Double-Null Evolution

A double-null foliation turns Einstein’s equations into a coupled hierarchy along two families of light rays. Some equations transport geometric data in the outgoing or incoming null direction; others are elliptic Hodge systems on the two-dimensional intersections. This division is the backbone of characteristic existence proofs, black-hole decay estimates, and trapped-surface formation arguments.

Required background. Analysis for Mathematical Relativity supplies energy, trace, and elliptic estimates; The Einstein Equation supplies the Bianchi equations; Hypersurface Normals and Tangential Projectors supplies induced geometry.

Helpful background. Causality and Global Structure helps interpret null domains, and PDEs and Green Functions for Gravitation supplies the hyperbolic estimates.

Optical functions and the null frame

Let $u$ and $\bar u$ solve the eikonal equation. Their level sets are outgoing and incoming null hypersurfaces, and

$$S_{u,\bar u}=\{u=\text{constant}\}\cap\{\bar u=\text{constant}\}$$

is a spacelike two-surface with metric $\gamma$. Choose future null vectors $e_4$ and $e_3$, normal to $S_{u,\bar u}$, and an orthonormal tangent frame $e_A$, with normalization

$$g(e_3,e_4)=-2, \qquad g(e_A,e_B)=\delta_{AB}.$$

The null second fundamental forms are

$$\chi_{AB}=g(\nabla_Ae_4,e_B), \qquad \underline\chi_{AB}=g(\nabla_Ae_3,e_B).$$

Their traces are the outgoing and incoming expansions; their trace-free parts $\hat\chi$ and $\underline{\hat\chi}$ are the shears. Torsion one-forms and the inaffinity coefficients complete the connection data. For the all-lowered Weyl tensor $W_{\mu\nu\rho\sigma}=g_{\mu\lambda}W^\lambda{}_{\nu\rho\sigma}$, define the components used below by

$$\begin{aligned} \alpha_{AB}&=W(e_A,e_4,e_B,e_4),& \beta_A&=\tfrac12 W(e_A,e_4,e_3,e_4),\\ \rho&=\tfrac14W(e_4,e_3,e_4,e_3),& \sigma&=\tfrac14(\star W)(e_4,e_3,e_4,e_3). \end{aligned}$$

The symbol $\star W$ means the spacetime Hodge dual on the first index pair, using the global orientation. The incoming components are $\underline\alpha_{AB}=W(e_A,e_3,e_B,e_3)$ and $\underline\beta_A=\tfrac12W(e_A,e_3,e_3,e_4)$. These definitions, together with the frame normalization, fix the curvature signs in the equations below; see Luk 2012, §2.2, equations (1)–(6).

There remains a boost freedom $e_4\mapsto a e_4$, $e_3\mapsto a^{-1}e_3$, with $a\gt0$. The expansions scale by $a$ and $a^{-1}$. Their magnitudes depend on normalization; each sign, and their product, is invariant. Any quantitative estimate must fix the lapse or frame normalization.

Transport equations along the generators

For affinely parametrized outgoing generators in vacuum, Raychaudhuri’s equation reads

$$\nabla_4\operatorname{tr}\chi +\frac12(\operatorname{tr}\chi)^2 =-\lvert\hat\chi\rvert^2.$$

With matter, the right side also contains $-8\pi G\,T_{44}$. If $\nabla_{e_4}e_4=\kappa e_4$, the right side additionally contains $+\kappa\operatorname{tr}\chi$. The equation is an ordinary differential equation along each generator once the shear is controlled. Its incoming counterpart transports $\operatorname{tr}\underline\chi$ along $e_3$.

For the affine frame, the trace-free part of the same optical equation is

$$\nabla_4\hat\chi +\operatorname{tr}\chi\,\hat\chi =-\alpha.$$

Here $\nabla_4$ is the covariant derivative projected onto the two-surfaces, including the induced connection on both tensor indices. A non-affine frame adds $+\kappa\hat\chi$ on the right. Other structure equations propagate torsion and the mixed expansions. The displayed couplings explain the hierarchy: curvature drives shear, while shear focuses the congruence.

