CoChat
Sign Up
Cited by 61Black Holes and Theoretical PhysicsCosmology and Gravitation TheoriesAstrophysical Phenomena and Observations

The DOI matches its Crossref record; 51 of the 51 references it cites were found in a registry or an index; none of the 26 sources its claims cite is checked for support yet.

Black holes on thick branes

Open access
Emparan, Roberto, Gregory, Ruth, Santos, Caroline·2001·Physical Review D
DOI 10.1103/physrevd.63.104022
Open this paper in CoChat — free accountOpen PDF

Abstract

The interplay between topological defects (branes) and black holes has been a subject of recent study, motivated in part by interest in brane-world scenarios. In this paper we analyze in detail the description of a black hole bound to a domain wall (a two-brane in four dimensions), for which an exact description in the limit of zero wall thickness has been given recently. We show how to smooth this singular solution with a thick domain wall. We also show that charged extremal black holes of a size (roughly) smaller than the brane thickness expel the wall, thereby extending the phenomenon of flux expulsion. Finally, we analyze the process of black hole nucleation on a domain wall, and argue that it is preferred over a previously studied mechanism of black hole nucleation away from the wall.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

The paper constructs and analyzes a black hole bound to a thick domain wall, showing how the singular thin-wall solution can be smoothed by a topological wall and how thermodynamics favors the black hole remaining on the wall. It also argues that sufficiently small extremal black holes expel the wall and that black-hole nucleation on the wall is preferred over a previously studied nucleation away from the wall.

Key findings

  • The thin wall–black-hole metric is built from the C-metric and Israel junction conditions; the resulting spacetime describes two oppositely charged black holes at antipodal points of a spherical domain wall, with wall tension \(\sigma = A/(2\pi G)\).
  • Thermodynamics gives \(S = 4\pi G M^2 [1 + 5\pi G^2 \sigma M + O(G^4 M^2 \sigma^2)]\), and \(S > S_0 = 4\pi G M^2\) for all allowed masses, so it is entropically favored for the black hole to stick to the wall.
  • For extremal black holes, flux expulsion occurs: if \(M^2 < 1/2\), the only possible horizon solution is \(X \equiv 0\), i.e. the wall is expelled; if \(M > 4/\pi\), no such expelling solution can exist. For the C-metric, a weaker bound \(A|\xi_2| > 1\) is derived.
  • Black-hole nucleation on the wall (“hole-punch”) has \(I - I_0 \simeq M\sigma\), which is positive and exponentially suppressed, but the black-hole mass \(M\) can be varied independently of wall tension. This is argued to be preferred over the prior mechanism, where \(M = (6\sqrt{3}\pi\sigma)^{-1}\) is fixed and large.

Methods

  • Analytical general-relativistic and field-theoretic study: uses the C-metric, Israel junction conditions, thin-wall approximation, Reissner-Nordström backgrounds, and a linearized gravitational back-reaction calculation in \(\epsilon = 3A/2\).
  • Constructs a Euclidean instanton for black-hole nucleation on the wall and computes the action difference \(I - I_0\), comparing it to previously studied nucleation inside the wall.
  • No observational or numerical sample is used; the work is analytic, with reference to prior numerical work on a domain wall through a Schwarzschild black hole.

Limitations

  • The paper states that the bulk cosmological constant is set to \(\Lambda = 0\) for simplicity, though it says sections III–V can be extended to non-zero \(\Lambda\), and global structure, thermodynamics, and instantons extend qualitatively to de Sitter branes.
  • The paper states that the five-dimensional analogue is obstructed by the lack of an explicit C-metric analogue, so the five-dimensional wall–black-hole metric remains unknown; nucleation in five dimensions cannot be conclusively analyzed.
  • The text makes evident that the expulsion bound \(M^2 < 1/2\) is weak and the true limit likely higher, requiring numerical work; it also notes that the semiclassical approximation breaks down when the black-hole mass reaches the Planck scale.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevd.63.104022

Cites

51 of 51 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

0 of 26 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The wall-black hole metric describes two black holes at antipodal points of a spherical domain wall.

    “The domain wall that we have constructed actually contains two black holes, sitting at antipodal points of a spherical domain wall.”
    • Also stated in: § II. THE BLACK HOLE-WALL METRIC A. Constructing the solution · plain
      “Therefore, the WBH metric actually describes two black holes at antipodal points of a spherical domain wall.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    In the λΦ4 kink model, the energy per unit area is 8πGσ = 8ε/3.

