Three-Dimensional Quantum Black Holes: A Primer

In the paper

“However, static braneworld black holes in higher-dimensions have been found numerically, e.g., [67][68][69][70][71][72], and the qualitative argument was shown to have flaws [69].”

What the sources say

• Partly supported

  The Summary reports numerical, approximate solutions for black holes localized on a brane, but only with horizons small relative to the bulk curvature scale. The Introduction mentions an argument against static localized black holes; neither it nor the Summary shows that the argument has flaws. The Summary also cautions that the imposed boundary conditions do not guarantee the absence of naked singularities. (LLM, conf=0.88)

  cites [67] Kudoh, 2003, Small localized black holes in brane world: Formulation and numerical method, Phys. Rev. D sourceExperimental — accuracy not yet measured
• ?Not checked — no full text of the sourcecites [68] Kudoh, 2004, Six-dimensional localized black holes: Numerical solutions, Phys. Rev. D source
• Supported

  The Introduction and summary reports that numerical work found static localized braneworld black holes, while noting that large ones had not been constructed. Section 2 challenges the argument that static solutions cannot exist, presenting a counterexample and explaining why the predicted enhanced Hawking radiation need not occur. (LLM, conf=0.91)

  cites [69] Fitzpatrick, 2006, On the existence and dynamics of braneworld black holes, J. High Energy Phys. source
• Disputed

  Citation [69] is attached both to the numerical-finding claim and to the claim about flaws in a qualitative argument. Yoshino’s Summary and discussion reports nonsystematic numerical errors and says its results raise strong doubts that the specified static braneworld black-hole solutions exist, rather than establishing them numerically. The Introduction notes differing opinions about the AdS/CFT argument, but does not show that the argument has flaws. (LLM, conf=0.86)

  Checked by a reasoning model — may be wrong.

  cites [70] Yoshino, 2009, On the existence of a static black hole on a brane, J. High Energy Phys. sourceExperimental — accuracy not yet measured
• Supported

  Section III reports the numerical construction of static five-dimensional RSII braneworld black holes. The Introduction identifies a flaw in the qualitative non-existence argument: it applies free-field intuition to a strongly coupled CFT, an extrapolation the paper says may not be justified. Page numbers are unavailable in our copy. (LLM, conf=0.89)

  cites [71] Figueras, 2011, Gravity and large black holes in Randall-Sundrum II braneworlds, Phys. Rev. Lett. source
• Partly supported

  Section 2.4 describes static, stable black droplets on a brane, and Section 5 reports numerically built equilibrium funnels in an AdS5 braneworld. Our copy does not say that the qualitative argument mentioned in the citing sentence was shown to have flaws. (LLM, conf=0.82)

  cites [72] Emparan, 2023, Holographic duals of evaporating black holes, J. High Energy Phys. sourceExperimental — accuracy not yet measured

In the paper

“Notably, δg tt > 0, indicating gravitational attraction [32,33].”

What the sources say

• Disputed

  Section 4 (eqs. 4.10, 4.13–4.15; page unavailable) gives a negative correction to g_tt for the attractive Newtonian potential and states that acceleration is −∇Φ. Section 6 (eq. 6.10; page unavailable) permits a positive correction for twisted fields, but does not identify that case as gravitational attraction. The claimed link between δg_tt > 0 and attraction is therefore contrary to the paper’s treatment. (LLM, conf=0.87)

  Checked by a reasoning model — may be wrong.

  cites [32] Souradeep, 1992, Quantum effects near a point mass in (2+1)-Dimensional gravity, Phys. Rev. D sourceExperimental — accuracy not yet measured
• Supported

  In §2, equations (11) and (15) give A² = 1 − 2ℓ/r, while equation (1) defines g_tt = −A². The paper states that ℓ is positive, so the correction δg_tt = 2ℓ/r is positive; the same section says test particles experience gravitational attraction in this geometry. Page numbers are unavailable in our copy. (LLM, conf=0.91)

  cites [33] Soleng, 1993, Inverse square law of gravitation in (2+1) dimensional space-time as a consequence of Casimir energy, Phys. Scr. source

In the paper

“From the mass identification (4.27), M takes values in the finite range [23,24],”

• Also stated in: § 4.2.1 Static quantum BTZ · plain

  “- 1 8G 3 ≤ M ≤ 1 24G 3 , (4.37”

What the sources say

• Disputed

  Section 3.3 (eq. 3.26; page unavailable) gives the limiting three-dimensional mass as 1/(4G₃), not 1/(24G₃). Section 3.2 treats the small-mass limit as positive and approaching zero, rather than −1/(8G₃). The source supports a finite limiting mass, but contradicts the stated endpoints. (LLM, conf=0.88)

  Checked by a reasoning model — may be wrong.

  cites [23] Emparan, 2000, Exact description of black holes on branes, J. High Energy Phys. sourceExperimental — accuracy not yet measured
• Supported

  Section 3.2 gives the lower endpoint, stating that the negative-mass localized solutions run from −1/(8G₃) to zero, and gives 1/(24G₃) as the maximum positive mass for localized black holes. Section 6 clarifies that the lower endpoint is pure AdS rather than a black hole. Thus the stated finite range matches the paper’s mass identification for localized solutions, though it does not bound BTZ black strings. (LLM, conf=0.87)

  cites [24] Emparan, 2000, Exact description of black holes on branes. 2. Comparison with BTZ black holes and black strings, J. High Energy Phys. source

In the paper

“In fact, there are several examples of higher-dimensional static braneworld black holes, both analytically and numerically constructed [67][68][69][70][71][72][201][202][203][204][205][206][207][208].”

What the sources say

• Supported

  This paper is an example of the numerically constructed higher-dimensional static braneworld black holes cited in the sentence. The Introduction describes a numerical construction in the Randall–Sundrum braneworld, and the Summary reports small localized black-hole solutions. The citation need not itself provide an analytical example within the cited group. (LLM, conf=0.92)

  cites [67] Kudoh, 2003, Small localized black holes in brane world: Formulation and numerical method, Phys. Rev. D source
• ?Not checked — no full text of the sourcecites [68] Kudoh, 2004, Six-dimensional localized black holes: Numerical solutions, Phys. Rev. D source
• Supported

  The Introduction reports numerically constructed localized static black holes, including Kudoh’s solutions. Section 2.1 constructs an analytic five-dimensional static braneworld solution, the uniformly warped black string. Together, these passages support the cited sentence’s broad statement about analytic and numerical examples. (LLM, conf=0.87)

  cites [69] Fitzpatrick, 2006, On the existence and dynamics of braneworld black holes, J. High Energy Phys. source
• Disputed

  The Introduction (page unavailable) describes an exact static brane black hole in four-dimensional spacetime, but says finding a solution in five dimensions has been difficult and earlier numerical results require caution. The Summary and discussion (page unavailable) says this paper’s numerical results raise strong doubts that the five-dimensional static brane-black-hole solution studied exists, rather than providing a constructed example. (LLM, conf=0.85)

