Vibrating Black Holes in f(R) gravity
aa r X i v : . [ g r- q c ] A ug Prepared for submission to JCAP
Vibrating Black Holes in f(R) gravity
Anne Marie Nzioki, a Rituparno Goswami, a Peter K. S. Dunsby b,c a Astrophysics & Cosmology Research Unit, School of Mathematics Statistics and Computer Science,University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa. b Astrophysics Cosmology & Gravity Centre and Department of Mathematics and Applied Mathe-matics, University of Cape Town, Rondebosch, 7701, South Africa. c South African Astronomical Observatory, Observatory, Cape Town, South Africa.E-mail: [email protected], [email protected], [email protected]
Abstract.
We consider general perturbations of a Schwarzschild black holes in the context of f ( R ) gravity. A reduced set of frame independent master variables are determined, which obey twoclosed wave equations - one for the transverse, trace-free (tensor) perturbations and the other forthe additional scalar degree of freedom which characterise fourth-order theories of gravity. We showthat for the tensor modes, the underlying dynamics in f ( R ) gravity is governed by a modified Regge-Wheeler tensor which obeys the same Regge-Wheeler equation as in General Relativity. We find thatthe possible sources of scalar quasinormal modes that follow from scalar perturbations for the lowermultipoles result from primordial black holes, while higher mass, stellar black holes are associatedwith extremely high multipoles, which can only be produced in the first stage of black hole formation.Since scalar quasi-normal modes are short ranged, this scenario makes their detection beyond therange of current experiments. ontents f ( R ) Gravity 33 Formalism 4 f ( R ) gravity 10 Einstein’s theory of General Relativity (GR) [1] is widely accepted to be one of the most successfulfundamental theories in modern physics. Despite it’s success, corrections to GR have been introducedrecently to accommodate recent observations from the number counts of clusters of galaxies [2], mea-surements of type Ia supernovae [3] and the Cosmic Microwave Background (CMB) anisotropies [4],which together seem to indicate that the energy density budget of the Universe comprises 5% ordinarymatter (baryons, radiation and neutrinos), while the rest, which does not interact electromagnetically,consists of 27% dark matter and 68% Dark Energy (DE) [5]. If GR is the correct theory of the gravita-tional action then its application to cosmology should incorporate these observations. Consequently,the simplest best fit model to our Universe is the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW)model, which is dominated by cold dark matter (CDM) and DE in the form of an effective cosmologi-cal constant, whose nature is still to be understood and is required to explain the late-time acceleratedexpansion of the Universe.One of the main motivations for exploring possible alternative theories of gravity arises from theobscure nature of DE candidates. One possibility is to conjecture that the apparent need for DE couldsimply be a consequence of the break down of Einstein’s equations on astrophysical and cosmologicalscales. One theory of modified gravity that has recently attracted a considerable amount of attention– 1 –s fourth order gravity (FOG), which admits cosmologies that accelerate at late times without thepresence of DE [6–11] and can and can account for the rotation curves for spiral galaxies without theneed for dark matter [12] (see [13–17] for detailed reviews).The tetrad description of spacetime includes the Newman-Penrose null tetrad method [18] andthe developed by Ehlers and Ellis [19–21] which includes both a full and‘semi-tetrad’ approach. The latter formalism is based on a 1+3 threading of the spacetime manifoldwith respect to a timelike congruence in such a way that tensorial objects encoding the physicscan be decomposed into their space and time parts, and has been a useful tool for understandingof many aspects of relativistic fluid flows in cosmology and relativistic astrophysics. In particularthe 1+3 approach to cosmological perturbation theory, developed by Ellis, Bruni and Dunsby [22–24],which built on early work by Hawking [25], Lyth and Mukherjee [26] and Ellis and Bruni [27], employskinematic and dynamical variables to describe scalar, vector and tensor perturbations which have botha clear physical and geometric meaning and remain valid in all coordinate systems. This approach hasbeen used to tackle problems in linear and non-linear perturbation theory and has been particularlysuccessful in describing the physics of the CMB [28–31]. More recently, linear perturbation theoryhas been developed for describing the cosmology of fourth order theories of gravity (FOG) using the1+3 covariant approach [32–36], providing important features that differentiate the structure growthscenarios in FOG from standard GR.A natural extension to the 1+3 approach, suitable for problems which have spherical symmetry,including the Schwarzschild solution, Lemaˆıtre-Tolman-Bondi (LTB) models and many classes ofBianchi models was developed by Clarkson and Barrett [37]. This approach involves a ‘semi-tetrad’where, in addition to the timelike vector field of the 1+3 approach, a spatial vector is introduced.In GR, this ‘1+1+2 formalism’ has been applied to the study of perturbations of locally rotationallysymmetric (LRS) spacetimes [37–47] and strong lensing studies [48]. It has also been introduced todescribe the properties of LRS spacetimes in the context of f ( R ) gravity [49–51].In GR, linear perturbations of black holes was first considered by Chandrasekhar using themetric approach together with the Newman-Penrose (NP) formalism [52] and more recently usingthe ‘semi-tetrad’ 1+1+2 covariant formalism by Clarkson and Barrett [37]. In the metric approach,perturbations of the Schwarzschild spacetime geometry are described by two wave equations, i.e.,the Regge -Wheeler equation for odd parity modes and the Zerilli equation in the even parity case.These wave equations are expressed as functions (and their derivatives) in the perturbed metric whichare not gauge-invariant, as a general coordinate transformation would not preserve the form of thewave equation. Using the 1+1+2 covariant approach, Clarkson and Barrett [37] demonstrated thatboth the odd and even parity perturbations may be unified in a covariant wave equation equivalentto the Regge -Wheeler equation. This wave equation is characterised by a single a covariant, frame-and gauge-invariant, transverse-traceless tensor. These results were extended to include couplings (atsecond order) to a homogeneous magnetic field leading to an accompanying electromagnetic signalalongside the standard tensor (gravitational wave modes) [39].There have been a number of recent investigations of the properties of black holes in FOG theoriesincluding an extensive study of the Schwarzschild de Sitter black hole in [53, 54]. Perturbations ofSchwarzschild black holes in f ( R, G ) gravity were considered in [55] and a stability analysis of theSchwarzschild black hole in the Einstein Frame was presented in [56].The aim of this paper is to apply the 1+1+2 approach to the analysis of general linear pertur-bation of a Schwarzschild black hole in f ( R ) gravity. We perform all our calculations in the JordanFrame, where the dynamics of the extra gravitational degree of freedom inherent in FOG theories isdetermined by the trace of the effective Einstein equations, leading to a linearised scalar wave equationfor the Ricci scalar. Gauge invariance is assured by constructing perturbation variables which satisfythe Stewart-Walker lemma [57] and we adopt the standard linearisation procedure by dropping allterms which are second order or higher in these variables. Harmonic functions can then be introducedin the background which results in two decoupled parities reflecting the invariance of the backgroundspacetime under parity transformation. The introduction of harmonics reduces the problem of findinga solution to one of simply solving a system of linear equations algebraically. After introducing theharmonic functions, the main objective is to find a reduced set of master variables which obey a closed– 2 –et of wave equations.The outline of this paper is as follows. In Section 2 we introduce f ( R ) theories of gravity andpresent the general equations for these theories. Then in Section 3 we outline the 1+3 and field1+1+2 covariant methods in f ( R ) gravity which provide a covariant (gauge invariant) descriptionof spacetime. In Section 4 we present the vacuum field equations linearised around a Schwarzschildblack hole background using the 1+1+2 formalism. We discuss the spherical and time harmonics,which, when applied to our system of equations allows us to write them as a set of ordinary differentialequations (ODEs) for each mode. Closed covariant and gauge-invariant wave equations for scalar andtensor modes are given in Section 5. In the case of tensors, this is just the Regge-Wheeler equationfor a master variable that describes the