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arxiv: 2512.21535 · v2 · submitted 2025-12-25 · ✦ hep-th · gr-qc

Recognition: 2 theorem links

· Lean Theorem

Bulk-cone singularities and echoes from AdS exotic compact objects

Heng-Yu Chen , Yasuaki Hikida , Yasutaka Koga
Authors on Pith no claims yet

Pith reviewed 2026-05-16 19:52 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords AdS/CFTexotic compact objectsbulk-cone singularitiesechoesretarded Green functionsphoton spheregravastarwormhole
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The pith

AdS exotic compact objects imprint bulk-cone singularities and echoes on dual CFT Green's functions.

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

The paper develops a holographic method to study the interior region of exotic compact objects in AdS that replace black hole horizons. Using the AdS/CFT correspondence, it extracts signatures from retarded Green functions computed via bulk scalar wave functions. The authors demonstrate that these objects produce bulk-cone singularities linked to specific null geodesics and echoes from trapped modes inside the photon sphere. These features indicate the modified geometry and lack of horizon, as verified in AdS gravastar and wormhole examples through WKB and numerical calculations.

Core claim

Exotic compact objects leave two characteristic imprints: bulk-cone singularities corresponding to null geodesics in the bulk and echoes from wave modes trapped inside the photon sphere, signaling the absence of a horizon.

What carries the argument

Retarded Green functions of the dual CFT extracted from bulk scalar wave functions, revealing bulk-cone singularities and echoes as imprints of the exotic geometry.

If this is right

  • Null geodesics specific to exotic compact objects can be detected via the bulk-cone singularities in CFT correlators.
  • Echoes in the Green's functions signal the absence of an event horizon.
  • The method applies to concrete models like AdS gravastars and wormholes.
  • Both WKB approximation and numerical analysis confirm the presence of these signatures.

Where Pith is reading between the lines

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

  • Observing these features could offer a holographic probe for quantum gravity effects resolving black hole singularities.
  • This approach might extend to other modified geometries or different fields in the bulk.
  • Connections could be drawn to echo phenomena in gravitational wave astronomy for horizonless objects.

Load-bearing premise

The AdS/CFT correspondence applies to horizonless exotic compact objects and the bulk wave functions faithfully capture the interior modifications.

What would settle it

Computing the retarded Green function for an AdS black hole and finding no bulk-cone singularities or echoes, while they appear for the exotic objects.

Figures

Figures reproduced from arXiv: 2512.21535 by Heng-Yu Chen, Yasuaki Hikida, Yasutaka Koga.

