Cauchy-Characteristic Evolved Waveforms

• The method integrates Cauchy (3+1) evolution with characteristic (null) evolution to produce unambiguous gravitational waveforms at null infinity by eliminating artificial boundary effects.
• It employs advanced spectral algorithms and rigorous error analysis to achieve high numerical stability and exponential convergence, with mismatches as low as 10⁻¹² for dominant modes.
• The approach ensures accurate alignment of BMS frames and full matching between interior and exterior solutions, vital for extracting physical waveforms in complex scenarios like binary black hole mergers.

Cauchy-Characteristic Evolved Waveforms are gravitational radiation signals computed in numerical relativity through the integration of Cauchy (spacelike) and characteristic (null) initial value problems. In this computational paradigm, spacetime is advanced using standard 3+1 (Cauchy) evolution in regions containing strongly gravitating, nonlinear dynamics (such as black hole interiors), while the radiation zone and asymptotic region are handled via characteristic evolution along outgoing null hypersurfaces. This hybrid approach enables accurate and gauge-invariant extraction of gravitational waveforms at future null infinity (𝒥⁺ or ℐ⁺), where physical radiation is unambiguously defined, by minimizing contamination from artificial outer boundaries, gauge ambiguities, and near-zone effects inherent to finite-radius extractions. The core machinery—encompassing both Cauchy-Characteristic Extraction (CCE) and Cauchy-Characteristic Matching (CCM)—has advanced substantially in numerical stability, error control, efficiency, and physical completeness, supporting high-fidelity waveform modeling for gravitational-wave astronomy and fundamental tests of general relativity.

1. Characteristic Evolution and Bondi–Sachs Formulation

Characteristic evolution reformulates the dynamics of general relativity by slicing spacetime along outgoing null hypersurfaces labelled by a retarded time coordinate $u$, with coordinates $(u, r, x^A)$. The most widely-adopted formalism is the Bondi–Sachs system, whose metric can be written as

$$ds^2 = -[e^{2\beta}(V/r) - r^2 h_{AB} U^A U^B] du^2 - 2e^{2\beta} du dr - 2r^2 h_{AB} U^B du dx^A + r^2 h_{AB} dx^A dx^B$$

with the determinant condition $\det(h_{AB}) = \det(q_{AB})$, where $q_{AB}$ is the metric of the unit 2-sphere. The independent metric fields $(\beta, V, U^A, h_{AB})$ encode the radiative and nonlinear gravitational degrees of freedom.

The advantage of the characteristic approach is that, with suitable compactification (for example, $x = r/(1+r)$), the computational grid extends to $\mathscr{I}^+$, allowing the unambiguous calculation of asymptotic quantities such as the Bondi mass, angular momentum, and—most crucially—the Bondi news function: $N = \frac{1}{4} Q^A Q^B \partial_u c_{AB}$ where $c_{AB}$ is the leading (radiative) coefficient in the asymptotic expansion of $h_{AB}$. This direct access to null infinity eliminates the need for extrapolating finite-radius data, a major source of systematic error in standard extraction techniques (0810.1903).

2. Cauchy–Characteristic Extraction and Matching

The two main paradigms for integrating Cauchy and characteristic evolution are:

• CCE (Cauchy–Characteristic Extraction): Cauchy data (metric and its derivatives) from a simulation with an artificial outer boundary are provided as inner boundary data on an extraction worldtube for the characteristic code. The characteristic evolution then propagates this data to $\mathscr{I}^+$, yielding the waveform. CCE removes outer boundary artifacts and is central to producing physically robust waveforms for detectors such as LIGO and LISA (Babiuc et al., 2010, Babiuc et al., 2011).
• CCM (Cauchy–Characteristic Matching): Data transfer is bidirectional. The interior Cauchy region and the exterior characteristic region overlap on the worldtube, each supplying boundary conditions to the other. This eliminates the need for artificial boundaries altogether and, in principle, yields a global solution to Einstein’s equations. Recent full 3D implementations demonstrate robust stability and improved physical fidelity in nonlinear dynamical spacetimes, such as binary black hole mergers, and facilitate the paper of late-time power-law tails in gravitational wave signals (Ma et al., 2023, Ma et al., 9 Dec 2024).

CCE and CCM both rely on precise worldtube data exchange, involving transformation between Cauchy (Cartesian/harmonic) and Bondi–Sachs (spherical null) gauges through spectral decomposition and coordinate Jacobians (Babiuc et al., 2010, Ma et al., 9 Dec 2024). The matching procedure includes careful treatment of both metric fields and gauge (BMS) transformations to ensure physical alignment and accurate waveform propagation.

3. Numerical Methods, Regularity, and Error Analysis

Modern CCE/CCM schemes utilize advanced spectral algorithms to integrate the characteristic equations (Handmer et al., 2014, Barkett et al., 2019, Moxon et al., 2021). Spectral representations—using Chebyshev/Gauss–Lobatto points radially and spin-weighted spherical harmonics angularly—yield exponential convergence and high efficiency. Analytic regularization, such as Laurent expansions and integration by parts, removes spurious logarithmic singularities at null infinity, ensuring C∞ smoothness and suppressing the artifacts that would spoil spectral convergence.

A central metric for waveform quality is the mismatch,

$$\mathcal{M}(h, h') = 1 - \max_{\Delta\phi} \mathcal{O}(h, h')$$

with $\mathcal{O}$ denoting the normalized inner product, typically in the frequency domain. High-resolution CCE waveforms for binary black holes exhibit mismatches as low as $10^{-12}$ for the dominant (2,2) mode at optimal extraction radii (Rashti et al., 18 Nov 2024), and mismatches $\lesssim 3\times 10^{-4}$ for broad aligned-spin parameter studies (Chu et al., 2015). However, mismatch increases for larger extraction radii or inclusion of higher multipoles.

