Raychaudhuri Equation and Geodesic Focusing

Derivation

Raychaudhuri Equation and Geodesic Focusing

D-401 Home PU-403 Threads energy · fields · force Depends on Parallel Transport and Geodesic Deviation, Riemann Tensor from the Covariant Commutator

Statement

For a smooth congruence of timelike geodesics with unit tangent field \(u^a\) (\(u^au_a=-1\), \(u^b\nabla_b u^a=0\)), the expansion scalar \(\theta=\nabla_a u^a\) obeys the Raychaudhuri equation \(\dfrac{d\theta}{d\tau}=-\tfrac{1}{3}\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b\); when the congruence is hypersurface-orthogonal (\(\omega_{ab}=0\)) and the Ricci term is non-negative (strong energy condition), every source term is non-positive, so an initially converging bundle (\(\theta_0<0\)) reaches a caustic \(\theta\to-\infty\) within proper time \(\tau\le 3/|\theta_0|\).

Why it matters

The Raychaudhuri equation is the engine of the Penrose–Hawking singularity theorems. It converts a local statement about matter (an energy condition on \(R_{ab}u^au^b\)) into a global statement about geometry (the inevitable convergence of nearby geodesics), showing that gravity, on average, focuses. This focusing produces conjugate points, and conjugate points forbid the existence of maximal-length curves, which is precisely the contradiction that forces geodesic incompleteness.

Beyond singularities the same equation governs the growth of cosmic structure (its trace is the second Friedmann equation), the area increase of black-hole horizons (the null version underlies the area theorem), and the behaviour of gravitational-lensing bundles. It is one equation that links energy, the tidal field, and the kinematics of a flowing family of worldlines.

Assumptions

The tangent field forms a smooth geodesic congruence.If worldlines cross or the field is non-differentiable, \(B_{ab}=\nabla_b u_a\) is ill-defined and \(\theta\) diverges spuriously rather than physically; if the curves are accelerated, an extra term \(\nabla_a a^a\) with \(a^a=u^b\nabla_bu^a\) reappears on the right. The tangents are unit-normalised, \(u^au_a=-1\).Drop it and \(u^a\nabla_b u_a\ne0\), so \(B_{ab}\) is no longer purely transverse and the tidy trace decomposition into \(\theta,\sigma,\omega\) fails. Spacetime is four-dimensional with a Levi-Civita (torsion-free, metric) connection.The \(\tfrac13\) coefficient is \(1/(n-1)\) for \(n\) dimensions; torsion adds antisymmetric-connection terms and spoils the Ricci-identity step used to introduce \(R_{ab}\). The strong energy condition \(R_{ab}u^au^b\ge0\) (equivalently \((T_{ab}-\tfrac12Tg_{ab})u^au^b\ge0\)) holds for the focusing corollary only.Fields that violate it — a cosmological constant, inflaton, or dark energy — make the Ricci term defocusing, so bundles can diverge and no caustic is guaranteed.

Derivation

\[ B_{ab}\equiv\nabla_b u_a,\qquad B_{ab}u^a=\tfrac12\nabla_b(u^au_a)=0,\qquad B_{ab}u^b=u^b\nabla_b u_a=0. \]

Define the tangent gradient. Normalisation kills the first contraction; the geodesic equation kills the second, so \(B_{ab}\) lives entirely in the 3-space orthogonal to \(u^a\). A

\[ h_{ab}=g_{ab}+u_au_b,\quad h^a{}_a=3;\qquad B_{ab}=\tfrac13\theta\,h_{ab}+\sigma_{ab}+\omega_{ab}. \]

Split the transverse tensor into trace \(\theta=h^{ab}B_{ab}=\nabla_au^a\), symmetric-traceless shear \(\sigma_{ab}=B_{(ab)}-\tfrac13\theta h_{ab}\), and antisymmetric twist \(\omega_{ab}=B_{[ab]}\). Purely algebraic and unique. A

\[ u^c\nabla_c B_{ab}=u^c\nabla_c\nabla_b u_a=u^c\big(\nabla_b\nabla_c u_a+R_{cba}{}^{d}u_d\big). \]

Transport \(B_{ab}\) along the flow and swap the outer derivatives with the Ricci identity \([\nabla_c,\nabla_b]u_a=R_{cba}{}^{d}u_d\) — this is where curvature enters. C

\[ u^c\nabla_b\nabla_c u_a=\nabla_b(u^c\nabla_c u_a)-(\nabla_b u^c)(\nabla_c u_a)=-B_{ac}B^{c}{}_{b}. \]

