Gravitationally collapsing stars in f(R) gravity
GGravitationally collapsing stars in f ( R ) gravity Suresh C. Jaryal ∗ and Ayan Chatterjee † Department of Physics and Astronomical ScienceCentral University of Himachal PradeshDharamshala, Kangra (HP), India 176215.
The gravitational dynamics of a collapsing matter configuration which is simultaneously ra-diating heat flux is studied in f ( R ) gravity. Two particular functional forms in f ( R ) gravity areconsidered to show that it is possible to envisage boundary conditions such that the end state ofthe collapse has a weak singularity and that the matter configuration radiates away all of its massbefore collapsing to reach the central singularity. Keywords:
Gravitational collapse. f ( R ) gravity. The general theory of relativity (GR) is an impeccably robust relativistic theory of gravity. It formsthe basis for our understanding of gravitational phenomenon at small as well as large scales [1, 2].However, it is well known that GR cannot be the ultimate theory of gravitation since it has a welldefined regime of validity; for example, understanding past and future spacetime singularities arebeyond the reach of GR. Suitable modification(s) of GR is(are) essential to understand singularities,or to resolve them. It is believed that a quantum theory of gravity may lead to solution to theproblems affecting GR [2, 3, 4, 5, 6]. Naturally, in absence of any consensus on the theory ofquantum gravity, modified theories of gravity with quantum corrections are also of interest. TheEinstein- Hilbert action may be thought of as only a low energy contribution and higher curvatureterms consistent with the diffeomorphism invariance may become relevant as one goes to higherenergies. Higher curvature corrections should leave imprints at low energy scales which becomeimportant for low energy physics too [5, 6, 7]. Out of these alternate theories, we shall study the f ( R ) model since it has been found interesting in the cosmological studies as well. These f ( R )gravity models are thought of as alternate to the dark energy models [8, 9, 10]. The standard wayto construct these f ( R ) theories is to replace the Einstein- Hilbert Lagrangian by a well defined ∗ suresh.fi[email protected] † [email protected] a r X i v : . [ g r- q c ] F e b unction of Ricci scalar, f ( R ) (for general relativity f ( R ) = R ) [11]. For a detailed review of themotivation, validity of various functional forms f ( R ), applications as well as shortcomings of thesegravity theories have been extensively analyzed [12, 13, 14, 15, 16, 17, 18].The purpose of the present work is to construct,in f ( R ) gravity, spacetimes formed due togravitational collapse which admit gravitationally weak spacetime singularities [19]. Usually, gravi-tationally collapsing (spherical) matter configurations have density and curvatures diverging at thecenter. However, it is possible to arrange the matter and geometric variables (without violatingenergy conditions) in a way that during the collapse, the configuration radiates away mass (inthe form of heat flux) at such a rate that the matter boundary never reaches its Schwarzschildhorizon. Hence, no horizon is formed at the star boundary and the matter center remains visibleto the observers at infinity. A further consequence of these conditions is that the matter centeradmits a weak singularity and all the mass is radiated away before the star collapses to the centralsingularity. This collapsing configuration is such that the energy density, radial and tangentialpressure, pressure anisotropy, heat flux, remain regular and positive throughout the collapse. Theluminosity and adiabatic index are also regular and positive, and admits maximum value whenthe star approaches the singularity. Thus, for an observer at infinity observing the collapse, thestar shall become extremely bright, reaching its maximum luminosity before turning off, indicatingthat it has radiated off all its mass. Solutions of such kind are not unknown and is possible in theNewtonian gravity as well. Consider a star in the Newtonian gravity which is extremely heavy tobe supported by the Pauli exclusion principle alone. So, when the gravitation contraction takesplace, thermal pressure may balance to some extent. But since there is no event horizon, the starshall continue to radiate all the gravitational energy to infinity. Hence all the matter contained inthe star shall be converted to thermal radiation and radiated off. We shall show here that config-urations of similar nature are also possible in f ( R ) gravity. The matter configuration we considerin this paper satisfy the equations of motion of f ( R ) gravity.From a theoretical point of view, gravitational collapse in the f ( R ) gravity is important, andrecently there has been an increasing interest to understand whether the nature of collapse isaltered in the modified gravity theories. For example, gravitational collapse in GR show that thecollapse outcome depends upon, among other quantities, the choices of mass profiles and velocityprofiles of the collapsing matter. In the context of inhomogeneous LTB models in GR, theseissues have been considered in great detail for various matter models including dust and viscousfluids [20]. In [21] the authors studied the stellar collapse of the homogeneous dust cloud in f ( R )gravity. Earlier studies have looked into various aspects of radiating stellar collapse under the f ( R )regime, for different forms of f ( R ) function, different matter and density distributions [22]-[31].The aspects of the numerical simulation of the gravitational collapse under f ( R ) regime has beenstudied in [32, 33]. Different type of f ( R ) models has been considered in the last few decades.However, only those f ( R ) models should be considered physically viable which are in agreementwith the standard cosmological observations. Here, we consider two models of f ( R ) gravity as( a ) f ( R ) = R + λR [34] and ( b ) f ( R ) ∼ R n +1 [27]. The interior collapsing spacetime is smoothlymatched with the exterior Vaidya spacetime over a timelike surface Σ [35, 36]. A well- known way tostudy spherical gravitational matter configurations (and stars) is by using the Karmarkar condition237, 38, 39].These conditions determine gravitational potentials for static and non- static system both ingeneral relativity as well as in modified gravity regimes [31][40]-[42]. We must mention here thatsimilar studies on f ( R ) gravity have been carried out in [31]. However, the solutions obtainedthere are restrictive in the sense that one off the metric function have been kept constant to derivethe values of other metric function. On the other hand we shall show that such restrictions arenot necessary. Karmarkar condition expresses relationship between metric functions. Naturally,these forms of metric functions are arbitrary, and dependent on one’s choice. Recently in [43], ithas been argued that this arbitrariness may be removed if metric functions are related to mattervariables. For example, it has been shown that a specific form of pressure anisotropy (differenceof the radial and tangential pressures, denoted by ∆ = p t − p r ), gives rise to unambiguous set ofgravitational potentials. In the present study, we shall use this particular approach to evaluate themetric functions.The stability of this model is also analyzed and it is found that a faraway observer will see asource whose luminosity is exponentially increasing until a time when it shuts off quickly. This isdue to the fact that the total mass of the star radiates linearly and, as the star reaches its maximumluminosity there is no mass left to radiate. The evolution of the temperature profiles during stellarcollapse is also studied since they play an important role in the study of transport processes inradiative gravitational collapse [44]-[52].The paper is organised as follows. In section 2, we give the field equations of f ( R ) gravityand the junction conditions for smooth matching of the interior and the exterior spacetimes acrossthe timelike hypersurface Σ. This section also includes the solutions of the f ( R ) field equationsalong with the explicit expressions for physical quantities. We inspect the physical relevance of ourexact solutions by verifying the energy conditions. The stability criteria and discussion about theluminosity and adiabatic index, radial and transverse velocity are carried out in section 2.1. Insection 2.2 we study the temperature profiles of the radiative stellar collapse. In section 3 includesthe discussion of the results accompanied with concluding remarks. The action for the f ( R ) gravity is obtained by replacing the standard Einstein- Hilbert Lagrangianby a well defined function of Ricci scalar [13] S = 12 (cid:90) √− g [ f ( R ) + 2 L M ( g µν , Ψ m )] d x , (1)where Ψ m refers collectively to all matter fields, L M is the Lagrangian density of the matter fieldsΨ m , g is the determinant of the metric tensor g µν , R is the Ricci scalar curvature and f ( R ) isthe generic function of Ricci scalar defining the theory under consideration and (using units with c = 1 = 8 πG ). Varying the action (1) with respect to the metric tensor g µν yields the following3eld equations: F ( R ) R µν − f ( R ) g µν − ( ∇ µ ∇ ν − g µν (cid:3) ) F ( R ) = T Mµν , (2)where F ( R ) = d f ( R ) /dR , and (cid:3) ≡ ∇ µ ∇ µ . This equation may also be rewritten as R µν − (1 / g µν R = F ( R ) − (cid:0) T Mµν + T Dµν (cid:1) , (3)where the left side of the equation (3) is the usual Einstein tensor, T Mµν and T Dµν are the energymomentum tensor and effective energy momentum tensor having the form as: T Mµν = ( p t + ρ ) u µ u ν + p t g µν + ( p r − p t ) X µ X ν + q µ u ν + q ν u µ , (4) T Dµν = (1 /
2) [ f ( R ) − R F ( R )] g µν + ( ∇ µ ∇ ν − g µν (cid:3) ) F ( R ) . (5)Here, ρ , p r and p t are the energy density, radial pressure and the tangential pressure respectively.Also, q µ u µ , X µ represents the radial heat flow vector, 4-velocity vector and spacelike 4-vectorrespectively, which satisfy u µ u µ = − X µ X µ = − u µ X µ = u µ q µ = 0.We now consider a general non- static shear free spherically symmetric spacetime metric givenby the following form ds = − a ( r ) dt + b ( r ) s ( t ) (cid:0) dr + r dθ + r sin θdφ (cid:1) . (6)The forms of u µ , X µ and q µ in terms of the metric (6) are u µ = a − δ µ ; X µ = ( b s ) − δ µ ; q µ = ( b s ) − X µ , (7)The magnitude of the expansion scalar Θ and Ricci scalar for the metric (6) have the formΘ = (cid:53) µ u ν = 3 ˙ sa s , (8) R = 6 s ¨ s + ˙ s a s − b s (cid:20) a (cid:48)(cid:48) a − b (cid:48) b + a (cid:48) b (cid:48) a b + 2 b (cid:48)(cid:48) b + 2 r (cid:18) a (cid:48) a + 2 b (cid:48) b (cid:19)(cid:21) . (9)The field equations in f ( R ) gravity for the metric (6), energy momentum tensor (4), (5) and (7)are ρ = F ( R ) s (cid:20) s a − b (cid:18) b (cid:48)(cid:48) b − b (cid:48) b + 4 r b (cid:48) b (cid:19)(cid:21) + (cid:18) f − R F (cid:19) + 3 ˙ s ˙ Fs a − b s (cid:20) F (cid:48)(cid:48) + F (cid:48) (cid:18) b (cid:48) b + 2 r (cid:19)(cid:21) , (10) p r = Fs (cid:20) − a (cid:0) s ¨ s + ˙ s (cid:1) + 1 b (cid:18) a (cid:48) b (cid:48) a b + 2 r (cid:18) a (cid:48) a + b (cid:48) b (cid:19) + b (cid:48) b (cid:19)(cid:21) − (cid:18) f − R F (cid:19) − ˙ Fa (cid:32) ¨ F ˙ F + 2 ˙ ss (cid:33) + F (cid:48) b s (cid:18) a (cid:48) a + 2 r + 2 b (cid:48) b (cid:19) , (11)4 t = Fs (cid:20) − a (cid:0) s ¨ s + ˙ s (cid:1) + 1 b (cid:18) a (cid:48)(cid:48) a + b (cid:48)(cid:48) b − b (cid:48) b + 1 r (cid:18) a (cid:48) a + b (cid:48) b (cid:19)(cid:19)(cid:21) − (cid:18) f − R F (cid:19) − ˙ Fa (cid:32) ¨ F ˙ F + 2 ˙ ss (cid:33) + 1 b s (cid:18) F (cid:48)(cid:48) + F (cid:48) (cid:18) a (cid:48) a + 1 r (cid:19)(cid:19) , (12) q = − a (cid:48) ˙ s ˙ Fa b s + 1 a b s (cid:32) ˙ F (cid:48) − ˙ F a (cid:48) a − ˙ s F (cid:48) s (cid:33) , (13)where prime and dot are the derivatives with respect to r and t respectively.Let us consider the junction conditions for the smooth matching of the interior manifold M − (equation (6) considered above) with the exterior manifold M + across timelike hypersurface Σ, at r = r b . As described in [53, 54], the junction conditions for the f ( R ) gravity requires the matchingof several geometric quantities other than the induced metric ( h ij ) and the extrinsic curvature ( K ij ).In fact, it has been established that in f ( R ) gravity, the following variables must be matched atthe boundary: [ h ij ] + − = 0 , (14) F ( R ) [ K ij − (1 / K h ij ] + − = 0 , (15)[ K ] + − = 0 , (16)( ∂F ( R ) /∂R ) [ ∂ τ R ] + − = 0 , (17)[ R ] + − = 0 , (18)where τ represents the proper time of the timelike hypersurface, and K is the trace of the ex-trinsic curvature. Out of these five conditions, for the set of f ( R ) theories under consideration,which are f ( R ) = R + λR , and f ( R ) = α (cid:48) R , it is sufficient to match the metric, the extrinsic cur-vature, the Ricci scalar, and the derivative of the Ricci as given above, determined from either sides.The Vaidya spacetime in the outgoing coordinate is taken to be our exterior spacetime M + [35] ds = − (cid:20) − M ( v ) r (cid:21) dv − dvd r + r (cid:0) dθ + sin θdφ (cid:1) , (19)For our later convenience, let us define the proper time, dτ = a ( r ) Σ dt . The junction condition asgiven (14), implies the following conditions r Σ ( v ) = ( r b s ) Σ , (20) (cid:18) dvdτ (cid:19) − = (cid:18) − M r + 2 d r dv (cid:19) Σ , (21)where τ represents the proper time defined on the hypersurface Σ. The normal vector fields to Σare given by n − l = [0 , ( b s ) Σ , , , n + l = (cid:20) − M r + 2 d r dv (cid:21) − Σ (cid:20) − d r dv δ l + δ l (cid:21) Σ . (22)5he extrinsic curvatures for metrics (6) and (19) are given by K − ττ = − (cid:20) a (cid:48) a b s (cid:21) Σ , K − θθ = (cid:20) r b s (cid:18) r b (cid:48) b (cid:19)(cid:21) Σ , (23) K + ττ = (cid:34) d vdτ (cid:18) dvdτ (cid:19) − − (cid:18) dvdτ (cid:19) M r (cid:35) Σ , K + θθ = (cid:20)(cid:18) dvdτ (cid:19) (cid:18) − M r (cid:19) r − r d r dτ (cid:21) Σ , (24) K − φφ = sin θK − θθ , K + φφ = sin θK + θθ . (25)Now, from the junction condition on K ij (because of the conditions that K must satisfy on the hy-persurface, matching K ij is enough), we get the following. From the equality for the θθ componentsat hypersurface Σ, and the equations (20) and (21), we obtain (cid:20) r b s (cid:18) r b (cid:48) b (cid:19)(cid:21) Σ = (cid:20)(cid:18) dvdτ (cid:19) (cid:18) − M r (cid:19) r − r d r dτ (cid:21) Σ , (26)and the total energy inside the boundary hypersurface Σ, given by the Misner- Sharp mass, denotedby 2 m (such that m = M on the matching hypersurface) [55, 56], where m Σ = (cid:20) r ˙ s b s a − r s b (cid:48) b − r s b (cid:48) (cid:21) Σ . (27)Now, again from the matching of the K + ττ = K − ττ component we have the following equation: − (cid:20) a (cid:48) a b s (cid:21) Σ = (cid:34) d vdτ (cid:18) dvdτ (cid:19) − − (cid:18) dvdτ (cid:19) M r (cid:35) Σ , (28)and, substituting the relation between proper and coordinate time along with the eqns. (20) and(27) into the eqn.(26) we have (cid:18) dvdτ (cid:19) Σ = (cid:18) r b (cid:48) b + r b ˙ sa (cid:19) − . (29)Now, differentiating (29) with respect to the τ and using eqns (27) and (29), we can rewrite (28).Further, comparing with equations (11) and (13) we have the following useful form (cid:0) p r + T Drr + b sT Dtr (cid:1) Σ = ( q b s ) Σ . (30)where, T Drr = (cid:18) f − R F (cid:19) + ˙ Fa (cid:32) ¨ F ˙ F + 2 ˙ ss (cid:33) − F (cid:48) b s (cid:18) a (cid:48) a + 2 r + 2 b (cid:48) b (cid:19) , (31) T Dtr = 1 a b s (cid:32) ˙ F (cid:48) − ˙ F a (cid:48) a − ˙ s F (cid:48) s (cid:33) , (32)6re the dark source terms. From equation (30), it is found that just like for general relativity, theradial pressure does not vanish at the boundary but, instead is proportional to the dissipative aswell as radiative dark source terms. The extra terms T Drr and T Dtr on the LHS of equation (30) arethe dark source term and may appear due to the higher order curvature geometry of the collapsingsphere [31].Let us now move to match K as given in (14). The expressions for the trace of extrinsiccurvatures on the either sides lead to the following matching condition on the hypersurface: (cid:2) p r + T Drr + b sT Dtr − q b s (cid:3) Σ = 2 [ M − m ] Σ , (33)and naturally, this condition is identically satisfied due to the abovementioned equations. Thematching of the Ricci and its proper time derivative gives the following conditions which are to besatisfied for the metric of the internal manifold (at the hypersurface Σ): s ¨ s + ˙ s = a b (cid:20) a (cid:48)(cid:48) a + 2 b (cid:48)(cid:48) b + a (cid:48) b (cid:48) a b + 2 r (cid:18) a (cid:48) a + 2 b (cid:48) b (cid:19) − b (cid:48) b (cid:21) , s ¨ s + s ... s = 0 . (34)The metric of the internal manifold must be chosen so as to satisfy the two conditions in (34). Todetermine metric functions according to all these junction