pith. sign in

arxiv: 2601.14315 · v2 · pith:W4NKOOT6new · submitted 2026-01-19 · ✦ hep-th · gr-qc

Static four-charge squashed black hole in five-dimensional STU-W²U supergravity and its thermodynamics

Di Wu , Shuang-Qing Wu This is my paper

Pith reviewed 2026-05-21 16:34 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords five-dimensional black holessupergravitysquashed horizonsKaluza-Kleinelectric chargesblack hole thermodynamicsasymptotically locally flat spacetimes
0
0 comments X

The pith

A simple exact solution describes a static four-charge squashed black hole in five-dimensional STU-W²U supergravity.

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

This paper gives an exact solution to the equations of D=5, N=2 supergravity coupled to three vector multiplets whose prepotential is fixed to V=STU−W²U≡1. The solution is a static Kaluza-Klein black hole whose horizon is a squashed three-sphere and which carries four independent electric charges. The geometry is asymptotically locally flat, with spatial infinity of topology R×S¹ that embeds into an S². Conserved charges are obtained by the counterterm method, and the thermodynamic quantities are shown to obey both the first law and the Bekenstein-Smarr relation once the length of the compact extra dimension is promoted to a thermodynamic variable.

Core claim

We present a remarkably simple expression for the exact solution to the D=5, N=2 supergravity coupled to three vector multiplets with the prepotential V=STU−W²U≡1, which represents a five-dimensional static Kaluza-Klein black hole with squashed S³ horizons and four independent electric charges. It is asymptotically locally flat and has a spatial infinity R×S¹↪S². We compute its conserved charges via the counterterm method and demonstrate that the thermodynamic quantities satisfy both the first law and Bekenstein-Smarr mass formula, provided the length of the compact extra dimension is treated as a thermodynamic variable.

What carries the argument

The exact metric and gauge-field ansatz constructed from the supergravity action with the fixed prepotential V=STU−W²U=1, which simultaneously supports four independent electric charges while preserving the squashed S³ horizon geometry.

If this is right

  • The first law of black-hole mechanics holds for this four-charge configuration in an asymptotically locally flat spacetime.
  • The Bekenstein-Smarr relation is recovered once the compact-circle length is included as a thermodynamic variable.
  • The construction supplies a new example of a Kaluza-Klein black hole with independent charges on a squashed horizon.
  • Earlier solutions carrying fewer charges are recovered as special cases of the same ansatz.

Where Pith is reading between the lines

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

  • The same reduction technique may generate analogous solutions in other five-dimensional gauged supergravities.
  • The role of the compact dimension length as a thermodynamic variable could be tested in other Kaluza-Klein black-hole families.
  • Phase-transition studies of multi-charge black holes in this background become feasible with the explicit solution in hand.

Load-bearing premise

The counterterm method correctly yields the conserved charges and treating the length of the compact extra dimension as a thermodynamic variable is enough to make the first law and Bekenstein-Smarr formula hold.

What would settle it

An explicit numerical evaluation of the horizon area, Hawking temperature, electric potentials, and mass that shows the first law fails when the extra-dimension length is omitted from the thermodynamic relation.

read the original abstract

In this paper, we present a remarkably simple expression for the exact solution to the $D = 5$, $\mathcal{N} = 2$ supergravity coupled to three vector multiplets with the prepotential $\mathcal{V} = STU -W^2U \equiv 1$, which represents a five-dimensional static Kaluza-Klein black hole with squashed $S^3$ horizons and four independent electric charges. It is asymptotically locally flat and has a spatial infinity $R \times S^1 \hookrightarrow S^2$. We compute its conserved charges via the counterterm method and demonstrate that the thermodynamic quantities satisfy both the first law and Bekenstein-Smarr mass formula, provided the length of the compact extra dimension is treated as a thermodynamic variable.

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

2 major / 1 minor

Summary. The paper presents a remarkably simple exact solution for a static four-charge Kaluza-Klein black hole with squashed S³ horizons in five-dimensional N=2 supergravity coupled to three vector multiplets with prepotential V = STU - W²U ≡ 1. The spacetime is asymptotically locally flat with spatial infinity R × S¹ ↪ S². Conserved charges are computed using the counterterm method, and the thermodynamic quantities are shown to satisfy the first law and Bekenstein-Smarr formula when the length L of the compact extra dimension is promoted to a thermodynamic variable.