evolution of a gauge and frame invariant transverse-traceless(TT) tensor.We then investigate the stability of these black hole to generic perturbations. Like in GR, initialtensor perturbations of the black hole eventually decay exponentially (ringing) at frequencies thatare characteristic of the black hole and independent of the source of the perturbation - a feature firstdiscovered by Vishveshwara in 1970 [58]. These quasinormal modes satisfy boundary conditions forpurely outgoing waves at infinity and purely ingoing waves at the black hole horizon. In addition tothese tensor modes, we also determine the quasinormal modes which arise from the additional scalardegree of freedom and discuss whether it is possible to use them to constrain f ( R ) gravity. In Section6 focuses on the method of solution to the perturbation equations using matrix methods where wedemonstrate the significance of the freedom of choice of frame basis. Finally in section 7 we presentor our conclusions.Unless otherwise specified, geometric units (8 πG = c = 1) will be used throughout this paper.The symmetrisation and the anti-symmetrisation over the indexes of a tensor T ab are defined as T ( ab ) = 12 ( T ab + T ba ) , T [ ab ] = 12 ( T ab − T ba ) . (1.1)over the indexes of the tensor. The symbol ∇ represents the usual covariant derivative and ∂ corre-sponds to partial differentiation. f ( R ) Gravity
One of the most widely studied modifications to General Relativity is f ( R ) gravity which is derivedfrom the following action: S = 12 Z dV (cid:2) √− g f ( R ) + 2 L M ( g ab , ψ ) (cid:3) , (2.1)where R is the Ricci scalar. This represents the simplest generalisation of the Einstein-Hilbert action.Demanding that the action (2.1) be invariant under a particular choice of symmetry guarantees thatthe resulting field equations also respect that symmetry. That being the case, since the Lagrangian isa function R only, and R is a generally covariant and a locally Lorentz invariant scalar quantity, thenthe field equations that follow are generally covariant and Lorentz invariant (2.1). After variationwith respect to the metric g ab are given by: δ S = − Z dV √− g (cid:20) f g ab δg ab − f ′ δR + T Mab δg ab (cid:21) , (2.2)where ′ denotes differentiation with respect to R , and T Mab is the matter energy momentum tensor(EMT) defined as T Mab = − √− g δ L M δg ab . (2.3)Writing the Ricci scalar as R = g ab R ab and assuming the connection is the Levi-Civita one, we canwrite f ′ δR ≃ δg ab ( f ′ R ab + g ab (cid:3) f ′ − ∇ a ∇ b f ′ ) , (2.4)– 3 –here the ≃ sign denotes equality up to surface terms and (cid:3) ≡ ∇ c ∇ c . By demanding that the actionbe stationary, so that δ S = 0 with respect to variations in the metric, one has finally f ′ (cid:18) R ab − g ab R (cid:19) = 12 g ab ( f − R f ′ ) + ∇ a ∇ b f ′ − g ab (cid:3) f ′ + T Mab . (2.5)It can be seen that for the special case f = R , the equations reduce to the standard Einstein fieldequations.It is convenient to write (2.5) in the form of effective Einstein equations as G ab = (cid:18) R ab − g ab R (cid:19) = ˜ T Mab + T Rab = T ab , (2.6)where we define T ab as the total EMT comprising˜ T Mab = T Mab f ′ (2.7)and T Rab = 1 f ′ (cid:20) g ab ( f − R f ′ ) + ∇ a ∇ b f ′ − g ab (cid:3) f ′ (cid:21) . (2.8)The components of the T ab can be considered to represent two effective “fluids” [6, 10, 12, 59]: thecurvature “fluid” (associated with T Rab ) and the effective matter “fluid” (associated with ˜ T Mab ). Thisallows us to adapt more easily techniques from the “covariant approach” (see, [21, 24, 27, 37, 60]), tostudy a wide range of problems in f ( R ) gravity that were originally devised for GR.The field equations (2.6) are fourth order in derivatives of the metric, which can be seen from theexistence of the ∇ a ∇ b f ′ term in (2.8). This result also follows directly from a ramification of Lovelock’stheorem [61, 62] which requires, in a four-dimensional Riemannian manifold, that the construction ofa metric theory of modified gravity admits higher than second order derivatives to the field equations.This feature is problematic in a Lagrangian based theory as it can lead to Ostrogradski instabilities[63] in the solutions of the field equations. In f ( R ) theories, however, these instabilities are absent[64], due to the existence of an equivalence with scalar-tensor theories.In order to help avoid confusion later, we point out that we use the superscripts M and R todenote quantities relating to the standard matter fluid and curvature fluid respectively and that theunbarred dynamic quantities with no superscripts are derived from the total effective EMT. The covariant approach we consider adopts a fluid-flow description of the matter content (includingany modifications to General Relativity) of spacetime. In the usual 1+3 splitting [19, 20, 65–67] ofspacetime, the fluid flow is determined at each point by the field vector u a , tangent to the flow lines.The vector u a is a timelike unit vector representing the normalised 4-velocity of the matter, hence u a u a = − . (3.1)The tensor h ab ≡ g ab + u a u b , (3.2)projects any tensor onto the hypersurface orthogonal to u a and has the following properties h ab u b = 0 , h ac h cb = h ab , h aa = 3 . (3.3)These constant time hypersurfaces represent the local rest 3-space associated with the observer.– 4 –he effective volume element for the rest space of the comoving observer is given by ε abc = ε abcd u d , where ε abc = ε [ abc ] and ε abc u c = 0 , (3.4)where ε abcd is the four-dimensional volume element ( ε abcd = p | det g | ) δ a δ b δ c δ d ] ) of the space-time manifold.Any projected rank-2 tensor S ab can be split as S ab = S h ab i + 13 S h ab + S [ ab ] , (3.5)where S = h ab S ab is the spatial trace, S h ab i is the orthogonally projected symmetric trace-free PSTFpart of the tensor defined as S h ab i = (cid:18) h c ( a h db ) − h ab h cd (cid:19) S cd , (3.6)and S [ ab ] is the antisymmetric part of this tensor. We use angle brackets to represent any PSTFtensors.The covariant derivatives for any tensor S a..bc..d are defined as the time derivative along u a :˙ S a..bc..d ≡ u f ∇ f S a..bc..d , (3.7)and the covariant spatial derivative defined in the local rest 3-spaces orthogonal to u a :D e S a..bc..d = h ej h al ... h bg h f c ... h id D j S l..gf..i , (3.8)with projection on all the free indices.Kinematical quantities are introduced by decomposing the covariant derivative of u a into itsirreducible parts: ∇ a u b = − u a ˙ u b + σ ab + ω ab + 13 Θ h ab . (3.9)where ˙ u b = u c ∇ c u b , Θ = D a u a , ω ab = D [ a u b ] , σ ab = D h a u b i , (3.10)are respectively, the four-acceleration, the expansion scalar which represents the local volume rate ofexpansion of the fluid, the antisymmetric vorticity tensor which describes the rigid rotation of matterrelative to a non-rotating frame, and the PSTF shear tensor that determines the distortion arisingin the matter flow, leaving the volume invariant. By construction, the following properties hold forthese kinematical quantities σ [ ab ] = ω [ ab ] = 0 , ω ab u b = σ ab u b = 0 , σ aa = 0 . (3.11)The total energy momentum tensor (EMT) T ab as defined in (2.6) can be decomposed relative to u a by splitting it into parts parallel and orthogonal to u a as follows: T ab = µ u a u b + q a u b + u a q b + p h ab + π ab ; (3.12)where µ is the total effective energy density relative to u a , p the total isotropic pressure, q a the totalenergy flux (momentum density) relative to u a and π ab the and PSTF total anisotropic stress, suchthat µ = T ab u a u b = µ M f ′ + µ R , p = 13 T ab h ab = p M f ′ + p R , (3.13) q a = − T bc u c h ba = q Ma f ′ + q Ra , π ab = T cd h c h a h db i = π Mab f ′ + π Rab . (3.14)– 5 –n equation of state needs to be specified to relate the matter thermodynamic variables.The derivative terms of the curvature EMT T Rab can be decomposed into time and spatial partsresulting in T Rab = 1 f ′ (cid:20) g ab ( f − R f ′ ) − ˙ f ′ (cid:18) h ab θ + σ ab + ω ab (cid:19) + 13 h ab D f ′ + D h a D b i f ′ + 12 ε abc ε cdf D d D f f ′ − u a (cid:16) h cb (D c f ′ ) ˙ + ˙ u c u b D c f ′ − ˙ f ′ ˙ u b (cid:17) + u b (cid:18) θ D a f ′ + σ ac D c f ′ + ω ac D c f ′ + u a ¨ f ′ − D a ˙ f ′ (cid:19) − g ab (cid:16) ˙ u c D c f ′ − θ ˙ f ′′ − ¨ f ′ + D f ′ (cid:17)i . (3.15)The locally free gravitational field is given by the Weyl curvature tensor C abcd defined by the equation C abcd = R abcd − g [ a [ c R b ] d ] + 13 R g [ a [ c g b ] d ] . (3.16)which can be split relative to u a into the ‘electric’ and ‘magnetic’ Weyl curvature parts as follows: E ab = C abcd , H aa = 0 , , (3.17)in analogy to the 1+3 split of the Maxwell field strength tensor [68].The dynamical relations for an arbitrary spacetime in the 1+3 formulation of FOG arise fromthe Ricci identities for the fundamental timelike vector field u a , that is,2 ∇ [ a ∇ b ] u c = R abcd u d , (3.18)and from contracting the second Bianchi identities ∇ [ e R ab ] cd = 0 . (3.19)resulting in a set of propagation and constraint equations when covariantly decomposed [69]. We