conditions, it is necessary to use someauxiliary conditions. We shall see below that these equations are consistent with a collapsing timedependent internal metric. In fact, one may argue that junction conditions indeed force such apossibility. Additionally, we must also ascertain the physical viability of the spacetime metric.From equations (11), (12) and (6), the pressure anisotropy factor ∆ = p t − p r has the form∆ = Fb s (cid:34) b (cid:48)(cid:48) b − b (cid:48) b + a (cid:48)(cid:48) a − a (cid:48) b (cid:48) a b − r (cid:18) a (cid:48) a + b (cid:48) b (cid:19)(cid:35) + F (cid:48)(cid:48) b s − F (cid:48) b s (cid:18) b (cid:48) b + 1 r (cid:19) . (35)The general expression for the shear free spacetime as given in (35) is has the complicated form.To find the solution of the metric functions and mathematical simplicity, we take an adhoc form ofthe pressure anisotropy ∆ to be:∆ = F (cid:48)(cid:48) b s − F (cid:48) b s (cid:18) b (cid:48) b + 1 r (cid:19) − Fb s (cid:20) a (cid:48) b (cid:48) a b − a (cid:48)(cid:48) a + a (cid:48) r a (cid:21) (36)Although, we have chosen this form of the anisotropy in pressure ∆ for the mathematical simplicity,later we will see that they represents the physically viable solutions of the potentials. Also, thischoice of ∆ is physically significant, such that ∆ is regular throughout the collapse. It must benoted that this choice of the anisotropy (36) reduces the total pressure anisotropic equation (35)as differential equation of only one function, given by0 = 1 s b (cid:32) b (cid:48)(cid:48) b − b (cid:48) b − b (cid:48) r b (cid:33) , (37)7he form of the function b ( r ) is b ( r ) = − C r + 2 C ] − , (38)where C and C are constant of integration.Let us now use the fact that under certain conditions, a ( n + 1)-dimensional space can beembedded into a pseudo Euclidean space of dimension ( n +2) [37]. Thus the necessary and sufficientcondition for any Riemannian space to be an embedding class I is the Karmarkar condition [38, 39], R rtrt R θφθφ = R rθrθ R tφtφ − R θrtθ R φrtφ . (39)The non vanishing components of the Riemann tensor for the metric (6) are R rtrt = a (cid:18) a (cid:48)(cid:48) a − b sa ¨ s − a (cid:48) a b (cid:48) b (cid:19) , (40) R θφθφ = r b s (cid:32) b a ˙ s − b (cid:48) rb − b (cid:48) b (cid:33) sin θ, (41) R rθrθ = r b s (cid:32) b a ˙ s − b (cid:48) rb − b (cid:48)(cid:48) b + b (cid:48) b (cid:33) , (42) R tφtφ = r a b (cid:18) a (cid:48) ra − b sa ¨ s + a (cid:48) a b (cid:48) b (cid:19) sin θ, (43) R θrtθ = r b sa a (cid:48) ˙ s, (44) R φrtφ = sin θR θrtθ . (45)Using the expressions for Riemannian tensors from eqns (40)- (45) into the eqn (39) we have0 = b ˙ s b (cid:32) a (cid:48)(cid:48) a − a (cid:48) a b (cid:48) b + a (cid:48) a − a (cid:48) ra (cid:33) − r b s ¨ s (cid:32) b (cid:48)(cid:48) b − b (cid:48) b − b (cid:48) rb (cid:33) + r aa (cid:48) b (cid:48)(cid:48) (cid:18) b (cid:48) b + 1 r (cid:19) − r aba (cid:48)(cid:48) (cid:32) b (cid:48) b + 2 b (cid:48) rb (cid:33) + raba (cid:48) (cid:32) b (cid:48) rb + 2 b (cid:48) b (cid:33) . (46)For a given form of metric function b ( r ) (38), the class I condition in equation (46) is nonlinear.A physical relevant collapsing model must satisfy (30) and (46) simultaneously. It must be notedthat simplest choice of solutions of (30) is a linear solution [57] s ( t ) = − C Z t, (47) C Z >
0. The form of the other metric function a ( r ) is obtained by using equation (38) and (47)into the class 1 condition (46) a ( r ) = 12 √ C C [ C ( C b ( r ) + 4 C C ) − C Z ] / (48)8here C and C are integration constants. Surprisingly, the quantity in the numerator inside thesquare root, arises naturally from the matching of the Ricci scalar (and it’s derivative), given in (34)These forms of the solutions of the gravitational potentials are same as obtained in [43] for shearfree spacetime. In [43], it has been shown that for the static case, Karmarkar condition togetherwith the pressure isotropy yields the Schwarzschild [58] like form of the metric functions. Also, ithas been shown that these set of gravitational potentials are the special class of those found in [59].Thus, although we have assume this particular form of ∆ (36) for the mathematical simplicity,represents the physically viable solutions.It is now instructive to rewrite the physical quantities of the matter cloud in terms of the metricvariables for a better understanding of the dynamics of spacetime during the collapse process. Theseexpressions have been written in detail in the Appendix. The boundary condition (30) in the viewof these equations in the Appendix, (66)-(68) becomes2 s ¨ s + ˙ s − x ˙ s + b s T Drt = y − T Drr , (49)where T Drr and T Drt are given by equations (31) and (32) respectively and the quantities x and y are x = (cid:18) a (cid:48) b (cid:19) Σ , y = (cid:32) a b (cid:34) b (cid:48) b + 2 r b (cid:48) b + 2 a (cid:48) b (cid:48) ab + 2 r a (cid:48) a (cid:35)(cid:33) Σ . (50)The metric functions a ( r ) and b ( r ) should not vanish during the collapsing phenomena, sinceotherwise the metric shall become degenerate. This also implies that their signatures remainunchanged. For second metric potential to be positive i.e. a ( r ) > C Z < C (cid:20) C C r + 2 C − C C (cid:21) . (51)This equation also implies that at the center of the cloud, r = 0, we must have C Z < C − C C C .The graphical representations of the physical quantities (65)-(68) shows that they are well de-fined throughout the stellar collapse for both the f ( R ) models. Figures 1a and 1b, 2a and 2b, 3aand 3b, 4a and 4b shows that the density, radial, tangential pressures and pressure anisotropy arepositive and regular throughout the