Pith. sign in

REVIEW 3 major objections 5 minor 58 references

Semilinear wave equations in homothetic hyperboloidal coordinates and tail decay

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Homothetic hyperboloidal coordinates turn semilinear wave tails into a fixed smooth profile with one decay rate at every compactified radius, removing the late-time gradient problem and reducing the step count from $O(U)$ to $O(\log U)$.

desk verdict A useful numerical method for tail computations whose strongest advertised claim—uniform exponential rates for every multipole—rests on an unproved self-similar ansatz, and whose own p=7 rates do not fully reproduce the conjecture. read the letter →

arxiv 2608.08863 v1 pith:BKD2WGGT submitted 2026-08-09 gr-qc cs.NAmath.APmath.NA

An{i}l Zenginoğlu , Sebastiano Bernuzzi , Andrea Nützi This is my paper · ORCID
classification gr-qccs.NAmath.APmath.NA MSC 35L0535B4065M70
keywords semilinearwaveequationslate-timetailshomothetichyperboloidalcoordinatesself-similardecaynullinfinitytailratespseudospectralmethodsMinkowskispacetime
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Late-time wave tails decay more slowly along future null infinity than along timelike worldlines at fixed radius, so a compactified numerical grid must represent two different decay laws separated by an increasingly sharp transition. The paper argues that this steepening is an artifact of coordinates adapted to time translations rather than to the tail itself. Using homothetic hyperboloidal coordinates adapted to the scaling symmetry, the tail approaches a fixed smooth radial profile with the same exponential decay rate at every compactified radius; because homothetic time is the logarithm of retarded time, reaching very late retarded times costs only logarithmic numbers of steps. The authors demonstrate the mechanism with 3+1-dimensional pseudospectral evolutions, reproduce the conjectured generic decay rates for several powers and angular modes, and tune a one-parameter family of cubic monopole data to cancel the leading null-infinity coefficient and expose a faster subleading tail.

What carries the argument

The central object is the homothetic hyperboloidal coordinate system $\tau=\ln((t^2-r^2)/(2Rt))$, $\rho=r/t$ on the future cone, with conformal factor $\Omega=1/t$; it is adapted to the dilatation operator $D=t\partial_t+r\partial_r$, the conformal metric is the static patch of de Sitter space, and future null infinity is fixed at $\rho=1$ with $u|_{I^+}=Re^{\tau}$. The mechanism is the assumed leading self-similar tail representation $\chi_\ell(u,\rho)=u^{-q}F_\ell(\rho)+R_\ell(u,\rho)$ with origin-regular $F_\ell=\rho^{\ell+1}K_\ell$ and uniform remainder bounds; substituting $u=2Re^{\tau}/(1+\rho)$ turns this into $\chi_\ell(\tau,\rho)=e^{-q\tau}P_\ell(\rho)+O(e^{-(q+\delta)\tau}\rho^{\ell+1})$, giving the same exponent at every fixed $\rho$ and a smooth profile $Q_\ell=P_\ell/\rho$ for the evolved conformal field. The fixed-radius decay $t^{-(q+\ell+1)}$ follows from the same formula by taking $\rho=r/t\to 0$.

What would settle it

Evolve a compactly supported $\ell=1$, $p=3$ mode to $\tau\approx 10$ at high radial resolution and compare the fitted local exponent $-\partial_\tau\ln|\chi_1|$ at two fixed homothetic radii, say $\rho=0.5$ and $\rho=1$, over a window of width $\Delta\tau=1$; if the exponents differ beyond numerical error, or if the normalized profiles $e^{\tilde q_1\tau}\chi_1(\tau,\rho)$ continue to change shape as $\tau$ grows, the uniform-profile claim fails. Equivalently, compute the same tail in standard retarded coordinates and test whether $u^{q}\chi_\ell(u,\rho)$ becomes independent of $u$ at fixed $\rho$; a residual dependence decaying more slowly than $u^{-\delta}$ would falsify the assumed self-similar representation.

