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Cited by 4Black Holes and Theoretical PhysicsCosmology and Gravitation TheoriesNoncommutative and Quantum Gravity Theories

The DOI matches its Crossref record; 42 of the 42 references it cites were found in a registry or an index; none of the 14 sources its claims cite is checked for support yet.

3D black holes on a 2-brane in 4D Minkowski space

Open access
Bazeia, D., Brito, F.A., Costa, F.G.·2011·Physics Letters B
DOI 10.1016/j.physletb.2011.09.066
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Abstract

Abstract We investigate three-dimensional black hole solutions in the realm of pure and new massive gravity in 2 + 1 dimensions induced on a 2-brane embedded in a flat four-dimensional spacetime. There is no cosmological constant neither on the brane nor on the four-dimensional bulk. Only gravitational fields are turned on and we indeed find vacuum solutions as black holes in 2 + 1 dimensions even in the absence of any cosmological solution. There is a crossover scale that controls how far the three- or four-dimensional gravity manifests on the 2-brane. Our solutions also indicate that local BTZ and SdS 3 solutions can flow to local four-dimensional Schwarzschild-like black holes, as one probes from small to large distances, which is clearly a higher dimensional manifestation on the 2-brane. This is similar to the DGP scenario where the effects of extra dimensions for large probed distances along the brane manifest.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

The paper finds vacuum 2+1-dimensional black-hole solutions induced on a 2-brane embedded in flat 4D Minkowski space, without a cosmological constant or matter, in both pure induced gravity and new massive gravity (NMG). It identifies a crossover scale controlling whether 3D BTZ/SdS₃ or 4D Schwarzschild-like behavior dominates, with numerical solutions showing flow between them.

Key findings

  • In pure induced gravity, the exact 3D black-hole solution is \(A_0(r)=c-r^2/(2r_0^2)\), with crossover scale \(r_0^2=\kappa_{(4)}^4/(32\kappa_{(3)}^4)\). For \(c=1\) this is de Sitter; for \(c=1-8G_3M\) it is a 3D de Sitter–Schwarzschild (SdS₃) black hole; for the AdS₃ embedding, \(c=-8G_3M\) gives a BTZ black hole. No cosmological constant is required a priori; the induced curvature acts as a “cosmic fluid” with \(p_{\rm curv}=-\rho_{\rm curv}\).
  • In NMG, exact non-trivial solutions are found for the \(n=0\) ansatz, with independent \(\alpha\)-type 3D black holes having non-coincident horizons \(r_\alpha=(4+\alpha^2)r_{0\pm}/(2\sqrt{2})\), \(\alpha>0\). The plus sign with \(c<0\) corresponds to BTZ solutions.
  • Numerical solutions show local BTZ/SdS₃ solutions can flow to four-dimensional Schwarzschild-like black holes as one probes from small to large distances. For the SdS₃ case, the effective dimension \(d(r)=-d\ln V(r)/d\ln r\) gives \(d(r\simeq0)\simeq-2\) at small distances and \(d(r\simeq6)\simeq1\) at large distances, with \(A(r)-c\sim -1/r\) and \(c=1.013\).
  • As the graviton mass \(m\to0\), the crossover scale becomes large and the solution approaches the Schwarzschild regime \(A_0^{(2)}(r)\simeq 1-GM/r\).

Methods

  • Theoretical setup: a 4D action \(S=S^{(4)}+S^{(3)}\) with a 2-brane at \(z=0\), induced metric \(q_{\mu\nu}=g_{\mu\nu}(r,z=0)\), and metric ansatz \(ds_4^2=-A(r,z)dt^2+A(r,z)^{-1}dr^2+r^2d\theta^2+dz^2\). Only vacuum gravitational fields are considered, so \(T_{ab}=0\).
  • Israel junction conditions at \(z=0\) relate \(A'(0^+)/A_0\) to the brane curvature energy density. In pure gravity this yields an exact integrable equation for \(A_0(r)\). In NMG, the action includes \(K=R_{\mu\nu}R^{\mu\nu}-\frac{3}{8}R^2\), leading to a fourth-order differential equation for \(A_0(r)\).
  • Exact NMG solutions use the ansatz \(A_0(r)=a r^n+b r^2+c\) for \(n=0\). Numerical flow solutions use dimensionless variables \(r/r_0\) and \(\alpha=mr_0\), with boundary conditions \(A(r_\infty)=1\), \(A'(r_\infty)=0\), \(A''(r_\infty)=0\), \(A'''(r_\infty)=1/10\), \(r_\infty=2\pi\), and \(\alpha=6.7\) for Figs. 1–2; Fig. 3 uses \(A'''(r_\infty)=-1/10\), \(\alpha=25\), and \(r_0=6\).

