Stability of relativistic Bondi accretion in Schwarzschild-(anti-)de Sitter spacetimes
aa r X i v : . [ g r- q c ] S e p Stability of relativistic Bondi accretion in Schwarzschild–(anti-)de Sitter spacetimes
Patryk Mach and Edward Malec
M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krak´ow
In a recent paper we investigated stationary, relativistic Bondi-type accretion in Schwarzschild–(anti-)de Sitter spacetimes. Here we study their stability, using the method developed by Mon-crief. The analysis applies to perturbations satisfying the potential flow condition. We prove thatglobal isothermal flows in Schwarzschild–anti-de Sitter spacetimes are stable, assuming the test-fluid approximation. Isothermal flows in Schwarzschild–de Sitter geometries and polytropic flows inSchwarzschild–de Sitter and Schwarzschild–anti-de Sitter spacetimes can be stable, under suitableboundary conditions.
I. INTRODUCTION
In [1] we have obtained a family of solutions describing spherically symmetric, steady accretion of perfect fluids inSchwarzschild–(anti-)de Sitter spacetimes. Here we study linear stability of these solutions. The proof follows strictlythe work of Moncrief [2], who showed linear stability of analogous solutions in the Schwarzschild case.In [1] we dealt with transonic solutions for polytropic, p = Kρ Γ , and “isothermal” equations of state of the form p = ke with k = 1 / , / , k one obtains analytic solutions in a closed form). Here p denotes thepressure, ρ the baryonic density, e is the energy density, and K and Γ are constants.In this paper we focus only on those transonic solutions that are subsonic far from the central black hole andsupersonic in its vicinity. Usually, there exists another branch of solutions that is subsonic for radii smaller than thesonic radius r ∗ , and supersonic for r > r ∗ . This branch has a natural interpretation of “wind”, instead of an accretionflow. We will not discuss the stability of such solutions.The strategy adopted in [1] was as follows. We fix some, usually finite, radius r = r ∞ of the boundary of the cloud,the boundary value of ρ or e (denoted as ρ ∞ and e ∞ ), and, in the case of polytropic equations of state, the polytropicexponent Γ and the boundary value of the local speed of sound a ∞ . For those parameters we search for a transonicsolution that can be continued inward, at least up to the horizon of the black hole. The condition that the solutionpasses through a sonic point fixes the boundary value of the radial velocity, but one has to verify that the flow issubsonic in the outer part of the accretion cloud.For a negative cosmological constant Λ and the assumed “isothermal” equations of state such solutions can bealso continued outward, up to infinite radii, where the energy density tends to zero (note that in the Schwarzschildspacetime e → e ∞ = 0 as r → ∞ [3]). This is not possible for Λ >
0, in which case the solutions obtained for the“isothermal” equations of state diverge at the cosmological horizon. Polytropic solutions can only be continued toarbitrarily large radii for Λ = 0.
II. NOTATION
In this paper we follow notation conventions of [1]. The flow satisfies the continuity equation ∇ µ ( ρu µ ) = 0 (1)and the energy–momentum conservation law ∇ µ T µν ≡ ∇ µ [( e + p ) u µ u ν + pg µν ] = 0 . (2)Here, as already stated in the Introduction, ρ denotes the baryonic density; e is the energy density; p is the pressure; u µ are components of the four-velocity of the fluid. In the following we will also use the specific enthalpy h = ( e + p ) /ρ .The local speed of sound will be denoted by a .We work in geometrical units with c = G = 1 and assume the signature of the metric g µν of the form ( − , + , + , +).Greek indices run through spacetime dimensions 0 , , ,
3. Latin indices are used for spatial dimensions only. Thedeterminant of the metric g µν will be denoted by det g . The projection tensor is defined as P νµ = δ νµ + u µ u ν . Throughout this paper we work in spherical coordinates ( t, r, θ, φ ). III. STATIONARY SOLUTIONS
