Perturbative Hyperboloidal Extraction

• Perturbative Hyperboloidal Extraction is a method that uses Cauchy data from numerical relativity as inner boundary conditions for a perturbative evolution on hyperboloidal slices, enabling direct waveform extraction at null infinity.
• It employs hyperboloidal coordinates and compactification techniques to include null infinity in the computational domain, eliminating the need for outer boundary conditions.
• The approach improves waveform accuracy over finite-radius extraction and robustly compares with characteristic methods in various astrophysical simulations.

Perturbative hyperboloidal extraction (PHE) is a framework to propagate to null infinity gravitational waves computed at timelike worldtubes in the interior of a 3+1 (Cauchy) numerical relativity simulations. In this method, numerical relativity data are used as the inner inflowing boundary of a perturbative time-domain Regge-Wheeler-Zerilli simulation in hyperboloidal coordinates that reaches null infinity, so that the asymptotic waveform can be read off directly at future null infinity $\mathscr{I}^+$ rather than inferred by finite-radius extrapolation (Bernuzzi et al., 7 Aug 2025). Its geometric basis lies in hyperboloidal evolution with scri-fixing coordinates, where spacelike slices asymptote to outgoing null hypersurfaces and include $\mathscr{I}^+$ as part of the computational domain (0808.0810).

1. Geometric foundations

The hyperboloidal idea is to replace ordinary Cauchy slices, which terminate at a finite artificial outer boundary or approach spatial infinity, with spacelike hypersurfaces that reach future null infinity. In the conformal formulation of the Einstein equations, the physical and rescaled metrics are related by

$$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$

with $\Omega=0$ defining the conformal boundary. A central result is that a conformal scale factor can be freely prescribed a priori in terms of coordinates in a well-posed hyperboloidal initial value problem such that the location of null infinity is independent of the time coordinate, and that with an appropriate choice of a single gauge source function each of the formally singular conformal source terms in the equations attains a regular limit at null infinity (0808.0810).

This scri-fixing viewpoint was developed numerically for hyperbolic PDEs on fixed backgrounds by combining a height-function time transformation, radial compactification, and conformal rescaling. In that construction, null infinity sits at a fixed coordinate location, the outer boundary is null infinity itself, there are no incoming characteristics at the boundary, and therefore no outer boundary condition is needed (Zenginoglu et al., 2010). The same geometric principle underlies black-hole perturbation theory on hyperboloidal slices: perturbations are regular when evaluated toward the event horizon and null infinity, and hyperboloidal surfaces naturally connect these regions (Macedo et al., 2024).

For waveform extraction, the significance of this geometry is direct. Gravitational radiation is unambiguously defined at null infinity, while standard finite-radius extraction is affected by gauge effects, finite-radius extraction error, and artificial outer boundary conditions. Hyperboloidal evolution addresses the outer boundary problem and the wave extraction problem by changing the foliation rather than by constructing approximate transparent boundary data (0808.0810).

2. Core perturbative construction

PHE specializes the hyperboloidal idea to perturbations of a Schwarzschild background evolved outside an extraction worldtube. The conceptual picture is:

1. run full 3+1 NR,
2. extract RWZ waveform data on a finite-radius sphere,
3. inject that data as an inner boundary condition for a 1+1 perturbative evolution,
4. evolve this data on hyperboloidal slices all the way to null infinity,
5. read off the asymptotic waveform directly at $\mathscr{I}^+$ (Bernuzzi et al., 7 Aug 2025).

The perturbative variables are the even- and odd-parity Regge-Wheeler-Zerilli master functions $\Psi^{(e/o)}_{\ell m}$. The strain is written as

$$h_+ - i h_\times = \sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} \frac{N_\ell}{D_L} \left(\Psi^{(e)}_{\ell m}(u)+ i\,\Psi^{(o)}_{\ell m}(u)\right)\,{}_{-2}Y_{\ell m}(\theta,\phi),$$

with $D_L$ the luminosity distance and $N_\ell=\sqrt{( \ell+2)!/(\ell-2)!}$ in the standard normalization (Bernuzzi et al., 7 Aug 2025).