    “which has an energy per unit area 8πGσ = 2ǫ ∞ -∞ sech 4 z = 8ǫ 3 .”
    Quote in the paperNot a substantial claim
  • Findingown data

    Charged extremal black holes of a size roughly smaller than the brane thickness expel the wall, extending the phenomenon of flux expulsion.

    “We also show that charged extremal black holes of a size (roughly) smaller than the brane thickness expel the wall, thereby extending the phenomenon of flux expulsion.”
    • Also stated in: § I. INTRODUCTION · boosted
      “We also explore the question of 'flux' expulsion for the wall, proving analytically that such expulsion occurs, and deriving bounds for the mass of the black hole (in wall units) at which it must occur.”
    Quote in the paperSubstantial claim
  • Findingnone given

    For mA = 1/(3√3), the black hole horizon and the acceleration horizon become coincident.

    “for mA = 1/3 √ 3 the black hole horizon and the acceleration horizon (which yields the cosmological deSitter horizon on the brane) become coincident”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The temperature is T = A|F'(-ξ2)|/(4π) = A|G'(ξ2)|/(4π).

    “The result is T = A|F ′ (-ξ 2 )| 4π = A|G ′ (ξ 2 )| 4π .”
    Quote in the paperSubstantial claim
  • It may state two separate claims as one. 30
  • Probably not a substantial claim. 7
  • The quote only partly states this claim. 2

  • It may state two separate claims as one.

    Findingreasoning

    The Euclidean instanton requires β = 4π/G'(ξ3) = 4π/|G'(ξ2)|; the second equality requires ξ1 - ξ2 - ξ3 + ξ4 = 0.

    “β = 4π G ′ (ξ 3 ) = 4π |G ′ (ξ 2 )| .(56)”
    • Also stated in: § VI. NUCLEATION OF BLACK HOLES ON WALLS · plain
      “Specifically, this equation requires ξ 1 -ξ 2 -ξ 3 + ξ 4 = 0.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    Hence I - I0 ≃ M/σ, which is positive; nucleation of black holes is exponentially suppressed.

    “Hence, I -I 0 ≃ M σ ,(63)”
    • Also stated in: § VI. NUCLEATION OF BLACK HOLES ON WALLS · plain
      “which, we observe, is a positive quantity.Nucleation of black holes is exponentially suppressed, as was to be expected.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingreasoning

    For 1 ≤ z ≤ 2, the plot in figure 3 would extend to yield the entropy (area) of the acceleration horizon, for a given black hole mass.

    “For 1 ≤ z ≤ 2, the plot in figure 3 would extend to yield the entropy (area) of the acceleration horizon, for a given black hole mass.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingnone given

    The wall-black hole space is compact, and the black holes are accelerated along with the wall.

    “As expected, the space is compact, and the black holes are accelerated along with the wall.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingnone given

    In the range 2 < z < ∞, m and S decrease monotonically with z, while T grows monotonically with it.

    “In the range 2 < z < ∞, m and S decrease monotonically with z, while T grows monotonically with it.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    To this order, only A_acc contributes; the black hole entropy ∼4πM^2 enhances the nucleation rate but enters only at the next order.

    “It may be observed that, to this order, only A acc contributes.The black hole entropy ∼ 4πM 2 contributes to enhance the nucleation rate, but enters only at the next order.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingnone given

    The domain wall eliminates the string from the spacetime and cuts off the acceleration horizon, turning it into a horizon of finite area.

    “Note that the domain wall not only eliminates the string from the spacetime, but it also cuts off the acceleration horizon and turns it into a horizon of finite area.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    For a small black hole (m, q ≪ 1/A), the geometry approaches the Reissner-Nordstrom solution, allowing the black hole mass to be identified approximately as m/G.

    “for a small black hole, i.e., m, q ≪ 1/A, the geometry approaches the Reissner-Nordstrom solution, which allows us to identify approximately the black hole mass as m/G.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    For small black holes, M = m/G (1 - 15/4 mA + O(m^2 A^2)) and S = 4πm^2/G (1 - 5mA + O(m^2 A^2)).

    “M = m G 1 - 15 4 mA + O(m 2 A 2 ) , S = 4πm 2 G 1 -5mA + O(m 2 A 2 ) .”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationcitation

    The obstruction to arbitrarily large black holes on the brane disappears for the asymptotically flat brane of [16], and in that case the horizon on the brane can grow arbitrarily large.