Watch

Extended reading notes

Core claim

The central claim is that an asymptotic radiative mode of a semilinear tail has the homothetic form $\chi_\ell(\tau,\rho)\sim e^{-\tilde q_\ell\tau}P_\ell(\rho)$ at every fixed $0<\rho\le 1$, with a time-independent radial profile and one common decay exponent across the compactified ball. In stationary hyperboloidal coordinates the same physics gives different exponents near null infinity and in the interior, producing an ever-narrowing transition layer. In homothetic coordinates the faster interior decay is not a competing exponent on the grid: it is recovered kinematically through the origin-regularity factor $\rho^{\ell+1}$ as $\rho=r/t\to 0$ along a fixed physical radius. Numerically, local logarithmic rates at 20 extraction radii agree across the compactified interval to about $10^{-5}$ in homothetic coordinates, while stationary runs show rates ranging from about 2 in the interior to about 1 at null infinity; fitted exponents for $p=3,5,7$ and $\ell=0,1,2$ match the conjectured generic rates $\tilde q_\ell=\max(p-2,\ell+1)$.

Load-bearing premise

The argument assumes, rather than proves, that every mode has a leading self-similar tail representation $\chi_\ell(u,\rho)=u^{-q}F_\ell(\rho)+R_\ell(u,\rho)$ with an origin-regular profile $F_\ell=\rho^{\ell+1}K_\ell$ and uniform remainder bounds; this is supported by small-data theory in the spherical case, but for higher multipoles it is exactly the unproved structure that produces the uniform exponential rate and the logarithmic cost, and the fixed-radius rate additionally needs $K_\ell(0)\neq 0$.

Editorial extensions

If this is right

  • For tail-dominated evolutions, reaching a fixed retarded time $U$ at null infinity costs $O(\log U)$ time steps in homothetic coordinates instead of $O(U)$ steps in stationary hyperboloidal coordinates.
  • The normalized late-time radial profile approaches a fixed shape, so a fixed spectral grid remains resolved indefinitely instead of requiring adaptive refinement near null infinity.
  • The generic null-infinity rates $\tilde q_\ell=\max(p-2,\ell+1)$ and the finite-radius rates $q_\ell=\tilde q_\ell+\ell+1$ are reproduced numerically for compactly supported data.
  • Tuning a one-parameter family of cubic monopole data can cancel the leading $e^{-\tau}$ coefficient at null infinity, leaving a resolved $e^{-2\tau}$ tail; generic-rate statements therefore hold on an open dense set, not for all smooth data.
  • Initial data with non-negligible support at null infinity decay as $q_{I^+}^{\mathrm{nc}}=\ell+1$ independent of the nonlinearity power, so tail-rate predictions must specify the data class.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to apply the same profile-collapse diagnostic to other self-similar late-time regimes: if normalized profiles at different $\tau$ collapse onto one curve, logarithmic step counts should follow.
  • For black-hole spacetimes, where the mass breaks exact scale invariance, the likely route is a hybrid scheme matching stationary horizon-penetrating coordinates to an asymptotically homothetic exterior, which the paper identifies as future work.
  • The bisection tuning of the leading tail coefficient suggests a general numerical technique for exposing subleading asymptotic terms in any multipole, beyond the cubic monopole example shown here.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces homothetic hyperboloidal coordinates for the numerical evolution of semilinear wave equations in Minkowski spacetime. It argues that in these coordinates late-time tails take the form chi_l(tau,rho) ~ e^{-q tau} P_l(rho) with the same decay exponent at every fixed compactified radius, so the radial profile remains smooth and late retarded times are reached in a number of time steps that grows logarithmically with u. The authors implement the formulation with a pseudospectral Dedalus ball basis, validate it with convergence and energy-balance tests, compare stationary and homothetic tail extraction for the cubic monopole, survey null-infinity rates for p=3,5,7 and ell=0,1,2, and present a tuned one-parameter family for the cubic monopole in which the leading e^{-tau} coefficient is cancelled and an e^{-2tau} tail is observed.