The geodesic equation \(u^c\nabla_c u_a=0\) annihilates the first bracket; the remaining product is \(B\!\cdot\!B\). Hence \(u^c\nabla_cB_{ab}=-B_{ac}B^{c}{}_{b}+R_{cba}{}^{d}u^cu_d\). B

\[ \frac{d\theta}{d\tau}=g^{ab}\,u^c\nabla_cB_{ab}=-B_{ac}B^{ca}-R_{cd}\,u^cu^d. \]

Trace with \(g^{ab}\) (metric-compatible, so it passes through \(\nabla_c\)); the Riemann term contracts to the Ricci tensor \(g^{ab}R_{cba}{}^{d}u_d=-R_{cd}u^d\) via its symmetries. B

\[ B_{ac}B^{ca}=\Big(\tfrac13\theta h_{ac}+\sigma_{ac}+\omega_{ac}\Big)\Big(\tfrac13\theta h^{ca}+\sigma^{ca}+\omega^{ca}\Big)=\tfrac13\theta^2+\sigma_{ab}\sigma^{ab}-\omega_{ab}\omega^{ab}. \]

Insert the decomposition. Symmetric×antisymmetric cross terms vanish; \(h_{ab}h^{ab}=3\) and \(h_{ab}\sigma^{ab}=0\); the twist enters with a sign because \(\omega^{ca}=-\omega^{ac}\). B

\[ \boxed{\;\frac{d\theta}{d\tau}=-\tfrac13\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b\;} \]

Substitute Step 6 into Step 5. This is the Raychaudhuri equation for a timelike geodesic congruence. A

\[ \omega_{ab}=0\ (\text{Frobenius}),\ \ \sigma_{ab}\sigma^{ab}\ge0,\ \ R_{ab}u^au^b\ge0\ (\text{SEC})\ \Longrightarrow\ \frac{d\theta}{d\tau}\le-\tfrac13\theta^2. \]

Hypersurface-orthogonality forces zero twist and it stays zero along the flow; shear-squared is manifestly non-negative; the strong energy condition fixes the Ricci sign. All three surviving terms then oppose expansion. C

\[ \frac{d}{d\tau}\!\left(\frac{1}{\theta}\right)=-\frac{1}{\theta^2}\frac{d\theta}{d\tau}\ge\frac13\ \Longrightarrow\ \frac{1}{\theta(\tau)}\ge\frac{1}{\theta_0}+\frac{\tau}{3}. \]

Divide the inequality by \(-\theta^2<0\) (flips it) and integrate. If \(\theta_0<0\) the right side reaches zero at finite \(\tau\), forcing \(\theta\to-\infty\): a caustic within \(\tau\le 3/|\theta_0|\). A

Result

\[ \frac{d\theta}{d\tau}=-\tfrac13\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b,\qquad \frac{1}{\theta(\tau)}\ge\frac{1}{\theta_0}+\frac{\tau}{3}. \]

Reading. The fractional rate of change of a comoving 3-volume, \(\theta\), is driven down by three effects that all focus — its own square (self-focusing of the flow), shear (anisotropic tidal stretching), and the Ricci curvature sourced by matter — and driven up only by rotation, which centrifugally resists collapse. Remove the rotation and satisfy the energy condition, and convergence, once begun, runs away to a caustic in bounded proper time.

Units check. With \(c=1\), \(\theta=\nabla_au^a\) carries dimension (length)\(^{-1}\); \(d\theta/d\tau\), \(\theta^2\), \(\sigma^2\), \(\omega^2\), and \(R_{ab}u^au^b\) are each (length)\(^{-2}\), so every term matches. Restoring \(c\), \(\theta\) is (time)\(^{-1}\) and the bound \(3/|\theta_0|\) is a proper time.

Limiting cases

Shear- and twist-free, vacuum (\(\sigma=\omega=0\), \(R_{ab}u^au^b=0\)): \(d\theta/d\tau=-\tfrac13\theta^2\), giving \(\theta(\tau)=\theta_0/(1+\theta_0\tau/3)\) — pure geometric focusing.
Isotropic homogeneous flow (FRW): \(\theta=3H\), \(\sigma=\omega=0\); the trace equation becomes the acceleration (second Friedmann) equation \(\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p)\).
Rigid rotation dominating (\(\omega^2\gg\theta^2,\sigma^2,R_{ab}u^au^b\)): \(d\theta/d\tau>0\), collapse is halted — the Gödel-type centrifugal barrier.
Null congruence: replace \(\tfrac13\to\tfrac12\) (2D screen space) and SEC by the null energy condition \(R_{ab}k^ak^b\ge0\); underlies the black-hole area theorem.

