Geodesic congruence

A geodesic congruence is a family of nonintersecting geodesics filling an open spacetime region. Its derivative tensor describes the relative motion of neighboring geodesics.

Requiring each member to be a geodesic specializes a congruence of curves in general relativity.

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.

For a unit timelike vector field $u^a{u}_a=-1$, the tensor $h_{ab}=g_{ab}+u_a{u}_b$ projects onto the three-dimensional subspace orthogonal to $u^a$. In particular, $h_{ab}{u}^a=h_{ab}{u}^b=0$ and $g^{ab}{h}_{ab}=3$ in four spacetime dimensions.

For a unit geodesic vector field $u^a$, differentiation of $u^a{u}_a=-1$ and the geodesic equation show that $B_{ab}{u}^a=B_{ab}{u}^b=0$. The tensor therefore acts entirely in the local rest space.

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.

With the curvature convention $[{\nabla }_{\mu },{\nabla }_{\beta }]X_{\alpha }=-R^{\rho }{}_{\alpha \mu \beta }{X}_{\rho }$, the deformation tensor along a unit timelike geodesic field satisfiesTaking its spatial trace and inserting the kinematic decomposition of a timelike geodesic congruence leads to the timelike Raychaudhuri equation.

The expansion scalar of a unit timelike congruence with tangent $u^a$ is $\theta ={\nabla }_a{u}^a$. It gives the fractional rate of change of an infinitesimal comoving spatial volume.

The shear tensor is the symmetric trace-free spatial part of ${\nabla }_b{u}_a$. It changes the shape of an infinitesimal comoving volume without changing that volume to first order.

The vorticity tensor is the antisymmetric spatial part of ${\nabla }_b{u}_a$. It vanishes exactly when the timelike congruence is locally hypersurface orthogonal.

A focal point of a hypersurface along an orthogonal geodesic is a point where a nonzero normal Jacobi field arising from variations of the initial point vanishes. Beyond the first focal point, that geodesic no longer locally maximizes proper time from a spacelike initial hypersurface.

A spacetime is timelike-geodesically complete when every inextendible affinely parametrized timelike geodesic has proper time extending without a finite endpoint in both directions.

Timelike geodesic incompleteness means that at least one inextendible timelike geodesic reaches an endpoint after finite proper time. Singularity theorems commonly establish this conclusion without asserting that curvature diverges.

A null geodesic congruence has an affinely parametrized null tangent $k^a$ satisfying $k^a{k}_a=0$ and $k^b{\nabla }_b{k}^a=0$.

For affine null tangent $U$, choose an auxiliary null vector $N$ with $U\cdot N=-1$ and parallel transport it along the rays. The screen-space projector is $P_{ab}=g_{ab}+U_a{N}_b+N_a{U}_b$. The optical matrix of a null congruence describes evolution of screen-projected deviation vectors. Its trace is the null expansion, its symmetric trace-free part is the null shear, and its antisymmetric part is the null twist. In four dimensions $\hat{B}=\theta P/2+\hat{\sigma }+\hat{\omega }$.

For a parallel-transported screen and affine null geodesic tangent $U$, the optical matrix satisfiesIts trace gives the null Raychaudhuri equation. In a two-dimensional screen, symmetric trace-free matrices have scalar squares and anticommute with antisymmetric matrices. The trace-free symmetric and antisymmetric parts consequently yield the null shear and twist transport equations without additional quadratic trace-free terms.

In the Ingoing Eddington-Finkelstein coordinates metric $d{s}^2=-f(r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$, where $f=1-2M/r$, the tangent $U=-E{\partial }_r$ is affine when $E>0$ is constant along each ray. The displayed radial partner obeys $N^2=0$ and $U\cdot N=-1$. Since ${\Gamma }^v{}_{rv}={\Gamma }^a{}_{rr}=0$ and ${\Gamma }^r{}_{rv}=-f^{\prime }/2$, its derivative ${\partial }_r{N}^r=f^{\prime }/(2E)$ cancels the connection term, giving ${\nabla }_{U}N=0$. Its screen is the angular two-plane, with optical tensor ${\hat{B}}_{AB}=-Er{\gamma }_{AB}$, null expansion $-2E/r$, and zero null shear and null twist.

If a null vector tangent satisfies ${\nabla }_{K}K=\kappa K$, then $U=hK$ is affine when $K(\log h)=-\kappa$. Choose $h=1$ on the initial transverse surface. The screen-space projector kills derivative terms proportional to $K$, so the initial optical tensor and its null expansion agree with those of $K$. In ingoing spherical coordinates, $K={\partial }_v+(f/2){\partial }_r$ has $\kappa =f^{\prime }/2$.

For an affine null geodesic congruence with future tangent $U$, choose a future unit timelike vector $T$ on a transverse section, put $E=-U\cdot T$, and set $N=T/E-U/(2E^2)$ there. It has $N^2=0$ and $U\cdot N=-1$. Parallel transport along each generator preserves these contractions by metric compatibility and ${\nabla }_{U}U=0$. The construction is local up to caustics and supplies the auxiliary normal for a screen-space projector.

If $k_a={\nabla }_{a}S$ is a null gradient, symmetry of the Hessian gives $k^a{\nabla }_a{k}_b=k^a{\nabla }_b{k}_a=\frac{1}{2}{\nabla }_b(k^2)=0$. Its integral curves are therefore affinely parametrized null geodesics, rather than merely null curves with a nonaffinity term.

