Pith. sign in

REVIEW 5 major objections 6 minor 35 references

Deforming charged black holes with dipolar differential rotation boundary

T0 review · 5 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper claims that in charged, deforming anti-de Sitter black holes with a dipolar differential rotation boundary, two small-horizon branches with different radii have identical horizon geometry, entropy, and scalar quasinormal-mode…

desk verdict Charged deforming black holes with an interesting degeneracy claim, but a sign inconsistency in the metric ansatz and absent numerical validation make it currently unconvincing. read the letter →

arxiv 1909.01628 v1 pith:56JM35OW submitted 2019-09-04 hep-th gr-qc

Tong-Tong Hu , Shuo Sun , Hong-Bo Li , Yong-Qiang Wang This is my paper · ORCID
classification hep-thgr-qc PACS 04.70.-s04.40.Nr
keywords AdS/CFTdualityblackholesquasinormalmodeschargeddeforminghorizondifferentialrotationboundaryDeTurckmethodgeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies four-dimensional anti-de Sitter black holes whose boundary rotates differentially with the dipolar profile $\Omega(\theta)=\varepsilon\cos\theta$, and adds electric charge to the bulk. It finds that charge removes the uncharged theory's minimum-temperature barrier: for any nonzero charge and any temperature, at least one horizon exists. The central discovery is a degeneracy: at temperatures admitting three horizon radii, the two smaller solutions have different radii yet the same horizon geometry, the same entropy, and the same quasinormal-mode frequencies. This means the boundary data and temperature do not fix the local size of a small horizon, even though they fix its observable features. The paper also maps how the charge $q$ controls temperature extrema, the entropy phase diagram, and the onset of scalar condensation.

What carries the argument

The DeTurck method is the core mechanism: adding the gauge-fixing vector $\xi^\mu = g^{\nu\rho}(\Gamma^\mu{}_{\nu\rho}[g]-\Gamma^\mu{}_{\nu\rho}[\tilde g])$ converts the Einstein equations into a determined elliptic system that can be solved numerically with a reference metric sharing the same boundary and horizon structure. The horizon-counting argument runs through the analytic temperature formula $T = [y_+^4 + \delta(-q^2+y_+^2(1+2y_+^2))]/(4\pi y_+^3)$, whose extrema in $y_+$ depend on $q$ and $\delta$. The degeneracy of the two small branches is demonstrated through equal isometric embeddings into hyperbolic 3-space, equal entropy from $S=(2\pi y_+^2 L^2/G_N)\int_0^1 dx\,(1-x^2)/\sqrt{2-x^2}\sqrt{U_3(x,0)U_5(x,0)}$, and equal quasinormal frequencies obtained from a massless scalar perturbation in Eddington-Finkelstein coordinates.

What would settle it

Recompute the two small branches at a representative point, for example $T=0.2585$, $q=0.07057$, $\varepsilon=1.6$, on successively finer grids while monitoring the DeTurck residual; if their entropies and quasinormal frequencies cease to match as resolution improves, the two-branch degeneracy is a numerical artifact. An independent spectral code should reproduce the stated horizon radii $y_+=0.3110$ and $y_+=0.0859$ with identical embedding curves for the degeneracy claim to stand.

Watch

Extended reading notes

Core claim

The authors construct numerical solutions of the Einstein-Maxwell equations in AdS$_4$ with the conformal boundary metric $ds^2_\partial = -dt^2 + d\theta^2 + \sin^2\theta\,(d\varphi + \varepsilon\cos\theta\,dt)^2$, using the DeTurck method to turn the Einstein equations into elliptic equations. They find that the temperature-horizon relation depends sharply on the charge: for $q=0$ there is a minimum temperature below which no horizon exists; for $0<q<1/6$ there are two temperature extrema; for $q=1/6$ the extrema coalesce at the Reissner-Nordstr\"om-AdS value $T_{\rm RN}=\sqrt{6}/(3\pi)$; and for $q>1/6$ there is no extremum, so a horizon exists at every temperature. In the temperature band with three horizon radii, the two small branches have horizon radii that differ by as much as a factor of four, yet their hyperbolic embeddings coincide, their entropy integrals give the same value, and their scalar quasinormal frequencies match for every azimuthal quantum number studied. The paper further reports that large-branch horizon deformation grows with the rotation parameter $\varepsilon$, small-branch deformation shrinks with $\varepsilon$, and scalar condensation appears when the azimuthal quantum number satisfies $m\ge 13$.

