Scalar fields and 3D Flat Space Cosmologies

Arjun Bagchi; Astha Kakkar; Saikat Mondal; Supratik Biswas

REVIEW 4 major objections 5 minor 70 references

Scalar fields and 3D Flat Space Cosmologies

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A hard wall at the cosmological horizon yields the quasi-normal mode spectrum of scalar fields in flat space cosmologies, matching the flat-space limit of BTZ and feeding one-loop partition functions.

desk verdict The direct-bulk strategy is the right instinct, but the proposed global mode function has a cusp at x=0 and fails the Klein-Gordon equation away from x>0, so the central QNM spectrum is unproven. read the letter →

arxiv 2506.02148 v2 pith:CLAXIQV5 submitted 2025-06-02 hep-th

Arjun Bagchi , Supratik Biswas , Astha Kakkar , Saikat Mondal This is my paper

classification hep-th MSC 83C8083C5781T20

keywords flatspacecosmologiesquasi-normalmodesthree-dimensionalgravitycosmologicalhorizonscalarperturbationone-looppartitionfunctionholographyBTZblackhole

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

Flat Space Cosmologies (FSC) are time-dependent solutions of three-dimensional Einstein gravity with zero cosmological constant, orbifolds of flat space with a cosmological horizon and the flat-space counterparts of BTZ black holes. This paper seeks to establish that a massive scalar field in an FSC background has a definite quasi-normal mode (QNM) spectrum, provided the cosmological horizon is treated as a hard wall that reflects waves and the spatial momentum is allowed to be complex. The resulting frequencies obey $\omega_{i,r}^2 = \sqrt{A_{n,N}^2 + B_{n,N}^2} \mp A_{n,N}$ with $A_{n,N} = (n^2 - N^2 \hat r_+^2 + m^2 r_0^2)/(2r_0^2)$ and $B_{n,N} = \hat r_+ n N / r_0$, and the same spectrum is recovered from the flat-space limit of the BTZ scalar analysis. If correct, this is the first direct bulk derivation of FSC quasi-normal modes, the basic data for linear response and one-loop thermodynamics in flat-space holography. The authors further feed the modes into the Denef-Hartnoll-Sachdev construction of the one-loop scalar partition function, obtaining closed forms in several limits and reproducing known answers in the large-mass limit.

What carries the argument

The carrying object is the exact scalar wavefunction in the FSC background, written in the coordinates $ds^2 = dy^2 - d\tau^2 + E^2\tau^2(dy+dx)^2$ outside the horizon, where the two commuting Killing vectors allow a basis of simultaneous eigenfunctions $\psi = J_{\pm i p/E}(\omega \tau) e^{ipx} e^{iny/r_0}$ with $\omega^2 = (p - n/r_0)^2 + m^2$ and $E = \hat r_+/r_0$. Three moves carry the argument: complexifying the momentum $p$, so boundedness at $|x| \to \infty$ quantizes $p = i E N$ and turns plane waves into decaying modes $e^{-EN|x|}J_N(\omega\tau)$; enforcing the hard wall at the horizon by the flux-balance condition $F^+_{n,p} + F^-_{n,p} = 0$, which selects the same integer $N$; and reducing $\omega^2 = (p - n/r_0)^2 + m^2$ algebraically to the closed spectrum $\omega^2_{i,r} = \sqrt{A_{n,N}^2 + B_{n,N}^2} \mp A_{n,N}$. The same mode data, inserted into the Denef-Hartnoll-Sachdev product over thermal frequencies, produce the one-loop partition function.

What would settle it

Solve the boundary-value problem for the scalar in the FSC background directly in the coordinates of (2.19a): impose flux balance $F^+ + F^- = 0$ at the horizon $\tau = 0$ and boundedness as $|x|, \tau \to \infty$, treating $\omega$ as the unknown complex frequency. A numerical sweep over the integers $n$, $N$ and several masses should return exactly $\omega^2_{i,r} = \sqrt{A_{n,N}^2 + B_{n,N}^2} \mp A_{n,N}$; any deviation, or the survival of extra modes, would refute the claimed spectrum. The boundary condition itself can be probed separately by computing the retarded scalar two-point function via a Wronskian mode sum and checking whether its poles coincide with (3.27), which would test whether the reflective wall is the condition the physics actually selects.