haveto include the trace of (2.5) R f ′ − f = − (cid:16) f ′′ D R + f ′′′ D a R D a R − f ′′′ ˙ R − f ′′ ¨ R + ˙ u c f ′′ D c R − f ′′ θ ˙ R (cid:17) . (3.20)in order to close the system of equations. The 1+3 covariant approach involves splitting spacetime into its temporal and spatial parts in such away that the local 3-space is orthogonal to the vector field u a which provides a timelike threading forthe spacetime. This can be naturally extended to give a 1+1+2 covariant decomposition of spacetimeby introducing the unit vector field n a in the local 3-space orthogonal to u a , such that n a n a = 1 , n a u a = 0 . (3.21)The 2-dimensional tensor N ab ≡ g ab + u a u b − n a n b , n a N ab = 0 = u a N ab , N aa = 2 . (3.22)projects onto the tangent 2-spaces (which we call ‘sheets’) orthogonal to both u a and n a . The volumeelement of the sheet is the totally anti-symmetric 2-tensor ε ab ≡ ε abc n c , (3.23)– 6 –here ε abc is the volume element of the 3-spaces.The covariant derivatives for any tensor S a..bc..d are defined as the time derivative ‘ . ’ along u a as given in (3.7), the spatial divergence ‘ˆ’ along n a in the surfaces orthogonal to u a ˆ S a..bc..d ≡ n f D f S a..bc..d , (3.24)and the projected covariant derivative ‘ δ a ’ on the sheet δ f S a..bc..d ≡ N f j N al ... N bg N hc ... N id D j S l..gh..i , (3.25)where again the projection applies to every free index. The spatial derivative ‘ D a ’ is as defined in(3.8).In the 1+1+2 splitting of spacetime, any 3-vector V a can be irreducibly split into a scalarcomponent, V , along n a and a 2-vector component on the sheet, V a , orthogonal to n a , i.e., V a = V n a + V a , where V ≡ V a n a and V a ≡ N ab V b , (3.26)Similarly, a PSTF 3-tensor, V ab , can be decomposed into 2-scalar, 2-vector and PSTF 2-tensor partsas V ab = V h ab i = V (cid:18) n a n b − N ab (cid:19) + 2 V ( a n b ) + V ab , (3.27)where V ≡ n a n b V ab = − N ab V ab , V a ≡ N ab n c V bc , V ab ≡ V { ab } ≡ (cid:18) N ( ac N b ) d − N ab N cd (cid:19) V cd . (3.28)The curly brackets denote the part of a tensor which is PSTF with respect to n a .It then follows that the 1+3 kinematical and Weyl quantities can be irreducibly split as˙ u a = A n a + A a , (3.29) ω a = Ω n a + Ω a , (3.30) σ ab = Σ (cid:18) n a n b − N ab (cid:19) + 2 Σ ( a n b ) + Σ ab , (3.31) E ab = E (cid:18) n a n b − N ab (cid:19) + 2 E ( a n b ) + E ab , (3.32) H ab = H (cid:18) n a n b − N ab (cid:19) + 2 H ( a n b ) + H ab . (3.33)The irreducible form of the covariant decomposition of the derivative of n a is ∇ a n b = −A u a u b − u a α b + (cid:18)
13 Θ + Σ (cid:19) n a u b + (Σ a − ε ac Ω c ) u b + n a a b + 12 φ N ab + ξ ε ab + ζ ab , (3.34)where along the spatial direction n a , φ = δ a n a is the expansion of the sheet, ζ ab = δ { a n b } is the shearof n a and a a = n c D c n a = ˆ n a its acceleration, while ξ = ε ab δ a n b is the vorticity associated with n a .Finally, the anisotropic fluid variables q a and π ab can be split as follows: q a = Q n a + Q a , (3.35) π ab = Π (cid:18) n a n b − N ab (cid:19) + 2Π ( a n b ) + Π ab . (3.36)– 7 – .2.1 Energy momentum tensor In terms of the 1+1+2 variables, the total energy momentum tensor (3.12) is given by T ab = µ u a u b + p h ab + 2 u ( a (cid:2) Q n b ) + Q a (cid:3) + Π (cid:18) n a n b − N ab (cid:19) + 2 Π ( a n b ) + Π ab . (3.37)Moreover, in terms of the 1+1+2 variables, the curvature fluid can be decomposed as follows: µ R = 1 f ′ (cid:20)
12 (
R f ′ − f ) − θf ′′ ˙ R + f ′′′ X + f ′′′ δ a R δ a R + f ′′ ˆ X + φf ′′ X − a a f ′′ δ a R + f ′′ δ a δ a R (cid:21) , (3.38) p R = 1 f ′ (cid:20)
12 ( f − R f ′ ) + 23 θ f ′′ ˙ R + f ′′′ ˙ R + f ′′ ¨ R − A f ′′ X − A a f ′′ δ a R −
23 ( φ f ′′ X + f ′′′ δ a R δ a R + f ′′ δ a δ a R + f ′′′ X + f ′′ ˆ X − a a f ′′ δ a R ) i , (3.39) Q R = − f ′ h f ′′′ ˙ R X + f ′′ (cid:16) ˙ X − A ˙ R (cid:17) − α a f ′′ δ a R i , (3.40) Q Ra = 1 f ′ (cid:20)(cid:18) θ −
12 Σ (cid:19) f ′′ δ a R + (cid:0) Σ a − ε ab Ω b (cid:1) f ′′ X + (cid:0) Σ ab + ε ab Ω (cid:1) f ′′ δ b R − ˙ R f ′′′ δ a R − f ′′ δ a ˙ R (cid:21) , (3.41)Π R = 1 f ′ (cid:20) (cid:16) f ′′′ X + 2 f ′′ ˆ X − A a f ′′ δ a R − φ f ′′ X − f ′′′ δ a R δ a R − f ′′ δ a δ a R (cid:17) − Σ f ′′ ˙ R (cid:21) , (3.42)Π Ra = 1 f ′ (cid:20) − Σ a f ′′ ˙ R + X f ′′′ δ a R + f ′′ δ a X − φ f ′′ δ a R + (cid:0) ξ ε ab − ζ ab (cid:1) f ′′ δ b R − (cid:0) Σ a + ε ab Ω b (cid:1) f ′′ ˙ R (cid:21) , (3.43)Π Rab = 1 f ′ h − Σ ab f ′′ ˙ R + ζ ab f ′′ X + f ′′′ δ { a R δ b } R + f ′′ δ { a δ b } R i , (3.44)where we have defined ˆ R = X . Additionally, the 1+1+2 split of the curvature trace equation (3.20)results in R f ′ − f = 3 (cid:16) f ′′ θ ˙ R − f ′′′ X − f ′′′ δ a R δ a R − ( A + φ ) f ′′ X − f ′′ ˆ X − f ′′ δ a δ a R + f ′′′ ˙ R + f ′′ ¨ R (cid:17) . (3.45) The three derivatives defined so far, dot - ‘ ˙ ’, hat - ‘ˆ’ and delta - ‘ δ a ’ satisfy the following commutationrelations when they act on scalars V :ˆ˙ V − ˙ˆ V = −A ˙ V + (cid:18)
13 Θ + Σ (cid:19) ˆ V + (cid:0) Σ a + ε ab Ω b − α a (cid:1) δ a V , (3.46) δ a ˙ V − ( δ a V ) ·⊥ = −A a ˙ V + (cid:0) α a + Σ a − ε ab Ω b (cid:1) ˆ V + (cid:18)
13 Θ −
12 Σ (cid:19) δ a V + (Σ ab + Ω ε ab ) δ b V , (3.47) δ a ˆ V − ( d δ a V ) ⊥ = − ε ab Ω b ˙ V + a a ˆ V + 12 φ δ a V + ( ζ ab + ξ ε ab ) δ b V , (3.48) δ [ a δ b ] V = ε ab (cid:16) Ω ˙
V − ξ ˆ V (cid:17) . (3.49)– 8 –-vectors V a : ˆ˙ V ¯ a − ˙ˆ V ¯ a = −A ˙ V ¯ a + (cid:18)
13 Θ + Σ (cid:19) ˆ V ¯ a + (Σ b + ε bc Ω c − α b ) δ b V a + A a (Σ b + ε bc Ω c ) V b + H ε ab V b , (3.50) δ a ˙ V b − ( δ a V b ) ·⊥ = −A a ˙ V b + ( α a + Σ a − ε ac Ω c ) ˆ V ¯ b + (cid:18)
13 Θ −
12 Σ (cid:19) ( δ a V b + V a A b )+ (Σ ac + Ω ε ac ) ( δ c V b + V c A b ) + 12 ( V a Q b − N ab V c Q c ) − (cid:18) φ N ac + ξ ε ac + ζ ac (cid:19) V c α b + H a ε bc V c , (3.51) δ a ˆ V b − ( d δ a V b ) ˆ ⊥ = − ε ac Ω c ˙ V ¯ b + a a ˆ V ¯ b + 12 φ ( δ a V b − V a a b ) + ( ζ ac + ξ ε ac ) ( δ c V b − V c a b ) − (cid:0) Ω ε a [ b + Σ a [ b (cid:1) (cid:0) Σ c ] + ε c ] d Ω d (cid:1) V c − (cid:20)(cid:18)
12 Σ −
13 Θ (cid:19) (Σ b + ε bc Ω c ) + 12 Π b + E b (cid:21) V a + N ab (cid:20)(cid:18)
12 Σ −
13 Θ (cid:19) (cid:0) Σ c + ε cd Ω d (cid:1) + 12 Π c + E c (cid:21) V c , (3.52) δ [ a δ b ] V c = "(cid:18)
13 Θ −
12 Σ (cid:19) − φ + 12 Π + E − µ V [ a N cb ] −V [ a (cid:20) − (cid:18)
13 Θ −
12 Σ (cid:19) (cid:16) Σ cb ] + Ω ε cb ] (cid:17) + 12 φ (cid:16) ζ cb ] + ξ ε cb ] (cid:17) + 12 Π cb ] + E cb ] (cid:21) + N c [ a (cid:20) − (cid:18)
13 Θ −
12 Σ (cid:19) (cid:0) Σ b ] d + Ω ε b ] d (cid:1) + 12 φ (cid:0) ζ b ] d + ξ ε b ] d (cid:1) + 12 Π b ] d + E b ] d (cid:21) V d − h(cid:16) Σ c [ a + Ω ε c [ a (cid:17) (cid:0) Σ b ] d + Ω ε b ] d (cid:1) − (cid:16) ζ c [ a + ξ ε c [ a (cid:17) (cid:0) ζ b ] d + ξ ε b ] d (cid:1)i V d + ε ab (cid:16) Ω ˙ V ¯ c − ξ ˆ V ¯ c (cid:17) , (3.53)where we have used both the bar ‘¯’ over the index and ‘ ⊥ ’ to denote projection onto the sheet.Analogous relations for second-rank tensors hold but are more complicated. The key variables of the 1+1+2 formalism of FOG are the irreducible set of geometric variables, { R, Θ , A , Ω , Σ , E , H , φ, ξ, A a , Ω a , Σ a , α a , a a , E a , H a , Σ ab , ζ ab , E ab , H ab } , (3.54)together with the set of irreducible thermodynamic matter variables, { µ M , p M , Q M , Π M , Q Ma , Π Ma , Π Mab } , (3.55)for a given equation of state. The full 1+1+2 equations for the above covariant variables can beobtained by applying the 1+1+2 decomposition procedure to the 1+3 equations, and in addition, bycovariantly splitting the Ricci identities for n a : R abc ≡ ∇ [ a ∇ b ] n c − R abcd n d = 0 , (3.56)where R abcd is the Riemann curvature tensor. By splitting this third-rank tensor using the two vectorfields u a and n a , we obtain the evolution equations (along u a ) and propagation equations (along n a )for α a , a a , φ , ξ and ζ ab . The full set of 1+1+2 equations for arbitrary spacetimes is given in [41] .