Significance. The proposed method addresses a real bottleneck in compactified tail computations: the radial steepening caused by observer-dependent decay rates. If the central claim holds, the homothetic formulation offers exponential efficiency in reaching late retarded times and removes the late-time gradient problem, which would be valuable for numerical relativity and for testing tail asymptotics. The paper's strengths include a clear coordinate construction, transparent balance laws, spectral convergence tests in both formulations, and numerical evidence (Fig. 4, Table I) that the method works for the cubic and quintic monopole and for higher modes. The tuned-monopole experiment is a genuinely interesting test of the 'generic but not universal' character of Rinne's conjecture. However, the central statement is analytically conditional: it relies on an assumed self-similar representation that is not proven for ell>=1, and the numerical verification of the uniform-in-radius property is currently limited to the cubic monopole. The p=7 rates also deviate by 5-6%, so the survey is only partially consistent with the conjectured rates.

major comments (3)
  1. [Sec. V C, Eqs. (63)-(65)] The uniform exponential rate and the smooth-profile claim are derived from an assumed leading asymptotic representation chi_l = u^{-q}F_l(rho)+R_l with uniform remainder estimates, including u du R_l = O(u^{-q-delta} rho^{l+1}). The paper explicitly says 'we assume that a leading asymptotic representation exists.' For l=0, the cited small-data estimate (56) supplies the leading power, but I do not see that it supplies the uniform derivative-bound (65); for l>=1 neither the representation nor the remainder estimate is established. Eq. (70), which is the basis for the abstract and Conclusions, is therefore conditional. The authors should either prove or cite a proof of (63)-(65), or downgrade the presentation to a conditional/heuristic statement and mark the ell>=1 efficiency claim as numerically motivated rather than established.
  2. [Sec. VI E, Table I and Fig. 4] The numerical support for the central claim is incomplete. Fig. 4 demonstrates the uniform decay rate only for the cubic monopole, and Table I reports fitted rates only at rho=1. No direct measurement is shown at several interior compactified radii for ell=1,2 or for p=5,7, so the key 'same rate at every fixed compactified radius' property is not verified for the higher multipoles that are central to the claimed advantage. In addition, the p=7 entries are 5.255-5.288 rather than the conjectured 5, i.e., 5-6% off; the manuscript should address whether this is a systematic error, an insufficiently long fitting window, or a genuine discrepancy.
  3. [Eqs. (70)-(71) and Sec. VII] The blanket statement that the tail approaches the same decay rate at every compactified radius is stronger than the derivation. Eq. (70) holds only at radii where P_l(rho) is nonzero, and the fixed-radius rate (71) additionally requires K_l(0) to be nonzero. Neither condition is established for the semilinear higher-multipole solutions considered here. The abstract and Section VII should be reworded to include these qualifications, or the nonvanishing conditions should be verified, at least numerically for the cases presented.
minor comments (5)
  1. [Sec. VI B, Fig. 4 caption] The text says 'medians over all 20 radii are 1.010 to the quoted precision, with a total spread of only 6e-6'; this phrasing is confusing because the top-right panel appears to show a larger spread over the plotted interval. Clarify whether the spread refers only to the late-time window and define what 'quoted precision' means.
  2. [Sec. VI D, Fig. 6] The notation p=0 for the linear equation is nonstandard because the nonlinearity is written as phi^p; a reader may expect p=0 to denote a constant source. Define p=0 explicitly as the linear case in the text and caption.
  3. [Sec. VI F, Eq. (77)] The fit model e^tau a00 = A1 + B e^{-tau} assumes the expansion (75), but no residual or goodness-of-fit statistic is reported for the tuned run. Since the claim of a clean e^{-2tau} window depends on this fit, reporting the residual would strengthen the evidence.
  4. [Sec. IV B, Eq. (40)] The conformal metric is identified with the static patch of de Sitter spacetime, but the Killing vector ∂_tau is not explicitly related to the de Sitter time orientation; a one-sentence clarification would help readers unfamiliar with horizon-fixing coordinates.
  5. [Appendix A, Eqs. (A1)-(A6)] The signs in the source terms S_hom and S_stat are written as '±' with a note that the sign is chosen according to focusing or defocusing nonlinearity; since the paper consistently uses the focusing sign in the main text, it would be clearer to remove the ambiguity by stating the implemented sign explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniform-rate result is explicitly conditional on the self-similar ansatz (63)-(65) and is benchmarked against external conjectures; no fitted parameter is relabeled as a prediction.