Watch

Extended reading notes

Core claim

The central claim is that scalar perturbations of the FSC, with the cosmological horizon acting as a reflective hard wall, have quasi-normal frequencies $\omega_i^2 = \sqrt{A_{n,N}^2 + B_{n,N}^2} - A_{n,N}$ and $\omega_r^2 = \sqrt{A_{n,N}^2 + B_{n,N}^2} + A_{n,N}$, where $n \in \mathbb{Z}$ is the momentum quantum number around the compact $y$-circle, $N \in \mathbb{Z}$ indexes the complexified momentum, and $\hat r_+, r_0$ are the FSC mass and angular-momentum parameters. Purely ingoing or purely outgoing conditions at the horizon produce trivial or divergent solutions, so the authors impose flux balance $F^+ + F^- = 0$ at the horizon; this forces the imaginary part of the momentum to be quantized as $p_i = \hat r_+ N / r_0$, and boundedness at $|x| \to \infty$ then selects the decaying wavefunctions $e^{-N\hat r_+ |x|/r_0} J_N(\omega \tau)$. The same wavefunctions and the same effective potential emerge by taking the $\ell \to \infty$ limit of the scalar analysis in the BTZ black hole, which the paper presents as a consistency check of the intrinsic flat-space computation. The QNM data are then used in the Denef-Hartnoll-Sachdev prescription to assemble the one-loop partition function, with closed forms in the $m^2 \to \infty$, $N=0$, and $m=0$ limits; the first of these matches known flat-gravity results in the literature.

Load-bearing premise

The whole mode spectrum rests on a boundary condition that is chosen rather than derived from the physics: the cosmological horizon is taken to be a perfectly reflecting wall, because the more obvious conditions of purely ingoing or purely outgoing waves give trivial or divergent solutions; if the physically correct behaviour at the horizon is something else, the spectrum changes.

Editorial extensions

If this is right

The spectrum (3.27) becomes a concrete bulk prediction: every pair of integers $(n,N)$ and mass $m$ yields a definite complex frequency, with the $N=0$ sector reducing to the real normal-mode frequencies $\omega_r^2 = (n/r_0)^2 + m^2$.
The flat-space limit of the BTZ intermediate-region scalar analysis reproduces the same wavefunctions, frequencies and effective potential, so the intrinsic FSC computation and the AdS-derived one describe the same physics.
The one-loop scalar partition function built by the DHS method from these modes reproduces the known flat-gravity and Selberg-zeta answers in the $m^2 \to \infty$ limit, tying the spectrum to thermodynamic data such as the FSC entropy.
Because the natural ingoing and outgoing horizon conditions yield no non-trivial modes, the reflective condition is not a refinement but a prerequisite: the FSC QNM spectrum exists only if the horizon is a hard wall.
The general one-loop answer extends previously known partition functions, which the paper reads as evidence that the scalar boundary conditions may be more general than the standard null-infinity boundary-condition class, possibly signalling an enhanced asymptotic symmetry algebra.

Reading between the lines

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

If the hard-wall spectrum is the physical one, the same frequencies should appear as the poles of the retarded scalar Green's function built by a Wronskian or mode-sum method; the paper does not perform this independent cross-check.
The paper itself states that it has no physical justification for why the $m^2 \to \infty$ limit matches earlier results, so the general partition function beyond that limit is unverified; a heat-kernel or Selberg-zeta computation of the same determinant would test it directly.
Because the asymptotic-symmetry analysis was performed in the standard boundary-condition frame while the QNM computation lives in the orbifold coordinates (2.19), re-deriving the surface charges in the latter frame could reveal whether the boundary conditions indeed enhance the symmetry algebra, as the paper itself suggests.
A dual check: Carrollian CFT correlators come in two branches, so the matching of FSC QNMs to thermal Green's-function poles may be branch-dependent; computing the two-dimensional delta-function-branch correlator and locating its poles would test the completeness of the bulk spectrum.