– 9 – Perturbations around a Schwarzschild black hole in f ( R ) gravity In this section we present the complete set of 1+1+2 covariant and gauge invariant evolution, propa-gation and constraint equations linearised around the Schwarzschild background in f ( R ) gravity. In the standard approach to investigating perturbations, any quantity T in the physical manifold M can be split into a background part T on the background manifold ¯ M and a small perturbation δ T . T = T + δ T (4.1)To define the perturbations a gauge choice has to be made. This essentially corresponds to a choice ofthe mapping Φ between the real spacetime defined by the manifold M and the fiducial (background)manifold ¯ M . The existence of arbitrary numbers of mappings corresponds to the gauge freedomof the theory and herein lies the problem of choosing the best way of constructing this mapping orcorrespondence - also known as the “fitting problem” in cosmology [27]. If a quantity is invariantunder this choice of mapping, then it is gauge invariant.An alternative definition of gauge invariance is described by the Stewart & Walker lemma [57]:This states that a variable is gauge invariant in M if and only if it eitheri. vanishes in ¯ M ,ii. is a constant scalar in ¯ M ,iii. is a constant linear combination of products of Kronecker deltas with constant coefficients.The definition of gauge invariance we use here is from the first two options. In this case the mappedquantity will be constant regardless of choice of mapping Φ.The covariant approach presented here is based on the introduction of a partial frame in thetangent space of each point. Once the frame has been chosen, a complete set of covariantly defined(i.e., gauge invariant) exact variables, all of which vanish in the background, are obtained. Thesevariables make up the equations describing the true spacetime. Since the true spacetime lacks thesymmetry of the background, there are a number of natural choices for the choice of frame vectors andit then follows that one is free to choose the frame to work in. Hereafter, the term ‘frame invariant’refers to invariance under the choice of frame vectors. The background spacetime we consider is spherically symmetric. Spherically symmetric spacetimes arerotationally symmetric about a preferred spatial direction with zero vorticity [70]. Since continuoussymmetry of isotropy at each point applies, of all 1+1+2 vectors and tensors vanish and the spacetimeis described by the covariantly defined scalars: { R, Θ , A , Ω , Σ , E , H , φ, ξ, µ M , p M , Q M , Π M } . (4.2)The further constraint that the vorticity vanishes Ω = ξ = 0 results in a zero magnetic Weyl curvaturescalar H = 0. Thus the variables { R, Θ , A , Σ , E , φ, µ M , p M , Q M , Π M } , (4.3)fully describe the spherically symmetric spacetime.If we consider the geometry of a vacuum ( µ M = p M = Q M = Π M = 0) spherically symmetricspacetime, then the set of scalars that describe spacetime reduces to { R, A , Θ , φ, Σ , E} . (4.4)– 10 –he condition of staticity implies that Θ and Σ vanish [51].If we impose further the conditions | f ′ (0) | < + ∞ , | f ′′ (0) | < + ∞ , | f ′′′ (0) | < + ∞ . (4.5) f (0) = 0 , R = 0 , f ′ (0) = 0 , (4.6)the system of equations for the variables reduces toˆ φ = − φ − E , (4.7)ˆ E = − φ E , (4.8)ˆ A = − A ( φ + A ) , (4.9)together with the constraint: E + A φ = 0 . (4.10)The parametric solutions for these variables are φ = 2 r r − mr , A = mr (cid:20) − mr (cid:21) − , E = 2 mr , (4.11)where m is the Schwarzschild mass. We now linearise the field equations (evolution, propagation and constraint) as given in [41] for FOG around a Schwarzschild background. The background is characterised by the variables {A , E , φ } and { ˆ A , ˆ E , ˆ φ } which are of zeroth-order. The remaining set of 1+1+2 variables { R, Θ , Σ , Ω , H , ξ, A a , Ω a , Σ a , α a , a a , E a , H a , Σ ab , E ab , H ab , ζ ab } , (4.12)are first-order variables which vanish in the background. These quantities are all of O ( ǫ ) with re-spect to the Schwarzschild radius which sets up the scale for perturbations for a vacuum sphericallysymmetric spacetime with vanishing Ricci scalar [51]. Keeping in mind that gauge invariance holdsfor the variables (4.12), we linearise the equations by neglecting the products of these variables alongwith their derivatives and the dot - ‘ ˙ ’ and delta - ‘ δ ’ derivatives of {A , E , φ } to obtain: Evolution equations :˙ φ = (cid:18)
23 Θ − Σ (cid:19) (cid:18) A − φ (cid:19) + δ a α a + f ′′ f ′ ( A ˙ R − ˙ X ) , (4.13)˙ ξ = (cid:18) A − φ (cid:19) Ω + 12 ε ab δ a α b + 12 H , (4.14)˙Ω = 12 ε ab δ a A b + A ξ , (4.15)˙Σ −
23 ˙Θ = − φ A − δ a A a − E − f ′′ f ′ (cid:16) δ R + ( φ + 2 A ) X − R (cid:17) , (4.16) As a reminder, the thermodynamic quantities in [41] are derived from the total effective EMT that comprises boththe standard matter and curvature fluid terms. – 11 – E = (cid:18)
32 Σ − Θ (cid:19) E + ε ab δ a H b + φ A f ′′ f ′ ˙ R , (4.17)˙ H = − ε ab δ a E b − ξ E , (4.18)˙Σ ¯ a − ε ab ˙Ω b = δ a A + (cid:18) A − φ (cid:19) A a − E a + f ′′ f ′ (cid:18) δ a X − φ δ a R (cid:19) , (4.19)˙ E ¯ a + 12 ε ab ˆ H b = 34 E (cid:0) ε ab Ω b + Σ a − α a (cid:1) − (cid:18) φ + A (cid:19) ε ab H b + 34 ε ab δ b H + 12 ε bc δ b H ca , (4.20)˙ H ¯ a = − E ε ab A b − ε ab δ b E −
12 ( φ − A ) ε ab E b + ε c { d δ d E ca } − E f ′′ f ′ ε ab δ b R , (4.21)˙ ζ { ab } = (cid:18) A − φ (cid:19) Σ ab + δ { a α b } − ε c { a H cb } , (4.22)˙Σ { ab } = δ { a A b } + A ζ ab − E ab + f ′′ f ′ δ { a δ b } R , (4.23) f ′′ f ′ δ a ˙ R = δ a Σ − δ a θ + 2 ε ab δ b Ω + 2 δ b Σ ab + φ (cid:0) Σ a + ε ab Ω b (cid:1) + 2 ε ab H b . (4.24) Propagation equations : ˆ φ = − φ − E + δ a a a − f ′′ f ′ (cid:16) X + φ X + δ R (cid:17) , (4.25)ˆ ξ = − φ ξ + 12 ε ab δ a a b , (4.26)ˆΩ = − δ a Ω a + ( A − φ ) Ω , (4.27)ˆ A − ˙Θ = − δ a A a − ( A + φ ) A + f ′′ f ′ h R − δ R − ˆ X − (3 A + φ ) X i , (4.28)ˆΣ −
23 ˆΘ = − φ Σ − δ a Σ a − ε ab δ a Ω b + f ′′ f ′ (cid:16) ˙ X − A ˙ R (cid:17) , (4.29)– 12 – E = − φ E − δ a E a − E f ′′ f ′ X , (4.30)ˆ H = − δ a H a − φ H − E Ω , (4.31)˙ a ¯ a − ˆ α ¯ a = (cid:18) φ + A (cid:19) α a − (cid:18) φ − A (cid:19) (cid:0) Σ a + ε ab Ω b (cid:1) + ε ab H b + f ′′ f ′ δ a ˙ R , (4.32)ˆΣ ¯ a − ε ab ˆΩ b = 12 δ a Σ + 23 δ a θ − ε ab δ b Ω − φ Σ a + (cid:18) φ + 2 A (cid:19) ε ab Ω b − δ b Σ ab + f ′′ f ′ δ a ˙ R , (4.33)ˆ A a − a = − δ a A − (cid:18) A − φ (cid:19) A a − A a a + 2 E a − f ′′ f ′ (cid:18) δ a X − φ δ a R (cid:19) . (4.34)ˆ E ¯ a = 12 δ a E − δ b E ab − E a a − φ E a + E f ′′ f ′ δ a R , (4.35)ˆ H ¯ a = 12 δ a H − δ b H ab + 32 E (cid:0) Ω a − ε ab Σ b (cid:1) − φ H a , (4.36)ˆ ζ { ab } = − φ ζ ab + δ { a a b } − E ab − f ′′ f ′ δ { a δ b } R , (4.37)ˆΣ { ab } = δ { a Σ b } − ε c { a δ c Ω b } − φ Σ ab − ε c { a H cb } , (4.38)˙ E { ab } − ε c { a ˆ H cb } = − ε c { a δ c H b } + (cid:18) φ + 2 A (cid:19) ε c { a H cb } − E Σ ab , (4.39)˙ H { ab } + ε c { a ˆ E cb } = ε c { a δ c E b } + 32 E ε c { a ζ cb } − (cid:18) φ + 2 A (cid:19) ε c { a E cb } . (4.40) f ′′ f ′ (cid:18) δ a X − φδ a R (cid:19) = − δ a φ + ε ab δ b ξ + δ b ζ ab − E a , (4.41) The trace equation : f ′′ ( ˆ X − ¨ R ) = 13 R f ′ − f ′′ (cid:2) δ R + ( φ + A ) X (cid:3) . (4.42) Constraint equations : δ a Ω a + ε ab δ a Σ b = (2 A − φ ) Ω + H , (4.43)In the above equations, X = ˆ R, f ′ = f ′ (0) and f ′′ = f ′′ (0).– 13 – .4 Gauge invariant variables Not all the set of covariant equations in the previous section are gauge invariant due to the isolatedzeroth-order background terms that appear in them. To fix this, we define three key variables bytaking the angular derivatives of the background variables {E , φ, A} , W a = δ a E , (4.44) Y a = δ a φ , (4.45) Z a = δ a A . (4.46)These new variables vanish in background and are therefore gauge invariant. Applying the commuta-tion relations (3.47) and (3.48) and substituting for the subsequent equations, we obtain the followinglinearised propagation and evolution equations for these new variables:˙ W a = 32 φ E (cid:0) α a + Σ a − ε ab Ω b (cid:1) + 32 E (cid:18) δ a Σ − δ a Θ (cid:19) + ε bc δ a δ b H c + A φ f ′′ f ′ δ a ˙ R , (4.47)˙ Y a = (cid:18) φ + E (cid:19) (cid:0) α a + Σ a − ε ab Ω b (cid:1) + δ a δ c α c + (cid:18) φ − A (cid:19) (cid:18) δ a Σ − δ a Θ (cid:19) + f ′′ f ′ (cid:16) A δ a ˙ R − δ a ˙ X (cid:17) , (4.48)ˆ W a = − φ W a − E Y a + 32 φ E a a − δ a δ b E b − E f ′′ f ′ δ a X , (4.49)ˆ Y a = − W a − φ Y a + (cid:18) φ + E (cid:19) a a + δ a δ b a b − δ a R + f ′′ f ′ (cid:20)(cid:18) A + 12 φ (cid:19) δ a X + 12 (cid:18) E − φ (cid:19) δ a R + 12 δ δ a R − δ a ¨ R (cid:21) , (4.50)ˆ Z a = − (cid:18) φ + 2 A (cid:19) Z a − A Y a + A ( φ + A ) a a + δ a ˙Θ − δ a δ b A b + f ′′ f ′ (cid:16) δ a ¨ R − A δ a ˙ X (cid:17) . (4.51)These equations add no new information to what has already been given in the previous sectionhowever, since they are gauge invariant, we can replace the equations (4.17), (4.13), (4.30), (4.25) and(4.28) with (4.47), (4.48), (4.49), (4.50) and (4.51) respectively.The following additional constraints are obtained by applying the commutation relation (3.48) to thenew variables (4.47)-(4.51) , ε ab δ a W b = 3 φ E ξ , (4.52) ε ab δ a Y b = (cid:0) φ + 2 E (cid:1) ξ , (4.53) ε ab δ a Z b = 2 A ( φ + A ) ξ . (4.54)It is also useful to replace (4.16) with δ a ˙Σ − δ a ˙ θ = − W a − A Y a − φ Z a − δ a δ b A b − f ′′ f ′ (cid:20) δ δ a R − δ a ¨ R + (cid:18) E − φ (cid:19) δ a R + ( φ + 2 A ) δ a X (cid:21) . (4.55) The following are the relevant commutation relations for the derivatives of first-order scalar, vectorand tensor quantities, T : – 14 – calars : ˙ˆ T − ˆ˙ T = A ˙ T , (4.56) δ a ˙ T − ( δ a T ) · = 0 , (4.57) δ a ˆ T − \ ( δ a T ) = 12 φ δ a T , (4.58) δ [ a δ b ] T = 0 ; (4.59) Vectors : ˙ˆ T ¯ a − ˆ˙ T ¯ a = A ˙ T ¯ a , (4.60) δ [ a δ b ] T c = (cid:18) φ − E (cid:19) N c [ a T b ] ; (4.61) Tensors : ˙ˆ T { ab } − ˆ˙ T { ab } = A ˙ T { ab } , (4.62) δ [ a δ b ] T cd = (cid:18) φ − E (cid:19) (cid:0) N c [ a T b ] d + N d [ a T b ] c (cid:1) . (4.63) In order to solve the equations, it is standard procedure to decompose the first order variables harmon-ically (see, [25, 71]). The perturbations can be described by a linear system of ODEs by introducingspherical and time harmonics.
