Existence of steady states of the massless Einstein-Vlasov system surrounding a Schwarzschild black hole
aa r X i v : . [ g r- q c ] F e b Existence of steady states of the masslessEinstein-Vlasov system surrounding aSchwarzschild black hole
H˚akan Andr´eassonMathematical SciencesChalmers University of TechnologyUniversity of GothenburgSE-41296 Gothenburg, Swedenemail: [email protected] 17, 2021
Abstract
We show that there exist steady states of the massless Einstein-Vlasov system which surround a Schwarzschild black hole. The steadystates are (thick) shells with finite mass and compact support. Fur-thermore we prove that an arbitrary number of shells, necessarily wellseparated, can surround the black hole. To our knowledge this is thefirst result of static self-gravitating solutions to any massless Einstein-matter system which surround a black hole. We also include a numer-ical investigation about the properties of the shells.
The Einstein-Vlasov system typically models self-gravitating particle en-sembles such as galaxies or clusters of galaxies. The particles in the formercase are stars and in the latter case they are galaxies. Clearly, the particlescarry mass in these two situations. In this work we are instead interestedin the case of massless particles, e.g. photons, and we show that there existself-gravitating ensembles of massless particles with finite mass and compactsupport surrounding a Schwarzschild black hole. To put our result in contextlet us briefly review some related results. Existence of steady states to theEinstein-Vlasov system in the case of massive particles was first established1n [23]. The steady states constructed in this work are spherically symmet-ric with a regular centre. Several simplifications and generalizations havesince then been obtained and we refer to [20] for a simplified and generalapproach, to [6] for the existence of highly relativistic static solutions and to[11] for the existence of stationary solutions in the axisymmetric case. Thereare several other existence results and also results about the properties ofthe static solutions in the literature, and we refer to [8] for a review and to[9, 2, 3] for more recent results.By relaxing the condition of a regular centre the case with aSchwarzschild black hole was considered in [21], where the existence of mas-sive static shells of Vlasov matter surrounding a black hole was shown. Adifferent method leading to a similar result was more recently given by Jabiriin [18]. For a fluid, the first result of a massive static shell surrounding ablack hole was obtained in [15]. If the matter model originates from quan-tum mechanics similar results need not be true as is for instance shownin [16], where the absence of static black hole solutions is shown for theEinstein-Dirac-Yang/Mills equations. It is argued in [16] that a reason forthe difference between the classical and the quantum mechanical case is thatclassical particles are prevented from falling into the black hole by the cen-trifugal barrier, whereas quantum particles can tunnel through this barrier.Solutions of the Einstein-Vlasov system can also model ensembles ofmassless particles, e.g. photons. The first mathematical study of the mass-less Einstein-Vlasov system is to our knowledge the work [24] by Rendall,where the dynamics of cosmological solutions is investigated. Only morerecently results about static solutions have been obtained. Akbarian andChoptuik constructed massless solutions with compact support numericallyin [1]. An existence proof was obtained in [10], where also a discussion aboutthe relation to Wheeler’s concept of geons is given. Gundlach studied theproblem by numeric and analytic tools in [17]. An important difference be-tween the massive and massless case is that the existence of massless staticsolutions requires that the solutions are highly relativistic in the sense thatthe compactness ratio 2
M/R is large. Here M is the ADM mass of thesolution and R its (areal) radial support. It is known that 2 M/R is alwaysbounded by 8 /
9, cf. [7]. (The classical result by Buchdahl [14] does notapply in this case although the bound is the same.) Numerically it has beenfound that a necessary lower bound is roughly 2
M/R > /
5, cf. [1, 10, 17],for the existence of massless static solutions whereas no such lower boundis needed in the massive case.In the present work we combine the methods from [21] and [10]. Weconsider the case with a Schwarzschild black hole in the centre and we show2hat there exist static massless shells of Vlasov matter with compact supportand finite mass which surround the black hole. Necessarily there is a gapbetween the black hole and the shell; the inner radius of the shell has to belarger than the radius of the photon sphere of the black hole. In our proofthe shell is placed far away from the photon sphere. This is a technicalcondition. Numerically, we find that there are situations when the shell canbe arbitrary close to the photon sphere, cf. Section 5. The shell solutionsare highly relativistic in the sense that 2
