Thin shells surrounding black holes in F(R) gravity
Abstract
In this article, we consider spherical thin shells of matter surrounding black holes in F(R) theories of gravity. We study the stability of the static configurations under perturbations that conserve the symmetry. In particular, we analyze the case of charged shells outside the horizon of noncharged black holes. We obtain that stable static thin shells are possible if the values of the parameters of the model are properly selected.
1 Introduction
The observations concerning the accelerated expansion of the Universe, the rotation curves of galaxies, and the anisotropy of the microwave background radiation can be explained within General Relativity by the presence of dark matter (\(\sim \) 25%) and dark energy (\(\sim \) 70%), besides the ordinary barionic matter (\(\sim \) 5%). In the concordance or \(\Lambda \)CDM model, the dark energy contribution comes in the form of a cosmological constant \(\Lambda \) and the cold dark matter in the form of nonrelativistic fluid, supplemented by an inflationary scenario driven by a scalar field called the inflaton. Although successful, this model is not free of difficulties, such as the extremely small observed value of \(\Lambda \) compared to the expected one if thought as originated from a vacuum energy in particle physics, or the unclear nature of dark matter (although several candidates exist). Other approaches can be adopted, such as modified gravity theories, in order to try to avoid these problems and explain the observed features of the Universe without dark matter and dark energy. Quantum gravity also provides motivation for modified gravity. One well known theory is F(R) gravity [1, 2, 3, 4] in which the Einstein–Hilbert Lagrangian is replaced by a function F(R) of the Ricci scalar R. In recent years, several solutions of the field equations in F(R) gravity have been found, including static and spherically symmetric black holes [1, 5, 6, 7, 8, 9, 10, 11, 12, 13], traversable wormholes [14, 15, 16], and branes [17].
The Darmois–Israel junction conditions [18, 19] provide the tools for matching two solutions across a hypersurface in General Relativity. These conditions allow for the study of thin shells of matter, by relating the energy–momentum tensor at the joining hypersurface with the spacetimes at both sides of it. The formalism has been broadly adopted in many different scenarios due to its flexibility and simplicity; among them it is used to model vacuum bubbles and thin layers around black holes [20, 21, 22, 23, 24, 25, 26, 27, 28], gravastars [29, 30, 31, 32], and wormholes [33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. In the case of highly symmetric configurations, the stability analysis is usually easy to perform, at least for perturbations preserving the symmetry.
The junction conditions in F(R) theories [43, 44] are more restrictive than in General Relativity. For nonlinear F(R), the trace of the second fundamental form should always be continuous at the matching hypersurface [44]. Except in the quadratic case, the curvature scalar R should also be continuous there [44]. In quadratic F(R) gravity, the hypersurface has in general, in addition to the standard energy–momentum tensor, an external energy flux vector, an external scalar pressure (or tension), and another energy–momentum contribution resembling classical dipole distributions. All these contributions have to be present [44, 45, 46] in order to have a divergencefree energy–momentum tensor, which guarantees local conservation. It was recently shown that these features are shared by any theory with a quadratic lagrangian [47]. The junction conditions in F(R) were recently applied to the construction of thinshell wormholes [48, 49, 50, 51] and bubbles [52, 53]. A particularly interesting example of a pure double layer in quadratic F(R) was found [52].
In this work, we construct spherical thin shells surrounding noncharged black holes by using the junction conditions in F(R) gravity and we analyze the stability of the static configurations under perturbations that preserve the symmetry. In Sect. 2, we review the general formalism for spherical geometries with constant curvature scalars at both sides of the shell. In Sect. 3, we perform the construction and we study the stability of charged thin shells outside the black hole event horizon. Finally, in Sect. 4, we present the conclusions of the paper. We adopt a system of units in which \(c=G=1\), with c the speed of light and G the gravitational constant.
2 Spherical thin shells: construction and stability
2.1 The same constant curvature scalar \(R_0\)
2.2 Different and constant curvature scalars \(R_1\) and \(R_2\)
Following the same procedure of the previous subsection, the stability of the static configurations is found from Eq. (23) in terms of the potential of Eq. (24), with its second derivative given by Eq. (25); again \(V''(a_0)>0\) correspond to the stable ones.
3 Charged thin shells
3.1 Curvature scalar \(R_0\) at both sides
In order to start the construction of the shell we choose a radius a satisfying Eq. (8), larger than the event horizon radius in \( {\mathcal {M}}_1 \), so the black hole is always present, and when \( R_0> 0 \), also smaller than the cosmological horizon radius coming from the original geometry for this region. On the other hand, this radius a should be large enough to avoid the presence of the event horizon and the singularity of the geometry used for the region \( {\mathcal {M}}_2 \). When \( R_0> 0 \), it also has to be smaller than the cosmological horizon of this outer region. As we mentioned in the previous section, it is necessary that \( F '(R_0)> 0 \) to avoid ghosts. It is also preferable that the matter on the shell satisfies the weak energy condition, in order to guarantee the presence of normal matter on \( \Sigma \). The energy density and the pressure are given by Eqs. (16) and (17), respectively, which fulfill the equation of state (18).