Significance. If verified, the result would add a new explicit multi-charge example to the limited set of exact solutions in five-dimensional STU-like supergravity with squashed horizons and non-standard asymptotics. The claimed simplicity of the solution expression and the extension of thermodynamic identities to include the compact-circle length as a variable are potential strengths for studies of black-hole thermodynamics in asymptotically locally flat settings.

major comments (2)
  1. [Solution presentation] The central claim that the given metric and gauge-field expressions constitute an exact solution requires explicit substitution into the supergravity field equations derived from the prepotential V = STU - W²U = 1; no such verification or derivation steps are supplied.
  2. [Thermodynamic identities] The assertion that the first law and Bekenstein-Smarr relation hold once L is treated as a thermodynamic variable rests on the counterterm charges producing exact cancellation of all boundary variations; the manuscript does not display the explicit on-shell variation or boundary-term analysis for the four-charge family.
minor comments (1)
  1. [Introduction] The notation 'V = STU - W²U ≡ 1' would benefit from a brief clarification of the normalization convention and the role of the auxiliary field W.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and have prepared revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Solution presentation] The central claim that the given metric and gauge-field expressions constitute an exact solution requires explicit substitution into the supergravity field equations derived from the prepotential V = STU - W²U = 1; no such verification or derivation steps are supplied.

    Authors: We acknowledge that the manuscript presents the solution without including the explicit substitution steps into the field equations. The expressions were obtained by solving the equations of motion derived from the prepotential V = STU - W²U ≡ 1, but for improved verifiability we will add an appendix in the revised manuscript that outlines the key substitution steps and confirms that the metric and gauge fields satisfy the supergravity equations identically. revision: yes

  2. Referee: [Thermodynamic identities] The assertion that the first law and Bekenstein-Smarr relation hold once L is treated as a thermodynamic variable rests on the counterterm charges producing exact cancellation of all boundary variations; the manuscript does not display the explicit on-shell variation or boundary-term analysis for the four-charge family.

    Authors: We agree that an explicit display of the on-shell variation and boundary-term analysis would clarify the thermodynamic results. The counterterm method yields charges whose variations cancel to produce the first law and Bekenstein-Smarr formula with L as a thermodynamic variable. In the revision we will include a dedicated subsection presenting the boundary variation computation for the general four-charge case, showing the precise cancellation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs an exact solution to the five-dimensional supergravity equations with the given prepotential for a static four-charge squashed Kaluza-Klein black hole, then applies the standard counterterm method to obtain conserved charges and verifies the first law and Bekenstein-Smarr relation after promoting the compact-circle length to a thermodynamic variable. No step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise collapses to a self-citation chain. The central claims rest on direct solution of the field equations and explicit (if non-standard) thermodynamic accounting, remaining independent of prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the paper relies on the standard N=2 supergravity framework in five dimensions and the given prepotential normalization. No explicit free parameters, ad-hoc axioms, or new invented entities are stated in the available text.

pith-pipeline@v0.9.0 · 5667 in / 1338 out tokens · 55017 ms · 2026-05-21T16:34:58.970411+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    We present a remarkably simple expression for the exact solution to the D=5, N=2 supergravity coupled to three vector multiplets with the prepotential V=STU−W²U≡1, which represents a five-dimensional static Kaluza-Klein black hole with squashed S³ horizons and four independent electric charges. It is asymptotically locally flat and has a spatial infinity R×S¹↪S².

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We compute its conserved charges via the counterterm method and demonstrate that the thermodynamic quantities satisfy both the first law and Bekenstein-Smarr mass formula, provided the length of the compact extra dimension is treated as a thermodynamic variable.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · 1 internal anchor

  1. [1]

    Then, we verify the Einstein equation and find that all their components vanish after we let:L 2 ∞ =ρ 2 0 f(−ρ 0) =ρ 2 0 + ρ0ρ1 +ρ 2

    +q2(q2 1 −p 2 1) q1 −q 2 . Then, we verify the Einstein equation and find that all their components vanish after we let:L 2 ∞ =ρ 2 0 f(−ρ 0) =ρ 2 0 + ρ0ρ1 +ρ 2. At this step, all equations of motion are completely satisfied. As our final step, we shall re-parameterize the integral con- stants to relate them to the solution parameters (ρ 1 =2mand qI). By s...

