pith. sign in

arxiv: 2606.26263 · v1 · pith:AMEZSGKAnew · submitted 2026-06-24 · ✦ hep-th

Charged and rotating near-horizon geometries in five dimensions

Alex Colling , Jun Liu This is my paper

Pith reviewed 2026-06-26 01:15 UTC · model grok-4.3

classification ✦ hep-th
keywords near-horizon geometriesfive-dimensional gravityEinstein-Maxwell theoryextremal horizonsSasakian structureChern-Simons termrotating black holescharged horizons
0
0 comments X

The pith

New charged and rotating near-horizon geometries in five-dimensional Einstein-Maxwell theory are the most general rotating extremal horizons with constant co-rotating electric field.

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

The paper constructs explicit analytic solutions for charged rotating near-horizon geometries in five dimensions, parametrized by charge and two independent angular momenta, and extends them to include a Chern-Simons term. These geometries live on spherical cross-sections that carry a Sasakian structure. The authors use that structure to prove a characterization theorem: the solutions are the most general rotating extremal horizons for which the co-rotating electric field is a non-zero constant, with no additional symmetry assumptions required. The solutions obey the entropy relations expected for charged Myers-Perry and rotating Reissner-Nordström-Tangherlini black holes yet do not reduce to the vacuum Myers-Perry horizon.

Core claim

The presented solutions are the most general rotating extremal horizons in five-dimensional Einstein-Maxwell theory (and its Chern-Simons extension) for which the co-rotating electric field is a non-zero constant. This characterization is obtained by exploiting the Sasakian structure on the spherical horizon cross-sections and holds without any symmetry assumptions. The same construction yields, in higher dimensions, a two-parameter family of such horizons generated by any Sasaki-Einstein manifold.

What carries the argument

The Sasakian structure on the spherical horizon cross-sections, which is used to derive the general form once the co-rotating electric field is required to be constant.

If this is right

  • The solutions satisfy the entropy relations expected for charged extremal Myers-Perry black holes and for rotating extremal Reissner-Nordström-Tangherlini black holes.
  • The construction extends directly to Einstein-Maxwell-Chern-Simons theory with arbitrary coupling constant.
  • Any Sasaki-Einstein manifold in higher dimensions produces a two-parameter family of charged rotating near-horizon geometries.
  • The new geometries remain distinct from the vacuum Myers-Perry horizon even when the charge is taken to zero.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The constant co-rotating electric field condition may serve as a useful selection rule for classifying near-horizon geometries in other dimensions or matter couplings.
  • The Sasakian assumption could be relaxed if other geometric structures on horizon cross-sections allow a similar reduction.
  • The mismatch with the vacuum Myers-Perry limit suggests these solutions may require non-vacuum matter or higher-derivative corrections to be realized as limits of full black-hole spacetimes.

Load-bearing premise

The horizon cross-sections are assumed to be spherical and to carry a Sasakian structure.

What would settle it

An explicit rotating extremal horizon solution with constant non-zero co-rotating electric field whose cross-section is not Sasakian or whose metric and fields fall outside the two-parameter family given in the paper.

Figures

Figures reproduced from arXiv: 2606.26263 by Alex Colling, Jun Liu.

Figure 1
Figure 1. Figure 1: Parameter range of the solutions. Regions in grey are not allowed. The two blue boundaries correspond to the same static U(1) × U(1) solution in (A.1). The red boundary is a new two-parameter family of static solutions. Extending either region I to κ → ∞ or region V to κ → −∞ recovers the same vacuum limit – the two-parameter family described in §4.2. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left Panel: normalised entropy and angular momentum for region III (orange) and region V (blue), with fixed |J1|/|J2| = 3/8 and λ = 0.01. Right Panel: a blow-up of the centre of the left panel [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left Panel: normalised entropy and angular momentum for region I (green) and region II (purple), with fixed |J1|/|J2| = 3/8 and λ = 0.01. The purple curve attains a static limit at zero angular momentum, but the green curve has a turning point at positive |J1|. Right Panel: a blow-up of the turning point of the green curve. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
read the original abstract

