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REVIEW 3 major objections 4 minor 43 references

Thermal Origin of Black Hole Quasinormal Modes

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper establishes that eikonal black hole quasinormal modes are the retarded response of a thermal system living on the photon ring, with the photon ring Lyapunov exponent acting as the temperature scale through an induced Rindler…

desk verdict A clean microscopic derivation of eikonal QNMs from an induced Rindler worldsheet, plus an elegant but conditional thermal-response story that overstates its independence. read the letter →

arxiv 2608.09797 v1 pith:4WKC6OJI submitted 2026-08-10 hep-th astro-ph.COgr-qchep-ph

D. Giataganas , G.F. Giudice , A. Kehagias , F. Quevedo , A. Riotto This is my paper · ORCID
classification hep-thastro-ph.COgr-qchep-ph
keywords quasinormalmodesphotonringLyapunovexponentRindlerworldsheetthermalfieldtheoryeikonallimitblackholeringdownopenquantumsystems
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

In the short-wavelength (eikonal) limit, the paper argues that the characteristic ringing of a black hole is the retarded response of a thermal quantum system living on the photon ring. A probe string in the plane-wave limit of the near-ring geometry acquires a Rindler worldsheet whose surface gravity equals the photon ring Lyapunov exponent $\lambda_L$, so the regular worldsheet state is thermal at $T_{\rm ind}=\lambda_L/(2\pi)$. From this thermal data the paper derives the eikonal quasinormal spectrum $\omega_{mn}=m\Omega_{\rm orb}-i\lambda_L(n+\frac12)$ in two complementary ways: as outgoing resonances of an unstable transverse string fluctuation, and as the pole tower of a causal response function after projecting onto the escape channel. If correct, this gives a first-principles reason why the ringing always decays, why the overtone ladder is evenly spaced with spacing $\lambda_L$, and why the tower starts at the universal half-integer offset $n+\frac12$.

What carries the argument

The load-bearing object is a probe string stretched along the stable transverse direction of the plane-wave limit around the photon ring, whose induced worldsheet metric is Rindler, $ds^2_{\rm ws}=-\lambda_L^2\sigma^2 d\tau^2+d\sigma^2$, with horizon at $\sigma=0$ and surface gravity $\kappa_{\rm ind}=\lambda_L$. Euclidean regularity fixes $T_{\rm ind}=\lambda_L/(2\pi)$. The spectrum is then carried by two linked mechanisms: the zero-momentum unstable transverse fluctuation, governed by the inverted oscillator with outgoing resonance poles $\omega_{cr,n}=-i\lambda_L(n+\frac12)$, and a projection that separates the near-ring sector from the complementary leakage channels, producing an absorptive self-energy whose retarded poles lie in the lower half-plane. The half-integer offset comes from the unitary half-density dilation representation on the projected escape coordinate, which fixes the boost weight $h=1/2$; this is the step that KMS thermality alone cannot determine.

What would settle it

A microscopic calculation of the projected retarded Green function from the string action that found the positive-time commutator decaying with exponents different from $(2\pi T_{\rm ind})(n+\frac12)$ would falsify the macroscopic derivation; observationally, a black hole whose leading eikonal QNM imaginary parts deviate from $-(n+\frac12)\lambda_L$ would contradict the claimed universality.

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Extended reading notes

Core claim

The central claim is that the eikonal black hole quasinormal-mode spectrum has a thermal origin: the photon ring induces a Rindler horizon on a probe string's worldsheet, with surface gravity $\kappa_{\rm ind}=\lambda_L$ and temperature $T_{\rm ind}=\lambda_L/(2\pi)$, and the QNMs are the retarded response poles of this thermal system. The unstable transverse string fluctuation supplies the microscopic mechanism, realized as an inverted harmonic oscillator whose outgoing resonances give $E_n=-i\lambda_L(n+\frac12)$ in the co-rotating frame. Macroscopically, the same tower arises as the pole structure of the causal response of the projected escape channel, where thermal detailed balance, spectral positivity, and an open-subsystem projection place the poles in the lower half of the complex-frequency plane. The half-integer offset is fixed by the unitary half-density representation of worldsheet boosts on the projected radial coordinate, giving effective weight $h=1/2$. Restoring the orbital motion yields $\omega_{mn}=m\Omega_{\rm orb}-i\lambda_L(n+\frac12)$, identical to the known eikonal spectrum, now obtained from thermal worldsheet data as in Eqs. (2.35) and (5.28).

