Axial and polar modes for the ring down of a Schwarzschild black hole with an r dependent mass-function
GGeneral Relativity and Gravitation manuscript No. (will be inserted by the editor)
Axial and polar modes for the ring down of aSchwarzschild black hole with an r dependentmass-function Peter O. Hess · Enrique L´opez-Moreno
Received: date / Accepted: date
Abstract
The axial and polar modes for the ring down of a Schwarzschildblack hole are calculated, by first deriving the Regge-Wheeler and Zerilliequations, respectively, and finally applying the Asymptotic Iteration Method(AIM). We were able to reach up to 500 iterations, obtaining for the first timeconvergence for a wide range of large damping modes. The
General Relativ-ity (GR) and a particular version of an extended model with an r -dependentmass-function are compared. This mass-function allows an analytical solutionfor the Tortoise coordinate. The example of the mass-function corresponds tothe leading correction for extended theories and serves as a starting point totreat other r -dependent parameter mass-functions. Keywords
General Relativity · axial modes · polar modes General Relativity (GR) is one of the best tested theories, which accountsfor the observations in the solar system [1] and also its prediction of gravita-tional waves [2] was confirmed recently [3,4]. The first indirect proof of thesewaves stems from the 1970’s [5], through the observation of changes in theorbital frequency of a neutron star binary. The black hole merger consists oftwo phases: the inspiral and the ring-down phase. For the description of the
DGAPA-PAPIIT IN100418P. O. HessInstituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Circuito Ex-terior, C.U., A.P. 70-543, 04510 M´exico D.F., MexicoandFrankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universit¨at, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany E-mail: [email protected]. L´opez-MorenoFacultad de Ciencias, Universidad Nacional Aut´onoma de M´exico, Mexico-City, Mexico a r X i v : . [ g r- q c ] J a n Peter O. Hess, Enrique L´opez-Moreno inspiral phase the result depends very much on the approximation used [2,8]or on a non-linear hydro-dynamic approach [9]. For the ring-down phase thesituation is ”simpler”, because only the stability of the black hole under met-ric perturbations has to be studied. The investigation of the ring-down phasecan be traced back to S. Chandrasekhar’s book on the mathematics of blackholes [10] and [11,12] (though, not the first). The equation for the calculationof the ring-down frequencies where treated in [13] for the polar modes and in[14] for the axial modes. These equations permit to calculate the frequencies,solving the eigenvalue problem of the equation, which is much simpler thanto implement a dynamical theory. There are several methods to solve theseequations, as for example the so-called
Asymptotic Iteration Method (AIM)[15,16], with an improved approach published in [17].For this reason, we restrict our analysis to the study of the ring-down modesonly and in addition to a non-rotating star, i.e., to the Schwarzschild metric.We follow closely the method described in the book of S. Chandrasekhar [10],Chapter 4 on the perturbations of a Schwarzschild black hole.An interesting questions is: What changes, when a r -dependent mass-function m ( r ) is used, instead of a constant mass parameter m ? There isa 1 /r leading order, due to the following arguments: a) 1 /r corrections areexcluded due to observations in the solar system [1]. b) Also 1 /r correctionsare excluded due to adjusting to the inspiral phase [29,30] in the first observedgravitational event. Thus, the leading corrections are of the order 1 /r . Othercorrections, as appearing in cosmological models (de Sitter), are not includedbut the path explained can be extended to it. It is probable that the constantmass m is substituted by a function in r as soon as GR is extended and,therefore, it is interesting to ask what kind of changes one can expect? Doesthe 1 /r correction to the metric still lead to stable modes? These are two ofthe motivations of this contribution.Another example is given in [18,19] where the pseudo-complex GeneralRelativity (pc-GR) is proposed, which adds in the vicinity of a black hole adistribution of dark energy, which is repulsive and halts the collapse of a star,before forming an event horizon and a singularity. First observable predictionsare published in [20,21]. More recent descriptions of this model can be foundin [22–24]. One reason for using this theory becomes obvious in the main bodyof the text: The mass-function in pc-GR stars with a 1 /r correction andwe show that it permits an analytical solution for the Tortoise coordinate,thus, it provides us with a controlled handling of the asymptotic limit of thesolutions. This property helps to understand the changes in the spectrum ofthe Quasinormal Modes (QNM) in the ring-down phase of a black hole and itsstability.The paper is organized as follows: In Section 2 the Schwarzschild limitwill be discussed. In Section 3 the easier to treat case of axial modes are de-termined and in Section 4 the polar modes. In Section 5 the Regge-Wheelerand Zerilli equations are derived and the numerical method to solve the dif-ferential equation is shortly explained. In Section 6 the asymptotic limit ofthe corresponding solution is calculated, using an analytical solution for the xial and polar ring down modes 3