Bianchi equations and elliptic control

In vacuum, the differential Bianchi identity $\nabla^\mu W_{\mu\nu\rho\sigma}=0$ supplies first-order propagation equations for the curvature components. In a null frame they pair neighboring components: an $e_3$ derivative of $\alpha$ couples to angular derivatives of $\beta$, an $e_4$ derivative of $\beta$ couples to angular derivatives of $\alpha$, and the chain continues through $\rho$ and $\sigma$ to the underlined components.

On each $S_{u,\bar u}$, the Codazzi and Gauss equations form elliptic systems. A representative schematic Codazzi equation is

$$\operatorname{div}\hat\chi -\frac12\,d\operatorname{tr}\chi =-\beta+\text{connection products}.$$

Hodge estimates turn control of the divergence and curl into control of angular derivatives. The proof cycle is therefore not pure transport:

1. integrate structure equations in one null direction;
2. use curvature fluxes from the Bianchi equations;
3. recover angular derivatives through elliptic estimates on $S_{u,\bar u}$;
4. commute derivatives and close a bootstrap for the coupled norms.

Luk states the data, norms, local theorem, and estimate hierarchy in Luk 2012, §§2.1–2.5 and §§3–4. His counterexample discussion in §§7.2–7.3 also shows why the available null structure and regularity cannot be discarded casually.

What the hierarchy does and does not determine

Two transversely intersecting initial null hypersurfaces provide both directions of transport. On one branch, the outgoing equations determine fields away from the corner; on the other, the incoming equations do the analogous job. Corner values join the two integrations. This is the geometric content behind the characteristic local-existence theorem.

The hierarchy does not make every component freely prescribable, nor does it by itself give a global solution. Caustics can destroy the foliation even while spacetime curvature remains finite. Conversely, a coordinate breakdown need not be geometric singularity. Global conclusions require flux bounds, control of the area radius and lapse, and a continuation criterion.

In Bondi-type characteristic evolution, the same triangular structure appears as hypersurface equations, an evolution equation for the conformal two-metric, and supplementary equations. Winicour presents that continuum hierarchy in Winicour 2012, §§3.1–3.3; its numerical realization belongs to Computational Relativity.

Solved diagnostic: flat and Schwarzschild expansions

Compute the future null expansions of round spheres in Minkowski spacetime for $e_4=\partial_t+\partial_r$ and $e_3=\partial_t-\partial_r$. Then compare the outgoing expansion on a Schwarzschild event-horizon section.

Solution

In Minkowski spacetime let $e_4=\partial_t+\partial_r$ and $e_3=\partial_t-\partial_r$. They satisfy $g(e_3,e_4)=-2$. Since a round sphere has area $A=4\pi r^2$,

$$\operatorname{tr}\chi=\frac{1}{A}e_4(A)=\frac{2}{r}, \qquad \operatorname{tr}\underline\chi=\frac{1}{A}e_3(A)=-\frac{2}{r}.$$

Thus ordinary spheres are untrapped. Since $e_4r=1$ and the shear vanishes, $e_4(2/r)+\tfrac12(2/r)^2=0$: the flat result also checks the coefficient in Raychaudhuri’s equation. On the future Schwarzschild event horizon away from the bifurcation sphere, an outgoing null generator leaves the area constant, so its outgoing expansion is zero, while the ingoing expansion is negative. At the bifurcation sphere both expansions vanish. A strictly trapped surface instead has both future expansions negative.

Next, hyperboloidal data pose a different initial-value problem on a spacelike slice that reaches conformal infinity.

References

• Luk, Jonathan. “On the Local Existence for the Characteristic Initial Value Problem in General Relativity.” International Mathematics Research Notices 2012, no. 20 (2012): 4625–4678. arXiv:1107.0898.
• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): 2. arXiv:0810.1903.