Load-bearing premise

The macroscopic thermal derivation depends on the assumption, stated in Sections 5.1 and Appendix A, that the nonlocal positive-time part of the retarded commutator for the projected escape channel decays with the same exponents as the thermal two-point kernel; the paper states this is not implied by the KMS condition and is not derived from the microscopic string action.

Editorial extensions

If this is right

  • If the central claim is correct, every eikonal ringdown is a thermal response: the damping rate of each overtone is set by the photon ring temperature $T_{\rm ind}=\lambda_L/(2\pi)$, with no separate input from the event-horizon temperature.
  • The universal half-integer offset $n+\frac12$ follows from the unitary half-density representation of boosts on the projected escape coordinate, so it should persist for any black hole whose photon ring is non-degenerate.
  • Causality, spectral positivity, and the passivity of the thermal state jointly require QNM poles to sit in the lower half of the complex-frequency plane, giving a first-principles reason why ringing decays rather than grows.
  • The two derivations, microscopic inverted-oscillator resonances and macroscopic thermal response, are complementary, so the same spectrum in Eqs. (2.35) and (5.28) can be recovered either from string fluctuations or from the causal correlator.

Reading between the lines

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

  • Editorial inference: The paper leaves the subleading eikonal regime open; a natural next step is to compute the first finite-eikonal correction to the projected self-energy and test whether the half-density weight $h=1/2$ is protected or shifted.
  • Editorial inference: Because the paper identifies worldsheet fluctuations with near-ring dissipation through a fluctuation-dissipation relation, numerical ringdown simulations could test the predicted proportionality between the early-time fluctuation spectrum and the QNM damping slope.
  • Editorial inference: If the induced Rindler temperature is physical rather than formal, a detector coupled to the worldsheet should register a thermal bath at $\lambda_L/(2\pi)$ even though the photon ring itself is not a horizon; this gives a sharper analogue-gravity test of the thermal interpretation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper argues that eikonal black-hole quasinormal modes have a thermal origin on the photon ring. A probe string in the Penrose-limit plane wave near the unstable circular null orbit acquires an induced Rindler worldsheet metric with surface gravity equal to the photon-ring Lyapunov exponent; Euclidean regularity of the regular worldsheet state gives an induced temperature T_ind = λ_L/(2π). The σ-independent transverse fluctuation is an inverted harmonic oscillator whose outgoing Gamow poles yield ω_mn = mΩ_orb − iλ_L(n+1/2), Eq. (2.35). The authors then attempt a macroscopic derivation: KMS thermality, Feshbach projection onto an escape channel, and the unitary half-density representation of Rindler boosts are used to show that the retarded response of an effective operator has the same pole tower, Eq. (5.28). The paper concludes that black-hole ringing is the retarded response of a thermal system living on the photon ring.

Significance. If the central claim is accepted, the paper offers a conceptually unifying picture in which the photon-ring Lyapunov exponent acts as an induced temperature saturating the MSS bound and the half-integer overtone offset is a kinematical consequence of unitary half-density boost representations. The microscopic derivation in Section 2 is a genuine strength: it is explicit, contains no free parameters, and reduces the eikonal QNM damping to an inverted-oscillator Gamow problem. The Feshbach/causality discussion in Section 4 usefully explains the absorptive sign of the width. The significance is currently limited by the status of the macroscopic derivation, which rests on an explicitly undereived response-theory assumption; the paper therefore establishes a consistent effective thermal description rather than two fully independent derivations of the spectrum.