  Checked by a reasoning model — may be wrong.

  cites [70] Yoshino, 2009, On the existence of a static black hole on a brane, J. High Energy Phys. sourceExperimental — accuracy not yet measured
• Supported

  Section III, “5D Static RSII Black Holes,” reports the paper’s numerical construction of five-dimensional static braneworld black holes. The Introduction also describes earlier numerical constructions and perturbative constructions of small black holes. This paper therefore supports the cited sentence as an example within its list of sources. (LLM, conf=0.91)

  cites [71] Figueras, 2011, Gravity and large black holes in Randall-Sundrum II braneworlds, Phys. Rev. Lett. source
• Supported

  Section 2.4 describes static black holes on branes, identifies the uniform funnel as an exact solution, and says static stable droplets can be found on the brane. Section 2.2 says droplet solutions can be constructed numerically. Together these provide analytical and numerical examples of the kind cited. (LLM, conf=0.89)

  cites [72] Emparan, 2023, Holographic duals of evaporating black holes, J. High Energy Phys. source
• Supported

  The Introduction reports numerically constructed static brane-localized black holes whose horizons are not much larger than the bulk curvature scale. Section IV describes an exact analytic brane-localized black-hole solution in the four-dimensional RS-II case. The paper reports these examples; it does not claim to have constructed a static large black hole itself. (LLM, conf=0.86)

  cites [201] Tanahashi, 2008, Time-symmetric initial data of large brane-localized black hole in RS-II model, J. High Energy Phys. source
• Supported

  The introductory discussion (page not supplied) reports numerically constructed static, nonsingular braneworld black holes and identifies an explicit black-string solution. The later discussion describes that black string as a suitable brane-plus-bulk solution for a range of parameters. Together, these support the cited sentence's broad claim about numerical and analytic examples. (LLM, conf=0.90)

  cites [202] Gregory, 2008, Classical and quantum gravity of brane black holes, J. High Energy Phys. source
• Partly supported

  Section 4 discusses the existence of Randall–Sundrum braneworld black holes, and the References list a work titled “Exact description of black holes on branes,” supporting the analytical-example part of the claim. The passages found do not establish that this paper constructs a braneworld black hole numerically or report several analytically and numerically constructed examples. (LLM, conf=0.78)

  cites [203] Figueras, 2011, Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua, Class. Quant. Grav. sourceExperimental — accuracy not yet measured
• Supported

  The Abstract states that the authors constructed large static Randall–Sundrum II braneworld black-hole solutions using a numerical method. The Conclusion describes independent numerical evidence for those solutions. This paper therefore supplies a numerical example in the cited group; the sentence does not require every cited paper to provide both an analytical and a numerical construction. (LLM, conf=0.91)

  cites [204] Abdolrahimi, 2013, Large Randall-Sundrum II Black Holes, Phys. Lett. B source

In the paper

“In particular, in scenarios of gravity induced due to matter loops [221,222], the Bekenstein-Hawking entropy is solely due to entanglement due to vacuum fluctuations [223][224][225], where the UV-cutoff dependence can be absorbed in a renormalization of Newton's constant [130].”

What the sources say

• ?Not checked — no full text of the sourcecites [221] Sakharov, 1967, Vacuum quantum fluctuations in curved space and the theory of gravitation, Dokl. Akad. Nauk Ser. Fiz. source
• Not checked yet

  Checked before — its earlier verdict is held back until a new check runs.

  cites [222] Visser, 2002, Sakharov’s induced gravity: A Modern perspective, Mod. Phys. Lett. A source
• Not checked yet

  Checked before — its earlier verdict is held back until a new check runs.

  cites [223] Jacobson, T. (1994). Black hole entropy and induced gravity. arXiv. source
• Disputed

  Section 4 (Eq. 4.30; page unavailable) says that, in the induced-gravity model studied, Bekenstein–Hawking entropy is not solely a statistical-mechanical entropy: it also requires a horizon term involving scalar-field fluctuations. The authors say that term is not related to counting states and removes divergences. The UV-cutoff renormalization clause is attached to a different citation in the citing sentence. (LLM, conf=0.90)

  Checked by a reasoning model — may be wrong.

  cites [224] Frolov, 1997, Statistical origin of black hole entropy in induced gravity, Nucl. Phys. B sourceExperimental — accuracy not yet measured
• Not checked yet

  Checked before — its earlier verdict is held back until a new check runs.

  cites [225] Frolov, 1997, Mechanism of generation of black hole entropy in Sakharov’s induced gravity, Phys. Rev. D source
• Supported

  The citation to this paper is attached to the UV-cutoff and Newton’s-constant clause, not the preceding claim that entropy is solely due to entanglement. Section 3 (pp. 13, 15) says the entropy’s quadratic divergence also enters the renormalization of the gravitational coupling, and that the divergences in entropy are the same ones that renormalize it. (LLM, conf=0.93)

  cites [130] Susskind, 1994, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D source

In the paper

“Notably, thus far, all 3 + 1-dimensional braneworld models that have been explored and exhibit island formation are described by massive gravity theories; islands disappear in the limit the graviton on the brane becomes massless [267] (see, e.g., [268] for a review).”

What the sources say

• Supported

  The Introduction describes the explored four-dimensional RS-brane island models as having massive gravitons, and §5 says islands are lost in the massless-graviton limit. The paper qualifies that result: in its RS model, the limit also makes G vanish, which the authors identify as the driver of the disappearance; they do not rule out islands in every possible massless-graviton theory. (LLM, conf=0.88)

  cites [267] Geng, 2020, Massive islands, J. High Energy Phys. source
• Disputed

  The Introduction acknowledges that an earlier Karch-Randall setup has massive gravity, but then describes island surfaces and a Page curve despite massless gravity on the brane. It also explicitly states that islands exist with massless gravity, contrary to the claim that they disappear when the graviton becomes massless. Page numbers are not available in our copy. (LLM, conf=0.90)

  Checked by a reasoning model — may be wrong.

  cites [268] Geng, H. (2023). Revisiting Recent Progress in the Karch-Randall Braneworld. arXiv. sourceExperimental — accuracy not yet measured

In the paper

“Generally, however, the thermodynamic volume (5.90) differs from the geometric volume[142][143][144].”