M/R is large. However, when theshell can be placed close to the photon sphere the ratio 2
M/R is largerthan, but close to, 2 /
3. Hence, the presence of a black hole reduces therequired lower bound of 2
M/R . Clearly, since the ratio 2
M/R of the shellis larger than 2 / R , to be sufficiently large, whereas R is required to be sufficiently small in [10]. In fact, the compactness ratio2 M/R → / both the limits R → R → ∞ .Let us finally mention that the linear massless Einstein-Vlasov systemhas been studied on a fixed black hole spacetime in [4]. The authors showthat solutions to the linear Einstein-Vlasov system on a Kerr backgroundsatisfy a Morawetz estimate. Our result shows that an analogous result can-not hold for the nonlinear Einstein-Vlasov system. On the other hand, themain purpose of [4] is to understand perturbations of black hole spacetimes.The steady states we construct require compact configurations and the mat-ter components cannot be made arbitrary small. Thus they should not berelevant when studying perturbations.The outline of the paper is as follows. In Section 2 we introduce themassless static Einstein-Vlasov system. In Section 3 we formulate the mainresults and in Section 4 we prove our main theorem. Section 5 is devoted toa numerical investigation of the properties of the solutions.3 The static Einstein -Vlasov system
The metric of a static spherically symmetric spacetime takes the followingform in Schwarzschild coordinates ds = − e µ ( r ) dt + e λ ( r ) dr + r ( dθ + sin θdϕ ) , where r ≥ , θ ∈ [0 , π ] , ϕ ∈ [0 , π ] . Asymptotic flatness is expressed by theboundary conditions lim r →∞ λ ( r ) = lim r →∞ µ ( r ) = 0 . We now formulate the static massless Einstein-Vlasov system. For an intro-duction to the Einstein-Vlasov system we refer to [8], [22] and [24]. Belowwe use units such that c = G = 1 where G is the gravitational constant and c is the speed of light. The static massless Einstein-Vlasov system is givenby the Einstein equations e − λ (2 rλ r −
1) + 1 = 8 πr ρ, (2.1) e − λ (2 rµ r + 1) − πr p, (2.2) µ rr + ( µ r − λ r )( µ r + 1 r ) = 8 πp T e λ , (2.3)together with the static Vlasov equation wε ∂ r f − ( µ r ε − Lr ε ) ∂ w f = 0 , (2.4)where ε = ε ( r, w, L ) = p w + L/r . The matter quantities are defined by ρ ( r ) = πr Z ∞−∞ Z ∞ ε ( r, w, L ) f ( r, w, L ) dLdw, (2.5) p ( r ) = πr Z ∞−∞ Z ∞ w ε ( r, w, L ) f ( r, w, L ) dLdw, (2.6) p T ( r ) = π r Z ∞−∞ Z ∞ Lε ( r, w, L ) f ( r, w, L ) dLdw. (2.7)The variables w and L can be thought of as the momentum in the radialdirection and the square of the angular momentum respectively.4he matter quantities ρ, p and p T are the energy density, the radialpressure and the tangential pressure respectively. The system of equationsabove are not independent and we study the reduced system (2.1)-(2.2)together with (2.4) and (2.5)-(2.6). It is straightforward to show that asolution to the reduced system is a solution to the full system.Define E = e µ ε, then the ansatz f ( r, w, L ) = Φ( E, L ) , (2.8)satisfies (2.4). By inserting this ansatz into (2.5)-(2.6) the system of equa-tions reduce to a system where the metric coefficients µ and λ alone are theunknowns. This has turned out to be an efficient method to construct staticsolutions and we will use this approach here. The following form of Φ willbe used Φ( E, L ) = ( E − E ) k + ( L − L ) l + , (2.9)where l ≥ / , k ≥ , L > , E > , and x + := max { x, } . In theNewtonian case with l = L = 0 , this ansatz leads to steady states with apolytropic equation of state.The aim in this work is to show that static shells of Vlasov matter existwhich surround a Schwarzschild black hole; in fact there can be arbitrarymany shells separated by vacuum surrounding the black hole. To prove ourresult we construct highly compact shells, i.e., shells for which the compact-ness ratio Γ := sup 2 m ( r ) r , (2.10)is large; roughly Γ ≥ . Here m is the Hawking mass defined for r ≥ M by m ( r ) = M + Z r M s ρ ( s ) ds, r ≥ M . From [7] it always holds that Γ < . The result in [7] concerns steady stateswith a regular centre but it is straightforward to show that it holds also inthe case with a Schwarzschild black hole at the centre. Remark 2.1
Numerically we are able to construct solutions where the shellis close to the photon sphere of the black hole. For such solutions it turns outthat Γ is larger than, but close to, / , cf. Section 5. Hence, the presenceof a black hole reduces the required lower bound of Γ . Indeed, recall that thenumerical studies in the regular case indicates that the required lower boundis larger in that case, cf. [10], [1] and [17].
