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arxiv: 2604.13516 · v2 · submitted 2026-04-15 · 🌀 gr-qc · hep-th

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Quasinormal Modes of pp-Wave Spacetimes and Zero Temperature Dissipation

Huayu Dai , Guangtao Zeng
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:05 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords quasinormal modespp-wave spacetimesKaigorodov metriczero temperature dissipationscalar perturbationsirregular singular pointsholographyAdS/CFT
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The pith

pp-Wave spacetimes exhibit zero-temperature dissipation for scalar modes when d is at least 3

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

The paper computes quasinormal modes of scalar perturbations on Kaigorodov pp-wave spacetimes, which are horizonless gravity duals of zero-temperature null fluids. For spacetime dimensions d of 3 or higher, every mode frequency has negative imaginary part, so the perturbations decay even though no event horizon or entropy is present. The leading term of the radial equation reduces to Bessel's equation of order d over d plus 2, which forces the spectrum to be gapped. In the two-dimensional case the equation becomes the Whittaker equation and the spectrum is exactly non-dissipative. Numerical results for d equals 3, 4 and 5 confirm a discrete tower of decaying modes together with linear stability of the background.

Core claim

The pp-wave deformation promotes the Poincaré horizon at r equals 0 to an irregular singular point of rank (d plus 2) over 2. This point acts as a geometric absorber for ingoing waves. As a result, when d is at least 3 every scalar quasinormal mode satisfies Im(ω_n) less than 0, establishing zero-temperature dissipation without horizon or entropy. At zeroth order the radial equation is Bessel's equation of order μ equals d over (d plus 2), proving the modes are gapped. For d equals 2 the equation reduces to the Whittaker equation and yields an exact non-dissipative spectrum with Im(ω) equals 0.

What carries the argument

The irregular singular point of rank (d+2)/2 at r=0, which functions as a geometric absorber for ingoing waves under the chosen boundary conditions

If this is right

  • All scalar quasinormal modes form a discrete gapped dissipative tower for d greater than or equal to 3.
  • Linear stability of the background is established by the negative imaginary parts of all frequencies.
  • Zero-temperature dissipation is realized in horizonless geometries dual to null fluids.
  • The two-dimensional case recovers the known non-dissipative spectrum of extremal BTZ black holes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The rank of the irregular singularity may control whether dissipation appears at zero temperature.
  • The same mechanism could operate in other horizonless AdS geometries used for holographic models.
  • The leading Bessel approximation supplies an analytic estimate for the gap that could be compared with lattice data in the dual field theory.

Load-bearing premise

That the irregular singular point at r=0 functions as an absorber for ingoing waves with the boundary conditions chosen in the analysis.

What would settle it

Discovery of any quasinormal mode with non-negative imaginary part for d greater than or equal to 3 would falsify the claim that dissipation occurs universally in these spacetimes.

read the original abstract

We compute the quasinormal mode spectrum of scalar perturbations on Kaigorodov pp-wave spacetimes, the horizonless gravity duals of zero temperature null fluids. The pp-wave deformation promotes the Poincar\'e horizon at $r=0$ to an irregular singular point of rank $(d+2)/2$, which acts as a geometric absorber for ingoing waves: rank~$0$ corresponds to thermal dissipation, rank~$1$ to quantum-critical (extremal black hole), and rank~$\geq 2$ to gapped, horizonless dissipation. For $d=2$ (extremal BTZ) the radial equation reduces to the Whittaker equation with exact non-dissipative spectrum $\mathrm{Im}(\omega)=0$; for $d \geq 3$ all modes satisfy $\mathrm{Im}(\omega_n) < 0$, establishing zero temperature dissipation without horizon or entropy. At zeroth order the radial equation becomes Bessel's equation of order $\mu=d/(d+2)$, proving all scalar QNMs are gapped. Numerical spectra for $d=3,4,5$ yield a discrete dissipative tower and confirm linear stability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript computes the quasinormal mode spectrum of scalar perturbations on Kaigorodov pp-wave spacetimes, horizonless gravity duals to zero-temperature null fluids. The pp-wave deformation is argued to promote the Poincaré horizon at r=0 to an irregular singular point of rank (d+2)/2 that acts as a geometric absorber for ingoing waves. For d=2 the radial equation reduces exactly to the Whittaker equation yielding a non-dissipative spectrum with Im(ω)=0. For d≥3 all modes satisfy Im(ω_n)<0, establishing zero-temperature dissipation without horizon or entropy. At zeroth order the radial equation reduces to Bessel's equation of order μ=d/(d+2), proving the spectrum is gapped; numerical spectra for d=3,4,5 confirm a discrete dissipative tower and linear stability.