We perform a decomposition of first order perturbations into scalar, vector and tensor modes inanalogy with the FLRW models [21, 23]. The perturbations of the Schwarzschild geometry fall intotwo distinct classes based on how they transform on the surfaces of spherically symmetry: even(electric) and odd (magnetic) modes . Given the spherical symmetry of the background, we cannaturally use spherical harmonics to expand the first order quantities. This being the case, thescalars can be expanded as a sum of even modes and the vectors and tensors can be expanded in sumsover both the even and odd modes. Moreover, the angular derivatives appearing in the equationsare effectively replaced by a harmonic component of the derivative. The presentation in this sectionfollows [37] where the harmonics were introduced in a covariant manner.We introduce the set of dimensionless spherical harmonic functions Q = Q ( ℓ,m ) , with m = − ℓ, · · · , ℓ , defined on the background as eigenfunctions of the spherical Laplacian operator such that δ Q = − ℓ ( ℓ + 1) r Q . (4.64)The function Q is defined in order to be covariantly constant along u a and n a ,ˆ Q = 0 = ˙ Q . (4.65)The function r is, up to an arbitrary constant, covariantly defined byˆ rr = 12 φ , ˙ r = 0 = δ a r , (4.66) Alternatively, as first presented in Chandrasekhar’s book [52], odd perturbations are called axial and even pertur-bations are called polar. – 15 –nd gives a natural length scale to the spacetime as seen when r is defined as r ≡ (cid:18) φ − E (cid:19) − / . (4.67)We stress that these relations and harmonics are used in expanding gauge invariant first-order quan-tities only.We now look successfully at the expansion of first order scalars, vectors and tensors in sphericalharmonics and the replacements which must be made in the equations. Scalar harmonics
We can now define the harmonic expansion of any first order scalar Ψ in terms of the functions Q asΨ = ∞ X ℓ =0 m = ℓ X m = − ℓ Ψ ( ℓ,m ) S Q ( ℓ,m ) = Ψ S Q, (4.68)where from now on we drop the sum over ℓ and m (implicit in the last equality) in the harmonicexpansions hereafter. We use the subscript S to indicate that a scalar spherical harmonic expansionhas been made.The replacements which must be made for scalars when expanding the equations in sphericalharmonics are Ψ = Ψ S Q , (4.69) δ a Ψ = r − Ψ S Q a , (4.70) ε ab δ b Ψ = r − Ψ S ¯ Q a . (4.71) Vector harmonics
The vector harmonics can be either of even (electric) or odd (magnetic) parity. The even parity vectorspherical harmonics for ℓ ≥ Q ( ℓ ) a = r δ a Q ( ℓ ) (4.72)where Q a is covariantly constant along u a and n a ˆ Q a = 0 = ˙ Q a . (4.73)The vector harmonic (4.72) is defined as an eigenfunction of the spherical Laplacian operator: δ Q a = (1 − ℓ ( ℓ + 1)) r − Q a , (4.74)and satisfies the properties δ a Q a = − ℓ ( ℓ + 1) r − Q , (4.75) ε ab δ a Q b = 0 . (4.76)Similarly, we define odd parity vector spherical harmonics as¯ Q ( ℓ ) a = r ε ab δ b Q ( ℓ ) ⇒ ˆ¯ Q a = 0 = ˙¯ Q a , δ ¯ Q a = (1 − ℓ ( ℓ + 1)) r − ¯ Q a , (4.77)¯ Q a being a solenoidal vector, δ a ¯ Q a = 0 , (4.78)and satisfies the property ε ab δ a ¯ Q b = ℓ ( ℓ + 1) r − Q . (4.79)– 16 –ote that Q a and ¯ Q a are parity inversions of one another other¯ Q a = ε ab Q b ⇔ Q a = − ε ab ¯ Q b , (4.80)where ε ab is a parity operator.Since the even and odd vector harmonics are orthogonal: Q a ¯ Q a = 0 (for each ℓ ), then any first-ordervector Ψ a may be expanded in terms of these harmonics asΨ a = ∞ X ℓ =1 Ψ ( ℓ ) V Q ( ℓ ) a + ¯Ψ ( ℓ ) V ¯ Q ( ℓ ) a = Ψ V Q a + ¯Ψ V ¯ Q a . (4.81)where the V indicates that a vector spherical harmonic expansion has been made.As in the scalar case, the replacements to be made for vectors when expanding the equations inspherical harmonics are Ψ a = Ψ V Q a + ¯Ψ V ¯ Q a , (4.82) ε ab Ψ b = − ¯Ψ V Q a + Ψ V ¯ Q a , (4.83) δ a Ψ a = − ℓ ( ℓ + 1) r − Ψ V Q , (4.84) ε ab δ a Ψ b = ℓ ( ℓ + 1) r − ¯Ψ V Q , (4.85) δ { a Ψ b } = r − (cid:0) Ψ V Q ab − ¯Ψ V ¯ Q ab (cid:1) , (4.86) ε c { a δ c Ψ b } = r − (cid:0) ¯Ψ V Q ab + Ψ V ¯ Q ab (cid:1) . (4.87) Tensor harmonics
We define even and odd parity tensor spherical harmonics for ℓ ≥ Q ab = r δ { a δ b } Q, ⇒ ˆ Q ab = 0 = ˙ Q ab , δ Q ab = (cid:2) φ − E − ℓ ( ℓ + 1) r − (cid:3) Q ab , (4.88)¯ Q ab = r ε c { a δ c δ b } Q , ⇒ ˆ¯ Q ab = 0 = ˙¯ Q ab , δ ¯ Q ab = (cid:2) φ − E − ℓ ( ℓ + 1) r − (cid:3) ¯ Q ab , (4.89)and posses the same orthogonal and parity property Q ab ¯ Q ab = 0 ,Q ab = − ε c { a ¯ Q cb } ⇔ ¯ Q ab = ε c { a Q cb } , as the vector case. Any first-order tensor Ψ ab can be expanded in terms of these harmonics asΨ ab = ∞ X ℓ =2 Ψ ( ℓ ) T Q ( ℓ ) ab + ¯Ψ ( ℓ ) T ¯ Q ( ℓ ) ab = Ψ T Q ab + ¯Ψ T ¯ Q ab . (4.90)For the tensors, the following replacements must be made when expanding the equations in sphericalharmonics: Ψ ab = Ψ T Q ab + ¯Ψ T ¯ Q ab , (4.91) ε c { a Ψ b } c = − ¯Ψ T Q ab + Ψ T ¯ Q ab , (4.92) δ b Ψ ab = (cid:20) − ℓ ( ℓ + 1) (cid:21) r − (cid:0) Ψ T Q a − ¯Ψ T ¯ Q a (cid:1) , (4.93) ε c { d δ d Ψ a } c = − (cid:20) − ℓ ( ℓ + 1) (cid:21) r − (cid:0) ¯Ψ T Q a + Ψ T ¯ Q a (cid:1) . (4.94)– 17 – dd and even parity perturbations Expanding the perturbations into spherical harmonics, leads to two independent set of equations withthe following variables:
Odd perturbations : V O ≡{ ¯ E T , H T , ¯Σ T , ¯ ζ T } , { ¯ E V , H V , ¯Σ V , Ω V , ¯ A V , ¯ α V , ¯ a V , ¯ X V , ¯ Y V , ¯ Z V } , {H S , Ω S , ξ S } ; (4.95) Even perturbations : V E ≡ {E T , ¯ H T , Σ T , ζ T } , {E V , ¯ H V , Σ V , ¯Ω V , A V , α V , a V , X V , Y V , Z V } , { Σ S , θ S R S } ; (4.96)We see in the equations that ‘parity switching’ occurs between some sets of variables where certainterms always appear alongside the factor ‘ ε ab ’ relative to other variables (e.g., H ab and Ω a appearalongside ‘ ε ab ’ relative to the variables E ab and Σ a , respectively). Since the background is static, we can resolve the perturbations into temporal harmonics. We do thisby performing a Fourier analysis of the time derivatives of the first order quantities by decomposingthem into their Fourier components. This corresponds to assuming a harmonic time dependence e iωτ for the first order variables.We define the time harmonic function T ( ω ) in the background by˙ T ( ω ) = i ω T ( ω ) , ˆ T ( ω ) = 0 = δ a T ( ω ) ; ˙ ω = 0 = δ a ω . (4.97)From the commutation relation between the dot - ‘ . ’ and hat - ‘ ˆ ’ derivatives the above-defined timeharmonic must satisfy ˆ˙ T + A ˙ T = 0 , (4.98)which in turn implies ˆ ω = −A ω , (4.99)in the background.Integrating (4.99) in terms of r , gives ω = σ (cid:18) − mr (cid:19) − / = 2 σφ r , (4.100)where σ is a constant. Then any first order variable Ψ in the equations may be expanded asΨ = X ω Ψ ( ω ) T ( ω ) = Ψ ( ω ) T ( ω ) , (4.101)and the dot - ‘ . ’ derivatives of these first order quantities can be replaced by factors of iω . In GR, the gravitational perturbations of Schwarzschild black holes are governed by a single second-order wave equation, namely the Regge -Wheeler equation [72], describing the odd perturbations andthe Zerilli equation [73] describing the even perturbations. Both the equations satisfy a Schr¨odinger-like equation and it was demonstrated in [74] that the effective potentials of these equations have thesame spectra. The aim of this section is to perform an analysis of the perturbation of the Schwarzschildblack hole in f ( R ) gravity and find a reduced set of master variables which obey a closed set of waveequations for these theories. – 18 – .1 Gravitational perturbations If we consider very large distances from the source ( A = φ = 0), the gravitational perturbationsshould be well approximated by a plane wave, with n a lying in the direction of propagation. Onimposing the condition that R vanishes at infinity, the plane gravitational waves are described by the1+1+2 transverse-traceless tensors E ab , H ab , Σ ab and ζ ab only, as in GR. Otherwise there is couplingwith the scalar waves which can produce other scalar and vector modes. The tensors E ab and H ab represent the tidal and gravitational waves effects in analogy with the propagation of electromagneticwaves. However, the wave equations for these two tensors do not close in the general frame.If we now consider the general case, apart from the four TT tensors, a number of other TT tensorscan be constructed from the δ - derivatives of vectors and scalars, for example, δ { a W b } , δ { a a b } , δ { a δ b } Ω,etc. The wave equations for these tensors can be calculated by applying the wave operator ¨Ψ { ab } − ˆˆΨ { ab } to that tensor Ψ ab [37]. The aim here is to calculate all such possible wave equations involvingthese tensors and systematically eliminating unwanted terms until a closed equation is obtained. Inparticular, calculating the wave operator for ζ ab and δ { a W b } , we notice that they contain similarterms.We consider the case of the wave operator for ζ ab , that is, ¨ ζ { ab } − ˆˆ ζ { ab } , where we apply thefollowing steps: − Take the dot- derivative across (4.22), for which the resulting evolution equations are substituted. − Substitute for a a from (4.49) and α a from (4.47) (while utilising the constraints (4.52),(4.24),(4.54), (4.41) and (4.53) to substitute for ξ, Σ , Z a even Y a and odd Y a respectively).What follows is an expression consisting of only δ { a W b } and ζ ab , for the odd harmonics and δ { a X b } , ζ ab and δ { a δ b } R for the even harmonics. We can recast this result as the wave equation,¨ M { ab } − ˆˆ M { ab } − A ˆ M { ab } + (cid:0) φ + E (cid:1) M ab − δ M ab = 0 , (5.1)where we have introduced the dimensionless, gauge-invariant, frame-invariant, transverse-tracelesstensor M ab defined as M ab = 12 φ r ζ ab − r E − δ { a W b } + f ′′ f ′ r δ { a δ b } R . (5.2)The even part of (5.2) is coupled to the curvature term and as a result we have to include the traceequation (4.42) to achieve closure. On the other hand, the curvature term vanishes for the odd partof M ab and this leaves the tensor in exactly the same form as in the GR case [37].We can expand (5.1) into scalar harmonics as¨ M − ˆˆ M − A ˆ M + (cid:20) ℓ ( ℓ + 1) r + 3 E (cid:21) M = 0 , (5.3)where we let M = { M T , M T } . In appropriate coordinates the wave equation (5.3) is the Regge -Wheelerequation . Both the odd and even parity parts of M ab satisfy the same wave equation (5.3).We convert to the parameter r using (5.4), the time harmonics in (5.3) and the fact that hatderivative of any scalar K for a static spacetime [38] isˆ M = 12 r φ dMdr , (5.4)to obtain κ M − mr (cid:20) m − rr (cid:21) dMdr + (cid:18) m − rr (cid:19) d Mdr + (cid:18) m − rr (cid:19) (cid:20) ℓ ( ℓ + 1) r − mr (cid:21) M = 0 . (5.5)– 19 –e then make a change to the ‘tortoise’ coordinate r ∗ , which is related to r by r ∗ = r + 2 m ln (cid:16) r m − (cid:17) , (5.6)thus, (5.5) can be written in the form (cid:18) d dr ∗ + κ − V T (cid:19) M = 0 , (5.7)with the effective potential V T V T = (cid:18) − mr (cid:19) (cid:20) ℓ ( ℓ + 1) r − mr (cid:21) , (5.8)which is the Regge -Wheeler potential for gravitational perturbations. The trace equation (4.42), which is a wave equation in the Ricci scalar R , corresponds to scalarmodes that are not present in standard GR but occur in f ( R ) theories of gravity due to the extrascalar degree of freedom. The equation constitutes the same generalised Regge -Wheeler equation formassive scalar perturbations on a LRS background spacetimes in GR with U = f ′ f ′′ , (5.9)as the effective mass of the scalar.To obtain the familiar Regge -Wheeler equation we first rescale R as R = r − R and use (4.7)and (4.66) to rewrite equation (4.42) in the form¨ R − ˆˆ R − A ˆ R − (cid:0)
E − U + δ (cid:1) R = 0 . (5.10)Proceeding as in the previous case, we introduce scalar spherical harmonics to (5.10) resulting in¨ R S − ˆˆ R S − A ˆ R S − (cid:20) E − ˜ U − ℓ ( ℓ + 1) r (cid:21) R S = 0 . (5.11)where ˜ U = C / (3 C ) with C and C as constants.Converting to the parameter r and then the tortoise coordinate, we get (cid:18) d dr ∗ + κ − V S (cid:19) R = 0 , (5.12)where V S = (cid:18) − mr (cid:19) (cid:20) ℓ ( ℓ + 1) r + 2 mr + ˜ U (cid:21) . (5.13)The expression (5.13) is the Regge -Wheeler potential for the scalar perturbations. The form of the wave equations (5.7) and (5.12) describing black hole perturbation is similar to a onedimensional Schr¨odinger equation and hence their potentials correspond to a single potential barrier.We consider the potential profile of the effective potentials V T and V S in a Schwarzschild black holecase for the gravitational and the scalar fields respectively. The Regge -Wheeler equations (5.7) and– 20 –5.12) can be made dimensionless by dividing through by the black hole mass m . In this way thepotentials (5.8) and (5.13) become V T = (cid:18) − r (cid:19) (cid:20) ℓ ( ℓ + 1) r − r (cid:21) , (5.14) V S = (cid:18) − r (cid:19) (cid:20) ℓ ( ℓ + 1) r + 2 r + u (cid:21) , (5.15)where we have defined (and dropped the primes), κ ′ = m κ , r ′ = rm , u = m ˜ U . (5.16)For the gravitational perturbations and the scalar perturbations with u = 0, the derivative of thepotential has two roots with one in the unphysical region r < r > u = 0, the potentialhas three extrema: one in the unphysical region r <
0, a local maximum at r max and local minimumat r min in the region r > < r max < r min .Fig 1 shows a plot of the potential for the gravitational field for different ℓ as a function ofthe Schwarzschild radial coordinate r in (a) and the tortoise coordinates r ∗ in (b). In this case thepotential decays exponentially near the horizon and as 1 /r at spatial infinity. Figure 1 . The potential for the gravitational field for ℓ = 2 , , r (a) and r ∗ (b). Fig 2 shows the potential profile for the scalar field for several values of u at ℓ = 2 in (a) andat ℓ = 4 in (b). We see that the effect of the massive term ˜ U is to move the asymptotic value of thepotential of scalar perturbations up by u and to cause the potential to approach the asymptotic valueslowly. Moreover, increasing the value of u causes the peak of the potential to broaden as the peakvalue decreases relative to the asymptotic value. The peak eventually disappears altogether when u exceeds a certain value. We now investigate the stability of the black hole to external perturbations which depends on theblack hole remaining bounded in time as it evolves. The asymptotic behaviour of the solutions to(5.7) is given as M ∼ e ± iκr ∗ , (5.17)both at the horizon and at spatial infinity. If we consider purely imaginary solutions such that we set κ = − iα , then the time dependence of the perturbations evolves like e αt , which is unstable owing to– 21 – igure 2 . The potential for the scalar field for different u as a function of r for ℓ = 2 (a) and ℓ = 3 (b). the fact that they grow exponentially with time. For regularity, we require the perturbation to falloff to zero at spatial infinity and therefore choose M ∼ e − αr ∗ . (5.18)If (5.18) is to be matched to the solution that goes to zero at the horizon, then ∂M/∂r ∗ < ∂ M/∂r ∗ < −∞ to ∞ . However, this is not the case since the potential is positivedefinite and as a result (5.7) never becomes negative in this range. Since the solutions cannot bematched, this rules out perturbations that grow exponentially with time. This proof of stability ofa black hole was first provided by [58]. Later on [75, 76] provided a more rigorous proof using theenergy integral. This can be derived by first considering the time dependent version of (5.7) (cid:18) ∂ ∂t − ∂ ∂r ∗ + V T (cid:19) M = 0 . (5.19)(recalling that the time dependence was replaced by the factor e iωt when we considered time harmon-ics). Multiplying (5.19) by the partial derivative of the complex conjugate M ∗ with respect to timeand then adding the resulting equation to its complex conjugate we get ∂∂r ∗ (cid:18) ∂M ∗ ∂t ∂M∂r ∗ + ∂M ∗ ∂t ∂M∂r ∗ (cid:19) = ∂∂t (cid:18) | ∂M∂t | + | ∂M∂r ∗ | + V T | M | (cid:19) . (5.20)After integration by parts over r ∗ from −∞ to ∞ , the left-hand side of (5.20) vanishes and we obtainthe energy integral, Z ∞−∞ (cid:18) | ∂M∂t | + | ∂M∂r ∗ | + V T | M | (cid:19) dr ∗ = constant . (5.21)Since V T is positive definite, the integral (5.21) bounds the integral of | ∂M/∂t | and it thereforeexcludes exponential growing solutions to (5.7). The above energy integral argument for stabilityfalls short of a complete proof as it does not rule out perturbations that grow linearly with t . Also,since we have only provided the bounds for integrals of M , the perturbation may still blow up as r → ∞ . The best proof of black hole stability was provided by Kay and Wald [77] which,unlike theenergy integral proof, proved that ψ remains pointwise bounded when (5.7) is evolved from a smooth,bounded initial data.The proof of stability for the scalar perturbations depends on ˜ U . The potential V S in (5.13) remainspositive definite subject to the condition ˜ U = C C ≥ . (5.22)– 22 –here could also be tachyonic instabilities associated with these modes if C ≤
0. Both these instabil-ities do not arise, however, as we have shown in [51] that the necessary conditions for the existence ofa Schwarzschild black hole solution in f ( R ) theories are consistent with the requirement that C > C > The gravitational quasinormal modes (QNMs) are solutions to the Regge -Wheeler equation (5.7)subject to the boundary conditions M ∼ ( e iκr ∗ for r ∗ → −∞ e − iκr ∗ for r ∗ → + ∞ . (5.23)These boundary conditions represent purely outgoing waves at infinity ( r ∼ r ∗ → ∞ ) and purelyingoing waves at the horizon ( r → m , r ∗ → −∞ ). In other words we want to discard unwantedcontributions at the event horizon and at spatial infinity as we do not want gravitational radiationentering the spacetime from infinity to continue to perturb the black hole, nor do we want wavescoming from the vicinity of the horizon.Obtaining solutions to (5.7) and (5.12) requires discrete values of the frequency parameter κ called quasinormal frequencies belonging to the quasinormal modes of the black hole. The quasinormalfrequencies have both a real and imaginary part which we write as κ = ℜ ( κ ) + ℑ ( κ ) . (5.24)Since QNMs are characterised by the parameters of the black hole [58], we expect the imaginarypart to be damped with time for each value of r ∗ due to energy being radiated to infinity or thehorizon. If we then consider that in (5.7) and (5.12) that the time dependence has been replaced bythe factor e iωt , we expect to have M ∼ e iκ ( t − r ∗ ) at spatial infinity. We see from this that ℑ ( κ ) < r ∗ → + ∞ . Thisoption for a negative imaginary part is excluded since the potential V T decays towards spatial infinityand therefore disallows these bound states. We can therefore only have ℑ ( κ ) > r ∗ → + ∞ on a hypersurface ofconstant time; the same holds for the horizon. This consequence of divergence is balanced out by thefact that it takes the signal an infinite time to reach, for example, spatial infinity.The scalar QNMs correspond to solutions of (5.12) with R ∼ ( e iχr ∗ for r ∗ → −∞ e − iχr ∗ for r ∗ → + ∞ , (5.25)where χ = p κ − ˜ U for the scalar field. For the choices ℑ ( κ ) ≈ κ ≤ ˜ U , there will be no energyradiating into infinity. The sign of χ is chosen so as to be in the same complex surface quadrant as κ .There have been numerous attempts to calculate QNMs to high accuracy using numerical andsemi-analytical methods. Difficulties arise from, for example, the admixture of the solutions such thatthe exponentially growing required solution gets contaminated by traces of the unwanted solutionwhich decreases exponentially as we approach the boundaries. In 1975, Chandrasekhar and Detweiler[74] computed numerically the first few modes and in 1985, Leaver [78] proposed the most accuratemethod to date. Other methods have been employed in [78–88]. Comprehensive reviews on black holeQNMs can be found in [89–92].For the scalar field perturbations, studies have shown that the mass of the field has crucialinfluence on the damping rate of the QNMs. Using the WKB approximation [93–95], it was found thatwhen the massive term u of the scalar field increases, the damping rate decreases. Later calculations,using the continued fraction