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.

Desk Editor's Note

read the letter

Arjun and co-workers take a direct route to FSC quasi-normal modes: solve the bulk scalar equation in region I, impose a reflective boundary at the cosmological horizon, complexify the momentum, and then feed the resulting frequencies into a DHS one-loop partition function. That is a sensible plan, and the paper does useful groundwork: the FSC background review is clean, the BTZ flat-space limit is worked out carefully, and the appendices are genuinely helpful. The m→∞ partition function matching [53,47] is a nice consistency check, though the authors themselves flag that they cannot justify why that limit is the right one.

The problem is that the workhorse mode (3.14) is not actually a solution of the Klein-Gordon equation (3.2a). For x>0 you choose p=iEN and get a decaying exponential; for x<0 you choose p=-iEN to keep the solution bounded. But a single frequency ω from (3.23) only cancels the equation on one side. On the other side, the (∂y−∂x)² term leaves a residual proportional to τ² nN sign(x) ψ, and at x=0 the |x| cusp produces a delta-function source. The authors mention the cusp in passing, but they do not address its consequences. Since x=0 is not a boundary in the FSC patch, there is no physical wall to justify absorbing that delta. So equations (3.14) and (3.27) are formal half-line constructions, not global mode solutions. The boundary condition is also ad hoc: a hard wall at the cosmological horizon is chosen because ingoing/outgoing conditions give trivial or divergent results, but no physical derivation is offered.

None of this means the paper is worthless. The background material and the limiting analysis are solid, and the strategy of complexifying momenta is worth pursuing. But the central claim—that (3.27) is the FSC scalar QNM spectrum—does not hold as stated. A referee should ask for either a genuine distributional treatment (including a physical interpretation of the x=0 source) or a different boundary condition that yields smooth global modes. The partition function section inherits the problem, so the matching to earlier results is weaker than it appears.

This is a paper for researchers working on flat-space holography and Carrollian CFT, and it deserves a serious referee despite the flaw. I would not cite the spectrum in its current form, but I would want to see a revised version.

Referee Report

4 major / 5 minor

Summary. The paper studies a massive scalar field on three-dimensional flat space cosmologies (FSC), proposing quasi-normal modes obtained by imposing a reflective hard-wall boundary condition at the cosmological horizon and complexifying the spatial momentum. The central results are the QNM spectrum in Eq. (3.27), a limiting match with the BTZ analysis in Sec. 3.4, and one-loop partition functions built from these modes via the Denef-Hartnoll-Sachdev prescription in Sec. 4. The paper claims this is the first direct bulk derivation of the FSC scalar QNM spectrum.

Significance. If correct, the QNM spectrum would be a useful addition to the flat-space holography program, providing a bulk observable that could be compared with Carrollian CFT correlators, and the partition-function construction would connect QNMs to one-loop determinants in 3D asymptotically flat gravity. The paper contains a substantial amount of useful background material and several explicit computations, including the BTZ and dS reviews in the appendices. However, the main derivation has a fundamental gap: the proposed mode functions do not actually solve the Klein-Gordon equation, so the central claim is not established.

major comments (4)