Limitations

  • The paper states no limitations section.
  • Evident from the text: the analysis is purely theoretical and classical, restricted to vacuum gravitational fields with no cosmological constant or matter, and no observational or experimental constraints are considered.
  • Evident from the text: the numerical flow results depend on chosen boundary conditions and parameters (e.g., \(r_\infty=2\pi\), \(\alpha=6.7\); \(\alpha=25\), \(r_0=6\)), and the exact NMG solutions are restricted to the \(n=0\) ansatz.

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

Integrity signals

Registry

Crossref · 10.1016/j.physletb.2011.09.066

Cites

42 of 42 references verified0 citation errors · 0 not checkable

Statistics

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Verification

Claims the paper makes

0 of 14 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    Fixing n=1, a=-GM and c=1 gives the Schwarzschild regime for very large three-dimensional black hole horizons r_α ≃ r0: A0^(2)(r) ≃ 1 - GM/r, with r ≃ r0.

    “This allows us to fix n = 1, a = -GM and c = 1 to find the Schwarzschild regime for very large three-dimensional black hole horizons r α ≃ r 0 A (2) 0 (r) ≃ 1 - GM r ,(45) with r ≃ r 0 ; recall that r 2 0 = κ 4 (4) /32κ 4 (3) .”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Assuming the metric describes a four-dimensional flat space with two timelike coordinates t and z which provides an AdS3 embedding, the solution changes to ds^2(3) = -(c + r^2/(2r0^2)) dt^2 + (c + r^2/(2r0^2))^{-1} dr^2 + r^2 dθ^2.

    “Assuming that our metric (10) now describes a four-dimensional flat space with two timelike coordinates t and z which provides an AdS 3 embedding results in the Eq. ( 16) with a changed sign”
    • Also stated in: § II. CIRCULARLY SYMMETRIC BLACK HOLE SOLUTION IN 3D INDUCED GRAVITY ON A 2-BRANE IN 4D FLAT SPACETIME · plain
      “As a consequence our previous solution changes and now reads ds 2 (3) = -c + r 2 2r 2 0 dt 2 + c + r 2 2r 2 0 -1 dr 2 + r 2 dθ 2 .(33)”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The metric (10) describes a four-dimensional flat space with two timelike coordinates t and z, providing an AdS3 embedding.
  • Findingreasoning

    The exact three-dimensional black hole solution living on the 2-brane is ds^2(3) = -(c - r^2/(2r0^2)) dt^2 + (c - r^2/(2r0^2))^{-1} dr^2 + r^2 dθ^2, where r0^2 = κ4^4/(32κ3^4) is the crossover scale and c is an integration constant.

    “This gives us an exact threedimensional black hole solution living on the 2-brane that can be expressed in the usual form ds 2 (3) = -c - r 2 2r 2 0 dt 2 + c - r 2 2r 2 0 -1 dr 2 + r 2 dθ 2 .(31)”
    Quote in the paperSubstantial claim
  • Findingown data

    Local BTZ and SdS3 solutions can flow to local four-dimensional Schwarzschild-like black holes as one probes from small to large distances.

    “Our solutions also indicate that local BTZ and SdS3 solutions can flow to local four-dimensional Schwarzschild like black holes, as one probes from small to large distances, which is clearly a higher dimensional manifestation on the 2-brane.”
    • Also stated in: § IV. CONCLUSIONS · boosted
      “Now, the local BTZ/SdS 3 solutions can flow to four-dimensional Schwarzschild like black holes, as one moves from small to large distances, which is clearly a higher dimensional manifestation on the 2-brane.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The paper found a class of exact non-trivial solutions for n=0, with r0^2- = 8/(α^2(α^2+4)^2 m^2) and r0^2+ = 8/((α^2+8)(α^2+4)^2 m^2), such that A0^(α)(r) = c ± 8(4+α^2)^2 r^2 / r0±^2.

