# Reissner–Nordström Anti-de Sitter Black Holes in Mimetic F(R) Gravity

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Mimetic $F\left(R\right)$ Gravity and Reissner–Nordström Black Holes

#### 2.1. The Mimetic $F\left(R\right)$ Gravity Theoretical Framework

#### 2.2. Motivation for Studying the Reissner–Nordström Black Holes

#### 2.3. General Study of the Solutions

#### 2.4. Mimetic $F\left(R\right)$ Reissner–Nordström Black Holes: A Study of the Solutions

- Case I: Both the first and second term are equal to zero, that is,$$\left(\frac{{g}_{\mu \nu}R}{4}-{R}_{\mu \nu}\right){F}^{\prime}\left(R\right)=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{g}_{\mu \nu}\left(\frac{3\lambda \left(\varphi \right)}{4}\right)=0\phantom{\rule{0.166667em}{0ex}}$$
- Case II: Both the first term are equal to the same constant, but with opposite signs, that is,$$\left(\frac{{g}_{\mu \nu}R}{4}-{R}_{\mu \nu}\right){F}^{\prime}\left(R\right)=\Gamma ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{g}_{\mu \nu}\left(\frac{3\lambda \left(\varphi \right)}{4}\right)=-\Gamma \phantom{\rule{0.166667em}{0ex}}$$

#### 2.4.1. Case I

- Scenario I: This scenario corresponds to $Q\ne 0$, and therefore the following constraints correspond to this scenario,$${F}^{\prime}\left({R}_{0}\right)=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}V\left(\varphi \right)=\Lambda ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\lambda \left(\varphi \right)=0\phantom{\rule{0.166667em}{0ex}}$$
- Scenario II: This scenario corresponds to $Q=0$ and it is described by the following constraints,$$Q=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}V\left(\varphi \right)=\Lambda ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\lambda \left(\varphi \right)=0\phantom{\rule{0.166667em}{0ex}}$$

#### 2.4.2. Case II

## 3. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Parameters $\mathit{M}$ and $\mathit{Q}$ in Terms of ${\mathit{r}}_{\mathbf{0}}$ and ${\mathit{r}}_{\mathbf{1}}$