Load-bearing premise

The load-bearing premise is that the numerical DeTurck solutions are fully converged and accurate, since the paper reports no convergence tests, residual norms, or error estimates for the computed metrics, entropies, or quasinormal frequencies.

Editorial extensions

If this is right

  • For any nonzero charge $q$, a charged deforming AdS black hole exists at any temperature, unlike the uncharged case where temperatures below $T_S=\sqrt{3}/(2\pi)$ admit no horizon.
  • In the three-horizon temperature band, the two small branches are thermodynamically and spectroscopically degenerate despite having different horizon radii, so the boundary metric and temperature do not uniquely determine the local horizon size.
  • The entropy phase diagram splits into three temperature regions for $0<q<1/6$: the large-branch entropy increases with $\varepsilon$, the small-branch entropy decreases, and the branches join when $T\le T_{\rm RN}$.
  • Scalar condensation, signalled by a negative real part of the quasinormal frequency, occurs for azimuthal quantum number $m\ge 13$ at sufficiently large $\varepsilon$.
  • Adjusting the parameter $\delta$ below 1 produces three horizon radii even below the uncharged minimum temperature, broadening the region in which the small-branch degeneracy can be studied.

Reading between the lines

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

  • If the two-branch degeneracy is exact rather than a numerical coincidence, it suggests an emergent symmetry or equivalence relating the two small solutions, possibly tied to the isometric embedding into hyperbolic space; the paper does not identify a mechanism.
  • A natural testable extension is to compute the full quasi-local stress tensor, angular momentum, and mass of the two small branches; if all charges and thermodynamic potentials also match, the degeneracy would be a genuine failure of uniqueness rather than a coincidence of entropy and geometry.
  • Since the paper reports no convergence tests or residual measures, a decisive check is to rerun the DeTurck solver at higher resolution and monitor the gauge-fixing residual; if the degeneracy persists to machine precision, it is robust.
  • Extending the construction to nonlinear electrodynamics or $f(R)$ gravity, as the authors propose to do, would test whether the degeneracy is special to Einstein-Maxwell theory or a generic feature of deformed AdS horizons.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper numerically constructs charged, asymptotically AdS_4 black holes with a dipolar differential rotation on the boundary, using the DeTurck method in Einstein-Maxwell theory. The authors study how the number of black hole horizons depends on the temperature T and charge q, and identify a temperature window for 0<q<1/6 in which three horizon radii exist for a fixed T. They report that in this window the two smaller-horizon solutions share the same horizon geometry, entropy, and quasinormal-mode frequencies despite having different horizon radii. The paper also studies isometric embeddings of the horizon into hyperbolic space and the entropy as a function of the boundary rotation parameter, and discusses the stability of the small branches under massless scalar perturbations.

Significance. If the main claims hold, the most notable result is the apparent degeneracy between the two small-horizon branches: different horizon radii would lead to identical geometric and dynamical properties. This would be a new phenomenon in the study of deformed AdS black holes and could have implications for holographic models with deformed boundaries. The paper follows an established numerical approach (the DeTurck method) and extends previous work on uncharged deforming black holes to the charged case. However, the significance is conditional: the internal sign inconsistency in the temperature formula and the absence of numerical convergence tests prevent a reliable assessment of the central claims.