5f the inner radius of the shell is denoted by R , we show that for highlyrelativistic shells there is a radius R such that f ( r, w, L ) = 0 in an interval[ R , R + ǫ ] , ǫ >
0. This fact makes it possible to glue a Schwarzschildsolution at r = R to the shell solution with support in [ R , R ]. If aSchwarzschild solution is not attached at r = R , then the ansatz (2.9)implies that Vlasov matter will occur again and there exists a radius R such that f > r > R and the solution is not asymptotically flat.This is a general feature of massless static solutions of the Einstein-Vlasovsystem obtained from an ansatz, cf. equation (2.8). In the massive case thesituation is different and solutions generated by the ansatz (2.9) alone givesrise to compactly supported solutions.In the massive case the existence of shells surrounding a Schwarzschildblack hole was settled in [22]. These shells are not highly relativistic. Toconstruct highly relativistic shells for which Γ is sufficiently large we adaptthe method developed in [6], which in turn was used to show existence ofmassless steady states with a regular centre in [10]. Let M > e µ ( r ) = 1 − M r . Note that the ansatz (2.9) implies that f = 0 whenever E > E . Accordinglywe let f = 0 in the interval [2 M , R ] , where R is the largest root to theequation (cid:0) − M r ) L r = E = 1 . (3.11)Of course, we need a condition on L which guarantees that the equationhas real roots. Remark 3.1
The parameter L can be removed. From above we see thatby replacing E by ˜ E = E / √ L we could consider the case L = 1 anduse E as free parameter, cf. [25, 17]. However, below we keep L as freeparameter and we fix E = 1 . An elementary computation shows that the maximum value of the lefthand side of (3.11) is L M , r = 3 M . This radius corresponds to the radius of the photonsphere of the black hole. We fix E = 1 and impose the condition that L > M =: L ∗ . Equation (3.11) then has three real roots and we denote the largest root by R . To carry out the proof of our main result we will take L large. Wehave the following result. Lemma 3.2
There exists a constant
C > , depending on M , such that − C √ L ≤ R √ L ≤ as L → ∞ . Proof:
The proof is a straightforward application of the solution formulafor cubic equations. Indeed, equation (3.11) is equivalent to r − L r + 2 M L = 0 , which we write as r + pr + q = 0 where p = − L and q = 2 M L . Set D = ( p + ( q = L ( − L
27 + M ) . Since we choose L > M it follows that D < r = u + v, r , = u + v ± u − v i √ , where u = ( − q/ i p | D | ) / and v = ( − q/ − i p | D | ) / . Note here that u + v and ( u − v ) i are real. The largest of these roots is R := r = u + v = 2( q | D | ) / cos α , where α = π q/ p | D | . We have 0 ≤ ( q/ p | D | = ≤ M √ L q − M L ≤ C √ L , L − L ∗ > L . Hence thereexists a constant C > π ≤ α ≤ π C √ L , for large L . This implies that √ − C √ L ≤ cos α ≤ √ , for some positive constant C . Moreover, we have2( q | D | ) / = 2( M L + L ( L − M )) / = 2 √ L √ . Hence we conclude that for large L − C √ L ≤ R √ L ≤ . ✷ We now consider the Einstein-Vlasov system on the domain r ≥ R andwe prescribe data for µ and λ at r = R by letting e µ ( R ) = e − λ ( R ) = 1 − M R . Using results from previous works we can assume that there exists a solutionto the Einstein-Vlasov system which exists on [ R , ∞ [ with the property thatΓ < / R is sufficiently large, i.e. we take L large, there is aradius R > R such that the energy density and the pressure componentsvanish at r = R . The shell [ R , R ] is thin in the sense that R R → R → ∞ . (3.12)However, the difference R − R does not need to decrease. Depending on theparameters in equation (2.9) the difference may even become unbounded, cf.Remark (3.4). Hence the shell is thin in the sense (3.12), which is differentfrom the usual notion of thin shells in general relativity.When the distribution function f has the form f ( r, w, L ) = Φ( E, L ) , (3.13)8here Φ = 0 whenever E > E , the matter quantities ρ and p becomefunctionals of µ , and we have ρ = 2 πr Z E e − µ √ L r Z r ( s − L Φ( e µ s, L ) s q s − − Lr dLds, (3.14) p = 2 πr Z E e − µ √ L r Z r ( s − L Φ( e µ s, L ) r s − − Lr dLds. (3.15)(3.16)Here we have kept the parameter E but in what follows we use that E = 1.By taking (2.9) for Φ these integrals can be computed explicitly in the caseswhen k = 0 , , , ... and l = 1 / , / , ... Let γ = − µ −