Tortoise coordinate. The spectrum of axial and polar modes are determined,within GR and its extension. We will show that the symmetry observed inGR, namely that the frequencies of the axial and polar modes are the same,is not maintained for an r -dependent mass-function not maintained, though,some similar structures are present. In Section 7 the Conclusions are drawn. Following the notation of [10–12], the length element is given by ds = e ν ( dt ) − e ψ ( dφ − ωdt − q dx − q dx ) − e µ ( dx ) − e µ ( dx ) , (1)where dx = r and dx = θ , the azimuth angle. The functions ν , φ , ω , q , q , µ and µ depend, in general, on t , x = r and x = θ , though, in thiscontribution we restrict to a spherical symmetry. The components of the Ricciand the Einstein tensor can be retrieved from the book by S. Chandrasekhar[10], in terms of the above functions.The metric perturbations are introduced around the generalized Schwarzschildmetric e ν = e − µ = (cid:18) − m ( r ) r (cid:19) = ∆r e µ = r , e ψ = r sin θ and ω = q = q = 0 , ∆ = r − m ( r ) r . (2)The m ( r ) is the parameter mass-function, which is m in the GR case butwill depend on r for a generalized form, as will be used later. It is preferableto define the dimensionless coordinate y = rm and the dimensionless mass-function m ( y ) = m ( r ) m (for simplicity, we use the same letter m for the massfunction).Perturbations are introduced in first order contributions for δω , δq , δq , δµ , δµ and δψ . Axial waves (negative parity) are related to the perturbationsof δω , δq and δq , while polar waves (positive parity) are related by theperturbations δν , δµ , δµ and δψ [10]. To follow this section, please consult the book of S. Chandrasekhar [10]. Thefunction ν ( r ), ψ ( r ), µ ( r ), µ ( r ), ω ( r ), q ( r ) and q ( r ) are maintained as gen-eral functions in r . In [10] the Riemann and Ricci tensor components are Peter O. Hess, Enrique L´opez-Moreno written in terms of these functions, as are the Einstein tensor components G µν . Only at the very last the explicit functions are substituted by their ex-pressions in the Schwarzschild metric, which in this contribution is of the formlisted in (2).Demanding the invariance of these components under variation, leads to δR = 0 and δR = 0 [10], which in turn results in the equations (using thesame notation as in [10]) (cid:0) e ψ + ν − µ − µ Q (cid:1) | = − e ψ − ν + µ − µ Q | (cid:0) e ψ + ν − µ − µ Q (cid:1) | = + e ψ − ν + µ − µ Q | , (3)where ” | k ” denotes the usual derivative with respect to the variable x k andthe Q ab are defined as Q ab = q a | b − q b | a , Q a = a a | − ω | a . (4)With the ansatz Q ( t, r, θ ) = ∆Q sin θ = ∆ ( q | − q | )sin θ , (5)using (2), we arrive at the equations1 r sin 3 θ ∂Q∂θ = − (cid:0) ω | − q | (cid:1) | ∆r sin 3 θ ∂Q∂r = + (cid:0) ω | − q | (cid:1) | . (6)Eliminating ω and assuming a time dependence of e iωt , we arrive at r ∂∂r (cid:18) ∆r ∂Q∂r (cid:19) + sin θ ∂∂θ (cid:18) θ ∂Q∂θ (cid:19) + ω r ∆ Q = 0 . (7)With the ansatz Q ( t, θ ) = Q ( r ) C − l +2 ( θ ) , (8)where C − l +2 is a Gegenbauer function, we obtain for the final equation ∆ ddr (cid:18) ∆r dQdr (cid:19) − µ ∆r Q + ω Q = 0with µ = 2 n = ( l − l + 2) . (9) xial and polar ring down modes 5 Further, setting Q ( r ) = rZ ( − ) (10)and defining ( r ∗ is the Tortoise coordinate) ddr ∗ = ∆r ddr , (11)we arrive finally at the Regge-Wheeler equation [14] (cid:18) d dr ∗ + ω (cid:19) Z ( − ) = V ( − ) Z ( − ) . (12)The Eq. (11) is the definition of the Tortoise coordinate, which for an r de-pendent m has not the simple form as the one exposed in the book of Chan-drasekhar [10] and in [11]. Later it will be shown that in certain cases alsoanalytic solutions may exist.The potential V ( − ) is derived in the Appendix A and is given by V ( − ) ( r ) = µ ∆r − ∆r ddr (cid:18) ∆r (cid:19) . (13)Using the definition of ∆ and applying the derivatives leads to V ( − ) ( r ) = ∆r (cid:2) ( µ + 2) r − m ( r ) + 2 m (cid:48) ( r ) r (cid:3) . (14)The prime indicates a derivation in r and (14) reduces to the result by Chan-drasekhar [10,11] when the derivative of m ( r ) is zero, i.e., when it is constant.The upper index ( − ) refers to axial (negative parity) modes.Thus, the derivation of the equation for the axial modes is in completeanalogy to the derivation presented in [10]. The changes in the formulas areminimal. In contrast to the axial modes, for obtaining the differential equation of thepolar modes the procedure is more involved. The final equation will have asimilar form as in (12), however, with a quite complex potential. S. Chan-drasekhar proved [10] that the frequencies of the polar oscillations are thesame as for axial modes, which is one of the reasons in most cases only theaxial modes are calculated. Using the extended mass-function, It will be shownthat axial and polar modes are not equal anymore, though, some similaritiescan be conjectured. For that reason, the axial and polar modes have to be