major comments (5)
  1. [3 (Eq. (3.1b) vs. Eq. (3.6))] There is a sign inconsistency in the q^2 term. The ansatz in Eq. (3.1b) contains Δ(y) = q^2(1-y^2)^2/(L^2 y_+^2) + (1-y^2)^2 + y_+^2(3-3y^2+y^4), so Δ(0)=1+3y_+^2+q^2/y_+^2 (with L=1). The standard surface gravity calculation in the RN-AdS limit then gives T=(1+3y_+^2+q^2/y_+^2)/(4π y_+), in conflict with Eq. (3.6), whose minus sign is required to reproduce the RN-AdS temperature. With the printed plus sign, T(y_+) has only a single extremum (a minimum) for q>0, so the three-horizon region for 0<q<1/6, on which the central degeneracy claim rests, would not exist. Please correct the sign in Eq. (3.1b) or explicitly state that the code uses a different Δ, and confirm that Eq. (3.6) is the temperature of the solutions actually constructed.
  2. [3 (Eq. (3.7))] The analytic expressions for T_min and T_max in Eq. (3.7) do not reproduce the values stated in the text. For q=0.07057, substituting s=√(1−36q^2)≈0.9059 into the printed formula gives T≈0.193, while the text and Fig. 1 quote T_min≈0.2735 and T_max≈0.4635 for this charge. The correct RN-AdS extrema obtained from T(y_+)=(1+3y_+^2−q^2/y_+^2)/(4πy_+) are T_min=(2+s)√6/(6π√(1+s)) and T_max=(2−s)√6/(6π√(1−s)), which do match the quoted numbers. The equations should be corrected and the derivation shown.
  3. [2 and 3] The paper reports no convergence tests, no residual or constraint-violation measures, and no error estimates for any of the numerical quantities (metric functions, entropy, quasinormal frequencies). This is especially serious for the central claim that the two small branches have identical horizon geometry, entropy, and QNMs: if the DeTurck solutions are not fully converged, the apparent degeneracy could be a numerical artifact. Please provide the numerical grid sizes and a representative convergence study (e.g., the DeTurck vector norm as a function of resolution) for the cases in Figs. 2, 4, 6, and 9.
  4. [3.3] The stability analysis is based on the sign of Re(ω): the text states that 'Re ω would appear a negative value ... which means we could obtain a stable deforming charged black hole solution with scalar condensation.' For the perturbation convention Φ∼e^{-iωt} in Eq. (3.3.2), an instability corresponds to Im(ω)>0; Re(ω)<0 is not a standard instability criterion, and 'stable ... with scalar condensation' is internally contradictory. The paper never reports Im(ω). The stability conclusions in this section should be reworked: compute Im(ω) for the modes in Fig. 9, or otherwise justify the criterion, and interpret the onset of the unstable mode.
  5. [3.1-3.3] The paper states repeatedly that the two small branches, e.g., y_+=0.0992 and y_+=0.1773, have the same horizon embedding, entropy, and quasinormal frequencies. Since y_+ is a coordinate parameter tied to the horizon scale and the boundary metric is fixed, it is not obvious how two solutions with different y_+ can be physically identical. No direct comparison of the full metric functions U_i(x,y) for the two branches is given, and no symmetry or coordinate transformation is identified to explain the degeneracy. Please provide evidence that the two solutions represent the same physical geometry, or clarify the precise sense in which they are degenerate.
minor comments (6)
  1. [Figure 2] The caption and the text disagree on the values of q in the bottom-right panel: the text lists q=0, 1.7068, 2.2684, 3.4299 for five colored lines, while the caption includes q=2.8363 as well.
  2. [Figure 1] The text describes the line for q=1/6 as 'orange' while the caption calls it 'red'; these labels should be made consistent.
  3. [3 (after Eq. (3.5))] The boundary condition A_t(x,1)=μ and A_t(1,y)=0 conflict at the corner (x=1,y=1); the treatment of corners in the numerical scheme should be described.
  4. [3.2] The statement 'we also find another family of small black hole solutions' is vague; the authors should specify the boundary conditions or parameter range in which this second family exists and how it relates to the first family.
  5. [Abstract and Introduction] The statement that there exists at least one horizon for an arbitrary temperature is only true for q≠0, since for q=0 there is no horizon for T<T_S; please qualify the statement accordingly.
  6. [References] Reference [18] is cited as arXiv:1906.06183 without a journal reference; if it has been published, please update the citation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the temperature branch structure and two-small-branch degeneracy are numerical outputs, not fitted inputs.

full rationale

None of the paper's load-bearing claims reduce to their inputs. The temperature function (3.6) is obtained from the surface gravity of the metric ansatz (3.1), and the three-branch horizon counting in Sec. 3 is an algebraic consequence of T(y_+) rather than a fitted relation. The central claim that two small branches share horizon geometry, entropy, and quasinormal-mode frequencies is presented as a numerical output of the DeTurck solutions (Figs. 4, 6, 9); no constant is adjusted to produce that degeneracy. Self-citations [18,19] are limited to background statements about the authors' earlier multipolar solutions and are not used to justify the new charged solutions or the degeneracy. The DeTurck method and the metric ansatz are attributed to [16,23-25], which are independent or externally established method references. A sign inconsistency between Eq. (3.1b) and Eq. (3.6) in the RN-AdS limit is a potential correctness defect, but it is not a circularity: the paper's derivation does not assume the conclusion. Therefore the circularity score is low.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the DeTurck numerical scheme and on the specific ansatz (3.1). No free parameters are fitted to data; the model parameters q, epsilon, T, delta, and mu are inputs, and the only hand-chosen constant is the embedding radius l=0.73, which is not physically load-bearing. The main physical claims depend on the completeness of the ansatz and on the reliability of the numerical solver.