What the sources say

• Disputed

  In §2 (Enthalpy), equation (8) gives the thermodynamic volume as 4πrₕ³/3 and says the naïve volume is the correct one for thermodynamic relations. Section 5 likewise calls the higher-dimensional thermodynamic volume the naïve result. Section 6 identifies a narrower exception: quantum corrections reduce the BTZ black hole’s volume below its classical value. These passages run contrary to the unqualified claim that the volumes generally differ. (LLM, conf=0.86)

  Checked by a reasoning model — may be wrong.

  cites [142] Dolan, 2011, The cosmological constant and the black hole equation of state, Class. Quant. Grav. sourceExperimental — accuracy not yet measured
• Supported

  The Conclusions state that the thermodynamic volume differs from the geometric volume when a black hole rotates, while they can coincide without rotation. The Introduction likewise distinguishes the two volumes for Kerr-AdS black holes. This supports the claim’s general, rather than universal, wording. (LLM, conf=0.94)

  cites [143] Cvetic, 2011, Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume, Phys. Rev. D source
• Supported

  Section 2.1 states that Taub-NUT is a case where thermodynamic and naive geometric volumes differ, and Section 2.3 specifies that its naive geometric volume is zero while its thermodynamic volume is nonzero. The paper also notes that the volumes coincide for static black holes, which is consistent with the claim’s qualifier “generally,” rather than “always.” (LLM, conf=0.87)

  cites [144] Johnson, 2014, Thermodynamic Volumes for AdS-Taub-NUT and AdS-Taub-Bolt, Class. Quant. Grav. source

In the paper

“Born out of studies in string theory, AdS/CFT is a non-perturbative candidate model of quantum gravity, where gravitational physics in a bulk d + 1-dimensional asymptotically AdS spacetime has a dual description in terms of a CFT living on the d-dimensional conformal boundary of AdS.”

What the sources say

• Supported

  Split into 2 atoms — 2 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [6] Maldacena, 1998, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. source

In the paper

“In any case, the Green function of the quantum scalar field in the BTZ background is in the Hartle-Hawking state, satisfying the Kubo-Martin-Schwinger (KMS) condition at the black hole temperature [42].”

What the sources say

• Supported

  Section III identifies the BTZ scalar-field Green function as a Hartle–Hawking Green function. Section III.B states that it satisfies the KMS condition outside the horizon at a local temperature related by the Tolman relation to the black hole temperature. (LLM, conf=0.94)

  cites [42] Lifschytz, 1994, Scalar field quantization on the (2+1)-dimensional black hole background, Phys. Rev. D source

In the paper

“The AdS/CFT correspondence [6], in its strongest form, describes a duality between a theory of gravity and conformal field theory at the level of their partition functions, summarized by Gubser, Klebanov, Polyakov and Witten (GKPW) [46,47]”

What the sources say

• Supported

  The citation to Maldacena identifies the AdS/CFT correspondence; the citing sentence separately attributes its partition-function formulation to GKPW [46,47]. In “General Idea” (p. 1), Maldacena conjectures that quantum M/string theory on AdS, with suitable boundary conditions, is dual to the corresponding conformal brane theory. (LLM, conf=0.91)

  cites [6] Maldacena, 1998, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. source
• Not checked yet

  Checked before — its earlier verdict is held back until a new check runs.

  cites [46] Gubser, 1998, Gauge theory correlators from noncritical string theory, Phys. Lett. B source
• Supported

  Section 2.3 explicitly identifies the CFT/AdS correspondence with equality between the CFT partition function and the supergravity or string-theory partition function, subject to specified boundary conditions. This supports the attributed duality at the level of partition functions. (LLM, conf=0.94)

  cites [47] Witten, 1998, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. source

In the paper

“Conjecture: Classical black holes which localize on a brane in AdS d+1 exactly map to d-dimensional quantum-corrected black holes including all orders of backreaction.”

What the sources say

• Partly supported

  The Introduction states the conjecture that classical black holes localized on an AdS brane correspond to quantum-corrected black holes in one fewer dimension. It also specifies that the correspondence captures quantum effects from all planar diagrams in the large-N limit; it does not claim all orders of backreaction without that limitation. (LLM, conf=0.86)

  cites [22] Emparan, 2002, Quantum black holes as holograms in AdS brane worlds, J. High Energy Phys. sourceExperimental — accuracy not yet measured

In the paper

“With this restriction, H(r) has a single positive root r + representing a black hole horizon, 22 and the C-metric describes a single 'slowly accelerating' black hole suspended away from the center of AdS 4 by a cosmic string attached at the horizon [91].”

What the sources say

• Partly supported

  Section 2.1 (page unavailable) gives an AdS₄ C-metric with H(r), and section 4.4 identifies r₊ as a black-hole horizon. But section 2.1 describes a geometry formed by gluing two copies at a tension-bearing brane, not a single slowly accelerating black hole suspended by a cosmic string. The copy does not establish that latter description. (LLM, conf=0.82)

  cites [91] Emparan, 2022, Holographic complexity of quantum black holes, J. High Energy Phys. sourceExperimental — accuracy not yet measured

In the paper

“The non-holographic stress-tensor may be expressed as an infinite sum over images, e.g.,”

What the sources say

• Supported

  Section 3.3, equations (3.29)–(3.33), expresses the free-CFT stress-tensor components in a rotating BTZ background as infinite sums. The following passage says the sums arise from the method of images and that their coefficients are complicated functions of M and J. The cited paper uses different equation numbers from the citing document’s (4.68). (LLM, conf=0.92)

  cites [25] Emparan, 2020, Quantum BTZ black hole, J. High Energy Phys. source
• Supported

  In “RSET for the rotating nonextremal BTZ black hole” (Eqs. 3.20–3.31; page unavailable), the paper writes multiple stress-tensor components as infinite sums with coefficients expressed through parameters of the black hole’s mass M and angular momentum J. The “Renormalized stress-energy tensor” section (Eq. 3.16; page unavailable) explicitly gives the image-sum form. Equation 4.68 is not an equation in this source and is not needed for the attributed point. (LLM, conf=0.90)

  cites [45] Casals, 2019, Quantum-corrected rotating black holes and naked singularities in (2+1) dimensions, Phys. Rev. D source

In the paper

“For x 1 = 1 this coincides with the familiar angular velocity of Kerr-AdS 4 (e.g., [95,96]).”

What the sources say

• Supported

  Section 2.1 (equations 2.6–2.8) gives the angular velocity of a four-dimensional Kerr–AdS black hole, distinguishing the value measured in a non-rotating frame at infinity from one measured in a rotating frame. The citation supports the reference to a familiar Kerr–AdS₄ angular velocity; x₁ = 1 is the citing authors’ specialization, not a parameter used in this source. (LLM, conf=0.85)

  cites [95] Gibbons, 2005, The First law of thermodynamics for Kerr-anti-de Sitter black holes, Class. Quant. Grav. source
• ?Could not verify

  The provided passages and the searched Additional information section discuss static charged shells in Reissner–Nordström spacetime, not Kerr-AdS₄ angular velocity. Searches for Kerr-AdS₄, Ω±, x₁, and angular velocity found no passage addressing the claimed coincidence. (LLM, conf=0.76)

  cites [96] Lemos, 2021, All fundamental electrically charged thin shells in general relativity: From star shells to tension shell black holes, regular black holes, and beyond, Phys. Rev. D sourceExperimental — accuracy not yet measured

In the paper

“Interestingly, the temporal component vanishes in the limit ℓ → 0 while the azimuthal component is independent of backreaction.This is consistent with three-dimensional dyonic defect in conical AdS 3 or a BTZ black hole [99].”

