Massive scalar field perturbation on Kerr black holes in dynamical Chern-Simons gravity
MMassive scalar field perturbation on Kerr black holes in dynamicalChern-Simons gravity
Shao-Jun Zhang ∗ Institute for Theoretical Physics & Cosmology,Zhejiang University of Technology, Hangzhou 310032, ChinaUnited Center for Gravitational Wave Physics,Zhejiang University of Technology, Hangzhou 310032, China (Dated: February 23, 2021)We study massive scalar field perturbation on Kerr black holes in dynamical Chern-Simonsgravity by performing a (2+1)-dimensional simulation. Object pictures of the wave dynamicsin time domain are obtained. The tachyonic instability is found to always occur for anynonzero black hole spin and any scalar field mass as long as the coupling constant exceeds acritical value. The presence of the mass term suppresses or even quenches the instability. Thequantitative dependence of the onset of the tachyonic instability on the coupling constant,the scalar field mass and the black hole spin is given numerically.
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I. INTRODUCTION
Past years have witnessed the great achievements in astronomical observations, especially thefirst-ever detection of gravitational wave (GW) [1–3]. With the continuous improvement of theGW detection ability, the study of black hole physics has entered a golden age, which enables usto test general relativity (GR) with unprecedented precision in the strong gravity regime. Despitethe great successes of GR in explaining various astrophysical and cosmological phenomena [4],there remain tension between it and quantum theory and cosmological observations. Therefore,to alleviate the tension, a variety of modified gravity theories (MOGs) have been proposed amongwhich some were found viable by employing the current available observational constraints, such asscalar-tensor theory, dynamical Chern-Simon gravity (dCSG), scalar-Einstein-Gauss-Bonnet theory(sEGB) and Lorentz-violating gravity [5].As a special MOG, dCSG has attracted lots of attention recently. In this theory, beyond the ∗ Electronic address: [email protected] a r X i v : . [ g r- q c ] F e b usual Einstein-Hilbert term, an additional dynamical scalar field is introduced to couple non-minimally with the gravitational Chern-Simons invariant (also called Pontrayagin density) [6, 7].This kind of coupling naturally arises in some candidate quantum gravity theories including stringtheory [8, 9] and loop quantum gravity [10–12], and also in effective field theories [13]. For areview, please refer to Ref. [14]. Amounts of effort have been devoted to study black hole physicsin dSCG and its astrophysical implications [15–31]. It is interesting to note that GR black holesolutions, the Kerr black holes, are also allowed in dSCG. However, dynamics of perturbations onthe same Kerr background is generally different from that in GR, which actually provides us amethod to distinguish dSCG and GR through the study of perturbation dynamics. Most recently,with the presence of such coupling and in Kerr background, it is found that the massless scalar fieldwill acquire an effective mass square which becomes negative in the vicinity of black hole horizon,resulting in the tachyonic instability and thus leading to the so-called spontaneous scalarization[32–34]. Actually, this novel phenomenon has long been observed in neutron stars but there theinstability is caused by the surrounding matter instead of the curvature [35]. It has also beenobserved and studied extensively most recently in sEGB theory where a similar coupling is presentbut between the scalar field and the Gauss-Bonnet invariant [36–49].The minority existing work on the tachyonic instability in dSCG [32–34] are all focused on thecase when the scalar field is massless. From the viewpoint of effective field theory, it is naturalto include a mass term or a more general self-interaction term for the scalar field, which has beenstudied a lot in sEGB theory [40–42, 46]. However, at linearized level, only