Raychaudhuri Equations

The Raychaudhuri equations evolve the expansion of a smooth timelike or null congruence. They are local geometric identities: focusing follows only after the parameterization, twist, curvature convergence, and initial expansion have been fixed. The complete geometric derivation is in The Raychaudhuri Identity; the physical assumptions are developed in Raychaudhuri Equations for Timelike and Null Congruences.

Timelike rest-space equation

In $D$ dimensions let $u^au_a=-1$, and define

$$h_{ab}=g_{ab}+u_au_b, \qquad a^a=u^b\nabla_bu^a, \qquad B_{ab}=h_a{}^ch_b{}^d\nabla_du_c.$$

The projected derivative has the unique decomposition

$$\begin{aligned} B_{ab}&=\frac{\theta}{D-1}h_{ab}+\sigma_{ab}+\omega_{ab},\\ \theta&=h^{ab}B_{ab}=\nabla_au^a,\\ \sigma_{ab}&=B_{(ab)}-\frac{\theta}{D-1}h_{ab}, &\omega_{ab}&=B_{[ab]}. \end{aligned}$$

Thus $\sigma_{ab}$ is spatial, symmetric, and trace free, while $\omega_{ab}$ is spatial and antisymmetric. Brackets have unit weight, and the norm convention has no extra factor of one half: $\sigma^2=\sigma_{ab}\sigma^{ab}$ and $\omega^2=\omega_{ab}\omega^{ab}$. Because the rest metric is positive definite, both norms are nonnegative.

With the site curvature convention, the full timelike equation is

$$u^a\nabla_a\theta =-\frac{\theta^2}{D-1} -\sigma_{ab}\sigma^{ab} +\omega_{ab}\omega^{ab} -R_{ab}u^au^b +\nabla_aa^a.$$

For geodesics $a^a=0$; for a locally hypersurface-orthogonal congruence $\omega_{ab}=0$. These are independent restrictions. In particular, the nongeodesic contribution is the divergence $\nabla_aa^a$, not the acceleration magnitude $a_aa^a$. The geodesic specialization and its curvature-convergence sign are checked against Guth and Turner 2017, Lecture 14, printed pp. 59–61, Eqs. (14.10)–(14.18).

Null screen-space equation

Let $k^a$ be null and choose an auxiliary null vector $\ell^a$ with $k\cdot\ell=-1$. For $D\ge3$,

$$q_{ab}=g_{ab}+k_a\ell_b+\ell_ak_b, \qquad \mathcal B_{ab}=q_a{}^cq_b{}^d\nabla_dk_c$$

projects onto the positive-definite $(D-2)$-dimensional screen. Define

$$\mathcal B_{ab} =\frac{\theta}{D-2}q_{ab}+\sigma_{ab}+\omega_{ab}, \qquad \theta=q^{ab}\mathcal B_{ab},$$

with screen-trace-free symmetric shear and antisymmetric twist. If

$$k^b\nabla_bk^a=\kappa k^a,$$

then the nonaffine null equation is

$$k^a\nabla_a\theta =\kappa\theta -\frac{\theta^2}{D-2} -\sigma_{ab}\sigma^{ab} +\omega_{ab}\omega^{ab} -R_{ab}k^ak^b.$$

An affine parameter has $\kappa=0$. Both $\theta$ and $\kappa$ change when the generator is rescaled, so an affine-distance bound is meaningful only after fixing an affine normalization. Null hypersurface generators are twist free before caustics. The affine specialization and curvature-convergence term agree with Kontou and Sanders 2020, § 2, p. 8, Eqs. (16)–(17); the $+\kappa\theta$ term is retained before specializing.

Focusing conditions and equality cases

For a geodesic, twist-free timelike congruence satisfying $R_{ab}u^au^b\ge0$,

$$\frac{d\theta}{d\tau}\le-\frac{\theta^2}{D-1}.$$

If $\theta_0\lt0$, a focal point occurs within proper time $(D-1)/\lvert\theta_0\rvert$, provided the smooth congruence exists that far. Equality throughout an interval requires $\sigma_{ab}=0$ and $R_{ab}u^au^b=0$ in addition to the stated geodesic and twist-free hypotheses. The equality solution is

$$\theta(\tau) =\frac{\theta_0} {1+\theta_0(\tau-\tau_0)/(D-1)}.$$

For an affinely parametrized, twist-free null congruence satisfying $R_{ab}k^ak^b\ge0$, replace $D-1$ by $D-2$. Initial $\theta_0\lt0$ then gives the affine bound $(D-2)/\lvert\theta_0\rvert$; equality additionally requires zero shear and $R_{ab}k^ak^b=0$. No such bound follows from the displayed nonaffine equation without first controlling $\kappa$.

Curvature convergence is not itself an energy condition. In Einstein gravity with $D\gt2$,

$$\begin{aligned} R_{ab}k^ak^b&=8\pi G_D T_{ab}k^ak^b,\\ R_{ab}u^au^b &=8\pi G_D\left(T_{ab}u^au^b+\frac{T}{D-2}\right) -\frac{2\Lambda}{D-2}. \end{aligned}$$

Thus the null energy condition implies null convergence, whereas the matter strong energy condition implies timelike convergence only when the cosmological term is absent or separately controlled. A focal point is degeneration of the congruence’s transverse Jacobi map; it is not by itself curvature blow-up or geodesic incompleteness Kontou and Sanders 2020, § 5.1, printed pp. 37–42.

Discriminating checks

For an ingoing shear-free light bundle in $D$-dimensional Minkowski spacetime, let its area scale as $A\propto(\lambda_*-\lambda)^{D-2}$. Using $\theta=d\ln A/d\lambda$ gives

$$\theta=-\frac{D-2}{\lambda_*-\lambda}, \qquad \frac{d\theta}{d\lambda} =-\frac{\theta^2}{D-2},$$

so the null coefficient, sign, equality solution, and finite affine focusing time are all fixed while curvature remains zero.

For a comoving flow in a $D$-dimensional FLRW spacetime with scale factor $A(t)$, $u=\partial_t$, $H=\dot A/A$, and

$$\theta=(D-1)H, \qquad \sigma_{ab}=\omega_{ab}=a^a=0, \qquad R_{tt}=-(D-1)(\dot H+H^2).$$

Substitution reduces both sides to $(D-1)\dot H$, checking the timelike coefficient and Ricci sign. As a nongeodesic flat-space check, Rindler observers have zero expansion, shear, and twist but nonzero acceleration; $\nabla_aa^a=0$, so every term still vanishes even though $a_aa^a$ does not.

References

• Guth, Alan H., lectures with notes by Andrew P. Turner. 8.962 General Relativity, Spring 2017. Massachusetts Institute of Technology, 2017. Open PDF.
• Kontou, Eleni-Alexandra, and Ko Sanders. “Energy Conditions in General Relativity and Quantum Field Theory.” Classical and Quantum Gravity 37 (2020) 193001. DOI; arXiv record.