We present new charged and rotating near-horizon geometries in five-dimensional Einstein-Maxwell theory in closed analytic form. The solutions can be parametrised by the charge and two independent angular momenta. We also generalise these near-horizon geometries to theories with an additional Chern-Simons term in the action multiplied by an arbitrary coupling constant. The new solutions have the same entropy relations as expected for charged versions of extremal Myers-Perry black holes and for rotating versions of extremal Reissner-Nordstr\"om-Tangherlini black holes, but they do not reduce to the Myers-Perry horizon in the vacuum limit. The horizon cross-sections are spherical and carry a Sasakian structure. We exploit this structure to prove a characterisation of our solutions: without any symmetry assumptions, they are the most general rotating extremal horizons for which the co-rotating electric field is a (non-zero) constant. We further extend this construction to higher dimensions, where we show that any Sasaki-Einstein manifold generates a two-parameter family of charged and rotating horizons.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs explicit charged and rotating near-horizon geometries in five-dimensional Einstein-Maxwell theory (and with Chern-Simons term), parametrized by charge and two independent angular momenta. The horizon cross-sections are spherical and carry a Sasakian structure, which is exploited to prove that these solutions are the most general rotating extremal horizons with constant non-zero co-rotating electric field, without symmetry assumptions. The solutions match expected entropy relations but do not reduce to Myers-Perry in the vacuum limit. The construction is extended to higher dimensions, where any Sasaki-Einstein manifold yields a two-parameter family of such horizons.

Significance. If the characterization holds, the closed analytic forms and the classification result under the Sasakian assumption on spherical cross-sections would provide a concrete addition to the study of extremal horizons in 5D Einstein-Maxwell theory. The explicit parametrization by charge and two angular momenta, the entropy relations, and the higher-dimensional generalization via Sasaki-Einstein manifolds are concrete strengths that facilitate further analysis.

major comments (1)
  1. [Abstract] Abstract (final paragraph): the claim that the solutions are 'the most general rotating extremal horizons for which the co-rotating electric field is a (non-zero) constant' without symmetry assumptions is established by assuming and exploiting the Sasakian structure on spherical horizon cross-sections. This structure is not shown to follow from the Einstein-Maxwell equations plus the constant-E condition, so the characterization applies only inside the Sasakian class.
minor comments (1)
  1. [Abstract] Abstract: the statement that the solutions 'do not reduce to the Myers-Perry horizon in the vacuum limit' is noted but left without further comment; a short remark on the physical or mathematical implications would improve context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the scope of our characterization result. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the claim that the solutions are 'the most general rotating extremal horizons for which the co-rotating electric field is a (non-zero) constant' without symmetry assumptions is established by assuming and exploiting the Sasakian structure on spherical horizon cross-sections. This structure is not shown to follow from the Einstein-Maxwell equations plus the constant-E condition, so the characterization applies only inside the Sasakian class.

    Authors: We agree that the Sasakian structure on the spherical horizon cross-sections is an assumption in our analysis, rather than a property derived from the Einstein-Maxwell equations together with the constant co-rotating electric field condition. The phrase 'without any symmetry assumptions' in the abstract is intended to indicate that we do not impose additional Killing vector fields or other isometries beyond those naturally present in the near-horizon geometry and the Sasakian structure itself. However, to make the claim fully precise, we will revise the abstract (and the corresponding statement in the introduction) to explicitly note that the characterization holds under the assumption of Sasakian structure. This revision clarifies the result without changing any of the technical content or proofs in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under explicit Sasakian assumption

full rationale

The paper states upfront that horizon cross-sections are spherical and carry a Sasakian structure, then exploits this to characterize solutions with constant co-rotating electric field. The construction begins from the Einstein-Maxwell equations (plus optional Chern-Simons term) under these geometric assumptions and yields explicit parametrized solutions. No step reduces a claimed prediction or generality result to a fitted parameter, self-citation chain, or definitional equivalence; the Sasakian structure is an input assumption rather than an output derived from the constant-E condition. The 'most general' claim is therefore scoped to the assumed class and does not collapse by construction. This is the standard honest case of a result that is independent of its own fitted values.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Einstein-Maxwell field equations (standard), the assumption that horizon cross-sections admit a Sasakian structure (domain assumption), and the choice of an ansatz that enforces constant co-rotating electric field (ad_hoc_to_paper). No new entities are postulated.

free parameters (2)
  • charge parameter
    One of the three parameters labeling the family; appears as an input to the analytic solution.
  • two angular momentum parameters
    Independent rotation parameters in the five-dimensional solution.
axioms (2)
  • standard math Einstein-Maxwell equations in five dimensions
    The theory whose solutions are constructed.
  • domain assumption Horizon cross-sections are spherical and Sasakian
    Invoked to prove the characterization without symmetry assumptions.