In the wave zone, each master function satisfies the standard 1+1 RWZ wave equation in tortoise coordinates,

$\Psi_{tt} - \Psi_{r_* r_*} + V(r)\,\Psi = 0,$

where $\mathscr{I}^+$0 is the $\mathscr{I}^+$1-dependent RWZ potential. The numerical relativity code provides RWZ waveform data on a finite-radius extraction worldtube at coordinate radius $\mathscr{I}^+$2, or equivalently $\mathscr{I}^+$3. These data are interpreted as incoming boundary data for the perturbative exterior problem through

$\mathscr{I}^+$4

and for actual NR data,

$\mathscr{I}^+$5

The perturbative domain is then the wave zone outside the extraction sphere, and the evolution propagates the signal outward to $\mathscr{I}^+$6 (Bernuzzi et al., 7 Aug 2025).

This construction is distinct from a full characteristic evolution of the Einstein equations. It is a perturbative worldtube-to-$\mathscr{I}^+$7 propagation scheme whose simplicity derives from using a linear RWZ evolution in the exterior region (Bernuzzi et al., 7 Aug 2025).

3. Hyperboloidal coordinates and compactification

The coordinate transformation used in PHE is the hyperboloidal layer construction of Zenginoğlu. Coordinates are transformed from $\mathscr{I}^+$8 to $\mathscr{I}^+$9 via

$$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$0

Here $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$1 is a height function that tilts slices upward so that they become asymptotically null, and $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$2 is a compactification factor mapping the infinite domain $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$3 to a finite coordinate range $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$4, with $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$5 marking the start of the hyperboloidal layer (Bernuzzi et al., 7 Aug 2025).

The defining requirement is that outgoing null rays are preserved in the layer,

$$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$6

which implies

$$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$7

where $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$8 is the boost function. The compactification is chosen so that $$g = \Omega^2 \tilde g, \qquad \Omega > 0 \ \text{in the interior},$$9 in the interior region and smoothly transitions in the layer. Thus the inner region is essentially standard Cauchy-like coordinates, the exterior layer is compactified hyperboloidal geometry, and $\Omega=0$0 is included at a finite coordinate location $\Omega=0$1 (Bernuzzi et al., 7 Aug 2025).

On hyperboloidal slices, the RWZ equation becomes

$\Omega=0$2

For $\Omega=0$3, this reduces to the standard RWZ equation. Because the domain is compactified to include $\Omega=0$4, no outer boundary condition is required at null infinity, and the waveform can be evaluated directly at $\Omega=0$5, i.e. at $\Omega=0$6 (Bernuzzi et al., 7 Aug 2025).

This is the perturbative counterpart of the broader hyperboloidal principle that the location of null infinity is known a priori and can be fixed at a coordinate surface. In the conformal Einstein setting, this is achieved by prescribing $\Omega=0$7 as a coordinate function and choosing the corresponding gauge source function so that $\Omega=0$8, which realizes the preferred conformal gauge at $\Omega=0$9 (0808.0810).

4. Numerical realization and relation to other perturbative hyperboloidal methods

The PHE evolution is performed in first-order-in-time, second-order-in-space form, with method of lines, using 4th-order Runge–Kutta, with 4th-order finite differences in space, and no ghost points or outer boundary data at $\mathscr{I}^+$0. Initial data are taken as zero for $\mathscr{I}^+$1 and $\mathscr{I}^+$2. The only boundary condition is the inner timelike one, implemented with a 4th-order prescription designed for stable initial-boundary value problems (Bernuzzi et al., 7 Aug 2025).

This numerical pattern fits into a wider family of hyperboloidal perturbative schemes. In Schwarzschild self-force calculations, hyperboloidal compactification turns frequency-domain problems into regular boundary-value problems on a finite interval, with the horizon and null infinity on the numerical grid and boundary conditions replaced by regularity conditions (Macedo et al., 2022). In Kerr perturbation theory, the 1+1 Teukolsky equation can be placed on a hyperboloidal, horizon-penetrating compactified domain, so that $\mathscr{I}^+$3 is future null infinity and relevant waveform quantities are available directly there (Markakis et al., 2023). In the hyperboloidal approach to quasinormal modes, outgoing boundary conditions at the horizon and infinity are converted into boundedness and regularity properties of the rescaled field on a compactified domain (Macedo et al., 2024).

These related developments clarify the place of PHE within perturbation theory. It is not merely a finite-radius correction formula. Rather, it uses the same geometric regularization that makes perturbative fields finite at the asymptotic boundaries, but applies it in a time-domain worldtube-driven extraction problem tailored to 3+1 numerical relativity (Bernuzzi et al., 7 Aug 2025).