    “We have found a class of exact non-trivial solutions for n = 0, assuming r 2 0-= 8/α 2 (α 2 + 4) 2 m 2 and r 2 0+ = 8/(α 2 + 8)(α 2 + 4) 2 m 2 , by properly using Eq.( 32) in place of Eq.( 40), being c a constant, such that our solutions become A (α) 0 (r) = c ± 8 (4 + α 2 ) 2 r 2 r 2 0± . (43”
    Quote in the paperSubstantial claim
  • It may state two separate claims as one. 10
  • Probably not a substantial claim. 5
  • The quote only partly states this claim. 1

  • It may state two separate claims as one.

    Findingcitation

    For c=1 the solution is just de Sitter spacetime, whereas for c=1-8G3M it is a three-dimensional de Sitter-Schwarzschild (SdS3) black hole, a conical singularity with associated mass M.

    “For c = 1 we have just a de Sitter spacetime, whereas for c = 1 -8G 3 M we have a three-dimensional de Sitter -Schwarzschild (SdS 3 ) black hole, a conical singularity with associated mass M [6].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [6]

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

  • It may state two separate claims as one.

    Findingown data

    The numerical results show a local SdS3 for r≃0, with effective dimension d(r≃0)≃-2 and potential A(r)-c ∼ -1/r^-2, c=191.59.

    “Left panel:The warp factor A(r) for small distances.We find a local SdS3 for r ≃ 0. Right panel: The precise power d(r ≃ 0) ≃ -2 of the 'potential' A(r) -c ∼ -1/r -2 , being c = 191.59.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingown data

    The numerical results show a local Schwarzschild-like solution for r≃6, with effective dimension d(r≃6)≃1 and potential A(r)-c ∼ -1/r, c=1.013.

    “Left panel: The warp factor A(r) for large distances.We approach a local Schwarzschild-like solution for r ≃ 6.Right panel: The precise power d(r ≃ 6) ≃ 1 of the 'potential' A(r) -c ∼ -1/r, being c = 1.013.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    p_curv = -ρ_curv, which shows that the 'cosmic fluid' acts as a cosmological constant.

    “Notice that p curv = -ρ curv , which shows that the 'cosmic fluid' acts as a cosmological constant.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The energy density and pressure for the brane curvature are ρ_curv = 1/(κ3^2) (Ȧ0/(2r)) and p_curv = -1/(κ3^2) (Ȧ0/(2r)).

    “ρ curv = 1 κ 2 (3) Ȧ0 2r ,(25) and p curv = - 1 κ 2 (3) Ȧ0 2r . (26”
    Quote in the paperSubstantial claim

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

    Premisecitation

    BTZ black holes in 2+1 dimensional gravity with a cosmological constant were first found long ago.

    “The three-dimensional black holes in 2+1 dimensional gravity with a cosmological constant such as BTZ black holes have been first found long ago [6][7][8].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [6]

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

    • cites [7]

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

    • cites [8]

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

  • It may state two separate claims as one.

    Interpretationreasoning

    For κ4^2 ≫ κ3^2 there is four-dimensional gravity dominance over three-dimensional gravity, such that r0 is very large and the solution (30) approaches the solution of a flat space.

    “Notice that for κ 2 (4) ≫ κ 2 there is a four-dimensional gravity dominance over the three-dimensional gravity, such that r 0 is very large and then the solution (30) approaches the solution of a flat space.”
    Quote in the paperNot a substantial claim

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

    Interpretationreasoning

    For massless gravitons (m→0), r ≃ r_α ≃ r0 ∼ 1/m → ∞, so the approximate solution approaches the constant c.

    “For a finite value of a this is particularly true for massless gravitons, m → 0, because r ≃ r α ≃ r 0 ∼ 1/m → ∞.”
    Quote in the paperSubstantial claim

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

    Premisecitation

    There are well-known exact solutions describing black holes on a 2-brane embedded in four dimensions found for branes in the Randall-Sundrum-like scenario.

    “One should also mention that there are well-known exact solutions describing black holes on a 2-brane embedded in four-dimensions found for branes in the Randall-Sundrum-like scenario [13,14].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [13]

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

    • cites [14]

      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 black hole solution does not require a cosmological constant a priori; instead, the induced curvature on the 2-brane gives rise to a 'cosmic fluid' that plays the role of a 'cosmological constant'.