## References

- Riess, A.G.; Filippenko, A.; Challis, P.; Clocchiatti, A.; Diercks, A.; Garnavich, P.; Gilliland, R.; Hogan, C.; Jha, S.; Kirshner, S.; et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J.
**1998**, 116, 1009–1038. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Accelerating cosmology in modified gravity: From convenient F(R) or string-inspired theory to bimetric F(R) gravity. Int. J. Geom. Methods Mod. Phys.
**2014**, 11, 1460006. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Introduction to modified gravity and gravitational alternative for dark energy. Int. J. Geom. Methods Mod. Phys.
**2007**, 4, 115–146. [Google Scholar] [CrossRef] - Capozziello, S.; Faraoni, V. Beyond Einstein Gravity; Springer: Berlin, Germany, 2010. [Google Scholar]
- Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models. Phys. Rep.
**2011**, 505, 59–144. [Google Scholar] [CrossRef] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified gravity and cosmology. Phys. Rep.
**2012**, 513, 1–189. [Google Scholar] [CrossRef] - Capozziello, S.; De Laurentis, M. Extended theories of gravity. Phys. Rep.
**2011**, 509, 167–321. [Google Scholar] [CrossRef] - Barrow, J.D.; Clifton, T. The power of general relativity. Phys. Rev. D
**2014**, 90, 029902. [Google Scholar] - Clifton, T.; Barrow, J.D. Exact cosmological solutions of scale-invariant gravity theories. Class. Quantum Gravity
**2006**, 23, 1. [Google Scholar] [CrossRef] - Gorbunov, D.S.; Rubakov, V.A. Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory. Contemp. Phys.
**2012**, 53, 361–396. [Google Scholar] - Linde, A. Inflationary Cosmology after Planck 2013. 2014; arXiv:1402.0526. [Google Scholar]
- Brandenberger, R.H. The Matter Bounce Alternative to Inflationary Cosmology. 2012; arXiv:1206.4196. [Google Scholar]
- Bamba, K.; Odintsov, S.D. Inflationary cosmology in modified gravity theories. Symmetry
**2015**, 7, 220–240. [Google Scholar] [CrossRef] - Mukhanov, V.F.; Feldman, H.A.; Brandenberger, R.H. Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions. Phys. Rep.
**1992**, 215, 203–333. [Google Scholar] [CrossRef] - Brandenberger, R.H.; Kahn, R. Cosmological perturbations in inflationary universe models. Phys. Rev. D
**1984**, 29, 2172. [Google Scholar] [CrossRef] - Brandenberger, R.H.; Kahn, R.; Press, W.H. Cosmological perturbations in the early universe. Phys. Rev. D
**1983**, 28, 1809. [Google Scholar] [CrossRef] - Sebastiani, L.; Myrzakulov, R. F(R) gravity and inflation. Int. J. Geom. Methods Mod. Phys.
**2015**, 12, 1530003. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration. Phys. Rev. D
**2003**, 68, 123512. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Bouncing cosmology with future singularity from modified gravity. Phys. Rev. D
**2015**, 92, 024016. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Quantitative analysis of singular inflation with scalar-tensor and modified gravity. Phys. Rev. D
**2015**, 91, 084059. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K.; Saridakis, E.N. Superbounce and loop quantum ekpyrotic cosmologies from modified gravity: F(R), F(G) and F(T) theories. Ann. Phys.
**2015**, 363, 141–163. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Matter bounce loop quantum cosmology from F(R) gravity. Phys. Rev. D
**2014**, 90, 124083. [Google Scholar] [CrossRef] - Bamba, K.; Capozziello, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci.
**2012**, 342, 155–228. [Google Scholar] [CrossRef] - Cai, Y.F.; Saridakis, E.N.; Setare, M.R.; Xia, J.Q. Quintom cosmology: Theoretical implications and observations. Phys. Rep.
**2010**, 493, 1–60. [Google Scholar] [CrossRef] - Sami, M. A primer on problems and prospects of dark energy. Curr. Sci.
**2009**, 97, 887. [Google Scholar] - Peebles, P.J.E.; Ratra, B. The cosmological constant and dark energy. Rev. Mod. Phys.
**2003**, 75, 559–606. [Google Scholar] [CrossRef] - Li, M.; Li, X.D.; Wang, S.; Wang, Y. Dark energy. Commun. Theor. Phys.
**2011**, 56, 525–604. [Google Scholar] [CrossRef] - Padmanabhan, T. Cosmological constant: The Weight of the vacuum. Phys. Rep.
**2003**, 380, 235–320. [Google Scholar] [CrossRef] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2015 results. XX. Constraints on inflation. 2015; arXiv:1502.02114. [Google Scholar]
- Shafi, Q.; Tanyildizi, S.H.; Un, C.S. Neutralino dark matter and other LHC predictions from quasi yukawa unification. Nucl. Phys.
**2015**, 900, 400–411. [Google Scholar] [CrossRef] - Oikonomou, V.K.; Vergados, J.D.; Moustakidis, C.C. Direct detection of dark matter-rates for various wimps. Nucl. Phys. B
**2006**, 773, 19–42. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Mukhanov, V. Mimetic dark matter. J. High Energy Phys.
**2013**, 2013, 135. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Mukhanov, V.; Vikman, A. Cosmology with mimetic matter. J. Cosmol. Astropart. Phys.
**2014**, 2014, 017. [Google Scholar] [CrossRef] - Golovnev, A. On the recently proposed mimetic dark matter. Phys. Lett. B
**2014**, 728, 39–40. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Mimetic F(R) gravity: Inflation, dark energy and bounce. Mod. Phys. Lett. A
**2014**, 29, 1450211. [Google Scholar] [CrossRef] - Matsumoto, J.; Odintsov, S.D.; Sushkov, S.V. Cosmological perturbations in a mimetic matter model. Phys. Rev. D
**2015**, 91, 064062. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Accelerating cosmology and phase structure of F(R) gravity with lagrange multiplier constraint: mimetic approach. Phys. Rev. D
**2016**, 92, 124024. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Viable mimetic F(R) gravity compatible with planck observations. Ann. Phys.
**2015**, 363, 503–514. [Google Scholar] [CrossRef] - Astashenok, A.V.; Odintsov, S.D.; Oikonomou, V.K. Modified gauss bonnet gravity with the lagrange multiplier constraint as mimetic theory. Class. Quantum Gravity
**2015**, 32, 185007. [Google Scholar] [CrossRef] - Myrzakulov, R.; Sebastiani, L.; Vagnozzi, S.; Zerbini, S. Static spherically symmetric solutions in mimetic gravity: Rotation curves and wormholes. Class. Quantum Gravity
**2016**, 33, 125005. [Google Scholar] [CrossRef] - Rabochaya, Y.; Zerbini, S. A note on a mimetic scalar-tensor cosmological model. Eur. Phys. J. C
**2016**, 76, 85. [Google Scholar] [CrossRef] - Raza, M.; Myrzakulov, K.; Momeni, D.; Myrzakulov, R. Mimetic attractors. Int. J. Theor. Phys.
**2016**, 55, 2558–2572. [Google Scholar] [CrossRef] - Momeni, D.; Moraes, P.H.R.S.; Gholizade, H.; Myrzakulov, R. Mimetic compact stars. 2015; arXiv:1505.05113. [Google Scholar]
- Myrzakulov, R.; Sebastiani, L. Spherically symmetric static vacuum solutions in Mimetic gravity. Gen. Relativ. Gravit.
