Cauchy-Characteristic Matching and Extraction

A timelike worldtube can connect a Cauchy interior to an outgoing characteristic exterior. The interface is not specified by copying coordinate components: one must construct a shared null frame, transform the metric and its derivatives, satisfy the worldtube constraints, and exchange only the characteristic fields entering each subdomain. One-way extraction and two-way matching differ precisely in whether information is returned to the Cauchy region.

Required background. Timelike Congruences and Spatial Projection supplies the observer and projection geometry used to construct the interface frame.

Helpful background. We use The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, Causality and Global Structure, and PDEs and Green Functions for Gravitation.

Timelike worldtube and shared null frame

Let $\Gamma$ be a timelike worldtube, foliated by spacelike two-surfaces $S_t$. On a Cauchy slice, let $T^\mu$ be its future unit normal and let $s^\mu$ be the outward unit spatial normal to $S_t$. The interface fixes the null boost by

$$\ell^\mu=\frac{T^\mu+s^\mu}{\sqrt2}, \qquad n^\mu=\frac{T^\mu-s^\mu}{\sqrt2}, \qquad \ell\cdot n=-1.$$

Thus $\ell$ is future outgoing and $n$ future ingoing. A later rescaling of the characteristic generator must be accompanied by the inverse rescaling of $n$ and by a corresponding change in the worldtube coordinate map. Independent normalizations on the two sides produce inconsistent radial derivatives and waveform amplitudes.

The spatial metric, lapse, and shift reconstruct the spacetime metric. The extrinsic curvature determines the time derivative of the spatial metric through $K_{ij}=-\tfrac12\mathcal L_T\gamma_{ij}$; spatial derivatives and the gauge evolution of lapse and shift supply the remaining first derivatives. Values of lapse and shift alone do not determine their derivatives. Together with the worldtube embedding, these fields determine its intrinsic geometry, outward derivatives, and the screen metric on $S_t$. A nonsingular Jacobian then transforms the metric, and differentiation of that transformation supplies the derivative data in characteristic coordinates $(u,r,y^A)$.

Data crossing the worldtube

For an outgoing Bondi–Sachs exterior, the screen determinant defines areal radius, the angular labels are transported along $\ell$, and $u$ is tied to the worldtube foliation. The transformed Cauchy data supply boundary values for $h_{AB}$, $\beta$, $U^A$, and $V$, plus the derivative combinations needed to start the radial hierarchy. Initial $h_{AB}$ data on one outgoing null hypersurface complete the characteristic problem.

These values are constrained. The induced two-metric, generator normalization, and worldtube embedding must agree. Normal and mixed projections of Einstein’s equation impose worldtube compatibility, while the Bondi supplementary equations must hold at the inner boundary so that Bianchi propagation can carry them outward. Prescribing arbitrary metric and normal-derivative data independently would overdetermine the interface.

In Cauchy–characteristic extraction, information travels one way: the Cauchy solution supplies worldtube data, and the characteristic region carries them to null infinity. It can evaluate asymptotic shear and news without finite-radius extrapolation, but it does not alter the Cauchy outer boundary.

In Cauchy–characteristic matching, outgoing Cauchy fields seed the characteristic region and incoming characteristic fields return as Cauchy boundary data. Each physical incoming mode is supplied once. Gauge and constraint modes also require a compatible prescription. The continuum data exchange and its historical implementations are separated carefully in Winicour 2012, §§ 5.1–5.8.

Flux and domain checks

An interface flux provides a discriminating test. For a common solution and a common energy current, opposite outward normals give equal and opposite interface fluxes. In two-way matching, omitting an incoming mode or prescribing it independently on both sides can break that balance. In one-way extraction the supplied metric and derivative data should still describe the same physical fields at the worldtube, but the algorithm does not enforce a combined energy estimate by returning exterior information to the interior. Gravity has no universal local positive energy density; a PDE stability proof must use the energy and boundary flux of the chosen reductions.

The coordinate map must also remain regular. The screen determinant must be nonzero, outgoing generators must avoid caustics between $\Gamma$ and the extraction endpoint, and the worldtube must remain timelike. Characteristic extraction at null infinity and its gauge requirements are reviewed in Winicour 2012, § 6.1.

This page states the continuum interface. Separate well-posedness results for a Cauchy system and a characteristic system do not by themselves prove well-posedness of their coupled matching problem: the interface conditions must close a common estimate without derivative loss. Initial null data must also agree with the worldtube data at their first cut; extraction does not determine the missing initial exterior radiation by itself. Interpolation, filtering, discrete stability, waveform error budgets, and production implementations belong to Computational Relativity.

Exercises

Use the reduced radial scalar model

$$(-\partial_t^2+\partial_r^2)\psi=0, \qquad \psi=F(t-r)+G(t+r).$$

At a worldtube $r=R$, define $w_{\mathrm{out}}=(\partial_t-\partial_r)\psi$ and $w_{\mathrm{in}}=(\partial_t+\partial_r)\psi$. Show how these fields distinguish one-way extraction from two-way matching and compute the reduced outward energy flux.

Solution

Direct differentiation gives

$$w_{\mathrm{out}}=2F'(t-r), \qquad w_{\mathrm{in}}=2G'(t+r).$$

The reduced outward flux is

$$\mathcal F =-\partial_t\psi\,\partial_r\psi =F'^2-G'^2 =\frac14\left(w_{\mathrm{out}}^2-w_{\mathrm{in}}^2\right).$$

One-way extraction reads $w_{\mathrm{out}}$ from the interior and evolves it outward; it sends no value of $w_{\mathrm{in}}$ back. Two-way matching must also return the exterior value of $w_{\mathrm{in}}$ as the single incoming boundary field for the Cauchy domain. If the two domains use different $w_{\mathrm{in}}$, their interface fluxes disagree even when the field value itself is continuous.

References

• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. arXiv:0810.1903. doi:10.12942/lrr-2012-2.