Peter O. Hess, Enrique L´opez-Moreno treated separately. Again, we will closely follow the path exposed in [10] andmainly mention key points, deviations and approximations.To obtain the Zerilli equation [13] for the polar modes, we proceed inthe same manner as in [10], section 24b and of [11]. The variations δR , δR , δR , δG and δR lead to the identical equations as given in [10].New functions are introduced, varying ν , µ , µ and ψ (equations (36)-(39) inchapter 24 of [10]), restricting to the quadrupole mode ( l = 2) of the multipoleexpansion: δν = N ( r ) P l (cos θ ) δµ = L ( r ) P l (cos θ ) δµ = (cid:2) T ( r ) P l + V ( r ) P l | θ | θ (cid:3) δψ = (cid:2) T ( r ) P l + V ( r ) P l | θ cot θ (cid:3) . (15)We are also lead to the relation (equation (43) in [10]) T − V + L = 0 , (16)which reduces the number of linear independent functions by one.The following steps in [10] will remain the same. One reason is that thegeneral structure is not affected by the modified metric term e ν , e.g., onlythe function ν and its derivative ν | r = ν (cid:48) appear and not their explicit depen-dence on m ( r ), it is implicit . In Eq. (48) of [10] a new function is defined insubstitution of V , namely X = nV = 12 ( l − l + 2) V . (17)A relation of the derivatives of these defined functions is given in (52)-(54)of [10], which we will repeat here, because the coefficients in these equationsdo depend on m ( r ) and, which is new, its radial derivative m (cid:48) ( r ): N | r = aN + bL + cXL | r = (cid:18) a − r + ν | r (cid:19) N + (cid:18) b − r − ν | r (cid:19) L + cXX | r = − (cid:18) a − r + ν | r (cid:19) N − (cid:18) b + 1 r − ν | r (cid:19) L − (cid:18) c + 1 r − ν | r (cid:19) X . (18)The coefficients and ν | r , have now new contributions due to the mass-function,and are given by xial and polar ring down modes 7 a = n + 1 r − m ( r ) b = − r − nr − m ( r ) + m ( r ) r ( r − m ( r )) + m ( r ) r ( r − m ( r )) + ω r ( r − m ( r )) − m (cid:48) ( r ) r − m ( r ) − m ( r ) m (cid:48) ( r )( r − m ( r )) + r ( m (cid:48) ( r )) ( r − m ( r )) c = − r + 1 r − m ( r ) + m ( r ) r ( r − m ( r )) + ω r ( r − m ( r )) + m (cid:48) ( r ) [ − m ( r ) + m (cid:48) ( r ) r ]( r − m ( r )) ν | r = m ( r ) r ( r − m ( r )) − m (cid:48) ( r ) r − m ( r ) . (19)In the second row of each factor the new contributions appear, if any, depend-ing on the derivatives of m ( r ). The results reduce to the one in [10] when m (cid:48) ( r ) is set to zero. In what follows, we write some of the expressions needed to derive theZerilli equations in explicit form, because of new contributions due to thedependence of m ( r ) on r :( L + X ) | r = − (cid:18) r − m ( r ) r ( r − m ( r )) + m (cid:48) ( r ) r − m ( r ) (cid:19) L − (cid:18) r − m ( r ) r ( r − m ( r )) + m (cid:48) ( r ) r − m ( r ) (cid:19) X = − (cid:18) r − m ( r ) r ( r − m ( r )) + m (cid:48) ( r ) r − m ( r ) (cid:19) ( L + X ) + nr V = − r ( r − m ( r )) { [2 r − m ( r ) + rm (cid:48) ( r )] L + [ r − m ( r ) + rm (cid:48) ( r )] X } X | r = − [ nr + 3 m ( r ) − rm (cid:48) ( r )] r ( r − m ( r )) N − ( n + 1) r − m ( r ) X − (cid:34) − m ( r ) r ( r − m ( r )) + (cid:0) m ( r ) + ω r (cid:1) r ( r − m ( r )) + m (cid:48) ( r ) r − m ( r ) − m ( r ) m (cid:48) ( r )( r − m ( r )) + r ( m (cid:48) ( r )) ( r − m ( r )) − nr − m ( r ) (cid:21) ( L + X ) (20)In order to obtain the Zerilli equation in GR, one defines a function Z (+) as a particular combination of N , V , X and L (see (58) and (59) in [10]). After Peter O. Hess, Enrique L´opez-Moreno that one calculates the first and second derivative with respect to r ∗ , usingthe above equations of derivatives for N , L and X .In [10,11] only the ansatz for Z (+) is presented, without a derivation. Here,we provide the foundation for this ansatz of the general Zerilli equation, whichincludes in the limit of a constant mass-function the GR.The combination of Z (+) in terms of the functions V , L and X is chosensuch that, when the second order derivative in r ∗ is applied, the only contri-butions left is solely proportional to Z (+) . Due to the new contributions in thederivatives of m ( r ), the ansatz for the linear combination changes to Z (+) = α ( r ) N + β ( r ) V + γ ( r )( L + X ) . (21)On how the functions α ( r ), β ( r ) and γ ( r ) are determined is explained inthe Appendix B. Also, the ansatz proposed by S. Chanbrasekhar [10] will