Finite-Radius Extrapolation

Waveforms measured on finite Cauchy spheres contain near-zone, gauge, tetrad, and discretization effects in addition to radiation. Extrapolation estimates the leading asymptotic coefficient by comparing several radii at a common retarded time. It is controlled only when the radius coordinate and time alignment are physically motivated, the sampled spheres lie in an asymptotic regime, and the inferred limit is stable against fit order and radius window.

Helpful background. Formulations of General Relativity supplies gauge and asymptotic variables; Differential Equations for Relativity supplies interpolation and least-squares methods; Black-Hole Geometry and Mechanics supplies the strong-field scale; Gravitational Radiation and Waveforms defines the waveform modes being extrapolated; and Computational Relativity supplies convergence and uncertainty practice.

The asymptotic fit

Suppose a complex strain mode $h_{\ell m}$ or curvature mode $\Psi_{4}^{\ell m}$ has been extracted on several coordinate spheres. First compute the areal radius

$$r_{\mathrm A}=\sqrt{\frac{A}{4\pi}},$$

rather than treating the coordinate label as a physical radius. Next define a common retarded time $u=t_{\mathrm{corr}}-r_*$, with the tortoise-like radius and corrected time declared explicitly. For an approximately Schwarzschild exterior one often uses

$$r_*=r_{\mathrm A}+2GM\ln\left(\frac{r_{\mathrm A}}{2GM}-1\right),$$

but this is an asymptotic modeling choice, not an identity in a dynamical gauge.

At fixed $u$, fit a rescaled strain mode as

$$r_{\mathrm A}h_{\ell m}(u,r_{\mathrm A}) = \sum_{k=0}^{N}\frac{H_{\ell m}^{(k)}(u)}{r_{\mathrm A}^{k}},$$

so $H_{\ell m}^{(0)}$ is the extrapolated waveform. A curvature expansion has the corresponding leading radial scaling. Dimensionless variants replace $1/r_{\mathrm A}$ by a wavelength or mass scale divided by radius. Boyle and Mroué construct and compare phase–amplitude and complex-waveform procedures in Boyle and Mroué 2009, §II, Eqs. (1)–(16), and §II.D, Figs. 3–6.

The fit assumes that all radii sample the same outgoing feature at the same $u$ and that a low-order inverse-radius expansion describes the remaining dependence. Time interpolation therefore belongs to the algorithm. Near merger, mode amplitude can approach zero and phase becomes poorly defined; fitting the complex mode may then be safer. During long inspiral, phase and amplitude fits can be better conditioned. The choice must be stated per mode and interval.

A two-radius calculation

At one fixed retarded time, write $F(r)=rh_{\ell m}=H+a/r$ and take $r_2=2r_1$. Eliminating $a$ gives

$$H=2F(2r_1)-F(r_1).$$

For $F(r_1)=1.02L$ and $F(2r_1)=1.01L$, where $L$ is a chosen length unit, the inferred limit is $H=L$. The arithmetic is simple; the assumption that both samples refer to the same outgoing feature is the difficult part.

This also exposes two error mechanisms. If the exact data contain $b/r^2$, the inferred limit is biased by $-b/(2r_1^2)$. If the two samples instead have independent errors of standard deviation $\sigma$, the inferred standard deviation is $\sqrt5\,\sigma$. Extrapolation removes a leading radial bias while amplifying noise. Additional radii and held-out-radius tests distinguish the two effects.

Order and radius are two controls

Use several nested radius windows, for example all resolved spheres, the outermost six, and the outermost four. For each window sweep fit orders $N=1$ through $5$ when enough radii remain. A small least-squares residual is not sufficient: higher order always has more freedom and can amplify interpolation noise while moving $H^{(0)}$ substantially. Record the design-matrix condition number or an equivalent conditioning diagnostic.