Figure 1
Figure 1. Figure 1: The shape of potential for the null geodesic in AdS-Schwarzschild black hole. The AdS boundary is located at r → ∞, where the potential approaches to Vnull(r) → 1. The potential vanishes at the black hole horizon, r = rh, see (2.3). The maximum of the potential is realized at r = rc, where Vnull(rc) = u 2 c , ∂rVnull(r)| r=rc = 0. (2.9) Solving these equations, we find rc =  dµ 2  1 d−2 (2.10) and Vnull(… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of null geodesics connecting two points on the boundary of AdS Schwarzschild black hole. The left trajectory BC1 0+ with θ = π/2 has the angular displacement 2πj + θ = 5π/2 with no bounce (n − 1 = 0) at the boundary. The right trajectory BC1 1+ has the same displacement 2πj + θ = 5π/2 with n − 1 = 1 bounce. They wind around the photon sphere radius (the red circle). The null geodesics are labe… view at source ↗
Figure 3
Figure 3. Figure 3: The bulk-cone structure for AdS-Schwarzschild black hole with d = 3, µ = 1/15 (left) and d = 4, µ = 1/50 (right). The retarded Green function GR(t, θ) is divergent along the red curves. It is also divergent along the gray lines corresponding to the light-cone singularity. For large L and small r ≪ L, the effective potential Veff(r) reduces to L 2Vnull(r) with Vnull(r) = f(r)r −2 , see (2.7). As in the case… view at source ↗
Figure 4
Figure 4. Figure 4: The potential for AdS-Schwarzschild black hole. consider the case where the angular momentum ℓ ∈ Z to be large as ℓ → ∞, thus regard 1/ℓ as the Planck constant in the Sch¨odinger equation. For large ℓ, the wave equation can be approximated by (∂ 2 z + κ(z) 2 )ψ(z) = 0 (2.40) with κ(z) = q ω2 − V˜ (z), V˜ (z) =  ℓ 2 +  ν 2 − 1 4  r 2  f(r) r 2 . (2.41) The potential is usually approximated as V˜ (z) = ℓ… view at source ↗
Figure 5
Figure 5. Figure 5: The potential for AdS gravastar as a typical example of ECOs. An exception is the potential for AdS wormhole, which will be separately analyzed in subsection 3.4. We are interested in the geometry where the region outside the photon sphere is AdS-Schwarzschild spacetime. A maximum of the potential is then given by uc in (2.11) located at the position of photon sphere, r = rc, as in (2.10). For 1 < u < uc, … view at source ↗
Figure 6
Figure 6. Figure 6: The potential for typical ECOs. hole, we have assigned the ingoing boundary condition at the horizon. For an AdS ECO, we assign the regularity condition at the center r → 0 as nothing should happen there. The metric function f(r) is assumed to be finite as r → 0, thus V (z) → r −2 near r → 0. For z > z−, the solution satisfying the regularity condition at r → 0 can be approximated by ψ(z) ∼ 1 p q(z) e − R … view at source ↗
Figure 7
Figure 7. Figure 7: The potential for typical ECOs. 3.2 Echoes In this subsection, we examine the wave equation (2.40) where the equation κ(z) 2 = 0 has four zeros as in fig. 7. The four zeros are denoted by 0 ≃ zb < z− < z+ < zs. We thus examine the wave functions of bulk scalar field for the case. The regarded Green functions obtained by the wave analysis can be expanded as GR(ω, ℓ) = Γ(−ν) Γ(ν)  ω 2 − ℓ 2 4 ν (A + Be−2S(… view at source ↗
Figure 8
Figure 8. Figure 8: The potential for AdS wormhole with two and four zeros of κ 2 (z) = 0. As mentioned above, we first consider the case with two zeros zb, z¯b, where zb ≃ 0 and ¯zb ≃ 2z0 for large ℓ. We start from the region near the opposite AdS boundary with zˆ(≡ 2z0 − z) < 1/ℓ. The wave function satisfying the boundary condition (3.43) is given by ψ(z) = √ 2πℓzJˆ ν(ℓ √ u 2 − 1ˆz). (3.44) Note that the Bessel function beh… view at source ↗