method by Leaver [96, 97], showed that as a result of the decreasingdamping rates, for certain values of u , there are QNM oscillations that are ‘almost’ purely real modeswith arbitrary long life. – 23 –n GR the possible sources of massive scalar QNMs are from the collapse of objects made up ofself-gravitating scalar fields (‘boson’ stars) [98–100], in situations where the massless field gains aneffective mass [101] or as scalar field dark matter [102]. In order to illustrate what these results meanfor f ( R ) theories of gravity we restrict our attention to the ℓ = 0 multipole of the field. From [96],the cut-off mass at which the QNMs disappear for these modes is approximately at m ˜ U = 0 . − . U as˜ U = C C >> L (5.26)where L is the smallest length scale on which Newtonian gravity has been observed. Recent results[104] place at L ∼ µm and using this we can set (5.26) as˜ U ≫ . × m − (5.27)Given these details, we can estimate that the mass of the black hole associated with the disappearanceof the QNMs BH mass ≪ µm . (5.28)Such a black hole could only have been formed from density fluctuations in the early universe [105, 106].Furthermore, if these primordial black hole are to be detected now, they would have to have an initialmass of subatomic scales ( ∼ − m ) [107]. These results apply to QNMs at lower overtones andeven then, QNMs are short-ranged, making their detection currently unfeasible [92]. The structure of the system of governing equations for the perturbations is made up of covariantand gauge invariant evolution, propagation and constraint equations. The true degrees of freedom ofthis system is governed by the reduced set of master variables M and R , which obey the tensorialequations (5.1) and (4.42), respectively. All other variables are then related to these master variablesby quadrature, plus frame degrees of freedom. Harmonic expansion of the perturbation equationsallows us, at any radial position from the black hole, to present the equations in matrix form. Theharmonic variables in (4.95) and (4.96) can then be treated as the basis of a 34-dimensional vectorspace V . We can then analyse the system of equations to obtain solutions. In this section, wepresent the procedure for this analysis, as set out in [37]. − After adopting spherical harmonic decomposition, the number of variables in the system ofequations is 34 in total. Let V denote the 34-dimensional vector consisting of these odd V O and even V E variables as presented in (4.95) and (4.96) respectively, such that V = (Odd variables | Even variables) = ( V O , V E ) . (6.1) − We use the time harmonics in these equations which results in: •
29 propagation equations which constitute a linear system of ODEsˆ V = P V , (6.2)where V is a vector consisting of the 29 elements of V which have a propagation equationand P is a 29 ×
34 propagation matrix in which the evolution equations, where the dotderivatives are replaced by iω , contain hat derivatives in them. •
25 algebraic relations between the variables, made up of 18 evolution equations as well as7 constraints. These, in matrix notation, take the form
F V = , (6.3)– 24 –here F is a 25 ×
34 matrix. Since the constraints propagate and evolve consistently, thismeans that the rows that make up the constraints are really linear combinations of the18 rows that make up the algebraic relations derived from the evolution equations (thisexcludes the constraint (4.54) since there is no evolution equation for Z a ). As a result, 6of the rows in F give no additional information, resulting in F being of rank 19. − So far, the formulation has resulted in 34 unknowns and 19 algebraic relations in the systemwhich corresponds to 34 −
19 = 15 degrees of freedom. This means that there are 15 variablesthat need to be solved for, which we denote by v , and write V = C v , (6.4)where C is a 34 ×
15 matrix of the form ←− −→ ←− −→ odd even . (6.5) − We now split the vector v into two parts: v = ( v D , v F ), the first one v D containing the 10variables which have an individual propagation equation and the second one v F the 15 −
10 = 5variables that do not. The latter part corresponds to 5 frame degrees of freedom. Inserting (6.4)into the propagation equation, (6.2) yields the underlying propagation equation for the solutionvector as ˆ v D = B v D + A v F , (6.6)where B is a 10 ×
10 matrix and A is 10 × − Finally, since we have the freedom to choose the 5 frame basis ( v F ), we find that there are only10 − v D The problem of finding a solution lies in deciding which variables to choose as the basis. To concurwith [37] for the GR case, we will choose the frame in which ¯ Y V = ¯ A V = 0 and as a result ξ S = Ω S =¯ a V = W V = ¯ Z V = Ω V = 0. The basis vector for the solution is chosen to be v = M T ˆ M T ! ; (6.7)– 25 –ccording to (6.4), the remaining variables in terms of this solution basis vector are given by ¯ E T H T ¯Σ T ¯ ζ T ¯ E V H V ¯Σ V Ω V ¯ A V ¯ α V ¯ a V W V ¯ Y V ¯ Z V H S Ω S ξ S = − J/ φ r − /φr (cid:0) − L + J + 8 r ω + 16 (cid:1) / iωφr − J/ iωφ r /iωr /iωφr /φr l/φr − l/iωφr − l/iωφr
00 00 0 l/iωφr
00 00 00 00 0 − Ll/iωφr
00 00 0 M T ˆ M T ! (6.8)where for the sake of brevity we have used the aliases J = 3 φ r − , (6.9) L = ℓ ( ℓ + 1) , (6.10) l = ( ℓ −
1) ( ℓ + 2) = L − . (6.11)(6.12) As in the odd case, we choose the frame A V = Y V = 0 (and hence Z V = 0). We will choose v = M T ˆ M T R S ˆ R S , (6.13)as the basis vector for the full solution. The expressions for the obtained solutions are large and soin the interest of brevity we introduce the variable M as a function of the basis variables such that M = 124 c C ( L l − A (4 L + 4 − c ) r ω ) {− i ω φ r [96 L l ( L + 1) − l ( L + 4)+3(8 L − − c ) c ) φ r ] C M T − iω A φ r c C ˆ M T − i ω φ r [(8 l ( L + 4) + (8 L − − c ) c ) φ r − L l ( L + 1)] C R S + (24 i ω A φ r ) c C ˆ R S o . (6.14)with the solution given by: – 26 – VEN M M T R S X S E T − LlJ +( J − J +4) ω r iωφ r (4 lc − l ( L +1)+ c )2(8 L +8+ c ) φ r − ∆18 Llr c C (4 L − − c )(3 r − C φr c C ¯ H T − ( J − L )( J +4 l )8 φ r − iωφ r T − Ll +( J − ω r ω φr liωc r − l − c ) C − r (3 r − C +6 C ω )]9 iωr c C J − C iωφr c C ζ T Lliωr φ φr L +4 − c ) C φr c C − C c C E V − Ll iωφ r lφr (4 L +4 − c )( c − L +8) C φr c C (4 L − − c ) C rc C ¯ H V l ( J − L +2 ω r )8 ω φr − l ( J − iωr (4+ c ) ( J − l − c ) C − r (3 r − C +6 ω C )]36 iωr c C ( J − AC iωr c C Σ V − Ll [(2 L − c +( J − E r ]2 ω φ r c liωφr Π6 iωφr c C [ c (8 A +(2 L − − c )(4 E− φ )) − l E ] C iωr (4 E− φ ) c C ¯Ω V − Ll ( J − A c ω r φ A (2 c r C +[8 l ( L +1)+(( L +10) c − c ω φ )] C ) iω rc C A [( J − c − l ( J +4)] C iωφrc C A V α V − Ll ((4 L − c ))4 ω φ r l (4 L − c ) iωφr c Ψ36 iωφr c C − [ A (4 L − c )(4 A + φ ) − ω r ] C iωc C a V − Ll ( J − iωc r φ − c r C +2(8 l (1+ L )+( L +10) c − c φ ω ) C rc C [8 l ( J +4) − c ( J − C rφc C W V Ll ( J − iωr φ ( J − L +1) C r c C − ( J − φ C rc C Y V Z V S Ll ( J − c − A −E ) r ]12 ω φr c χiωr c C − [(32 l ( J +4) − J − c )( A −E )+9 φ (8 − J + c ) c ] C iω c C φ θ S − Llℓ ( J − A −E ) ω rφc Γ iωrc C E (12 A r c +[6 l ( J +8) − ( J − c ] r − c ) C iω (4+ J )(4 A + φ ) rc C whereΓ = c C + 4( A − E )[(2 l ( J + 4) + 3( L + 2) c ) C + 8 c C φ − E ] r + 2[(4 + J ) E + 3 c r ] ω c C r , (6.15) χ = 1 / { L ( c + 8) + 8 − c ) c C + 64( A − E ) r ( l ( J + 4) C + c C r ) − J + 4)(8 L + 8 − c ) C − c C − C E (16 + 8 L + 3 c − ω φ )+ 8 A C (2 + lω φ )) r ] c } , (6.16)Π = (2 l − c )(4 L + 4 − c ) c C + 2 E [2 c C r + (2 l (4 + J ) + 3(2 + L ) c ) r C + 2 c ( − − L + c ) C ω ] , (6.17)Ψ = 3(4 L − c )[4( J − L )( L + 1) + ( J + 2 L ) c − c ] C − c r (3 r ) − L − c ) C + 2(8 − L + c ) C ω ] , (6.18)∆ = 3 r { Llc C + 6 Ll ( − − L + c ) φ C + [16( L + 1) + 8(2( L − L − c + ( L + 1) c ] ω C } + 6 Ll (4 + 4 L − c )(6 + 6 L − c ) C − lLc r ( C + 6 C ω ) . (6.19) We used the 1+1+2 covariant approach to GR to give a detailed analysis of linear perturbations ofSchwarzschild black holes in f ( R ) gravity.Since the background only involves scalar quantities, all vector and tensor quantities are gaugeinvariant under linear perturbations (as a consequence of the Stewart and Walker Lemma).We were able to obtain a frame invariant TT tensor M T which satisfies the Regge-Wheelerequation irrespective of parity and demonstrated that for the tensor modes, the underlying dynamicsin f ( R ) gravity is governed by a modified Regge-Wheeler tensor which obeys the same Regge-Wheelerequation as in GR. In order to close the system a scalar wave equation for the Ricci scalar must beincluded which corresponds to the propagation of the additional scalar degree of freedom not presentin GR. Since the Regge-Wheeler equation governs the odd (axial) perturbations.The main difference between GR and f ( R ) gravity is the appearance of scalar perturbations rep-resenting the propagation of the additional gravitational degree of freedom not present in GR. This– 27 –xtra mode introduces a ghost problem but this can be avoided in f ( R ) theories of gravity if the con-dition ∂f /∂R > f ( R ) theories are consistent with the requirement that ∂f /∂R > ∂ f /∂R > f ( R ) was studied.For the (QNMs) that follow from the scalar perturbations, we find that possible sources of scalarQNMs for the lower multipoles are from primordial Black Holes. Higher mass, stellar black holes areassociated with extremely high multipoles, which can only be produced in the first stage of black holeformation. Since the scalar QNMs are short ranged, this scenario makes their detection beyond therange of current experiments.Finally, we find the solutions to the perturbation equations by introducing harmonics to thesystem of linearised equations. The harmonic decomposition reduced the system into a linear systemof algebraic equations which simplified things and we were able to find the solution of the systemusing matrix methods, while employing the freedom to choice of frame vectors. Acknowledgements
AMN would like to thank Chris Clarkson for helpful discussions. RG acknowledges the NationalResearch Foundation (South Africa) and the University of KwaZulu-Natal for financial support.