the mass term canalter the onset of the tachyonic instability and the induced spontaneous scalarization. So, in thepresent paper, we would like to extend the study of the tachyonic instability of Kerr black holein dCSG to the case when the scalar field is massive. We will see later that the inclusion of massterm will alter the object picture of the wave dynamics and the onset of the tachyonic instabilityconsiderably.This paper is organized as follows. In Sec. II, we give a brief introduction of the dCSG theoryand write out the scalar field perturbation equation. In Sec. III, we describe our numerical methodfor solving the scalar field perturbation equation. In Sec. IV, we present our numerical results.The last section is devoted to summary and discussions. II. DYNAMICAL CHERN-SIMONS GRAVITY AND MASSIVE SCALAR FIELDPERTURBATION
The action of a general dynamical Chern-Simons gravity is [14, 32–34] S = 12 κ (cid:90) dx √− g ( R −
2Λ + αf (Φ) ∗ RR + L Φ ) , L Φ = − ∇ µ Φ ∇ µ Φ − V (Φ) , where the scalar field Φ is non-minimally coupled to the Chern-Simons invariant ∗ RR ≡ (cid:15) αβγδ R ναβµ R µγδν , (1)with the coupling constant α . f (Φ) is a function of the scalar field and Λ is the cosmologicalconstant. From the action, one can derive the equations of motion ∇ Φ = dVd Φ − α ∗ RR dfd Φ , (2) R µν − g µν R + Λ g µν = αT CSµν + T Φ µν , (3) T CSµν = − ∇ σ f (cid:15) σαβ ( µ ∇ β R αν ) − ∇ α ∇ β f ∗ R α ( µν ) β ,T Φ µν = 12 ∇ µ Φ ∇ ν Φ − g µν V (Φ) − g µν ∇ ρ Φ ∇ ρ Φ . The theory admits GR black hole solutions with constant scalar profile Φ = Φ , if V (Φ ) = 0 , dVd Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ = 0 , dfd Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ = 0 . (4)In the following, we will consider a simple case by choosing Λ = 0 and V (Φ) = m Φ so thatthe scalar field is massive with mass m Φ without self-interaction. Also, we assume the couplingfunction f (Φ) to take a general form as f (Φ) = 12 β (cid:16) − e − β Φ (cid:17) , (5)where β > f (Φ) reduces to a quadratic form considered inRef. [32].We are going to study the wave dynamics of scalar field perturbations on the background ofKerr black holes in the linear regime. The metric in the Boyer-Lindquist coordinates is ds = − ∆ ρ ( dt − a sin θdφ ) + ρ ∆ dr + ρ dθ + sin θρ ( adt − ( r + a ) dϕ ) , (6)where ∆ ≡ r − M r + a and ρ ≡ r + a cos θ . In this case, the scalar perturbation equation(2) in the Kerr background reduces to ∇ Φ = ( m − α ∗ RR )Φ , (7) ∗ RR = 96 aM r cos θ (3 r − a cos θ )( r − a cos θ )( r + a cos θ ) , where the Chern-Simons (CS) invariant is valued in the background. As one can see, the scalar fieldacquires an effective mass square m = m − α ∗ RR , which is position-dependent and approachesto m at infinity, with the sign depending on the coupling constant α . When α = 0, the aboveequation describes wave propagation of a free scalar field in the Kerr background which has beenstudied thoroughly and stability depends on the value of the physical mass m Φ : When m Φ = 0,no instability is observed [50–52]; While for m Φ (cid:54) = 0, superradiant instability may occur [53–59].For m Φ = 0 and α >
0, tachyonic instability is found to exist for any nonzero spin as long as α exceeds a critical value α c , and α c decreases as a is increased [32]. Scalarized rotating black holesolution, which is expected to be the end-state of this tachyonic instability, has been constructed inRef. [34] in the so-called ”decoupling limit”. It should be noted that the scalar field perturbationequation (10) is invariant under the transformation α → − α, θ → π − θ. (8)So the sign of α can be absorbed into the CS invariant by redefining the θ -coordinate, and thus thesituation with α < α >
0, as has been confirmed numericallyand analytically in Ref. [33]. This is different from the case in sEGB, where α > α < α > α and the mass m Φ on wave dynamics in thedCSG theory. III. NUMERICAL METHOD
