Distribution functions for a family of general-relativistic Hypervirial models in collisionless regime
aa r X i v : . [ g r- q c ] J u l Distribution functions for a family of general-relativistic Hypervirial models incollisionless regime
Henrique Matheus Gauy ∗ and Javier Ramos-Caro † Departamento de F´ısica, Universidade Federal de S˜ao Carlos, S˜ao Carlos, 13565-905 SP, Brazil (Dated: July 2, 2018)By considering the Einstein-Vlasov system for static spherically symmetric distributions of matter,we show that configurations with constant anisotropy parameter β have, necessarily, a distributionfunction (DF) of the form F = l − β ξ ( ε ), where ε = E/m and l = L/m are the relativistic energyand angular momentum per unit rest mass, respectively. We exploit this result to obtain DFsfor the general relativistic extension of the Hypervirial family introduced by Nguyen and Lingam(2013), which Newtonian potential is given by φ ( r ) = − φ o / [1 + ( r/a ) n ] /n ( a and φ o are positivefree parameters, n = 1 , , ... ). Such DFs can be written in the form F n = l n − ξ n ( ε ). For odd n , wefind that ξ n is a polynomial of order 2 n + 1 in ε , as in the case of the Hernquist model ( n = 1), forwhich F ∝ l − (2 ε −
1) ( ε − . For even n , we can write ξ n in terms of incomplete beta functions(Plummer model, n = 2, is an example). Since we demand that F ≥ ξ n leads to restrictions for the values of φ o . For example, for theHernquist model we find that 0 ≤ φ o ≤ /
3, i.e. an upper bounding value less than the one obtainedfor Nguyen and Lingam (0 ≤ φ o ≤ PACS numbers: 04.40.-b, 04.70.Bw, 98.62.Hr
I. INTRODUCTION
Globular clusters, galactic bulges and dark matterhaloes have been usually modeled as many-particle sys-tems endowed by spherical symmetry. Although theNewtonian theory of gravitation is usually chosen asone of the paradigms of galactic dynamics, the ideaof formulating these models in the general relativis-tic realm has been gaining interest in recent decades[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] becomingone of the topical problems in stelar dynamics and rela-tivistic astrophysics.If one adopts a statistical standpoint to analyze suchself-gravitating configurations, it is advisable to performthe description by considering the Einstein-Vlasov sys-tem, in order to provide, in a self-consistent fashion, themetric, the energy-momentum tensor and the distribu-tion function (DF). In the context of galactic dynam-ics, usually based on Newtonian gravity, these theoret-ical constructions are called as dynamical models: theset composed by DF, potential and density (see [15, 16]for example). In this paper, adopting the general rela-tivistic paradigm, we also shall call the solutions of theEinstein-Vlasov system as dynamical models.On one hand, the DF or probability density function,can be considered as a concept involving all the relevantphysical information about the system. Once the DF isknown we can have access to astrophysical observables as,for example, the projected density and the light-of-sightvelocity, provided by photometric and kinematic mea-surements. On the other hand, the DF is a dynamical ∗ [email protected] † [email protected] entity governed by a kinetic equation which determinesthe statistical evolution of the configuration. For systemsin collisionless regime it obeys the Vlasov equation, some-times called as collisionless Boltzmann equation. In thecase of many-particle self-gravitating systems, the term“collisionless” is devoted to situations where the gravi-tational encounters are not significant in the evolution.Important examples are galaxies and clusters of galaxies,whose life time is lesser than the corresponding relax-ation time. But for smaller systems as stellar clusters,galactic bulges and haloes, encounters might play a sig-nificant role in the evolution and the DF is said to obeythe Fokker-Planck equation, which contains a collisionterm characterized by the so-called diffusion coefficients .Usually, they are computed