Raychaudhuri Equations for Timelike and Null Congruences

The Raychaudhuri equations are local kinematic identities for the expansion of smooth timelike or null congruences. They separate isotropic expansion, shear, vorticity, acceleration or nonaffinity, and Ricci curvature. Focusing theorems arise only after additional signs are imposed; no energy condition or global singularity conclusion is built into the identities themselves.

Required background. Congruence geometry supplies expansion, shear, twist, and spatial projection. Riemann curvature supplies the covariant-derivative commutator.

Helpful background. Special-relativistic causal normalization and smooth tensor calculus help with the contractions; global causal structure distinguishes later incompleteness results from these local identities.

Timelike deformation tensor

Let $u^a$ be a smooth future-directed unit timelike field in $D\ge2$ dimensions, $u^au_a=-1$, with acceleration $a^a=u^b\nabla_bu^a$. The rest-space projector is $h_{ab}=g_{ab}+u_au_b$. Decompose the spatial derivative as

$$h_a{}^ch_b{}^d\nabla_du_c =\frac{\theta}{D-1}h_{ab}+\sigma_{ab}+\omega_{ab},$$

where $\theta=\nabla_au^a$, $\sigma_{ab}$ is spatial symmetric trace-free, and $\omega_{ab}$ is spatial antisymmetric. Taking a derivative along $u^a$, commuting covariant derivatives with

$$[\nabla_a,\nabla_b]V^c=R^c{}_{dab}V^d,$$

and tracing the resulting matrix Riccati equation gives

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

Here $\dot\theta=u^a\nabla_a\theta$. The signs follow from the site curvature convention and positive spatial norms. MIT derives the geodesic four-dimensional form and its focusing inequality Guth–Turner 2017, Lecture 14, pp. 59–61, Eqs. (14.10)–(14.18), PDF.

For geodesics $a^a=0$. For a congruence orthogonal to a family of spacelike hypersurfaces, Frobenius’ theorem gives $\omega_{ab}=0$. Only after both restrictions can the shear and Ricci terms be used directly as nonpositive contributions under timelike convergence.

Proof from the matrix equation

In the geodesic case define $B_{ab}=\nabla_bu_a$ on the rest space. Differentiate along the flow:

$$u^c\nabla_cB_{ab} =-B_{ac}B^c{}_b-R_{acbd}u^cu^d.$$

The first term follows by differentiating $u^c\nabla_cu_a=0$; the second follows by commuting derivatives. Trace with $h^{ab}$. The algebraic decomposition gives

$$B_{ab}B^{ba} =\frac{\theta^2}{D-1} +\sigma_{ab}\sigma^{ab} -\omega_{ab}\omega^{ab}.$$

Antisymmetry creates the minus sign in $B_{ab}B^{ba}$ and hence the plus sign for vorticity in Raychaudhuri’s equation. Contracting the curvature term yields $R_{ab}u^au^b$. This proves the geodesic identity before any inequality is invoked.

Kontou and Sanders state the timelike and null identities in the energy-condition setting and make the convergence assumptions explicit Kontou and Sanders 2020, § 2, p. 7, Eqs. (16)–(17), PDF.

Null screen-space equation

Let $k^a$ be a null tangent. Choose an auxiliary null vector $\ell^a$ with $k\cdot\ell=-1$ and screen projector

$$q_{ab}=g_{ab}+k_a\ell_b+\ell_ak_b.$$

Assume $D\gt2$, so the screen has positive dimension $D-2$. Project $\nabla_bk_a$ and decompose it into $\theta q_{ab}/(D-2)$, screen shear $\sigma_{ab}$, and screen twist $\omega_{ab}$. For affine generators, $k^b\nabla_bk^a=0$, the trace equation is

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

For a nonaffine parameter satisfying $k^b\nabla_bk^a=\kappa k^a$, the right-hand side additionally contains $+\kappa\theta$ under this definition of expansion. The numerical value of null expansion rescales with $k^a$, so focusing bounds require a fixed affine normalization. Surface-orthogonal null normals have zero screen twist before caustics.

In four dimensions the coefficients are $1/3$ for timelike congruences and $1/2$ for null congruences. Confusing the rest-space dimension with the screen dimension is a common source of wrong focusing times.

Expansion is also tied directly to volume. If $\mathcal V$ is the infinitesimal spatial volume transported by a timelike congruence, then $\theta=d\ln\mathcal V/d\tau$ before caustics. For a null bundle with screen area $A$, $\theta=d\ln A/d\lambda$. Shear changes shape at fixed first-order volume, whereas expansion changes the determinant. This supplies an independent derivation of the trace: take the determinant of the Jacobi map, differentiate its logarithm, and then differentiate once more using the matrix evolution equation.

The auxiliary null vector $\ell^a$ is not unique. Changes of $\ell^a$ that preserve $k\cdot\ell=-1$ alter the representative screen basis, but the projected expansion and shear of a fixed null hypersurface transform consistently. Physical statements should therefore use the screen geometry or area evolution, not unprojected components that depend on the auxiliary choice.

Independent limiting checks

For a shear-free radial null bundle in four-dimensional Minkowski spacetime, cross-sectional area scales as $A\propto\lambda^2$. Hence $\theta=d\ln A/d\lambda=2/\lambda$. Raychaudhuri gives

$$\frac{d\theta}{d\lambda}=-\frac{2}{\lambda^2} =-\frac12\theta^2,$$

with shear, twist, and Ricci curvature zero. This checks the null coefficient and sign.

For a homogeneous isotropic timelike congruence in $D=4$, $\theta=3H$ and $\sigma=\omega=a=0$. The equation reduces to $3\dot H=-3H^2-R_{ab}u^au^b$, consistent with the geometric Ricci contraction for that flow. A rotating congruence in flat spacetime supplies the opposite check: positive $\omega_{ab}\omega^{ab}$ can oppose convergence, so the vorticity-free hypothesis in a focusing theorem is substantive.

Scope of the identity

The decomposition is valid only while the congruence and screen remain smooth. At a caustic, the vector field becomes multivalued and $\theta$ can diverge; one then continues individual geodesics, not the same smooth congruence chart. Raychaudhuri predicts this breakdown under suitable inequalities but does not itself imply curvature blow-up, geodesic incompleteness, or any global causal theorem.

Regularity is part of that claim. The derivation assumes a Levi-Civita connection and enough differentiability to commute derivatives and evolve the deformation tensor. Torsion would add terms, and a distributional curvature layer requires a weak or junction formulation. Likewise, a forced timelike congruence is covered only when the acceleration divergence is retained rather than silently set to zero.

Exercises

In $D$ dimensions, a flat shear-free twist-free null bundle has area $A\propto\lambda^{D-2}$. Verify the affine null Raychaudhuri equation.

Solution

$\theta=d\ln A/d\lambda=(D-2)/\lambda$, so $d\theta/d\lambda=-(D-2)/\lambda^2$. The only surviving term on the right is

$$-\frac{\theta^2}{D-2} =-\frac{(D-2)^2}{(D-2)\lambda^2} =-\frac{D-2}{\lambda^2},$$

which agrees exactly.

References

• Guth, Alan H., lectures with notes by Andrew P. Turner. 8.962 General Relativity, Spring 2017. MIT (2017). 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.