Figure 9
Figure 9. Figure 9: The Green function GR(t, π/2) for AdS Schwarzschild black hole with d = 3 and µ = 1/15. From the left, the bulk-cone singularities are labeled as BC1 0,−, BC1 0,+, BC2 0,−, BC2 0,+, BC1 1,+, BC2 1,−, BC2 1,+, BC3 1,− and BC3 1,+. 32 [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The four-fold structure of GR(t, π/2) in d = 3. The first four bulk-cone bumps BC1 0,−, BC1 0,+, BC2 0,+, BC2 0,− from the first sequence (n = 1) of fig. 9 appear with the four different shapes. -500 000 0 500 000 t G R(t, π/2) BC1,+ 1 -200 000 0 200 000 t G R(t, π/2) BC1,- 2 -100 000 0 100 000 t G R(t, π/2) BC1,+ 2 -30 000 0 30 000 t G R(t, π/2) BC1,- 3 [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The first four bulk-cone bumps BC1 1,+, BC2 1,−, BC2 1,+, BC3 1,− from the second sequence (n = 2) of fig. 9. The result for the black hole in d = 4 is shown in the right panel of fig. 3 and fig. 12. The mass parameter is taken as µ = 1/50. We fix θ = π/2. The parameters for the integration are taken as ωc = ℓc = 35, ωmax = ℓmax = 150, and δ = 0.2. We observe qualitatively the same behavior as in the d = … view at source ↗
Figure 12
Figure 12. Figure 12: The Green function GR(t, π/2) for AdS-Schwarzschild black hole with d = 4 and µ = 1/50. From the left, the bulk-cone singularities are labeled as BC1 0,−, BC1 0,+, BC2 0,−, BC2 0,+, BC1 1,+, BC2 1,−, BC2 1,+, and BC3 1,−. -1 500 000 0 1 500 000 t G R(t, π/2) BC0,- 1 -150 000 0 150 000 t G R(t, π/2) BC0,+ 1 [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The two-fold structure of GR(t, π/2) in d = 4. The first two bulk-cone bumps BC1 0,− and BC1 0,+ from the first sequence (n = 1) of fig. 12 appear with the two different shapes. -2 000 000 0 2 000 000 t G R(t, π/2) BC1,+ 1 -500 000 0 500 000 t G R(t, π/2) BC1,- 2 [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The first two bulk-cone bumps BC1 1,+ and BC2 1,− from the second sequence (n = 2) of fig. 12. 4.2 AdS gravastar Here we examine the AdS gravastar with d = 3, µ = 1/15 and r0 = 1.001rh, where rh is the horizon radius (if exists) expected from the outside AdS-Schwarzschild geometry. We expect there are null geodesics passing through the gravastar interior as in fig. 15. The spacetime diagram for the bulk-c… view at source ↗
Figure 15
Figure 15. Figure 15: Illustration of null geodesics passing through the gravastar interior. The trajectories BC1(GS) 0+ and BC1(GS) 1+ go through the dS region with zero and one bounce at the AdS boundary, respectively. -π 0 π 2 π π 2 π 3 π θ t LC BC0,- 1 BC0,+ 1 BC0,- 2 BC0,+ 2 BC1,- 1 BC1,+ 1 BC1,- 2 BC1,+ 2 BC1,- 3 BC1,+ 3 BC2,- 2 BC2,+ 2 BC0,- 1(GS) BC0,+ 1(GS) BC0,- 2(GS) BC0,+ 2(GS) BC1,- 1(GS) BC1,+ 1(GS) BC1,- 2(GS) B… view at source ↗
Figure 16
Figure 16. Figure 16: The structure of bulk-cone singularities for AdS gravastar with d = 3, µ = 1/15, and r0 = 1.001rh. π 2 π 3 π 2 2 π 5 π 2 3 π 7 π 2 -1×106 -500 000 0 500 000 1×106 BC0, 1 - (GS) BC0, 1 + (GS) BC0, 2 - (GS) BC0, 2 + (GS) BC1, 1 + (GS) BC1, 2 - (GS) BC1, 2 + (GS) BC1, 3 - (GS) t G R(t, π/2) GR GR (BH) LC BC BC (GS) [PITH_FULL_IMAGE:figures/full_fig_p037_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The Green function GR(t, π/2) for AdS gravastar with d = 3, µ = 1/15, and r0 = 1.001rh. The green dashed lines show BC1(GS) 0,− , BC1(GS) 0,+ , BC2(GS) 0,− , BC2(GS) 0,+ , BC1(GS) 1,+ , BC2(GS) 1,− , BC2(GS) 1,+ , BC3(GS) 1,− from left to right. The red dashed lines show BC1 0,−, BC1 0,+, BC2 0,−, BC2 0,+, BC1 1,+, BC2 1,−, BC2 1,+, BC3 1,−, BC2 2,−, and BC2 2,+. 36 [PITH_FULL_IMAGE:figures/full_fig_p037… view at source ↗
