Direct Wave in Black-Hole Ringdown

• Direct wave is a prompt signal component in gravitational-wave ringdown, distinct from quasinormal modes and tied to the plunging source's dynamics.
• Empirical extraction using numerical relativity and rational filtering reveals discrepancies in frequency and damping compared to horizon-mode predictions.
• Beyond ringdown, 'direct' refers to immediate responses in chaotic scattering and first-principles simulations, illustrating its context-dependent usage across wave physics.

Searching arXiv for the key "Direct Wave" usages and the latest black-hole ringdown papers. “Direct wave” does not denote a single universal object across the research literature. In the most technically developed recent usage, it denotes a distinct non–quasinormal-mode component of binary-black-hole merger radiation, interpreted as prompt signal from a plunging source rather than as linear Kerr ringing. In closely related wave-scattering usage, “direct” or “prompt” processes denote short-path, non-equilibrated contributions that sit on top of universal chaotic scattering statistics. In adjacent methodological literatures, “direct” instead labels first-principles simulation, direct measurement, or direct field-targeted design. The term is therefore best understood contextually, with the black-hole ringdown usage providing the sharpest contemporary physical definition (Dyer et al., 23 Jun 2026, Kankani et al., 2 Jul 2026).

1. Scope and principal usages

In the cited literature, the term appears in several non-equivalent ways. The common element is immediacy: either prompt propagation from source to observer, or the elimination of an intermediate inference layer.

Domain	Meaning of “direct wave” or closest equivalent	Representative papers
Binary-black-hole ringdown	Non-QNM prompt post-merger component	(Dyer et al., 23 Jun 2026, Kankani et al., 2 Jul 2026, Chung et al., 4 Jun 2026)
Chaotic wave scattering	Prompt response from imperfect coupling or short paths	(Méndez-Sánchez et al., 2011)
Adjacent methodological usage	Direct measurement, direct field targeting, or direct first-principles treatment	(Nguyen et al., 2019, Wen et al., 2021, Liu et al., 2021, Cao et al., 2023, Li et al., 12 Dec 2025)

This distribution of meanings suggests that “direct wave” is not a field-independent technical noun. In gravitational-wave ringdown it names a specific signal component; in scattering theory it refers to prompt, non-equilibrated contributions; and in several neighboring literatures “direct” modifies the methodology rather than the wave itself (Méndez-Sánchez et al., 2011, Li et al., 12 Dec 2025).

2. Direct wave in binary-black-hole ringdown

In recent black-hole merger studies, the direct wave is a component of the post-merger signal that exists alongside the usual quasinormal modes (QNMs). In the Green-function picture used for black-hole perturbations, QNMs arise from poles of the Green’s function and late-time tails arise from a branch cut, whereas the direct wave is associated with the two quarter-circular arcs of the contour and is “the signal that propagates directly from the source to the observer” (Dyer et al., 23 Jun 2026).

This physical distinction is central. A QNM is an intrinsic oscillation of the remnant Kerr black hole, with complex frequency fixed by the remnant mass and spin. By contrast, the direct wave is source-linked: it is tied to the plunging object and therefore depends on the particle trajectory as well as on the background spacetime. In numerical-relativity analyses it is treated as a non-QNM residual uncovered after applying rational filters designed to remove known QNM content from the waveform (Kankani et al., 2 Jul 2026).

For the gravitational-wave strain expanded in spin-weight $-2$ spherical harmonics,

$$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$

the direct-wave contribution is modeled phenomenologically as

$$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$

and the full ringdown model is

$h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$

The usual “horizon mode” ansatz identifies

$\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$

with Kerr horizon quantities

$$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$

For the dominant $(2,2)$ harmonic, this predicts oscillation near $2\Omega_{\rm H}$ and damping set by $\kappa$ (Dyer et al., 23 Jun 2026).

A complementary extraction route uses SXS numerical-relativity strain data and rational filters. One implementation removes 7 prograde QNMs, 2 retrograde QNMs, and the $(l=3,m=2,n=0)$ prograde QNM required to address spherical-spheroidal mixing in SXS data; the filtered residual is then interpreted as the direct-wave component (Kankani et al., 2 Jul 2026).

3. Empirical extraction and the horizon-interpretation debate

The direct wave has been identified in numerical-relativity ringdowns using start-time-dependent Bayesian mode-content analysis, Gaussian-process noise modeling, evidence-based sequential mode selection, amplitude-stability checks, free-frequency inference, and rational filtering. In the CCE:0004 simulation, the $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$0 direct wave was reported as confidently identified with significance $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$1 for start times up to $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$2, with amplitude approximately a factor of ten below the fundamental QNM; at $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$3, the supplement gives $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$4 (Dyer et al., 23 Jun 2026).