**2015**, 47, 89. [Google Scholar] [CrossRef] - Momeni, D.; Myrzakulov, R.; Godekli, E. Cosmological viable Mimetic F(R) and F(R, T) theories via Noether symmetry. Int. J. Geom. Methods Mod. Phys.
**2015**, 12, 1550101. [Google Scholar] [CrossRef] - Leon, G.; Saridakis, E.N. Dynamical behavior in mimetic F(R) gravity. J. Cosmol. Astropart. Phys.
**2015**, 2015, 031. [Google Scholar] [CrossRef] - Momeni, D.; Altaibayeva, A.; Myrzakulov, R. New modified mimetic gravity. Int. J. Geom. Methods Mod. Phys.
**2014**, 11, 1450091. [Google Scholar] [CrossRef] - Capozziello, S.; Makarenko, A.N.; Odintsov, S.D. Gauss-Bonnet dark energy by Lagrange multipliers. Phys. Rev. D
**2013**, 87, 084037. [Google Scholar] [CrossRef] - Capozziello, S.; Francaviglia, M.; Makarenko, A.N. Higher-order Gauss-Bonnet cosmology by Lagrange multipliers. Astrophys. Space Sci.
**2014**, 349, 603–609. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Instabilities and anti evaporation of Reissner Nordstrom black holes in modified F(R) gravity. Phys. Lett. B
**2014**, 735, 376–382. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Anti-evaporation of schwarzschild-de sitter black holes in F(R) gravity. Class. Quantum Gravity
**2013**, 30, 125003. [Google Scholar] [CrossRef] - Sebastiani, L.; Momeni, D.; Myrzakulov, R.; Odintsov, S.D. Instabilities and, anti, evaporation of Schwarzschild de Sitter black holes in modified gravity. Phys. Rev. D
**2013**, 88, 104022. [Google Scholar] [CrossRef] - Clifton, T.; Barrow, J.D. The power of general relativity. Phys. Rev. D
**2005**, 72, 103005. [Google Scholar] [CrossRef] - De Laurentis, M.; Capozziello, S. Black holes and stellar structures in F(R) -gravity. 2012; arXiv:1202.0394. [Google Scholar]
- Clifton, T. Spherically symmetric solutions to fourth-order theories of gravity. Class. Quantum Gravity
**2006**, 23, 7445. [Google Scholar] [CrossRef] - Faraoni, V. Horizons and singularity in clifton’s spherical solution of F(R) vacuum. In Cosmology, Quantum Vacuum and Zeta Functions; Springer: Berlin, Germany, 2011; Volume 137, pp. 173–181. [Google Scholar]
- Faraoni, V. Black hole entropy in scalar-tensor and F(R) gravity: An overview. Entropy
**2010**, 12, 1246–1263. [Google Scholar] [CrossRef] - Pun, C.S.J.; Kovacs, Z.; Harko, T. Thin accretion disks in F(R) modified gravity models. Phys. Rev. D
**2008**, 78, 024043. [Google Scholar] [CrossRef] - Briscese, F.; Elizalde, E. Black hole entropy in modified gravity models. Phys. Rev. D
**2008**, 77, 044009. [Google Scholar] [CrossRef] - Mazharimousavi, S.H.; Halilsoy, M. Black hole solutions in F(R) gravity coupled with non-linear Yang-Mills field. Phys. Rev. D
**2011**, 84, 064032. [Google Scholar] [CrossRef] - Moon, T.; Myung, Y.S.; Son, E.J. F(R) black holes. Gen. Relativ. Gravit.
**2011**, 43, 3079–3098. [Google Scholar] [CrossRef] - Olmo, G.J.; Rubiera-Garcia, D. Palatini F(R) black holes in nonlinear electrodynamics. Phys. Rev. D
**2011**, 84, 124059. [Google Scholar] [CrossRef] - Cai, R.G.; Cao, L.-M.; Hu, Y.-P.; Ohta, N. Generalized misner-sharp energy in F(R) gravity. Phys. Rev. D
**2009**, 80, 104016. [Google Scholar] [CrossRef] - Hollenstein, L.; Lobo, F.S.N. Exact solutions of F(R) gravity coupled to nonlinear electrodynamics. Phys. Rev. D
**2008**, 78, 124007. [Google Scholar] [CrossRef] - Sheykhi, A. Higher-dimensional charged F(R) black holes. Phys. Rev. D
**2012**, 86, 024013. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Effective action for conformal scalars and anti-evaporation of black holes. Int. J. Mod. Phys. A
**1999**, 14, 1293–1304. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Quantum evolution of Schwarzschild-de Sitter, Nariai, black holes. Phys. Rev. D
**1999**, 59, 044026. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Quantum dilatonic gravity in, D = 2, -dimensions, D = 4, -dimensions and, D = 5, -dimensions. Int. J. Mod. Phys. A
**2001**, 16, 1015–1108. [Google Scholar] [CrossRef] - Israel, W. Event horizons in static vacuum space-times. Phys. Rev.
**1967**, 164, 1776–1779. [Google Scholar] [CrossRef] - Hartnoll, S.A.; Herzog, C.P.; Horowitz, G.T. Holographic Superconductors. J. High Energy Phys.
**2008**, 12, 015. [Google Scholar] [CrossRef] - Hartnoll, S.A.; Herzog, C.P.; Horowitz, G.T. Building a Holographic Superconductor. Phys. Rev. Lett.
**2008**, 101, 031601. [Google Scholar] [CrossRef] [PubMed] - Hawking, S.W. Particle creation by black holes. Commun. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] [CrossRef] - Bousso, R.; Hawking, S.W. Anti, evaporation of Schwarzschild-de Sitter black holes. Phys. Rev. D
**1998**, 57, 2436–2442. [Google Scholar] [CrossRef] - Hendi, S.H. The Relation between F(R) gravity and Einstein-conformally invariant Maxwell source. Phys. Lett. B
**2010**, 690, 220–223. [Google Scholar] [CrossRef] - Sherkatghanad, Z.; Brandenberger, R.H. The effect of primordial non-gaussianities on the seeds of super-massive black holes. 2015; arXiv:1508.00968. [Google Scholar]
- Barrow, J.D. Gravitational memory? Phys. Rev. D
**1992**, 46, R3227–R3230. [Google Scholar] [CrossRef] - Oikonomou, V.K. A note on gravitational memory in F(R)-theories and their equivalent scalar-tensor theories. Astrophys. Space Sci.
**2014**, 352, 925–935. [Google Scholar] [CrossRef]