5. Accuracy, demonstrated applications, and comparison with other extraction methods

PHE was validated on a range of 3+1 NR simulations: gravitational collapse of rotating neutron stars, binary black hole mergers and scattering, and binary neutron star mergers. The method generally performs at least as well as extrapolation and often better, especially when wave-zone extraction radii are not very large (Bernuzzi et al., 7 Aug 2025).

The comparison with finite-radius extraction and $\mathscr{I}^+$4-extrapolation is straightforward. Standard finite-radius extraction suffers from strong systematics, especially for early inspiral or precursor features and in simulations with limited wave-zone resolution. A common workaround is to extrapolate waveforms in $\mathscr{I}^+$5. Against that baseline, PHE provides asymptotic waveforms without extrapolation in radius because the waveform is read off directly at $\mathscr{I}^+$6 (Bernuzzi et al., 7 Aug 2025).

The comparison with Cauchy-characteristic extraction (CCE) is more nuanced. CCE propagates data from a worldtube to $\mathscr{I}^+$7 using a characteristic evolution of the full Einstein equations and is therefore more first-principles than simple extrapolation, but it is still sensitive to worldtube choice, reconstruction or integration of News to strain, and implementation details. PHE was found to be a simple yet effective alternative that often agrees well with CCE for “optimal” worldtube choices, and in some cases is more robust than CCE when characteristic reconstruction of strain introduces drift or integration ambiguity (Bernuzzi et al., 7 Aug 2025).

The documented applications exhibit both agreement and limitations. In rotating neutron star collapse, PHE performs well and sometimes better than CCE for moderate worldtube radii. In binary black hole circular mergers, the dominant $\mathscr{I}^+$8 mode matches well with CCE up to merger, while finite-radius extraction shows visibly larger phase errors. In a binary neutron star inspiral of about ten orbits to merger, PHE and CCE agree very well, and the phase differences of PHE relative to CCE are small, at the level of $\mathscr{I}^+$9 rad up to merger (Bernuzzi et al., 7 Aug 2025).

6. Limitations, misconceptions, and extensions

A common misconception is to treat PHE as a fully nonlinear asymptotic evolution scheme. The method is explicitly linear perturbation theory on a Schwarzschild background. It does not include nonlinear propagation effects such as nonlinear tails, Christodoulou memory, and nonlinear mode coupling. In particular, the $\Psi^{(e/o)}_{\ell m}$0 memory mode in binary black hole mergers cannot be correctly captured by linear PHE (Bernuzzi et al., 7 Aug 2025).

Another misconception is that hyperboloidal extraction necessarily requires evolving the full Einstein equations on hyperboloidal slices from the strong-field region to $\Psi^{(e/o)}_{\ell m}$1. The broader literature shows several distinct levels of implementation. One may evolve the conformal Einstein equations directly with $\Psi^{(e/o)}_{\ell m}$2 at a fixed coordinate location (0808.0810); evolve test fields or perturbations on compactified hyperboloidal backgrounds (Zenginoglu et al., 2010, Markakis et al., 2023); or use a hybrid strategy in which full 3+1 numerical relativity supplies inner data and a perturbative hyperboloidal exterior carries them to null infinity (Bernuzzi et al., 7 Aug 2025). This suggests that PHE occupies an intermediate position between raw finite-radius extraction and full hyperboloidal or characteristic formulations.

The natural extension identified for PHE is second-order perturbative hyperboloidal evolution, where the master equations acquire quadratic source terms schematically of the form

$\Psi^{(e/o)}_{\ell m}$3

Such an extension could capture mode coupling and memory at perturbative order while retaining hyperboloidal access to $\Psi^{(e/o)}_{\ell m}$4 (Bernuzzi et al., 7 Aug 2025). Related developments in nonlinear metric formulations support that trajectory: strong hyperboloidal compactification in the spherical DF-GHG formulation has extracted scalar-field quasi-normal ringing and tail decay directly at future null infinity from numerical data (Peterson et al., 3 Jun 2025), and free hyperboloidal evolution of the Einstein-Maxwell-Klein-Gordon system on compactified hyperboloidal slices has likewise reached null infinity and extracted signals there (Álvares et al., 22 May 2025).

Within contemporary numerical relativity, the principal significance of PHE is therefore methodological. It combines the simplicity and robustness of perturbative evolution with the correct asymptotics of hyperboloidal compactification, providing a linear, worldtube-to-$\Psi^{(e/o)}_{\ell m}$5 wave-propagation method that is simpler than CCE but offers substantially improved asymptotic waveform access over raw finite-radius extraction (Bernuzzi et al., 7 Aug 2025).