by taking into account anequilibrium DF that is solution of the Vlasov equation.In other words, the task of describing the evolutionof globular clusters in collision regime, starts with theknowledge of the corresponding stationary DF in colli-sionless regime. Such a DF must determine, in a self-consistent manner, the associated energy-momentum andmetric tensors under equilibrium conditions. In thisline we will focus the principal subject of the presentpaper: providing adequate DFs, solutions of Einstein-Vlasov equations, for certain self-gravitating sphericallysymmetric configurations of astrophysical interest in gen-eral relativity. For such purpose, the well known ρ to f approach of Newtonian gravity [17, 18, 19, 20, 21, 22, 23],which obtains the DF starting from the potential-densitypair, by inversion, can also be used in the General Rela-tivity realm. Here, we will show that for certain sphericaldistributions this procedure can be performed analyti-cally.A wide variety of astrophysical configurations can berepresented as spherical systems with pressure anisotropy(the so-called anisotropic models), as confirmed by anumber of authors in the last three decades [24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,42, 43, 44, 45]. They are characterized by an anisotropyparameter β measuring the quotient between the radialpressure P r and the tangential (or azimuthal) pressure P θ . In particular, for β constant (i.e. independent ofthe radial coordinate r ), it can be proven that the DF isproportional to L − β (see section III B), as in the case ofthe hypervirial models [45], for which β = (2 − n ) /
2, with n = 1 , , ... , admitting some cases of interest. For n = 1(the Hernquist model), since lim L → F = ∞ , radial or-bits are much more abundant that closed orbits and weexpect most of the matter distribution to be located inthe inner region of the system. For n >
2, the situation isthe opposite: the DF increases with L , leading to config-urations with an overabundance of closed orbits and wedo not expect a large mass concentration near the center.The case n = 2 (Plummer model) is the only isotropicmodel of this family, where the mass distribution tendsto be homogeneous. These features, along with the in-teresting property of satisfy the virial theorem locally,makes the hipervirial family a set of models appropriateto represent galaxies and dark matter halos, from both aNewtonian [46] and relativistic [44, 45] point of view.Apart from the characteristics mentioned above,the relativistic hypervirial models introduced byNguyen and Lingam [45] have the remarkable property ofhaving the same constant anisotropy parameter as theirNewtonian counterparts. Here we will exploit this fact toderive analytical expressions for the associated general-relativistic DFs determining the energy-momentum ten-sor and other basic settings making such models physi-cally realizable configurations. In particular, it is worthmentioning that the requirement that the DFs be positiveleads to diminish the upper bounds of the free parame-ters (see section IV), compared with the ones obtainedfrom energy conditions [45]. In this sense, the require-ment that the DFs be positive can be interpreted as astatement more fundamental than the imposition of en-ergy conditions (an interesting analysis can also be foundin [47]).The paper is organized as follows: In Sec. II we com-ment some general features of the relativistic extensionof Hernquist solution, focusing on the requirements thatmust hold to obtain physically realizable configurations,from the perspective of energy conditions. We will showthat they impose an upper bound of 4 / φ o . However this upper limit decreases to2 / c = 1, where c is the speed oflight. Greek indices µ, ν run from 0 to 3. When using isotropic coordinates ( t, r, θ, ψ ) we introduce the follow-ing associations for indices: 0 → t , 1 → r , 2 → θ and3 → ψ . Thus the symbol T rr will denote T , as well as P equals to P t , for example. II. A GENERAL-RELATIVISTIC VERSION FORTHE HERNQUIST MODEL