major comments (3)
  1. [§5.1 and Appendix A, Eq. (A.11)] The macroscopic pole derivation is conditional on the assumption, stated at Eq. (A.11), that the nonlocal positive-time part of the projected commutator has the same decay exponents as the thermal Wightman kernel. The manuscript correctly notes that this is "not implied by the KMS condition" and is not derived from the microscopic Nambu-Goto theory. This matters because the retarded response is controlled by the spectral density ρ(ω), while KMS only relates G> and G< and does not fix the analytic structure of ρ(ω). As written, Eqs. (5.5)-(5.9) and (A.13)-(A.17) therefore do not independently derive the pole tower; they show that a 1/sinh kernel, together with the stated assumption, reproduces it. The abstract's claim that the spectrum is "explicitly derive[d] ... macroscopically" is accordingly overstated. Please either derive the assumption from the microscopic worldsheet theory or explicitly reframe the macroscopic part as a consistency check of an effective description.
  2. [§4.2] The open-system step relies on Feshbach projectors P and Q, but the manuscript states that these projectors are not obtained from an explicit mode-by-mode decomposition of the gauge-fixed Nambu-Goto theory. Since the projected effective Hamiltonian H_eff = H_PP + Σ(ω) and hence the pole positions in the thermal derivation depend on the choice of P and Q, the absence of an explicit construction leaves the identification of the leakage channels uncontrolled. This does not affect the microscopic Gamow derivation, but it is a second reason why the macroscopic thermal-response calculation is not yet a first-principles derivation of the pole positions.
  3. [§5.3 and Appendix A] The derivation of h=1/2 from unitary dilations on L²(R+,dχ) is kinematically correct once the reduced coordinate χ is identified with the expanding branch of the inverted oscillator. However, this identification and the assignment of half-density covariance to the projected escape operator O_esc are made by hand; the manuscript itself says the operator is not assumed to be a fundamental local Nambu-Goto field. The half-integer offset in the thermal derivation is therefore an effective representation of the microscopic Gamow result rather than an independent computation. Section 6 should state this limitation more plainly, or the authors should derive the projected operator from the microscopic string theory.
minor comments (4)
  1. [§4.2 heading] The heading reads "F eshbach projection" and should be "Feshbach projection".
  2. [§5.1, Eq. (5.5)] The co-rotating frequency ω_b is used in Eq. (5.5) but is defined only later in Eq. (5.8); define it at first use.
  3. [§5.2] The discussion of a "fermion-like thermal monodromy" should clarify that this is a monodromy of the effective half-density kernel, not a statement about the spin-statistics of the worldsheet fields.
  4. [References] Reference [21] is cited as "2605.29923" without an arXiv prefix; if it is a preprint, please provide the full identifier and its current status.

Circularity Check

1 steps flagged · score 6.0 of 10

The macroscopic thermal-response derivation assumes the decay exponents that produce the QNM tower (App. A, Eq. A.11); the microscopic Gamow derivation is independent, so the circularity is partial.

  1. self definitional [Appendix A, after Eq. (A.10) and Eq. (A.11); see also Section 5.1, Eqs. (5.4)-(5.7)]
    "We assume that the nonlocal positive-time part of the retarded commutator has the same decay exponents as (A.12), although its coefficients may differ. This assumption is not implied by the KMS condition and is not derived here from the microscopic Nambu–Goto theory."

    The exponents in Eq. (A.12) are the pole positions: 1/sinh(kappa tau/2) expands as 2 sum_n exp[-kappa(n+1/2) tau], which Fourier transforms to poles at -i kappa (n+1/2) (Eqs. A.13-A.16). By assuming the retarded commutator has the same decay exponents, the paper assumes the exact set of poles that Eq. (A.17) and Eq. (5.28) then present as the derived QNM spectrum. KMS fixes only the thermal scale kappa = 2 pi T_ind and the analytic strip, not the retarded exponents; the paper concedes this in the quoted sentence. Appendix A itself labels the result 'a consistency check ... rather than an independent derivation of the pole spectrum.' Thus the macroscopic thermal derivation reduces by construction to its own output: the assumed exponents are the predicted poles.

full rationale

The microscopic derivation in Section 2.3 is self-contained and not circular: the Penrose-limit metric and the probe-string embedding are given explicitly, the worldsheet metric is pulled back to Rindler form, the zero-mode equation is an inverted oscillator with frequency equal to the photon-ring Lyapunov exponent computed from tidal data, and the Gamow resonance condition yields the eikonal QNM tower (2.35). The half-density weight h = 1/2 in Section 5.3 is likewise derived from the unitary dilation representation on L^2(R_+, dchi), independent of QNM data. The circularity is confined to the advertised second, macroscopic derivation: the retarded pole tower is obtained by Fourier transforming the thermal Wightman kernel after assuming that the commutator's nonlocal part has the same decay exponents (Eq. A.11), an assumption the paper explicitly says is not implied by KMS and not derived from the Nambu-Goto theory. Appendix A further states that this is a consistency check, not an independent derivation. Therefore the abstract's claim of 'two complementary perspectives' overstates the epistemic status of the thermal-response derivation. Self-citations [18] and [21] are not load-bearing because the Rindler worldsheet and tidal data are rederived in Section 2. Overall score 6: one of the two advertised derivations reduces to its assumed input, while the central spectrum still has independent microscopic support.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data; lambda_L and Omega_orb are computed from the photon ring geometry. The central claim rests on several explicitly stated assumptions, most notably the response-theory assumption in Appendix A and the existence of an unprojected boost-covariant completion in Section 3.