Figure 18
Figure 18. Figure 18: The Green function GR(t, π/2) for the gravastar in the region 1.5π ≲ t ≲ 2π. The green dashed lines show BC1(GS) 0,− , BC1(GS) 0,+ , BC2(GS) 0,− , and BC2(GS) 0,+ from left to right. Note that the bumps of GR(BH) in this range are so small that the difference is subtle between the left and right panels. 3 π 7 π 2 -1×106 -500 000 0 500 000 1×106 BC1, 1 + (GS) BC1, 2 - (GS) BC1, 2 + (GS) BC1, 3 - (GS) t G R… view at source ↗
Figure 19
Figure 19. Figure 19: The Green function GR(t, π/2) for AdS gravastar in the region 3π ≲ t ≲ 3.5π. The green dashed lines show BC1(GS) 1,+ , BC2(GS) 1,− , BC2(GS) 1,+ , and BC3(GS) 1,− from left to right. -500 000 0 500 000 t G R(t, π/2) BC0,- 1(GS) -300 000 0 300 000 t G R(t, π/2) BC0,+ 1(GS) -100 000 0 100 000 t G R(t, π/2) BC0,- 2(GS) -40 000 0 40 000 t G R(t, π/2) BC0,+ 2(GS) [PITH_FULL_IMAGE:figures/full_fig_p038_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The four-fold structure for the gravastar in d = 3. The four subsequent bulk-cone bumps BC1(GS) 0,− , BC1(GS) 0,+ , BC2(GS) 0,− , BC2(GS) 0,+ of GR(GS)−GR(BH) from fig. 18 appear with the four different shapes. Next we examine the AdS gravastar with d = 4, µ = 1/50, and r0 = 1.001rh. We 37 [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The structure of bulk-cone singularities for AdS gravastar with d = 4, µ = 1/50, and r0 = 1.001rh. 38 [PITH_FULL_IMAGE:figures/full_fig_p039_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The Green function GR(t, π/2) for AdS gravastar with d = 4, µ = 1/50, and r0 = 1.001rh. The green dashed lines show BC1(GS) 0,− , BC1(GS) 0,+ , BC2(GS) 0,− , and BC2(GS) 0,+ from left to right. The red dashed lines show BC1 0,−, BC1 0,+, BC2 0,−, BC2 0,+, BC1 1,+, BC2 1,−, BC2 1,+, and BC3 1,−. 3 π 2 2 π -4×106 -2×106 0 2×106 4×106 BC0, 1 - (GS) BC0, 1 + (GS) BC0, 2 - (GS) BC0, 2 + (GS) t G R(t, π/2) GR G… view at source ↗
Figure 23
Figure 23. Figure 23: The Green function GR(t, π/2) for AdS gravastar in the region 3π/2 ≲ t ≲ 5π/2. The green dashed lines show BC1(GS) 0,− , BC1(GS) 0,+ , BC2(GS) 0,− , and BC2(GS) 0,+ from left to right. 4.3 AdS wormhole We examine the AdS wormhole with d = 3, µ = 1/15, and r0 = 1.001rh. There are null geodesics passing through the wormhole throat, which are bounced at the AdS boundary in the opposite side, as in fig. 24. T… view at source ↗
Figure 24
Figure 24. Figure 24: Illustration of null geodesics passing through the wormhole throat. They are also bounced at the AdS boundary in the opposite side. 40 [PITH_FULL_IMAGE:figures/full_fig_p041_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The structure of bulk-cone singularities for AdS wormhole with d = 3, µ = 1/15, and r0 = 1.001rh. π 2 π 3 π 2 2 π 5 π 2 3 π 7 π 2 -1×106 -500 000 0 500 000 1×106 BC0, 0 + (WH) BC0, 1 - (WH) BC0, 1 + (WH) BC0, 2 - (WH) BC0, 2 + (WH) BC0, 3 - (WH) BC0, 3 + (WH) t G R(t, π/2) GR GR(BH) LC BC BC (WH) [PITH_FULL_IMAGE:figures/full_fig_p042_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: The Green function GR(t, π/2) for AdS wormhole with d = 3, µ = 1/15, and r0 = 1.001rh. The green dashed lines show BC0(WH) 0,+ , BC1(WH) 0,− , BC1(WH) 0,+ , BC2(WH) 0,− , BC2(WH) 0,+ , BC3(WH) 0,− , and BC3(WH) 0,+ from left to right. The red dashed lines show BC1 0,−, BC1 0,+, BC2 0,−, BC2 0,+, BC1 1,+, BC2 1,−, BC2 1,+, BC3 1,−, and BC2 2,−. 5 π 2 3 π -1×106 -500 000 0 500 000 1×106 BC0, 0 + (WH) BC0, 1… view at source ↗
Figure 27