Given null vectors $k$ and $n$ with $k\cdot n=-1$, the screen-space projector isIt annihilates $k$ and $n$ and, in $D$ spacetime dimensions, projects onto the $(D-2)$-dimensional spacelike space transverse to them.

For $B_{ab}={\nabla }_b{k}_a$, the optical tensor is the screen projection ${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{B}_{cd}$. Its trace, symmetric trace-free part, and antisymmetric part are the expansion, shear, and twist of the null congruence.

The null expansion is $\theta =P^{ab}{\hat{B}}_{ab}$. It is the logarithmic derivative of an infinitesimal transverse area: $\theta =A^{-1}dA/d\lambda$.

For a four-dimensional spherical metric $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, a radial null vector normal $K$ to a round sphere has null expansion ${\theta }_K=\frac{1}{2}{q}^{AB}{\mathcal{L}}_K{q}_{AB}=2K(r)/r$, where $q_{AB}=r^2{\gamma }_{AB}$. The angular metric changes only by a common scale, so radial congruences have zero null shear; hypersurface orthogonality also gives zero null twist. The convention here is the trace, not the screen-averaged expansion.

For a four-dimensional spherically symmetric metric $d{s}^2=-2fdUdV+r^{2}d{\Omega }^2$, the affine null gradients are $f^{-1}{\partial }_V$ and $f^{-1}{\partial }_U$. Their null expansions are $2r_{,V}/(fr)$ and $2r_{,U}/(fr)$. Their null shear and null twist vanish because the transverse metric changes only by a common scale and the covectors are exact gradients.

The Null Raychaudhuri equation and Einstein field equations give ${\partial }_U(f^{-1}{r}_{,U})=-r{R}_{UU}/(2f)=-4\pi r{T}_{UU}/f$, with an exchanged-coordinate identity in $V$. Under the null energy condition, each rescaled radial derivative is nonincreasing along its corresponding future null direction.

In a regular double-null region satisfying the null energy condition, an initially negative $r_{,U}$ on a Cauchy hypersurface remains negative in its future domain of dependence. If both spherical null expansions are negative at one sphere, the exchanged-coordinate focusing inequality preserves the other sign along its future outgoing null direction. Every later sphere that still exists there remains a trapped surface.

The null shear is the symmetric trace-free part ${\hat{\sigma }}_{ab}={\hat{B}}_{(ab)}-\theta P_{ab}/(D-2)$. It changes the shape of a transverse beam while preserving its area to first order.

For an affine null congruence in four spacetime dimensions with a parallel-transported screen, the displayed equation follows by taking the symmetric trace-free part of the Sachs optical equations. The Ricci contribution to the optical tidal tensor is $P_{ab}{R}_{cd}{U}^c{U}^d/2$, so it changes the expansion but contributes no screen-trace-free source. The projected Weyl tensor supplies the tidal source of shear.

The null twist is ${\hat{\omega }}_{ab}={\hat{B}}_{[ab]}$. It vanishes for null generators orthogonal to a hypersurface by the Frobenius theorem.

For an affine null geodesic congruence, $B_{ab}={\nabla }_b{U}_a$ annihilates $U$ in both slots. Relative to a null partner, its antisymmetric part is its screen projection plus a term of the form $U_{[a}{q}_{b]}$. Hence zero null twist implies $U\wedge dU=0$, and Frobenius theorem makes $U_a$ locally proportional to a hypersurface gradient. Conversely, if $U_a=h{\nabla }_{a}S$, its antisymmetric derivative has a factor of $U$, so the screen projection vanishes. Geodesicity is essential to the converse from vanishing screen twist to the full Frobenius condition.

For an affinely parametrized four-dimensional null geodesic congruence, the screen-projected evolution of the null twist isConsequently zero twist remains zero along every generator on which the optical description remains regular.

For an affinely parametrized null congruence in $D$ spacetime dimensions,

For an affinely parametrized twist-free null geodesic congruence, let $A$ be local transported screen area and ${\sigma }^2={\hat{\sigma }}_{AB}{\hat{\sigma }}^{AB}/2$. The Null Raychaudhuri equation gives ${\theta }^{\prime }=-{\theta }^2/2-2{\sigma }^2-R_{ab}{k}^a{k}^b$ and $A^{\prime }/A=\theta$. Combining these gives the displayed equation. Positive shear focuses the beam and has a negative sign. For nonuniform expansion this is a local area identity, not the same differential equation for the entire integrated horizon area.

In flat spacetime take a twist-free beam's two screen Jacobi factors to be $1+\lambda$ and $1-\lambda$. They solve the zero-curvature geodesic deviation equation. Their product is the displayed area, and subtracting the trace from their optical matrix gives the displayed nonnegative shear. Direct differentiation yields $(A^{1/2}{)}^{\prime \prime }=-(1-{\lambda }^2{)}^{-3/2}=-{\sigma }^2{A}^{1/2}$, an explicit check of the focusing sign.

For a hypersurface-orthogonal null congruence under the null energy condition in $D$ spacetime dimensions, an initially negative expansion ${\theta }_0$ diverges to $-\infty$ within affine distance at most $(D-2)/|{\theta }_0|$.

A conjugate point to a spacelike surface occurs where normal geodesics develop a focal degeneracy: the transverse Jacobi field map from initial points on the surface ceases to be invertible. Collapse of the transverse area makes the null expansion diverge. Beyond a first such point, the normal null geodesic cannot continue as a generator of the achronal boundary of the surface's future.