What the sources say

• Partly supported

  Section 3.1 (eq. 3.15; page unavailable) gives a temporal gauge potential proportional to ℓ, supporting its vanishing as ℓ → 0. It gives no azimuthal component or three-dimensional dyonic defect. The same section says the charged solution resembles RN-AdS rather than a charged BTZ black hole, so it does not support the claimed interpretation. (LLM, conf=0.87)

  cites [99] Feng, Y., Ma, H., Mann, R.B., Xue, Y., and Zhang, M. (2024). Quantum Charged Black Holes. arXiv. sourceExperimental — accuracy not yet measured

In the paper

“The superextremal black hole solutions are possible due to the combined non-linear effects of the rotation and quantum backreaction, and are consequently dubbed nonperturbative rotating black holes [135].”

What the sources say

• Supported

  The Quantum Penrose inequality section states that the rotating black holes past the classical extremality bound exist because of the combined, non-linear effects of rotation and backreaction, and explicitly calls them “nonperturbative rotating black holes.” (LLM, conf=0.95)

  cites [135] Frassino, A.M., Hennigar, R.A., Pedraza, J.F., and Svesko, A. (2024). Quantum inequalities for quantum black holes. arXiv. source

In the paper

“To see this, consider the boundary CFT stress-tensor, which can be computed in full using holographic renormalization methods, leading to [202]”

What the sources say

• Supported

  In “Why do we see simply a Schwarzschild-AdS 4 geometry?” (eqs. 7–11; page not provided), the authors say they evaluate the full boundary stress tensor using holographic renormalization and give the stated result in eq. 11. (LLM, conf=0.94)

  cites [202] Gregory, 2008, Classical and quantum gravity of brane black holes, J. High Energy Phys. source

In the paper

“For example, for even-dimensional homogeneous and isotropic FRW cosmologies, the trace anomaly for conformal quantum fields is known [178,179].”

What the sources say

• Partly supported

  Section 2.1 describes homogeneous, isotropic FRW universes in four dimensions, and Section 2.2 gives the trace anomaly for the conformal field theory considered there. These passages support the claim for four-dimensional cosmologies, but the paper does not establish it for even dimensions generally. Page numbers are unavailable in our copy. (LLM, conf=0.86)

  cites [178] Hawking, 2001, Trace anomaly driven inflation, Phys. Rev. D sourceExperimental — accuracy not yet measured
• Supported

  The Introduction (page unavailable) gives the trace anomaly for even-dimensional conformal field theories in curved spacetime. The paper also treats conformally flat backgrounds, a class that includes homogeneous and isotropic FRW cosmologies; it does not need to name FRW explicitly for the claim to follow. (LLM, conf=0.86)

  cites [179] Herzog, 2013, Stress Tensors from Trace Anomalies in Conformal Field Theories, Phys. Rev. D source

In the paper

“Quantum AdS black holes also undergo thermal phase transitions [124,169].In fact, large backreaction effects can trigger new transitions unseen by their classical counterparts.”

What the sources say

• Partly supported

  The canonical-ensemble comparison reports a thermal Hawking–Page-like transition between a BTZ black string and thermal AdS branes. The paper analyzes classical braneworld solutions; these passages do not establish that quantum backreaction triggers a new transition absent from classical counterparts. (LLM, conf=0.84)

  cites [124] Kudoh, 2004, Thermodynamics of four-dimensional black objects in the warped compactification, Phys. Rev. D sourceExperimental — accuracy not yet measured
• Supported

  The paper’s Abstract states that quantum-field backreaction can trigger new thermal phase transitions in a three-dimensional quantum-corrected AdS black hole. The ‘Phase transitions of the quantum BTZ black hole’ section says reentrant transitions occur only with large backreaction and identifies a zeroth-order transition with no classical counterpart. (LLM, conf=0.93)

  cites [169] Frassino, 2024, Reentrant phase transitions of quantum black holes, Phys. Rev. D source

In the paper

“The classical BTZ black hole also undergoes a HP phase transition [167,168].To wit, the canonical free energy F BTZ = -T log Z of a static BTZ black hole is F BTZ = M -T S = - π 2 ℓ 2 3 2G 3 T 2 , (5.99) for temperature T = r + /2πℓ 2 3 .Comparing to the free energy of thermal AdS, F AdS = M AdS = -1/8G 3 , a first-order phase transition occurs at a temperature T HP = 1 2πℓ 3 .”

What the sources say

• Supported

  Split into 2 atoms — 2 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [167] Maldacena, 1998, AdS(3) black holes and a stringy exclusion principle, J. High Energy Phys. source
• Partly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [168] Birmingham, 2003, Relaxation in conformal field theory, Hawking-Page transition, and quasinormal normal modes, Phys. Rev. D sourceExperimental — accuracy not yet measured

In the paper

“Fitzpatrick, Randall, and Wiseman also argue that the dynamical instability dual to semi-classical evaporation is unlikely to occur [69].”

What the sources say

• Partly supported

  The Introduction and summary calls dynamically unstable static solutions the less likely option, and §3.3 finds a Gregory–Laflamme instability very unlikely in the region examined. But the Introduction explicitly describes the proposed instability as unrelated to Hawking radiation, so the characterization of it as dual to semi-classical evaporation is not supported. (LLM, conf=0.88)

  cites [69] Fitzpatrick, 2006, On the existence and dynamics of braneworld black holes, J. High Energy Phys. sourceExperimental — accuracy not yet measured

In the paper

“Previously thought to be a problem only quantum gravity would solve, the paradox can be addressed in semi-classical gravity, for which the extremization prescription (7.4) plays a prominent role [254][255][256].”

What the sources say

• Supported

  Section 2.3 says the quantum extremal surface calculated in section 2.2 explains the Page curve and Hayden–Preskill decoding criterion for black-hole evaporation. The paper also says these results follow from the surface being an extremum of generalized entropy, supporting the claimed role of extremization in addressing the paradox within a semiclassical treatment. (LLM, conf=0.88)

  cites [254] Penington, 2020, Entanglement Wedge Reconstruction and the Information Paradox, J. High Energy Phys. source
• Partly supported

  The paper defines quantum extremal surfaces by extremizing generalized entropy (§4.1) and says a semiclassical calculation recovers the expected Page curve (JHEP12(2019)063). But it explicitly says this does not resolve the information paradox, because the calculation does not explain the required correlations between early and late Hawking radiation. The source therefore supports the role of extremization in addressing an aspect of the paradox, not an unqualified solution. (LLM, conf=0.86)

  cites [255] Almheiri, 2019, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, J. High Energy Phys. sourceExperimental — accuracy not yet measured
• Supported

  Section 1.3 explains that including a replica-wormhole saddle in the gravitational path integral yields a Page curve consistent with unitarity, addressing the information paradox in the models studied. Sections 1.3 and 2.2 describe the quantum extremal surface prescription and show that its extremality condition follows from extremizing the generalized entropy. (LLM, conf=0.90)

  cites [256] Almheiri, 2020, Replica Wormholes and the Entropy of Hawking Radiation, J. High Energy Phys. source

In the paper

“Indeed, while some bulk black holes get stuck on the brane and never shrink ('black droplets') [71] and hence dynamically stable, there are other bulk black solutions that evaporate due to a Gregory-Laflamme instability [72], as originally suggested in [22,34].”

