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A physicist whose work explains how the convergence of neighboring freely falling paths depends on spacetime geometry and on their relative motion. The evolution of the expansion scalar of a geodesic congruence separates the effects of the shear tensor of a timelike congruence, rotation and curvature. The resulting focusing arguments are central to gravitational singularity theorems: they help establish what follows from assumptions about gravity without having to solve the Einstein field equations for an exactly symmetric universe.^[ref]

Born rigidity is a restriction on a body's motion: infinitesimally neighboring material points keep the same proper distance when compared in their local instantaneous rest frame. It concerns the projected spatial strain of a four-velocity field. Equivalently, its expansion scalar and shear tensor of a timelike congruence vanish. A rotating motion need not admit one global simultaneity slice for the entire body, so this definition is local.

For rectilinear motion with constant positive rear proper acceleration $a_0$, points at fixed proper distance $\xi$ have the metric in Rindler coordinatesTheir proper acceleration is $a(\xi )=a_0/(1+a_0\xi /c^2)$. Thus a front at $\xi =L>0$ accelerates less than the rear; the proper distance stays $L$ even while the laboratory length changes.^[ref]

This explains the kinematic issue in Bell's spaceship paradox: prescribing equal accelerations and fixed laboratory separation is a different motion. Born rigidity does not describe an instantaneous material response to a push; stresses in a real body still propagate at a finite speed of sound.

In four spacetime dimensions, the deformation tensor decomposes aswhere $\theta =B^{ab}{h}_{ab}$ is the expansion scalar, ${\sigma }_{ab}=B_{(ab)}-\theta h_{ab}/3$ is the symmetric trace-free shear tensor of a timelike congruence, and ${\omega }_{ab}=B_{[ab]}$ is the vorticity tensor of a timelike congruence.

For the normal flow with zero shift vector, $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ obeys $H_{\mathrm{loc}}=N^{-1}{\partial }_t\log a_{\mathrm{loc}}=-K/3$ in the negative extrinsic curvature of a spatial hypersurface convention. The expansion scalar is $\theta =-K=3H_{\mathrm{loc}}$.

Write $D=u^a{\nabla }_a$, $\theta ={\nabla }_a{u}^a$ for the expansion scalar, $a_b=u^a{\nabla }_a{u}_b$ for the four-acceleration, and $P_b{}^c={\delta }_b{}^c+u_b{u}^c$ for the spatial projector of a timelike congruence. The normalization of the four-velocity gives $u^b{a}_b=0$ and $u^b{\nabla }_a{u}_b=0$. The product rule and metric compatibility give the covariant derivative of the perfect fluid in general relativity stress-energy tensor:Its tensor contraction with the four-velocity isand its projection with the spatial projector of a timelike congruence isThus stress-energy conservation implies the relativistic perfect-fluid energy equation and the relativistic Euler equation:Conversely, the two displayed equations set both projections of $F_b$ to zero. Since $F_b=P_b{}^c{F}_c-u_b{u}^c{F}_c$, they imply $F_b=0$. This establishes equivalence, including points where $\rho +p=0$, without dividing by that quantity.

Set $h=\det h_{ij}$. Contract the extrinsic curvature with a negative shift with $h^{ij}$. The Jacobi determinant derivative formula and metric compatibility giveConsequentlyThe local volume element is $\sqrt{h}{d}^{3}x$, so for zero shift vector the proper-time derivative along the normal is $d/ds=N^{-1}{\partial }_t$ andHere $\theta$ is the expansion scalar. Writing $a_{\mathrm{loc}}=h^{1/6}$ defines a local linear scale from this volume, whence the local volume Hubble parameter isThe factor of three converts volume expansion into linear expansion, and $N$ converts coordinate time into proper time. This agrees with the usual Hubble parameter in a homogeneous FLRW metric.

For a volume-shape decomposition the unimodular spatial metric is ${\gamma }_{ij}=a_{\mathrm{loc}}^{-2}{h}_{ij}$, with determinant one. If instead the printed positive power $a_{\mathrm{loc}}^2{h}_{ij}$ is taken literally, its determinant is $a_{\mathrm{loc}}^{12}$; it is a conformal rescaling but not the unimodular shape metric. The trace calculation does not need that rescaling.

A timelike geodesic vector field is a smooth unit timelike field $u^a$ whose integral curves obey $u^b{\nabla }_b{u}^a=0$. Its expansion scalar, shear tensor of a timelike congruence, and vorticity tensor of a timelike congruence evolve according to the timelike Raychaudhuri equation.