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Four-charge static non-extremal black holes in the five-dimensional $\mathcal{N}=2$, $STU-W^2U$ supergravity

Di Wu, Shuang-Qing Wu

TL;DR

The paper extends the well-known STU construction in five-dimensional $\mathcal{N}=2$ supergravity by introducing a fourth vector multiplet through the STU-W^2U pre-potential, enabling static non-extremal black holes with four independent electric charges. An explicit ansatz and scalar parametrization yield a complete solution set, including special (BPS and equal-charge) cases and a general four-charge configuration, with mass, entropy, temperature, charges, and potentials satisfying the differential first law and Smarr relation. The authors further generalize to squashed-horizon geometries and to AdS$_5$ via a gauged extension, highlighting robustness and applicability to broader contexts, including holographic setups. These results broaden the landscape of exact higher-dimensional black holes in ungauged and gauged supergravity, offering new platforms for studying black hole thermodynamics and potential dual field theories.

Abstract

We construct, for the first time, new static non-extremal five-dimensional black hole solutions (without or with squashed horizons) endowing with four different electric charge parameters in the $D = 5$, $\mathcal{N} = 2$ supergravity coupled to three vector multiplets with a specific pre-potential $\mathcal{V} = STU -W^2U \equiv 1$. When the fourth charge parameter disappears, the solution simplify reduces to the three-charge static black hole solution previously presented in ref. [1], which belongs to the solution to the $D = 5$, $\mathcal{N} = 2$ supergravity coupled to two vector multiplets (also notably known as the $STU$ model). We parameterize the model in such a simple fashion that not only can one easily recover the static three-charge solution but also it is very convenient to study their thermodynamical properties of the obtained black hole solutions in the case without a squashing horizon. We then show that the thermodynamical quantities perfectly obey both the differential first law and integral Smarr formula of thermodynamics. Finally, we also extend to present its generalizations with squashed horizons or including a nonzero cosmological constant.

Four-charge static non-extremal black holes in the five-dimensional $\mathcal{N}=2$, $STU-W^2U$ supergravity

TL;DR

The paper extends the well-known STU construction in five-dimensional supergravity by introducing a fourth vector multiplet through the STU-W^2U pre-potential, enabling static non-extremal black holes with four independent electric charges. An explicit ansatz and scalar parametrization yield a complete solution set, including special (BPS and equal-charge) cases and a general four-charge configuration, with mass, entropy, temperature, charges, and potentials satisfying the differential first law and Smarr relation. The authors further generalize to squashed-horizon geometries and to AdS via a gauged extension, highlighting robustness and applicability to broader contexts, including holographic setups. These results broaden the landscape of exact higher-dimensional black holes in ungauged and gauged supergravity, offering new platforms for studying black hole thermodynamics and potential dual field theories.

Abstract

We construct, for the first time, new static non-extremal five-dimensional black hole solutions (without or with squashed horizons) endowing with four different electric charge parameters in the , supergravity coupled to three vector multiplets with a specific pre-potential . When the fourth charge parameter disappears, the solution simplify reduces to the three-charge static black hole solution previously presented in ref. [1], which belongs to the solution to the , supergravity coupled to two vector multiplets (also notably known as the model). We parameterize the model in such a simple fashion that not only can one easily recover the static three-charge solution but also it is very convenient to study their thermodynamical properties of the obtained black hole solutions in the case without a squashing horizon. We then show that the thermodynamical quantities perfectly obey both the differential first law and integral Smarr formula of thermodynamics. Finally, we also extend to present its generalizations with squashed horizons or including a nonzero cosmological constant.
Paper Structure (25 sections, 47 equations)

This paper contains 25 sections, 47 equations.

Table of Contents

  1. Introduction
  2. $D = 5$, $\mathcal{N} = 2$ ungauged supergravity and $STU$ model
  3. Basic framework
  4. The $STU$ model
  5. Static non-extremal $STU$ black hole
  6. Four-charge static non-extremal black hole within the $STU-W^2U$ model
  7. The $STU-W^2U$ model: Motivation and structure
  8. Lagrangian and field equations
  9. Scalar field parametrization:
  10. Bosonic Lagrangian:
  11. Dual field strengths:
  12. Gauge field equations:
  13. Einstein equations:
  14. Scalar field equations:
  15. Static non-extremal $STU-W^2U$ black hole
  16. ...and 10 more sections