full rationale

The paper's central derivation in Sec. V C is not circular because it is transparently conditional: Eq. (63) assumes chi_l = u^{-q}F_l(rho)+R_l with remainder estimates (65), and Eqs. (67)-(70) merely rewrite this assumed form in homothetic coordinates. The conclusion that every fixed rho has the same exponential rate q is algebraically contained in the assumption, but the paper does not claim to prove the assumption there; it labels it 'we assume that a leading asymptotic representation exists' and separately provides numerical evidence (Fig. 4, Table I). The spherical case is anchored to the external small-data bound (56) from Szpak et al. [3,47], and the higher-multipole rates are checked against Rinne's conjectures [29] rather than being fitted to produce them. The only parameter tuning, lambda* in Sec. VI F, is openly a bisection to cancel the leading coefficient (Eqs. 75-77), followed by an independent local-rate measurement; this is a constructed nongeneric example, not a prediction disguised as a fit. Self-citations ([15],[32],[45],[51]) provide the coordinate framework and are re-derived or used as background; they are not invoked as an unverified uniqueness theorem, and no load-bearing claim reduces to an author-only citation. The genuine weakness is rigor, not circularity: the origin-regular profile and uniform derivative remainder estimate (64)-(65) are unproved for l>=1, and Table I reports rates only at rho=1, so the uniform interior profile is directly confirmed only for the cubic monopole. That is an open assumption and a numerical-validation gap, not a circular reduction of the paper's own equations.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on cited asymptotic theorems for spherical tails, on Rinne's conjecture for higher multipoles, on an assumed self-similar tail representation with derivative estimates that is not proven for l>=1, and on a nonzero origin contribution K_l(0). The only numerical quantity tuned to enforce a result is lambda* used to cancel the leading monopole coefficient; the fit windows for exponent extraction are hand-picked. No new physical entities are introduced.

free parameters (2)
  • lambda* (monopole mixing amplitude) = 0.0007546 at (Nr,Ntheta,Nphi)=(144,10,4), resolution-dependent
    Tuned by bisection to make the fitted leading e^{-tau} coefficient A1 vanish at I+; this is the numerical handle for the claimed nongeneric cancellation and is fit to the numerical signal, not independently predicted.
  • Late-time fit windows [tau1, tau2] = e.g., [4.5,7] for p=3,5; [3,4.5] or [3,5] for p=7; [6.8,7.0] for cancellation runs
    The quoted exponents qfit and the p=7 excess are least-squares fits over manually chosen windows; no uncertainty model is given, and the 5-6% p=7 excess is partly window-dependent.
assumptions (5)
  • standard math Small-data decay estimate (56): |phi(t,r)| <= C / ((1+t+r)(1+t-r)^{p-2}) for spherical data, from [3,47]
    Used in Sec V A to show the spherical homothetic profile is smooth and to fix q0=p-2; the inequality is cited, not re-derived.
  • domain assumption Rinne's conjectured multipole rates (61): qtilde_l=max(p-2,l+1) at I+ and q_l=max(l+p-1,2l+2) at fixed radius
    The paper uses these as benchmarks for higher multipoles in Sec V B and Sec VI E; they are not proven theorems for the semilinear problem.
  • ad hoc to paper Leading self-similar tail representation (63)-(65): chi_l=u^{-q}F_l(rho)+R_l with F_l=rho^{l+1}K_l and remainder estimates R_l=O(u^{-q-delta}rho^{l+1}), u du R_l=O(u^{-q-delta}rho^{l+1}), uniformly in rho
    Sec V C assumes this structure to derive the uniform homothetic rate; for l>=1 it is not established, and it is the property that makes the method efficient.
  • domain assumption K_l(0) != 0 in the origin expansion
    Eq (71) uses K_l(0)!=0 to recover the fixed-radius rate q_l; if K_l(0)=0 the leading coefficient vanishes and a fuller expansion controls the decay.
  • domain assumption Compactly supported or rapidly decaying initial data for generic rates
    The conjectured generic rates and the numerical tests in Sec VI E assume data with negligible support at I+; Sec VI D shows that boundary-supported data decay with l+1 independent of p.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Semilinear wave equations in homothetic hyperboloidal coordinates and tail decay." pith.science (2026). https://pith.science/paper/BKD2WGGT