pith-pipeline@v0.9.1-grok · 5705 in / 1510 out tokens · 23842 ms · 2026-06-26T01:15:25.841924+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

50 extracted references · 35 linked inside Pith

  1. [1]

    R. C. Myers and M. J. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys.172, 304, 1986

    1986

  2. [2]

    F. R. Tangherlini, Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cim.27, 636–651, 1963

    1963

  3. [3]

    J. Kunz, F. Navarro-Lerida and A. K. Petersen, Five-dimensional charged rotating black holes, Phys. Lett. B614, 104–112, 2005, [arXiv:gr-qc/0503010]

    Pith/arXiv arXiv 2005

  4. [4]

    G. T. Horowitz and J. E. Santos, Smooth extremal horizons are the exception, not the rule, JHEP02, 169, 2025, [arXiv:2411.07295 [hep-th]]

    arXiv 2025

  5. [5]

    A. N. Aliev, Rotating black holes in higher dimensional Einstein-Maxwell gravity, Phys. Rev. D74, 024011, 2006, [arXiv:hep-th/0604207]

    Pith/arXiv arXiv 2006

  6. [6]

    Deshpande and O

    R. Deshpande and O. Lunin, Rotating Einstein-Maxwell black holes in odd dimensions, JHEP06, 066, 2025, [arXiv:2411.01795 [hep-th]]

    arXiv 2025

  7. [7]

    Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity, Phys. Rev. Lett.95, 161301, 2005, [arXiv:hep-th/0506029]

    Pith/arXiv arXiv 2005

  8. [8]

    H. K. Kunduri, J. Lucietti and H. S. Reall, Near-horizon symmetries of extremal black holes, Class. Quant. Grav.24, 4169–4190, 2007, [arXiv:0705.4214 [hep-th]]

    Pith/arXiv arXiv 2007

  9. [9]

    Dunajski and J

    M. Dunajski and J. Lucietti, Intrinsic rigidity of extremal horizons, J. Diff. Geom.132, 179–201, 2026, [arXiv:2306.17512 [gr-qc]]

    arXiv 2026

  10. [10]

    Colling, D

    A. Colling, D. Katona and J. Lucietti, Rigidity of the extremal Kerr–Newman horizon, Lett. Math. Phys.115, 19, 2025, [arXiv:2406.07128 [gr-qc]]

    arXiv 2025

  11. [11]

    Colling, Symmetries of extremal horizons, 2025, [arXiv:2512.10200 [gr-qc]]

    A. Colling, Symmetries of extremal horizons, 2025, [arXiv:2512.10200 [gr-qc]]

    arXiv 2025

  12. [12]

    Emparan, Rotating circular strings, and infinite nonuniqueness of black rings, JHEP 03, 064, 2004, [arXiv:hep-th/0402149]

    R. Emparan, Rotating circular strings, and infinite nonuniqueness of black rings, JHEP 03, 064, 2004, [arXiv:hep-th/0402149]

    Pith/arXiv arXiv 2004

  13. [13]

    H. K. Kunduri and J. Lucietti, Constructing near-horizon geometries in supergravities with hidden symmetry, JHEP07, 107, 2011, [arXiv:1104.2260 [hep-th]]

    Pith/arXiv arXiv 2011

  14. [14]

    H. K. Kunduri and J. Lucietti, Classification of near-horizon geometries of extremal black holes, Living Rev. Rel.16, 8, 2013, [arXiv:1306.2517 [hep-th]]

    Pith/arXiv arXiv 2013

  15. [15]

    H. K. Kunduri and J. Lucietti, A Classification of near-horizon geometries of extremal vacuum black holes, J. Math. Phys.50, 082502, 2009, [arXiv:0806.2051 [hep-th]]. 39

    Pith/arXiv arXiv 2009

  16. [16]

    Hollands and A

    S. Hollands and A. Ishibashi, All vacuum near horizon geometries in arbitrary dimensions, Annales Henri Poincare10, 1537–1557, 2010, [arXiv:0909.3462 [gr-qc]]

    Pith/arXiv arXiv 2010

  17. [17]

    Rasheed, The Rotating dyonic black holes of Kaluza-Klein theory, Nucl

    D. Rasheed, The Rotating dyonic black holes of Kaluza-Klein theory, Nucl. Phys. B454, 379–401, 1995, [arXiv:hep-th/9505038]

    Pith/arXiv arXiv 1995

  18. [18]