We will apply the numerical method as Refs. [45, 46, 50, 51, 60] to solve the scalar perturbationequation (7). In this method, the tortoise coordinate r ∗ and Kerr azimuthal coordinate φ ∗ areutilized, which are defined through the transformation dr ∗ = r + a ∆ dr, dφ ∗ = dφ + a ∆ dr. (9)In the new coordinates ( t, r ∗ , θ, φ ∗ ), the semi-infinite radial domain outside the horizon r ∈ ( r + , ∞ )is mapped to infinite range r ∗ ∈ ( −∞ , + ∞ ) and the scalar perturbation equation can be writtenas (cid:2) ∆ a sin θ − ( r + a ) (cid:3) ∂ t Φ + ( r + a ) ∂ r ∗ Φ + 2 r ∆ ∂ r ∗ Φ − M ar∂ t ∂ φ ∗ Φ+2 a ( r + a ) ∂ r ∗ ∂ φ ∗ Φ + ∆ (cid:20) θ ∂ θ (sin θ∂ θ Φ) + 1sin θ ∂ φ ∗ Φ (cid:21) = ( m − α ∗ RR )∆ ρ Φ . (10)Taking into account the axial symmetry of the Kerr spacetime, the scalar perturbation can bedecomposed as Φ( t, r ∗ , θ, φ ∗ ) = (cid:88) m Ψ( t, r ∗ , θ ) e imφ ∗ , (11)where m is the well-know azimuthal number. With this ansatz and by introducing a new variableΠ ≡ ∂ t Ψ, finally the perturbation equation can be cast into a form of two coupled first-order partialdifferential equations ∂ t Ψ = Π ,∂ t Π = 1Σ (cid:34) − iamM r Π + ( r + a ) ∂ r ∗ Ψ + (cid:0) iam ( r + a ) + 2 r ∆ (cid:1) ∂ r ∗ Ψ+∆ ∂ θ Ψ + ∆ cot θ∂ θ Ψ − (cid:18) ∆ m sin θ + m − α ∗ RR (cid:19) Ψ (cid:35) , (12)where Σ ≡ ( r + a ) − ∆ a sin θ .Equations with the form as (12) are suitable for the method of line [61]. Precisely, the derivativesin r ∗ and θ directions are approximated by finite differences, and the evolution in the time directionis implemented with the fourth-order Runge-Kutta integrator. Also, we impose physical boundaryconditions, namely ingoing wave at the horizon and outgoing wave at infinity, following [62]. Inpractical calculations, one has to truncate the infinite radial computational domain to a finite rangeand put boundary conditions at the outer edges, thus inevitably resulting spurious wave reflectionsfrom the outer edges. To overcome this “outer-boundary problem”, one can simply push the outeredges to very large values so that the spurious reflections will not affect the observed signal fora sufficiently long evolution time. At the poles in the angular direction θ = 0 and π , we imposephysical boundary conditions Ψ | θ =0 ,π = 0 for odd m while ∂ θ Ψ | θ =0 ,π = 0 for even m [51].As Ref. [49], the initial data of the scalar perturbation is considered to be a Gaussian distributionlocalized outside the horizon at r ∗ = r ∗ and has time symmetry,Ψ( t = 0 , r ∗ , θ ) ∼ Y (cid:96)m e − ( r ∗− r ∗ )22 σ , (13)Π( t = 0 , r ∗ , θ ) = 0 . (14)where Y (cid:96)m is the θ -dependent part of the spherical harmonic function and σ is the width of theGuassian distribution. In the following, without loss of generality, we take r ∗ = 20 M . Also, weset M = 1 so that all quantities are measured in units of M . Observers are assumed to locate at r ∗ = 30 M and θ = π .One should note that the Kerr spacetime is not spherically symmetric except when a = 0, sothe mode-mixing phenomenon occurs [32, 52, 63, 64]: a pure initial (cid:96) - multipole will excite othermultipoles with the same m as it evolves. Taking into account this phenomenon and for simplicity,in the following we will only consider axisymmetric perturbations with (cid:96) = m = 0. IV. RESULTS
We have performed the time evolution of the scalar perturbations for various values of spin a and scalar field mass m Φ , and found that instability always occurs as long as the coupling constant α exceeds a critical value α c . Representative examples are given In Figs. 1 and 2, with similarpictures for other values of parameters. α = α = α c = α = α = - - - t l og | Ψ | (a) a = 0 . α = α = α c = α = α = - - - t l og | Ψ | (b) a = 0 . FIG. 1: (color online) Time evolution of the scalar perturbation for a = 0 . . m Φ = 0 .
5. The initial multipole we considered is (cid:96) = m = 0. Time is in units of M . m Φ = m Φ = m Φ = m Φ = - - - t l og | Ψ | (a) a = 0 . , α = 30 m Φ = m Φ = m Φ = m Φ = - - - t l og | Ψ | (b) a = 0 . , α = 3 . FIG. 2: (color online) Time evolution of the scalar perturbation. The parameters are fixed as ( a = 0 . , α =30) ( left ) and ( a = 0 . , α = 3 .