@misc{pith2026260808863,
  author       = {Pith},
  title        = {Pith review of: Semilinear wave equations in homothetic hyperboloidal coordinates and tail decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKD2WGGT}},
  note         = {Machine review of arXiv:2608.08863}
}
read the original abstract

Late-time wave tails decay at different rates along future null infinity and along timelike worldlines at finite radius. A compactified numerical evolution must represent both the slower decay at null infinity and the faster interior decay, producing an increasingly sharp transition between the two regimes. We address this difficulty for semilinear wave equations in Minkowski spacetime using homothetic hyperboloidal coordinates adapted to the scaling structure of the tail. In these coordinates, the tail approaches a smooth radial profile with the same decay rate at every compactified radius. The formulation therefore avoids the steepening of the radial profile seen in stationary hyperboloidal evolutions, and it reaches late times in a number of steps that grows only logarithmically with retarded time. We demonstrate this approach using pseudospectral simulations in 3+1 dimensions and reproduce the generic decay rates conjectured by Rinne. We also provide numerical evidence consistent with a nongeneric codimension-one cancellation of the leading tail coefficient at null infinity, resulting in a faster decay rate.

Figures

Figures reproduced from arXiv: 2608.08863 by the authors.

Figure 1
Figure 1. FIG. 1. Penrose diagrams of Minkowski spacetime for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Coefficient-space self-convergence for the stationary hyperboloidal formulation (top row) and the homothetic formu [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Conformal energy-balance for short linear evolutions. Left: normalized energy and the integrated flux prediction for [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Uniform exponential decay rate of the cubic monopole in homothetic coordinates, compared with stationary hyper [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Computational cost and tail-fit accuracy for the cubic monopole at [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Local decay rates at [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Generic and nongeneric cubic monopole tails at [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 29 canonical work pages

  1. [1]

    Convergence tests Details of the numerical implementation are discussed in App. A. Here, we verify that the radial discretization is spectrally accurate by performing a convergence test on the final-time numerical solution for both the stationary and homothetic formulations. We use the same regular, axisymmetric, origin-centered initial data in all runs, ...

    work page

  2. [2]

    For the energy balance, we set the nonlinear term to zero and evolve the linear equation in both coordinate systems

    Energy-balance diagnostics The integrated identities (28) and (55) provide consistency checks for the numerical evolutions. For the energy balance, we set the nonlinear term to zero and evolve the linear equation in both coordinate systems. We evolve the same regularℓ= 2,m= 1 pulse in the stationary and homothetic formulations, as in [29], with ψ(0, ρ, θ,...

    work page 2000

  3. [3]

    In a direct implementation, regularity is obtained through cancellations between radial and angular terms

    Regularity at the origin The second-order equations (33) and (40) have coefficients of the form 1 ρ (· · ·), 1 ρ2 (· · ·). In a direct implementation, regularity is obtained through cancellations between radial and angular terms. To make the origin regularity explicit in the numerical formulation, we rewrite the equation to make the Laplacian manifest, ∆U...

    work page

  4. [4]

    R. H. Price, Phys. Rev.D5, 2439 (1972)

    1972

  5. [5]