[Sec. 3.1, Eq. (3.14)] The wavefunctions ψ_{n,N}=e^{-EN|x|}J_N(ωτ)e^{in y/r0} are not solutions of the Klein-Gordon equation (3.2a). For x≠0, substitution gives a residual τ^2[m^2+n^2/r0^2+E^2N^2-ω^2-2iEN n sign(x)/r0]ψ; using (3.23) this vanishes only for x>0, while for x<0 it equals 4iEN n τ^2/r0 ψ. At x=0 the cusp in |x| produces a delta-function source. Thus no single frequency from (3.27) admits a global mode function on region I (τ>0, x∈R), and the text's own acknowledgment that the functions are not differentiable at x=0 does not address this. The QNM spectrum (3.27) is therefore not established.
[Sec. 3.3, Eq. (3.27)] The definition B_{n,N}=r̂+ n N / r0 is inconsistent with the preceding result (3.25). From (3.24), P_n=-n/r0, Q_N=r̂+N/r0 and 2ω_rω_i=2P_nQ_N, so (ω_rω_i)^2=r̂+^2 n^2 N^2 / r0^4. With ω_r^2=sqrt(A^2+B^2)-A and ω_i^2=sqrt(A^2+B^2)+A, one obtains (ω_rω_i)^2=B^2. Hence the correct B is r̂+ n N / r0^2, not r̂+ n N / r0. As printed, (3.27) does not solve (3.25) and has incompatible dimensions (A ~ L^{-2}, B ~ L^{-1} in the printed version).
[Sec. 3.2, Eq. (3.19)] The reflective boundary condition F^+ + F^- =0 at the cosmological horizon is introduced as a choice. Appendix A.1 shows that purely ingoing or outgoing conditions lead to divergent solutions, but it does not establish that the hard-wall condition is the physically correct or unique choice for an FSC. Since the entire QNM spectrum (3.27) and the subsequent partition functions depend on this condition, the paper needs either a derivation of this boundary condition from the geometry or symmetries, or a discussion of how the spectrum changes under alternative conditions.
[Sec. 4, Eqs. (4.14)-(4.18)] The comparison with [47,53] is not a validation of the spectrum. The match is obtained only in the m→∞ limit, after dropping a Tolman factor (footnote 2), and the paper states that it cannot justify this limit. Moreover [53] shares an author with the present paper, so the check is not fully independent. The conclusion in Sec. 5 that the construction yielded a matching should be tempered accordingly.

minor comments (5)

[Throughout] The text has several typos: 'Banados-Tietelboim-Zanelli' in the abstract, 'subtelties' in Sec. 5, 'Toleman' in footnote 2, and 'commutating' in Sec. 3.
[Fig. 3] Figure 3 shows the cusp at x=0 but the caption calls it a decaying solution; the non-differentiability should be discussed in the main text rather than only in a parenthetical remark.
[Eq. (3.16)] The general solution adds the oscillatory and decaying pieces with different frequencies but does not specify how the relative coefficients are determined or whether the sum satisfies the wave equation.
[Eq. (3.27)] The notation ω^2_{i,r} is confusing: it could be read as the square of a frequency with subscripts i and r, while later ω_i and ω_r denote the imaginary and real parts. Please clarify.
[Table 1] In Table 1, the FSC QNM formula uses n and N but these are not defined in the table; add a definition or refer to Sec. 3.3.

Circularity Check

0 steps flagged · score 2.0 of 10

The FSC QNM spectrum is obtained by solving the Klein-Gordon equation with a stated hard-wall condition and complex momenta, not by fitting to or defining the partition function; the only self-referential element is a non-load-bearing consistency check against a same-group paper.

full rationale

The central derivation in Sections 3.1-3.3 solves the covariant scalar wave equation on the FSC background, imposes the reflective boundary condition F+ + F- = 0 at the cosmological horizon, and obtains the quantization p = i r̂+ N/r0 and hence the frequency relation (3.23), which is then algebraically rearranged into the QNM spectrum (3.27). No parameter is fitted to the one-loop partition function; rather, the QNMs are used as inputs to the DHS prescription in Section 4. The comparison with [53] (same first author) and [47] is presented as a matching in a particular limit, not as the source of the QNM spectrum, and the authors explicitly acknowledge the unexplained factor and the lack of justification for the m→∞ regime. The hard-wall boundary condition is an assumption rather than a consequence of the target result, so its physical status is a justification or correctness issue, not a circularity issue. A separate concern about the distributional validity of the e^{-EN|x|} cusp ansatz would also be a correctness issue, not a self-reference issue. Overall, no load-bearing step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The computation is built on standard mathematical tools (Bessel functions, hypergeometric equations, DHS factorization) and on the FSC background from prior literature. The paper-specific assumptions are the ad hoc hard-wall boundary condition and the complexification of momentum with the accompanying quantization; these are not derived from first principles. No new entities are introduced.