0) ( right ). The initial multipole we considered is (cid:96) = m = 0. Time is in unitsof M . In Fig. 1, time evolutions of the axisymmetric scalar perturbation are plotted with fixed spinand scalar field mass. From the figure, one can see that instability will be triggered once thecoupling constant α exceeds a critical value α c , and α c decreases as the spin a is increased. For α > α c , larger α makes the instability to appear earlier and more violent. In Fig. 2, we fix thespin and the coupling constant to study the influence of the scalar field mass on the time evolutionof the perturbations. From the figure, one can see that increasing m Φ will suppress the instabilityand delay its appearance, and the instability will cease to appear if m Φ is further increased, whichimplies that α c increases as m Φ is increased.Physically, the influences of the coupling constant and the scalar field mass on the stability canbe understood qualitatively from the profiles of the effective mass square m = m − α ∗ RR , asshown in Fig. 3. The profiles exhibit odd parity under the transformation θ → π − θ as we havealready mentioned above in Eq. (8). We should note that the effective mass square is a positiveconstant m = 0 . when α = 0. As α is increased and exceeds some value α , m will becomenegative in vicinity of the horizon for θ ∈ [0 , π ) and become more negative when α is furtherincreased. We note that α < α c , which implies that small negative m < m is negative enough ( α > α c ) can the instability bedeveloped. With the further increase of α , the instability appears earlier and becomes more violentfor which m will become more negative. Moreover, from the analytical definition of the effectivemass square, it is explicit that the influence of the scalar field mass m Φ is opposite to that of thecoupling constant. (a) a = 0 . a = 0 . FIG. 3: (color online) Profiles of the effective mass square m for a = 0 . . m Φ = 0 . The more complete picture of the influences of the parameters ( a, α, m Φ ) on the onset of thetachyonic instability is summarized in Fig. 4, from which the above mentioned phenomena can beseen more clearly. When m Φ = 0, the scalar field becomes massless and its wave dynamics hasbeen studied in Ref. [32] by adopting a different numerical strategy. Our results for this particularcase are in good agreement with those there. We should also note that in the extreme limit a → α c increases with the increasing of m Φ . UnstableStable m Φ = m Φ = m Φ = a α FIG. 4: (color online) Boundary between stable and unstable regions in a − α plane for different scalar fieldmasses. The initial multipole we considered is (cid:96) = m = 0. V. SUMMARY AND DISCUSSIONS
In this work, within the framework of dCSG theory, we studied carefully the time evolutionof the massive scalar field perturbation on Kerr background by performing a (2 + 1)-dimensionalsimulation. We found that tachyonic instability always occurs for any nonzero spin a and anyscalar field mass m Φ as long as the coupling constant α exceeds a critical value α c . The value of α c depends on the values of a and m Φ . For fixed m Φ , α c decreases as a is increased; While for fixed a , α c increases as m Φ is increased, which means the scalar field mass m Φ will suppress the instabilityor even quench the instability if m Φ is large enough. Physically, as shown in Fig. 3, the influencesof the parameters α and m Φ on the onset of the tachyonic stability can be explained qualitativelyfrom the behaviors of the effective mass square m = m − α ∗ RR .Although, we have obtained object pictures of the time evolution of the scalar field perturbationand the quantitative influences of the parameters ( a, α, m Φ ) on the onset of the tachyonic instability,there remains several interesting issues. From Figs. 1 and 2, one can see that, if instability is nottriggered ( α < α c ), the scalar field perturbation will exhibit oscillatory behavior at late time. Suchbehavior has been observed in GR ( α = 0) with analytical expression given in Refs. [65, 66],Ψ ∼ cos( m Φ t ) t − / . (15)And it has also been observed in sEGB [46]. From the figures, it is interesting to see that thecoupling constant α nearly has no influence on such oscillatory behavior. How to understand thisphenomenon needs more careful studies on the late time tail. In this work, we only study the timeevolution of the scalar field perturbation in linearized level. The appearance of instability indicatesthe possible existence of scalarized black holes as an end-state. To gain better understanding ofthe fate of this instability, a full non-linear evolution of the perturbation and the construction ofscalarized black holes are called for. As perturbations with higher azimuthal number m normallytrigger more moderate instability, so in this work we only focus on axisymmetric perturbations with m = 0. For perturbations with m (cid:54) = 0, beyond the tachyonic instability, there may appear anothertype of instability for massive scalar field perturbations, the well-known superradiant instability[67]. Unfortunately, as the growing time of the superradiant instability is usually very large, theobservation of this instability requires a long stable time evolution of the perturbations, which willbe a great challenge for numerical calculations. We leave these questions for further investigations.0 Acknowledgments
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