Gauss-Bonnet black holes supporting massive scalar field configurations: The large-mass regime
aa r X i v : . [ g r- q c ] J a n Gauss-Bonnet black holes supporting massive scalar field configurations: Thelarge-mass regime
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: January 8, 2021)It has recently been demonstrated that black holes with spatially regular horizons can sup-port external scalar fields (scalar hairy configurations) which are non-minimally coupled to theGauss-Bonnet invariant of the curved spacetime. The composed black-hole-scalar-field system ischaracterized by a critical existence line α = α ( µr H ) which, for a given mass of the supported scalarfield, marks the threshold for the onset of the spontaneous scalarization phenomenon [here { α, µ, r H } are respectively the dimensionless non-minimal coupling parameter of the field theory, the propermass of the scalar field, and the horizon radius of the central supporting black hole]. In the presentpaper we use analytical techniques in order to explore the physical and mathematical properties ofthe marginally-stable composed black-hole-linearized-scalar-field configurations in the eikonal regime µr H ≫ α = α ( µr H ) of the system which separates bare Schwarzschild black-holespacetimes from composed hairy (scalarized) black-hole-field configurations. I. INTRODUCTION
The mathematically elegant no-hair theorems presented in [1–3] have revealed the physically important fact that,within the framework of classical general relativity, spherically symmetric black holes with regular horizons cannotsupport external static matter configurations which are made of scalar fields with minimal coupling to gravity. Asexplicitly proved in [4, 5], the intriguing no-hair property of static black holes can also be extended to the physicalregime of scalar matter fields which are characterized by a non-trivial (non-minimal) coupling to the Ricci curvaturescalar of the corresponding spherically symmetric spacetimes.Interestingly, later developments [6–11] have revealed the intriguing fact that spatially regular hairy matter configu-rations which are made of scalar fields with non-minimal couplings to the Gauss-Bonnet curvature invariant G may besupported in curved black-hole spacetimes. In particular, it has been proved [9–11] that, in extended Scalar-Tensor-Gauss-Bonnet theories whose actions contain a non-trivial field-curvature coupling term of the form f ( φ ) G [12], blackholes with regular horizons may support scalar fields with non-trivial spatial profiles (see [13–15] for the physicallyrelated model of spontaneously scalarized charged black-hole spacetimes which owe their existence to a non-trivialcoupling between the external scalar field and the electromagnetic field tensor of the central supporting charged blackhole).In a physically realistic field theory, the spontaneous scalarization phenomenon should be characterized by a non-trivial coupling function f ( φ ) whose mathematical form allows the existence of bare (non-scalarized) black-hole so-lutions in the weak-coupling regime [9–11]. Specifically, the physically important studies presented in [9–11] haveconsidered Scalar-Tensor-Gauss-Bonnet theories whose coupling functions are characterized by the limiting behavior f ( φ → ∝ αφ in the weak-field regime. Here the physical parameter α is the dimensionless coupling constant ofthe non-trivial field theory [see Eq. (10) below].Intriguingly, it has recently been proved [16] that, for non-minimally coupled massive scalar fields, the composedblack-hole-field system is characterized by a critical existence-line α = α ( µr H ) which separates bare Schwarzschildblack holes from hairy (scalarized) black-hole-field solutions of the field equations (here µ is the proper mass ofthe supported scalar field and r H is the horizon radius of the central black hole). In particular, the existence-lineof the system corresponds to linearized marginally-stable scalar field configurations which are supported by centralSchwarzschild black holes. [In the physics literature [17, 18], the supported linearized scalar field configurations areusually called scalar ‘clouds’ in order to distinguish them from self-gravitating (non-linear) hairy matter configu-rations]. Interestingly, the numerical results presented in [16] have revealed the fact that, for a given value of thedimensionless coupling parameter α , the horizon radius (mass) of the central supporting black hole is a monotonicallydecreasing function of the mass of the supported scalar field.The main goal of the present paper is to explore, using analytical techniques, the physical and mathematical proper-ties of the composed Schwarzschild-black-hole-nonminimally-coupled-linearized-massive-scalar-field cloudy configura-tions. In particular, using a WKB analysis in the dimensionless large-mass µr H ≫ α = α ( µr H ) of thecomposed Schwarzschild-black-hole-massive-scalar-field system. Interestingly, the derived resonance formula [see Eq.