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Abstract： f（R） gravity is a straight extension of general relativity，in which the Lagrangian is an arbitrary function of the curvature scalar R.We derive an exact spherically symmetric vacuum solution to the field equations of the metric f（R） gravity with the condition F（r）=1+αrF（r）≡df（R（r））dr and the term αr indicates a minor correction to general relativity.Moreover，we consider the scalar perturbations on this black hole background spacetimes.Using the six-order WKB （Wentzel-Kramers-Brillouin） method，we discuss the relations between quasinormal modes of this black hole and its parameters which show that the black hole is stable.
Key words： f（R） gravity； black hole solution； the six-order WKB method
CLC number： P 142 Document code： A Article ID： 1000-5137（2018）04-0406-06
摘 要： f（R）引力是一個直接拓展广义相对论的修正引力理论，它的拉格朗日量是一个仅含曲率标量R的任意函数f（R）.在F（r）=1+αr 的条件下F（r）≡df（R（r））dr和αr是一个对广义相对论小的修正量，导出了度规f（R）引力理论中场方程的精确球对称真空解.此外，考虑了这个黑洞背景时空中的标量场扰动.用六阶WKB（Wentzel-Kramers-Brillouin）方法，讨论了拟正则模和这个黑洞的参数之间的关系，得出这个黑洞是稳定的结论.
关键词： f（R）引力； 黑洞解； 六阶WKB方法
1 Introduction
In 1998，the observations for Supernovae typeIa （SNela） show that our universe is in the stage of accelerated expansion.This observable consequence is confirmed by the subsequent observations，such as cosmic microwave background （CMB），large scale structure （LSS） and baryon acoustic oscillations （BAO）.In context of general relativity （GR），this phenomenon is explained by a spatially homogeneous and gravitationally repulsive energy component，dubbed dark energy.However，it is well known that there are some puzzles in these dark energy models such as the fine-tune problem，the coincidence problem，and the nature of dark energy problem.Thus，the cosmic acceleration may originate from some modification of gravity to GR such as Lovelock theories[1]，string theory[2] or f（R） theories[3].
In f（R） theories，the curvature scalar R of the Lagrangian in the Einstein Hilbert action is replaced by an arbitrary function of R，f（R），which is one of the simplest modifications to GR.The Lagrangian for GR is a special case of f（R） gravity theories，f（R）=R-2Λ （Λ is the cosmological constant.）.Different from GR，f（R） theories can reproduce the two accelerated expansions of cosmological history：inflation and late-time cosmic acceleration.Nevertheless，subjected to local tests of gravity and cosmological constraints，the viable f（R） theories must close to GR （f（R）=R2Λ），in which f（R） must satisfy the condition f（R）=T+δ（R），and δ（R） must be small in recent era[4].According to this requirement，Crames et al.[5] proposed that F（r）=1+αrF（r）≡df（R（r））dr，and the term αr indicates a minor correction to general relativity，and found a f（R） global monopole solution under this condition.Then this solution is extensively investigated.The strong lensing effect for the f（R） global monopole was discussed[6].A rotating f（R） global monopole solution[7] was considered by the coordinate transformation.Moreover，the stability of the f（R） black hole with a solid deficit angle was studied by Wentzel-Kramers-Brillouin （WKB） method[8].However，the investigation of the spherically symmetric vacuum solution in the metric f（R） gravity and its dynamical property is lacking. In this paper，we consider the spherically symmetricvacuum solution to the field equations of the metric f（R） gravity with F（r）=1+αr and its dynamical property.Firstly，we derive a new black hole solution to the field equation of the f（R） gravity in the metric formalism analytically.Secondly，we use one of semi-analytic methods the six-order WKB method[9] to calculate quasinormal modes （QNMs） of this black hole for scalar perturbations.The relations between QNMs of the f（R） black hole and its parameters are discussed.Based on these information，we conclude that the new f（R） black hole is stable.