Error budgets must account for:

• Characteristic code truncation error
• Extraction/worldtube radius dependence
• Cauchy evolution resolution
• Gauge artifacts and initial-data mismatch
• Numerical noise in mode expansions

Careful Richardson extrapolation and convergence testing of both amplitude and phase versus grid spacing and extraction radius are standard practices for robust error quantification (Chu et al., 2015, Rashti et al., 18 Nov 2024). The systematic use of CCE enables unambiguous cross-code and cross-methodology comparisons, facilitating community-wide validation (Babiuc et al., 2010, Babiuc et al., 2011).

4. BMS Frame, Gauge Fixing, and Physical Consistency

Physical interpretability of the extracted waveform depends crucially on the fixity of the BMS frame—the Bondi–Metzner–Sachs symmetry residual at null infinity, encompassing both the 10-parameter Poincaré group and infinite-dimensional supertranslations. Numerical relativity outputs are often contaminated by unphysical drifts in the center-of-mass, boosts, or supertranslations, necessitating a post-processing transformation to a physically meaningful frame.

Recent work computes asymptotic Poincaré charges (linear momentum, angular momentum, boosts, center of mass) directly from the waveform data at $\mathscr{I}^+$ and applies an iterative fitting procedure to remove spurious translation, boost, and supertranslation components (Mitman et al., 2021). The preferred "PN BMS frame" is found by minimizing the $L^2$ difference between the numerical waveform and a high-order post-Newtonian model across several orbits, robustly aligning both the oscillatory and memory content. This refinement is critical for hybridization of analytic models with NR waveforms, reducing mismatches, and constraining systematic errors in model calibration.

The correction procedure outperforms traditional Newtonian center-of-mass based schemes, especially in cases with high mass ratios or spins, where Newtonian methods leave residual drifts (Mitman et al., 2021).

5. Gauge Invariance, Memory Effects, and Physical Content

CCE produces gauge-invariant waveforms by construction. Direct extraction of Newman–Penrose scalars (e.g., $\Psi_4$) at finite radius and their extrapolation are susceptible to gauge contamination, especially for non-oscillatory or memory ($m=0$) modes, and for strong-field coordinate gauge choices. CCE, by propagating data to $\mathscr{I}^+$ in Bondi coordinates, produces waveforms that are robust against gauge differences and properly encode displacement and nonlinear memory effects (Taylor et al., 2013). This is crucial for interpreting waveform memory, momentum flux, and for systematic studies of late-time ringdown tails that probe nonlinearities and backscatter in black hole mergers (Ma et al., 9 Dec 2024, Ma et al., 10 Sep 2024).

The method also enables the uniform calculation of radiation fluxes associated with the full BMS algebra—including energy, linear and angular momentum, and supermomentum—through coordinate transformation to an asymptotically inertial frame and direct computation of BMS charges (Handmer et al., 2016, Moxon et al., 2020).

6. Mathematical Structure, Hyperbolicity, and Stability

The PDE structure of CCE/CCM systems is characterized by a hierarchy of (often nonlinear) hypersurface equations along outgoing null directions. The principal part of these systems in Bondi–like gauges can, however, be merely weakly hyperbolic rather than symmetric or strongly hyperbolic (Giannakopoulos et al., 2023). Weak hyperbolicity can lead to a loss of derivative control, affecting stability and convergence properties—numerical experiments show that (unless data is controlled in a sufficiently strong norm, e.g., $H^1$) the convergence order may fall below that of the formal numerical scheme and instabilities can develop, particularly in CCM when matching from strongly hyperbolic Cauchy IBVPs to weakly hyperbolic characteristic CIBVPs.

Addressing these issues requires careful choice of numerical norms (for example, the $q$ norm including an extra angular derivative), and careful interface design to ensure control over all the relevant solution components (Giannakopoulos et al., 2023). Improvements such as the use of partially flat gauges, direct transformation to inertial frames, and analytic removal of pure-gauge logarithms have enabled exponential convergence, robust stability, and high efficiency in practical spectral implementations (Barkett et al., 2019, Moxon et al., 2021).

7. Extensions, Applications, and Future Directions

The CCE and CCM frameworks have been generalized to include additional physical degrees of freedom, such as scalar fields (Einstein–Klein–Gordon systems), for studies of beyond-GR theories. Scalar field contributions appear as source terms in the hypersurface equations, modify Weyl scalar asymptotics, and produce both gravitational and scalar memory effects; CCE methods accurately capture these effects and the distinct power-law decay of ringdown tails at null infinity, enabling robust tests of fundamental physics (Ma et al., 10 Sep 2024).

Applications span from high-precision waveform banks for NextGen observatories (LISA, Cosmic Explorer, Einstein Telescope), cross-code calibration, to the systematic paper of late-time dynamics and nonlinear memory in BBH mergers (Rashti et al., 18 Nov 2024, Ma et al., 9 Dec 2024). The public release of high-accuracy waveform catalogs, such as from GR-Athena++ and SXS/SpEC, supports these goals.

Ongoing mathematical analysis and algorithmic development aim to resolve remaining questions about well-posedness, interface stability, and optimal gauge choices in fully generic setups, ensuring the reliability and fidelity required for next-generation gravitational-wave science.