(24) below] would provide a simple analytical explanation for the numerically observed [16] monotonic behavior of thefunction r H = r H ( µ ; α ) along the critical existence-line of the system. II. DESCRIPTION OF THE SYSTEM
We shall study analytically the discrete resonant spectrum which characterizes the composed Schwarzschild-black-hole-linearized-massive-scalar-field configurations in the physical regime of large field masses. As shown numericallyin [9–11, 16], the spatially regular cloudy field configurations owe their existence to their non-trivial coupling to theGauss-Bonnet invariant
G ≡ R µνρσ R µνρσ − R µν R µν + R of the curved spacetime. The black-hole spacetime ischaracterized by the spherically-symmetric curved line element [19] ds = − h ( r ) dt + 1 h ( r ) dr + r ( dθ + sin θdφ ) , (1)where h ( r ) = 1 − r H r . (2)Here r H = 2 M is the horizon radius of the central supporting Schwarzschild black hole of mass M .The composed black-hole-field system is characterized by the action [9, 10, 16, 20] S = 12 Z d x √− g h R − ∇ α φ ∇ α φ − µ φ + f ( φ ) G i , (3)where the radius-dependent Gauss-Bonnet curvature invariant of the Schwarzschild black-hole spacetime is given by G = 12 r r . (4)The scalar function f ( φ ) in (3) controls the non-minimal coupling between the Gauss-Bonnet invariant of the curvedspacetime and the massive scalar field. As shown in [9, 10, 16], in order to guarantee the existence of bald (non-scalarized) black-hole solutions in the field theory, this coupling function should have the universal leading-orderquadratic behavior f ( φ ) = 18 ηφ (5)in the linearized regime. The physical parameter η , which controls the strength of the non-trivial quadratic couplingbetween the massive scalar field and the Gauss-Bonnet curvature invariant, has the dimensions of length .Using the functional expression [21] φ ( r, θ, φ ) = X lm ψ lm ( r ) r Y lm ( θ ) e imφ (6)for the non-minimally coupled static scalar field and defining the tortoise radial coordinate y by the relation [22] drdy = h ( r ) , (7)one finds that the spatial behavior of the supported massive scalar field configurations in the Schwarzschild black-holespacetime (1) is determined by the Schr¨odinger-like ordinary differential equation [9, 10, 16] d ψdy − V ψ = 0 , (8)where [9, 10, 16] V ( r ) = (cid:16) − r H r (cid:17)h l ( l + 1) r + r H r + µ − αr r i . (9)Here α ≡ ηr (10)is the dimensionless non-trivial coupling parameter of the composed black-hole-massive-scalar-field system.The Schr¨odinger-like equation (8) with its effective radial potential (9), supplemented by the physically motivatedboundary conditions of exponentially decaying scalar eigenfunctions at spatial infinity and a spatially regular functionalbehavior at the black-hole horizon [9, 10, 16], ψ ( r → ∞ ) ∼ r − e − µr → ψ ( r = r H ) < ∞ , (11)determine the discrete resonant spectrum { α n ( µ, r H ) } n = ∞ n =0 which characterizes the composed cloudy black-hole-nonminimally-coupled-linearized-massive-scalar-field configurations. In particular, the fundamental resonant mode, α = α ( µ, r H ), determines the critical existence-line of the field theory in the curved black-hole spacetime. III. THE DISCRETE RESONANT SPECTRUM OF THE COMPOSEDBLACK-HOLE-LINEARIZED-MASSIVE-SCALAR-FIELD SYSTEM: A WKB ANALYSIS