3 WKB method and numerical result
The WKB approximative method was firstly applied by Schutz and Will[10] to the problem of scattering around black holes.This method is based on matching of the asymptotic WKB solutions at spatial infinity and the event horizon with the Taylor expansion near the top of the effective potential barrier through the two turning points.In another word，WKB method can determine the complex frequencies of a black hole if its perturbation equation can be described by a Schrdinger-like equation.There are three WKB computational schemes：the lowest approximation[10]，the third-order improvements[11] and the sixth-order corrections[9].The accuracy of the sixth-order method is better than those of the former two for a smaller multiple index l.Here，we adopt the sixth-order WKB method to numerically calculate the solutions to the basic equation for resonant perturbations of a new f（R） black hole.The QNMs is as follows[9]i（ω2-V0）-V″ 0+∑6i=2Λi=n+12.
（22） Our numerical results of QNMs for scalar perturbations are listed in Table 1-3.As a reminder，the oscillating quasi-period and the damping time scale are shown in these tables.In Table 1，the complex frequencies vary with the parameter rs.The real parts of the quasinormal frequencies decrease with rs increasing.But the imaginary parts increase with rs.It means that the larger rs is，more quickly the f（R） black hole oscillates and the oscillation of this black hole decays.In Table 2，the relation between the frequencies of QNMs and the parameter α is listed.We find that the period of the oscillations increases with α increasing.While the damping rate gradually decreases as α increases.It corresponds to the case that with an increase of α，the oscillation and the decay of the perturbation both decrease.In Table 3，it is shown that the quasinormal frequencies vary with different l.The real parts and the imaginary parts of the quasinormal frequencies both increase as l increases.It is to say that the larger l is，the more slowly the perturbation oscillates and the oscillation decays.It is worth to note the quasinomal frequencies do not vary with the parameter U0 or β.It is to say that，for different U0 or β，the real part and the imaginary part of the quasinormal frequencies are constant.From these results，we conclude that the new f（R） black hole is stable under the scalar perturbations. 4 Conclusion
In this work，we obtain a spherically symmetric solution in the context of the metric f（R） gravity with F（r）=1+αr.Using the six-order WKB method，we calculate quasinormal frequencies of the scalar perturbations in this new f（R） black hole spacetime.The relations between the quasinormal frequencies and the parameters of this spacetimes are obtained.It is shown that the real parts of the quasinormal frequencies decrease while the imaginary parts increase with rs increasing；with an increase of α，the oscillation and the decay of the perturbation both decrease；the larger l is，the more slowly the perturbation oscillates and the oscillation decays.The most interesting result is that the real part and the imaginary part of the quasinormal frequencies keep constant for different U0 or β.From these results，we conclude that the new f（R） black hole is stable.
References：
[1] Camanho X O，Edelstein J D.Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity [J].Journal of High Energy Physics，2010，4（4）：1-31.
[2] Scherk J.An introduction to the theory of dual models and strings [J].Reviews of Modern Physics，1975，47（1）：123-164.
[3] Felice A，Tsujikawa S.f（R） theories [J].Living Reviews in Relativity，2010，13（1）：3.
[4] Pogosian L，Silvestri A.The pattern of growth in viable f（R） cosmologies [J].Physical Review D，2007，77（2）：023503.
[5] Carames T R P，Fabris J C，Mello E R，et al.f（R） global monopole revisited [J].European Physical Journal C，2017，77（7）：496.
[6] Man J，Cheng H.Analytical discussion on strong gravitational lensing for a massive source with a f（R） global monopole [J].Physical Review D，2015，92（2）：024004.
[7] Morais J P，Bezerra V B.Gravitational field of a rotating global monopole in f（R） theory [J].Modern Physics Letters A，2012，27（31）：1250178.
[8] Graca J P M，Vieira H S，Bezerra V B.Scalar and spinor QNMs of a black hole with a global monopole in f（R） gravity [J].General Relativity & Gravitation，2016，48（4）：1-13.
[9] Konoplya R A.Quasinormal behavior of the D-dimensional Schwarzschild black hole and higher order WKB approach [J].Physical Review D，2003，68（2）：1557-1567.