collapse for both f ( R ) = R + λR and f ( R ) = R n +1 / ( n + 1)with λ = 0 . n = 1. As seen from the figures 5a and 5b, for both the f ( R ) models, the heat fluxincrease as the collapse starts and remains positive throughout the collapse.9 a) (b) Figure 1: (a) and (b) shows the plots of the density ρ (65) w.r.t. time t and radial r coordinatesfor f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively. For both f ( R ) models, it remains regular as well as positive throughout the collapse. (a) (b) Figure 2: (a) and (b) shows the plots of the radial pressure p r (66) w.r.t. time t and radial r coordinates for f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.For both f ( R ) models, it remains regular as well as positive throughout the collapse.10 a) (b) Figure 3: (a) and (b) shows the plots of the tangential pressure p t (67) w.r.t. time t and radial r coordinates for f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.For both f ( R ) models, it remains regular as well as positive throughout the collapse. (a) (b) Figure 4: (a) and (b) shows the plots of the pressure anisotropy ∆ (36) w.r.t. time t and radial r coordinates for f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.For both f ( R ) models, it remains regular and positive throughout the collapse.11 a) (b) Figure 5: (a) and (b) shows the plots of the radial heat flux q (68) w.r.t. time t and radial r coordinates for f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.For both the f ( R ) models, it remains positive throughout the collapse. (a) (b) Figure 6: (a) Plot of the expansion scalar Θ (69) w.r.t. time t and radial r coordinates. At thebeginning of collapse Θ has zero value and it starts decreasing and remains negative throughoutthe collapse. (b) Plot of the mass of the collapsing star (70) w.r.t. time t , and it shows that massradiates linearly. 12 related quantity of importance in this study is the total luminosity visible to an observer atinfinity, which may be defined in terms of the mass loss from the boundary surface: L ∞ = − (cid:18) dmdv (cid:19) Σ = (cid:34) r s b p r (cid:18) r b (cid:48) b + r b ˙ sa (cid:19) (cid:35) Σ , (52)where we have used the equations (11), (27) and (28). Now, as soon as the black hole is formed,by definition, the luminosity of the surface is zero. From the above equation, this implies thatsufficient condition for the formation of a black hole is (cid:20) r b (cid:48) b + r b ˙ sa (cid:21) Σ = 0 . (53)Naturally, for any static observer at asymptotic infinity, the redshift diverges at the time of forma-tion of the black hole. (a) (b) Figure 7: (a) and (b) shows the plots of luminosity (52) at r = r Σ = 1, w.r.t. time t for f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.To show that these spacetime solutions are physically viable, we show that they satisfy the energyconditions as well. Indeed, all the energy conditions namely weak (W), null (N), dominant (D) andstrong (SEC) hold good for the collapsing star. In the following we list these conditions [58, 60] E1 : ( ρ + p r ) − q ≥ E2 : ρ − p r ≥ E3 : ρ − p r − p t + (cid:113) ( ρ + p r ) − q ≥ E4 : ρ − p r + (cid:113) ( ρ + p r ) − q ≥ a) (b) (c) Figure 8: (a), (b) and (c) shows the plots of energy conditions E E E t andradial r coordinates for f ( R ) = R + λR with λ = 0 .
001 respectively.
E5 : ρ − p r + 2 p t + (cid:113) ( ρ + p r ) − q ≥ E6 : p t + (cid:113) ( ρ + p r ) − q ≥ E7 : ρ > p r > p t >
0, and ρ (cid:48) < p (cid:48) r < p (cid:48) t < E1 , E2 , E3 and E7 are enough to validate the physicalconditions existing inside the star.For the radiating collapsing stellar models in f ( R ) gravity, figures 8a,8b and 8c show that theenergy conditions are positive and regular throughout the interior of the star for the f ( R ) = R + λR model with λ = 0 . f ( R ) = R n +1 / ( n + 1) model with n = 1. (a) (b) (c) Figure 9: (a), (b) and (c) shows the plots of energy conditions E E E t andradial r coordinates for f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.14 .1 Stability Criteria The study of dynamical instability (stability) of spherical stellar system shows that for adiabaticindex Γ < / > /
3) the stellar system becomes unstable (stable) as the weight of the stellarsystem increase much faster (remains less than) than that of its pressure [61]. Also, the causalitycondition imposes certain constraints on the dynamics of the stellar system such that inside the star,the radial V r and the transverse V t components of the speed of sound should be less than the speedof the light ( c = 1), so that 0 ≤ V r ≤ ≤ V t ≤ eff = (cid:20) ∂ (ln p r ) ∂ (ln ρ ) (cid:21) Σ = (cid:20)(cid:18) ˙ p r p r (cid:19) (cid:18) ˙ ρρ (cid:19)(cid:21) Σ (54) V r = d p r dρ , V t = d p t dρ (55)Although stability may be understood from the behaviour of the pressure and density variables, (a) (b) Figure 10: (a) and (b) shows the plots of the effective adiabatic index (54) at r = r Σ , w.r.t. time t for f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.the quantities in (54) and (55) are considered to be better to establish stability. For f ( R ) = R + λR model with λ = 0 . / f ( R ) = R n +1 / ( n + 1) model with n = 1, similar behavior ofluminosity is obtained as that of for the first model. Figure 10b shows that the effective adiabaticindex is constant function of time, and is positive and less than 4 /
3, which implies it representsthe collapsing scenario. As we have shown graphically that the star radiates all its mass beforereaching at the singularity. So, there are no trapped surfaces formed during the collapse. Whichimplies that neither the black hole nor naked singularity are the end state of the collapse.