That evidence does not settle the stronger claim that the direct wave is a faithful probe of remnant horizon properties. In the same study, when the direct-wave frequency is allowed to float, the recovered value is only close to the horizon-mode prediction and drifts by $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$5 over time. The resulting assessment is deliberately qualified: the horizon mode is a useful phenomenological model, but the direct wave does not yet function as a precise direct probe of event-horizon geometry (Dyer et al., 23 Jun 2026).

A sharper critique was then made using comparable-mass SXS simulations spanning remnant spins from about $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$6 to $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$7. There, the direct-wave frequency extracted from rationally filtered strain data was found not to be generically correlated with the horizon frequency $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$8. For smaller remnant spins in the sample, the direct-wave frequency is much larger than $$r\mathfrak{h}(t,\theta,\phi)=M\sum_\beta \mathfrak{h}^\beta(t)\,{}_{-2}Y_\beta(\theta,\phi),$$9; for higher remnant spins it becomes much smaller; and the apparent agreement near $$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$0 is interpreted as an incidental crossing rather than a physical horizon-frequency relation (Kankani et al., 2 Jul 2026).

The damping behavior is even less compatible with a simple horizon interpretation. Writing

$$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$1

the real part is described as quasi-stable, but the imaginary part changes substantially over short post-merger intervals. This implies that a single damped sinusoid with constant complex frequency can only fit over sufficiently short windows, and that “the damping time is set by the surface gravity” is not a useful characterization for the comparable-mass mergers studied (Kankani et al., 2 Jul 2026).

The high-spin case is particularly decisive in this critique. For SXS:BBH:0178, with $$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$2, the instantaneous direct-wave frequency lies significantly below $$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$3, the evolving horizon-based model tracks the wrong real frequency, and its imaginary part evolves too slowly compared with the filtered numerical-relativity direct wave. The authors therefore conclude that a horizon-tied evolving-frequency model fails for large remnant spin, even though stronger frame dragging might have suggested the opposite expectation (Kankani et al., 2 Jul 2026).

4. Hawking-area inference and the GW250114 controversy

Direct-wave phenomenology was used to define a Kerr-equivalent near-merger area measurement for GW250114. In that construction, the post-start-time waveform is modeled as

$$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$4

with a direct-wave term

$$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$5

Interpreting $$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$6 and $$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$7, the fitted horizon quantities are mapped to a Kerr-equivalent area through

$$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$8

Applied to GW250114, analyses initiated $$rh_{\rm DW}(t,\theta,\phi)=M\sum_\beta C^\beta_{\rm DW} e^{-i\omega^\beta_{\rm DW}(t-t_0)}\,{}_{-2}Y_\beta(\theta,\phi),$$9–$h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$0 before the peak-amplitude time yielded an area posterior overlapping the late-time ringdown area and the Kerr prediction based on $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$1 and $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$2. In that start-time window, the posterior for $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$3 had substantial support near zero, with $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$4, so no strong evidence against Hawking’s area law was found at leading order (Chung et al., 4 Jun 2026).

The same study also emphasized its own limitations. The inferred quantity is an effective Kerr-equivalent area, not the exact dynamical event-horizon area during merger; the interpretation is start-time dependent; a localized fluctuation appears at $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$5; and resolving the genuinely positive first-order area increment for a GW250114-like event was estimated to require direct-wave signal-to-noise ratio of order $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$6 (Chung et al., 4 Jun 2026).

The critique in (Kankani et al., 2 Jul 2026) directly challenges the parameter identification underlying this area inference. If fitted direct-wave frequency and damping are not reliable measurements of $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$7 and $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$8, then $h_{\rm ringdown}=h_{\rm QNM}+h_{\rm DW}.$9 need not equal the true Kerr horizon area even when computed exactly from the fit. Using numerical-relativity remnants, the authors show that direct-wave-based area estimates can become inconsistent with the actual remnant area and can produce apparent violations of Hawking’s area law even though general relativity is perfectly valid in the simulation. They note that consistency appears only in a limited region, roughly $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$0, and even there can result from compensating errors in $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$1 and $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$2 rather than from correct horizon inference (Kankani et al., 2 Jul 2026).

For GW250114 specifically, both sides of the debate focus on the remnant spin near $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$3. One interpretation treats the near-agreement between fitted direct-wave frequency and horizon frequency as evidence that a near-merger horizon probe is possible; the other treats it as a coincidence associated with the incidental crossing near $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$4 (Chung et al., 4 Jun 2026, Kankani et al., 2 Jul 2026).