Breaks when

Energy condition violated. A cosmological constant or inflaton gives \(R_{ab}u^au^b=-8\pi G\,\rho_\Lambda<0\); the Ricci term now defocuses, \(\theta\) can stay positive forever, and no caustic forms — this is exactly how inflation and dark-energy acceleration evade the theorem.
Rotation present. With \(\omega_{ab}\ne0\) the positive \(+\omega^2\) term can balance the focusing terms; a hypersurface-orthogonal assumption is required, and any twist invalidates the simple inequality \(d\theta/d\tau\le-\tfrac13\theta^2\).
At the caustic itself. When \(\theta\to-\infty\) the geodesics cross, \(B_{ab}=\nabla_bu_a\) is singular, and the equation describes a coordinate/congruence breakdown, not necessarily a curvature singularity — the focusing is real but its interpretation requires the full singularity theorem plus global causal structure.

Failure modes

Sign-flipping the twist. Writing \(-\omega^2\) instead of \(+\omega^2\); the antisymmetry \(\omega^{ca}=-\omega^{ac}\) in Step 6 is what makes rotation defocusing — getting the sign wrong reverses the physics.
Using the wrong energy condition. Applying the weak or dominant condition to the timelike equation. Focusing needs the strong condition \(R_{ab}u^au^b\ge0\), i.e. \(\rho+3p\ge0\), not \(\rho\ge0\).
Confusing \(\tfrac13\) with \(\tfrac12\). Using the timelike coefficient for null bundles (or vice versa); it is \(1/(D-2)\) with \(D-2\) the dimension of the transverse screen.
Assuming a caustic is a curvature singularity. A caustic where geodesics cross is a conjugate point; the metric can be perfectly regular there (e.g. focal points of a flat-space light bundle).
Dropping the acceleration term illegitimately. Applying the geodesic form to a fluid with pressure gradients, where worldlines are accelerated and \(\nabla_a a^a\) survives.

Discussion

The Raychaudhuri equation is a Riccati equation for \(\theta\): the quadratic \(-\tfrac13\theta^2\) self-term is what makes focusing catastrophic rather than gradual. A converging bundle feeds its own convergence, and the blow-up in finite proper time is the generic behaviour of \(\dot y=-y^2\). Shear only worsens this — it appears squared and negative — which encodes the physical fact that tidal distortion, however oriented, ultimately aids collapse. Only rotation, entering with the opposite sign, can arrest it, and the Frobenius theorem tells us rotation is precisely the obstruction to the congruence being orthogonal to a family of spacelike slices.

The decisive move is Step 3: the Ricci identity trades a second derivative of the flow for the curvature tensor, so a purely kinematic quantity (how a volume element evolves) becomes coupled to matter through Einstein's equation \(R_{ab}=8\pi G(T_{ab}-\tfrac12 Tg_{ab})\). The strong energy condition then reads \(R_{ab}u^au^b=8\pi G(\rho+3p)/\,\)... more precisely \(4\pi G(\rho+3p)\ge0\): ordinary matter, including radiation, focuses; a large negative pressure does not. This single inequality is the physical hypothesis of the Hawking–Penrose theorems.

Physically, a conjugate point along a timelike geodesic is where an infinitesimally neighbouring geodesic reconverges — the Jacobi-field analogue of a focal point in optics. Between a point and its conjugate point the geodesic locally maximises proper time; beyond it, a longer nearby curve exists. The singularity theorems weaponise this: assuming geodesic completeness together with an energy condition guarantees conjugate points, but a maximal geodesic (which the causal structure of a trapped region demands) cannot contain one — the contradiction forces incompleteness, the technical definition of a singularity.

The equation is exact and nonperturbative; no linearisation or weak-field assumption enters. This is why it survives into strong-gravity regimes where Newtonian intuition fails, and why its null cousin — with \(R_{ab}k^ak^b\ge0\) and \(\theta\ge0\) forced monotone on a future horizon — proves that classical black-hole area never decreases, the geometric input to black-hole thermodynamics. The same focusing lemma, run on the wormhole throat, shows that traversable wormholes demand exotic matter that violates the null energy condition, tying the entire family of "no-go" results to the sign of one contraction of the curvature.