12 log L r , we then have the following lemma from [10]. Lemma 3.3
Let k = 0 , , , ... and let l = 1 / , / , / , ... then there arepositive constants π jk,l , j = 1 , , such that when γ ≥ ρ = π k,l r l ( L r ) l +2 ( e γ − l + k +3 / P l +5 / − k ( e γ ) , (3.17) p = π k,l r l ( L r ) l +2 ( e γ − l + k +5 / P l +3 / − k ( e γ ) , (3.18) If γ < then all matter components vanish. Here P n ( e γ ) is a polynomial ofdegree n and P n > . In order to simplify the technical details we consider only the case k = 0and l = 1 / Remark 3.4
In the case k = 0 and l = 1 / the support of the shell [ R , R ] satisfy R − R ≤ C , independently of R as shown below, whereas for othervalues of the parameters we claim that R − R ∼ R q − − lq , where q = k + l + 5 / , cf. Section 5. This relation is obtained by performingthe analysis below in the general case, cf. [6]. t = e γ , we then have (with k = 0 and l = 1 / ρ ( r ) = π r (cid:0) L r (cid:1) / ( t − (cid:0) t + 6 t + 4 t + 215 (cid:1) , (3.19)and p ( r ) = π r (cid:0) L r (cid:1) / ( t − (cid:0) t + 9 t + 860 (cid:1) . (3.20)Let R be large and define δ := (cid:0) π (cid:1) / . (3.21)The argument below will be carried out in the interval I := [ R , R ] where R < R + 50 δ =: R ∗ . Since R will be taken large, R /R is as close to 1as we wish. We now formulate the main results in this work. Theorem 3.5
Let M ≥ be the ADM mass of a Schwarzschild blackhole. Then there exist static solutions with finite ADM mass to the masslessspherically symmetric Einstein-Vlasov system surrounding the black hole.The matter components are supported on a finite interval [ R , R ] , where R > M , and spacetime is asymptotically flat. Remark 3.6
The arguments below lead to a solution for which µ ( r ) hasa finite limit µ ( ∞ ) as r → ∞ . In order to obtain an asymptotically flatsolution we rescale by letting ˜ E := e µ ( ∞ ) and ˜ µ ( r ) := µ ( r ) − µ ( ∞ ) . Remark 3.7
The regularity of the solution depends on the parameters k and l , cf. equations (3.17) and (3.18). For the values of k and l that weconsider in this work the matter quantities are continuously differentiable. Remark 3.8
As mentioned in the introduction, note that our result holdsalso in the case when M = 0 but that the family of solutions obtained in thiswork is different from the family of solutions obtained in [10]. In the presentsituation we require the inner radius R to be large whereas in [10] R isrequired to be small. Clearly, when M > it is not possible to take R smallsince necessarily R > M . Both families share the property that Γ → / in the extreme limits, i.e., when R → as in [10] or when R → ∞ as inthe present case. Remark 3.9
Having a black hole with one shell surrounding it, we canstart from this solution and add another shell with the strategy in the proof.Hence, our result implies that an arbitrary number of shells can surroundthe black hole.
10n view of the discussion in Section 2, Theorem 3.5 is a consequence ofthe following result.
Theorem 3.10
Consider a static solution to the massless Einstein-Vlasovsystem, corresponding to the ansatz (2.9) with k = 0 and l = 1 / , with datagiven at r = R . For R sufficiently large, there exists R ≤ R + 50 δ suchthat the matter components are supported in the interval [ R , R ] and vanishat r = R . Here δ is given by (3.21). Furthermore, Γ → as R → ∞ . The proof of our main result will follow from a chain of lemmas.
Proof of Theorem 3.10 . First we establish convenient formulas for thematter terms. Although L is our free parameter, we will instead use R as free parameter since R → ∞ as L → ∞ in view of Lemma 3.2. Nextwe note that R ∗ /R ≤ C/R . Hence, by Lemma 3.2 we have for any r ∈ [ R , R ∗ ] 1 − α ( R ) ≤ L r ≤ α ( R ) , (4.22)where α ( R ) ≥ α ( R ) → R → ∞ . By the definition of R wehave that γ ( R ) = 0. Moreover, γ ′ ( r ) = − µ ′ ( r ) + 1 r . (4.23)Now, since ρ = p = 0 and m ( r ) = M for r ≤ R , we have that that µ ′ ( R ) < /R . Here we used that M /R < / e λ ( r ) = 11 − m ( r ) r . The last relation is a consequence of the Einstein equation (2.1). This impliesthat γ ′ ( R ) >
0. Thus γ ( r ) > R and the aimis to show that there exists R < R ∗ such that γ ( R ) = 0. Hence γ > I . An upper bound on γ follows from(4.23). We have since µ ′ ( r ) ≥ γ ( r ) = γ ( r ) − γ ( R ) ≤ log rR = log(1 + r − R R ) , which implies that 0 ≤ γ ≤ α ( R ) on I if R is sufficiently large. Hence for r ∈ I γ ( r ) ≤ e γ ( r ) − ≤ γ ( r )(1 + α ( R )) .