    work page

  2. [2]

    Tomizawa and H

    S. Tomizawa and H. Ishihara, Chapter 2. Exact solutions of higher dimensional black holes, Prog. Theor. Phys. Supp.189, 7 (2011)

    work page 2011

  3. [3]

    Ishihara and K

    H. Ishihara and K. Matsuno, Kaluza-Klein black holes with squashed horizons, Prog. Theor. Phys.116, 417 (2006)

    work page 2006

  4. [4]

    Wang, A rotating Kaluza-Klein black hole with squashed horizons, Nucl

    T. Wang, A rotating Kaluza-Klein black hole with squashed horizons, Nucl. Phys. B756, 86 (2006)

    work page 2006

  5. [5]

    Yazadjiev, Dilaton black holes with squashed horizons and their thermodynamics, Phys

    S.S. Yazadjiev, Dilaton black holes with squashed horizons and their thermodynamics, Phys. Rev. D74, 024022 (2006)

    work page 2006

  6. [6]

    Nedkova and S.S

    P.G. Nedkova and S.S. Yazadjiev, Magnetized black hole on Taub-Nut instanton, Phys. Rev. D85, 064021 (2012)

    work page 2012

  7. [7]

    Nedkova and S.S

    P.G. Nedkova and S.S. Yazadjiev, New magnetized squashed black holes-Thermodynamics and Hawking radiation, Eur. Phys. J. C73, 2377 (2013)

    work page 2013

  8. [8]

    Knoll and P.G

    C. Knoll and P.G. Nedkova, Charged rotating dilaton black holes with Kaluza-Klein asymptotics, Phys. Rev. D93, 064052 (2016)

    work page 2016

  9. [9]

    Tomizawa and A

    S. Tomizawa and A. Ishibashi, Charged black holes in a rotating Gross-Perry-Sorkin monopole background, Classical Quantum Gravity25, 245007 (2008)

    work page 2008

  10. [10]

    Matsuno, H

    K. Matsuno, H. Ishihara, T. Nakagawa, and S. Tomizawa, Ro- tating Kaluza-Klein multi-black holes with G ¨odel parameter, Phys. Rev. D78, 064016 (2008)

    work page 2008

  11. [11]

    Stelea, K

    C. Stelea, K. Schleich, and D. Witt, Squashed black holes in G¨odel universes, Phys. Rev. D78, 124006 (2008)

    work page 2008

  12. [12]

    Tomizawa, H

    S. Tomizawa, H. Ishihara, K. Matsuno, and T. Nakagawa, Squashed Kerr-G ¨odel black holes: Kaluza-Klein black holes with rotations of black hole and universe, Prog. Theor. Phys. 121, 823 (2009)

    work page 2009

  13. [13]

    Nakagawa, H

    T. Nakagawa, H. Ishihara, K. Matsuno, and S. Tomizawa, Charged rotating Kaluza-Klein black holes in five dimensions, Phys. Rev. D77, 044040 (2008)

    work page 2008

  14. [14]

    Compactified black holes in five-dimensional U(1)**3 ungauged supergravity

    S. Tomizawa, Compactified black holes in five-dimensional U(1)3 ungauged supergravity, arXiv:1009.3568

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    Cai, L.-M

    R.-G. Cai, L.-M. Cao, and N. Ohta, Mass and thermodynamics of Kaluza-Klein black holes with squashed horizons, Phys. Lett. B639, 354 (2006)

    work page 2006

  16. [16]

    Kurita and H

    Y . Kurita and H. Ishihara, Mass and free energy in thermody- namics of squashed Kaluza-Klein black holes, Classical Quan- tum Gravity24, 4525 (2007)

    work page 2007

  17. [17]

    Kurita and H

    Y . Kurita and H. Ishihara, Thermodynamics of squashed Kaluza-Klein black holes and black strings–A comparison of reference backgrounds, Classical Quantum Gravity25, 085006 (2008)

    work page 2008

  18. [18]

    Saito, H

    M. Saito, H. Noda, and T. Tashiro, Kaluza-Klein black holes with squashed horizons and Hawking-Horowitz mass, Int. J. Mod. Phys. A24, 2357 (2009)

    work page 2009

  19. [19]