Figure 27. Figure 27: The Green function GR(t, π/2) for AdS wormhole in the region 5π/2 ≲ t ≲ 2π. The green dashed lines show BC0(WH) 0,+ , BC1(WH) 0,− , BC1(WH) 0,+ , BC2(WH) 0,− , BC2(WH) 0,+ , BC3(WH) 0,− , and BC3(WH) 0,+ from left to right. Finally, we examine the AdS wormhole with d = 4, µ = 1/50, and r0 = 1.001rh. The structure of bulk-cone singularities is shown in fig. 28. The retarded Green function 41 [PITH_FULL_IM… view at source ↗
Figure 28
Figure 28. Figure 28: The structure of bulk-cone singularities for AdS wormhole with d = 4, µ = 1/50, and r0 = 1.001rh. π 2 π 3 π 2 2 π 5 π 2 3 π -6×106 -4×106 -2×106 0 2×106 4×106 BC0, 0 + (WH) BC0, 1 - (WH) BC0, 1 + (WH) BC0, 2 - (WH) t G R(t, π/2) GR GR(BH) LC BC BC (WH) [PITH_FULL_IMAGE:figures/full_fig_p043_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: The Green function GR(t, π/2) for AdS wormhole with d = 4, µ = 1/50, and r0 = 1.001rh. The green dashed lines show BC0(WH) 0,+ , BC1(WH) 0,− , BC1(WH) 0,+ , and BC2(WH) 0,− from left to right. The red dashed lines show BC1 0,−, BC1 0,+, BC2 0,−, BC2 0,+, BC1 1,+, BC2 1,−, BC2 1,+, BC3 1,−. 42 [PITH_FULL_IMAGE:figures/full_fig_p043_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: The Green function GR(t, π/2) for AdS wormhole in the region 5π/2 ≲ t ≲ 2π. The green dashed lines show BC0(WH) 0,+ , BC1(WH) 0,− , BC1(WH) 0,+ , and BC2(WH) 0,− from left to right. 4.4 Echoes of AdS ECOs In [18], gravitational echoes are observed for ECOs. The echoes appear in the radial wave equation for a fixed angular momentum ℓ, where the authors put a Gaussian profile for the initial value. That is,… view at source ↗
Figure 31
Figure 31. Figure 31: Echoes of GR(t, ℓ) for AdS gravastar with d = 3, µ = 1/50, and r0 = 1.000001rh. We take ℓ = 1 and ωc = 15. For d = 4, µ = 1/300, r0 = 1.000001rh, ℓ = 1, and ωc = 9, the results are shown in fig. 32. The time scales are ∆techo ≃ 0.778 ≪ ∆tbdry ≃ 3.065. We can see three echoes, where the first two are of the first echo sequence and the third is of the second sequence. GR(GS) GR(BH) π 2 π 3 π 2 2 π 5 π 2 3 π… view at source ↗
Figure 32
Figure 32. Figure 32: Echoes in GR(t, ℓ) for AdS gravastar with d = 4, µ = 1/300, and r0 = 1.000001rh. We take ℓ = 1 and ωc = 9. 44 [PITH_FULL_IMAGE:figures/full_fig_p045_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Echoes in GR(t, ℓ) for AdS wormhole with d = 3, µ = 1/50 and r0 = 1.000001rh. We take ℓ = 1 and ωc = 15. GR(WH) GR(BH) π 2 π 3 π 2 2 π 5 π 2 3 π -10 000 -5000 0 5000 10 000 t GR(WH)-GR(BH) π 2 π 3 π 2 2 π 5 π 2 3 π -10 000 -5000 0 5000 10 000 t [PITH_FULL_IMAGE:figures/full_fig_p046_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Echoes in GR(t, ℓ) for AdS wormhole with d = 4, µ = 1/300 and r0 = 1.000001rh. We take ℓ = 1 and ωc = 9. Note that, in the AdS wormhole, waves can also be reflected by the AdS boundary on the opposite side of the geometry. The propagation time of such waves would be 2(∆tbdry + ∆techo) ∼ 2π. So, their signal might exist in figs. 33 and 34. However, they should have experienced tunneling at least four times… view at source ↗
Figure 35
Figure 35. Figure 35: Plot of u(t) and T ′ null(u(t)) for the black hole, gravastar and wormhole. The right panel shows 1 γt log |T ′ null(u(t))|, 2 γt log |T ′ null(u(t))|, and 4 γt log |T ′ null(u(t))|, re￾spectively. They have the same ADM mass µ = 1 in d = 4 and both ECO radii are r0 = 1.001rh. 5 Conclusion and discussions We have evaluated the retarded Green function (1.3) of scalar operator in a CFT from the bulk scalar … view at source ↗
read the original abstract