    H. K. Kunduri and J. Lucietti, Static near-horizon geometries in five dimensions, Class. Quant. Grav.26, 245010, 2009, [arXiv:0907.0410 [hep-th]]

    Pith/arXiv arXiv 2009

  19. [19]

    J. L. Blázquez-Salcedo, J. Kunz and F. Navarro-Lerida, Angular momentum - area - proportionality of extremal charged black holes in odd dimensions, Phys. Lett. B727, 340–344, 2013, [arXiv:1309.2088 [gr-qc]]

    Pith/arXiv arXiv 2013

  20. [20]

    Tomizawa and S

    S. Tomizawa and S. Mizoguchi, General Kaluza-Klein black holes with all six independent charges in five-dimensional minimal supergravity, Phys. Rev. D87, 024027, 2013, [arXiv:1210.6723 [hep-th]]

    Pith/arXiv arXiv 2013

  21. [21]

    H. K. Kunduri and J. Lucietti, Extremal Sasakian horizons, Phys. Lett. B713, 308–312, 2012, [arXiv:1204.5149 [hep-th]]

    Pith/arXiv arXiv 2012

  22. [22]

    Komar, Covariant conservation laws in general relativity, Phys

    A. Komar, Covariant conservation laws in general relativity, Phys. Rev.113, 934–936, 1959

    1959

  23. [23]

    D. N. Page, Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity, Phys. Rev. D28, 2976–2982, 1983

    1983

  24. [24]

    G. J. Galloway and R. Schoen, A Generalization of Hawking’s black hole topology theorem to higher dimensions, Commun. Math. Phys.266, 571–576, 2006, [arXiv:gr-qc/0509107]

    Pith/arXiv arXiv 2006

  25. [25]

    Lucietti, Two remarks on near-horizon geometries, Class

    J. Lucietti, Two remarks on near-horizon geometries, Class. Quant. Grav.29, 235014, 2012, [arXiv:1209.4042 [gr-qc]]

    Pith/arXiv arXiv 2012

  26. [26]

    Hollands and S

    S. Hollands and S. Yazadjiev, A Uniqueness theorem for 5-dimensional Einstein-Maxwell black holes, Class. Quant. Grav.25, 095010, 2008, [arXiv:0711.1722 [gr-qc]]

    Pith/arXiv arXiv 2008

  27. [27]

    H. K. Kunduri and J. Lucietti, The first law of soliton and black hole mechanics in five dimensions, Class. Quant. Grav.31, 032001, 2014, [arXiv:1310.4810 [hep-th]]

    Pith/arXiv arXiv 2014

  28. [28]

    Hanaki, K

    K. Hanaki, K. Ohashi and Y. Tachikawa, Comments on charges and near-horizon data of black rings, JHEP12, 057, 2007, [arXiv:0704.1819 [hep-th]]

    Pith/arXiv arXiv 2007

  29. [29]

    H. S. Reall, Higher dimensional black holes and supersymmetry, Phys. Rev. D68, 024024, 2003, [arXiv:hep-th/0211290]

    Pith/arXiv arXiv 2003

  30. [30]

    Hajian, A

    K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, NHEG Mechanics: Laws of Near Horizon Extremal Geometry (Thermo)Dynamics, JHEP03, 014, 2014, [arXiv:1310.3727 [hep-th]]

    Pith/arXiv arXiv 2014

  31. [31]

    Johnstone, M

    M. Johnstone, M. M. Sheikh-Jabbari, J. Simon and H. Yavartanoo, Extremal black holes and the first law of thermodynamics, Phys. Rev. D88, 101503, 2013, [arXiv:1305.3157 [hep-th]]. 40

    Pith/arXiv arXiv 2013

  32. [32]

    Mizoguchi and N

    S. Mizoguchi and N. Ohta, More on the similarity between D = 5 simple supergravity and M theory, Phys. Lett. B441, 123–132, 1998, [arXiv:hep-th/9807111]

    Pith/arXiv arXiv 1998

  33. [33]

    Foscolo and M

    L. Foscolo and M. Haskins, New G2-holonomy cones and exotic nearly kähler structures on S6 andS 3 ×S 3, Annals of Mathematics185, 59–130, 2017, [arXiv:1501.07838 [math.DG]]

    Pith/arXiv arXiv 2017

  34. [34]

    J. C. Breckenridge, R. C. Myers, A. W. Peet and C. Vafa, D-branes and spinning black holes, Phys. Lett. B391, 93–98, 1997, [arXiv:hep-th/9602065]