When \( R_0 <0 \), for \( Q <Q_ {c} \), there is only an unstable solution composed by normal matter. For larger values of Q , there are two solutions made of exotic matter, one of them is stable while the other, close to the event horizon of the black hole, is unstable.

For \( R_0> 0 \) and \(  Q  <Q_ {c} \) there is an unstable solution constituted by normal matter. For values \(  Q > Q_ {c} \) and a restricted range of charge, there are three solutions composed by exotic matter, one of them is stable. For values of Q much larger than \( Q_c \), there is only an unstable solution close to the event horizon of the black hole.
3.2 Curvature scalars \(R_1 \ne R_2\)
The radius a of the shell is properly chosen, in the same way as done in the previous subsection, and it should satisfy Eq. (8). The surface energy density is obtained by Eq. (27), the pressure by Eq. (28) and the external tension/pressure by Eq. (30). These three equations determine, together with Eq. (8), the equation of state at the shell (31). We also have that \( {\mathcal {T}}_\mu = 0 \) and, because \( [R]\ne 0 \), the nonnull tensor \( {\mathcal {T}}_{\mu \nu } \), with a dipolar density \( {\mathcal {P}}_{\mu \nu } \) given by Eq. (34).

For \( R_1 > R_2 \) and values of Q close to \( Q_c \), there are two solutions, one stable and the another unstable, both made of normal matter. For larger values of Q , unstable solutions constituted by exotic matter predominate. Only for a restricted range of Q there is a stable solution consisting of exotic matter.

For \( R_1 < R_2 \) and charge values \(  Q  < Q_c \), there is an unstable solution made of normal matter. For \(  Q  > Q_c \) and for a broad range of values of charge, there are two solutions composed by exotic matter, one of them stable.
4 Conclusions
In this work, we have studied spherically symmetric thin shells of matter around black holes within F(R) theories of gravity. We have adopted constant curvature scalars at both sides of the shell and we have analyzed two scenarios: one in which the curvature scalars are equal to the same value \( R_0 \), and the other in which the values of the curvature scalars \( R_1 \) and \( R_2 \) are different. The case with both curvature scalars equal to \( R_0 \) does not impose any limitations on the function F(R) . The matter at the shell should fulfill the equation of state \( \sigma  2p = 0 \) that relates the surface energy density \(\sigma \) and the pressure p. When the curvature scalars are different, it is necessary to restrict the analysis to quadratic F(R) . Then, for \( R_1 \ne R_2 \) the shell is composed by matter that satisfies the equation of state \( \sigma  2p = {\mathcal {T}} \), which depends on the external tension/pressure \({\mathcal {T}}\); it also requires the presence of the vector \( {\mathcal {T}}_\mu = 0 \) and the tensor \( {\mathcal {T}} _{\mu \nu } \ne 0 \) contributions. In particular, we have constructed a thin shell of matter surrounding a static noncharged black hole with mass \( M_1 \). The geometry outside the shell corresponds to a solution with mass \( M_2 \) and charge Q.
In the case with the same value of the curvature scalar at both sides of the shell, the behavior of the solutions is determined by the sign of \( R_0 \), and it is always possible to find unstable solutions with normal matter. On the other hand, stable solutions constituted by normal matter are not found; those that are stable are present, but always built with exotic matter, within a small range of Q when \( R_0> 0 \) or for large values of Q when \( R_0 < 0 \). This result is similar to the one obtained in Ref. [53] for bubbles, with the main difference being that for the shells around black holes there is an extra solution, unstable and constituted by exotic matter, with values of the shell radius close to the event horizon one.
For different values of the curvature scalar at the regions separated by the shell, the solutions have a distinct behavior depending on the relation between \( R_1 \) and \( R_2 \). Only in the case with \( R_1> R_2 \) it is possible to find stable solutions constituted by normal matter for Q close to \( Q_c \). The rest of the solutions are unstable and made of exotic matter, with the exception of a small range of Q for which the shell is stable but, once again, it is composed by exotic matter. When \( R_1 <R_2 \), we have only found unstable solutions constituted by normal matter for \( Q < Q_c \), or by exotic matter for \(Q > Q_c \). The only stable solution for this case is made of exotic matter for a broad range of values of the charge. These results are similar to those found in Ref. [53] in the case of \( R_1 \ne R_2 \), with the difference that, in shells surrounding black holes, an additional unstable solution emerges close to the event horizon. Compared with the case of the Ref. [53], it is more complicated to find an appropriate set of parameters that allow the construction of a stable shell around the black hole, made of normal matter and without charge. However, we have found that it is possible to build such a case, for a limited range of values of \( R_1 \) and \( R_2 \), and we have shown an example with \( R_1 = 0.39 \) and \( R_2 = 0.3 \).