11n particular this implies that (recall t = e γ )1 ≤ t + 6 t + 4 t + 215 ≤ α ( R ) , and similarly for the corresponding polynomial in p ( r ). Putting these esti-mates together we conclude that π rγ ( r )(1 − α ( R )) ≤ ρ ( r ) ≤ π rγ ( r )(1 + α ( R )) , and similarly for p ( r ). Since for all arguments below it is sufficient to havea lower and an upper bound on ρ and p we assume for simplicity that ρ ( r ) = π rγ ( r ) , (4.24)and p ( r ) = π rγ ( r ) , (4.25)for r ∈ I . Lemma 4.1
Let δ be as above. Then γ ′ ( r ) ≥ r for r ∈ [ R , R + δ ] =: I . Proof:
We have by the mean value theorem that for any σ ≤ δγ ( R + σ ) = γ ( R + σ ) − γ ( R ) = σγ ′ ( ξ ) ≤ δR , where ξ ∈ [ R , R + σ ], since γ ( r ) ≤ /r . Hence, ρ ( r ) ≤ π r δ R on I . We get for σ ≤ δm ( R + σ ) ≤ M + 4 π δ R Z R + σR η dη ≤ M + 4 π δ ( R + σ ) R σ. By taking R large we thus obtain m ( R + σ ) R + σ ≤ π δ σ + α ( R ) ≤ π δ + α ( R ) ≤
15 + α ( R ) . r ∈ I m ( r ) r e λ ( r ) ≤
13 + α ( R ) . Moreover, we have for r ∈ I that4 πr p ( r ) e λ ( r ) ≤ πr p ( r )( 53 + α ( R )) ≤ π δ α ( R )) ≤
19 + α ( R ) . Hence, for r ∈ I we get for sufficiently large R µ ′ ( r ) ≤ ( 13 + 19 + α ( R )) 1 r ≤ r . This completes the proof of the lemma. ✷ The lemma implies that for σ ≤ δγ ( R + σ ) ≥ γ ( R ) + σ inf ≤ s ≤ σ γ ′ ( R + s ) ≥ σ R + σ ) . Let σ ∗ := δ, and define γ ∗ = σ ∗ R + σ ∗ ) . Remark 4.2
The notation σ ∗ is in this case superfluous but for generalparameter values it is motivated, cf. [6]. Clearly we have that γ ( R + σ ∗ ) ≥ γ ∗ . The result in the following lemma shows that γ will reach the γ ∗ level againat a larger radius. Lemma 4.3
There exists a radius r > R + σ ∗ such that γ ( r ) = γ ∗ andsuch that γ ( r ) > γ ∗ for R + σ ∗ < r < r . Moreover, r ≤ R + 11 δ .Proof: Since γ ( R + σ ∗ ) ≥ γ ∗ , and since Lemma 4.1 gives that γ ′ ( R + σ ∗ ) > , the radius r must be strictly greater than R + σ ∗ . Let [ R + σ ∗ , R + σ ∗ + ∆] , for some 0 < ∆ < δ, be the smallest interval such that γ ≥ γ ∗ on13he interval. We will show that there in fact γ ( R + σ ∗ + ∆) = γ ∗ . We havefrom (4.24) m ( R + σ ∗ + ∆) ≥ π Z R + σ ∗ +∆ R + σ ∗ σ ∗ R + σ ∗ ) r dr ≥ π σ ∗ ( R + σ ∗ )∆ . (4.26)Hence, m ( R + σ ∗ + ∆) R + σ ∗ + ∆ ≥ π σ ∗ ( R + σ ∗ ) R + σ ∗ + ∆ ∆ ≥ π σ ∗ ∆ , where we used that R ≥ δ , so that ( R + σ ∗ ) / ( R + σ ∗ + ∆) ≥ / R . Since for r ≥ R necessarily m ( r ) r < , we conclude that there is a ∆ such that∆ ≤ π σ ∗ = 4081 π δ = 40 · / π ≤ δ, with the property that γ ( R + σ ∗ + ∆) = γ ∗ , since otherwise we obtain acontradiction. ✷ Next we show an important property of the solution at the radius r = r when R is sufficiently large. Lemma 4.4
Let r be as above. If R is sufficiently large then m ( r ) r ≥ . Proof:
We consider the fundamental equation (10) in [5]. In our case witha black hole at the center it takes the form( m ( r ) r + 4 πrp ( r )) e µ ( r )+ λ ( r ) − M R = 1 r Z rR πη e µ + λ ( ρ + p + 2 p T ) dη. (4.27)Here we used that µ ( R ) + λ ( R ) = 0 and that p ( R ) = 0. This equation isa consequence of the generalized Oppenheimer-Tolman-Volkov equation. Inthe present massless case we have p + 2 p T = ρ , so that ρ + p + 2 p T = 2 ρ . Ifwe take r = r we then get m ( r ) r e ( µ + λ )( r ) ≥ πr Z r R η e µ + λ ρ dη − πr pe ( µ + λ )( r ) . (4.28)14ere we dropped the term involving M due to sign. Note that it becomesarbitrary small for R large. Next we write Z r R πηρe λ dη = Z r R ( − ddr e − λ ) dη + Z r R me λ η dη ≥ r − M R − s − m ( r ) r = 1 − s − m ( r ) r − (cid:0) − r − M R (cid:1) = 2 m ( r ) r (1 + q − m ( r ) r ) − α ( R ) . (4.29)In the inequality above we dropped the second term due to sign. We notethat this term is as small as we wish for a relatively thin shell. The reasonwe point this out is that the chain of inequalities in this paragraph is closeto a chain of equalities for very thin shells. This is essential to understandwhy for a thin shell Γ approaches the limit 8 / µ is increasing Y m ( r ) r e ( µ + λ )( r ) ≥ πr Z r R η e µ + λ ρ dη − πr p ( r ) e ( µ + λ )( r ) ≥ πe µ ( R ) R r Z r R ηe λ ρ dη − πr p ( r ) e ( µ + λ )( r ) . Let us introduce the notation P := 4 πr p ( r ). Using (4.29) we