Earlier studies have shown that relaxation effects are important to understand dissipative gravita-tional collapse [45, 46, 47, 50, 64]. To study the temperature profiles, we consider the transportequation for the metric (6) given by [44, 45, 65] τ h νµ ˙ q ν + q µ = − k (cid:0) h νµ ∇ ν T + T ˙ u µ (cid:1) (56) τ ( qbs ) ,t + q a b s = − k ( aT ) ,r bs , (57)where, α > β > γ > σ >
0, & h µν = g µν + u µ u ν . Also, τ c = ( α/γ ) ( T ) − σ , k = γ τ c T , τ = τ c ( β γ ) /α (58)where τ c is the mean collision time, k is thermal conductivity and τ represents the relaxation timerespectively [45][52]. The quantity τ measures the strength of relaxational effects and is called thecausality index. The values τ = 0 or β = 0 represents the noncausal temperature profile. Usingconditions in equation (58), the the causal heat transport equation (57) becomes βT − σ ( qbs ) ,t + q a b s = − α ( aT ) ,r bs T − σ . (59)The noncausal solution of the heat transport equation (59), with β = 0 i.e. τ = 0 are [52]( a T ) = − α (cid:90) a q b s dr + G ( t ) , σ = 0 (60)ln ( a T ) = − α (cid:90) q b s dr + G ( t ) . σ = 4 (61)16 a) (b) Figure 11: (a) & (b) shows the plots of temperature profiles of the collapsing stellar system w.r.t.radial coordinate r for σ = 0 and f ( R ) = R + λR with λ = 0 .
001 and f ( R ) = R n +1 / ( n + 1) with n = 1 respectively.The causal solution of the above heat transport equation (59) are [52]( a T ) = − α (cid:20) β (cid:90) a b s ( q b s ) ,t dr + (cid:90) a q b s dr (cid:21) + G ( t ) , σ = 0 (62)( a T ) = − βα exp (cid:18) − (cid:90) q b s α dr (cid:19) (cid:90) a b s ( q b s ) ,t dr exp (cid:18)(cid:90) q b s α dr (cid:19) + G ( t ) exp (cid:18) − (cid:90) q b s α dr (cid:19) , σ = 4 (63)where G ( t ) appears as a function of integration and is determined by following boundary condition (cid:0) T (cid:1) Σ = (cid:18) L ∞ πδr b s (cid:19) Σ . (64)where L ∞ is the total luminosity for an observer at infinity given by (52) and δ > f ( R ) models, both the temperatures are same atthe boundary. At the later stages of gravitational collapse, due to the relaxation effects, the stellarsystem deviates from thermodynamical equilibrium, and so, the causal and noncausal temperatureprofiles differ inside the interior of the star. This behavior can be seen from the figures 11a and11b that with β >
0. 17
Discussion of the results
In this paper, we investigated the dynamics of a collapsing stellar system in f ( R ) gravity. Theinterior spacetime has been smoothly matched with the Vaidya metric across a timelike hypersur-face. Incidentally, as has been noted earlier too, the matching conditions for the f ( R ) gravity ishighly restrictive, since the geometric variables which are to matched here not only includes inducedmetric and the extrinsic curvatures, but also the trace of the extrinsic curvatures, and the Ricciscalar along with it’s time derivative. However, we have shown that all these matching conditionscan be carried out consistently, leading to a spacetime solution which admits a collapsing scenarioin which the matter cloud radiates heat flux, in such a manner that the entire matter is radiatedout without forming a black hole. Although similar solutions have been reported earlier for GR,our solution incorporates these features into the collapsing models in the f ( R ) gravity, and morespecifically for two particular theories: f ( R ) = R + λR , and f ( R ) = R n +1 / ( n + 1)For both these f ( R ) models, we have analyzed the physical quantities, energy density (65),radial pressure (66) and tangential pressure (67), pressure anisotropy (36) and it can be seen fromthe Figs. 1a and 1b, 2a and 2b, 3a and 3b, 4a and 4b that they are regular and positive throughoutthe collapse. From Fig. 5a and 5b it is clear that the radial heat flux (68) is finite and positivethroughout collapse. In particular, for f ( R ) = R + λR model with λ = 0 . / f ( R ) = R n +1 / ( n + 1) model with n = 1, similar behavior of the luminosity is obtainedas well. Figure 10b shows that the effective adiabatic index is constant function of time, and ispositive and less than 4 /
3, which implies it represents the collapsing scenario. As we have showngraphically that the star radiates all its mass before reaching at the singularity. So, there are notrapped surfaces formed during the collapse. Which implies that neither the black hole nor nakedsingularity are the end state of the collapse. Also, from figures 9a,9b and 9c, the energy conditionsare also satisfied for f ( R ) = R n +1 / ( n + 1) model with n = 1. These graphs show that these modelunder consideration are physically viable. Also, the results obtained here reduces to those obtainedfor the general relativity regime for f ( R ) = R [43].Let us now comment on the nature of the central singularity. First, we note that the Ricciscalar (9) together with (47) imply that it diverges at t = 0, when all the matter has been radiatedaway. So, naturally the question arises if this central curvature singularity is gravitationally strong.18f that is so, this would lead to an example of a naked singularity. The sufficient condition forthe singularity to be weak is that the curvature scalars go as t − , which is precisely the case here.So, this solution represents a physically viable model where the spherically symmetric collapsingmatter cloud undergoes gravitational collapse, which alongside also radiates away mass in the formof heat flux. The flux is radiated at such a rate that no horizon is ever formed and the centralsingularity is naked but gravitationally weak in nature. Acknowledgments
The author AC is supported by the DST-MATRICS scheme of government of India throughMTR/2019/000916 and by the DAE-BRNS Project No. 58/14/25/2019-BRNS.