5. Prompt-response analogues in other wave theories

The closest physical analogue outside black-hole ringdown is the theory of direct processes in chaotic wave scattering. There, direct processes are prompt responses associated with imperfect coupling, direct reflection at the antenna, or short paths that do not sample the chaotic cavity ergodically. They are encoded by a nonzero average scattering matrix,

$\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$5

and in the one-channel lossless case they modify the phase distribution from the uniform circular-ensemble law to the Poisson kernel,

$\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$6

With absorption, the same direct contribution can mimic changes in the absorption strength $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$7, which is why the paper emphasizes that direct processes introduce a second timescale and alter the meaning of weak and strong absorption: $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$8 and $\omega^\beta_{\rm DW}=m\Omega_{\rm H}-i\kappa,$9, with $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$0 (Méndez-Sánchez et al., 2011).

Elsewhere in wave physics, “direct” more often marks first-principles treatment or direct observation rather than a prompt post-merger component. Direct numerical simulations of capillary wave turbulence solve the full 3D two-phase Navier–Stokes equations and recover a stationary direct capillary cascade with spectra consistent with

$$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$1

together with the time-scale ordering $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$2 over the inertial interval (Deike et al., 2014). Direct numerical simulation of weak gravitational-wave turbulence similarly shows a dual cascade, with a direct energy cascade toward high wavenumber and an inverse cascade of wave action, compatible with

$$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$3

respectively (Galtier et al., 2021). In high-energy-density physics, XFEL phase-contrast imaging has directly resolved shock-wave splitting in diamond, revealing an elastic precursor traveling at $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$4 and a plastic wave slowing from $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$5 to $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$6 (Makarov et al., 2022).

These uses suggest a broader semantic pattern: in wave physics, “direct” frequently denotes either prompt propagation not mediated by an equilibrated or modal response, or observation and simulation that avoid indirect reconstruction (Deike et al., 2014, Makarov et al., 2022).

6. Adjacent methodological uses of “direct” in wave-related research

In adjacent literatures, “direct” often modifies the measurement or design paradigm rather than naming a prompt wave component. In quantum optics, direct measurement of a photon’s transverse spatial wave function reconstructs

$$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$7

point by point through system–pointer coupling and post-selection. A modified strong-measurement scheme inserts a liquid crystal plate after the coupling so that the signal branch is transformed into the post-selected momentum mode, removing the factor $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$8 from the right-hand side of the reconstruction formula and yielding a magnification $$\Omega_{\rm H}=\frac{M_f^{-1}\chi_f}{2\left(1+\sqrt{1-\chi_f^2}\right)},\qquad \kappa=\frac{M_f^{-1}\sqrt{1-\chi_f^2}}{2\left(1+\sqrt{1-\chi_f^2}\right)}.$$9; for parameters from the 2011 Lundeen experiment the paper estimates $(2,2)$0 and $(2,2)$1 (Wen et al., 2021). A complementary metrological treatment reformulates direct tomography as estimation of a complex phase $(2,2)$2, with Heisenberg-limited precision available in principle through GHZ-like or Dicke-state pointer resources (Nguyen et al., 2019).

In computational wave and transport modeling, directness can denote algorithmic coupling rather than spatial decomposition. Direct Waveform Inversion (DWI) for layered media reconstructs velocity and density recursively from shallow to deep by enforcing wavefield causality in the time-space domain, using the earliest unresolved event to infer the next interface and local reflectivity rather than solving a global full-waveform optimization problem (Liu et al., 2021). The direct unified wave-particle (DUWP) method likewise couples an NS solver and DSMC directly at the algorithmic level: when $(2,2)$3, a fraction

$(2,2)$4

of particles is absorbed into the wave part, while the remaining particle part is advanced kinetically; the resulting multiscale Boltzmann equation mixes the full Boltzmann collision operator and the linearized operator with weights determined by $(2,2)$5 (Cao et al., 2023).

In inverse design, direct wave-shaping (DWS) topology optimization for elastic metasurfaces bypasses intermediary concepts such as generalized Snell’s law, target impedance profiles, or prescribed diffraction orders, and instead optimizes the monolithic geometry against the desired wavefield itself. With movable, deformable, interactable elliptical voids, the framework was used to realize large-angle longitudinal-to-transverse beam steering, wavelength-multiplexed steering, and reflective and transmissive metalenses with numerical apertures exceeding $(2,2)$6 (Li et al., 12 Dec 2025).

Taken together, these adjacent usages reinforce the contextual nature of the term. In black-hole ringdown, the direct wave is a physically distinct prompt signal component; in scattering theory it is a prompt, non-equilibrated contribution; and in methodological settings, “direct” denotes elimination of intermediary modeling, reconstruction, or conversion layers rather than a unique wave object (Cao et al., 2023, Li et al., 12 Dec 2025).