In the present section we shall use analytical techniques in order to study the discrete resonant spectrum { α n ( µ, r H ) } n = ∞ n =0 of the dimensionless scalar-Gauss-Bonnet coupling parameter which characterizes the composedblack-hole-linearized-massive-scalar-field configurations in the large-mass regime µr H ≫ max { , l } . (12)We first point out that, in terms of the tortoise coordinate y [see Eq. (7)], the Schr¨odinger-like radial differentialequation (8) has a mathematical form which is amenable to a standard WKB analysis. In particular, a standard second-order WKB analysis for the spatially regular bound-state resonances of the Schr¨odinger-like ordinary differentialequation (7) yields the well-known quantization condition [23–26] Z y + y − dy p − V ( y ; α ) = (cid:0) n − (cid:1) · π ; n = 1 , , , ... . (13)The two boundaries { y − , y + } of the WKB integral relation (13) are determined by the classical turning points ofthe radial binding potential (9) [that is, V ( y − ) = V ( y + ) = 0]. The integer n is the resonance parameter whichcharacterizes the discrete bound-state resonant modes of the composed black-hole-nonminimally-coupled-massive-scalar-field system.Taking cognizance of the differential relation (7), one can express the WKB integral relation (13), which characterizesthe composed black-hole-linearized-massive-scalar-field configurations, in the form Z r + r − dr p − V ( r ; α ) h ( r ) = (cid:0) n − (cid:1) · π ; n = 1 , , , ... . (14)The radial turning points { r − , r + } of the binding potential (9) are determined by the two polynomial relations1 − r H r − = 0 (15)and l ( l + 1) r + r H r + µ − αr r = 0 . (16)As we shall now show explicitly, the WKB integral relation (14) can be studied analytically in the large-mass regime(12). In particular, defining the dimensionless radial coordinate x ≡ r − r H r H , (17)one can expand the effective binding potential of the composed black-hole-massive-field system in the form V [ x ( r )] = − (cid:16) αr − µ (cid:17) · x + (cid:16) αr − µ (cid:17) · x + O ( x ) . (18)The near-horizon radial potential (18) has the form of an effective binding potential. In particular, from (18) onefinds that the two turning points { x − , x + } of the WKB integral relation (14) are given by the simple dimensionlessfunctional expressions x − = 0 (19)and x + = αr − µ αr − µ . (20)Taking cognizance of Eqs. (17), (18), (19), and (20), one finds that, in the large-mass regime (12), the WKB integralequation (14) can be approximated by [27] q α − µ r Z x + dx s x − x + = (cid:0) n −
14 ) · π ; n = 1 , , , ... . (21)Interestingly, and most importantly for our analysis, the integral on the l.h.s of Eq. (21) can be evaluated analyticallyto yield the WKB resonance relation α − µ r p α − µ r = n −
14 ; n = 1 , , , ... . (22)The solution of the polynomial equation (22) for the dimensionless coupling parameter α of the composed black-hole-massive-field theory is given by the rather cumbersome expression α n = µ r + 14( n + 34 ) + 2 r µ r ( n + 34 ) + 49( n + 34 ) ; n = 0 , , , ... . (23)In the large-mass µr H ≫ n + 1 regime, the resonance spectrum (23) can be approximated by the compact analyticalrelation α n = µ r · h √ n + ) µr H i ; n = 0 , , , ... . (24)The discrete resonance spectrum (24) of the non-minimal coupling parameter α characterizes the cloudy Schwarzschild-black-hole-massive-scalar-field configurations in the eikonal large-mass regime (12). IV. SUMMARY
The recently published highly interesting works [9–11, 16] have explicitly proved that, in some field theories, blackholes may support external matter configurations (hair) made of scalar fields, a phenomenon which is known bythe name black-hole spontaneous scalarization. In particular, it has been demonstrated numerically [9–11, 16] thatspatially regular (massless as well as massive) scalar fields with nontrivial couplings to the Gauss-Bonnet curvatureinvariant may be supported by central black holes with regular horizons.Intriguingly, the numerical results presented in [9–11, 16] have revealed the fact that the dimensionless physicalparameter α , which controls the non-trivial coupling between the Gauss-Bonnet invariant of the curved spacetimeand the supported scalar matter configurations, is characterized by a discrete resonant spectrum { α n } n = ∞ n =0 whichcorresponds to black holes that support spatially regular nonminimally coupled linearized scalar field configurations.In the present paper we have used analytical techniques in order to explore the physical properties of the sponta-neously scalarized hairy black-hole spacetimes in the regime of cloudy (linearized) supported field configurations. Inparticular, we have derived the compact WKB analytical formula (24) for the discrete resonant spectrum which char-acterizes the non-trivial coupling parameter α of the composed black-hole-massive-scalar-field theory in the physicalregime µr H ≫ µr Hmax ( α ) = r α + 278 − r