    “This obstruction obviously disappears for the asymptotically flat brane of [16], and in that case the horizon on the brane can grow arbitrarily large.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [16]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Study design and conductnone given

    The hole-punch instanton is completely non-singular (save for the smoothable singularity at the wall) and compact.

    “The hole-punch instanton thus constructed (see Fig. 4) is completely non-singular (save for the smoothable singularity at the location of the wall), and it is also compact”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    In the configuration where the black hole is enclosed by the wall, the mass is fixed to M = (6√3πσ)^-1 and cannot be varied independently of the wall tension; the geometry is Schwarzschild with the wall at r = 3M.

    “However, there is a crucial difference: in the configuration where the black hole is enclosed by the wall, the mass of the black hole is fixed to be M = (6 √ 3πσ) -1 and hence cannot be varied independently of the tension of the wall.In fact, the geometry is that of the Schwarzschild solution, with the wall sitting at a fixed radius r = 3M.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    Flux expulsion is not dependent on the vortex being local; it occurs for global strings and for pure flux p-branes.

    “Such a flux expulsion is not dependent on the vortex being local, it occurs for global strings [23] and for pure flux p-branes [22]”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [23]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [22]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    The transition to this configuration involves a quantum fluctuation where the initial domain wall is annihilated and recreated with the black holes on it; the probability is given in first approximation by exp[-(I - I0)].

    “Again, the transition to this configuration involves a quantum fluctuation where the initial domain wall is annihilated, to be recreated back with the black holes on it.The probability for this to happen is given, in first approximation, by exp[-(I -I 0 )]”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationnone given

    The final state of the hole-punch mechanism is an accelerating spherical domain wall with a pair of black holes grafted at antipodal points; the black holes do not swallow the brane, so the holes do not grow.

    “The final state will be precisely the one we have been describing in the previous sections: an accelerating spherical domain wall with a pair of black holes grafted at antipodal points on the wall.The black holes do not swallow the brane, hence the holes do not grow.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Interpretationnone given

    The results in sections III, IV and V can be easily extended to non-zero Λ.

    “the results in sections III, IV and V can be easily extended to non-zero Λ.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationcitation

    Black holes on the wall will neither swallow up the brane nor slide off of it.

    “as argued in [26], the black holes will neither swallow up the brane, nor slide off of it.”
    • Also stated in: § II. THE BLACK HOLE-WALL METRIC A. Constructing the solution · plain
      “Below we will give an alternative, thermodynamic argument for this feature.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [26]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one. The quote only partly states this claim.

    Interpretationreasoning

    Domain walls can only nucleate inside them black holes of a certain (large) size; the process of [27] can only lead to the formation of very large black holes and therefore will be heavily suppressed.

    “As a result, the process of [27] can only lead to the formation of very large black holes, and therefore will be heavily suppressed.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    Domain walls could nucleate black holes at a finite distance from them; a spherical domain wall accelerating in empty Minkowski space may tunnel to a configuration where it encloses a black hole (two black holes, one on each side).

    “It was found that domain walls could nucleate black holes at a finite distance from them.That is, a spherical domain wall that is accelerating in an otherwise empty Minkowski space (such as we have described in eqs.(11)(12)(13)) may tunnel to a configuration where it encloses a black hole.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it.

    Premisecitation

    Large black holes in the four-dimensional RS brane-world have the shape of a pancake, with the horizon flattened and having a small extent away from the brane.

    “There, it was argued that large black holes in the four dimensional RS brane-world have the shape of a pancake, the horizon being flattened and having a small extent away from the brane.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [16]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    A neutral black horizon is smaller, hence hotter, than an acceleration horizon, so the thermal equilibrium equation has no solutions if q = 0.

    “A neutral black horizon is smaller, hence hotter, than an acceleration horizon, and as a consequence the above equation has no solutions if q = 0.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Study design and conductcitation

    The action can be computed by plugging the explicit solution into I = -1/(16π)∫d^4x√g(R - F^2) + σ∫d^3x√h; on-shell it can be simplified to I = -1/4(A_acc + A_bh). Both methods yield the same result.