free parameters (2)
  • delta = 1 in most of the paper; <1 in Secs. 3.1-3.2
    A parameter in the temperature formula (3.6) that the authors adjust to obtain three horizon branches below Tmin; it is not determined by the equations and is effectively a tuning parameter.
  • Embedding radius l = 0.73
    Fixed by hand for the hyperbolic embedding in Sec. 3.1; it only affects the scale of the visualization plots, not the physics.
assumptions (4)
  • domain assumption The DeTurck method converges to solutions of the original Einstein equations, i.e., the gauge-fixing term in (2.10) vanishes for the computed solutions.
    The method relies on [23-25]; the paper does not report the DeTurck vector norm or residual.
  • domain assumption The metric ansatz (3.1) with six functions U_i(x,y) and the stated boundary conditions captures all stationary, axisymmetric charged black hole solutions with the dipolar differential rotation boundary.
    No uniqueness or completeness argument is given; the phase diagram and degeneracy claims are made within this ansatz.
  • domain assumption The Hawking temperature of the deformed solutions is given by (3.6), which depends only on y+, q, and delta.
    This follows from the ansatz and the boundary condition U1=1 at the horizon; it is used to count horizon branches.
  • ad hoc to paper A negative real part of the quasinormal frequency Re omega indicates scalar condensation or instability.
    The paper states this in Sec. 3.3 without displaying Im omega; standard stability analysis relies on the sign of Im omega.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Deforming charged black holes with dipolar differential rotation boundary." pith.science (2026). https://pith.science/paper/56JM35OW

@misc{pith2026190901628,
  author       = {Pith},
  title        = {Pith review of: Deforming charged black holes with dipolar differential rotation boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56JM35OW}},
  note         = {Machine review of arXiv:1909.01628}
}
abstract

Motivated by the recent studies of the novel asymptotically global AdS$_4$ black hole with deforming horizon, we consider the action of Einstein-Maxwell gravity in AdS spacetime and construct the charged deforming AdS black holes with differential boundary. In contrast to deforming black hole without charge, there exists at least one value of horizon for an arbitrary temperature. The extremum of temperature is determined by charge $q$ and divides the range of temperature into several parts. Moreover, we use an isometric embedding in the three-dimensional space to investigate the horizon geometry. We also study the entropy and quasinormal modes of deforming charged AdS black hole. It is interesting to find there exist two families of black hole solutions with different horizon radius for a fixed temperature, but these two black holes have same horizon geometry and entropy. Due to the existence of charge $q$, the phase diagram of entropy is more complicated.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 18 canonical work pages

  1. [16]

    Stirring a black hole,

    J. Markeviit and J. E. Santos, “Stirring a black hole,” JHEP 1802, 060 (2018) doi:10.1007/JHEP02(2018)060 [arXiv:1712.07648 [hep-th]]. – 16 –

    arXiv 2018

  2. [1]

    Event Horizons in Static Vacuum Space-Times,

    W. Israel, “Event Horizons in Static Vacuum Space-Times,” Phys. Rev. 164, 1776 (1967)

    work page 1967

  3. [2]

    Introducing the black hole,

    R. Ruffini and J. A.Wheeler, “Introducing the black hole,” Phys. Today 24, 30 (1971)

    work page 1971

  4. [3]

    Stationary Black Holes: Uniqueness and Beyond,

    P. T. Chrusciel, J. Lopes Costa and M. Heusler, “Stationary Black Holes: Uniqueness and Beyond,” Living Rev. Rel. 15, 7 (2012) [arXiv:1205.6112 [gr-qc]]

    arXiv 2012

  5. [4]

    Carter, C

    B. Carter, C. De Witt, and B. S. De Witt, in Proceedings of 1972 Session of Ecole dEte De Physique Theorique, (Gordon and Breach, New York, 1973)

    work page 1972

  6. [5]

    J. M. Maldacena, The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113 (1999)

    work page 1999

  7. [6]

    J. M. Maldacena, Adv.Theor. Math. Phys. 2, 231 (1998) [hep-th/9711200]

    arXiv 1998

  8. [7]