Two variations answer different questions. Raising $N$ tests truncation of the assumed asymptotic series. Removing inner radii tests entry into the asymptotic regime. If successive orders agree only when the innermost spheres are removed, near-zone contamination was important. If outer-only fits become noisy, the radial lever arm is too short. The preferred result should occupy a stability region in both axes rather than be chosen solely because one residual is smallest.

Cauchy resolution is a third axis. Repeat the complete extraction and fit at no fewer than three evolution resolutions. Interpolating many radii from one under-resolved evolution does not create independent information. Aligning waveforms by an optimized time or phase shift can hide a physical or gauge difference, so report both the alignment rule and unaligned diagnostic. The broader extraction systematics are classified in Bishop and Rezzolla 2016, §§4–7.

Benchmarks and an independent limit

A transparent implementation benchmark supplies synthetic radial data with known coefficients through a chosen order and adds controlled noise. It checks the time interpolation, radial scaling, complex least-squares solve, and recovery of $H^{(0)}$. A physical benchmark uses a linear outgoing perturbation of Schwarzschild or a Teukolsky wave sampled on multiple radii. Compare the extrapolated amplitude and phase with the known asymptotic solution across the resolution sequence.

For a production binary waveform, Cauchy–characteristic extraction provides a more independent comparison because it evolves worldtube data to future null infinity instead of assuming a polynomial radial form. Agreement between extrapolation and characteristic extraction within their separately estimated errors supports the result. Disagreement should be resolved as a function of mode, time, worldtube location, and extrapolation window; neither method is automatically the reference at finite numerical resolution.

An uncertainty budget can keep four components visible: Cauchy truncation, finite-radius fit variation, time/frame alignment, and the difference from characteristic extraction where available. These components are correlated, so quadrature addition requires justification. For small subdominant modes, absolute errors can be more informative than a relative error that diverges at zero crossings.

Reproduction and failure criteria

The environment record should pin the input waveform checksums, extraction implementation, floating-point precision, interpolation and least-squares libraries, harmonic convention, tetrad, radius definition, time correction, and software revision. Inputs include every extraction sphere, discarded time interval, radius window, polynomial order, weighting, and alignment rule. Outputs should retain the unextrapolated modes, fit coefficients, residuals, conditioning, leave-one-radius-out results, and convergence tables.

A clean future rerun would take a versioned linear-wave or binary dataset, sweep nested radius windows and orders $1$–$5$ at three Cauchy resolutions, and emit comparisons with a declared analytic waveform or characteristic result. This page supplies a reproducibility specification, not evidence that an absent program has run.

Reject the asymptotic claim when results fail to stabilize under both order and window variation, when the fit is ill-conditioned, when inner and outer windows imply inconsistent limits, when time alignment dominates the phase difference, or when increasing Cauchy resolution does not reduce the discrepancy. Extrapolation overfit is the central failure mode: a smooth curve and tiny residual can coexist with a wrong intercept at $1/r_{\mathrm A}=0$.

Check your understanding

Orders $N=1,2,3$ give nearly identical extrapolated phases using radii $80GM$–$200GM$. Order $N=5$ lowers the fit residual by a factor of one hundred but moves the phase by $0.08$ radians; removing the two inner radii makes its value move again. Which estimate is defensible?

Solution

The stable low-order cluster is the defensible estimate, with its order and window spread reported. The fifth-order residual measures interpolation of the supplied finite-radius samples, not accuracy of the intercept at infinity. Its sensitivity to removing inner radii is evidence of overfit or an inadequate asymptotic window. Confirm the choice across Cauchy resolutions and, if possible, against characteristic extraction before assigning the final systematic uncertainty.

Continue with Cauchy–Characteristic Extraction.

References

• Bishop, Nigel T., and Luciano Rezzolla. 2016. “Extraction of Gravitational Waves in Numerical Relativity.” Living Reviews in Relativity 19, 2. arXiv.
• Boyle, Michael, and Abdul H. Mroué. 2009. “Extrapolating Gravitational-Wave Data from Numerical Simulations.” Physical Review D 80, 124045. arXiv.