Common misconceptions. Focusing is not "gravity being attractive" in a Newtonian sense — it is the statement that the average tidal field, weighted by the energy condition, converges nearby worldlines; a purely tidal (Weyl/shear) field with zero Ricci still focuses, through the \(-\sigma^2\) term. And \(\theta<0\) at one instant does not by itself imply a singularity: without an energy condition and the twist-free hypothesis, \(\theta\) can recover.

Worked examples

\[ \text{FRW dust: } u^a=(\partial_t)^a,\ \theta=\nabla_a u^a=3\frac{\dot a}{a}=3H,\ \sigma_{ab}=\omega_{ab}=0. \]

Comoving observers in a homogeneous, isotropic universe form a shear- and twist-free geodesic congruence; the expansion is three times the Hubble rate. A

\[ \frac{d\theta}{d\tau}=3\dot H,\qquad -\tfrac13\theta^2=-3H^2,\qquad R_{ab}u^au^b=4\pi G(\rho+3p). \]

Evaluate each term; the Ricci projection for a perfect fluid follows from Einstein's equation. B

\[ 3\dot H=-3H^2-4\pi G(\rho+3p)\ \Rightarrow\ \frac{\ddot a}{a}=\dot H+H^2=-\frac{4\pi G}{3}(\rho+3p). \]

Rearrange using \(\ddot a/a=\dot H+H^2\); this is the second Friedmann equation, obtained purely from Raychaudhuri. A

\[ \rho=0.3\,\rho_c,\ \rho_c=\frac{3H_0^2}{8\pi G},\ p=0\ \Rightarrow\ \frac{\ddot a}{a}=-\frac{4\pi G}{3}\,\rho=-0.15\,H_0^2. \]

Insert present matter density; with \(H_0=2.20\times10^{-18}\,\mathrm{s^{-1}}\), \(H_0^2=4.84\times10^{-36}\,\mathrm{s^{-2}}\). A

\[ \frac{\ddot a}{a}=-0.15\,H_0^2\approx-7.3\times10^{-37}\ \mathrm{s^{-2}}<0. \]

Reading. Matter alone decelerates the expansion; only adding a component with \(\rho+3p<0\) (dark energy) can make \(\ddot a/a>0\), as observed today.

Units check. \((\mathrm{s^{-1}})^2=\mathrm{s^{-2}}\), matching \(\ddot a/a\).

\[ \text{Converging bundle: } \omega_{ab}=0,\ \theta_0=-3.0\times10^{-4}\ \mathrm{s^{-1}},\ R_{ab}u^au^b\ge0. \]

Take a hypersurface-orthogonal geodesic congruence (say infalling dust) with a small initial contraction and matter satisfying the strong energy condition. A

\[ \frac{d\theta}{d\tau}\le-\tfrac13\theta^2\ \Rightarrow\ \frac{1}{\theta(\tau)}\ge\frac{1}{\theta_0}+\frac{\tau}{3}. \]

All source terms are non-positive, so the focusing inequality and its integral (Steps 8–9) apply. B

\[ \theta\to-\infty\ \text{when}\ \frac{1}{\theta_0}+\frac{\tau}{3}=0\ \Rightarrow\ \tau_{\max}=\frac{3}{|\theta_0|}=\frac{3}{3.0\times10^{-4}\,\mathrm{s^{-1}}}. \]

The right-hand bound hits zero at finite proper time; solve for the latest possible caustic. A

\[ \tau_{\max}=1.0\times10^{4}\ \mathrm{s}\approx2.8\ \text{hours}. \]

Reading. A caustic (conjugate point) must form within about \(10^4\) s of proper time; any shear or positive Ricci curvature only brings it sooner. This bounded-time focusing is the geometric seed of the singularity theorems.

Units check. \(3/(\mathrm{s^{-1}})=\mathrm{s}\), a proper time, as required.