    Witten, Anti-de Sitter space and holography

    E. Witten, Anti-de Sitter space and holography . Adv. Theor. Math. Phys. 2, 253 (1998) [hep-th/9802150]

    arXiv 1998

Show all 35 references
  1. [8]

    Aharony, S

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

    2000 arXiv

  2. [9]

    Deformed hyperbolic black holes,

    Y. Chen, Y. K. Lim and E. Teo, “Deformed hyperbolic black holes,” Phys. Rev. D 92, no. 4, 044058 (2015) doi:10.1103/PhysRevD.92.044058 [arXiv:1507.02416 [gr-qc]]

    2015 arXiv

  3. [10]

    Nontrivial, static, geodesically complete, vacuum space-times with a negative cosmological constant,

    T. Levi-Civita, “Nontrivial, static, geodesically complete, vacuum space-times with a negative cosmological constant,” Rend. Acc. Lincei, 26, 307 (1917), 27, 3, 183, 220, 240, 283, 343 (1918), 28, 3, 101 (1919)

    1917

  4. [11]

    Nontrivial, static, geodesically complete, vacuum space-times with a negative cosmological constant,

    H. Weyl, “Nontrivial, static, geodesically complete, vacuum space-times with a negative cosmological constant,” Ann. Phys. 54, 117 (1917)

    1917

  5. [12]

    Rotating, charged, and uniformly accelerating mass in general relativity,

    J. F. Plebanski and M. Demianski, “Rotating, charged, and uniformly accelerating mass in general relativity,” Annals Phys. 98, 98 (1976). doi:10.1016/0003-4916(76)90240-2

    1976 doi

  6. [13]

    Gnecchi, K

    A. Gnecchi, K. Hristov, D. Klemm, C. Toldo and O. Vaughan, JHEP 1401, 127 (2014) doi:10.1007/JHEP01(2014)127 [arXiv:1311.1795 [hep-th]]

    2014 arXiv

  7. [14]

    Klemm, Phys

    D. Klemm, Phys. Rev. D 89, no. 8, 084007 (2014) doi:10.1103/PhysRevD.89.084007 [arXiv:1401.3107 [hep-th]]

    2014 arXiv

  8. [15]

    Black holes with bottle-shaped horizons,

    Y. Chen and E. Teo, “Black holes with bottle-shaped horizons,” Phys. Rev. D 93, no. 12, 124028 (2016) doi:10.1103/PhysRevD.93.124028 [arXiv:1604.07527 [gr-qc]]

    2016 arXiv

  9. [17]

    Superradiant instabilities of asymptotically anti-de Sitter black holes,

    S. R. Green, S. Hollands, A. Ishibashi and R. M. Wald, “Superradiant instabilities of asymptotically anti-de Sitter black holes,” Class. Quant. Grav. 33, no. 12, 125022 (2016) doi:10.1088/0264-9381/33/12/125022 [arXiv:1512.02644 [gr-qc]]

    2016 arXiv

  10. [18]

    S. Sun, T. T. Hu, H. B. Li and Y. Q. Wang, arXiv:1906.06183 [hep-th]

    1906 arXiv

  11. [19]

    H. B. Li, T. T. Hu, B. S. Song, S. Sun and Y. Q. Wang, JHEP 1906, 126 (2019) doi:10.1007/JHEP06(2019)126 [arXiv:1903.11967 [hep-th]]

    2019 arXiv

  12. [20]

    Attempts at vacuum counterexamples to cosmic censorship in AdS,

    T. Crisford, G. T. Horowitz and J. E. Santos, “Attempts at vacuum counterexamples to cosmic censorship in AdS,” JHEP 1902, 092 (2019) doi:10.1007/JHEP02(2019)092 [arXiv:1805.06469 [hep-th]]

    2019 arXiv

  13. [21]

    Deforming black holes in AdS,

    G. T. Horowitz, J. E. Santos and C. Toldo, “Deforming black holes in AdS,” JHEP 1811, 146 (2018) doi:10.1007/JHEP11(2018)146 [arXiv:1809.04081 [hep-th]]

    2018 arXiv

  14. [22]

    New black holes in D = 5 minimal gauged supergravity: Deformed boundaries and frozen horizons,

    J. L. Blzquez-Salcedo, J. Kunz, F. Navarro-Lrida and E. Radu, “New black holes in D = 5 minimal gauged supergravity: Deformed boundaries and frozen horizons,” Phys. Rev. D97, no. 8, 081502 (2018) doi:10.1103/PhysRevD.97.081502 [arXiv:1711.08292 [gr-qc]]