278 for α ≫ existence-line which characterizes the hairy Schwarzschild-black-hole-massive-scalar-field configurations.The α -dependent critical line (25) for the masses of the supported non-minimally coupled scalar fields marks, inthe large-mass µr H ≫ α and for a given mass (radius) of the central supporting black hole, the hairy black-hole-nonminimally-coupled-massive-scalar-field configurations are characterized by the mass inequality µ ( α ) ≤ µ max ( α ).Interestingly, the analytically derived formula (25) for the critical existence-line of the system implies, in agreementwith the important numerical results presented in [16], that, for a given value of the dimensionless coupling parameter α , the mass M (horizon radius r H ) of the central supporting black hole is a monotonically decreasing function of themass µ of the nonminimally coupled scalar field. ACKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I would like to thank Yael Oren, Arbel M. Ongo,Ayelet B. Lata, and Alona B. Tea for helpful discussions. [1] J. D. Bekenstein, Phys. Rev. D , 1239 (1972).[2] C. A. R. Herdeiro and E. Radu, Int. J. Mod. Phys. D , 1542014 (2015).[3] T. P. Sotiriou, Class. Quant. Grav. , 214002 (2015); T. P. Sotiriou and V. Faraoni, Phys. Rev. Lett. , 081103 (2012).[4] A. E. Mayo and J. D. Bekenstein and, Phys. Rev. D , 5059 (1996).[5] S. Hod, Phys. Lett. B , 521 (2017); S. Hod, Phys. Rev. D , 124037 (2017).[6] E. Babichev and C. Charmousis, JHEP , 106 (2014).[7] C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. , 221101 (2014).[8] T. P. Sotiriou and S.-Y. Zhou, Phys. Rev. Lett. , 251102 (2014); T. P. Sotiriou and S.-Y. Zhou, Phys. Rev. D ,124063 (2014).[9] D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. , 131103 (2018).[10] H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Phys. Rev. Lett. , 131104 (2018).[11] P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. Lett. , 011101 (2019).[12] Here φ is the non-minimally coupled scalar field.[13] C. A. R. Herdeiro, E.Radu, N. Sanchis-Gual, and J. A. Font, Phys. Rev. Lett. , 101102 (2018).[14] P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu, and N. Sanchis-Gual, Class. Quant. Grav. , 134002(2019).[15] S. Hod, Phys. Lett. B , 135025 (2019).[16] C. F. B. Macedo, J. Sakstein, E. Berti, L. Gualtieri, H. O. Silva, and T. P. Sotiriou, Phys. Rev. D , 104041 (2019).[17] S. Hod, Phys. Rev. D , 104026 (2012)[arXiv:1211.3202]; S. Hod, The Euro. Phys. Journal C , 2378 (2013)[arXiv:1311.5298]; S. Hod, Phys. Rev. D , 024051 (2014) [arXiv:1406.1179].[18] C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. , 221101 (2014).[19] We shall use natural units in which 8 πG = c = 1.[20] One may also include a quartic self-interaction term of the form − λφ / λ .[21] The parameters { l, m } (with l ≥ − l ≤ m ≤ l ) in the scalar field decomposition (6) are respectively the spherical andazimuthal harmonic indices. For brevity, we shall henceforth omit these integer indices.[22] Note that the differential relation (7) maps the semi-infinite radial regime r ∈ [ r H , ∞ ] to the corresponding infinite regime y ∈ [ −∞ , ∞ ].[23] L. D. Landau and E. M. Liftshitz, Quantum Mechanics , 3rd ed. (Pergamon, New York, 1977), Chap. VII.[24] J. Heading,
An Introduction to Phase Integral Methods (Wiley, New York, 1962).[25] C. M. Bender and S. A. Orszag,
Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York,1978), Chap. 10.[26] The phase shift of π in the WKB integral relation (13) reflects the fact that the radially-dependent potential (9) ofthe composed black-hole-massive-field system has a classical turning point at y − = −∞ . The WKB wave field in theclassically allowed region [ y − , y + ] should therefore be matched only once (at the outer classical turning point y = y + ) tothe corresponding WKB wave field in the classically forbidden region y > y + of the black-hole spacetime. The standardmatching procedure of the wave field across the outer classical turning point y = y + [see Eq. (20) below] yields the familiarphase shift of π in the second-order WKB resonance relation (13) [23–25].[27] Here we have used the strong inequality 1 /x + ≫ nn