bederived, in the limit of m ( r ) = m .In a first step, the first derivative ( dZ (+) dr ∗ ) and second derivative ( d Z (+) dr ∗ )have to be calculated, using (11). The factors proportional to ω are deter-mined and a solution of α ( r ) and β ( r ) is found. In a second step a differentialequation for γ ( r ) is set up, where the solution will depend on the mass-functionused. For a constant mass the GR solution of [10,11] is recovered. This is arather lengthy, but straightforward, calculation done in the Appendix B andC. but better done with MATHEMATICA [25,26].In the first step, for α ( r ) and β ( r ) we obtain (see Appendix B) α ( r ) = 0 β ( r ) = r . (22)In the second step, we take the ω independent term , obtained after havingapplied (cid:104) d dr ∗ + ω (cid:105) to Z (+) , which leads to the V (+) N N ( r ) + V (+) V rV ( r ) + V (+) LX γ ( r )( L ( r ) + X ( r )) , (23)where V (+) N depends on α ( r ) = 0, β ( r ) = r and γ ( r ).Because α ( r ) = 0, the Z (+) is only a combination in V ( r ) and ( L ( r )+ X ( r )),the factor of N ( r ) has to vanish. This condition leads to γ ( r ) = − r ( nr + 3 m ( r ) − rm (cid:48) ( r )) (cid:18) m (cid:48) ( r ) − rm (cid:48)(cid:48) ( r ) n (cid:19) . (24)For the linear combination in V ( r ) and ( L ( r ) + X ( r )) to be written as V (+) Z (+) , the two potential factors in (23) have to be equal, which leads tothe condition xial and polar ring down modes 9 G ( r ) = 12 r γ ( r ) (cid:8) ( − γ ( r ) ( r − m ( r ))( − m ( r ) + r (2 + m (cid:48) ( r )))+2 rγ ( r )( r − m ( r ))( m ( r )(23 − m (cid:48) ( r ))+ r ( − γ (cid:48) ( r ) − m (cid:48) ( r ) + rm (cid:48)(cid:48) ( r )) + r (8 m ( r ) (6 γ (cid:48) ( r ) − rγ (cid:48)(cid:48) ( r ))+4 rm ( r )( − m (cid:48) ( r ) − γ (cid:48) ( r )(5 + 2 m (cid:48) ( r ))+2 rγ (cid:48)(cid:48) ( r ) − rm (cid:48)(cid:48) ( r )) + r (2(1 + 4 γ (cid:48) ( r ) − m (cid:48) ( r ))(1 + m (cid:48) ( r )) − rγ (cid:48)(cid:48) ( r ) + r (1 + 2 m (cid:48) ( r )) m (cid:48)(cid:48) ( r ))) } = 0 . (25)As noted above, for a constant mass m ( r ) = m , this equation is identicallyfulfilled for the expression given in [10,11]. In Appendix C we will see thatthis solution satisfies the condition (25) for a wide range of r , save near r = m , with a small error, though. This shows that the Zerilli equation can beconstructed approximately.Finally, the Zerilli equation acquires the form (cid:20) d dr ∗ + ω (cid:21) Z (+) = V (+) Z (+) , (26)which describes the polar modes with an r dependent mass-function m ( r ). Aswe will see further below, the axial and polar modes, though different, stillshare similar structures.In terms of the mass-function m ( r ), using (24) the potential for the polarmodes is given by V (+) ( r ) =(( r − m ( r ))(18( m ( r )) + 3 r ( m ( r )) (6 n − m (cid:48) ( r ) + 5 rm (cid:48)(cid:48) ( r ) − r m (cid:48)(cid:48)(cid:48) ( r )) + r (2 n (1 + n ) − m (cid:48) ( r )) + ( m (cid:48) ( r )) (8 + 6 n + 3 rm (cid:48)(cid:48) ( r ))+ y ( m (cid:48)(cid:48) ( r )( − n (6 + n ) + 2 rm (cid:48)(cid:48) ( r )) + 2 nrm (cid:48)(cid:48)(cid:48) ( r )) − m (cid:48) ( r )( − n + r ((3 + n ) m (cid:48)(cid:48) ( r ) + rM (cid:48)(cid:48)(cid:48) ( r ))))+2 r m ( r )(3 n + 9( m (cid:48) ( r )) + r ( m (cid:48)(cid:48) ( r )( − n − rm (cid:48)(cid:48) ( r ))+(3 − n ) rm (cid:48)(cid:48)(cid:48) ( r )) + m (cid:48) ( r )( − n + r ( m (cid:48)(cid:48) ( r ) + 2 rm (cid:48)(cid:48)(cid:48) ( r )))))) / ( r (3 m ( r ) + r ( n − m (cid:48) ( r ))) ) . (27)In what follows, the dimensionless coordinate y = rm = y eh − ξ (28)is used, where y eh is the position of the event horizon and the variable ξ has therange [0 , ξ = 0, then y = y eh and when ξ tends to 1, the coordinate y tends to + ∞ . The particular mass-function used, corresponds to an eventhorizon at y eh = . The reason for using the coordinate ξ with a compact support lies in theuse of the AIM method, explained further below. It guarantees a better con-vergence of an iterative equation. In this section the final form of the differential equations, used for the axialand polar modes, will be derived.In subsection 5.2 the
Asymptotic Iteration Method (AIM) is resumed, whichsolves a differential equation of second order. In what follows, the Regge-Wheeler/Zerilli equation is rewritten in a form, which is practical for the AIM.For the function m ( r ) the particular form, defining y = rm , m ( r ) = m m ( y ) m ( y ) = 1 − y , (29)is used, which exhibits an event horizon at y = .5.1 Rewriting the differential equationFirst, the explicit form of the differential equation is derived, noting that theRegge-Wheeler and Zerilli equation can be, in general, written as (cid:20) d dr ∗ + ω − V ( ± ) ( r ) (cid:21) Z ( ± ) = 0 . (30)Let us use, for a moment, the more general mass-function m ( y ) = (cid:16) − b y (cid:17) ,which for b = acquires the one of (29). It was used in [23] for the studyof phase transitions from GR to