assumptions (4)

domain assumption FSC is a solution of 3D Einstein gravity with zero cosmological constant and is a shifted-boost orbifold of Minkowski space.
Section 2.2-2.3, based on [49,50]. The background geometry is taken as given.
ad hoc to paper The reflective (hard-wall) boundary condition at the cosmological horizon: F^+ + F^- = 0 (eq. 3.19).
Introduced in Section 3.2 to obtain non-trivial bounded solutions; no derivation from the geometry or a microscopic model. This is the load-bearing physical premise.
ad hoc to paper Complexification of momentum p = p_r + i p_i with quantization p_i = E N (N in Z) to ensure boundedness at |x|→infinity and regularity at τ→0.
Section 3.1-3.2. This is a mathematical device; the physical interpretation of complex momenta is not discussed.
standard math DHS method for one-loop determinants (Weierstrass factorization; relation to QNMs), including the non-static generalization of [63].
Section 4, eq. (4.8); cited from [54,63]. Assumed as known.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Scalar fields and 3D Flat Space Cosmologies." pith.science (2026). https://pith.science/paper/CLAXIQV5

@misc{pith2026250602148,
  author       = {Pith},
  title        = {Pith review of: Scalar fields and 3D Flat Space Cosmologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLAXIQV5}},
  note         = {Machine review of arXiv:2506.02148}
}

abstract

Flat Space Cosmologies (FSC) are time-dependent solutions in Einstein gravity in three-dimensional (3D) spacetimes with zero cosmological constant. These are orbifolds of 3D flat space that have a cosmological horizon and can be thought of as analogs of the Banados-Tietelboim-Zanelli (BTZ) black holes of AdS$_3$. We study scalar perturbations about these FSC solutions and explore the spectrum of quasi-normal modes (QNMs) crucially treating the cosmological horizon as a hard wall and extending to complex momenta. We connect this intrinsic analysis with the flatspace limit of the corresponding analysis in the BTZ black hole. The FSC QNMs are then utilized to build the scalar one-loop partition function by methods pioneered by Denef, Hartnoll and Sachdev in various simplifying limits and compared with existing answers in the literature.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

70 extracted references · 19 canonical work pages

[53]

Bagchi, C

A. Bagchi, C. Keeler, V. Martin and R. Poddar,A generalized Selberg zeta function for flat space cosmologies, JHEP 04 (2024) 066 [2312.06770]

arXiv 2024
[1]

LIGO Scientific Collaboration and Virgo Collaboration collaboration, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett.116 (2016) 061102

work page 2016
[2]

Regge and J.A

T. Regge and J.A. Wheeler,Stability of a schwarzschild singularity, Phys. Rev. 108 (1957) 1063

work page 1957
[3]

Zerilli,Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys

F.J. Zerilli,Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett.24 (1970) 737

work page 1970
[4]

Zerilli,Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics, Phys

F.J. Zerilli,Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D2 (1970) 2141

work page 1970
[5]

Zerilli,Perturbation analysis for gravitational and electromagnetic radiation in a reissner-nordstroem geometry, Phys

F.J. Zerilli,Perturbation analysis for gravitational and electromagnetic radiation in a reissner-nordstroem geometry, Phys. Rev. D9 (1974) 860. – 38 –

work page 1974
[6]

Chandrasekhar,The mathematical theory of black holes(1985)

S. Chandrasekhar,The mathematical theory of black holes(1985)

1985
[7]

Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936

C.V. Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936

work page 1970

Show all 70 references

[8]

Press,Long Wave Trains of Gravitational Waves from a Vibrating Black Hole, Astrophys

W.H. Press,Long Wave Trains of Gravitational Waves from a Vibrating Black Hole, Astrophys. J. Lett.170 (1971) L105

1971
[9]

Kokkotas and B.G

K.D. Kokkotas and B.G. Schmidt,Quasinormal modes of stars and black holes, Living Rev. Rel. 2 (1999) 2 [gr-qc/9909058]

1999 arXiv
[10]

Berti, V

E. Berti, V. Cardoso and A.O. Starinets,Quasinormal modes of black holes and black branes, Class. Quant. Grav.26 (2009) 163001 [0905.2975]

2009 arXiv
[11]