get m ( r ) r e ( µ + λ )( r ) ≥ e µ ( R ) R r m ( r ) r (1 + q − m ( r ) r ) − P e ( µ + λ )( r ) − α ( R ) . (4.30)Letting Y := m ( r ) r , we obtain by using that e − λ ( r ) = p − Y ( r ) Y ≥ e µ ( R ) − µ ( r ) (cid:18) R r (cid:19) Y √ − Y √ − Y − P − α ( R ) e − ( µ + λ )( r ) . (4.31)15ext we show a couple of properties of the solutions that we need to proceedwith the argument. From Lemma 4 we have that γ approaches γ ∗ from aboveand therefore γ ′ ( r ) ≤ , which implies that µ ′ ( r ) ≥ r , (4.32)in view of (4.23). Furthermore we have from (4.25) that p ( r ) = π r γ ∗ = π r δ R + δ ) . Now recall that µ ′ = ( mr + 4 πrp ) e λ . We will show that Y = m ( r ) /r ≥ /
4. Assume the contrary, i.e., assume
Y < , (4.33)then e λ ( r ) = 11 − Y < , and thus4 πr p ( r ) e λ ( r ) ≤ π r δ ( R + δ ) = 160 r ( R + δ ) ≤ . (4.34)Here we used that δ = 1 / π and that r ≤ R + 11 δ so that r ( R + δ ) ≤
65 for R large . Hence if (4.33) holds then µ ′ ( r ) = ( m ( r ) r + 4 πr p ( r )) e λ ( r ) ≤ ( 12 + 150 ) 1 r , which is a contradiction to (4.32) and we obtain that Y ≥ . Let us next consider the difference µ ( R ) − µ ( r ) . We have µ ( r ) − µ ( R ) = Z r R ( mη + 4 πηp ) e λ dη. γ ′ ( r ) ≤ /r we have for R ≤ r ≤ r ≤ R + 11 δγ ( r ) ≤ log rR ≤ δR , and we get by using (4.25), and that e λ ≤ πr p ( r ) e λ ( r ) ≤ π r γ ( r ) ≤ (cid:0) rR (cid:1) ≤ C. Hence, since m/r ≤ / Z r R ( mη + 4 πηp ) e λ dη ≤ Z r R Cη dη ≤ C log r R ≤ CR . As a consequence we obtain that1 − α ( R ) ≤ e µ ( R ) − µ ( r ) (cid:18) R r (cid:19) ≤ α ( R ) . From this estimate we also have that e − ( µ + λ )( r ) is bounded independentlyof R and using that Y ≥ / ≥ √ − Y √ − Y − P − α ( R ) . (4.35)Let us introduce the notation B = 4 P + α ( R ). We can assume that B ≥ √ − Y , and then squaring both sides we obtain Y ≥ − B √ − Y (1 + √ − Y )3 , (4.36)where we dropped the term involving B due to sign. Using that Y ≥ / √ − Y (1 + √ − Y )3 ≤ √ √ ≤ . Taking R large we derive from (4.36) the inequality Y ≥ − P. e λ ( r ) ≥ Y ≥ − ≥ . ✷ Remark 4.5
The proof above is very similar to the corresponding proof in[6]. However, for shells for which the inner radius R → , as in [6], thepressure term P → and one can conclude from the argument in the lemmaabove that as R → the compactness ratio Γ → / . In the present casethe situation is slightly different. However, as soon as we know that thereis a radius R such that ρ ( R ) = p ( R ) = 0 , and such that R /R → as R → ∞ , then we can use the argument above, with r replaced by R , toshow that the compactness ratio Γ → as R → ∞ . This shows the lastclaim in Theorem 3.10. Inspired by an idea of T. Makino introduced in [19], we show that γ neces-sarily must vanish close to the point r if R is sufficiently large. Let x := m ( r ) rγ ( r ) . Using that m ′ ( r ) = 4 πr ρ, and that µ ′ ( r ) = ( m/r + 4 πr ) e λ it follows that rx ′ = 4 πr ργ − x + x − γx − xγ + 4 πxr p ( r ) γ (1 − γx ) . In our case r > R and γ > γ ( r ) = 0 for some R ∈ I . Since γ > ρ, p ≥ rx ′ ≥ − x + x − γx − xγ = x − γx ) − x + 2 x − γx ) − xγ . (4.37)Take R sufficiently large so that m ( r ) /r ≥ / r ∈ [ r , r / , then since m is increasing in r we get m ( r ) r ≥ m ( r ) r = r r m ( r ) r ≥ ·
25 = 38 . Now by the definition of x it follows that x − γx = mγr e λ = mγr (1 − m/r ) ≥ γ , when mr ≥ . r , r / , x − γx ) − xγ ≥ , so that on this interval rx ′ ≥ x − γx ) − x ≥ x − x, (4.38)where we used that 11 − γx = 11 − m/r ≥ mr ≥ . The upper bound γ ≤ log(1 + r − R R )implies that x ( r ) = m ( r ) r γ ( r ) ≥ R r − R . (4.39)In particular x ( r ) x ( r ) − / ≤ , for R large. Solving (4.38) yields x ( r ) ≥ (cid:18) − r (4 x ( r ) / − r x ( r ) / (cid:19) − , on r ∈ [ r , r / , and we get that x ( r ) → ∞ as r → R , where R ≤ r x ( r ) x ( r ) − / . (4.40)In view of (4.39) we estimate R ≤ r + ( r − R ) r R − ( r − R ) . For sufficiently large R we have R / ≥ ( r − R ) ≥
0, recall that r − R ≤ δ , and we obtain R ≤ r + ( r − R ) r R ≤ r + 3( r − R ) ≤ R + 50 δ. This completes the proof of the theorem. ✷ Numerical results