    Nedkova and S.S

    P.G. Nedkova and S.S. Yazadjiev, On the thermodynamics of 5d black holes on ALF gravitational instantons, Phys. Rev. D84, 124040 (2011)

    work page 2011

  20. [20]

    S.-Q. Wu, D. Wen, Q.-Q. Jiang, and S.-Z. Yang, Thermody- namics of five-dimensional static three-charge STU black holes with squashed horizons, Phys. Lett. B726, 404 (2013)

    work page 2013

  21. [21]

    X.-D. Zhu, D. Wu, S.-Q. Wu, and S.-Z. Yang, New forms and thermodynamics of the neutral rotating squashed black hole in five-dimensional vacuum Einstein gravity theory, Gen. Relat. Grav.48, 154 (2016)

    work page 2016

  22. [22]

    Peng, Revisiting the ADT mass of the five-dimensional ro- tating black holes with squashed horizons, Eur

    J.-J. Peng, Revisiting the ADT mass of the five-dimensional ro- tating black holes with squashed horizons, Eur. Phys. J. C77, 706 (2017)

    work page 2017

  23. [23]

    Peng and S.Q

    J.J. Peng and S.Q. Wu, Extremal Kerr/CFT correspondence of five-dimensional rotating (charged) black holes with squashed horizons, Nucl. Phys. B828, 273 (2010)

    work page 2010

  24. [24]

    Ishihara and J

    H. Ishihara and J. Soda, Hawking radiation from squashed Kaluza-Klein black holes: A window to extra dimensions, Phys. Rev. D76, 064022 (2007)

    work page 2007

  25. [25]

    S.-B. Chen, B. Wang, and R.-K. Su, Hawking radiation in a rotating Kaluza-Klein black hole with squashed horizons, Phys. 6 Rev. D77, 024039 (2008)

    work page 2008

  26. [26]

    S.-W. Wei, R. Li, Y .-X. Liu, and J.-R. Ren, Anomaly analysis of Hawking radiation from Kaluza-Klein black hole with squashed horizon, Eur. Phys. J. C65, 281 (2010)

    work page 2010

  27. [27]

    Matsuno and K

    K. Matsuno and K. Umetsu, Hawking radiation as tunnel- ing from squashed Kaluza-Klein black hole, Phys. Rev. D83, 064016 (2011)

    work page 2011

  28. [28]

    X. He, B. Wang, S.-B. Chen, R.-G. Cai, and C.-Y . Lin, Quasi- normal modes in the background of charged Kaluza-Klein black hole with squashed horizons, Phys. Lett. B665, 392 (2008)

    work page 2008

  29. [29]

    X. He, B. Wang, and S.-B. Chen, Quasinormal modes of charged squashed Kaluza-Klein black holes in the G ¨odel Uni- verse, Phys. Rev. D79, 084005 (2009)

    work page 2009

  30. [30]

    Liu, S.-B

    Y . Liu, S.-B. Chen, and J.-L. Jing, Strong gravitational lensing in a squashed Kaluza-Klein black hole spacetime, Phys. Rev. D81, 124017 (2010)

    work page 2010

  31. [31]

    S.-B. Chen, Y . Liu, and J.-L. Jing, Strong gravitational lensing in a squashed Kaluza-Klein G¨odel black hole, Phys. Rev. D83, 124019 (2011)

    work page 2011

  32. [32]

    Ji, S.-B

    L.-Y . Ji, S.-B. Chen, and J.-L. Jing, Strong gravitational lensing in a rotating Kaluza-Klein black hole with squashed horizons, J. High Energy Phys.03(2014) 089

    work page 2014

  33. [33]

    Sadeghi, A

    J. Sadeghi, A. Banijamali, and H. Vaez, Strong gravitational lensing in a charged squashed Kaluza-Klein black hole, Astro- phys. Space Sci.343, 559 (2013)

    work page 2013

  34. [34]

    Sadeghi, J

    J. Sadeghi, J. Naji, and H. Vaez, Strong gravitational lensing in a charged squashed Kaluza-Klein G¨odel black hole, Phys. Lett. B728, 170 (2014)

    work page 2014

  35. [35]

    Kimura, K

    M. Kimura, K. Murata, H. Ishihara, and J. Soda, On the stability of squashed Kaluza-Klein black holes, Phys. Rev. D77, 064015 (2008)

    work page 2008

  36. [36]

