Generalized Deviation for Accelerated and Null Curves

Relative acceleration contains more than curvature when the reference curves are forced, and it must be projected differently when the curves are null. For a commuting family of accelerated timelike curves, the Jacobi equation acquires the force-gradient term $+\nabla_\xi a$. For an affinely parametrized null congruence, only the two-dimensional screen projection represents transverse beam separation. These modifications separate genuine tidal curvature from acceleration control and longitudinal relabeling.

Required background. Affine and Variational Geodesics supplies geodesic families and affine parameters; The Riemann Curvature Tensor supplies the curvature commutator; Observers, Rest Spaces, and Measured Components supplies timelike and screen projections.

Helpful background. Events, Intervals, and Minkowski Geometry reviews timelike and null vectors; Tensor Fields and Natural Operations reviews Lie brackets and projected tensors; The Newtonian Gravitational Field and Potential supplies the distinction between a field and its gradient.

Accelerated neighboring worldlines

Let $x(\tau,s)$ be a smooth family with tangent $u=\partial_\tau x$, connecting vector $\xi=\partial_s x$, and $[u,\xi]=0$. Assume a torsion-free metric connection and use the same parameter convention throughout the family. Define the four-acceleration field on the family by

$$a\equiv\nabla_u u.$$

The commutator calculation used for geodesic deviation now gives

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

In the site’s component ordering,

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

The first term is tidal curvature; the second is differential forcing across the family. The derivation reduces to the standard equation in Reall 2013, §6.5, pp. 60–62, PDF and Carroll 1997, §3, pp. 86–88, PDF when $a=0$.

If $a$ depends on velocity, $\nabla_\xi a$ denotes the full covariant variation induced by the family, including the change in $u$, not merely a coordinate spatial derivative. For timelike curves, $a$ is physical four-acceleration when $\tau$ is proper time on every member of the family. With a different parameter, $\nabla_u u$ also contains normalization and parameter terms; it cannot be read directly as an accelerometer measurement.

Covariant separation and accelerated measuring axes

Even with proper time, equal parameter values need not remain simultaneous in the reference observer’s rest space. Normalization and $[u,\xi]=0$ imply $u\cdot\nabla_u\xi=\tfrac12\nabla_\xi(u^2)=0$, so

$$\frac{d}{d\tau}(u\cdot\xi)=a\cdot\xi.$$

An initially spatial connecting vector therefore need not stay spatial. This is why the covariant quantity $D^2\xi/D\tau^2$ cannot automatically be identified with the second derivative of laboratory distance.

For a concrete check, use a nonrotating Fermi–Walker triad along the reference curve, $\nabla_u e_{\hat i}=a_{\hat i}u$, and set $\xi^{\hat i}=e_{\hat i}\cdot\xi$. At an event where $u\cdot\xi=0$, differentiating the triad as well as the connecting vector gives

$$\frac{d^2\xi^{\hat i}}{d\tau^2} =-\mathcal E^{\hat i}{}_{\hat j}\xi^{\hat j} +e_{\hat i}\cdot\nabla_\xi a +a^{\hat i}a_{\hat j}\xi^{\hat j}.$$

The last term is an accelerated-frame contribution. A rotating triad or a different simultaneity prescription adds further terms. The covariant deviation equation remains correct; interpreting its projection as instrument data requires this extra step.

Why acceleration gradients can imitate tides

Flat spacetime provides the decisive check. There $R=0$, yet neighboring rockets with different thrust programs can approach or separate because $\nabla_\xi a\ne0$. Conversely, a covariantly uniform acceleration field has $\nabla_\xi a=0$ at the comparison event and produces no instantaneous differential acceleration, even though every accelerometer reads nonzero.

The distinction is operational. To infer curvature from an accelerated platform, one must measure its acceleration field, calibrate how it varies across the baselines, and account for both the projected $\nabla_\xi a$ term and the chosen axes and synchronization. Calling every relative acceleration “tidal gravity” would wrongly assign flat-spacetime control gradients to the Riemann tensor. For spinning, extended, charged, or self-interacting probes, $a$ may also depend on structure or external fields; the compact formula remains kinematically true for a smooth family, but a separate dynamical model is needed to predict the acceleration variation.

Null deviation and the screen space

Let $k^\mu$ be the tangent to an affinely parametrized null geodesic, $k^2=0$ and $\nabla_k k=0$. Choose an auxiliary null vector $n^\mu$ with $k\mathbin{\cdot}n=-1$, parallel-transport it along the reference ray, and define the screen projector

$$q^\mu{}_\nu =\delta^\mu{}_\nu+k^\mu n_\nu+n^\mu k_\nu.$$

The screen is two-dimensional and orthogonal to both $k$ and $n$. For a transverse light bundle, require $k\cdot\xi=0$ initially, as for neighboring generators of the same wavefront. This condition is preserved: $\nabla_k(k\cdot\xi)=\tfrac12\nabla_\xi(k^2)=0$ for the commuting null family. It excludes a component along $n$ that would otherwise enter the projected Jacobi equation as an additional term.