    “The latter can be computed directly by plugging the explicit form of the solution into I = - 1 16π d 4 x √ g(R -F 2 ) + σ d 3 x √ h .(57)”
    • Also stated in: § VI. NUCLEATION OF BLACK HOLES ON WALLS · plain
      “Alternatively, one can use the on-shell equation [28] I = - 1 4 (A acc + A bh ) ,(58)”
    • Also stated in: § VI. NUCLEATION OF BLACK HOLES ON WALLS · plain
      “Both methods yield, of course, the same result, but the latter is perhaps slightly simpler.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [28]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Interpretationnone given

    Issues such as global structure, thermodynamics and instantons as studied here extend qualitatively to all cases where the geometry induced on the brane is deSitter, even if Λ is positive or negative.

    “issues such as global structure, thermodynamics and instantons as studied here, extend qualitatively to all cases where the geometry induced on the brane is deSitter, even if Λ is positive or negative.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

    Interpretationnone given

    If there were a negative cosmological constant in the bulk but a positive cosmological constant induced on the brane, the cosmological brane horizon would be larger than in the present setting, and the upper limit on the size of the black holes would be higher.

    “On the other hand, if there were a negative cosmological constant in the bulk, but a positive cosmological constant induced on the brane, the cosmological brane horizon would be larger than in the present setting, and the upper limit on the size of the black holes would be higher.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    In [27], the action difference for nucleation of a neutral black hole inside the wall is I - I0 = 11√3/36 M/σ ≈ 0.53 M/σ.

    “In [27] it was found that the action difference for nucleation of a neutral black hole inside the wall is I -I 0 = 11 √ 3 36 M σ ≈ 0.53 M σ ,(64)”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    If the horizons are identified, the black holes are joined by a wormhole; this happens if the black holes are nucleated as a pair.

    “If the horizons are identified, then the black holes are joined by a wormhole-this indeed happens if the black holes are nucleated as a pair.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    Thermodynamics provides a definition for the mass of the black hole, leading to the conclusion that it is entropically preferred to have the black hole on the wall, rather than away from it.

    “We have used thermodynamics to provide a definition for the mass of the black hole, which led us to conclude that it is entropically preferred to have the black hole on the wall, rather than away from it.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Interpretationnone given

    The phenomenon of flux expulsion appears to be essentially dimension independent.

    “On the other hand, the phenomenon of flux expulsion appears to be essentially dimension independent.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one. The quote only partly states this claim.

    Premisecitation

    For a (nearly) extremal charged black hole inside the wall, M is fixed as M = Q = 1/(8πσ), and I - I0 = 3/(32πσ^2) = 3/4 M/σ, which is smaller than (63).

    “In that case, M is again fixed in terms of σ, M = Q = 1/(8πσ), andfoot_4 I -I 0 = 3 32πσ 2 = 3 4 M σ ,(65)”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    If one tries to pull a small extremal black hole out of the brane, it will not experience the same elastic restoring force that non-extremal black holes suffer, so they might be able to slip off the brane into the bulk.

    “If one tries to pull a small extremal black hole out of the brane, it will not experience the same elastic restoring force that non-extremal black holes suffer, so, apparently they might be able to slip off of the brane into the bulk.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationnone given

    In the four dimensional setting, a domain wall can nucleate black holes of arbitrarily small size, which are preferred over larger black holes.

    “In contrast, we have argued that, in the four dimensional setting, a domain wall can nucleate black holes of arbitrarily small size-which are preferred over larger black holes.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    This result is exact, whereas corrections to (63) will be quite important since mA ≃ 2πMσ = 1/4, not a small number.

    “However, this result is exact, whereas corrections to (63) (which can be seen to lower its value) will be quite important since if we are supposed to be taking mA ≃ 2πMσ = 1/4, not a small number.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    In the hole-punch process, the black hole mass M is a parameter that can be varied independently of σ; therefore (63) can be made arbitrarily small.

    “In contrast, in the hole-punch process the black hole mass M is a parameter that can be varied independently of σ.Therefore, (63) can be made arbitrarily small 4 .”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationnone given

    The hole-punch process can form nearly extremal black holes much smaller than this, which will be less suppressed.

    “And, in any event, the hole-punch process can form nearly extremal black holes much smaller than this, which will be less suppressed.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    In [30], an instanton was presented which mediated the nucleation of a domain wall in five dimensions with a 'black object' on it; on the brane and on any surface at constant radius away from the brane, the four-geometry is the Nariai instanton S^2 × S^2.