’t Hooft,Dimensional reduction in quantum gravity, Conf

G. ’t Hooft,Dimensional reduction in quantum gravity, Conf. Proc.C930308 (1993) 284 [gr-qc/9310026]

1993 arXiv
[12]

Susskind,The World as a hologram, J

L. Susskind,The World as a hologram, J. Math. Phys.36 (1995) 6377 [hep-th/9409089]

1995 arXiv
[13]

Maldacena,The Large N limit of superconformal field theories and supergravity, Int

J.M. Maldacena,The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200]

1999 arXiv
[14]

Horowitz and V.E

G.T. Horowitz and V.E. Hubeny,Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D62 (2000) 024027 [hep-th/9909056]

2000 arXiv
[15]

Bañados, C

M. Bañados, C. Teitelboim and J. Zanelli,Black hole in three-dimensional spacetime, Phys. Rev. Lett.69 (1992) 1849

1992
[16]

Bañados, M

M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli,Geometry of the 2+1 black hole, Phys. Rev. D48 (1993) 1506

1993
[17]

Birmingham, I

D. Birmingham, I. Sachs and S.N. Solodukhin,Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett.88 (2002) 151301 [hep-th/0112055]

2002 arXiv
[18]

Strominger,On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [1312.2229]

A. Strominger,On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [1312.2229]

2014 arXiv
[19]

T. He, V. Lysov, P. Mitra and A. Strominger,BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [1401.7026]

2015 arXiv
[20]

Strominger and A

A. Strominger and A. Zhiboedov,Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [1411.5745]

2016 arXiv
[21]

Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory, 1703.05448

A. Strominger,Lectures on the Infrared Structure of Gravity and Gauge Theory, 1703.05448

arXiv
[22]

Pasterski,Lectures on celestial amplitudes, Eur

S. Pasterski,Lectures on celestial amplitudes, Eur. Phys. J. C81 (2021) 1062 [2108.04801]

2021 arXiv
[23]

Bagchi, R

A. Bagchi, R. Basu, A. Kakkar and A. Mehra,Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [1609.06203]

2016 arXiv
[24]

Bagchi, S

A. Bagchi, S. Banerjee, R. Basu and S. Dutta,Scattering Amplitudes: Celestial and Carrollian, Phys. Rev. Lett.128 (2022) 241601 [2202.08438]

2022 arXiv
[25]

Donnay, A

L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi,Carrollian Perspective on Celestial Holography, Phys. Rev. Lett.129 (2022) 071602 [2202.04702]

2022 arXiv
[26]

Bagchi, P

A. Bagchi, P. Dhivakar and S. Dutta,AdS Witten diagrams to Carrollian correlators, JHEP 04 (2023) 135 [2303.07388]

2023 arXiv
[27]

Donnay, A

L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi,Bridging Carrollian and celestial holography, Phys. Rev. D107 (2023) 126027 [2212.12553]. – 39 –

2023 arXiv
[28]

Mason, R

L. Mason, R. Ruzziconi and A. Yelleshpur Srikant,Carrollian amplitudes and celestial symmetries, JHEP 05 (2024) 012 [2312.10138]

2024 arXiv
[29]

Saha,Carrollian approach to 1 + 3D flat holography, JHEP 06 (2023) 051 [2304.02696]

A. Saha,Carrollian approach to 1 + 3D flat holography, JHEP 06 (2023) 051 [2304.02696]

2023 arXiv
[30]

Nguyen and P

K. Nguyen and P. West,Carrollian Conformal Fields and Flat Holography, Universe 9 (2023) 385 [2305.02884]

2023 arXiv
[31]

Bagchi, P

A. Bagchi, P. Dhivakar and S. Dutta,Holography in Flat Spacetimes: the case for Carroll, 2311.11246

arXiv
[32]

Bagchi, A

A. Bagchi, A. Banerjee, P. Dhivakar, S. Mondal and A. Shukla,The Carrollian Kaleidoscope, 2506.16164

arXiv
[33]

Bondi, H

M. Bondi, H. van der Burg and A. Metzner,Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond.21 (1962)

1962
[34]

Sachs,Gravitational waves in general relativity

R.K. Sachs,Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond.A270 (1962) 103

1962
[35]

