# Embedding Gauss–Bonnet Scalarization Models in Higher Dimensional Topological Theories

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. HCS Gravity

_{a}are relics of the Higgs scalar (with A

^{2}= A

_{μ}A

^{μ}The density (13) can be cast in a more useful form by dropping a total derivative term, which results in the equivalent expression (Note that ${\epsilon}^{\mu \nu \rho \sigma}{\epsilon}_{abcd}{R}_{\mu \nu}^{ab}{R}_{\rho \sigma}^{cd}=4{L}_{GB}$).

## 3. The BH Solutions

#### 3.1. Einstein-GB-Scalar Field BHs

#### 3.1.1. Generic Solutions

#### 3.1.2. Scalarized BHs

#### 3.2. The Scalar-Vector Model: Perturbative Solutions

## 4. Further Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Herdeiro, C.A.R.; Radu, E. Asymptotically flat black holes with scalar hair: A review. Int. J. Mod. Phys. D
**2015**, 24, 1542014. [Google Scholar] [CrossRef][Green Version] - Astefanesei, D.; Herdeiro, C.; Oliveira, J.; Radu, E. Higher dimensional black hole scalarization. J. High Energy Phys.
**2020**, 9, 186. [Google Scholar] [CrossRef] - Blázquez-Salcedo, J.L.; Herdeiro, C.A.R.; Kunz, J.; Pombo, A.M.; Radu, E. Einstein-Maxwell-scalar black holes: The hot, the cold and the bald. Phys. Lett. B
**2020**, 806, 135493. [Google Scholar] [CrossRef] - Silva, H.O.; Sakstein, J.; Gualtieri, L.; Sotiriou, T.P.; Berti, E. Spontaneous scalarization of black holes and compact stars from a Gauss–Bonnet coupling. Phys. Rev. Lett.
**2018**, 120, 131104. [Google Scholar] [CrossRef] [PubMed][Green Version] - Doneva, D.D.; Yazadjiev, S.S. New Gauss–Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories. Phys. Rev. Lett.
**2018**, 120, 131103. [Google Scholar] [CrossRef][Green Version] - Antoniou, G.; Bakopoulos, A.; Kanti, P. Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss–Bonnet Theories. Phys. Rev. Lett.
**2018**, 120, 131102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Antoniou, G.; Bakopoulos, A.; Kanti, P. Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss–Bonnet Theories. Phys. Rev. D
**2018**, 97, 084037. [Google Scholar] [CrossRef][Green Version] - Minamitsuji, M.; Ikeda, T. Scalarized black holes in the presence of the coupling to Gauss–Bonnet gravity. Phys. Rev. D
**2019**, 99, 044017. [Google Scholar] [CrossRef][Green Version] - Brihaye, Y.; Ducobu, L. Hairy black holes, boson stars and non-minimal coupling to curvature invariants. Phys. Rev. B
**2019**, 795, 135–143. [Google Scholar] [CrossRef] - Macedo, C.F.B.; Sakstein, J.; Berti, E.; Gualtieri, L.; Silva, H.O.; Sotiriou, T.P. Self-interactions and Spontaneous Black Hole Scalarization. Phys. Rev. D
**2019**, 99, 104041. [Google Scholar] [CrossRef][Green Version] - Doneva, D.D.; Staykov, K.V.; Yazadjiev, S.S. Gauss–Bonnet black holes with a massive scalar field. Phys. Rev. D
**2019**, 99, 104045. [Google Scholar] [CrossRef][Green Version] - Andreou, N.; Franchini, N.; Ventagli, G.; Sotiriou, T.P. Spontaneous scalarization in generalised scalar-tensor theory. Phys. Rev. D
**2019**, 99, 124022, Erratum in: Phys. Rev. D**2020**, 101, 109903. [Google Scholar] [CrossRef][Green Version] - Minamitsuji, M.; Ikeda, T. Spontaneous scalarization of black holes in the Horndeski theory. Phys. Rev. D
**2019**, 99, 104069. [Google Scholar] [CrossRef][Green Version] - Blázquez-Salcedo, J.L.; Kahlen, S.; Kunz, J. Critical solutions of scalarized black holes. Symmetry
**2020**, 12, 2057. [Google Scholar] [CrossRef] - Guo, H.; Kuang, X.M.; Papantonopoulos, E.; Wang, B. Topology and spacetime structure influences on black hole scalarization. arXiv
**2020**, arXiv:2012.11844. [Google Scholar] - Bakopoulos, A.; Kanti, P.; Pappas, N. Large and ultracompact Gauss–Bonnet black holes with a self-interacting scalar field. Phys. Rev. D
**2020**, 101, 084059. [Google Scholar] [CrossRef] - Peng, Y. Spontaneous scalarization of Gauss–Bonnet black holes surrounded by massive scalar fields. Phys. Rev. D
**2020**, 807, 135569. [Google Scholar] [CrossRef] - Liu, H.S.; Lu, H.; Tang, Z.Y.; Wang, B. Black Hole Scalarization in Gauss–Bonnet Extended Starobinsky Gravity. arXiv
**2020**, arXiv:2004.14395. [Google Scholar] - Cardoso, V.; Foschi, A.; Zilhao, M. Collective scalarization or tachyonization: When averaging fails. Phys. Rev. Lett.