[10] Schutz B F，Will C M.Black hole normal modes：A semi-analytic approach [J].Astrophysical Journal，1985，29（2）：L33-L36.
[11] Seidel E，Iyer S.Black-hole normal modes：A WKB approach.II.Schwarzschild black holes [J].Physical Review D，1990，41（2）：3632-3636.
（責任编辑：顾浩然）
Key words： f（R） gravity； black hole solution； the six-order WKB method
CLC number： P 142 Document code： A Article ID： 1000-5137（2018）04-0406-06
摘 要： f（R）引力是一個直接拓展广义相对论的修正引力理论，它的拉格朗日量是一个仅含曲率标量R的任意函数f（R）.在F（r）=1+αr 的条件下F（r）≡df（R（r））dr和αr是一个对广义相对论小的修正量，导出了度规f（R）引力理论中场方程的精确球对称真空解.此外，考虑了这个黑洞背景时空中的标量场扰动.用六阶WKB（Wentzel-Kramers-Brillouin）方法，讨论了拟正则模和这个黑洞的参数之间的关系，得出这个黑洞是稳定的结论.
关键词： f（R）引力； 黑洞解； 六阶WKB方法
1 Introduction
In 1998，the observations for Supernovae typeIa （SNela） show that our universe is in the stage of accelerated expansion.This observable consequence is confirmed by the subsequent observations，such as cosmic microwave background （CMB），large scale structure （LSS） and baryon acoustic oscillations （BAO）.In context of general relativity （GR），this phenomenon is explained by a spatially homogeneous and gravitationally repulsive energy component，dubbed dark energy.However，it is well known that there are some puzzles in these dark energy models such as the fine-tune problem，the coincidence problem，and the nature of dark energy problem.Thus，the cosmic acceleration may originate from some modification of gravity to GR such as Lovelock theories[1]，string theory[2] or f（R） theories[3].
In f（R） theories，the curvature scalar R of the Lagrangian in the Einstein Hilbert action is replaced by an arbitrary function of R，f（R），which is one of the simplest modifications to GR.The Lagrangian for GR is a special case of f（R） gravity theories，f（R）=R-2Λ （Λ is the cosmological constant.）.Different from GR，f（R） theories can reproduce the two accelerated expansions of cosmological history：inflation and late-time cosmic acceleration.Nevertheless，subjected to local tests of gravity and cosmological constraints，the viable f（R） theories must close to GR （f（R）=R2Λ），in which f（R） must satisfy the condition f（R）=T+δ（R），and δ（R） must be small in recent era[4].According to this requirement，Crames et al.[5] proposed that F（r）=1+αrF（r）≡df（R（r））dr，and the term αr indicates a minor correction to general relativity，and found a f（R） global monopole solution under this condition.Then this solution is extensively investigated.The strong lensing effect for the f（R） global monopole was discussed[6].A rotating f（R） global monopole solution[7] was considered by the coordinate transformation.Moreover，the stability of the f（R） black hole with a solid deficit angle was studied by Wentzel-Kramers-Brillouin （WKB） method[8].However，the investigation of the spherically symmetric vacuum solution in the metric f（R） gravity and its dynamical property is lacking. In this paper，we consider the spherically symmetricvacuum solution to the field equations of the metric f（R） gravity with F（r）=1+αr and its dynamical property.Firstly，we derive a new black hole solution to the field equation of the f（R） gravity in the metric formalism analytically.Secondly，we use one of semi-analytic methods the six-order WKB method[9] to calculate quasinormal modes （QNMs） of this black hole for scalar perturbations.The relations between QNMs of the f（R） black hole and its parameters are discussed.Based on these information，we conclude that the new f（R） black hole is stable.
3 WKB method and numerical result
The WKB approximative method was firstly applied by Schutz and Will[10] to the problem of scattering around black holes.This method is based on matching of the asymptotic WKB solutions at spatial infinity and the event horizon with the Taylor expansion near the top of the effective potential barrier through the two turning points.In another word，WKB method can determine the complex frequencies of a black hole if its perturbation equation can be described by a Schrdinger-like equation.There are three WKB computational schemes：the lowest approximation[10]，the third-order improvements[11] and the sixth-order corrections[9].The accuracy of the sixth-order method is better than those of the former two for a smaller multiple index l.Here，we adopt the sixth-order WKB method to numerically calculate the solutions to the basic equation for resonant perturbations of a new f（R） black hole.The QNMs is as follows[9]i（ω2-V0）-V″ 0+∑6i=2Λi=n+12.