**Table 1.**The Scenarios I and II for the Mimetic $F\left(R\right)$ Gravity Reissner–Nordström anti-de Sitter Black Hole.

Scenario | Constraints |
---|---|

Scenario I | ${F}^{\prime}\left({R}_{0}\right)=0$, $V\left(\varphi \right)=\Lambda $, $\lambda \left(\varphi \right)=0$, $F\left({R}_{0}\right)=\Lambda $ |

Scenario II | ${F}^{\prime}\left({R}_{0}\right)\ne 0$, $V\left(\varphi \right)=\Lambda $, $\lambda \left(\varphi \right)=0$, $F\left({R}_{0}\right)=\frac{{R}_{0}{F}^{\prime}\left({R}_{0}\right)}{2}+\Lambda $ |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Oikonomou, V.K.
Reissner–Nordström Anti-de Sitter Black Holes in Mimetic *F*(*R*) Gravity. *Universe* **2016**, *2*, 10.
https://doi.org/10.3390/universe2020010

**AMA Style**

Oikonomou VK.
Reissner–Nordström Anti-de Sitter Black Holes in Mimetic *F*(*R*) Gravity. *Universe*. 2016; 2(2):10.
https://doi.org/10.3390/universe2020010

**Chicago/Turabian Style**

Oikonomou, V. K.
2016. "Reissner–Nordström Anti-de Sitter Black Holes in Mimetic *F*(*R*) Gravity" *Universe* 2, no. 2: 10.
https://doi.org/10.3390/universe2020010