From Stress-Energy Conditions to Curvature Convergence

Raychaudhuri’s equations contain Ricci contractions, not stress-energy contractions. Einstein’s equation supplies the bridge. Null contraction removes both the trace term and the cosmological term, so NEC maps directly to null convergence. Timelike contraction retains the trace-reversed matter tensor and a cosmological shift, so SEC—not merely WEC—maps to timelike convergence only with explicit bookkeeping for $\Lambda$.

Required background. We use congruence expansion, Einstein’s equation, and the metric definition of stress-energy.

Helpful background. The causal contractions use special relativity and tensor algebra. Converting focusing to an incompleteness conclusion belongs to global structure.

Trace reversal in general dimension

In $D>2$ dimensions take

$$G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}.$$

Tracing gives

$$-\frac{D-2}{2}R+D\Lambda=8\pi G\,T,$$

and hence

$$R_{ab} =8\pi G\left(T_{ab}-\frac{T}{D-2}g_{ab}\right) +\frac{2\Lambda}{D-2}g_{ab}.$$

Reall fixes the trace and cosmological signs in the same convention Reall 2013, § 6.7, pp. 64–65, Eqs. (6.32)–(6.35), PDF. The restriction $D>2$ is real: trace reversal divides by $D-2$, and two-dimensional Einstein curvature has a special degeneracy.

Null convergence

Contract with a null vector $k^a$. Since $g_{ab}k^ak^b=0$,

$$R_{ab}k^ak^b=8\pi G\,T_{ab}k^ak^b.$$

Both the trace piece and geometric cosmological term vanish. For $G>0$, the null energy condition

$$T_{ab}k^ak^b\ge0$$

is therefore equivalent, within classical Einstein gravity, to the null convergence condition

$$R_{ab}k^ak^b\ge0.$$

This is an algebraic equivalence at an event. It does not prove NEC, and it does not prove focusing without affine, twist, shear, and initial-expansion hypotheses. Kontou and Sanders state the NEC and its geometric translation explicitly Kontou and Sanders 2020, § 2.1, pp. 9–10, Eqs. (26)–(27), PDF.

If $\Lambda$ is instead moved to an effective stress tensor, its null contraction is still zero. Null convergence is insensitive to a pure metric-proportional source at this algebraic step.

The equivalence also preserves strictness: a null direction with $T_{ab}k^ak^b=0$ has zero Ricci contraction even when $\Lambda\ne0$. Shear can still drive that bundle toward focusing, while Weyl curvature can generate the shear without appearing directly in the traced Raychaudhuri equation. Ricci convergence is therefore not the whole optical dynamics.

Timelike convergence

For a unit timelike vector $u^a$, $u^au_a=-1$, trace reversal gives

$$R_{ab}u^au^b =8\pi G\left(T_{ab}u^au^b+\frac{T}{D-2}\right) -\frac{2\Lambda}{D-2}.$$

The matter inequality

$$\left(T_{ab}-\frac{T}{D-2}g_{ab}\right)u^au^b\ge0$$

for every timelike $u^a$ is SEC. When $\Lambda=0$, SEC maps to $R_{ab}u^au^b\ge0$. When positive $\Lambda$ remains on the geometric side, matter SEC alone is insufficient: the negative $2\Lambda/(D-2)$ contribution can make the Ricci contraction negative. The exact convergence condition is

$$8\pi G\left(T_{ab}u^au^b+\frac{T}{D-2}\right) \ge\frac{2\Lambda}{D-2}.$$

Kontou and Sanders formulate SEC and timelike convergence in the $\Lambda=0$, $G=1$ convention Kontou and Sanders 2020, § 2.1, pp. 8–9, Eqs. (20)–(22), PDF. Restoring $G$ and $\Lambda$ is necessary before applying the statement to a different convention.

Why WEC is not enough

WEC says $T_{ab}u^au^b\ge0$, but timelike Ricci convergence depends also on $T/(D-2)$. Positive measured energy density can be outweighed by sufficiently negative pressure through the trace. In four dimensions, a perfect fluid comoving with $u^a$ has

$$T=-\rho+3p, \qquad T_{ab}u^au^b+\frac12T=\frac12(\rho+3p).$$

Thus comoving timelike convergence with $\Lambda=0$ requires $\rho+3p\ge0$. Requiring it for every timelike direction also yields $\rho+p\ge0$, the full perfect-fluid SEC.

Positive vacuum energy gives $p=-\rho$ and $T_{ab}^{(\Lambda)}=-\rho g_{ab}$. It satisfies WEC and saturates NEC, but $\rho+3p=-2\rho<0$. If treated geometrically, the same defocusing appears as the explicit negative $\Lambda$ term. Moving $\Lambda$ across the equation changes the label, not the Ricci contraction.

For a perfect fluid seen by a noncomoving unit observer $u^a$, the SEC contraction also contains the positive boost-dependent factor $(\rho+p)(v\cdot u)^2$. Requiring the inequality for all $u^a$, not only the comoving choice, yields both $\rho+3p\ge0$ and $\rho+p\ge0$ in four dimensions. This is why one comoving acceleration equation cannot replace the quantified SEC definition.

Proof boundary and independent checks

The translation assumes classical Einstein gravity with constant $G$ and $\Lambda$. Higher-curvature terms, torsion, or nonminimal matter rearrangements change which tensor is algebraically related to $R_{ab}$. An “effective stress tensor” may absorb geometric derivatives, but its energy conditions need not have the same material interpretation.

Three checks discriminate common errors. Null contraction must erase $\Lambda$. In vacuum with positive $\Lambda$, timelike contraction must be negative, $R_{ab}u^au^b=-2\Lambda/(D-2)$. In $D=4$, $\Lambda=0$, a dust source gives $R_{ab}u^au^b=4\pi G\rho>0$ for the comoving velocity. All follow from the same trace-reversed equation.

Dimensional analysis adds a fourth check. In units $c=1$, $GT_{ab}$, $R_{ab}$, and $\Lambda g_{ab}$ all have dimensions $L^{-2}$. A convergence inequality is invariant under positive constant rescaling of a null vector, while its numerical contraction scales quadratically. Only the sign is normalization independent.

Finally, convergence is observer quantified. Timelike convergence must hold for every relevant unit timelike $u^a$, not only for a fluid’s rest velocity, and null convergence for every null direction. Verifying one contraction may suffice for a symmetry-reduced theorem along a chosen congruence, but it does not establish the tensor energy condition globally. Conversely, a theorem about one specified bundle needs the inequality only along its generators, so a violation elsewhere need not invalidate that local argument.

Exercises

In four dimensions, evaluate null and comoving timelike Ricci contractions for a perfect fluid with $\Lambda$ retained geometrically.

Solution

For any null $k^a$, $T_{ab}k^ak^b=(\rho+p)(v\cdot k)^2$, so

$$R_{ab}k^ak^b=8\pi G(\rho+p)(v\cdot k)^2.$$

For $u^a=v^a$, trace reversal gives

$$R_{ab}u^au^b=4\pi G(\rho+3p)-\Lambda.$$

The different pressure combinations explain why NEC governs null convergence while SEC governs timelike convergence.

References

• Kontou, Eleni-Alexandra, and Ko Sanders. “Energy Conditions in General Relativity and Quantum Field Theory.” Classical and Quantum Gravity 37 (2020): 193001. DOI; arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.