assumptions (6)
  • domain assumption The Penrose limit along the photon ring retains the tidal data that fixes the eikonal QNM spectrum.
    Section 2 uses the pp-wave limit (2.3)-(2.6) as the complete near-ring geometry; the tower of overtones is extracted from this quadratic data.
  • domain assumption The regular two-sided state on the Rindler worldsheet is the physical state, fixing T_ind via Euclidean smoothness.
    Section 2.1, Eqs (2.18)-(2.20): standard Euclidean regularity, but assumes the worldsheet vacuum is the Hartle-Hawking-like state.
  • ad hoc to paper The sigma-independent zero mode of the transverse fluctuation is the unique channel reproducing the eikonal QNM tower.
    Section 2.2: modes with |q|<1 are unstable but the paper asserts that q != 0 modes should not be identified with the spacetime QNM overtones.
  • ad hoc to paper The worldsheet system admits a regular two-sided, boost-covariant completion to which the Bisognano-Wichmann theorem applies.
    Section 3, paragraph after Eq (3.3): stated as an assumption, needed for KMS thermality of H_R.
  • ad hoc to paper The Feshbach projectors P and Q separate the near-ring escape channel from leakage channels.
    Section 4.2: 'not obtained here from an explicit mode-by-mode decomposition' of the complete Nambu-Goto theory.
  • ad hoc to paper The positive-time part of the retarded commutator decays with the same exponents as the thermal Wightman kernel.
    Section 5.1 and Appendix A: explicitly 'not implied by the KMS condition and not derived here' from the microscopic theory.

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Pith. "Pith review of Thermal Origin of Black Hole Quasinormal Modes." pith.science (2026). https://pith.science/paper/4WKC6OJI

@misc{pith2026260809797,
  author       = {Pith},
  title        = {Pith review of: Thermal Origin of Black Hole Quasinormal Modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4WKC6OJI}},
  note         = {Machine review of arXiv:2608.09797}
}
read the original abstract

When a black hole rings after a merger, it emits gravitational waves at characteristic frequencies known as quasinormal modes (QNMs). In the eikonal limit, these modes are governed by the unstable circular light orbits that form the photon ring. In this work, we demonstrate that the ringing of a black hole has a precise thermal interpretation. A probe string propagating in the near-ring geometry acquires an induced Rindler horizon on its worldsheet, with a temperature set by the Lyapunov exponent of the photon ring. Out of this structure, the black hole QNMs emerge as thermal excitations, so that the characteristic ringing of a black hole is the retarded response of a thermal system living on the photon ring. We explicitly derive the QNM spectrum from two complementary perspectives: microscopically, via unstable transverse worldsheet fluctuations, and macroscopically, through the pole structure of the causal response function of an open thermal quantum system.

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Works this paper leans on

43 extracted references · 19 canonical work pages

  1. [1]

    Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature227(1970) 936

    C.V. Vishveshwara,Scattering of Gravitational Radiation by a Schwarzschild Black-hole, Nature227(1970) 936

    work page 1970

  2. [2]

    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

    work page 1971

  3. [3]

    Chandrasekhar and S.L

    S. Chandrasekhar and S.L. Detweiler,The quasi-normal modes of the Schwarzschild black hole,Proc. Roy. Soc. Lond. A344(1975) 441

    1975

  4. [4]

    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]

    arXiv 1999

  5. [5]

    Nollert,Topical Review: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars,Class

    H.-P. Nollert,Topical Review: Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars,Class. Quant. Grav.16(1999) R159

    work page 1999

  6. [6]

    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]

    arXiv 2009

  7. [7]