Significance. If the boundary-condition selection at the irregular singular point and the control of higher-order corrections are rigorously justified, the result would be significant: it supplies an explicit analytic and numerical example of dissipation at zero temperature in a horizonless spacetime, without invoking entropy or a horizon. The exact d=2 reduction and the clean Bessel approximation at leading order are attractive features that could inform holographic models of non-equilibrium null fluids.

major comments (3)
  1. [Boundary conditions at the irregular singular point r=0] The central claim that Im(ω_n)<0 for all scalar modes when d≥3 rests on the irregular singular point at r=0 functioning as a geometric absorber. For rank (d+2)/2 ≥2 the local solutions involve essential singularities and Stokes sectors; the manuscript must therefore specify the precise linear combination of solutions that selects the physical ingoing branch (see the discussion following the radial equation in the abstract and the numerical implementation for d=3,4,5). Without this explicit selection the sign of Im(ω_n) is not guaranteed.
  2. [Zeroth-order radial equation] The reduction of the radial equation to Bessel's equation of order μ=d/(d+2) at zeroth order is invoked to prove that all scalar QNMs are gapped. Because this is an approximation, the manuscript should supply either an error estimate for the neglected terms or a demonstration that those terms cannot shift any mode into Im(ω)≥0 (abstract and the paragraph containing the Bessel reduction).
  3. [Numerical spectra for d=3,4,5] The numerical spectra for d=3,4,5 are presented as confirmation of the dissipative tower. These results inherit the same boundary-condition choice at r=0; the paper must therefore document the numerical method, convergence tests, and how the ingoing condition is imposed at the irregular point to exclude the possibility that discretization or truncation artifacts mask a zero or positive-Im mode.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments identify areas where additional rigor and documentation are needed to support the claims of zero-temperature dissipation. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Boundary conditions at the irregular singular point r=0] The central claim that Im(ω_n)<0 for all scalar modes when d≥3 rests on the irregular singular point at r=0 functioning as a geometric absorber. For rank (d+2)/2 ≥2 the local solutions involve essential singularities and Stokes sectors; the manuscript must therefore specify the precise linear combination of solutions that selects the physical ingoing branch. Without this explicit selection the sign of Im(ω_n) is not guaranteed.

    Authors: We agree that an explicit description of the ingoing boundary condition is required for rigor. In the revised manuscript we will add a dedicated subsection that specifies the physical branch at the irregular singular point of rank (d+2)/2. This will include the relevant Stokes sectors, the asymptotic form of the local solutions, and the precise linear combination chosen to enforce absorption, thereby confirming that all modes satisfy Im(ω_n)<0 for d≥3. revision: yes

  2. Referee: [Zeroth-order radial equation] The reduction of the radial equation to Bessel's equation of order μ=d/(d+2) at zeroth order is invoked to prove that all scalar QNMs are gapped. Because this is an approximation, the manuscript should supply either an error estimate for the neglected terms or a demonstration that those terms cannot shift any mode into Im(ω)≥0.

    Authors: The referee is correct that the Bessel reduction is leading-order. We will include in the revision a perturbative error estimate for the higher-order terms in the pp-wave deformation and a demonstration that these corrections cannot move any frequency into the upper half-plane, preserving both the gap and the dissipative character of the spectrum. revision: yes

  3. Referee: [Numerical spectra for d=3,4,5] The numerical spectra for d=3,4,5 are presented as confirmation of the dissipative tower. These results inherit the same boundary-condition choice at r=0; the paper must therefore document the numerical method, convergence tests, and how the ingoing condition is imposed at the irregular point to exclude the possibility that discretization or truncation artifacts mask a zero or positive-Im mode.

    Authors: We will expand the numerical section to document the method (a shooting algorithm adapted to the irregular singularity), convergence tests under changes in grid resolution and truncation order, and the explicit implementation of the ingoing condition via the selected asymptotic branch at r=0. These additions will rule out numerical artifacts and confirm the dissipative tower. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation reduces metric wave equation to standard special functions via direct substitution

full rationale

The paper begins with the scalar wave equation on the explicit Kaigorodov pp-wave metric, substitutes the ansatz for time and transverse dependence, and obtains a radial ODE. At leading order this ODE is exactly Bessel's equation of order μ = d/(d+2) by algebraic rearrangement of the metric coefficients; the spectrum properties (Im ω_n < 0 for d ≥ 3) are then read off from the known analytic continuation and asymptotic behavior of Bessel functions under the stated ingoing boundary condition at the irregular singular point r = 0. No parameter is fitted to data and relabeled a prediction, no self-citation supplies a uniqueness theorem or ansatz, and the boundary condition is an explicit choice of linear combination of local solutions rather than a redefinition of the output. The numerical spectra are independent verifications, not inputs. The derivation chain is therefore self-contained against external mathematical facts about Bessel functions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard general relativity and holographic duality assumptions without introducing new fitted parameters or invented entities beyond the background metric.

axioms (2)
  • domain assumption The Kaigorodov pp-wave metric satisfies Einstein's equations with negative cosmological constant and serves as the gravity dual to zero-temperature null fluids.
    Invoked to justify the spacetime as the correct background for the perturbations.
  • domain assumption Ingoing wave boundary conditions at the irregular singular point r=0 define the quasinormal modes.
    Standard choice for dissipative spectra in asymptotically AdS geometries.

pith-pipeline@v0.9.0 · 5507 in / 1436 out tokens · 52788 ms · 2026-05-10T13:05:44.916744+00:00 · methodology

discussion (0)

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