Existence Theory for the Characteristic Initial-Value Problem

Characteristic data place part of Einstein evolution directly on lightlike hypersurfaces. The resulting problem is not an ordinary Cauchy problem: the hypersurface normal is also tangent, so some field equations become transport constraints along the null generators. With two transversely intersecting null hypersurfaces, compatible smooth data determine a vacuum spacetime in a neighborhood of their intersection, uniquely up to a diffeomorphism that fixes the initial hypersurfaces.

Required background. Analysis for Mathematical Relativity supplies Sobolev estimates and compatibility conditions; The Einstein Equation supplies the Bianchi identity and vacuum equations; Hypersurface Normals and Tangential Projectors supplies the induced geometry of null hypersurfaces.

Helpful background. Causality and Global Structure clarifies domains of dependence, while PDEs and Green Functions for Gravitation reviews characteristics of hyperbolic equations.

Data on two null hypersurfaces

Let $N_0$ and $\bar N_0$ be smooth null hypersurfaces meeting transversely in a spacelike two-surface $S$. Locally choose optical functions $u$ and $\bar u$ so that $N_0=\{u=0\}$ and $\bar N_0=\{\bar u=0\}$. Their gradients obey the eikonal equations

$$g^{\mu\nu}\partial_\mu u\,\partial_\nu u=0, \qquad g^{\mu\nu}\partial_\mu \bar u\,\partial_\nu \bar u=0.$$

The intersections $S_{u,\bar u}$ carry a positive-definite two-metric $\gamma_{AB}$. Its conformal class contains the two radiative degrees of freedom, but it cannot be prescribed independently of every other metric coefficient. Raychaudhuri’s equation, the Codazzi equations, and the remaining null constraints transport expansions, torsion, and connection coefficients along the generators. A normalization of the two null normals and coordinate choices must also be fixed.

At the corner $S$, data induced from $N_0$ and $\bar N_0$ must agree. This includes the same corner metric and compatible normalization. For higher regularity, derivatives tangent to $S$ must agree, and mixed derivatives must satisfy the field equations and differentiated constraints. Derivatives in the two different null directions need not equal one another: they describe different initial information. Smooth functions prescribed separately on the two branches are therefore not automatically admissible characteristic data. The formulation-level division into free and constrained fields is developed on Characteristic Data and the Null Evolution Hierarchy; here the issue is existence from such data.

Local existence and geometric uniqueness

A useful form of the local theorem is the following. Let $N_0$ and $\bar N_0$ be smooth, transversely intersecting null hypersurfaces with smooth spacelike intersection $S$. Suppose smooth vacuum characteristic data are given in a fixed admissible gauge, satisfy the null constraint hierarchy on each branch, and satisfy all compatibility conditions at $S$. Then there is a neighborhood of $S$ to the future of both hypersurfaces carrying a smooth Lorentzian metric with

$$\operatorname{Ric}(g)=0$$

that induces the prescribed data. Any two such developments are locally isometric by an isometry whose pullback fixes the initial data. Rendall proves this by reducing the characteristic problem near the corner to a standard spacelike Cauchy problem; the theorem and reduction occupy Rendall 1990, pp. 221–239.

This statement is deliberately local. It concerns a neighborhood of a regular codimension-two corner, not the entire future domain of dependence, a complete null cone with a vertex, or a global portion of null infinity. This limitation matters: a cone vertex introduces additional regularity conditions because the two-surfaces shrink to a point, while low-regularity characteristic data require estimates not supplied by the smooth reduction theorem.