pc-GR. The relation of y = rm to a variable ξ with a compact support for the range of integration is defined as in (28),where y eh ( b ) is the position of the event horizon as a function of the parameter b . It is the solution of the condition for the event horizon in the Schwarzschildcase, i.e., y − y + b . (31)Using the Wolfram MATHEMATICA code [25,26], the solution is xial and polar ring down modes 11 y eh ( b ) = 12 + 12 √ b (cid:0) b + √ b − b (cid:1) + (cid:16) b + (cid:112) b − b (cid:17) + 12 − b (cid:0) b + √ b − b (cid:1) − (cid:16) b + (cid:112) b − b (cid:17) + 2 √ (cid:32)(cid:20)(cid:18) b ( b + √ b − b ) + (cid:0) b + √ b − b (cid:1) (cid:19)(cid:21) (cid:33) . (32)The second order derivative with respect to the Tortoise coordinate is m d dr ∗ = ∆r ddr ∆r ddr = (cid:18) − y m ( y ) (cid:19) d dy + (cid:18) − y m ( y ) (cid:19) y ( m ( y ) − ym (cid:48) ( y )) ddy . (33)The prime refers to a derivative in y .Defining the dimensionless expressions˜ ω = m ω , ˜ V ( y ) = m V ( r ) , (34)the differential equation acquires the form (cid:34)(cid:18) − y m ( y ) (cid:19) d dy + (cid:18) − y m ( y ) (cid:19) y ( m ( y ) − ym (cid:48) ( y )) ddy + ˜ ω − ˜ V ( ± ) ( y ) (cid:21) Z ( y ) = 0 . (35)The next step is to substitute y by ξ . We also define m ( ξ ) as m ( y ( ξ )) and m (cid:48) ( ξ ) as the same as m (cid:48) ( y ) (in order to avoid more definitions of functions,we use the same letter m ). Doing so, leads after some manipulations to d dξ − − ξ ) − (1 − ξ ) y eh ( b ) ( m ( ξ ) − ( y eh / (1 − ξ )) m (cid:48) ( ξ )) (cid:16) − − ξ ) y eh ( b ) m ( ξ ) (cid:17) ddξ ( y eh ( b )) (1 − ξ ) (cid:16) ˜ ω − ˜ V ( ± ) ( ξ ) (cid:17)(cid:16) − − ξ ) y eh ( b ) m ( ξ ) (cid:17) Z ( ξ ) = 0 . (36)Remember that the prime refers to the derivative with respect to y and not to ξ .) Note also, that V ( ± ) ( r ) = V ( ± ) ( y ( ξ )) /m , where V ( ± ) ( y ( ξ )) is dimension-less.In practical calculations the function (29) is used, which results in thefunction m ( ξ ) m ( ξ ) = 1 − (1 − ξ ) . (37)For the potentials V ( ± ) in terms of ξ , we obtain, the expressions enlistedin [26].5.2 AIMThe AIM was introduced by H. Ciftci, R. L. Hall, and N. Saad [15,16], forsolving second order differential equations of the form [17] f (cid:48)(cid:48) ( ξ ) = λ ( ξ ) f (cid:48) ( ξ ) + s ( ξ ) f ( ξ ) , (38)with ξ as the variable.Deriving both sides p times, p being an integer, leads to an equivalentdifferential equation f ( p +1) ( ξ ) = λ p − ( ξ ) f (cid:48) ( ξ ) + s p − ( ξ ) f ( ξ ) , (39)with λ p ( ξ ) = λ (cid:48) p − ( ξ ) + s p − ( ξ ) + λ ( ξ ) λ p − ( ξ ) s p ( ξ ) = s (cid:48) p − ( ξ ) + s ( ξ ) λ p − ( ξ ) . (40)Convergence is achieved, when the ratio of s p ( x ) and λ p ( x ) does not change,with p the iteration number. Once achieved, the Quantization Condition reads s p ( ξ ) λ p − ( ξ ) − s p − ( ξ ) λ p ( ξ ) = 0 . (41)This expression depends on ξ , which can be chosen arbitrarily, and has to beresolved for the frequencies ω . Because of its ξ -dependence, it is a very subtle xial and polar ring down modes 13 task to obtain convergence rapidly. The problem was resolved partially in [17],expanding the λ p and s p in a Taylor series around a point ξ , defining λ p ( ρ ) = ∞ (cid:88) i =0 c ip ( ξ − ρ ) i s p ( ρ ) = ∞ (cid:88) i =0 d ip ( ξ − ρ ) i . (42)Substituting this into (40) leads to a new recursion relation for the coeffi-cients c ip and d ip and a new quantization condition d p c p − − d p − c p = 0 . (43)The clear advantage of (43) lies in the fact that (43) depends only on thefrequencies and is a polynomial in the frequencies. Thus, the determination ofthe ω -spectrum is restricted to solve (43). However, the result still depends onthe point of expansion ξ in the Taylor series (42).In order to obtain a ”quick” convergence, the following rules should beobserved, which are the results of the experience of others [17] and ours: – A compact support for the range of the coordinate should be used, i.e.the coordinate ξ , which is zero at the event horizon and approaches 1 for r → ∞ . In a Taylor expansion, this prevents too large deviations to thereal potential function at the limits ξ = 0 and ξ = 1. I.e., it is importantto describe the potential well near the limits. – The asymptotic behavior for r ∗ → ±∞ of the wave function should beextracted as exactly as possible. Analytic solutions are of course the best. – For the expansion around a point ξ , the maximum or minimum of thepotential is recommended as a starting point. However, shifting it to itsvicinity at larger values may give better convergence. The following crite-rion helps, namely that with an increasing number of iterations the lowerfrequencies are not changing any more for low values of − ω I . – Using MATHEMATICA, only rational numbers are allowed. In case ofirrational numbers, it is recommended to approximate them by rationalones, otherwise MATHEMATICA develops numerical instabilities.