It is well known that there is an equivalence between any F(R) gravity theory and a properly taken scalar–tensor theory [1, 2]; in particular, quadratic F(R) is equivalent to Brans–Dicke theory with a parameter \(\omega =0\), with a potential \(V(\phi )= 2 \Lambda +(\phi ^2 2\phi 3)/(4\alpha )\), where the scalar field \(\phi \) and the curvature scalar are related by \(\phi =2\alpha R1\). Then, it is worthy to note that the results obtained here can be translated to the corresponding scalar–tensor theory.
Notes
Acknowledgements
This work has been supported by CONICET and Universidad de Buenos Aires.
References
 1.T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82, 451 (2010)ADSCrossRefGoogle Scholar
 2.A. De Felice, S. Tsujikawa, Living Rev. Relativ. 13, 3 (2010)ADSCrossRefGoogle Scholar
 3.S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011)ADSMathSciNetCrossRefGoogle Scholar
 4.S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Phys. Rep. 692, 1 (2017)ADSMathSciNetCrossRefGoogle Scholar
 5.T. Clifton, J.D. Barrow, Phys. Rev. D 72, 103005 (2005)ADSMathSciNetCrossRefGoogle Scholar
 6.T. Multamäki, I. Vilja, Phys. Rev. D 74, 064022 (2006)ADSMathSciNetCrossRefGoogle Scholar
 7.S. Capozziello, A. Stabile, A. Troisi, Class. Quantum Gravity 25, 085004 (2008)ADSCrossRefGoogle Scholar
 8.A. de la CruzDombriz, A. Dobado, A.L. Maroto, Phys. Rev. D 80, 124011 (2009) [Erratum: Phys. Rev. 83, 029903 (2011)]Google Scholar
 9.T. Moon, Y.S. Myung, E.J. Son, Gen. Relativ. Gravit. 43, 3079 (2011)ADSCrossRefGoogle Scholar
 10.L. Sebastiani, S. Zerbini, Eur. Phys. J. C 71, 1591 (2011)ADSCrossRefGoogle Scholar
 11.Z. Amirabi, M. Halilsoy, S. Habib Mazharimousavi, Eur. Phys. J. C 76, 338 (2016)ADSCrossRefGoogle Scholar
 12.S. Nojiri, S.D. Odintsov, Class. Quantum Gravity 30, 125003 (2013)ADSCrossRefGoogle Scholar
 13.S. Nojiri, S.D. Odintsov, Phys. Lett. B 735, 376 (2014)ADSMathSciNetCrossRefGoogle Scholar
 14.F.S.N. Lobo, M.A. Oliveira, Phys. Rev. D 80, 104012 (2009)ADSMathSciNetCrossRefGoogle Scholar
 15.A. DeBenedictis, D. Horvat, Gen. Relativ. Gravit. 44, 2711 (2012)ADSCrossRefGoogle Scholar
 16.T. Harko, F.S.N. Lobo, M.K. Mak, S.V. Sushkov, Phys. Rev. D 87, 067504 (2013)ADSCrossRefGoogle Scholar
 17.S. Chakraborty, S. SenGupta, Eur. Phys. J. C 75, 11 (2015)ADSCrossRefGoogle Scholar
 18.G. Darmois, Mémorial des Sciences Mathématiques, Fascicule XXV, Chap. V (GauthierVillars, Paris, 1927)Google Scholar
 19.W. Israel, Nuovo Cimento B 44, 1 (1966) [Erratum: Nuovo Cimento 48, 463 (1967)]Google Scholar
 20.P.R. Brady, J. Louko, E. Poisson, Phys. Rev. D 44, 1891 (1991)ADSMathSciNetCrossRefGoogle Scholar
 21.M. Ishak, K. Lake, Phys. Rev. D 65, 044011 (2002)ADSMathSciNetCrossRefGoogle Scholar
 22.S.M.C.V. Gonçalves, Phys. Rev. D 66, 084021 (2002)ADSMathSciNetCrossRefGoogle Scholar
 23.F.S.N. Lobo, P. Crawford, Class. Quantum Gravity 22, 4869 (2005)ADSCrossRefGoogle Scholar
 24.E.F. Eiroa, C. Simeone, Phys. Rev. D 83, 104009 (2011)ADSCrossRefGoogle Scholar