**2020**, 124, 221104. [Google Scholar] [CrossRef] [PubMed] - Ventagli, G.; Lehébel, A.; Sotiriou, T.P. Onset of spontaneous scalarization in generalized scalar-tensor theories. Phys. Rev. D
**2020**, 102, 024050. [Google Scholar] [CrossRef] - Guo, H.; Kiorpelidi, S.; Kuang, X.M.; Papantonopoulos, E.; Wang, B.; Wu, J.P. Spontaneous holographic scalarization of black holes in Einstein-scalar-Gauss–Bonnet theories. Phys. Rev. D
**2020**, 102, 084029. [Google Scholar] [CrossRef] - Doneva, D.D.; Staykov, K.V.; Yazadjiev, S.S.; Zheleva, R.Z. Multiscalar Gauss–Bonnet gravity: Hairy black holes and scalarization. Phys. Rev. D
**2020**, 102, 064042. [Google Scholar] [CrossRef] - Heydari-Fard, M.; Sepangi, H.R. Thin accretion disk signatures of scalarized black holes in Einstein-scalar-Gauss–Bonnet gravity. arXiv
**2020**, arXiv:2009.13748. [Google Scholar] - Bakopoulos, A. Black holes and wormholes in the Einstein-scalar-Gauss–Bonnet generalized theories of gravity. arXiv
**2020**, arXiv:2010.13189. [Google Scholar] - Blázquez-Salcedo, J.L.; Doneva, D.D.; Kunz, J.; Yazadjiev, S.S. Radial perturbations of the scalarized Einstein-Gauss–Bonnet black holes. Phys. Rev. D
**2008**, 98, 084011. [Google Scholar] [CrossRef][Green Version] - Silva, H.O.; Macedo, C.F.B.; Sotiriou, T.P.; Gualtieri, L.; Sakstein, J.; Berti, E. Stability of scalarized black hole solutions in scalar-Gauss–Bonnet gravity. Phys. Rev. D
**2019**, 99, 064011. [Google Scholar] [CrossRef][Green Version] - Blázquez-Salcedo, J.L.; Doneva, D.D.; Kahlen, S.; Kunz, J.; Nedkova, P.; Yazadjiev, S.S. Axial perturbations of the scalarized Einstein-Gauss–Bonnet black holes. Phys. Rev. D
**2020**, 101, 104006. [Google Scholar] [CrossRef] - Blázquez-Salcedo, J.L.; Doneva, D.D.; Kahlen, S.; Kunz, J.; Nedkova, P.; Yazadjiev, S.S. Polar quasinormal modes of the scalarized Einstein-Gauss–Bonnet black holes. Phys. Rev. D
**2020**, 102, 024086. [Google Scholar] [CrossRef] - Hod, S. Spontaneous scalarization of Gauss–Bonnet black holes: Analytic treatment in the linearized regime. Phys. Rev. D
**2019**, 100, 064039. [Google Scholar] [CrossRef][Green Version] - Hod, S. Gauss–Bonnet black holes supporting massive scalar field configurations: The large-mass regime. Eur. Phys. J. C
**2019**, 79, 966. [Google Scholar] [CrossRef] - Konoplya, R.A.; Zhidenko, A. Analytical representation for metrics of scalarized Einstein-Maxwell black holes and their shadows. Phys. Rev. D
**2019**, 100, 044015. [Google Scholar] [CrossRef][Green Version] - Hod, S. Onset of spontaneous scalarization in spinning Gauss–Bonnet black holes. Phys. Rev. D
**2020**, 102, 084060. [Google Scholar] [CrossRef] - Cunha, P.V.P.; Herdeiro, C.A.R.; Radu, E. Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor–Gauss–Bonnet Gravity. Phys. Rev. D
**2019**, 123, 011101. [Google Scholar] [CrossRef][Green Version] - Collodel, L.G.; Kleihaus, B.; Kunz, J.; Berti, E. Spinning and excited black holes in Einstein-scalar-Gauss–Bonnet theory. Class. Quant. Grav. Phys. Rev. D
**2020**, 37, 075018. [Google Scholar] [CrossRef][Green Version] - Doneva, D.D.; Yazadjiev, S.S. On the dynamics of the nonrotating and rotating black hole scalarization. arXiv
**2021**, arXiv:2101.03514. [Google Scholar] - Dima, A.; Barausse, E.; Franchini, N.; Sotiriou, T.P. Spin-induced black hole spontaneous scalarization. Phys. Rev. D
**2020**, 125, 231101. [Google Scholar] - Herdeiro, C.A.R.; Radu, E.; Silva, H.O.; Sotiriou, T.P.; Yunes, N. Spin-induced scalarized black holes. Phys. Rev. Lett.