The general static isotropic metric, in isotropic coor-dinates ( t, r, θ, ψ ), can be written as [48]d s = − A ( r )d t + B ( r ) (cid:0) d r + r d θ + r sin θ d ψ (cid:1) . (1)Also, it can be expressed as a generalized version of theSchwarzschild metric, by defining A ( r ) = (cid:20) − f ( r )1 + f ( r ) (cid:21) , B ( r ) = [1 + f ( r )] , (2)in which the special case f = − GM/ r represents theSchwarzschild solution, with a Newtonian limit φ = − GM/r . In general, if one chooses f ( r ) = − φ ( r ) / φ ( r ) is any spherical solution of Poisson equa-tion, it gives rise, in the limit c → ∞ , to a Newto-nian potential φ . This fact sketches a simple procedureto construct general relativistic extensions of previouslyknown Newtonian solutions, as shown by several authors[7, 45, 49, 50, 51]. Here we first focus on the general rel-ativistic extension of the Hernquist potential, one of themodels obtained in [45]. Then we choose f as f ( r ) = − φ ( r )2 , φ ( r ) = − φ o r/a ) , (3)where φ o and a are positive parameters representing themaximum value of | φ | (at the center of the spherical con-figuration) and a scaling radius, respectively. Note thatthis metric describes an asymptotically flat space-timewith a Ricci scalar given by R = 4 φ o a ( r + a ) [ a ( φ o − − r ] r h r + a (cid:16) − φ o (cid:17)i h r + a (cid:16) φ o (cid:17)i , from which we note that there are two singularities,(i) r = 0 , (ii) r = a (cid:18) φ o − (cid:19) , (4)the second one depending on the free parameters a and φ o . It is easy to see that, for φ o ≤
2, singularity (ii)disappears. Also, it can be shown that, for 0 < φ o ≤
1, we have
R < φ o = 1, we find R = − a ( r + a ) ( r + a/ − ( r + 3 a/ − which means that both singularities,(i) and (ii), disappear. For all other cases, φ o = 1, wefind always a singularity at origin, r = 0.Energy conditions help us to state the range of valuesfor φ o leading to physically realizable configurations. In - - - - - - - r Ž R Ž - - - - r Ž R Ž - - - r Ž R Ž Figure 1: We show Ricci scalar for different values of parameter φ o . In particular we plot ˜ R = ( a / φ o ) R as a function of˜ r = r/a . For 0 < φ o ≤
1, we have ˜
R < R for 1 < φ o ≤
2, which is positive only nearthe singularity r = 0. For φ o > R is negative in a prominent region of its domain. order to use such conditions, we need the explicit formof the stress-energy tensor, which can be determined viaEinstein field equations. We find that the non vanishingcomponents of the stress-energy tensor can be written interms of f : T tt = 4 f πGφ o ar (1 + f ) (1 − f ) , (5) T rr = 2 f πGφ o ar (1 + f ) (1 − f ) , (6) T θθ = T ψψ sin θ = f πGφ o ar (1 + f ) (1 − f ) . (7)So, it is easy to state that weak energy condition, − T tt ≥
0, is satisfied if φ o ≥
0. Strong energy condition, T = − T tt + T rr + T θθ + T ψψ ≥
0, leads to4 f (1 + f ) (1 − f ) ≥ , which requires that 0 ≤ φ o ≤
2. Dominant energy condi-tion, given by (cid:12)(cid:12)(cid:12)(cid:12) T rr T tt (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) T θθ T tt (cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12) T ψψ T tt (cid:12)(cid:12)(cid:12)(cid:12) ≤ , is satisfied if φ o < /
3. In summary, we have to choosethe parameter φ o so that0 ≤ φ o < / , (8)in order to fulfill weak, dominant and strong energy con-ditions. This means that physically realizable configura-tions described by (2)-(3) have only one singularity, atthe center r = 0. We shall see, in Sec. IV, by analyzingthe behavior of the corresponding distribution function,that we have to choose φ o ≤ / r > III. SELF GRAVITATION EQUATIONS FORSTATIC ISOTROPIC DISTRIBUTIONS OFMATTER
In this section we show a detailed derivation of re-lations which help us to obtain the DF describing theconfiguration associated with the metric of (1), (2) and(3). At first, we shall deal with functions A ( r ) and B ( r )representing asymptotically flat space-times, in general,and then we consider the particular case in which suchfunctions are given by (2) and (3).The relation between the stress-energy tensor, T µν ,and the the DF, F ( x µ , P ν ) (here P µ = d x µ / d τ is the4-momentum vector and τ is the proper time), associ-ated with a self-gravitating system, is given by T µν = Z P µ P ν F√− g d P (9)where g = det ( g µν ) and we choose P t >