With a parallel-propagated orthonormal screen basis $e_A^\mu$, $A=1,2$, the physical transverse separation is $\zeta^A=e^A{}_\mu\xi^\mu$. Components along $k$ disappear on projection; adding $(a+b\lambda)k$ to a Jacobi field changes its affine-origin or affine-scale labeling without changing the transverse bundle. The orthogonality assumption and the screen construction are explicit in Perlick 2010, §2.3, printed pp. 13–14, Eqs. (9)–(18), PDF.

Projecting the null Jacobi equation gives

$$\frac{d^2\zeta^A}{d\lambda^2} =-\mathcal R^A{}_B\zeta^B, \qquad \mathcal R_{AB} =R_{\mu\alpha\nu\beta} e_A^\mu k^\alpha e_B^\nu k^\beta.$$

The screen tidal matrix is symmetric. Its trace is $R_{\mu\nu}k^\mu k^\nu$ and drives isotropic focusing; its trace-free Weyl part drives shear. It applies to exact null geodesics and to light rays within geometric optics. The second-order Jacobi equation itself remains regular at a caustic when curvature is smooth; the inverse Jacobi map, ray-density amplitude, and single-valued optical description can fail there. Diffraction, polarization-wave corrections, and global multiple-image questions require additional information.

Projection and parameter checks

Setting $a=0$ in the accelerated equation exactly recovers ordinary timelike geodesic deviation, verifying the force term’s normalization. Setting $R=0$ and $\nabla_\xi a=0$ gives affine relative motion, as special relativity requires. For null curves, rescaling $k\mapsto c k$ by a constant and $\lambda\mapsto\lambda/c$ changes both sides consistently; a position-dependent rescaling is non-affine and adds parameter terms.

The screen trace check is also discriminating: contracting $\mathcal R_{AB}$ over the two screen directions yields the null Ricci contraction, while the Weyl contribution remains trace-free. A purported optical tidal matrix that violates this split has an index, sign, or projection error.

Exercises

1. Flat-space differential thrust. At an event in Minkowski spacetime let $a^x=\kappa x$ for a one-dimensional rocket family. What is the separation equation?

Solution

Since $R=0$ and $\nabla_\xi a^x=\kappa\xi^x$ at that event, $D^2\xi^x/D\tau^2=\kappa\xi^x$. For $\kappa>0$ the rockets separate. The effect is entirely a thrust gradient, not spacetime curvature.

2. Geodesic reduction. Suppose every curve in the accelerated family has $a^\mu=0$. Which assumptions are still needed before writing the usual Jacobi equation?

Solution

The connection must be torsion-free, the family smooth, the geodesic parameter affine and consistently chosen, and the connecting and tangent fields must commute. Under these assumptions $\nabla_\xi a=0$ and the equation becomes $D^2\xi/D\tau^2=R(u,\xi)u$.

3. Null trace. Decompose $\mathcal R_{AB}=\tfrac12\mathcal R^C{}_C\delta_{AB}+\mathcal R^{\rm TF}_{AB}$. What do the two parts do to an initially circular beam?

Solution

The trace changes both screen directions equally and therefore changes the beam area at leading order. The trace-free part stretches one principal axis while compressing the other, producing shear. Rotation requires additional information about the congruence and is not generated by a symmetric instantaneous screen tidal matrix alone.

4. A rigid accelerated array. At the common moment of rest of a Rindler array in flat spacetime, let the proper acceleration along the spatial baseline be $a^{\hat x}=1/\rho$. Compare the projected covariant relative acceleration with the second derivative of separation components in the reference Fermi–Walker triad.

Solution

At this event, $e_{\hat x}\cdot\nabla_\xi a=-\xi^{\hat x}/\rho^2$ and curvature vanishes. Thus $e_{\hat x}\cdot D^2\xi/D\tau^2=-\xi^{\hat x}/\rho^2$. The accelerating triad adds $(a^{\hat x})^2\xi^{\hat x}=+\xi^{\hat x}/\rho^2$, giving $d^2\xi^{\hat x}/d\tau^2=0$ at the event. The negative covariant force-gradient term is therefore compatible with an instantaneously rigid array; it is not evidence of curvature.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” arXiv:gr-qc/9712019 (1997), §3, pp. 85–88. PDF.
• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7, 9 (2004; revised arXiv version 2010), §§2.1–2.4. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013, §6.5, pp. 60–62. PDF.