The Regge-Wheeler equation for the axial modes and Zerilli equations for the polar modes are solved, with the help of the AIM. In a first step the asymptoticlimit is discussed and taken into account in the definition of the Z ( ± ) -functions,which leads to the final form of the differential equation to solve. We will present calculations with different iteration numbers, which allowsto judge the convergence of the iteration method and see trends for largeiteration numbers.6.1 The asymptotic limitThe wave solution must satisfy the condition Ψ → (cid:26) e + iωr ∗ , for r ∗ → + ∞ ( r → ∞ ) e − iωr ∗ , for r ∗ → −∞ ( r → m ) . (44)The time dependence for both limits is e − iωt . This implies that for a complex ω = ω R + iω I , the time dependence has the form e − iωt = e − iω R t e + ω I t , (45)i.e., for an exponential decreasing function the imaginary part of the frequency( ω I ) has to be negative ( ω I < ω I solution appears.The integrated relation of the Tortoise coordinate, defined in (11), y ∗ = r ∗ m to the variable y = rm is given by y ∗ = (cid:90) dy (cid:16) − m ( y ) y (cid:17) . (46)For m ( y ) = 1, i.e. GR, the solution is well known, namely y ∗ = y + 2ln (cid:16) y − (cid:17) , (47)For m ( y ) = (cid:16) − y (cid:17) , there is surprisingly also a solution y ∗ = y + 2ln (cid:18) y − (cid:19) +2ln (3) −
32 + 9(12 − y ) − arctan (cid:104) y √ (cid:105) √ , (48)which was obtained, using MATHEMATICA [25]. When some of the parame-ters in m ( y ) are changed, no solution can be found. That only this particularansatz of m ( r ) provides an analytic solution for the Tortoise coordinate andnot any other with a different parametrization, is quite a surprise which wewould like to understand. xial and polar ring down modes 15 Changing to the variable ξ = 1 − y , the relation of the Tortoise coordinateto ξ is y ∗ = 32(1 − ξ ) + 2ln( ξ − ξ )+2ln(3) − − − ξ )4 ξ − arctan (cid:104) − ξ √ − ξ ) (cid:105) √ . (49) r ∗ → + ∞ We again define y = rm ( y ∗ = r ∗ m ) and substitute y ∗ by (49). We also define˜ ω = m ω . (50)For y ∗ → + ∞ , the ξ tends to 1, i.e, terms proportional to − ξ ) and ln(1 − ξ )dominate. This leads to the asymptotic form ( e iωr = e i (cid:101) ωy ∗ ) e + i (cid:101) ωy ∗ → e i ω − ξ ) (1 − ξ ) − i ˜ ω . (51)The other terms in (49) can be neglected, because they are either constant orapproach a constant value in this limit. r ∗ → −∞ In this case, the y approaches the event horizon.Using the above obtained expression for y ∗ , we obtain the additional terms,taking into account that terms proportional to ξ and ln( ξ ) dominate, namely e i (cid:101) ω (1 − ξ )4 ξ ξ − i (cid:101) ω . (52)The e i and e − i (cid:101) ω arctan (cid:2) (2+ ξ ) √ − ξ ) (cid:3) √ contributions are skipped because the first isa constant and the second has the limit e − i (cid:101) ω arctan √ for ξ = 0, thus, it doesnot effect the asymptotic limit. Thus, extracting the complete asymptotic limits, the new ansatz for the wavefunction is Z ( ± ) ( ξ ) = e i ω − ξ ) (1 − ξ ) − i ˜ ω e i (cid:101) ω (1 − ξ )4 ξ ξ − i (cid:101) ω P ( ± ) ( ξ ) , (53)with the new wave-function P ( ± ) ( ξ ), whose differential equations are set upand solved with the help of the AIM [26]. m ( r ) = m )is shown. Because in GR, the axial and polar modes are equal, it is sufficientto plot only this potential. In the lower row, left panel, the potential V ( − ) in pc-GR is depicted and in the right panel the potential V (+) for the polarmodes in pc-GR. As noted, all these potentials have similar characteristics, leading to the conjecture that the frequency spectrum of axial and polar modesstill may share some common structure | (cid:101) ω I | small), there is still no convergence obtained for the high damping modes.In Figure 3 the axial modes are depicted for 400 iterations. Above, the GR(left panel) is compared to pc-GR (right panel) for a large range of − (cid:101) ω I . Inthe lower row the same is plotted but for a restricted range of − (cid:101) ω I . The leftpanel (GR) reproduces the Figure 2 in [27] and of Figure 5 in [28], where alsodistinct methods to resolve the differential equation are resumed.Comparing GR with pc-GR, the structure shares still some common fea-tures, namely that the figure reminds at a fish with its head to the right andits tail to the left. However, the head is moving further to the left when thenumber of iterations is increased, thus, it is not a physical property and hasto be rejected. The left panel in Fig. 3 shows the result for GR and the rightone for pc-GR. Very interesting is, that the structure of a raising branch forpc-GR from low to large damping modes is also stable. This branch can beapproximated by continuum and represents a definite feature of pc-GR, whichis not present in GR.At low damping, convergence is obtained and the frequencies in pc-GR arecomparable in size to GR. This changes for the large damping modes, wherethe frequencies are significantly larger in pc-GR than in GR.6.4 Spectrum of the Zerilli equation: polar modesIn Figure 4 the polar frequency modes in pc-GR are depicted. The structureis similar to the one for the axial modes (see (3)), as expected when compar-ing the two potentials. In Figure 4 the red dots correspond to 200 iterations,the green dots to 300 and the blue dots to 400 iterations. Note, that conver-gence is clearly obtained for low values of − (cid:101) ω I and up to 30 the convergenceis also acceptable. Thus the feature of a raising curve for large damping isconfirmed. The ”fish head”, however, has moved further to the right, showingits unphysical nature. xial and polar ring down modes 17 xi V ( x i ) ξ V ( ξ ) ξ V ( ξ ) Fig. 1
