Congruences and Kinematical Decomposition

A congruence replaces one distinguished curve by a smoothly varying family of curves. Its derivative records how neighboring worldlines or light rays separate, distort, and rotate. The central achievement of this chapter is to decompose that derivative into expansion, shear, and vorticity, then derive the exact evolution identities obeyed by those pieces. These statements are purely geometric: no field equation, energy condition, or singularity theorem is assumed.

The timelike and null constructions look similar only after their different transverse spaces are handled correctly. A unit timelike tangent has an ordinary positive-definite rest space. A null tangent lies inside its own orthogonal complement, so its transverse geometry is a quotient or a two-dimensional screen chosen with an auxiliary null direction.

Helpful background. Calculus and Linear Algebra for Relativity supplies traces, determinants, and matrix evolution; The Riemann Curvature Tensor fixes the curvature operator; Affine and Variational Geodesics supplies affine geodesics, affine parameters, and the curve-variation setup; Jacobi Fields and Geodesic Deviation supplies the connecting fields, deviation equation, and Jacobi maps used along the chapter’s null-optics route; Frames, Coframes, and Anholonomy supplies transverse frame components.

Families of neighboring curves

Let $u^\mu$ be the future-directed unit tangent to a timelike congruence. The rest-space projector is $h_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu$. Projecting both indices of $\nabla_\nu u_\mu$ gives the deformation tensor $B_{\mu\nu}$. Its trace, symmetric trace-free part, and antisymmetric part are respectively the expansion $\theta$, shear $\sigma_{\mu\nu}$, and vorticity $\omega_{\mu\nu}$. The remaining piece of the full derivative is the acceleration $a^\mu=u^\nu\nabla_\nu u^\mu$.

For a null congruence with tangent $k^\mu$, there is no unit normalization. An auxiliary null vector $\ell^\mu$ satisfying $k\cdot\ell=-1$ defines

$$q_{\mu\nu}=g_{\mu\nu}+k_\mu\ell_\nu+\ell_\mu k_\nu,$$

which projects onto a two-dimensional screen in four spacetime dimensions. Rescaling $k\mapsto e^f k$ and $\ell\mapsto e^{-f}\ell$ leaves $q_{\mu\nu}$ unchanged but rescales the optical tensor. Any numerical expansion therefore belongs to a specified generator normalization and parameter.

Choose a route

• Start with Timelike Congruences and Spatial Projection to construct the rest-space derivative and relative-separation equation.
• Continue to Expansion, Shear, and Vorticity for the irreducible decomposition and its local volume, shape, and rotation meanings.
• Read The Raychaudhuri Identity to see how curvature and acceleration govern the expansion without importing gravitational dynamics.
• For light rays, begin instead with Null Congruences and Screen Geometry and its normalization freedom.
• Then use Optical Tensors and Sachs Evolution to relate a Jacobi map, beam area, and curvature-driven distortion.
• Finish with Caustics, Focal Points, and Volume Evolution to distinguish failure of congruence coordinates from singular spacetime curvature.

The first three pages form a timelike route. The last three form a null-optics route. Readers concerned mainly with hypersurface normals need the projector and vorticity criteria; readers preparing for lensing need the screen, Jacobi-map, and caustic pages.

Four flat-spacetime checks

Flat curvature does not force all kinematical quantities to vanish. In an inertial Cartesian congruence, $u=\partial_t$, every derivative of $u$ vanishes. In the future light cone, the Milne congruence $u=\partial_\tau$ has

$$\theta=\frac{3}{\tau}, \qquad \sigma_{\mu\nu}=\omega_{\mu\nu}=a^\mu=0.$$

Its expansion decreases even though the Riemann tensor is zero. Uniformly rotating observers provide the complementary check: their congruence is Born-rigid, so expansion and shear vanish, while vorticity and acceleration do not. Rindler observers are hypersurface orthogonal and Born-rigid; their nonzero acceleration is essential to the Raychaudhuri balance.

For outgoing radial null rays in Minkowski spacetime, an affine tangent can be chosen so that $\theta=2/r$, while shear and twist vanish. The beam expands because the area of a sphere is proportional to $r^2$, not because spacetime is curved. Reversing the rays makes the same area collapse at the origin, producing a caustic in flat spacetime.

These examples separate curvature, coordinates, and the choice of congruence. They also supply independent checks on every coefficient in the timelike and null trace decompositions. The convention-controlled derivations follow Kar and SenGupta 2007, §§II.A–II.E, pp. 5–14, with curvature signs translated to the convention used here.

Convention-sensitive distinctions

Timelike vectors are normalized by $u^\mu u_\mu=-1$, and their transverse dimension is three. Null generators have boost freedom, and their screen dimension is two. Consequently the trace terms in four dimensions are $\theta h_{\mu\nu}/3$ and $\theta q_{\mu\nu}/2$, respectively.

Expansion is a fractional volume rate, not a coordinate divergence detached from a chosen tangent field. Vorticity is the obstruction to local hypersurface orthogonality, not simply the angular velocity of coordinate lines. A divergence of $\theta$ signals degeneration of the chosen separation map; it does not by itself establish curvature blow-up, geodesic incompleteness, or inextendibility.

The Raychaudhuri and Sachs equations in this chapter are kinematical identities. Replacing their Ricci terms by matter variables, imposing energy inequalities, or drawing global conclusions belongs to later treatments of gravitational dynamics and global structure. Perlick’s Jacobi-tensor formulation makes the distinction between optical evolution and lensing interpretation precise Perlick 2004, §§2.3–2.5.

Check your understanding

Inside the future cone of Minkowski spacetime, write

$$ds^2=-d\tau^2+\tau^2\gamma_{ij}dx^i dx^j,$$

where $\gamma_{ij}$ is the unit hyperbolic metric. For $u=\partial_\tau$, determine $B_{ij}$ and $\theta$, then check the geodesic Raychaudhuri identity.

Solution

The induced rest metric is $h_{ij}=\tau^2\gamma_{ij}$. Since the shift vanishes and $u$ is geodesic,

$$B_{ij}=\frac12\partial_\tau h_{ij} =\tau\gamma_{ij} =\frac1{\tau}h_{ij}.$$

Thus $\theta=h^{ij}B_{ij}=3/\tau$, while the shear and vorticity vanish. Flatness gives $R_{\mu\nu}u^\mu u^\nu=0$, so

$$\dot\theta=-\frac{3}{\tau^2} =-\frac{\theta^2}{3},$$

which is exactly the timelike identity. The changing spatial metric describes the separation of inertial rays issuing from one event; it is not evidence of spacetime curvature.

What this chapter enables

After completing either route, you should be able to construct the correct transverse projector, decompose a congruence derivative, derive its trace evolution, track null rescalings, and diagnose a caustic through a Jacobi determinant. These tools feed directly into Hypersurfaces, Extrinsic Curvature, and Embeddings and later into geometric optics. Physical focusing assumptions, Einstein-equation substitutions, and global singularity conclusions are deliberately left to their proper volumes.

References

• Kar, Sayan, and Soumitra SenGupta. “The Raychaudhuri Equations: A Brief Review.” Pramana 69 (2007), 49–76. arXiv:gr-qc/0611123.
• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7, 9 (2004). doi:10.12942/lrr-2004-9.