Data availability
This manuscript has no associated data. This is a theoretical study and does not contain anyexperimental data.
References [1] S. W. Hawking and G. R. Ellis, The Large Scale Structure of Spacetime. Cambridge UniversityPress, Cambridge 1975.[2] Robert M. Wald, General Relativity. University of Chicago Press, 1984.[3] P. S. Joshi, Gravitational Collapse and Spacetime Singularities. Cambridge University Press,Cambridge, England, 2007.[4] R. M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics,Chicago Lectures in Physics. University of Chicago Press, Chicago, IL, 10, 1995.[5] I. Buchbinder, S. Odintsov and I. Shapiro, Effective action in quantum gravity. IOP, Bristol,UK, 1992.[6] M. Reuter and F. Saueressig, Quantum Gravity and the Functional Renormalization Group:The Road towards Asymptotic Safety. Cambridge University Press, 2019.[7] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory. Vol. 1: Introduction, Cam-bridge Monographs in Mathematical Physics, 1987.[8] S. Capozziello, V. F. Cardone, S. Carloni and A. Troisi, Int. J. Mod. Phys. D 12, 1969 (2003).[9] S. Nojiri and S. D. Odintsov, Phys. Rev. D 68 (2003) 123512.[10] S.M. Carroll, V. Duvvuri, M. Trodden, M.S. Turner, Phys. Rev. B 70, 043528 (2004) .[11] H.A. Buchdahl, Month. Not. Royal Astron. Soc. 150 (1970) 1; J.D. Barrow, A.C. Ottewill, J.Phys. A: Math. Gen. 16 (1983) 2757. 1912] T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rep. 513, 1 (2012).[13] T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).[14] S. Capozziello, M. De Laurentis, S. D. Odintsov and A. Stabile, Phys. Rev. D (2011)064004.[15] S. Capozziello, M. De Laurentis, I. De Martino, M. Formisano and S. D. Odintsov, Phys. Rev.D (2012) 044022.[16] A. V. Astashenok, S. Capozziello and S. D. Odintsov, JCAP (2013) 040.[17] S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011).[18] S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Phys. Rep. 692, 1 (2017).[19] C. J. S. Clarke, “The Analysis of space-time singularities,” CUP, 1993.[20] A. Chatterjee, A. Ghosh and S. Jaryal, Phys. Rev. D 102, no.6, 064048 (2020).[21] J. A. R. Cembranos, A. de la Cruz-Dombriz and B. Montes N´u˜nez, J. Cosmol. Astropart.Phys. 04, 021 (2012).[22] K. Bamba, S. Nojiri and S. D. Odintsov, Phys. Lett. B 698, 451 (2011).[23] E. V. Arbuzova and A. D. Dolgov, Phys. Lett. B 700, 289 (2011).[24] M. Sharif, H.R. Kausar, Astrophys. Space Sci. 331, 281 (2011).[25] R. Goswami, A.M. Nzioki, S.D. Maharaj, S.G. Ghosh, Phys. Rev. D 90, 084011 (2014).[26] S. Chakrabarti and N. Banerjee, Gen. Relativ. Gravit. 48, 57 (2016).[27] S. Chakrabarti and N. Banerjee, Eur. Phys. J. Plus. 131, 144 (2016).[28] A. V. Astashenok, K. Mosani, S. D. Odintsov and G. C. Samanta, Int. J. Geom. Meth. Mod.Phys. (2019) no.03, 1950035[29] G. Abbas, M. S. Khan, Z. Ahmad and M. Zubair, Eur. Phys. J. C , no.7, 443 (2017).[30] S. Chakrabarti, R. Goswami, S. D. Maharaj and N. Banerjee, Gen. Relativ. Gravit. 50, 148(2018).[31] G. Abbas and H. Nazar, Int. J. Mod. Phys. A 34(33), 1950220 (2019).[32] A. Borisov, B. Jain, P. Zhang, Phys. Rev. D 85, 063518 (2012).[33] J. Guo, D. Wang, A.V. Frolov, Phys. Rev. D 90, 024017 (2014).[34] A.A. Starobinsky, Phys. Lett. B. 91, 99 (1980).2035] P.C. Vaidya, Proc. Indian Acad. Sci. A 33, 264 (1951).[36] N. O. Santos, MNRAS, 216, 403 (1985).[37] L. P. Eisenhart, Riemannian Geometry. Princeton University Press, Princeton, 1925.[38] J. Eiesland, Trans. Am. Math. Soc. 27, 213 (1925).[39] K.R. Karmarkar, Proc. Indian Acad. Sci. A 27, 56 (1948).[40] K.N. Singh, N. Pant, Eur. Phys. J. C 76, 524 (2016).[41] N.F. Naidu, M. Govender, S.D. Maharaj, Eur. Phys. J. C 78, 48 (2018).