0. The phase-space domain associated with a particle of rest mass m is determined by the shell condition, g µν P µ P ν = − m , (10)from which we can express P t as a function of the re-maining phase-space coordinates: P t = P t ( P i , x µ ). Ad-ditionally, neglecting the effect of gravitational encoun-ters in the system, we demand that F must satisfy thecollisionless Boltzmann equation [52], P µ ∂ F ∂x µ − Γ λµν P µ P ν ∂ F ∂ P λ = 0 . (11)Such DF, through relation (9) and the Einstein fieldequations, R µν − g µν R/ − πGT µν , determines thespace-time geometry by the set of relations Rg µν − R µν = 16 πG Z P µ P ν F√− g d P , (12)which we denote here as the self-gravitation equations ,in the sense that they define, in a self-consistent fash-ion (obeying simultaneously Einstein’s equations andcollisionless Boltzmann equation, or, equivalently, theEinstein-Vlasov system), the evolution of the system.Relation (11) is equivalent to demand that d F / d τ = 0[53], i.e. F can be regarded as an integral of motion. Ifthe system is endowed by spherical symmetry (or cylin-drical or any other) the Jeans theorems guarantee that F can be expressed as a function the other integrals, which,for the spherical case, are the general relativistic exten-sions of energy E and angular momentum L . In thispaper we are focusing on this case.Motion of free falling test particles in the staticisotropic space-time described by (1) have one constantof motion, the rest mass m , and three integrals of mo-tion. The first of them, an energy-like integral of mo-tion, is the t -component of the covariant 4-momentumvector, P t . The second one is the azimuthal angular mo-mentum like integral, P ψ , and the third one is the gen-eral relativistic version of the total angular momentum, q P θ + P ψ / sin θ . For the sake of simplicity, we adoptthe notation P t = − E, P ψ = L z , P θ + P ψ sin θ = L , (13)and equations of motion for a free falling test particle canbe cast as m dtdτ = P t = EA ( r ) , (14a) m dψdτ = P ψ = L z r B ( r ) sin θ , (14b) m dθdτ = P θ = ± r B ( r ) r L − L z sin θ , (14c) m drdτ = P r = ± s E A ( r ) B ( r ) − L r B ( r ) − m B ( r ) , (14d)remembering that phase space coordinates are con-strained by the shell condition. Thus, equations (10)and (14) will be the base for constructing the distribu-tion function. A. The self-gravitation equations
Since g µν does not depend on 4-momentum, equation(9) can be written as T µν = √− g Z P µ P ν F d P , where the integral is defined in all the phase space do-main where F >
0. Since we are dealing with a DF thatis function of the integrals of motion, E , L z , L and m (which, through the shell condition (10), can be inter-preted as an integral of motion), it is convenient to makea transformation from coordinates ( P t , P r , P θ , P ψ ) to co-ordinates ( m, E, L z , L ). At this point we must be carefulwith the transformations of P r and P θ since, according to (14-c) and (14-d), they have two forms, one for eachchoosing of sign. Thus, we write P r + = s L m ( r ) − L r B ( r ) , P r − = −P r + , (15)where L m ( r ) = s B ( r ) r (cid:18) E A ( r ) − m (cid:19) , (16)and P θ + = 1 r B ( r ) r L − L z sin θ , P θ − = −P θ + . (17)Therefore we have to write T µν = √− g (cid:20)Z P µ P ν F d P t d P r + d P θ + d P ψ + Z P µ P ν F d P t d P r − d P θ + d P ψ + Z P µ P ν F d P t d P r + d P θ − d P ψ + Z P µ P ν F d P t d P r − d P θ − d P ψ (cid:21) . In particular, the expression for components T rr and T θθ requires a replacement of P r and P θ by P r + , P r − , P θ + and / or P θ − , according to the variables of integration.For example, in the above expression, the term involvingd P r + d P θ − requires that we set P r → P