Problems

Verify that \(h_{ab}=g_{ab}+u_au_b\) is the projector orthogonal to \(u^a\): show \(h_{ab}u^b=0\), \(h^a{}_b h^b{}_c=h^a{}_c\), and \(h^a{}_a=3\) in four dimensions.
Solution
\(h_{ab}u^b=g_{ab}u^b+u_a(u_bu^b)=u_a+u_a(-1)=0\). Idempotency: \(h^a{}_bh^b{}_c=(\delta^a_b+u^au_b)(\delta^b_c+u^bu_c)=\delta^a_c+u^au_c+u^au_c+u^au_c(u_bu^b)=\delta^a_c+2u^au_c-u^au_c=\delta^a_c+u^au_c=h^a{}_c\). Trace: \(h^a{}_a=\delta^a_a+u^au_a=4-1=3\).
Derive the null Raychaudhuri equation. For an affinely parametrised null geodesic congruence \(k^a\) (\(k^ak_a=0\), \(k^b\nabla_bk^a=0\)) the transverse screen is 2-dimensional. State the resulting equation and identify the coefficient of \(\theta^2\).
Solution
Repeating Steps 1–7 with \(k^a\) null: the transverse metric is now 2D so its trace is \(h^a{}_a=2\), giving \(S_{ab}S^{ab}=\tfrac12\theta^2+\sigma^2\). Hence \(\dfrac{d\theta}{d\lambda}=-\tfrac12\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}k^ak^b\). The coefficient of \(\theta^2\) is \(1/(D-2)=1/2\) in four dimensions. With the null energy condition \(R_{ab}k^ak^b\ge0\) and \(\omega=0\), the focusing bound becomes \(1/\theta\ge1/\theta_0+\lambda/2\).
For a flat FRW universe filled with radiation (\(p=\rho/3\)), use the trace of Raychaudhuri to find \(\ddot a/a\) in terms of \(\rho\), and comment on the deceleration relative to dust.
Solution
\(\rho+3p=\rho+\rho=2\rho\). The second Friedmann equation \(\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p)=-\tfrac{8\pi G}{3}\rho\). This is twice the dust value \(-\tfrac{4\pi G}{3}\rho\) at equal energy density: radiation pressure contributes positively to \(\rho+3p\), so radiation decelerates the expansion more strongly than pressureless matter.
A shear- and twist-free geodesic congruence in vacuum has \(\theta_0=+2.0\times10^{-3}\,\mathrm{s^{-1}}\) (initially diverging). Solve \(d\theta/d\tau=-\tfrac13\theta^2\) exactly and find \(\theta\) after \(\tau=500\,\mathrm{s}\).
Solution
Separating, \(\int d\theta/\theta^2=-\tfrac13\int d\tau\) gives \(-1/\theta=-\tau/3-1/\theta_0\), i.e. \(\theta(\tau)=\dfrac{\theta_0}{1+\theta_0\tau/3}\). With \(\theta_0=2.0\times10^{-3}\,\mathrm{s^{-1}}\): \(\theta_0\tau/3=(2.0\times10^{-3})(500)/3=0.333\). So \(\theta(500)=\dfrac{2.0\times10^{-3}}{1.333}=1.5\times10^{-3}\,\mathrm{s^{-1}}\). The expansion decays but never reaches a caustic because \(\theta_0>0\); the denominator only grows.
Two hypersurface-orthogonal geodesic congruences share \(\theta_0=-1.0\times10^{-3}\,\mathrm{s^{-1}}\) and satisfy the SEC, but one is shear-free while the other has constant \(\sigma_{ab}\sigma^{ab}=\sigma^2=4.0\times10^{-7}\,\mathrm{s^{-2}}\) (approximately constant over the interval, with \(R_{ab}u^au^b\) negligible). Estimate the earliest focusing time for each and compare.
Solution
Shear-free: bound \(\tau_{\max}=3/|\theta_0|=3/(1.0\times10^{-3})=3.0\times10^{3}\,\mathrm{s}\). With shear, treat \(d\theta/d\tau\le-\tfrac13\theta^2-\sigma^2\). Approximating near the start with \(\theta^2\approx\theta_0^2=1.0\times10^{-6}\,\mathrm{s^{-2}}\): \(\tfrac13\theta_0^2=3.3\times10^{-7}\), so the total driving rate \(\tfrac13\theta^2+\sigma^2\approx3.3\times10^{-7}+4.0\times10^{-7}=7.3\times10^{-7}\,\mathrm{s^{-2}}\), roughly \(2.2\times\) the shear-free rate. The shear term more than doubles the initial focusing rate, so the caustic arrives substantially sooner — well under \(3.0\times10^{3}\,\mathrm{s}\). Shear always accelerates focusing because it enters as \(-\sigma^2\le0\); it can never delay a caustic.