[42] M. Govender, A. Maharaj, K. N. Singh and N. Pant, Mod. Phys. Lett. A 35, no.20, 2050164(2020).[43] S.C. Jaryal, Eur. Phys. J. C 80, 683 (2020).[44] R. Maartens, Class. Quantum Gravity 12, 1455 (1995).[45] J. Martinez, Phys. Rev. D 53, 6921 (1996).[46] A. Di Prisco, L. Herrera, M. Esculpi, Class. Quantum Grav. 13, 1053 (1996).[47] L. Herrera, A. Di Prisco, J.L. Hernandez-Pastora, J. Martin and J. Martinez, Class. QuantumGrav. 14, 2239 (1997).[48] L. Herrera and N. O. Santos, Mon. Not. R. Astron. Soc. 287, 161 (1997).[49] M. Govender, S.D. Maharaj, R. Maartens, Class. Quant. Grav. 15, 323 (1998).[50] M. Govender, R. Maartens, S.D. Maharaj, Mon. Not. Roy. Astron. Soc. 310, 557 (1999).[51] R. Maartens, M. Govender, S.D. Maharaj, Gen. Relativ. Grav. 31, 815 (1999).[52] M. Govender and K. Govinder, Phys. Lett. A283, 71 (2001).[53] N. Deruelle, M. Sasaki and Y. Sendouda, Prog. Theor. Phys. , 237-251 (2008).[54] J. M. M. Senovilla, Phys. Rev. D , 064015 (2013).[55] C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571 (1964).[56] M. E. Cahill, G.C. McVittie, J. Math. Phys. 11, 1382 (1970).[57] A. Banerjee, S. Chatterjee, N. Dadhich, Mod. Phys. Lett. A 17, 2335 (2002).[58] K. Schwarzschild, Phys.-Math. Klasse, 189 (1916).[59] J. Ospino, L.A. N´ u ˜ n ez, Eur. Phys. J. C 80, 166 (2020).2160] R. Chan MNRAS, 288, 589 (1997). ; R, Chan, MNRAS, 299, 811 (1998).[61] S. Chandrasekhar, Astrophys. J. 140, 417 (1964).[62] L. Herrera, Phys. Lett. A 165, 206 (1992).[63] G. Pinheiro, R. Chan, Gen. Relat. Gravit.43, 1451 (2011).[64] L. Herrera and N. O. Santos, Phys. Rev. D 70, 084004 (2004); L. Herrera, G. Le Denmat, N.O. Santos, A. Wang, Int. J. Mod. Phys. D 13, 583 (2004); L. Herrera, Int. J. Mod. Phys. D15, 2197 (2006).[65] W. Israel and J. Stewart, Ann. Phys.(N.Y.), 118, 341 (1979). Appendix
In this appendix, we give the detail expressions of the physical quantities of the collapsing mattercloud in terms of the metric functions. More precisely, we give the values for (10)-(13), and otherquantities like the expansion scalar (8) and the Misner- Sharp mass function (27). ρ = 6 F C C S C Z t (cid:2) C − C C (cid:0) C + C r (cid:1)(cid:3) + 3 ˙ s ˙ Fs a − F (cid:48)(cid:48) b s + f − R F − F (cid:48) b s (cid:20) b (cid:48) b + 2 r (cid:21) , (65) p r = F C C S C Z t (cid:104) C C C (cid:0) C + 4 C C r − C r (cid:1) − C (cid:0) C − C r (cid:1) − C C C (cid:0) C + C r (cid:1) (cid:105) − (cid:18) f − R F (cid:19) − ˙ Fa (cid:32) ¨ F ˙ F + 2 ˙ ss (cid:33) + F (cid:48) b s (cid:18) a (cid:48) a + 2 r + 2 b (cid:48) b (cid:19) , (66) p t = F C C S C Z t (cid:2) C C (cid:0) C r − C (cid:1) + 2 C C C C (cid:0) − C r + 8 C C r + 28 C (cid:1)(cid:3) + F (cid:48) r b s + 2 F C C C S C Z t (cid:104)(cid:0) C r + 2 C (cid:1) (cid:2) C C C (cid:0) C r − C (cid:1) + C Z (cid:0) C − C r (cid:1)(cid:3)(cid:105) + F (cid:48) a (cid:48) a b s + 2 F C C C C S C Z t (cid:104)(cid:0) C r + 2 C (cid:1) (cid:2) C Z (cid:0) C r − C (cid:1) − C C C (cid:0) C r − C (cid:1)(cid:3)(cid:105) + F (cid:48)(cid:48) b s − F C C C S C Z t (cid:104)(cid:0) C r + 2 C (cid:1) (cid:0) C C C − C Z (cid:1)(cid:105) − (cid:18) f − R F (cid:19) − ˙ Fa (cid:32) ¨ F ˙ F + 2 ˙ ss (cid:33) , (67) q = − C / C / C Z t √ rF C (cid:104) C C − C C r +2 C (cid:105)(cid:20) C (cid:16) C C − C C r +2 C (cid:17) − C Z (cid:21) / + 1 a b s (cid:34) ˙ F (cid:48) − ˙ F a (cid:48) a − ˙ s F (cid:48) s (cid:35) , (68)22 = 6 √ C C t (cid:114) C (cid:16) C C − C C r +2 C (cid:17) − C Z , (69) m = 8 tr C C C Z ( C r + 2 C ) (cid:34) C C (cid:0) C r + 2 C (cid:1) − C C r + 2 C ) (cid:0) C C C − C Z (cid:1) − C C (cid:35) , (70) S = C (cid:0) C C − C / (cid:0) C + C r (cid:1)(cid:1) − C Z ..