We consider Schwarzschild–(anti-)de Sitter spacetimes with the line element ds = − (cid:18) − mr − Λ3 r (cid:19) dt + dr − mr − Λ3 r + r (cid:0) dθ + sin θdφ (cid:1) . (3)In [1] we obtained analytic solutions for equations of state of the form p = ke , where k = 1 / , /
2, and 1. Thefollowing simple formulae hold in all above cases: ρ = A/ ( r u r ), e = Bρ k , h = (1 + k ) Bρ k . Here A and B areconstant. The local speed of sound is also constant: a = k .The solution for k = 1 / X = − mr + Λ3 r + (1 − m ) r (3 m ) cos ( π −
13 arc cos " m (cid:0) − mr − Λ3 r (cid:1) (1 − m ) r ,X = − mr + Λ3 r + (1 − m ) r (3 m ) cos ( π " m (cid:0) − mr − Λ3 r (cid:1) (1 − m ) r . Then the square of the radial component of the four-velocity of the solution branch that is subsonic outside the sonicpoint can be written as ( u r ) = (cid:26) X , r ≥ m,X , r < m. (4)The sonic point is defined as a location where a = ( u r /u t ) . For k = 1 / r ∗ = 3 m , irrespectively ofthe value of the cosmological constant.For k = 1 / r ∗ = q | Λ | sinh (cid:20) ar sinh (cid:18) m √ | Λ | √ (cid:19)(cid:21) , Λ < , m/ , Λ = 0 , q cos h π + arc cos (cid:16) m √ Λ4 √ (cid:17)i , < Λ < / (9 m ) . The radial component of the velocity can be expressed as u r = B ∗ r + q B ∗ r − mr + Λ3 r , r ≥ r ∗ ,B ∗ r − q B ∗ r − mr + Λ3 r , r < r ∗ , (5)where B ∗ = − r ∗ r mr ∗ − . For k = 1 one obtains ( u r ) = 1 − mr − Λ3 r (cid:16) rr h (cid:17) − , (6)where the areal radius of the black hole horizon is given by r h = √ | λ | sinh h ar sinh (cid:16) m p | Λ | (cid:17)i , Λ < , m, Λ = 0 , √ Λ cos h π + arc cos (cid:16) m √ Λ (cid:17)i , < Λ < / (9 m ) . For Λ > r < √ Λ cos (cid:20) π −
13 arc cos (cid:16) m √ Λ (cid:17)(cid:21) (note that we are only dealing with static Schwarzschild–de Sitter spacetimes with Λ < / (9 m )). For Λ < r = 0), with ρ → r → ∞ .In [1] we also investigated polytropic solutions, both for positive and negative values of Λ. Unlike “isothermal”solutions, they cannot be expressed in a closed form, but they can be easily computed numerically, by solving algebraicequations only. As already mentioned in the Introduction, they can be extended to arbitrarily large radii only forΛ = 0. The absence of global polytropic solutions is not very surprising for Λ > <
0. There is an interesting mechanism that is responsible for thisfact, and it is discussed in [1]. The stability analysis presented below applies also for polytropic solutions, modulo areservation concerning boundary conditions.
IV. MONCRIEF’S METHOD
The technique used in [2] allows for a linear stability analysis against isentropic perturbations for which the vorticitytensor ω µν = P αµ P βν [ ∇ β ( hu α ) − ∇ α ( hu β )]vanishes. We say that such perturbations satisfy the potential flow condition.It is easy to show that for an isentropic flow, for which dh = dp/ρ , Eqs. (1) and (2) yield u µ ∇ ( hu ν ) + ∂ ν h = 0 . Using the above equation, one gets ω µν = ∇ ν ( hu µ ) − ∇ µ ( hu ν ) = ∂ ν ( hu µ ) − ∂ µ ( hu ν ) . Thus, if ω µν = 0, the vector field hu µ can be expressed (locally) as a gradient of a potential, i.e., hu µ = ∂ µ ψ . In thiscase, the normalization condition for the four-velocity, u µ u ν = −
1, yields h = p − ∂ µ ψ∂ µ ψ and u µ = ∂ µ ψ/ √− ∂ ν ψ∂ ν ψ .Most notably, the continuity equation (1) can be simplified to the scalar equation ∇ µ (cid:16) ρh ∇ µ ψ (cid:17) = 0 . (7)It can be also shown that stationary, isentropic Bondi-type flows satisfy the potential flow condition, i.e., ω µν = 0.The equation governing linear perturbations of the potential δψ can be obtained directly from Eq. (7). It reads1 √− det G ∂ µ (cid:16) √− det GG µν ∂ ν δψ (cid:17) = 0 , (8)where G µν denotes the Lorentzian metric G µν = ρah (cid:2) g µν + (cid:0) − a (cid:1) u µ u ν (cid:3) . The inverse matrix and the square root of the absolute value of its determinant read G µν = ahρ (cid:20) g µν − (cid:18) a − (cid:19) u µ u ν (cid:21) , √− det G = ρ ah p − det g. Equation (8) has a form of a wave equation for δψ , with respect to the metric G µν . There is a related energy–momentum tensor T νµ = 12 (cid:18) G να ∂ µ δψ∂ α δψ − δ νµ G αβ ∂ α δψ∂ β δψ (cid:19) and the perturbation energy measure E = − Z Ω d x √− det GT tt = Z Ω d x √− det G (cid:20) − G tt ( ∂ t δψ ) + 12 G ij ∂ i δψ∂ j δψ (cid:21) , where the integration is carried out over a space-like, t = const region Ω. In the following, we are interested inperturbations on the spherically symmetric, steady accretion flow on the background metric given by Eq. (3). Theenergy of perturbations comprised in a region between two radii r and r can be written as E ( r ,r ) = Z r r dr Z π dθ Z π dφ √− det G (cid:20) − G tt ( ∂ t δψ ) + 12 G rr ( ∂ r δψ ) (cid:21) . The time derivative of E ( r ,r ) can be easily computed: ddt E ( r ,r ) = 2 Z π dθ Z π dφ √− det GT rt (cid:12)(cid:12)(cid:12)(cid:12) r r (9)= Z π dθ Z π dφ √− det G (cid:2) G rt ( ∂ t δψ ) + G rr ∂ t δψ∂ r δψ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) r r . The strategy of proving the linear stability is standard, but there are certain adjustments, specific to the accretionproblem. The first step is to note that it is enough to focus on perturbations that are located outside the sonicpoint. Intuitively, this is because linear acoustic perturbations cannot escape from the supersonic region with r < r ∗ .Formally, the surface r = r ∗ is a null surface with respect to the metric G µν , located outside the event horizon ofthe black hole [2]. Let r ∞ denote the radius of the outer boundary of the accretion cloud. As pointed earlier, forΛ < r ∞ = ∞ , but in general r ∞ has to befinite. The energy of the perturbations located outside the sonic horizon is equal to E ( r ∗ ,r ∞ ) . In the second step of theproof one has to show that E ( r ∗ ,r ∞ ) is positive definite. The last step consists of establishing conditions, under which dE ( r ∗ ,r ∞ ) /dt ≤
0. We say that the accretion flow is linearly stable, provided that the above conditions are satisfied.Positivity of E ( r ∗ ,r ∞ ) follows by a direct inspection. Working in coordinates (3), we have √− det GG tt = − p det g ρ ( u t ) a h − a (cid:16) u r u t (cid:17) (cid:0) − mr − Λ3 r (cid:1) , √− det GG rr = p det g ρ ( u t ) a h " a − (cid:18) u r u t (cid:19) (remember that a > ( u r /u t ) for r > r ∗ ).There are two boundary terms in the expression for dE ( r ∗ ,r ∞ ) /dt given by Eq. (9). The inner boundary term (for r = r ∗ ) is clearly nonpositive. Note that √− det GG rr vanishes at r = r ∗ , while √− det GG rt = p det g ρh (cid:18) a − (cid:19) u t | u r | ≥ . The outer boundary term (at r = r ∞ ) is more troublesome. If r ∞ < ∞ , one has to assume that ∂ t δψ = 0 and ∂ r δψ = 0 at r = r ∞ — otherwise no result concerning stability can be established. However, if the above conditionsare satisfied, then dE ( r ∗ ,r ∞ ) /dt ≤
0, and the accretion flow is linearly stable.For r ∞ = ∞ one has to inspect the asymptotic (for r → ∞ ) behavior of δψ . In our case, this is only possible forΛ <
0, and “isothermal” equations of state p = ke with k = 1 / , / ,
1. The asymptotic behaviour of δψ is differentin these three cases, and it is different from that described in [2]. This issue is discussed below. V. ASYMPTOTIC BEHAVIOR
In what follows we assume Λ < p = ke with k = 1 / , / , ∂ t δψ and ∂ r δψ are given by requiring that the energy of perturbations E ( r ∗ , ∞ ) is finite. They turn out to be marginally sufficient for the purpose of the stability proof, but the actualbehavior of δψ — controlled by Eq. (8), governing the evolution of the perturbations — is characterized by a fasterfalloff.We restrict ourselves to perturbations whose asymptotic behavior can be characterized by δψ = O (cid:18) r n (cid:19) , ∂ t δψ = O (cid:18) r n (cid:19) , ∂ r δψ = O (cid:18) r n +1 (cid:19) , (10)as r → ∞ , where n > δψ must falloff atleast as 1 /r n . It is important that (10) must hold for all times t and all angles θ and φ .For Λ = 0 both √− det GG tt and √− det GG rr have the same asymptotic behaviour of O ( r ) (irrespectively of theassumed equation of state). The condition of the finiteness of E ( r ∗ , ∞ ) implies — assuming Eq. (10) — that ∂ t δψ = O (cid:18) r / ǫ (cid:19) , ∂ r δψ = O (cid:18) r / ǫ (cid:19) , as r → ∞ , with ǫ >