r + , when calculating T rr , and it will require P θ → P θ − , when computing T θθ .Note that in all cases, the Jacobian of the transformationis (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( P t , P r , P θ , P ψ ) ∂ ( m, E, L, L z ) (cid:12)(cid:12)(cid:12)(cid:12) = mL P r + P θ + AB r sin θ , and the domain of integration is given by the relations − L sin θ ≤ L z ≤ L sin θ ≤ L ≤ L m ,m √ A ≤ E ≤ m, ≤ m ≤ ∞ . (18)The bounds for E arise from the shell condition and fromthe escape energy, which can be elucidated from relation(14-d). At r → ∞ we have A = B = 1, since we are as-suming that (1) represents an asymptotically flat metric,and we have |P r | = p E − m , r → ∞ Then the escape energy, at r → ∞ , is E = m (rememberthat we chose energy to be positive), corresponding to thevalue |P r | = 0. Thus, we can state that particles withenergy larger than m can not belong to the configuration.It can be shown, from (1), that components of thestress-energy tensor that could be non-vanishing are T tt , T rr , T θθ and T ψψ , whereas the other components vanishin any case (i.e. for an arbitrary DF). This fact canbe checked directly from (9), except for the case of T tψ ,which does not vanish trivially. However, since the stress-energy tensor is a function only of radius r , it is requiredthat the DF has the form F ( m, E, L, L z ) = F ( m, E, L ) , leading to T tψ = 0 and simplified expressions for thenon-vanishing components: T tt = 4 πr A / B / ∞ Z m Z m √ A L m Z E mL FP r + d L d E d m,T rr = 4 πr A / B / ∞ Z m Z m √ A L m Z P r + mL F d L d E d m,T θθ = 2 πr A / B / ∞ Z m Z m √ A L m Z mL FP r + d L d E d m, and T ψψ = T θθ / sin θ . In many applications it is com-mon to assume that the mass for every constituent of thesystem is the same. This lead us to replace F ( m, E, L )by F ( E, L ), which now satisfies the following simplifiedform: T tt = 4 πmr A / B / m Z m √ A L m Z E L F ( E, L ) P r + d L d E, (19) T rr = 4 πmr A / B / m Z m √ A L m Z P r + L F ( E, L )d L d E, (20) T θθ = 2 πmr A / B / m Z m √ A L m Z L F ( E, L ) P r + d L d E. (21)The above relations, remembering that T µν =[ g µν ( R/ − R µν ] / (8 πG ), can be regarded as the self-gravitation equations in the case of a general staticisotropic metric. Then, by defining the functions A and B in eq. (1), in principle, we can determine F ( E, L )through equations (19), (20) and (21). A similar expres-sion is shown in [54] for a metric in the standard form.
B. Models with P θ = kP r In this section we assume that the configuration can beregarded as a fluid with a dynamics described in terms ofthe energy density ρ , the radial pressure P r and the tan-gential pressure P θ (or P ϕ ). In this context it is useful todistinguish between isotropic ( P r = P θ ) and anisotropicsystems ( P r = P θ ), by introducing the anisotropy param-eter β = 1 − P θ P r . (22)Thus, isotropic fluids are represented by β = 0 andanisotropic fluids are characterized by a function β ( r )which, in general, does not vanish. Here we focus in thecase in which the anisotropy parameter is a real con-stant, β = 1 − k , i.e. fluids such that P θ = kP r . We willshow that this particular class of systems with constantanisotropy are characterized by a distribution function ofthe form F = ξ ( E ) L k − .At first, remember that ρ , P r and P θ are related withthe stress-energy tensor by the relations ρ = − T tt , P r = T rr , P θ = T θθ = T ϕϕ , which, by using (19), (20) and (21), can be written as ρ = 4 πmr ( BA ) m Z m √ A L m Z E L F ( E, L ) P r + d L d E, (23) P r = 4 πmr √ BA m Z m √ A L m Z P r + L F ( E, L )d L d E, (24) P θ = 2 πmr B √ A m Z m √ A L m Z L F ( E, L ) P r + d L d E. (25)Note that, by choosing F ( E, L ) = ξ ( E ) L k − (with k aconstant) in the above equations we can write P θ = kP r .Also we can prove that by setting P θ = kP r , then theDF, necessarily, must have the form ξ ( E ) L k − .Let us write the statement P θ = kP r by using (24)-(25)and taking into account, for now, only the integral withrespect to L : Z L m L F p L m − L d L = 2 k Z L m L F p L m − L d L. Now, we can integrate by parts the right hand side of theabove expression,2 L m Z L F p L m − L d L = L m Z L F p L m − L d L − L m Z L p L m − L ∂ F ∂L d L − L m lim L → (cid:0) L F (cid:1) . It can be shown that lim L → ( L F ) = 0, for any F ( E, L )satisfying (19), (20) and (21) (see appendix B for a de-tailed proof). Then, we can write L m Z L p L m − L (cid:20) k − F −