    2018 arXiv

  15. [23]

    A New approach to static numerical relativity, and its application to Kaluza-Klein black holes,

    M. Headrick, S. Kitchen and T. Wiseman, “A New approach to static numerical relativity, and its application to Kaluza-Klein black holes,” Class. Quant. Grav. 27, 035002 (2010) doi:10.1088/0264-9381/27/3/035002 [arXiv:0905.1822 [gr-qc]]

    2010 arXiv

  16. [24]

    Wiseman, Numerical construction of static and stationary black holes, in Black holes in higher dimensions , G

    T. Wiseman, Numerical construction of static and stationary black holes, in Black holes in higher dimensions , G. Horowitz ed., Cambridge University Press, Cambridge U.K. (2012)

    2012

  17. [25]

    Numerical Methods for Finding Stationary Gravitational Solutions,

    O. J. C. Dias, J. E. Santos and B. Way, “Numerical Methods for Finding Stationary Gravitational Solutions,” Class. Quant. Grav. 33, no. 13, 133001 (2016) doi:10.1088/0264-9381/33/13/133001 [arXiv:1510.02804 [hep-th]]

    2016 arXiv

  18. [26]

    Beitrage zur Einsteinischen Gravitationtheorie,

    L. Flamm, “Beitrage zur Einsteinischen Gravitationtheorie,” Physikalische. Zeitschrift 17 (1916) 448-454 (in particular p. 450)

    1916

  19. [27]

    Surface Geometry of Charged Rotating Black Holes,

    L. Smarr, “Surface Geometry of Charged Rotating Black Holes,” Phys. Rev. D 7 (1973) 289

    1973

  20. [28]

    Isometric Embeddings of Riemannian manifolds into Euclidean spaces,

    A. Friedman, “Isometric Embeddings of Riemannian manifolds into Euclidean spaces,” Rev. Mod. Phys. 37 (1965) 201

    1965

  21. [29]

    Embedding various relativistic Riemannian spaces in Pseudo-Euclidean spaces,

    J. Rosen, “Embedding various relativistic Riemannian spaces in Pseudo-Euclidean spaces,” Rev. Mod. Phys. 37 (1965) 204

    1965

  22. [30]

    Local isometric embedding of Riemannian manifolds and Einsteins theory of gravitation

    H. Goenner, “Local isometric embedding of Riemannian manifolds and Einsteins theory of gravitation”, in General Relativity and Gravitation: One hundred years after the birth of Einstein, Edited by A.Held, Vol.1, Plenum Press, 1980

    1980

  23. [31]

    Global embedding of the Kerr black hole event horizon into hyperbolic 3-space,

    G. W. Gibbons, C. A. R. Herdeiro and C. Rebelo, “Global embedding of the Kerr black hole event horizon into hyperbolic 3-space,” Phys. Rev. D 80, 044014 (2009) doi:10.1103/PhysRevD.80.044014 [arXiv:0906.2768 [gr-qc]]. – 17 –

    2009 arXiv

  24. [34]

    Quasinormal modes of AdS black holes and the approach to thermal equilibrium,

    G. T. Horowitz and V. E. Hubeny, “Quasinormal modes of AdS black holes and the approach to thermal equilibrium,” Phys. Rev. D 62, 024027 (2000) doi:10.1103/PhysRevD.62.024027 [hep-th/9909056]. and its application to Kaluza-Klein black holes,” Class.Quant.Grav. 27 (2010) 035002...

    2000 arXiv

  25. [35]

    Boundary Conditions for Kerr-AdS Perturbations,

    O. J. C. Dias and J. E. Santos, “Boundary Conditions for Kerr-AdS Perturbations,” JHEP 1310, 156 (2013) doi:10.1007/JHEP10(2013)156 [arXiv:1302.1580 [hep-th]]

    2013 arXiv

  26. [36]

    Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS,

    V. Cardoso, O. J. C. Dias, G. S. Hartnett, L. Lehner and J. E. Santos, “Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS,” JHEP 1404, 183 (2014) doi:10.1007/JHEP04(2014)183 [arXiv:1312.5323 [hep-th]]

    2014 arXiv

  27. [37]

    Quasinormal modes of black holes and black branes,

    E. Berti, V. Cardoso and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26, 163001 (2009) doi:10.1088/0264-9381/26/16/163001 [arXiv:0905.2975 [gr-qc]]. – 18 –

    2009 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.