0. This suffices to show that dE ( r ∗ ,r ∞ ) /dt ≤ ∂ t δψ and ∂ r δψ have at most power-law falloffs ∂ t δψ = O (1 /r n ), ∂ r δψ = O (1 /r n ′ ), without specifying the relation between n and n ′ . Then, it follows form the condition E ( r ∗ , ∞ ) < ∞ , that ∂ t δψ = O (1 /r / ǫ ) and ∂ r δψ = O (1 /r / ǫ ′ ), where ǫ, ǫ ′ >
0. This is sufficient to conclude that dE ( r ∗ , ∞ ) /dt ≤ = 0 without providing additional information on ∂ r δψ and ∂ t δψ . A. Asymptotic behavior for p = e/ For the equation of state p = e/ u r = O ( r ), ρ = O (1 /r ), ρ/h = O (1 /r ) (this follows straightforward from the explicit solutions given in Sec. 3). Thus, √− det GG rr = O ( r ), √− det GG tt = O (1 /r ), and √− det GG rt = O ( r ). The requirement of the finiteness of the energy E ( r ∗ , ∞ ) yields,assuming Eq. (10), ∂ r δψ = O (1 /r / ǫ ), and ∂ t δψ = O (1 /r / ǫ ). This and Eq. (9), allows to show that dE ( r ∗ , ∞ ) /dt ≤
0. Indeed, we have √− det GG rt ( ∂ t δψ ) = O (1 /r ǫ ), and √− det GG rr ∂ t δψ∂ r δψ = O (1 /r ǫ ).A faster falloff of δψ can be inferred from Eq. (8) (in addition to the requirement that E ( r ∗ , ∞ ) < ∞ ). In the caseof the equation of state p = e/
3, Eq. (8) can be satisfied up to a leading order in the asymptotic expansion, providedthat δψ = O (1 /r ). These statements require an explanation. Equation (8) can be written as G tt ∂ t δψ + 2 G tr ∂ tr δψ + 1 √− det G ∂ r (cid:16) √− det GG rt (cid:17) ∂ t δψ (11)+ 1 √− det G ∂ r (cid:16) √− det GG rr ∂ r δψ (cid:17) + ha r ρ ∆ Ω δψ = 0 , where ∆ Ω denotes the Laplacian on the 2-sphere. Assume now the asymptotic behavior of the solution in the form(10). Then, taking into account the asymptotic falloff of the unperturbed solution for p = e/
3, one can check thatthe leading order term in Eq. (11) is1 √− det G ∂ r (cid:16) √− det GG rr ∂ r δψ (cid:17) = ( n − O (cid:0) r − n +2 (cid:1) . Clearly, in order to satisfy Eq. (11) in the leading order, we have to demand that n = 1. This yields the claimed falloff δψ = O (1 /r ). The above reasoning may by iterated, possibly yielding a stronger falloff condition. An analogousprocedure can be also repeated for the cases with p = e/ p = e , that are discussed briefly in the followingparagraphs. B. Asymptotic behavior for p = e/ In this case ρ = O (1 /r ), ρ/h = O (1 /r ), and u r = O ( r ). Also √− det GG rr = O ( r ), √− det GG tt = O (1 /r ),and √− det GG rt = O ( r ). Here the situation is similar to that for p = e/
3. Linear stability can be proved forperturbations satisfying E ( r ∗ , ∞ ) < ∞ and Eq. (10). Then ∂ r δψ = O (1 /r ǫ ), ∂ t δψ = O (1 /r ǫ ), and thus both termsappearing in Eq. (9), √− det GG rt ( ∂ t δψ ) and √− det GG rr ∂ t δψ∂ r δψ , behave asymptotically as 1 /r ǫ and 1 /r ǫ ,respectively.Again, additional information provided by Eq. (8) yield a faster falloff, namely δψ = O (1 /r ). For the sake ofbrevity, we omit the details of this calculation. C. Asymptotic behavior for p = e In the case of ultra-hard equation of state p = e , Eq. (8) is equivalent to a standard wave equation onthe Schwarzschild–anti-de Sitter spacetime. A similar problem — evolution of scalar fields in asymptoticallySchwarzschild–anti-de Sitter spacetimes — was investigated recently in [4].For p = e , ρ/h = const. We have √− det GG rr = O ( r ), √− det GG tt = O ( r ), G tr = 0. Finiteness of the energynorm yields ∂ r δψ = O (1 /r / ǫ ), and thus ∂ t δψ = O (1 /r / ǫ ). The above bounds are again marginally sufficient forthe stability — one gets √− det GG rr ∂ t δψ∂ r δψ = O (1 /r ǫ ).The asymptotic behavior enforced by Eq. (8) is δψ = O (1 /r ). VI. SUMMARY
We investigated conditions under which the Bondi-type accretion in Schwarzchild–(anti-)de Sitter spacetimes isstable. Stationary solutions were obtained in [1] for “isothermal” equations of state of the form p = ke , with k = 1 / , / ,
1. Polytropic solutions, corresponding to equations of state p = Kρ Γ , were also investigated, althoughthey cannot be written in a closed form. All these solutions appear linearly stable, provided that the perturbationsvanish at the outer boundary of the cloud. If the accretion cloud can formally extend up to infinity — this happensfor the investigated “isothermal” equations of state and Λ < ACKNOWLEDGMENTS