**2021**, 126, 011103. [Google Scholar] [CrossRef] - Berti, E.; Collodel, L.G.; Kleihaus, B.; Kunz, J. Spin-induced black-hole scalarization in Einstein-scalar-Gauss–Bonnet theory. Phys. Rev. Lett.
**2021**, 126, 011104. [Google Scholar] [CrossRef] - Doneva, D.D.; Collodel, L.G.; Krüger, C.J.; Yazadjiev, S.S. Black hole scalarization induced by the spin: 2+1 time evolution. Phys. Rev. D
**2020**, 102, 104027. [Google Scholar] [CrossRef] - Buonanno, A.; Gasperini, M.; Ungarelli, C. A class of nonsingular gravidilaton backgrounds. Mod. Phys. Lett. A
**1997**, 12, 1883–1889. [Google Scholar] [CrossRef][Green Version] - Tchrakian, D.H. Chern–Simons Gravities (CSG) and Gravitational Chern–Simons (GCS) Densities in All Dimensions. Phys. Atom. Nucl.
**2018**, 81, 930. [Google Scholar] [CrossRef][Green Version] - Radu, E.; Tchrakian, D.H. Gravitational Chern–Simons, and Chern–Simons Gravity in All Dimensions. Phys. Part. Nucl. Lett.
**2020**, 17, 753–759. [Google Scholar] [CrossRef] - Utiyama, R. Invariant theoretical interpretation of interaction. Phys. Rev.
**1956**, 101, 1597–1607. [Google Scholar] [CrossRef] - Kibble, T.W.B. Lorentz invariance and the gravitational field. J. Math. Phys.
**1961**, 2, 212–221. [Google Scholar] [CrossRef][Green Version] - Witten, E. (2+1)-Dimensional Gravity as an Exactly Soluble System. Nucl. Phys. B
**1988**, 311, 46. [Google Scholar] [CrossRef] - Chamseddine, A.H. Topological Gauge Theory of Gravity in Five-dimensions and All Odd Dimensions. Phys. Lett. B
**1989**, 233, 291. [Google Scholar] [CrossRef] - Chamseddine, A.H. Topological gravity and supergravity in various dimensions. Nucl. Phys. B
**1990**, 346, 213. [Google Scholar] [CrossRef] - Deser, S.; Jackiw, R.; Templeton, S. Topologically Massive Gauge Theories. Ann. Phys.
**1982**, 140, 372, reprinted in Ann. Phys.**1988**, 185, 406; reprinted in Ann. Phys.**2000**, 281, 409. [Google Scholar] [CrossRef] - Tchrakian, T. Notes on Yang–Mills-Higgs monopoles and dyons on Rsup D, and Chern–Simons-Higgs solitons on Rsup D-2: Dimensional reduction of Chern–Pontryagin densities. J. Phys. A
**2011**, 44, 343001. [Google Scholar] [CrossRef][Green Version] - Radu, E.; Tchrakian, T. New Chern–Simons densities in both odd and even dimensions. arXiv
**2011**, arXiv:1101.5068. [Google Scholar] - Tchrakian, D.H. Higgs-and Skyrme-Chern–Simons densities in all dimensions. J. Phys. A
**2015**, 48, 375401. [Google Scholar] [CrossRef][Green Version] - Tchrakian, D.H. A remark on black holes of Chern–Simons gravities in 2n + 1 dimensions: n = 1, 2, 3. Int. J. Mod. Phys. A
**2020**, 35, 2050022. [Google Scholar] [CrossRef][Green Version] - Delgado, J.F.M.; Herdeiro, C.A.R.; Radu, E. Spinning black holes in shift-symmetric Horndeski theory. J. High Energy Phys.
**2020**, 4, 180. [Google Scholar] [CrossRef] - Sotiriou, T.P.; Zhou, S.Y. Black hole hair in generalized scalar-tensor gravity. Phys. Rev. Lett.
**2014**, 112, 251102. [Google Scholar] [CrossRef][Green Version] - Sotiriou, T.P.; Zhou, S.Y. Black hole hair in generalized scalar-tensor gravity: An explicit example. Phys. Rev. D
**2014**, 90, 124063. [Google Scholar] [CrossRef][Green Version] - Horndeski, G.W. Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys.
**1974**, 10, 363. [Google Scholar] [CrossRef] - Wald, R.M. Black hole entropy is the Noether charge. Phys. Rev. D
**1993**, 48, 3427. [Google Scholar] [CrossRef] [PubMed][Green Version] - Barton, S.; Hartmann, B.; Kleihaus, B.; Kunz, J. Spontaneously vectorized Einstein-Gauss–Bonnet black holes. arXiv
**2021**, arXiv:2103.0165. [Google Scholar]