In this section we present some numerical results. In the analytic proofwe required L , or equivalently R , to be large. One aim is to investigatefor which L solutions can be obtained numerically. Recall that necessarily L > L ∗ , and one question is if solutions can be constructed for L valuesarbitrary close to L ∗ , i.e., if the shell can occur arbitrary close to the photonsphere of the black hole. It turns out that this depends on the parametersand on the mass of the black hole, but for a large class of solutions it ispossible. Hence, the condition in the proof that L is large is mainly atechnical condition. Moreover, based on our numerical findings togetherwith the arguments in the proof, we conjecture that the compactness ratiofor any shell surrounding a black hole satisfies Γ > /
3. Note that 2 M /r =2 / R , R ] of the solution satisfies R − R ∼ R q − − lq , where q = k + l + 5 /
2. Hence, the thickness of the shells depend on theparameter values k and l . This investigation is presented in Subsection 5.2.In Subsection 5.3 we construct solutions with several shells. We do thisby using two different strategies. From the set up in Section 2 we know thathaving a compactly supported solution with ADM mass M , where M inthis case is the total mass of the black hole and the shell(s) surrounding theblack hole, we can choose L > L ∗ (sufficiently large) and find the largestroot R to equation (3.11). We can then pose data at r = R and numericallyconstruct an additional shell. This strategy thus follows the analytic proofin the previous section.The second strategy is different in the sense that it does not follow thestrategy of the proof. We start with a black hole and then we use one ansatzfunction for the shells, in particular we do not change L for the differentshells. In this way a solution consisting of several shells, separated by vac-uum, is generated by solving equation (3.11) only once. After a number ofshells there is no longer a vacuum region separating the neighboring shellsand a Schwarzschild solution has to be glued before this happens. The solu-tion obtained in this way is very similar to the multi-peak solutions obtainedin the massive case [12]. In the latter case there is however no need to gluea Schwarzschild solution since the massive solutions have compact support.20 .1 A black hole with one shell First we compute solutions for the parameter values k = 0 and l = 1 / M = 0 . L ∗ = 27 M = 27 / ≈ .
69. First we try to put the shell closeto the photon sphere of the black hole by choosing L = 1 .
7. However,as Figure 1 shows there is no vacuum region after the first peak (and thusthere is no proper shell) and it is not possible to obtain an asymptoticallyflat solution. (In all figures the energy density ρ is displayed on the verticalaxis and the area radius r on the horizontal axis.) The choice L = 1 . r = 3 M = 0 .
75, and the inner radius of the shell is located at R = 0 . .
74. The shell is depicted in Figure2. The radius of the photon sphere preceding the shell is denoted by r ∗ .If we increase the parameter value k , to k = 1 instead of k = 0, wehave to increase L to obtain a proper shell, i.e., the shell must be placedfurther away from the photon sphere of the black hole. Indeed, in Figure3 we have used the same parameter values as in Figure 2 with the onlychange that k = 1 instead of k = 0. We see that in this case a propershell is not obtained. We have to increase L to L = 5 . R = 2 .
04 and the compactness ratio is Γ = 0 .
80. Hence, the parameter k has a considerable influence on how close to the photon sphere the shell canbe placed. We point out that the parameter l has a similar impact but wehave not made a systematic study of the dependence on k and l .If we increase the mass of the black hole the situation changes and theshell can be placed closer to the photon sphere. Again this depends on theparameter values of k and l , but generally the shells can be placed closer tothe photon sphere when M is larger. If M is taken sufficiently large theshell can be placed arbitrary close to the photon sphere. This results in alow value of the compactness ratio Γ. Since 2 M /r = 2 / > / R → ∞ , the compactness ratio Γ → / R small in order to get a low value of Γ.We choose M = 2 .