What the sources say

• Partly supported

  Section III describes static RSII black holes whose horizons intersect the brane and reports evidence that they should have no dynamical instabilities, at least under axisymmetric perturbations. It does not call them “black droplets” or establish that they never shrink; the citing sentence also presents qualified stability evidence as an unqualified conclusion. The available passages have no page numbers. (LLM, conf=0.86)

  cites [71] Figueras, 2011, Gravity and large black holes in Randall-Sundrum II braneworlds, Phys. Rev. Lett. sourceExperimental — accuracy not yet measured

In the paper

“A notable exception, however, is the classical rotating BTZ black hole, where the backreaction is mild enough such that the inner horizon does not become singular, indicating a violation in strong cosmic censorship [298,299].”

What the sources say

• Partly supported

  The Abstract and Introduction report that smooth classical linear perturbations leave the inner horizon regular enough to violate strong cosmic censorship, but only for sufficiently near-extremal rotating BTZ black holes. Section 5.1 says vacuum-polarization backreaction does not render that horizon singular under the conditions studied. The citing sentence omits the near-extremal restriction and presents the absence of a singularity more generally than the paper establishes. (LLM, conf=0.88)

  cites [298] Dias, 2019, The BTZ black hole violates strong cosmic censorship, J. High Energy Phys. sourceExperimental — accuracy not yet measured
• Partly supported

  Section 6 reports that, as the BTZ black hole approaches extremality, the classical stress tensor becomes arbitrarily regular across the Cauchy horizon, implying a violation of strong cosmic censorship. It does not establish that backreaction leaves the inner horizon nonsingular for rotating BTZ black holes generally; the finding is qualified by proximity to extremality. (LLM, conf=0.84)

  cites [299] Hollands, 2020, Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime, Class. Quant. Grav. sourceExperimental — accuracy not yet measured

In the paper

“Such perturbations display characteristic patterns of damped oscillations, dubbed quasi-normal modes (QNMs), that capture deviations away from equilibrium [314,315].”

What the sources say

• Supported

  The Introduction calls QNMs the characteristic oscillations of black holes and says their complex frequencies encode the decay timescale of perturbations. Section 2 identifies QNMs as modes of linear perturbations of black-hole backgrounds, while the discussion of boundary conditions says they decay exponentially with time. Together, these passages support the description of damped oscillations as deviations from a stationary background. (LLM, conf=0.92)

  cites [314] Berti, 2009, Quasinormal modes of black holes and black branes, Class. Quant. Grav. source
• Supported

  The introductory passage describes a perturbed black hole returning toward its unperturbed background through a period of damped oscillations dominated by quasinormal modes. Section IV identifies the quasinormal spectrum of a stable black hole as frequencies describing damped oscillations, and discusses both Schwarzschild and Kerr black holes. Page numbers are not supplied in our copy. (LLM, conf=0.93)

  cites [315] Konoplya, 2011, Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys. source

In the paper

“A detailed study of QNMs of the quantum BTZ black hole provides some preliminary insights [317].”

What the sources say

• Supported

  The Abstract says the paper computes the qBTZ quasinormal-mode spectrum and analyzes how its leading modes and overtones behave. Section 6 (Discussion) describes the work as an initial exploration, consistent with the citing sentence’s characterization of its insights as preliminary. (LLM, conf=0.94)

  cites [317] Cartwright, C., Gürsoy, U., Pedraza, J.F., and Planella Planas, G. (2024). Perturbing a quantum black hole. arXiv. source

In the paper

“Traditionally this accomplished either by the method of background subtraction, or, in the case of asymptotically AdS spacetimes including a local counterterm action [18,345].”

What the sources say

• Supported

  The introductory discussion contrasts background subtraction with a counterterm-subtraction approach for AdS spacetimes. Section III, “Counterterm Action,” explains that boundary counterterms cancel divergences in the gravitational action without a reference background. The paper notes limits to this approach for some boundary geometries, but supports the cited sentence’s description of the two methods. (LLM, conf=0.91)

  cites [18] Emparan, 1999, Surface terms as counterterms in the AdS / CFT correspondence, Phys. Rev. D source
• Supported

  The Introduction describes subtraction using a reference spacetime and removal of divergences by adding local counterterms. Section 2, “Defining the Stress Tensor,” explicitly contrasts the reference-spacetime approach with the paper’s local boundary-counterterm procedure for asymptotically AdS spacetimes. The supplied passages do not give page numbers. (LLM, conf=0.94)

  cites [345] Balasubramanian, 1999, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. source

In the paper

“Meanwhile, a dS 3 slicing, where -1 < λ < 0, obeys A > L -1 4 , such that the bulk solution is interpreted as two black holes separated by an acceleration horizon [344], and the system is not in thermal equilibrium.”

What the sources say

• Partly supported

  The Causal Structure section distinguishes the A > 1/ℓ regime, and the Conclusions interpret its massive solutions as two black holes approaching an acceleration horizon. The cited passages do not establish the dS3 slicing or the range −1 < λ < 0, and a search for thermal equilibrium and horizon temperatures found no statement that this system is out of thermal equilibrium. Page numbers are unavailable in our copy. (LLM, conf=0.86)

  cites [344] Dias, 2003, Pair of accelerated black holes in anti-de Sitter background: AdS C metric, Phys. Rev. D sourceExperimental — accuracy not yet measured

In the paper

“One way to circumvent this problem is to work in a regime in which the black hole is slowly accelerating, where A < L -1 4 [91].This translates to A < A √ 1 + λ, i.e., λ > 0, or, an AdS 3 slicing on the brane.”

What the sources say

• ?Could not verify

  Section 2.1, “Metric and parameters” (page unavailable), describes an AdS₃ brane and a four-dimensional bulk, but does not give the claimed acceleration bound. Searches for “slowly accelerating,” an acceleration horizon, and the λ condition found no passage connecting A < L₄⁻¹ to λ > 0 or presenting it as a way around a non-equilibrium problem. (LLM, conf=0.70)

  cites [91] Emparan, 2022, Holographic complexity of quantum black holes, J. High Energy Phys. sourceExperimental — accuracy not yet measured

In the paper

“Notably, the parameters ϵ and γ do not affect the geometry of the solution, as they do not appear in any curvature invariants [90], and sometimes dubbed 'kinematical parameters'.”