    Konoplya and A

    R.A. Konoplya and A. Zhidenko,Quasinormal modes of black holes: From astrophysics to string theory,Rev. Mod. Phys.83(2011) 793 [1102.4014]

    arXiv 2011

  8. [8]

    Berti et al.,Black hole spectroscopy: from theory to experiment,Class

    E. Berti et al.,Black hole spectroscopy: from theory to experiment,Class. Quant. Grav.43 (2026) 123001 [2505.23895]. [9]LIGO Scientific, Virgocollaboration,Observation of Gravitational Waves from a Binary Black Hole Merger,Phys. Rev. Lett.116(2016) 061102 [1602.03837]

    arXiv 2026

Show all 43 references
  1. [10]

    Berti, V

    E. Berti, V. Cardoso and C.M. Will,On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA,Phys. Rev. D73(2006) 064030 [gr-qc/0512160]

    2006 arXiv

  2. [11]

    M. Isi, M. Giesler, W.M. Farr, M.A. Scheel and S.A. Teukolsky,Testing the no-hair theorem with GW150914,Phys. Rev. Lett.123(2019) 111102 [1905.00869]

    2019 arXiv

  3. [12]

    Ferrari and B

    V. Ferrari and B. Mashhoon,New approach to the quasinormal modes of a black hole,Phys. Rev. D30(1984) 295

    1984

  4. [13]

    Mashhoon,Stability of charged rotating black holes in the eikonal approximation,Phys

    B. Mashhoon,Stability of charged rotating black holes in the eikonal approximation,Phys. Rev. D31(1985) 290

    1985

  5. [14]

    Cardoso, A.S

    V. Cardoso, A.S. Miranda, E. Berti, H. Witek and V.T. Zanchin,Geodesic stability, Lyapunov exponents and quasinormal modes,Phys. Rev. D79(2009) 064016 [0812.1806]. [15]Event Horizon Telescopecollaboration,First M87 Event Horizon Telescope Results. I. The Shadow of the Supermas...

    2009 arXiv

  6. [16]

    Gralla, D.E

    S.E. Gralla, D.E. Holz and R.M. Wald,Black Hole Shadows, Photon Rings, and Lensing Rings,Phys. Rev. D100(2019) 024018 [1906.00873]

    2019 arXiv

  7. [17]

    Johnson et al.,Universal interferometric signatures of a black hole’s photon ring,Sci

    M.D. Johnson et al.,Universal interferometric signatures of a black hole’s photon ring,Sci. Adv.6(2020) eaaz1310 [1907.04329]

    2020 arXiv

  8. [18]

    Giataganas, A

    D. Giataganas, A. Kehagias and A. Riotto,Quasinormal modes and universality of the Penrose limit of black hole photon rings,JHEP09(2024) 168 [2403.10605]

    2024 arXiv

  9. [19]

    Maldacena, S.H

    J. Maldacena, S.H. Shenker and D. Stanford,A bound on chaos,JHEP08(2016) 106 [1503.01409]

    2016 arXiv

  10. [20]

    Hadar, D

    S. Hadar, D. Kapec, A. Lupsasca and A. Strominger,Holography of the photon ring,Class. Quant. Grav.39(2022) 215001 [2205.05064]. – 25 –

    2022 arXiv

  11. [21]

    Giataganas, G.F

    D. Giataganas, G.F. Giudice, A. Kehagias, F. Quevedo and A. Riotto,Black Hole Photon Rings Saturate the Quantum Chaos Bound,2605.29923

    arXiv

  12. [22]

    M. Blau, D. Frank and S. Weiss,Fermi coordinates and Penrose limits,Class. Quant. Grav. 23(2006) 3993 [hep-th/0603109]

    2006 arXiv

  13. [23]

    Kapec and A

    D. Kapec and A. Sheta,pp-waves and the hidden symmetries of black hole quasinormal modes,Class. Quant. Grav.42(2025) 155002 [2412.08551]

    2025 arXiv

  14. [24]

    Unruh,Notes on black hole evaporation,Phys

    W.G. Unruh,Notes on black hole evaporation,Phys. Rev. D14(1976) 870

    1976

  15. [25]

    Bisognano and E.H

    J.J. Bisognano and E.H. Wichmann,On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys.16(1975) 985

    1975

  16. [26]