    A note on late-time tails of spherical nonlinear waves

    P. Bizon, T. Chmaj, and A. Rostworowski, Phys. Rev. D78, 024044 (2008), arXiv:0804.0903 [math-ph]

    work page Pith review arXiv 2008

  6. [6]

    Linear and nonlinear tails II: exact decay rates in spherical symmetry

    N. Szpak, P. Bizon, T. Chmaj, and A. Rostworowski, J. Hyperbol. Diff. Equat.6, 107 (2009), arXiv:0712.0493 [math-ph]

    work page Pith review arXiv 2009

  7. [7]

    Harms, S

    E. Harms, S. Bernuzzi, and B. Br¨ ugmann, Class.Quant.Grav.30, 115013 (2013), arXiv:1301.1591 [gr-qc]

    arXiv 2013

  8. [8]

    Albanesi, S

    S. Albanesi, S. Bernuzzi, T. Damour, A. Nagar, and A. Placidi, Phys. Rev. D108, 084037 (2023), arXiv:2305.19336 [gr-qc]

    arXiv 2023

Show all 58 references
  1. [9]

    De Amicis, S

    M. De Amicis, S. Albanesi, and G. Carullo, Phys. Rev. D110, 104005 (2024), arXiv:2406.17018 [gr-qc]

    2024 arXiv

  2. [10]

    Islam, G

    T. Islam, G. Faggioli, G. Khanna, S. E. Field, M. van de Meent, and A. Buonanno, Phys. Rev. D112, 024061 (2025), arXiv:2407.04682 [gr-qc]

    2025

  3. [11]

    Islam, G

    T. Islam, G. Faggioli, and G. Khanna, Phys. Rev. D113, 124025 (2026), arXiv:2511.21898 [gr-qc]

    2026 arXiv

  4. [12]

    Alnasheet, V

    Q. Alnasheet, V. Cardoso, F. Duque, and R. Panosso Macedo, Phys. Rev. D112, 044066 (2025), arXiv:2508.20238 [gr-qc]

    2025 arXiv

  5. [13]

    J. A. L. Vega, A. Svyatkovskyy Kholyavka, S. Datta, and X. J. Forteza, (2026), arXiv:2606.02146 [gr-qc]

    2026 arXiv

  6. [14]

    De Amiciset al., Phys

    M. De Amiciset al., Phys. Rev. Lett.135, 171401 (2025), arXiv:2412.06887 [gr-qc]

    2025

  7. [15]

    S. Ling, S. Shah, and S. S. C. Wong, Phys. Rev. D112, 024008 (2025), arXiv:2503.19967 [gr-qc]

    2025 arXiv

  8. [16]

    Ling and S

    S. Ling and S. S. C. Wong, (2026), arXiv:2603.20379 [gr-qc]

    2026

  9. [17]

    Zengino˘ glu, Class

    A. Zengino˘ glu, Class. Quant. Grav.25, 175013 (2008), arXiv:0803.2018 [gr-qc]

    2008 arXiv

  10. [18]

    Zengino˘ glu, Class

    A. Zengino˘ glu, Class. Quant. Grav.25, 145002 (2008), arXiv:0712.4333 [gr-qc]

    2008 arXiv

  11. [19]

    Zengino˘ glu, D

    A. Zengino˘ glu, D. Nunez, and S. Husa, Class. Quant. Grav.26, 035009 (2009), arXiv:0810.1929 [gr-qc]

    2009 arXiv

  12. [20]

    Zenginoglu, Class

    A. Zenginoglu, Class. Quant. Grav.27, 045015 (2010), arXiv:0911.2450 [gr-qc]

    2010 arXiv

  13. [21]

    Jasiulek, Class.Quant.Grav.29, 015008 (2012), arXiv:1109.2513 [gr-qc]

    M. Jasiulek, Class.Quant.Grav.29, 015008 (2012), arXiv:1109.2513 [gr-qc]

    2012 arXiv

  14. [22]