A stronger local result follows when the data are smooth and regular on two entire truncated null hypersurfaces with spherical sections, satisfy the null constraints there, and have the controlled geometry and finite norms specified in Luk 2012, Theorems 1–2 and §§2.3–2.5. With normalized initial null coordinates, suppose the branches extend over $0\leq\bar u\leq I_1$ at $u=0$ and $0\leq u\leq I_2$ at $\bar u=0$. There is an $\epsilon\gt0$, depending on those data norms, for which a development exists in the union

$$\bigl([0,\epsilon]_u\times[0,I_1]_{\bar u}\bigr) \ \cup\ \bigl([0,I_2]_u\times[0,\epsilon]_{\bar u}\bigr).$$

Thus a thin future neighborhood can cover both initial branches, not just their corner. This still does not give the whole rectangle $[0,I_2]\times[0,I_1]$. The special null structure of Einstein’s equations is essential to this improvement.

Why the reduction works

The proof has four conceptual stages.

1. Integrate the null constraints. Starting with the freely specified conformal two-geometry and corner quantities, solve ordinary differential equations along the generators for the remaining initial fields. Failure here means there was never a complete Einstein data set.
2. Reduce to an auxiliary spacelike problem. Extend suitable fields off the characteristic surfaces and use an auxiliary spacelike Cauchy problem for the reduced equations. The extension and uniqueness argument recover the prescribed characteristic traces near $S$. This is a reduction theorem, not a claim that one may freely invent Einstein Cauchy data between the null branches.
3. Solve a reduced hyperbolic system. In a coordinate or frame gauge, the principal part becomes a system of wave equations. Standard energy estimates give a solution for a short time.
4. Recover Einstein’s equations. The contracted Bianchi identity gives a homogeneous subsidiary system for the gauge and constraint violations. Their initial values vanish, so uniqueness for the subsidiary system makes the reduced solution a vacuum solution.

Gauge uniqueness in stage 3 is not yet geometric uniqueness. To compare two developments, put them into a common gauge or solve wave-map equations from one to the other; uniqueness of the reduced system then produces the local isometry. This is why the theorem must not claim equality of coordinate components.

One null hypersurface is not the same problem

Data on a single null hypersurface generally do not enclose a local domain by themselves. One needs additional information: a second intersecting null hypersurface, data on a timelike worldtube, or regularity conditions at a cone vertex. In the worldtube–nullcone problem, boundary data on the worldtube and null data on each outgoing cone must agree where they meet. Winicour describes this hierarchy and its continuum interpretation in Winicour 2012, §§2.1 and 3.1–3.3.

Nor does characteristic evolution evade constraints. It reorganizes them into transport equations. Prescribing every metric coefficient freely would overdetermine the problem, while prescribing only the conformal two-metric without corner normalization leaves the geometry underdetermined.

Solved diagnostic: agreement at the corner

Solve $\partial_u\partial_{\bar u}\phi=0$ with $\phi(0,\bar u)=f(\bar u)$ and $\phi(u,0)=g(u)$. What compatibility condition is necessary, and must $f'(0)$ equal $g'(0)$?

Solution

Suppose a scalar model has $\partial_u\partial_{\bar u}\phi=0$, with $\phi(0,\bar u)=f(\bar u)$ on $N_0$ and $\phi(u,0)=g(u)$ on $\bar N_0$. The general solution is $\phi=A(u)+B(\bar u)$. At $S$ it requires

$$f(0)=A(0)+B(0)=g(0).$$

If $f(0)\ne g(0)$, both branch data are smooth but no continuous solution realizes them. When the values agree, one may take $\phi(u,\bar u)=g(u)+f(\bar u)-f(0)$. There is no requirement that $f'(0)=g'(0)$: the derivatives belong to different null directions. The mixed derivative vanishes because of the PDE. Einstein’s corner conditions are the tensorial, nonlinear analogue, with differentiated constraints governing the higher jets.

The next page develops the double-null evolution hierarchy that makes these constraints and propagation equations explicit.

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.
• Rendall, Alan D. “Reduction of the Characteristic Initial Value Problem to the Cauchy Problem and Its Applications to the Einstein Equations.” Proceedings of the Royal Society A 427 (1990): 221–239. DOI.
• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): 2. arXiv:0810.1903.