Bagchi,Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys

A. Bagchi,Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett.105 (2010) 171601 [1006.3354]

2010 arXiv
[36]

Bagchi and R

A. Bagchi and R. Fareghbal,BMS/GCA Redux: Towards Flatspace Holography from Non-Relativistic Symmetries, JHEP 10 (2012) 092 [1203.5795]

2012 arXiv
[37]

Duval, G.W

C. Duval, G.W. Gibbons and P.A. Horvathy,Conformal carroll groups and BMS symmetry, Classical and Quantum Gravity31 (2014) 092001

2014
[38]

Bagchi, S

A. Bagchi, S. Detournay, R. Fareghbal and J. Simón,Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett.110 (2013) 141302 [1208.4372]

2013 arXiv
[39]

Barnich,Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [1208.4371]

G. Barnich,Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [1208.4371]

2012 arXiv
[40]

Bagchi, R

A. Bagchi, R. Basu, D. Grumiller and M. Riegler,Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett.114 (2015) 111602 [1410.4089]

2015 arXiv
[41]

Jiang, W

H. Jiang, W. Song and Q. Wen,Entanglement Entropy in Flat Holography, JHEP 07 (2017) 142 [1706.07552]

2017 arXiv
[42]

Bagchi, D

A. Bagchi, D. Grumiller and W. Merbis,Stress tensor correlators in three-dimensional gravity, Phys. Rev. D93 (2016) 061502 [1507.05620]

2016 arXiv
[43]

Barnich, A

G. Barnich, A. Gomberoff and H.A. Gonzalez,The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D86 (2012) 024020 [1204.3288]

2012 arXiv
[44]

Bagchi, S

A. Bagchi, S. Detournay and D. Grumiller,Flat-Space Chiral Gravity, Phys. Rev. Lett.109 (2012) 151301 [1208.1658]

2012 arXiv
[45]

Afshar, A

H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel,Spin-3 Gravity in Three-Dimensional Flat Space, Phys. Rev. Lett.111 (2013) 121603 [1307.4768]

2013 arXiv
[46]

Gonzalez, J

H.A. Gonzalez, J. Matulich, M. Pino and R. Troncoso,Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [1307.5651]

2013 arXiv
[47]

Barnich, H.A

G. Barnich, H.A. Gonzalez, A. Maloney and B. Oblak,One-loop partition function of three-dimensional flat gravity, JHEP 04 (2015) 178 [1502.06185]

2015 arXiv
[48]

Hartong,Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104 [1511.01387]

J. Hartong,Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104 [1511.01387]. – 40 –

2016 arXiv
[49]

Cornalba and M.S

L. Cornalba and M.S. Costa,A new cosmological scenario in string theory, Phys. Rev. D66 (2002) 066001

2002
[50]

Cornalba and M

L. Cornalba and M. Costa,Time-dependent orbifolds and string cosmology, Fortschritte der Physik 52 (2004) 145–199

2004
[51]

Bagchi, P

A. Bagchi, P. Nandi, A. Saha and Zodinmawia,BMS Modular Diaries: Torus one-point function, JHEP 11 (2020) 065 [2007.11713]

2020 arXiv
[52]

Bagchi, S

A. Bagchi, S. Mondal, S. Pal and M. Riegler,BMS modular covariance and structure constants, JHEP 11 (2023) 087 [2307.00043]

2023 arXiv
[54]

Denef, S.A

F. Denef, S.A. Hartnoll and S. Sachdev,Black hole determinants and quasinormal modes, Class. Quant. Grav.27 (2010) 125001 [0908.2657]

2010 arXiv
[55]

Barnich and G

G. Barnich and G. Compere,Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav.24 (2007) F15 [gr-qc/0610130]

2007 arXiv
[56]

Brown and M

J.D. Brown and M. Henneaux,Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys.104 (1986) 207

1986
[57]

Riegler,Flat space limit of higher-spin Cardy formula, Phys

M. Riegler,Flat space limit of higher-spin Cardy formula, Phys. Rev. D91 (2015) 024044 [1408.6931]

2015 arXiv
[58]