What the sources say

• ?Could not verify

  The section on the physical interpretation of the C-metric gives its metric in terms of A, m, e and the cosmological constant, but does not discuss parameters ε and γ. Searches for those parameters, curvature invariants and “kinematical parameters” found no passage addressing the claim in our copy. (LLM, conf=0.76)

  cites [90] Podolsky, 2002, Accelerating black holes in anti-de Sitter universe, Czech. J. Phys. sourceExperimental — accuracy not yet measured

In the paper

“Rather, one realization is that of a D5-probe brane intersecting a stack of D3 branes [16,327], leading to a doubly holographic interpretation (Section 3.1).”

What the sources say

• Supported

  Section 3.1 describes a probe D5 brane intersecting D3 branes and occupying an AdS4 subspace of the D3 near-horizon geometry. Section 2 describes the resulting gravity setup as holographically dual to a CFT with a boundary or defect, then invokes AdS holography again to reduce that description further. This supports the citing sentence’s doubly holographic interpretation. (LLM, conf=0.91)

  cites [16] Karch, 2001, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, J. High Energy Phys. source
• Supported

  The Introduction describes Karch and Randall’s D3/D5 system, with a D5-brane treated as a probe in the near-horizon geometry of a stack of D3-branes. It also says they speculated that AdS/CFT would “act twice” in this system, supporting the claimed doubly holographic interpretation. (LLM, conf=0.91)

  cites [327] DeWolfe, 2002, Holography and defect conformal field theories, Phys. Rev. D source

In the paper

“Similarly, a (4) (which we did not explicitly compute; see[19]) has the coefficient in front replaced with 1/2 when d = 4.”

What the sources say

• Supported

  Appendix B, equations (B.1)–(B.2), writes a^(4) with a coefficient b^(4) and explicitly gives b^(4)(d = 4) = 1/2. This directly supports the stated replacement when d = 4. (LLM, conf=0.95)

  cites [19] Solodukhin, 2001, Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence, Commun. Math. Phys. source

In the paper

“Alternatively, the intersection term naturally arises using deformed braneworlds[84].”

What the sources say

• ?Could not verify

  The discussion of the two-dimensional dilaton-gravity action (p. 2) and its higher-dimensional black-hole interpretation (p. 4) does not address deformed braneworlds or an intersection term. Searches for those terms in our copy found no passage supporting the attributed claim. (LLM, conf=0.76)

  cites [84] Callan, 1992, Evanescent black holes, Phys. Rev. D sourceExperimental — accuracy not yet measured

In the paper

“The homology constraint for the RT surfaces depends on which perspective in double holography is being employed[262].”

What the sources say

• Supported

  Section 2 explicitly says that the homology constraint for bulk RT surfaces changes depending on whether entropy is computed from the boundary or brane perspective. Section 3 further explains that the two perspectives apply different homology constraints. (LLM, conf=0.94)

  cites [262] Neuenfeld, 2022, Homology conditions for RT surfaces in double holography, Class. Quant. Grav. source

In the paper

“There are, however, accelerating black holes in AdS3, cf.[187][188][189][190][191].”

What the sources say

• Supported

  Section 2 describes the 2+1-dimensional metric as an accelerating black hole and explicitly considers the Λ < 0 sector, corresponding to AdS3. The paper therefore supports the existence claim. (LLM, conf=0.94)

  cites [187] Astorino, 2011, Accelerating black hole in 2+1 dimensions and 3+1 black (st)ring, J. High Energy Phys. source
• Supported

  Section 4 explicitly describes an AdS black hole spacetime with an acceleration parameter. Section 6 identifies the solution as an accelerating BTZ spacetime and discusses its black hole horizons, supporting the claim that accelerating black holes exist in AdS3. (LLM, conf=0.94)

  cites [188] Xu, 2012, Accelerating BTZ spacetime, Class. Quant. Grav. source
• Partly supported

  Section 4 discusses AdS₃ BTZ black-hole solutions with a finite acceleration parameter, supporting the existence of the black-hole solutions the claim refers to. But the Conclusions caution that, unlike the accelerating particles studied in the paper, there is no established meaningful sense in which these three-dimensional black holes themselves accelerate. The unqualified word “accelerating” overstates the paper’s conclusion. (LLM, conf=0.86)

  cites [189] Gregory, 2022, On acceleration in three dimensions, J. High Energy Phys. sourceExperimental — accuracy not yet measured

In the paper

“Note that the A → ∞ limit of the C-metric with a de Sitter slicing on the x = 0 hypersurface is a double-Wick rotation of the hyperbolic AdS 4 metric (see Appendix C of [26] for details).”

What the sources say

• Supported

  Appendix C states that the ℓ → 0 limit of the AdS₄ C-metric is related by a double Wick rotation to the hyperbolic AdS₄ black hole. It also identifies the x = 0 boundary geometry as conformally equivalent to conical dS₃. Section 4.1 defines A = 1/ℓ, so this is the A → ∞ limit described in the claim. (LLM, conf=0.94)

  cites [26] Emparan, 2022, Black holes in dS3, J. High Energy Phys. source

In the paper

“Stringy realizations of the KR construction were uncovered in[16,[326][327][328] (and more recently,[59]).”

What the sources say

• Supported

  The Conclusions state that the paper gives a stringy realization of the boundary-CFT/AdS-brane duality conjecture in reference [2]. The References identify [2] as Karch and Randall’s “Locally localized gravity,” linking the realization to the KR construction. (LLM, conf=0.88)

  cites [16] Karch, 2001, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, J. High Energy Phys. source
• Supported

  The Abstract identifies the paper’s contribution as a string realization of the gravity-localizing AdS4 brane in AdS5. The Positive Tension section says intersecting branes are sufficient to realize four-dimensional gravity, while noting that the full supergravity solution remains to be found. This supports citing the paper as a stringy realization of the KR construction. (LLM, conf=0.91)

  cites [326] Karch, 2001, Localized gravity in string theory, Phys. Rev. Lett. source
• Supported

  The Introduction (page not supplied) describes the D3/D5-brane system as a string-theory realization of an AdS4 brane inside AdS5. Section 3.2 then studies the D5-brane action for the Karch–Randall system, supporting this paper’s citation as a stringy realization of the construction. (LLM, conf=0.88)

  cites [327] DeWolfe, 2002, Holography and defect conformal field theories, Phys. Rev. D source

In the paper

“The argument of[53,260] requires DGP couplings be turned on, couplings which can affect the growth rate of entanglement entropy of subregions of the dual CFT[263].”