**Figure 1.**Left panel: The asymptotic value of the scalar field ${\varphi}_{\infty}$ and the mass M of the solutions are shown as a function of the value of the scalar field at the horizon ${\varphi}_{H}$. The solutions have a fixed value of the horizon radius ${r}_{H}$ and of the coupling constant $\alpha $. Right panel: The profile of a typical scalarized solution (marked with a blue square in the left panel and Figure 2) is shown as a function of the compactified coordinate $1-{r}_{H}/r$.

**Figure 2.**Several quantities of interest are shown as a function of the ratio $\alpha /{M}^{2}$ for the set of scalarized BHs.

**Figure 3.**Left panel: The asymptotic value of the scalar field ${\varphi}_{\infty}$, the value of the scalar field at the horizon ${\varphi}_{H}$ and the charge of the vector field ${Q}_{e}$ are shown as a function of the ration $\alpha /{r}_{H}^{2}$ for solutions of the scalar-vector model in the probe limit. Right panel: The profile of a typical solution of the scalar-vector model in a fixed Schwarzschild BH background is shown as a function of the compactified coordinate $1-{r}_{H}/r$. For all solutions, the vector field vanishes both at the horizon and at infinity.

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**MDPI and ACS Style**

Herdeiro, C.; Radu, E.; Tchrakian, D.H.
Embedding Gauss–Bonnet Scalarization Models in Higher Dimensional Topological Theories. *Symmetry* **2021**, *13*, 590.
https://doi.org/10.3390/sym13040590

**AMA Style**

Herdeiro C, Radu E, Tchrakian DH.
Embedding Gauss–Bonnet Scalarization Models in Higher Dimensional Topological Theories. *Symmetry*. 2021; 13(4):590.
https://doi.org/10.3390/sym13040590

**Chicago/Turabian Style**

Herdeiro, Carlos, Eugen Radu, and D. H. Tchrakian.
2021. "Embedding Gauss–Bonnet Scalarization Models in Higher Dimensional Topological Theories" *Symmetry* 13, no. 4: 590.
https://doi.org/10.3390/sym13040590