Fareghbal and A

R. Fareghbal and A. Naseh,Aspects of Flat/CCFT Correspondence, Class. Quant. Grav.32 (2015) 135013 [1408.6932]

2015 arXiv
[59]

Hawking and D.N

S.W. Hawking and D.N. Page,Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys.87 (1983) 577

1983
[60]

Bagchi, S

A. Bagchi, S. Detournay, D. Grumiller and J. Simon,Cosmic Evolution from Phase Transition of Three-Dimensional Flat Space, Phys. Rev. Lett.111 (2013) 181301 [1305.2919]

2013 arXiv
[61]

Denef, S.A

F. Denef, S.A. Hartnoll and S. Sachdev,Quantum oscillations and black hole ringing, Phys. Rev. D 80 (2009) 126016 [0908.1788]

2009 arXiv
[62]

Martin and A

V.L. Martin and A. Svesko,Normal modes in thermal AdS via the Selberg zeta function, SciPost Phys. 9 (2020) 009 [1910.11913]

2020 arXiv
[63]

Castro, C

A. Castro, C. Keeler and P. Szepietowski,Tweaking one-loop determinants in AdS3, JHEP 10 (2017) 070 [1707.06245]

2017 arXiv
[64]

Grumiller, W

D. Grumiller, W. Merbis and M. Riegler,Most general flat space boundary conditions in three-dimensional Einstein gravity, Class. Quant. Grav.34 (2017) 184001 [1704.07419]

2017 arXiv
[65]

Strominger, w1+∞ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries, Phys

A. Strominger, w1+∞ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries, Phys. Rev. Lett.127 (2021) 221601 [2105.14346]

2021 arXiv
[66]

Aharony, S.S

O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz,Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111]

2000 arXiv
[67]

Birmingham,Choptuik scaling and quasinormal modes in the anti-de sitter space/conformal-field theory correspondence, Phys

D. Birmingham,Choptuik scaling and quasinormal modes in the anti-de sitter space/conformal-field theory correspondence, Phys. Rev. D64 (2001) 064024. – 41 –

2001
[68]

Birmingham, I

D. Birmingham, I. Sachs and S.N. Solodukhin,Relaxation in conformal field theory, hawking-page transition, and quasinormal or normal modes, Physical Review D67 (2003)

2003
[69]

D.-P. Du, B. Wang and R.-K. Su,Quasinormal modes in pure de Sitter space-times, Phys. Rev. D 70 (2004) 064024 [hep-th/0404047]

2004 arXiv
[70]

Lopez-Ortega, Quasinormal modes of D-dimensional de Sitter spacetime, Gen

A. Lopez-Ortega, Quasinormal modes of D-dimensional de Sitter spacetime, Gen. Rel. Grav. 38 (2006) 1565 [gr-qc/0605027]. – 42 –

2006 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [53]

Bagchi, C

A. Bagchi, C. Keeler, V. Martin and R. Poddar,A generalized Selberg zeta function for flat space cosmologies, JHEP 04 (2024) 066 [2312.06770]

arXiv 2024

[2] [1]

LIGO Scientific Collaboration and Virgo Collaboration collaboration, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett.116 (2016) 061102

work page 2016

[3] [2]

Regge and J.A

T. Regge and J.A. Wheeler,Stability of a schwarzschild singularity, Phys. Rev. 108 (1957) 1063

work page 1957

[4] [3]

Zerilli,Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys

F.J. Zerilli,Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett.24 (1970) 737

work page 1970

[5] [4]

Zerilli,Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics, Phys

F.J. Zerilli,Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D2 (1970) 2141

work page 1970

[6] [5]

Zerilli,Perturbation analysis for gravitational and electromagnetic radiation in a reissner-nordstroem geometry, Phys

F.J. Zerilli,Perturbation analysis for gravitational and electromagnetic radiation in a reissner-nordstroem geometry, Phys. Rev. D9 (1974) 860. – 38 –

work page 1974

[7] [6]

Chandrasekhar,The mathematical theory of black holes(1985)

S. Chandrasekhar,The mathematical theory of black holes(1985)

1985

[8] [7]

Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936

C.V. Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature 227 (1970) 936

work page 1970