trajectory. That is why tidal effects survive freely falling coordinates, as emphasized by the elementary laboratory argument in Reall 2013, §1.4, pp. 4–5, PDF.

The derivation is infinitesimal in the family parameter. It allows a nonzero initial $D\xi/D\tau$: the actual relative velocity $\delta s\,D\xi/D\tau$ is then of the same infinitesimal order as the separation $\delta s\,\xi$. It does not describe a nearby particle whose relative velocity remains finite as $\delta s\to0$; that requires a generalized deviation equation. Terms beyond first order in the family separation, finite-size forces, spin, applied force, and self-force also require extensions. Nor does one chosen observer recover every Riemann component: the equation samples only contractions containing two copies of that observer’s four-velocity.

Independent sign and limit checks

In flat spacetime $R=0$, so $D^2\xi/D\tau^2=0$: initially comoving inertial lines retain constant separation. In a stationary weak field with $g_{00}=-(1+2\Phi)$, one finds $R^i{}_{0j0}=\partial_i\partial_j\Phi$ at leading order. The component equation then becomes

$$\ddot\xi^i=-(\partial_i\partial_j\Phi)\xi^j,$$

exactly the difference between the Newtonian accelerations $-\boldsymbol\nabla\Phi$ at neighboring points. This check fixes the physical sign without relying on a memorized version of the Jacobi equation.

Exercises

1. Flat-space family. Suppose $R^\mu{}_{\nu\alpha\beta}=0$ and $D\xi^\mu/D\tau=v^\mu$ initially. Find $\xi^\mu(\tau)$ in a parallel frame.

Solution

The Jacobi equation gives $D^2\xi^\mu/D\tau^2=0$. In a parallel frame its solution is $\xi^\mu(\tau)=\xi^\mu(0)+\tau v^\mu$. Constant separation is the special case $v^\mu=0$.

2. Preserved orthogonality. Prove that $u\mathbin{\cdot}\xi=0$ remains true for a proper-time geodesic family.

Solution

Use $\nabla_u\xi=\nabla_\xi u$, metric compatibility, and $\nabla_u u=0$: $\nabla_u(u\mathbin{\cdot}\xi)=u\mathbin{\cdot}\nabla_\xi u=\tfrac12\nabla_\xi(u^2)=0$. Hence its initially zero value is preserved.

3. Diagnose a sign error. A proposed convention gives $\ddot\xi^i=+(\partial_i\partial_j\Phi)\xi^j$. Test it for $\Phi=-GM/r$ along a radial separation.

Solution

For the point-mass potential the radial Hessian eigenvalue is $-2GM/r^3$. The proposed sign predicts radial compression, whereas two radially separated freely falling probes separate because the inner one falls faster. The correct equation has the minus sign and gives $\ddot\xi^{\hat r}=+2GM\xi^{\hat r}/r^3$.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” arXiv:gr-qc/9712019 (1997), §3, pp. 85–88. PDF.
• Guth, Alan H., lectures; Andrew P. Turner, notes. 8.962 General Relativity, Spring 2017. Massachusetts Institute of Technology, 2017, §13.1, pp. 54–57. PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013, §§1.4 and 6.5, pp. 4–5 and 60–62. PDF.