What the sources say

• ?Not checked — no full text of the sourcecites [53] Israel, 1966, Singular hypersurfaces and thin shells in general relativity, Nuovo Cim. B source
• ?Could not verify

  Section 2 describes a dilaton-gravity action on a dynamical Planck brane, and §3.2 says the black hole–bath coupling is turned on. Neither passage identifies a DGP coupling or says the paper’s argument requires one. I searched for “DGP couplings,” “Dvali Gabadadze Porrati,” and brane-localized gravity terms without finding a passage that establishes the attributed requirement. The sentence separately cites [263] for the effect on entanglement-entropy growth. (LLM, conf=0.72)

  cites [260] Almheiri, 2020, The Page curve of Hawking radiation from semiclassical geometry, J. High Energy Phys. sourceExperimental — accuracy not yet measured
• Supported

  The Introduction states that adding gravitational couplings to the brane affects the growth rate of holographic entanglement entropy of subregions in the dual CFT. This supports the clause attached to this source’s citation, [263]; the separate assertion about what the arguments of [53,260] require is not attributed to this paper. (LLM, conf=0.94)

  cites [263] Lee, 2022, Bounds on gravitational brane couplings and tomography in AdS3 black hole microstates, J. High Energy Phys. source

In the paper

“Next, a static black hole stuck to the brane will neither eat the brane or slide off it.”

What the sources say

• ?Not checked — no full text of the sourcecites [89] Plebanski, 1976, Rotating, charged, and uniformly accelerating mass in general relativity, Ann. Phys. source

In the paper

“Thus, the matter contributions enter at order O(ℓ) while higher-curvature corrections appear at O(ℓ 2 ).”

In the paper

“Proceeding, the typical value of I CFT is |I CFT | ∼ c 3 , while the typical value of the gravitational part of the action (4.15)”

• Also stated in: § Karch-Randall braneworld holography · plain

  “is |I Bgrav | ∼ L 3 G 3 .”

In the paper

“As it happens, there are no conical singularities in the restricted range 0 ≤ x ≤ x 1 ; for µ > 0 the remaining conical singularities live in the range x < 0.”

In the paper

“Notice further, in the limit of small backreaction (ν < 1), the 1/r term in (4.26) is proportional to ℓ ∼ cG 3 ≫ L P (4.23), such that the horizon size of the quantum BTZ black hole is large compared to the Planck length.”

In the paper

“The tensionless limit corresponds to ℓ → ∞, where ℓ 3 → L 4 by virtue of (4.3).”

In the paper

“We emphasize, however, that the 1/r correction in g tt (2.12) is perturbative, while it is exact in (4.26).”

In the paper

“Applying the formalism of Section 3, the induced brane action is (setting d = 3 in (3.13))”

• Also stated in: § Karch-Randall braneworld holography · plain

  “with induced Newton's constant G 3 = G 4 2L 4 (4.16) and effective (AdS 3 ) brane cosmological constant and curvature scale,”

• Also stated in: § Karch-Randall braneworld holography · plain

  “Λ 3 = - 1 L 2 3 , 1 L 2 3 = 2 L 2 4 1 - L 4 ℓ ,(4.17)”

In the paper

“Here the singularity at r = 0 sits behind the horizon at r = r+ , the largest positive root of the metric function H(r), where the time-translation Killing vector ζi = η∂ i t goes null.”

In the paper

“Expanding for small (ℓ/ℓ 3 ) gives”

• Also stated in: § Karch-Randall braneworld holography · plain

  “2c 3 G 3 = L 4 ≈ ℓ 1 - ℓ 2 2ℓ 2 3 + 3 8 ℓ 4 ℓ 4 3 + ... ,(4.21)”

• Also stated in: § Karch-Randall braneworld holography · plain

  “and c 3 ∼ ℓ G 3 , which observe enters at linear order in ℓ.”

In the paper

“The desired parameter range follows from taking x 1 > 0 and x 1 ∈ (0, 1] for κ = +1 ,”

• Also stated in: § Horizons and bulk regularity · plain

  “x 1 ∈ (0, ∞) for κ = -1, 0 .”

In the paper

“The solution is valid for all 0 ≤ ℓ < ∞, however, we will be primarily interested in the case where the brane is near the AdS 4 boundary (before it has been removed via the ETW brane).”

In the paper

“From (4.21) it follows”

• Also stated in: § Karch-Randall braneworld holography · plain

  “ℓ ∼ c 3 ℏG 3 = c 3 L P ≫ L P ,(4.23)”

• Also stated in: § Karch-Randall braneworld holography · plain

  “since the holographic cutoff CFT 3 obeys c 3 ≫ 1.”

In the paper

“We see for the range of x 1 , the function G ′ (x 1 ) = -”

• Also stated in: § Horizons and bulk regularity · plain

  “and that ∆ϕ is independent of ℓ and ℓ 3 .Moreover, ∆ϕ grows monotonically from 0 to 2π.”

In the paper

“Succinctly, the qBTZ solution is characterized by the two branches (4.38), however, a finer subdivision of the two branches is [25] Branch 1a:”

• Also stated in: § 4.2.1 Static quantum BTZ · plain

  “κ = +1 , 0 < x 1 < 1 , Branch 1b: κ = -1 , 0 < x 1 < √ 3 , Branch 2: κ = -1 , √ 3 < x 1 < ∞ .”

What the sources say

• Not checked yetcites [25] Emparan, 2020, Quantum BTZ black hole, J. High Energy Phys. source

In the paper

“Since the backreaction is turned off as ℓ → 0, the metric (4.26) in this limit may be interpreted as 'classical'; indeed the 1/r correction is eliminated and the resulting geometry solves the three-dimensional vacuum Einstein equations.”

In the paper

“Thus, for small ℓ < ℓ 3 , we see g eff ≈ ν ≪ 1, and backreaction effects are 'small', and the induced theory is effectively characterized by the action (4.19).”

In the paper

“In the tensionless limit, ℓ → ∞, then σ b = 0, cutting the bulk in half through the equator.”

In the paper

“Unlike the classical BTZ black hole, the quantum BTZ (4.26) has a curvature singularity at r = 0, as evidenced by the Kretschmann invariant,”

• Also stated in: § 4.2.1 Static quantum BTZ · plain

  “R ijkl R ijkl = 12 ℓ 4 3 + 6F (M ) 2 ℓ 2 r6 . (4.35”

In the paper

“Decomposing the holographic stress-tensor in an expansion in ℓ 2 , i.e.,”

• Also stated in: § 4.2.1 Static quantum BTZ · plain

  “Substituting in the metric (4.26) we obtain for the renormalized stress-energy tensor,”

In the paper

“All negative mass solutions occur for κ = +1 and for x 1 ∈ (0, 1], while all positive mass solutions occur for κ = -1 and x 1 ∈ (0, ∞).”