First row: Potential for the axial modes in GR. Second row: The potential for theaxial modes ( V ( − ) , left panel) and the polar modes ( V (+) , right panel). In Fig. 5 a comparison of axial to polar modes within pc-GR is shown. Inthe upper row, with a wide range of − (cid:101) ω I , the structure seems to be similarup to large values of − (cid:101) ω I . In the lower row a Zoom to small values of − (cid:101) ω I isdepicted. The (cid:101) ω R polar modes are in general larger in pc-GR than in GR.That there is a branch in the frequency spectrum which has larger frequen-cies (cid:101) ω R in pc-GR than in GR is of importance: The real part of the frequencyis given by (cid:101) ω R = m ω R . Using the frequency ν = 250Hz and transforming it tounits in km − , one obtains an ω R = 5 .
24 10 − km − . In the first gravitational - - - - - - - Im ω R e ω - - - - Im ω R e ω - - - Im ω R e ω Fig. 2
Spectrum of the axial modes for 200 iterations. The horizontal axis corresponds tominus the imaginary part of the frequency, which has to be positive in order to representa damped, stable oscillation. No negative values are present, rendering the system stable.The vertical axis depicts the real part of the frequency. The first row shows the axial modesin GR, while the second row depicts the frequencies in pc-GR. The left panel shows thefrequency distribution for a larger range of − (cid:101) ω I while the right panel is restricted to smalldamping modes.xial and polar ring down modes 19 - - Im ω R e ω - - - - Im ω R e ω - - - Im ω R e ω - - - Im ω R e ω Fig. 3
Spectrum of the axial modes for 400 iterations. The horizontal axis correspondsto minus the imaginary part of the frequency and the vertical axis to the real part of thecomplex frequency. In each row, the left panel is for GR and the right one for pc-GR. Note,that compared to 200 iterations, the onset of the ”head” is shifted far to the right in − (cid:101) ω I ,showing that this part is of no physical significance. For more details, in the lower row azoom to a restricted range is shown.0 Peter O. Hess, Enrique L´opez-Moreno - - - Im ω R e ω - - - Im ω R e ω Fig. 4
Spectrum of the polar modes for 200 (red dots), 300 (green dots) and 400 (blue dots)iterations. The horizontal axis corresponds to minus the imaginary part of the frequency,which has to be positive for a damped oscillation. No negative values are present, renderingthe system stable. The vertical axis depicts the real part of the frequency. The left panelshows the frequency distribution for a larger range of − (cid:101) ω I while the right panel is restrictedto small damping modes. Note, how the ”fish-head” moves further to the right when theiteration number is increased. event observed a mass of the united system of about m = 60 solar masseswas reported. This gives a value of (cid:101) ω R = 0 .
47. This is the kind of order wealso obtain. However, he mass m was obtained recurring to the GR, i.e., itis theory based . When in a distinct theory a larger real frequency (cid:101) ω R is ob-tained, keeping ω R the same, a larger mass is deduced. This also implies alarger release in energy and a larger deduced luminous distance, as suggestedin [8].Where the observed distribution of frequencies lies, is a matter of the dy-namics of a black hole merger. If it corresponds to a low damping process,the differences between GR and pc-GR are probably too small to be detected.However, if the range of frequencies is in the large damping region, pc-GRpredicts larger masses than GR, resulting in larger distances of the mergerevent. One way to detect a difference is to search, for case of large dampingmodes, for simultaneous light events in the same region of the sky where themerger is observed. If consistently this light event is at larger distances thanthe deduced event, using GR, then it will be in favor of the existence of addi-tional terms in the metric. This depends also on the requirement that a lightevent is produced, requiring some mass distribution near to the event, as anaccretions disc. xial and polar ring down modes 21 - - - Im ω R e ω - - - Im ω R e ω - - - Im ω R e ω - - - Im ω R e ω Fig. 5
Comparison of axial to polar modes in pc-GR (left and right panel, respectively),depicting 300 (red dots), 400 (green dots) and 500 (blue dots) iterations. The upper rowshows a large range in − (cid:101) ω I and the lower row is a zoom to the lowest region. Note thestructural similarity of the axial to polar modes, suggesting an equivalence, though notperfect, as in GR. Axial and polar modes where calculated within the
General Relativity (GR)and possible extensions, involving a parametric mass-function m ( r ), whoseleading term correction is proportional to 1 /r . In particular the pseudo-complex General Relativity (pc-GR), leads to such an extension of the pa- rameter mass-function. The Regge-Wheeler equation for the axial modes andthe Zerilli equation for the polar modes were derived, with their correspond-ing potentials. After having constructed γ ( r ), it was shown that axial andpolar modes, though different, still share some common features. The modeswere found to be stable, implying that the corrections to the metric lead toconsistent results.Adding a further r -dependence to the mass-function leads to a branch offrequencies at high damping, resulting in larger deduced masses than in GR,while for low damping no large differences are observed.When the frequencies distribution in a merger lies in the large dampingregion, the deduced masses in the extended version are larger, implying also alarger distance to a gravitational wave event. If one detects at the same time ofthis event a light emission, the two observations result in a different distanceusing GR.The present results serve as a starting point to understand the changesinvolved in the frequency distribution of the ring down modes in extendingthe theory of GR, which results in a parametric mass-function m ( r ) in themetric components.In the Appendices explicit derivation of the ansatz for Z (+) is given, whichincludes the one proposed by S. Chandrasekhar in [10,11].In a future publication, we will address the pc-Kerr metric, with howevermore involved equations. This will pose a problem to the numerical methodused. Acknowledgments
Financial support from DGAPA-PAPIIT (IN100421) is acknowledged.