（22） Our numerical results of QNMs for scalar perturbations are listed in Table 1-3.As a reminder，the oscillating quasi-period and the damping time scale are shown in these tables.In Table 1，the complex frequencies vary with the parameter rs.The real parts of the quasinormal frequencies decrease with rs increasing.But the imaginary parts increase with rs.It means that the larger rs is，more quickly the f（R） black hole oscillates and the oscillation of this black hole decays.In Table 2，the relation between the frequencies of QNMs and the parameter α is listed.We find that the period of the oscillations increases with α increasing.While the damping rate gradually decreases as α increases.It corresponds to the case that with an increase of α，the oscillation and the decay of the perturbation both decrease.In Table 3，it is shown that the quasinormal frequencies vary with different l.The real parts and the imaginary parts of the quasinormal frequencies both increase as l increases.It is to say that the larger l is，the more slowly the perturbation oscillates and the oscillation decays.It is worth to note the quasinomal frequencies do not vary with the parameter U0 or β.It is to say that，for different U0 or β，the real part and the imaginary part of the quasinormal frequencies are constant.From these results，we conclude that the new f（R） black hole is stable under the scalar perturbations. 4 Conclusion
In this work，we obtain a spherically symmetric solution in the context of the metric f（R） gravity with F（r）=1+αr.Using the six-order WKB method，we calculate quasinormal frequencies of the scalar perturbations in this new f（R） black hole spacetime.The relations between the quasinormal frequencies and the parameters of this spacetimes are obtained.It is shown that the real parts of the quasinormal frequencies decrease while the imaginary parts increase with rs increasing；with an increase of α，the oscillation and the decay of the perturbation both decrease；the larger l is，the more slowly the perturbation oscillates and the oscillation decays.The most interesting result is that the real part and the imaginary part of the quasinormal frequencies keep constant for different U0 or β.From these results，we conclude that the new f（R） black hole is stable.
References：
[1] Camanho X O，Edelstein J D.Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity [J].Journal of High Energy Physics，2010，4（4）：1-31.
[2] Scherk J.An introduction to the theory of dual models and strings [J].Reviews of Modern Physics，1975，47（1）：123-164.
[3] Felice A，Tsujikawa S.f（R） theories [J].Living Reviews in Relativity，2010，13（1）：3.
[4] Pogosian L，Silvestri A.The pattern of growth in viable f（R） cosmologies [J].Physical Review D，2007，77（2）：023503.
[5] Carames T R P，Fabris J C，Mello E R，et al.f（R） global monopole revisited [J].European Physical Journal C，2017，77（7）：496.
[6] Man J，Cheng H.Analytical discussion on strong gravitational lensing for a massive source with a f（R） global monopole [J].Physical Review D，2015，92（2）：024004.
[7] Morais J P，Bezerra V B.Gravitational field of a rotating global monopole in f（R） theory [J].Modern Physics Letters A，2012，27（31）：1250178.
[8] Graca J P M，Vieira H S，Bezerra V B.Scalar and spinor QNMs of a black hole with a global monopole in f（R） gravity [J].General Relativity & Gravitation，2016，48（4）：1-13.
[9] Konoplya R A.Quasinormal behavior of the D-dimensional Schwarzschild black hole and higher order WKB approach [J].Physical Review D，2003，68（2）：1557-1567.
[10] Schutz B F，Will C M.Black hole normal modes：A semi-analytic approach [J].Astrophysical Journal，1985，29（2）：L33-L36.
[11] Seidel E，Iyer S.Black-hole normal modes：A WKB approach.II.Schwarzschild black holes [J].Physical Review D，1990，41（2）：3632-3636.
（責任编辑：顾浩然）