0, which implies that L ∗ = 27 M = 108. The photonsphere is located at r = 3 M = 6 .
0. We find that the choice L = 108 . r Figure 1: Not a proper shell ( k = 0 , l = 1 / r Figure 2: A proper shell ( k = 0 , l = 1 / L = 1 . , Γ = 0 .
74 and r ∗ = 0 . R = 6 .
1. The shell is depicted in Figure 5, where k = 0 and l = 1 /
2. Thecompactness ratio Γ for the shell is merely 0 .
68. If k is increased to k = 1,then we need to increase L , to L = 113, in order to obtain a proper shell,which results in a shell with an inner radius of R = 6 .
9, and a compactnessratio Γ = 0 . We claim in Remark (3.4) that the thickness of the shells, R − R , dependon the parameters k and l as R − R ∼ R q − − lq , (5.41)where q = k + l + 5 /
2. In Figure (6), (7) and (8) we have computed shellsfor L = 15, L = 75 and L = 375 in the three cases k = 0 , l = 1 / r Figure 3: Not a proper shell ( k = 1 , l = 1 / r Figure 4: A proper shell ( k = 1 , l = 1 / L = 5 . , Γ = 0 . r ∗ = 0 . r -3 Figure 5: A proper shell ( k = 0 , l = 1 / L = 108 . , Γ = 0 .
68 and r ∗ = 6 . .55 3.6 3.65 3.7 3.75 3.8 3.85 3.9 r (a) L = 15 r (b) L = 75 r (c) L = 375 , Γ = 0 . Figure 6: Three shells with constant thickness ( τ = 0) k = 1 , l = 1 / k = 0 , l = 3 /
2. The quantity τ := q − − lq , takes the following values in these cases, 0 , / − /
4, respectively. When τ = 0, the width of the shell is independent on R and this is confirmed inFigure (6) where the thickness R − R ≈ .
29 for any of these shells. Thisis the case we considered in the proof. In the second case τ = 1 /
4, and thewidth of the shell thus increases as R grows. We compute the ratio R − R R τ , (5.42)for the three shells in Figure (7) and we find that it is approximately con-stant, roughly 0 .
67. In the last case τ = − /
4, and the width decreases as R increases. This can be seen in Figure (8) where the ratio (5.42) is roughly0 .
46 for each shell. Hence, we find numerical support for the claim (5.41).Moreover, Γ is computed in the three cases where L = 375, to confirm thatΓ → as R → ∞ , cf. Figures 6c, 7c and 8c.24 .4 3.6 3.8 4 4.2 4.4 4.6 4.8 r (a) L = 15 r (b) L = 75
19 19.5 20 20.5 r -3 (c) L = 375 , Γ = 0 . Figure 7: Three shells with growing thickness ( τ = 1 / r (a) L = 15 r (b) L = 75 r (c) L = 375 , Γ = 0 . Figure 8: Three shells with decreasing thickness ( τ = − / .3 A black hole with several nested shells Here we construct solutions where the black hole is surrounded by severalshells. We choose the black hole mass M = 1 . L ∗ = 27 .
0. We puta shell close to the photon sphere by the choice L = 28 .
0. We obtain asolution with one shell surrounding the black hole with ADM mass of 1 . L ∗ = 45 . L = 48 for the second shell.The solution we obtain is depicted in Figure 9 where we have taken k = 0and l = 1 /
2. The radius of the photon sphere surrounding the black hole is r ∗ = 3 . r ∗ = 4 .
2. Similar as above, if we increase the parameter values k , then we have to take larger values of L to obtain proper shells. Clearly,the procedure can be continued and an arbitrary number of shells can beconstructed which surround the black hole.Next, we again start with a black hole but we only use one ansatz func-tion for the shells, in particular we only solve equation (3.11) once and L is fixed. We then obtain a solution consisting of several nested shells, sepa-rated by vacuum. In Figure 10 such a solution is depicted where M = 1 . L = 48, k = 0 and l = 1 /
2. After the fifth peak there is no longer a vac-uum region separating the neighboring shells (which is difficult to see in thepicture since ρ is very small) and a Schwarzschild solution has to be gluedafter the fourth peak in order to obtain an asymptotically flat solution. Thesolution obtained in this way is very similar to the multi-peak solutions ob-tained in the massive case [12]. In the latter case there is however no needto glue a Schwarzschild solution since the massive solutions have compactsupport. We also include a few cases with other choices of the parameters k and l . In Figure 11 we have used the same parameters as in Figure 10with the only change that k = 1. In this case there are no proper shells andan asymptotically flat solution is not possible to obtain. By increasing L proper shells are obtained as depicted in Figure 12. Finally we consider thecase l = 3 / Acknowledgement:
The author would like to thank Gerhard Rein foruseful discussions.
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