Appendix A: Axial potential
Using Q = rZ ( − ) gives for the first term in (9) ∆ ddr ∆r dQdr = ∆ ddr ∆r (cid:2) r ddr Z ( − ) + Z ( − ) (cid:3) = ∆ ddr (cid:104) ∆r dZ ( − ) dr + ∆r Z ( − ) (cid:105) = ∆ (cid:104) ddr (cid:0) ∆r (cid:1) Z ( − ) + ∆r dZ ( − ) dr (cid:105) + ∆ (cid:104) r ∆r dZ ( − dr (cid:105) = ∆ (cid:104) ddr (cid:0) ∆r (cid:1) Z ( − ) + ∆r dZ ( − ) dr (cid:105) − (cid:0) ∆r (cid:1) dZ ( − ) dr + ∆r ddr ∆r dZ ( − ) dr . (54)Using that ∆r ddr = ddr ∗ , we arrive finally at r d Z ( − ) dr ∗ + ∆ ddr (cid:18) ∆r (cid:19) Z ( − ) . (55) xial and polar ring down modes 23 This has to be substituted into the differential equation (9), leading to r d dr ∗ Z ( − ) + ∆ ddr (cid:18) ∆r (cid:19) Z ( − ) − µ ∆r Z ( − ) + rω Z ( − ) = 0 . (56)Dividing by r and reordering the terms in this differential equation, leadsto (12), with the potential given in (14). Appendix B: Ansatz for the polar mode wave function
The MATHEMATICA code, used to derive the equations in this section, canbe retrieved from [26].The general ansatz (21) is used, also valid for a constant mass-function,for which the expression in [10,12] is recovered.Using (21), namely Z (+) ( r ) = α ( r ) N ( r ) + β ( r ) V ( r ) + γ ( r ) LX ( r ) (57)and applying the operator d dr ∗ to Z (+) ( r ), we found [26] that the factor of N ( r ) does not depend on the frequency squared ω , only the factors of V ( r )and LX ( r ) do. We use the definitions LX ( r ) = L ( r ) + X ( r ) X ( r ) = nV ( r ) . (58)Concentrating only on the component proportional to ω , one should ob-tain for the factor of V ( r ) and LX ( r ) the result − ω ( β ( r ) V ( r ) + γ ( r ) LX ( r )).The factor obtained after the application of d dr ∗ onto V ( r ) is ( nα ( r ) − β ( r )),which must be equated to − β ( r ). This demands α ( r ) = 0 . (59)This automatically reduces (57) to Z (+) ( r ) = β ( r ) V ( r ) + γ ( r ) LX ( r ).For the factor of LX ( r ), restricting to the one proportional to ω , andusing (59) we obtain − β ( r ) + nγ ( r ) + 2 rβ (cid:48) ( r ) = nγ ( r ) (the prime refers tothe derivative in r ), which leads to the differential equation β (cid:48) = 1 r β , (60)with the solution β ( r ) = r . (61) y - - r Fig. 6
The left hand side depicts the two functions V ( r ) (green online) and V LX ( r ) (redonline) in a wide range of r . The right panel shows the difference ( V ( r ) − V LX ( r )) in alimited region in r. The γ ( r ) function was determined earlier and is given in (24) As seen inAppendix C, this solution satisfies the equation G ( r ) = 0 (25) in a wide rangeof r , save near the point r = m , implying that the Zerilli equation can beapproximately constructed, with a quite small error. Appendix C: Comparison of V (+) V with V (+) LX , using (24) for γ ( r ) In this appendix we analyze the functions V (+) ( r ), V (+) ( r ) and their differ-ence, which is proportional to the condition (25).On the left panel of Fig. 6 the two potentials V (+) V and V (+) LX are comparedfor a wide range of r . The potentials were extended to r < in order toappreciate the agreement of both potentials, using the γ ( r ) as given in (24).The two functions agree very well in a wide range of r . The only differenceappears near r = , which is shown in a reduced scale on the right hand sideof the figure, where, the difference G ( r ) = (cid:16) V (+) V − V (+) LX (cid:17) is depicted. All inall, the expression for γ ( r ) works very well, however with the grain of salt thatthe Zerilli equation is not identically satisfied. References
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