    R´ acz and G

    I. R´ acz and G. Z. T´ oth, Class.Quant.Grav.28, 195003 (2011), arXiv:1104.4199 [gr-qc]

    2011 arXiv

  15. [23]

    Bernuzzi, A

    S. Bernuzzi, A. Nagar, and A. Zenginoglu, Phys.Rev.D86, 104038 (2012), arXiv:1207.0769 [gr-qc]

    2012 arXiv

  16. [24]

    Zengino˘ glu, G

    A. Zengino˘ glu, G. Khanna, and L. M. Burko, Gen.Rel.Grav.46, 1672 (2014), arXiv:1208.5839 [gr-qc]

    2014 arXiv

  17. [25]

    Harms, S

    E. Harms, S. Bernuzzi, A. Nagar, and A. Zenginoglu, Class.Quant.Grav.31, 245004 (2014), arXiv:1406.5983 [gr-qc]

    2014 arXiv

  18. [26]

    Csuk´ as, I

    K. Csuk´ as, I. R´ acz, and G. Z. T´ oth, Phys. Rev. D100, 104025 (2019), arXiv:1905.09082 [gr-qc]

    2019 arXiv

  19. [27]

    Zengino˘ glu, J.Comput.Phys.230, 2286 (2011), arXiv:1008.3809 [math.NA]

    A. Zengino˘ glu, J.Comput.Phys.230, 2286 (2011), arXiv:1008.3809 [math.NA]

    2011 arXiv

  20. [28]

    Zengino˘ glu and L

    A. Zengino˘ glu and L. E. Kidder, Phys. Rev.D81, 124010 (2010), arXiv:1004.0760 [gr-qc]

    2010 arXiv

  21. [29]

    Hilditch, E

    D. Hilditch, E. Harms, M. Bugner, H. R¨ uter, and B. Br¨ ugmann, Class. Quant. Grav.35, 055003 (2018), arXiv:1609.08949 [gr-qc]

    2018 arXiv

  22. [30]

    Gautam, A

    S. Gautam, A. Va˜ n´ o-Vi˜ nuales, D. Hilditch, and S. Bose, Phys. Rev. D103, 084045 (2021), arXiv:2101.05038 [gr-qc]

    2021 arXiv

  23. [31]

    Peterson, S

    C. Peterson, S. Gautam, I. Rainho, A. Va˜ n´ o-Vi˜ nuales, and D. Hilditch, Phys. Rev. D108, 024067 (2023), arXiv:2303.16190 [gr-qc]

    2023 arXiv

  24. [32]

    Rinne, Nonlinearity38, 105026 (2025), arXiv:2507.00674 [cs.NA]

    O. Rinne, Nonlinearity38, 105026 (2025), arXiv:2507.00674 [cs.NA]

    2025

  25. [33]

    Reddy, S

    A. Reddy, S. Gautam, and P. Kumar, (2026), arXiv:2606.02051 [gr-qc]

    2026 arXiv

  26. [34]

    Gundlach, R

    C. Gundlach, R. H. Price, and J. Pullin, Phys.Rev.D49, 883 (1994), arXiv:gr-qc/9307009 [gr-qc]

    1994 arXiv

  27. [35]

    Donninger and A

    R. Donninger and A. Zengino˘ glu, Anal. Part. Diff. Eq.7, 461 (2014), arXiv:1304.4135 [math.AP]

    2014 arXiv

  28. [36]

    A. Y. Burtscher and R. Donninger, arXiv e-prints , arXiv:1511.08600 (2015), arXiv:1511.08600 [math.AP]

    2015 arXiv

  29. [37]

    Bonk and R

    A. Bonk and R. Donninger, arXiv e-prints , arXiv:2603.01924 (2026), arXiv:2603.01924 [math.AP]

    2026

  30. [38]

    Bizon and A

    P. Bizon and A. Zenginoglu, Nonlinearity22, 2473 (2009), arXiv:0811.3966 [math.AP]

    2009 arXiv

  31. [39]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett.10, 66 (1963)