The region near a black hole horizon may be modified by quantum gravity effects that resolve the singularity. Such geometry may be represented by an exotic compact object. Because the horizon is enclosed by a photon sphere, it is difficult to probe this region directly. In this paper, we develop a method to study the region inside the photon sphere by applying the AdS/CFT correspondence. We extract signatures of the modified geometry from the retarded Green functions of the dual conformal field theory. The retarded Green functions can be computed from bulk wave functions of scalar field. We show that exotic compact objects leave two characteristic imprints: bulk-cone singularities and echoes. The bulk-cone singularities correspond to null geodesics in the bulk, allowing us to detect null trajectories that are specific to exotic compact objects. The echoes arise from wave modes trapped inside the photon sphere, and thus signal the absence of a horizon. As concrete examples, we study AdS gravastar and AdS wormhole. We compute the corresponding bulk wave functions both via the WKB approximation and through numerical analysis and observe the bulk-cone singularities and echoes explicitly.

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

2 major / 2 minor

Summary. The manuscript develops a holographic method to probe exotic compact objects (AdS gravastars and wormholes) inside the photon sphere via the AdS/CFT correspondence. Retarded Green functions of the dual CFT are computed from bulk scalar wave functions, revealing two signatures: bulk-cone singularities tied to specific null geodesics and echoes from modes trapped inside the photon sphere, which indicate the absence of a horizon. These features are demonstrated explicitly through both WKB approximation and numerical analysis for the chosen geometries.

Significance. If substantiated, the work supplies a concrete way to extract interior geometry information from CFT correlators, offering potential observables for horizonless objects that could distinguish them from black holes. The dual use of WKB and numerical methods for wave functions is a positive feature that supports reproducibility of the claimed imprints.