 25.E.F. Eiroa, C. Simeone, Int. J. Mod. Phys. D 21, 1250033 (2012)ADSCrossRefGoogle Scholar
 26.E.F. Eiroa, C. Simeone, Phys. Rev. D 87, 064041 (2013)ADSCrossRefGoogle Scholar
 27.S.W. Kim, J. Korean Phys. Soc. 61, 1181 (2012)ADSCrossRefGoogle Scholar
 28.M. Sharif, S. Iftikhar, Astrophys. Space Sci. 356, 89 (2015)ADSCrossRefGoogle Scholar
 29.M. Visser, D.L. Wiltshire, Class. Quantum Gravity 21, 1135 (2004)ADSCrossRefGoogle Scholar
 30.N. Bilić, G.B. Tupper, R.D. Viollier, J. Cosmol. Astropart. Phys. 02, 013 (2006)ADSCrossRefGoogle Scholar
 31.F.S.N. Lobo, A.V.B. Arellano, Class. Quantum Gravity 24, 1069 (2007)ADSCrossRefGoogle Scholar
 32.P. MartinMoruno, N. Montelongo Garcia, F.S.N. Lobo, M. Visser, J. Cosmol. Astropart. Phys. 03, 034 (2012)ADSCrossRefGoogle Scholar
 33.E. Poisson, M. Visser, Phys. Rev. D 52, 7318 (1995)ADSMathSciNetCrossRefGoogle Scholar
 34.E.F. Eiroa, G.E. Romero, Gen. Relativ. Gravit. 36, 651 (2004)ADSCrossRefGoogle Scholar
 35.F.S.N. Lobo, P. Crawford, Class. Quantum Gravity 21, 391 (2004)ADSCrossRefGoogle Scholar
 36.G.A.S. Dias, J.P.S. Lemos, Phys. Rev. D 82, 084023 (2010)ADSCrossRefGoogle Scholar
 37.V. Varela, Phys. Rev. D 92, 044002 (2015)ADSMathSciNetCrossRefGoogle Scholar
 38.E.F. Eiroa, Phys. Rev. D 78, 024018 (2008)ADSCrossRefGoogle Scholar
 39.N. Montelongo Garcia, F.S.N. Lobo, M. Visser, Phys. Rev. D 86, 044026 (2012)ADSCrossRefGoogle Scholar
 40.E.F. Eiroa, C. Simeone, Phys. Rev. D 81, 084022 (2010) [Erratum: Phys. Rev. 90, 089906 (2014)]Google Scholar
 41.S. Habib Mazharimousavi, M. Halilsoy, Z. Amirabi, Phys. Rev. D 89, 084003 (2014)ADSCrossRefGoogle Scholar
 42.E.F. Eiroa, C. Simeone, Phys. Rev. D 91, 064005 (2015)ADSMathSciNetCrossRefGoogle Scholar
 43.N. Deruelle, M. Sasaki, Y. Sendouda, Prog. Theor. Phys. 119, 237 (2008)ADSCrossRefGoogle Scholar
 44.J.M.M. Senovilla, Phys. Rev. D 88, 064015 (2013)ADSCrossRefGoogle Scholar
 45.J.M.M. Senovilla, Class. Quantum Gravity 31, 072002 (2014)ADSCrossRefGoogle Scholar
 46.J.M.M. Senovilla, J. Phys. Conf. Ser. 600, 012004 (2015)CrossRefGoogle Scholar
 47.B. Reina, J.M.M. Senovilla, R. Vera, Class. Quantum Gravity 33, 105008 (2016)ADSCrossRefGoogle Scholar
 48.E.F. Eiroa, G. Figueroa Aguirre, Eur. Phys. J. C 76, 132 (2016)ADSCrossRefGoogle Scholar
 49.E.F. Eiroa, G. Figueroa Aguirre, Phys. Rev. D 94, 044016 (2016)ADSMathSciNetCrossRefGoogle Scholar
 50.M. ZaeemulHaq Bhatti, A. Anwar, S. Ashraf, Mod. Phys. Lett. A 32, 1750111 (2017)CrossRefGoogle Scholar
 51.S. Habib Mazharimousavi, Eur. Phys. J. C 78, 612 (2018)CrossRefGoogle Scholar
 52.E.F. Eiroa, G. Figueroa Aguirre, J.M.M. Senovilla, Phys. Rev. D 95, 124021 (2017)ADSMathSciNetCrossRefGoogle Scholar
 53.E.F. Eiroa, G. Figueroa Aguirre, Eur. Phys. J. C 78, 54 (2018)ADSCrossRefGoogle Scholar
 54.K.A. Bronnikov, M.V. Skvortsova, A.A. Starobinsky, Gravit. Cosmol. 16, 216 (2010)ADSMathSciNetCrossRefGoogle Scholar
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