    Bisognano and E.H

    J.J. Bisognano and E.H. Wichmann,On the Duality Condition for Quantum Fields,J. Math. Phys.17(1976) 303

    1976

  17. [27]

    Crispino, A

    L.C.B. Crispino, A. Higuchi and G.E.A. Matsas,The Unruh effect and its applications,Rev. Mod. Phys.80(2008) 787 [0710.5373]

    2008 arXiv

  18. [28]

    Hawking,Particle Creation by Black Holes,Commun

    S.W. Hawking,Particle Creation by Black Holes,Commun. Math. Phys.43(1975) 199

    1975

  19. [29]

    Barton,Quantum mechanics of the inverted oscillator potential,Annals Phys.166(1986) 322

    G. Barton,Quantum mechanics of the inverted oscillator potential,Annals Phys.166(1986) 322

    1986

  20. [30]

    Feshbach,Unified theory of nuclear reactions,Annals Phys.5(1958) 357

    H. Feshbach,Unified theory of nuclear reactions,Annals Phys.5(1958) 357

    1958

  21. [31]

    Feshbach,A Unified theory of nuclear reactions

    H. Feshbach,A Unified theory of nuclear reactions. 2.,Annals Phys.19(1962) 287

    1962

  22. [32]

    Fano,Effects of Configuration Interaction on Intensities and Phase Shifts,Phys

    U. Fano,Effects of Configuration Interaction on Intensities and Phase Shifts,Phys. Rev.124 (1961) 1866

    1961

  23. [33]

    Pusz and S.L

    W. Pusz and S.L. Woronowicz,Passive States and KMS States for General Quantum Systems,Commun. Math. Phys.58(1978) 273

    1978

  24. [34]

    vibrations

    C.J. Goebel,Comments on the “vibrations” of a Black Hole,APJL172(1972) L95

    1972

  25. [35]

    H. Yang, F. Zhang, A. Zimmerman, D.A. Nichols, E. Berti and Y. Chen,Branching of quasinormal modes for nearly extremal Kerr black holes,Phys. Rev. D87(2013) 041502 [1212.3271]

    2013 arXiv

  26. [36]

    Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700

    M.L. Bellac,Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press (3, 2011), 10.1017/CBO9780511721700

    2011 doi

  27. [37]

    Kubo,Statistical mechanical theory of irreversible processes

    R. Kubo,Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems,J. Phys. Soc. Jap.12(1957) 570

    1957

  28. [38]

    Martin and J.S

    P.C. Martin and J.S. Schwinger,Theory of many particle systems. 1.,Phys. Rev.115(1959) 1342

    1959

  29. [39]

    Haag, N.M

    R. Haag, N.M. Hugenholtz and M. Winnink,On the Equilibrium states in quantum statistical mechanics,Commun. Math. Phys.5(1967) 215

    1967

  30. [40]

    Srednicki,Quantum Field Theory, Cambridge University Press (2007)

    M. Srednicki,Quantum Field Theory, Cambridge University Press (2007)

    2007

  31. [41]

    Takagi,Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimension,Prog

    S. Takagi,Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimension,Prog. Theor. Phys. Suppl.88(1986) 1

    1986

  32. [42]

    Arrechea, C

    J. Arrechea, C. Barcel´ o, L.J. Garay and G. Garc ´ ıa-Moreno,Inversion of statistics and thermalization in the Unruh effect,Phys. Rev. D104(2021) 065004 [2101.11933]. – 26 –

    2021 arXiv

  33. [43]

    Hegde, V

    S.S. Hegde, V. Subramanyan, B. Bradlyn and S. Vishveshwara,Quasinormal Modes and the Hawking-Unruh Effect in Quantum Hall Systems: Lessons from Black Hole Phenomena, Phys. Rev. Lett.123(2019) 156802 [1812.08803]

    2019 arXiv

  34. [44]

    Arnaudo, C

    P. Arnaudo, C. Iossa, R. Karlsson and B. Withers,OPE = QNM,2607.24909

    arXiv

  35. [45]

    Barrat, D.N

    J. Barrat, D.N. Bozkurt, E. Marchetto, A. Miscioscia and E. Pomoni,Analytic thermal bootstrap in momentum space: From thermal OPE to QNMs,2607.24919. – 27 –

    arXiv

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