References [1] A. Einstein,
Annalen der Physik , 769 (1916).[2] S. W. Allen, A. E. Evrard, A. B. Mantz,
ARA&A , 409 (2011).[3] S. Perlmutter, et al. , Astrophys. J. , 565 (1999).[4] J. Dunkley, et al. , Astrophys. J. , 52 (2011).[5] Planck Collaboration, et al. , ArXiv e-prints arXiv:astro-ph.CO/1303.5062 (2013).[6] S. Capozziello, L. Z. Fang,
International Journal of Modern Physics D , 483 (2002).[7] S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D , 043528 (2004).[8] S. Nojiri, S. D. Odintsov, Phys. Rev. D , 123512 (2003).[9] A. A. Starobinsky, Soviet Journal of Experimental and Theoretical Physics Letters , 157 (2007).[10] S. Capozziello, S. Carloni, A. Troisi, Recent Res.Dev.Astron.Astrophys. , 625 (2003).[11] N. Goheer, J. Larena, P. K. S. Dunsby, Phys. Rev. D , 061301 (2009).[12] S. Capozziello, V. F. Cardone, A. Troisi, JCAP , 1 (2006).[13] S. Nojiri, S. Odintsov, Int. J. Geom. Meth. Mod. Phys. , 115 (2007).[14] T. P. Sotiriou, V. Faraoni, Rev. Mod. Phys. , 451 (2010).[15] A. De Felice, S. Tsujikawa, Living Rev.Rel. , 3 (2010).[16] S. Nojiri, S. D. Odintsov, Phys.Rept. , 59 (2011).[17] T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis,
Phys.Rept. , 1 (2012).[18] E. Newman, R. Penrose,
Journal of Mathematical Physics , 566 (1962).[19] J. Ehlers, Abh. Math.-Naturwiss. Kl. Akad. Wiss. Lit. Mainz , 792 (1961). Reprinted as GoldenOldie: Gen. Rel. Gravit. Contributions to the Relativistic Mechanics of Continuous Media. 25:1225–1226, 1993.[20] G. Ellis, General Relativity and Cosmology , R. Sachs, ed. (Academic Press, New York, U.S.A., 1971),pp. 104–182.[21] G. F. R. Ellis, H. van Elst,
NATO Adv.Study Inst.Ser.C.Math.Phys.Sci. , M. Lachi`eze-Rey, ed. (1999),vol. 541, pp. 1–116. – 28 –
22] G. F. R. Ellis, M. Bruni, J. Hwang, Phys. Rev. D , 1035 (1990).[23] M. Bruni, P. K. S. Dunsby, G. F. R. Ellis, Astrophys. J. , 34 (1992).[24] P. K. S. Dunsby, M. Bruni, G. F. R. Ellis, Astrophys.J. , 54 (1992).[25] S. W. Hawking, Astrophys. J. , 544 (1966).[26] D. H. Lyth, M. Mukherjee, Phys. Rev. D , 485 (1988).[27] G. F. R. Ellis, M. Bruni, Phys. Rev. D , 1804 (1989).[28] P. K. S. Dunsby, Classical and Quantum Gravity , 3391 (1997).[29] A. Challinor, A. Lasenby, Phys. Rev. D , 023001 (1998).[30] T. Gebbie, P. K. S. Dunsby, G. F. R. Ellis, Annals of Physics , 321 (2000).[31] R. Maartens, T. Gebbie, G. F. R. Ellis, Phys. Rev. D , 083506 (1999).[32] S. Carloni, P. K. S. Dunsby, A. Troisi, Phys. Rev. D , 024024 (2008).[33] K. N. Ananda, S. Carloni, P. K. S. Dunsby, Phys. Rev. D , 024033 (2008).[34] K. N. Ananda, S. Carloni, P. K. S. Dunsby, Classical and Quantum Gravity , 235018 (2009).[35] M. Abdelwahab, S. Carloni, P. K. S. Dunsby, Classical and Quantum Gravity , 135002 (2008).[36] A. Abebe, M. Abdelwahab, ´A. de la Cruz-Dombriz, P. K. S. Dunsby, Classical and Quantum Gravity , 135011 (2012). See also: A. Abebe, R. Goswami, and P. K. S. Dunsby. Simultaneous expansionand rotation of shear-free universes in modified gravity. In J. Beltr an Jim enez, J. A. RuizCembranos, A. Dobado, A. L opez Maroto, and A. De la Cruz Dombriz (ed.), American Institute ofPhysics Conference Series , 1458:307-310, July 2012.[37] C. A. Clarkson, R. K. Barrett,
Classical and Quantum Gravity , 3855 (2003).[38] G. Betschart, C. A. Clarkson, Classical and Quantum Gravity , 5587 (2004).[39] C. A. Clarkson, M. Marklund, G. Betschart, P. K. S. Dunsby, Astrophys. J. , 492 (2004).[40] R. B. Burston, A. W. C. Lun, ArXiv General Relativity and Quantum Cosmology e-printsarXiv:gr-qc/0611052 (2006).[41] C. Clarkson, Phys. Rev. D , 104034 (2007).[42] R. B. Burston, A. W. C. Lun, Classical and Quantum Gravity , 075003 (2008).[43] J. P. Zibin, Phys. Rev. D , 043504 (2008).[44] P. Dunsby, N. Goheer, B. Osano, J.-P. Uzan, JCAP , 17 (2010).[45] R. Goswami, G. F. R. Ellis, General Relativity and Gravitation , 2157 (2011).[46] R. Goswami, G. F. R. Ellis, General Relativity and Gravitation , 2037 (2012).[47] G. F. Ellis, R. Goswami, General Relativity and Gravitation , 2123 (2013).[48] B. de Swardt, P. K. S. Dunsby, C. Clarkson, ArXiv General Relativity and Quantum Cosmologye-prints arXiv:gr-qc/1002.2041 (2010).[49] A. M. Nzioki, S. Carloni, R. Goswami, P. K. S. Dunsby, Phys. Rev. D , 084028 (2010).[50] A. M. Nzioki, P. K. S. Dunsby, R. Goswami, S. Carloni, Phys. Rev. D , 024030 (2011). See also: A.M. Nzioki, P. K. S. Dunsby, R. Goswami, and S. Carloni. Strong gravitational lensing in f(R) gravity.In J. Beltr´an Jim´enez, J. A. Ruiz Cembranos, A. Dobado, A. L´opez Maroto, and A. De la CruzDombriz (ed.), American Institute of Physics Conference Series , 503 (2012).[51] A. M. Nzioki, R. Goswami, P. K. S. Dunsby, Phys. Rev. D , 064050 (2014). See also: A. M. Nzioki,P. K. S. Dunsby, R. Goswami, and S. Carloni. Strong gravitational lensing in f(R) gravity. In J.Beltr´an Jim´enez, J. A. Ruiz Cembranos, A. Dobado, A. L´opez Maroto, and A. De la Cruz Dombriz(ed.), American Institute of Physics Conference Series , 503 (2012).[52] S. Chandrasekhar,
The mathematical theory of black holes (Oxford: Clarendon Press, 1983).[53] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, S. Zerbini,
JCAP , 10 (2005). – 29 –
54] A. de la Cruz-Dombriz, ´A. Dobado, A. L. Maroto,
Phys. Rev. D , 124011 (2009).[55] T. De Felice, Antonio. Suyama, T. Tanaka, Phys. Rev. D , 104035 (2011).[56] T. Myung, Y. S. Moon, . . Son, E. J. 83, Phys. Rev. D , 124009 (2011).[57] J. Stewart, M. Walker, Proc.Roy.Soc.Lond.
A341 , 49 (1974).[58] C. V. Vishveshwara,
Nature (London) , 936 (1970).[59] S. Carloni, P. Dunsby, A. Troisi,
Phys. Rev. D
D77 , 024024 (2008).[60] P. K. S. Dunsby, B. A. C. C. Bassett, G. F. R. Ellis,
Classical and Quantum Gravity , 1215 (1997).[61] D. Lovelock, Journal of Mathematical Physics , 498 (1971).[62] D. Lovelock, Journal of Mathematical Physics , 874 (1972).[63] M. Ostrogradsky, Memoires de l’Academie Imperiale des Science de Saint-Petersbourg , 385 (1850).[64] R. Woodard, The Invisible Universe: Dark Matter and Dark Energy , L. Papantonopoulos, ed. (2007),vol. 720 of
Lecture Notes in Physics, Berlin Springer Verlag , p. 403.[65] W. Kundt, M. Tr¨umper,
Akad. Wiss. Lit. Mainz, Abhandl. Math.-Nat. Kl. , 969 (1962).[66] G. F. R. Ellis, Cargese Lectures in Physics , E. Schatzman, ed. (1973), vol. 6 of
Cargese Lectures inPhysics , p. 1.[67] R. Maartens, Phys. Rev. D , 463 (1997).[68] R. Maartens, B. A. Bassett, Classical and Quantum Gravity , 705 (1998).[69] H. van Elst, Extensions and applications of 1+3 decomposition methods in general relativisticcosmological modelling, Ph.D. thesis, Astronomy Unit, Queen Mary and Westfield College, Universityof London, 1996 (1996).[70] H. van Elst, G. F. R. Ellis, Classical and Quantum Gravity , 1099 (1996).[71] J. M. Bardeen, Phys. Rev. D , 1882 (1980).[72] T. Regge, J. A. Wheeler, Physical Review , 1063 (1957).[73] F. J. Zerilli,
Physical Review Letters , 737 (1970).[74] S. Chandrasekhar, S. Detweiler, Royal Society of London Proceedings Series A , 441 (1975).[75] S. L. Detweiler, J. R. Ipser, Astrophys. J. , 685 (1973).[76] R. M. Wald,
Journal of Mathematical Physics , 1056 (1979).[77] B. S. Kay, R. M. Wald, Classical and Quantum Gravity , 893 (1987).[78] E. W. Leaver, Royal Society of London Proceedings Series A , 285 (1985).[79] H. P. Nollert, Phys. Rev. D , 5253 (1993).[80] H. P. Nollert, B. G. Schmidt, Phys. Rev. D , 2617 (1992).[81] B. Mashhoon, Third Marcel Grossmann Meeting on General Relativity , H. Ning, ed. (1983), pp.599–608.[82] V. Ferrari, B. Mashhoon,
Physical Review Letters , 1361 (1984).[83] V. Ferrari, B. Mashhoon, Phys. Rev. D , 295 (1984).[84] B. F. Schutz, C. M. Will, Astrophys. Phys. J. Lett. , L33 (1985).[85] S. Iyer, C. M. Will, Phys. Rev. D , 3621 (1987).[86] S. Iyer, Phys. Rev. D , 3632 (1987).[87] N. Fr¨oman, P. O. Fr¨oman, N. Andersson, A. H¨okback, Phys. Rev. D , 2609 (1992).[88] L. Motl, A. Neitzke, ArXiv High Energy Physics - Theory e-prints arXiv:hep-th/0301173 (2003).[89] H. P. Nollert,
Classical and Quantum Gravity , 159 (1999).[90] K. Kokkotas, B. Schmidt, Living Reviews in Relativity , 2 (1999). – 30 –
91] E. Berti, V. Cardoso, A. O. Starinets,
Classical and Quantum Gravity , 163001 (2009).[92] R. A. Konoplya, A. Zhidenko, Reviews of Modern Physics , 793 (2011).[93] B. R. Iyer, S. Iyer, C. V. Vishveshwara, Classical and Quantum Gravity , 1627 (1989).[94] L. E. Simone, C. M. Will, Classical and Quantum Gravity , 963 (1992).[95] R. A. Konoplya, Physics Letters B , 117 (2002).[96] A. Ohashi, M.-a. Sakagami,
Classical and Quantum Gravity , 3973 (2004).[97] R. A. Konoplya, A. V. Zhidenko, Physics Letters B , 377 (2005).[98] M. Colpi, S. L. Shapiro, I. Wasserman,
Physical Review Letters , 2485 (1986).[99] R. Friedberg, T. D. Lee, Y. Pang, Phys. Rev. D , 3640 (1987).[100] E. Seidel, W.-M. Suen, Physical Review Letters , 1659 (1991).[101] R. A. Konoplya, R. D. B. Fontana, Physics Letters B , 375 (2008).[102] A. Cruz-Osorio, F. S. Guzm´an, F. D. Lora-Clavijo,
JCAP , 29 (2011).[103] T. Clifton, Phys. Rev. D , 024041 (2008).[104] A. A. Geraci, S. J. Smullin, D. M. Weld, J. Chiaverini, A. Kapitulnik, Phys. Rev. D , 022002(2008).[105] S. Hawking, Commun.Math.Phys. , 199 (1975).[106] J. H. MacGibbon, B. R. Webber, Phys. Rev. D , 3052 (1990).[107] S. Hawking, Mon.Not.Roy.Astron.Soc. , 75 (1971)., 75 (1971).