    “The novelty here is that the black hole solution does not require a cosmological constant a priori.Instead, the induced curvature on the 2-brane gives rise to a 'cosmic fluid' that plays the role of a 'cosmological constant'.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: There is no cosmological constant on the brane or in the four-dimensional bulk.

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

    Premisecitation

    Massive propagating gravitons can appear in some proposals, including topologically massive gravity and new massive gravity.

    “However, massive propagating gravitons can appear in some proposals, among them, the case where one adds gravitational Chern-Simons terms as first considered in [4], which has been named topologically massive gravity (TMG), and also within the more recent context, where new massive gravity (NMG) has been put forward by Bergshoeff, Hohm and Townsend [1]”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [4]

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

    • cites [1]

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

  • It may state two separate claims as one.

    Interpretationnone given

    At small distances, in the UV regime, BTZ and three-dimensional de Sitter-Schwarzschild (SdS3) black holes appear, three-dimensional gravity is recovered, and an improved UV regime is achieved.

    “whereas on the the other side, for small distances, i.e., in the UV regime, BTZ and three-dimensional de Sitter -Schwarzschild (SdS 3 ) black holes appear, the three-dimensional gravity is recovered and then an improved UV regime is achieved.”
    Quote in the paperSubstantial claim

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

    Premisecitation

    In the NMG context, a class of interesting solutions has recently been found.

    “However, in the NMG context, one has also recently found a class of interesting solutions -see e.g.[9].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [9]

      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.

    Study design and conductown data

    The position of the boundary r∞ is not relevant to the numerical results.

    “The boundary r∞ is assumed to be a sufficiently large scale -indeed, this is not necessary since several tests revealed that the position of this boundary is not relevant.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Premisereasoning

    The induced scalar curvature R(3) is made out of the three-dimensional metric and has no dependence on the transversal fourth coordinate z.

    “It follows that the induced scalar curvature R (3) is made out of this three-dimensional metric and has no dependence with the transversal fourth coordinate z.”
    Quote in the paperNot a substantial claim
15 of 15

  • The authors state this as an assumption.

    The metric (10) describes a four-dimensional flat space with two timelike coordinates t and z, providing an AdS3 embedding.

    “Assuming that our metric (10) now describes a four-dimensional flat space with two timelike coordinates t and z which provides an AdS 3 embedding results in the Eq. ( 16) with a changed sign”

  • The authors state this as an assumption.

    There is no cosmological constant on the brane or in the four-dimensional bulk.

    “There is no cosmological constant neither on the brane nor on the four-dimensional bulk.”
    “There is no cosmological constant neither on the brane nor on the four-dimensional bulk.”

  • The authors state this as an assumption.

    Only gravitational fields are turned on, so the matter field Lagrangians vanish and T_ab=0.

    “Only gravitational fields are turned on”
    “Once we shall only investigate vacuum solutions for this theory, the matter field Lagrangians L m = 0 and L (4) m = 0, so that we have T ab = 0. (7”
3 of 3

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Stated horizon radius omits its dependence on c

    Minor

    Setting the quadratic metric function in Eq. (43) to zero gives a horizon radius proportional to √|c|. The stated radius holds only when |c| = 1, although c is presented as an unrestricted constant.

1 of 1

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Metadata

Authors
Bazeia, D., Brito, F. A., & Costa, F. G.
Affiliations
  • Universidade Federal da Paraíba (BR)
  • Universidade Federal de Campina Grande (BR)
Corresponding author
Brito, F.A.
ORCID
Venue
Physics Letters B
Publisher
Elsevier BV
Published
2011-10
Volume
704
Issue
5
Pages
631-636
ISSN
0370-2693 (ISSN-L)
DOI
10.1016/j.physletb.2011.09.066
OpenAlex
W1595125642
Semantic Scholar
240a688b3d362a537d671e09e667f64ac7a89b21
Cited by
4
References
25
Citations by year
2025: 1 · 2017: 1 · 2015: 1
FWCI
0.17
Type
journal article
Language
en
Open access
Licence
Locations
  • Physics Letters B (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • CNPq
  • CAPES
  • PNPD/PROCAD — CAPES
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 › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
Keywords
2-branes, massive gravity, new massive gravity, 2+1 dimensions, BTZ black hole, Schwarzschild black hole, crossover scale, braneworld gravity

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