    1963

  32. [40]

    Frauendiener, Living Rev.Rel.3, 4 (2000)

    J. Frauendiener, Living Rev.Rel.3, 4 (2000)

    2000

  33. [41]

    R. H. Gowdy, Journal of Mathematical Physics22, 675 (1981)

    1981

  34. [42]

    Conformally regular ADM evolution equations,

    V. Moncrief, “Conformally regular ADM evolution equations,” (2000), talk at Santa Barbara, http://online.itp.ucsb.edu/online/numrel00/moncrief

    2000

  35. [43]

    Fodor and I

    G. Fodor and I. Racz, Phys. Rev. Lett.92, 151801 (2004), arXiv:hep-th/0311061

    2004 arXiv

  36. [44]

    Friedrich, Comm

    H. Friedrich, Comm. Math. Phys.91, 445 (1983)

    1983

  37. [45]

    Panosso Macedo, Phil

    R. Panosso Macedo, Phil. Trans. Roy. Soc. Lond. A382, 20230046 (2024), arXiv:2307.15735 [gr-qc]

    2024 arXiv

  38. [46]

    Va˜ n´ o-Vi˜ nuales and T

    A. Va˜ n´ o-Vi˜ nuales and T. Valente, Gen. Rel. Grav.56, 135 (2024), arXiv:2408.08952 [gr-qc]

    2024 arXiv

  39. [47]

    Zengino˘ glu, Gen

    A. Zengino˘ glu, Gen. Rel. Grav.57, 75 (2025), arXiv:2502.08581 [gr-qc]

    2025 arXiv

  40. [48]

    N¨ utzi, (2025), arXiv:2510.01964 [gr-qc]

    A. N¨ utzi, (2025), arXiv:2510.01964 [gr-qc]

    2025

  41. [49]

    M. K. Parikh, Phys. Lett. B546, 189 (2002), arXiv:hep-th/0204107

    2002 arXiv

  42. [50]

    Szpak, arXiv e-prints , arXiv:0907.4287 (2009), arXiv:0907.4287 [math-ph]

    N. Szpak, arXiv e-prints , arXiv:0907.4287 (2009), arXiv:0907.4287 [math-ph]

    2009 arXiv

  43. [51]

    Szpak, Journal of Mathematical Physics51, 082901 (2010), arXiv:0909.1264 [math-ph]

    N. Szpak, Journal of Mathematical Physics51, 082901 (2010), arXiv:0909.1264 [math-ph]

    2010 arXiv

  44. [52]

    Angelopoulos, S

    Y. Angelopoulos, S. Aretakis, and D. Gajic, Adv. Math.323, 529 (2018), arXiv:1612.01566 [math.AP]

    2018 arXiv

  45. [53]

    Luk and S.-J

    J. Luk and S.-J. Oh, (2024), arXiv:2404.02220 [gr-qc]

    2024 arXiv

  46. [54]

    N¨ utzi,Maurer-Cartan perturbation theory and scattering amplitudes in general relativity, Ph.D

    A. N¨ utzi,Maurer-Cartan perturbation theory and scattering amplitudes in general relativity, Ph.D. thesis, Zurich, ETH 23 (2023)

    2023

  47. [55]

    Zengino˘ glu and M

    A. Zengino˘ glu and M. Tiglio, Phys. Rev.D80, 024044 (2009), arXiv:0906.3342 [gr-qc]

    2009 arXiv

  48. [56]

    K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet, and B. P. Brown, Physical Review Research2, 023068 (2020), arXiv:1905.10388 [astro-ph.IM]

    2020 arXiv

  49. [57]

    Lecoanet, G

    D. Lecoanet, G. M. Vasil, K. J. Burns, B. P. Brown, and J. S. Oishi, Journal of Computational Physics: X3, 100012 (2019)

    2019

  50. [58]

    Vasil, D

    G. Vasil, D. Lecoanet, K. Burns, J. Oishi, and B. Brown, (2018), arXiv:1804.10320 [math.NA]

    2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.