    “Nevertheless, in [30] an instanton was presented which mediated the nucleation of a domain wall in five dimensions, with a “black object” on it.On the brane, as well as on any surface at constant radius away from the brane, the four-geometry is that of the Nariai instanton S 2 × S 2”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [30]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisecitation

    The size of the black objects nucleated via the instanton of [30] is fixed by the wall tension; the mass of the black object is not an independent parameter.

    “The size of the black objects nucleated via this instanton is, as was the case in [27], fixed by the wall tension.This is, the mass of the black object is not an independent parameter.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [30]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

36 of 36

  • The paper relies on this without support.

    For cosmic strings or pure flux p-branes, flux expulsion occurs for sufficiently small mass black holes.

    “For the cosmic string, or a pure flux p-brane, a phenomenon of 'flux expulsion' occurs for sufficiently small mass black holes”

  • The paper relies on this without support.

    For M^2 > 1/2, if a stable expelling solution exists, near the horizon it monotonically relaxes to a kink solution as one moves away from the horizon.

    “If a stable expelling solution exists, we expect that in the region near the horizon, it monotonically relaxes to a kink solution as we move away from the horizon”

  • The authors state this as an assumption.

    No boundary conditions are imposed for the wavefunctions of the corresponding universes.

    “we are assuming noboundary conditions for the wavefunctions of the corresponding universes”

  • The paper relies on this without support.

    The black holes are nearly extremal, so M ≃ Q.

    “(recall that the black holes are nearly extremal, so M ≃ Q)”

  • The authors state this as an assumption.

    It is reasonable to assume that a similar process will be possible in five dimensions, and small black holes will be nucleated on the brane.

    “It seems reasonable to assume that a similar process will be possible in five dimensions, and small black holes will be nucleated on the brane.”

  • The authors state this as an assumption.

    A five-dimensional analogue of the C-metric spacetime exists.

    “Even if it is fairly safe to assume, on physical grounds, that such a spacetime must exist, no explicit construction of it has been found.”

  • The paper relies on this without support.

    The gauge potential for the Maxwell field is unaltered by the presence of the wall.

    “Since the gauge potential for the Maxwell field is unaltered by the presence of the wall, its energy momentum tensor remains formally the same”

  • The paper relies on this without support.

    E, B, C, D effectively take their background values.

    “we guess that E, B, C, D will effectively take their background values”
8 of 8

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • In three dimensions, the electric gauge field At has a logarithmic dependence, At ∼ q log(r), producing a logarithmic correction to the three-dimensional blackening factor of classical charged BTZ.

    “Indeed, in three dimensions, the (electric) gauge field A t has a logarithmic dependence, A t ∼ q log(r), producing a logarithmic correction to the three-dimensional blackening factor of classical charged BTZ [101].”

    From Three-Dimensional Quantum Black Holes: A Primer (2024) · cites this paper as [101]

  • The classical geometry (l = 0) is not that of the charged BTZ metric.

    “Notably, the classical geometry (ℓ = 0) is not that of the charged BTZ metric [101], we will return to this point momentarily.”

    From Three-Dimensional Quantum Black Holes: A Primer (2024) · cites this paper as [101]

Citations

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. K. Akama, 1983, Gauge Theory and Gravitation, Proceedings of the International Symposium, Nara, Japan, 1982
  2. A. Lukas, 1999, Phys. Rev. D, 086001

Cited by

Sorted by citations

Related

Metadata

Authors
Emparan, R., Gregory, R., & Santos, C.
Affiliations
University of the Basque Country (ES)
ORCID
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2001-04-26
Volume
63
Issue
10
ISSN
  • 0556-2821 (ISSN-L)
  • 1089-4918
DOI
10.1103/physrevd.63.104022
OpenAlex
W2115150940
Semantic Scholar
46250c3d30a039346d22d97d16a082c951b9c80b
Cited by
61
References
51
Citations by year
2025: 3 · 2024: 4 · 2023: 2 · 2017: 1 · 2015: 3 · 2014: 3 · 2013: 3 · 2012: 2
FWCI
3.16
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • RECERCAT (Consorci de Serveis Universitaris de Catalunya) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • Dipòsit Digital de la Universitat de Barcelona (Universitat de Barcelona) (submittedVersion, public-domain)
  • arXiv (Cornell University)
Funders
Royal Society
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Astrophysical Phenomena and Observations
Keywords
black holes, thick domain walls, two-branes, braneworld scenario, topological defects, exact solutions, flux expulsion, black hole nucleation

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

A language model drew the claims from the paper on 2026-11-10, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).

A second model labelled each claim on 2026-11-10 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review