    Pith/arXiv arXiv 1997

  35. [35]

    Sparks, Sasaki-Einstein Manifolds, Surveys Diff

    J. Sparks, Sasaki-Einstein Manifolds, Surveys Diff. Geom.16, 265–324, 2011, [arXiv:1004.2461 [math.DG]]

    Pith/arXiv arXiv 2011

  36. [36]

    Geiges, Normal contact structures on3-manifolds, Tohoku Mathematical Journal49, 415 – 422, 1997

    H. Geiges, Normal contact structures on3-manifolds, Tohoku Mathematical Journal49, 415 – 422, 1997

    1997

  37. [37]

    Belgun, On the metric structure of non-kähler complex surfaces, Mathematische Annalen317, 1–40, 2000

    F. Belgun, On the metric structure of non-kähler complex surfaces, Mathematische Annalen317, 1–40, 2000

    2000

  38. [38]

    C. P. Boyer, K. Galicki and P. Matzeu, On eta-Einstein Sasakian geometry, Commun. Math. Phys.262, 177–208, 2006, [arXiv:math/0406627]

    Pith/arXiv arXiv 2006

  39. [39]

    J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S2 ×S 3, Adv. Theor. Math. Phys.8, 711–734, 2004, [arXiv:hep-th/0403002]

    Pith/arXiv arXiv 2004

  40. [40]

    Figueras, H

    P. Figueras, H. K. Kunduri, J. Lucietti and M. Rangamani, Extremal vacuum black holes in higher dimensions, Phys. Rev. D78, 044042, 2008, [arXiv:0803.2998 [hep-th]]

    Pith/arXiv arXiv 2008

  41. [41]

    J. P. Gauntlett, N. Kim and D. Waldram, Supersymmetric AdS3, AdS2 and Bubble Solutions, JHEP04, 005, 2007, [arXiv:hep-th/0612253]

    Pith/arXiv arXiv 2007

  42. [42]

    Kim and J.-D

    N. Kim and J.-D. Park, Comments on AdS(2) solutions of D=11 supergravity, JHEP09, 041, 2006, [arXiv:hep-th/0607093]

    Pith/arXiv arXiv 2006

  43. [43]

    Couzens, J

    C. Couzens, J. P. Gauntlett, D. Martelli and J. Sparks, A geometric dual of c-extremization, JHEP01, 212, 2019, [arXiv:1810.11026 [hep-th]]

    Pith/arXiv arXiv 2019

  44. [44]

    J. P. Gauntlett, D. Martelli and J. Sparks, Fibred GK geometry and supersymmetricAdS solutions, JHEP11, 176, 2019, [arXiv:1910.08078 [hep-th]]

    arXiv 2019

  45. [45]

    Mizoguchi and S

    S. Mizoguchi and S. Tomizawa, New approach to solution generation using SL(2,R)-duality of a dimensionally reduced space in five-dimensional minimal supergravity and new black holes, Phys. Rev. D84, 104009, 2011, [arXiv:1106.3165 [hep-th]]

    Pith/arXiv arXiv 2011

  46. [46]

    G. T. Horowitz, M. Kolanowski and J. E. Santos, Almost all extremal black holes in AdS are singular, JHEP01, 162, 2023, [arXiv:2210.02473 [hep-th]]

    arXiv 2023

  47. [47]

    G. T. Horowitz, M. Kolanowski, G. N. Remmen and J. E. Santos, Extremal Kerr Black Holes as Amplifiers of New Physics, Phys. Rev. Lett.131, 091402, 2023, [arXiv:2303.07358 [hep-th]]. 41

    arXiv 2023

  48. [48]

    G. T. Horowitz, M. Kolanowski, G. N. Remmen and J. E. Santos, Sudden breakdown of effective field theory near cool Kerr-Newman black holes, JHEP05, 122, 2024, [arXiv:2403.00051 [hep-th]]

    arXiv 2024

  49. [49]

    Li and J

    C. Li and J. Lucietti, Transverse deformations of extreme horizons, Class. Quant. Grav. 33, 075015, 2016, [arXiv:1509.03469 [gr-qc]]

    Pith/arXiv arXiv 2016

  50. [50]

    Dunajski, J

    M. Dunajski, J. Gutowski and W. Sabra, Einstein–Weyl spaces and near-horizon geometry, Class. Quant. Grav.34, 045009, 2017, [arXiv:1610.08953 [hep-th]]. 42

    Pith/arXiv arXiv 2017