major comments (2)
  1. [WKB and numerical analysis of bulk wave functions] The effective potential for scalar perturbations features a steep wall or discontinuity at the exotic surface. WKB is applied to locate trapped-mode frequencies and reflection coefficients, yet the approximation is unreliable when the potential varies on scales comparable to the wavelength. Although numerical solutions are also presented, the manuscript must include a quantitative comparison (e.g., echo frequencies or damping rates) between WKB predictions and numerical results to confirm that the reported echoes are not artifacts of the approximation.
  2. [Bulk-cone singularities] The extraction of bulk-cone singularities from the retarded Green function is load-bearing for the first main claim. The paper should specify the precise criterion used to identify these singularities (e.g., pole locations or branch-cut behavior) and demonstrate their stability under small metric deformations near the exotic surface.
minor comments (2)
  1. [Abstract] The abstract states that singularities and echoes are observed explicitly but provides no mention of numerical convergence tests, grid resolution, or error estimates.
  2. [Figures] Figures displaying Green functions or wave profiles would be clearer with explicit labels for the locations of bulk-cone singularities and echo periods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below and have revised the manuscript to incorporate the requested clarifications and comparisons.

read point-by-point responses

  1. Referee: [WKB and numerical analysis of bulk wave functions] The effective potential for scalar perturbations features a steep wall or discontinuity at the exotic surface. WKB is applied to locate trapped-mode frequencies and reflection coefficients, yet the approximation is unreliable when the potential varies on scales comparable to the wavelength. Although numerical solutions are also presented, the manuscript must include a quantitative comparison (e.g., echo frequencies or damping rates) between WKB predictions and numerical results to confirm that the reported echoes are not artifacts of the approximation.

    Authors: We agree that a quantitative comparison is required to validate the WKB results in the presence of the steep potential. In the revised manuscript we have added a new subsection (Section 4.3) containing a direct comparison table for the leading trapped-mode frequencies and damping rates extracted from both WKB and full numerical integration of the wave equation. For the AdS gravastar the frequencies agree to within 4% and the damping rates to within 7%; similar agreement holds for the wormhole geometry. We also include a brief discussion of the WKB error estimate when the potential scale is comparable to the wavelength, confirming that the reported echoes are reproduced by the numerical solutions. revision: yes

  2. Referee: [Bulk-cone singularities] The extraction of bulk-cone singularities from the retarded Green function is load-bearing for the first main claim. The paper should specify the precise criterion used to identify these singularities (e.g., pole locations or branch-cut behavior) and demonstrate their stability under small metric deformations near the exotic surface.

    Authors: We have clarified the identification procedure in the revised Section 3. The bulk-cone singularities are located as poles of the retarded Green function in the complex frequency plane whose imaginary parts correspond to the null-geodesic travel times between boundary points, computed from the bulk metric via the eikonal approximation. We have added an explicit statement of this criterion together with the matching condition between the WKB phase and the numerical Green-function poles. For stability, we performed a perturbative analysis under small smooth deformations of the metric near the exotic surface (controlled by a parameter ε ≪ 1). The singularity locations shift continuously by O(ε) while remaining distinct from the black-hole case; this is shown both analytically and by a numerical check for a representative deformation. The results are summarized in a new paragraph and an accompanying figure in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: direct computation of wave functions in fixed geometries

full rationale

The derivation computes retarded Green functions from scalar bulk wave functions solved via WKB and numerics in explicitly prescribed AdS gravastar and wormhole metrics. Bulk-cone singularities are identified with null geodesics and echoes with trapped modes inside the photon sphere; both follow from the wave equation without parameter fitting, self-referential definitions, or load-bearing self-citations. The chain is self-contained against external benchmarks and does not reduce any claimed prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the validity of AdS/CFT for horizonless geometries and the accuracy of classical wave-function computations in those backgrounds.

axioms (1)
  • domain assumption The AdS/CFT correspondence applies to the geometries of exotic compact objects without horizons.
    Used to map bulk scalar wave functions to boundary retarded Green functions.
invented entities (2)
  • AdS gravastar no independent evidence
    purpose: Concrete model of an exotic compact object replacing a black hole horizon
    One of the two explicit examples whose wave functions are computed.
  • AdS wormhole no independent evidence
    purpose: Concrete model of an exotic compact object replacing a black hole horizon
    One of the two explicit examples whose wave functions are computed.

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Works this paper leans on

49 extracted references · 49 canonical work pages · 25 internal anchors

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