Characteristic and Null Formulations

Characteristic formulations evolve Einstein geometry along light rays. Their main simplification is also their main constraint: because the normal to a null hypersurface is tangent to it, several field equations become transport equations along generators. Radiative conformal screen data are free, while area, connection, gauge, and supplementary fields are reconstructed hierarchically from corner or worldtube values.

Helpful background. This chapter draws on Timelike Congruences and Spatial Projection, The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, Causality and Global Structure, and PDEs and Green Functions for Gravitation.

Enter null evolution

On each null hypersurface choose a future-directed generator $\ell^\mu$ and an auxiliary future null vector $n^\mu$ with

$$\ell\cdot n=-1.$$

Their two-dimensional orthogonal screen carries

$$q_{\mu\nu} =g_{\mu\nu}+\ell_\mu n_\nu+n_\mu\ell_\nu.$$

The boost freedom

$$\ell\mapsto e^\alpha\ell, \qquad n\mapsto e^{-\alpha}n$$

leaves $q_{\mu\nu}$ fixed but rescales expansion and shear and shifts the normal-bundle connection. Every page therefore declares generator orientation, normalization, and affine or nonaffine parameter before comparing null data.

The conformal class of $q_{AB}$ contains two gravitational polarizations. Its tracefree generator derivative is shear. The area scale and expansion satisfy Raychaudhuri; other components of Einstein’s equation transport lapse-like, shift-like, and connection fields. This free-versus-constrained split is the organizing principle of the chapter.

Route through the chapter

The six pages move from intrinsic null data to coupled evolution and interfaces.

1. Geometry and Data on Null Hypersurfaces defines the null basis, screen metric, expansion, shear, twist, and nonaffinity. It derives their boost weights and identifies conformal screen data as free.
2. Double-Null Formulations introduces two transverse null foliations, their two second fundamental forms, and the normal-bundle connection. It shows how incoming and outgoing transport hierarchies share one corner.
3. The Bondi-Sachs Formulation fixes areal radius and the screen determinant. Vacuum equations determine $\beta$, $U^A$, and $V$ radially before evolving the radiative two-metric in retarded time.
4. Characteristic Data and the Null Evolution Hierarchy states the smooth local two-sheet data theorem, its hypotheses, proof architecture, and corner compatibility. Sharp function-space theory is handed to Mathematical Relativity.
5. Caustics and Breakdown of Null Coordinates distinguishes focal Jacobian loss and generator crossovers from curvature singularities. A converging Minkowski cone gives the decisive counterexample.
6. Cauchy-Characteristic Matching and Extraction constructs a shared worldtube null frame and compares one-way waveform extraction with two-way exchange of incoming and outgoing characteristic fields.

Winicour develops the worldtube–nullcone problem, Bondi hierarchy, matching interface, and initial null data in Winicour 2012, §§ 2.1, 3.1–3.3, 5.1–5.8, 6.1, and 6.6. This chapter uses that continuum architecture without importing implementation-performance claims.

The same equations in different domains

A regular cone with a vertex, a null hypersurface launched from a timelike worldtube, and two intersecting null sheets use related transport equations but different seed data.

At a cone vertex, the screen collapses smoothly only under special parity and leading-order regularity conditions. At a worldtube, finite screen data are supplied as constrained boundary values. In a double-null problem, two sets of conformal data meet at a nondegenerate spacelike corner and must agree there. Confusing these domains leads to either missing data or overprescription.

The smooth two-sheet theorem is local. In harmonic gauge, compatible positive screen data and corner functions determine a unique local vacuum metric in fixed coordinates near the transverse intersection. Rendall proves the model wave theorem and Einstein application in Rendall 1990, §§ 2–3 and 5, pp. 223–234, especially Theorems 1 and 3. It does not guarantee global caustic avoidance.

At large radius, Bondi–Sachs variables identify asymptotic shear and news. Full BMS charges and detector-facing waveform conventions require additional asymptotic and observational structures. At finite radius, a Cauchy–characteristic interface requires compatible tensor and derivative data, not only a continuous metric.

Consistency checks

Boost check. Rescale $\ell$ and $n$ inversely. The screen metric and vanishing of shear remain invariant, while all boost-weighted quantities transform as declared.

Determinant check. In Bondi gauge, $\det h=\det q$ removes the area mode. The flat assignment $\beta=0$, $U^A=0$, $V=r$, and $h_{AB}=q_{AB}$ must solve every level of the hierarchy.

Corner check. The two initial null sheets must induce the same screen metric and cross-normalization at their intersection. Two separately smooth datasets can fail to define any continuous spacetime when those restrictions disagree.

Jacobian check. Monitor both $\det q_{AB}$ and injectivity of the generator map. A chart can fail in flat spacetime at a focal point or crossover.

Flux check. Across a Cauchy–characteristic worldtube, an outgoing field is supplied to the exterior and an incoming field is supplied to the interior exactly once. Opposite interface orientations must give equal and opposite fluxes.

Review the chapter

Consider outgoing Minkowski spacetime in Bondi coordinates,

$$ds^2=-du^2-2du\,dr+r^2d\Omega_2^2.$$

1. Identify the Bondi–Sachs fields and the free radiative data.
2. With $\ell=\partial_r$, compute the screen expansion and state what changes under $\ell'=a\ell$, $n'=a^{-1}n$.
3. Explain what additional information is needed to turn one null hypersurface into a local evolution problem.
4. State why this chart cannot simply be continued through a caustic and how a Cauchy–characteristic worldtube can avoid that domain.

Solution

The flat fields are

$$\beta=0,\qquad U^A=0,\qquad V=r,\qquad h_{AB}=q_{AB}.$$

The determinant gauge holds, $\partial_rh_{AB}=0$, and the asymptotic shear coefficient and news vanish. The conformal screen has no nontrivial radiative data.

The physical screen metric is $r^2q_{AB}$, so

$$\theta_{(\ell)}=\frac2r, \qquad \sigma_{AB}=0.$$

Under the constant boost, $\theta_{(\ell') }=2a/r$ while the screen metric and zero-shear statement are unchanged. The inverse rescaling of $n$ preserves $\ell\cdot n=-1$.

One null sheet supplies conformal screen history but not all transverse information. A second transverse null sheet with compatible corner data, or a timelike worldtube with compatible boundary fields, seeds the remaining transport equations.

At a focal point the screen Jacobian vanishes; at a crossover the generator map is globally noninjective even if each branch retains a nonzero local Jacobian. No coordinate continuation repairs that failed labeling. A worldtube can terminate the Cauchy domain before the outgoing exterior rays focus; if the exterior itself remains caustic free, characteristic evolution can then reach null infinity. This is a domain choice, not a guarantee supplied by the field equations.

References

• 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 of London A 427, no. 1872 (1990): 221–239. doi:10.1098/rspa.1990.0009.
• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. arXiv:0810.1903. doi:10.12942/lrr-2012-2.