In the paper

“The x = 0 brane amounts to surfaces of fixed”

• Also stated in: § Karch-Randall braneworld construction · plain

  “σ = σ b obeying cosh(σ b ) = ℓ 3 L 4 = 1 + ℓ 2 3 ℓ 2 ,(4.13)”

• Also stated in: § Karch-Randall braneworld construction · plain

  “and the brane geometry is AdS 3 with curvature radius ℓ 3 , i.e., a Karch-Randall brane.”

In the paper

“The lower bound, meanwhile, occurs for κ = +1 and x 1 = 1 (where µ = 0 according to (4.7)), and M = 0 for either x 1 = 0 or κ = 0.”

In the paper

“Notice ⟨T ij ⟩ 2 has a non-zero trace, a consequence of breaking the conformal symmetry due to the cutoff ℓ.”

In the paper

“In fact, the solution may be categorized into physically distinct branches:”

• Also stated in: § 4.2.1 Static quantum BTZ · plain

  “Branch 1: -1 < -κx 2 1 < 3 , Branch 2: 3 < -κx 2 1 < ∞ .”

In the paper

“The most advantageous geometric feature of the C-metric (4.1), for any µ and κ, is that the x = 0 timelike hypersurface is umbilic.That is, the extrinsic curvature K ij of the hypersurface is proportional to the induced metric h ij on x = 0.”

In the paper

“with ℓ ∼ L 4 ≪ ℓ 3 .In this limit, the induced theory is”

• Also stated in: § Karch-Randall braneworld holography · plain

  “such that the higher-curvature expansion can be viewed as an expansion in an effective cutoff scale ℓ.”

In the paper

“The line element for the brane metric is now”

• Also stated in: § 4.2.1 Static quantum BTZ · hedged

  “Here we have suggestively identified the three-dimensional mass”

• Also stated in: § 4.2.1 Static quantum BTZ · plain

  “M ≡ - κ 8G 3 ℓ L 4 η 2 = - 1 2G 3 κx 2 1 (3 -κx 2 1 ) 2 , G 3 ≡ G 3 L 4 ℓ = G 3 √ 1 -ν 2 , (4.27)”

In the paper

“Note further M = 0 for either x 1 = 0 (for both κ = ±1) or x 1 → ∞ (and κ = +1).”

In the paper

“Further, for the range of µ and κ we are interested in, G(x) has three distinct zeros, each of which leads to a distinct conical singularity.”

In the paper

“When µ = 0 the geometry is more complicated, but the C-metrics have the nice feature that the surface x = 0 is always totally umbilic.”

In the paper

“Peeling off ℓ/16πG 3 r3 , identify the form function”

In the paper

“and form functions”

In the paper

“Further, near the ring singularity, there exists the possibility of closed timelike curves.Indeed, consider the norm of the axial Killing vector ∂ φ of (4.53) is”

In the paper

“With the coordinate change (4.48), the brane metric does not quite have the canonical asymptotic form of a rotating AdS black hole.To do so, one introduces a normalized radial coordinate r such that [25].”

What the sources say

• Not checked yetcites [25] Emparan, 2020, Quantum BTZ black hole, J. High Energy Phys. source

In the paper

“Despite complicating the geometry by including rotation, the x = 0 hypersurface remains umbilic, such that K ij = ℓ -1 h ij (4.11) and a Karch-Randall brane at x = 0 has tension (4.12).”

In the paper

“A gauge has been chosen for the gauge potential (4.74) such that it remains regular at the largest root of H(r) = 0, denoted r + , and x = x 1 .”

In the paper

“The Maxwell field strength also has junction conditions to obey.In particular, let n a denote the normal to the brane at x = 0 (pointing towards increasing values of x) and let e a i be a basis for tangent vectors to the brane.Then, projecting F ab onto the brane such that F ij ≡ F ab e a i e b j and f i ≡ F ab e a i n b , one has the following junction conditions [97] for a purely tensional brane”

What the sources say

• Not checked yetcites [97] Taylor, M. (2000). More on counterterms in the gravitational action and anomalies. arXiv.

In the paper

“The U (1) gauge field A a for the charged C-metric (4.71)”

In the paper

“As for the neutral black hole, real roots of G(x) correspond to symmetry axes of the Killing vector ∂ a ϕ , and we work in the restricted regime 0 ≤ x ≤ x 1 .”

In the paper

“As an exact solution to the full semi-classical theory (4.19), the metric (4.53) is known as the rotating quantum BTZ black hole.”

In the paper

“Despite µ = 0, a BTZ black hole lives on the brane at x = 0 with mass M ≥ 0 in the canonically normalized coordinates ( t, r, φ), however, the geometry is uncorrected from its classical counterpart since the quantum stress tensor vanishes (on account of F (M ) = 0 for all M ≥ 0).”

In the paper

“Here the mass is identified to be”

In the paper

“For simplicity, setting rotation to zero, the line element is of the same form as (4.1), except the metric functions (4.2) receive an additional term:”

In the paper

“Meanwhile, the angular velocity Ω ′ relative to a rotating frame at spatial infinity is”

In the paper

“Then, treating ℓ < ℓ 3 such that L 4 ∼ ℓ, the effective theory on the brane (4.19) is now”

In the paper

“Firstly, notice that the Killing vector ∂ ϕ of the bulk C-metric (4.40) no longer has vanishing norm at a zero x i of G(x).Rather, it is the Killing vector”

In the paper

“Altogether, having imposed bulk regularity conditions, the brane geometry (4.42) in the canonically normalized coordinates ( t, r, φ) is”

In the paper

“Similarly, the magnetic charge Q g is”

In the paper

“By evaluating the Kretschmann scalar invariant R abcd R abcd , there is a curvature singularity when r 2 Σ = r 2 + a 2 x 2 = 0, i.e., where r = x = 0.This is the familiar ring singularity of Kerr black holes.”

In the paper

“To ensure ∂ 2 ϕ ≥ 0, requires an additional restriction of the solution parameters: (i) 1 -ã2 ≥ 0, and (ii) η > 0, which implies -κx 2 1 > ã2 -3.Notice the second condition is always implied by (i) if κ = -1.”

In the paper

“This occurs when”

In the paper

“The O(ℓ 2 ) components ⟨T ij ⟩ 2 (4.32) are cumbersome to express in canonically normalized coordinates, however, in naive coordinates ⟨T ij ⟩ 2 is not traceless, indicating the presence of the a UV cutoff breaking the conformal symmetry of the CFT 3 .”

In the paper

“Lastly, as usual, the x = 0 hypersurface is totally umbilic, such that the Israel junction conditions fix the brane tension to be τ = (2πG 4 ℓ) -1 .”

In the paper

“Substituting the metric (4.85) into the stress-tensor at leading order in ℓ is equivalent to the neutral static qBTZ stress-tensor (4.33), while the effects of charge arise at O(ℓ 2 ).”