    Ishihara, M

    H. Ishihara, M. Kimura, R.A. Konoplya, K. Murata, J. Soda, and A. Zhidenko, Evolution of perturbations of squashed Kaluza-Klein black holes: escape from instability, Phys. Rev. D77, 084019 (2008)

    work page 2008

  37. [37]

    Nishikawa and M

    R. Nishikawa and M. Kimura, Stability analysis of squashed Kaluza-Klein black holes with charge, Classical Quantum Gravity27, 215020 (2010)

    work page 2010

  38. [38]

    Matsuno and H

    K. Matsuno and H. Ishihara, Geodetic precession in squashed Kaluza-Klein black hole spacetimes, Phys. Rev. D80, 104037 (2009)

    work page 2009

  39. [39]

    Long, J.-C

    F. Long, J.-C. Wang, S.-B. Chen, and J.-L. Jing, Shadow of a rotating squashed Kaluza-Klein black hole, J. High Energy Phys.10(2019) 269

    work page 2019

  40. [40]

    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 (1995)

    work page 1995

  41. [41]

    Larsen, Rotating Kaluza-Klein black holes, Nucl

    F. Larsen, Rotating Kaluza-Klein black holes, Nucl. Phys. B 575, 211 (2000)

    work page 2000

  42. [42]

    Gal’tsov and N.G

    D.V . Gal’tsov and N.G. Scherbluk, Improved generating tech- nique for D=5 supergravities and squashed Kaluza-Klein black holes, Phys. Rev. D79, 064020 (2009)

    work page 2009

  43. [43]

    Comp ´ere, S

    G. Comp ´ere, S. de Buyl, E. Jamsin, and A. Virmani, G 2 dual- ities in D=5 supergravity and black strings, Classical Quantum Gravity26, 125016 (2009)

    work page 2009

  44. [44]

    Comp ´ere, S

    G. Comp ´ere, S. de Buyl, S. Stotyn, and A. Virmani, A gen- eral black string and its microscopics, J. High Energy Phys.11 (2010) 133

    work page 2010

  45. [45]

    Tomizawa, Y

    S. Tomizawa, Y . Yasui, and Y . Morisawa, Charged rotating Kaluza-Klein black holes generated by G 2(2) transformation, Classical Quantum Gravity26, 145006 (2009)

    work page 2009

  46. [46]

    Mizoguchi and S

    S. Mizoguchi and S. Tomizawa, New approach to solution gen- eration usingSL(2,R)duality of a dimensionally reduced space in five-dimensional minimal supergravity and new black holes, Phys. Rev. D84, 024009 (2011)

    work page 2011

  47. [47]

    Mizoguchi and S

    S. Mizoguchi and S. Tomizawa, FlippedSL(2,R)duality in five-dimensional supergravity, Phys. Rev. D86, 024022 (2012)

    work page 2012

  48. [48]

    Tomizawa and S

    S. Tomizawa and S. Mizoguchi, General Kaluza-Klein black holes with all six independent charges in five-dimensional min- imal supergravity, Phys. Rev. D87, 024027 (2013)

    work page 2013

  49. [49]

    Gibbons, R

    G. Gibbons, R. Kallosh, and B. Kol, Moduli, Scalar Charges, and the First Law of Black Hole Thermodynamics, Phys. Rev. Lett.77, 4992 (1996)

    work page 1996

  50. [50]

    Wu and S.-Q

    D. Wu and S.-Q. Wu, Four-charge static non-extremal black holes in the five-dimensionalN=2,STU−W 2Usupergravity, to appear inJHEP, arXiv:2510.13655

    work page arXiv

  51. [51]

    Wu and S.-Q

    D. Wu and S.-Q. Wu, Thermodynamics and topological classi- fications of static non-extremal four-charge AdS black hole in the five-dimensionalN=2,STU−W 2Ugauged supergravity, arXiv:2510.20164

    work page arXiv

  52. [52]

    Mann and C

    R.B. Mann and C. Stelea, On the gravitational energy of the Kaluza Klein monopole, Phys. Lett. B634, 531 (2006)

    work page 2006

  53. [53]

    Astefanesei and E

    D. Astefanesei and E. Radu, Quasilocal formalism and black ring thermodynamics, Phys. Rev. D73, 044014 (2006)

    work page 2006