L ∂ F ∂L (cid:21) d L = 0 , which has to be satisfied for every L m (or for every E ).Therefore,2 ( k − F −
L ∂ F ∂L = 0 ⇒ F = ξ ( E ) L k − . Finally, we can state the following theorem:
Proposition 1
Let k be a constant and F a distributionfunction that satisfies the self-gravitation equations forstatic spherically symmetric configurations. Then P θ = kP r if and only if F ( E, L ) = ξ ( E ) L k − . Thus, models with constant anisotropy β are char-acterized by a distribution function proportional to ξ ( E ) L − β . In the next sections we show that the Hern-quist model, as well as the so-called hipervirial models,belong to this class of systems. IV. DISTRIBUTION FUNCTION FORGENERAL-RELATIVISTIC HERNQUIST MODEL
Here we show how to derive a relativistic DF for arelativistic Hernquist model, given by (2)-(3) by usingthe self-gravitation equations (19)-(21). Since the factor √ A appears repeatedly in eqs. (19)-(21), it is importantto note that (2)-(3) imply f ( r ) = − √ A ( r )1+ √ A ( r ) , r > a ( φ o − √ A ( r )1 − √ A ( r ) , < r ≤ a ( φ o − ≤ φ o < / a [( φ o / − < − a/ r sat-isfying 0 < r ≤ a [( φ o / − r satisfy r > a [( φ o / − f , consistent with all the energy conditions, is f ( r ) = 1 − p A ( r )1 + p A ( r ) , r > . (26)This means that relations (5)-(7), by introducing (26),can now be rewritten as T tt = (1 − √ A ) (1 + √ A ) πGφ o arA , (27) T θθ = T rr r = (1 − √ A ) (1 + √ A ) πGr φ o a √ A , (28) This form is particularly useful when compared with thecorresponding equations obtained from (19), (20) and(21). Indeed we find that P θ = P r /
2. By using theresult of Proposition 1, this fact implies that F ( E, L ) = ξ ( E ) L − , where ξ ( E ) is a function to be found by comparing theright hand side of eqs. (27), (28) with the right handside of (19)-(21). After some calculations we obtain tworelations for ξ : m Z m √ A E ξ ( E )d E = A / (1 − √ A ) π mGφ o a , (29) m Z m √ A ξ ( E ) (cid:2) E − m A (cid:3) d E = A (1 − √ A ) π mGφ o a . (30)From (29) we find ξ ( E ) = 34 m π Gφ o a (cid:18) Em − (cid:19) (cid:18) Em − (cid:19) , which is consistent with relation (30).For the sake of simplicity, we introduce the dimension-less energy ε and the dimensionless angular momentum l , as ε ≡ E/m, l ≡ L/m, (31)and thus we can write the explicit analytic form of theDF corresponding to the general relativistic extension ofHernquist model, as a function of ε and l : F ( ε, l ) = ξ o l − (2 ε −
1) ( ε − , (32)with ξ o = 3 (cid:0) m π Gφ o a (cid:1) − . (33)Note that such DF is negative for E < m/
2, so, in prin-ciple, we would have to restrict its domain to values ofenergy larger than m/
2. In the next section we show thata natural way to do this is by constraining the values ofthe free parameter φ o . In figure 2 we plot the behav-ior of the DF given by (32), once φ o has been chosenadequately. Constraining the values for φ o Self-gravitation equations (19)-(21) impose some re-strictions to the stress-energy tensor (not necessarilyequivalent to energy conditions), when one demands that
F ≥
0. They can be summarized as, T µν ≥ , (34a) T ≤ , (34b) E (cid:144) m F Ž Figure 2: Dimensionless DF, ˜ F = ξ − o F , for the general rela-tivistic extension of Hernquist potential as a function of E/m ,for different values of
L/m : 0.2 (blue), 0.5 (violet), 1 (yellow),1.5 (green).
Indeed these restrictions are stronger than the weak, null,dominant and strong energy conditions. When they areapplied to the stress-energy tensor given by (5)-(7), wefind the following inequality0 ≤ f ≤ / , (35)which in terms of the radial coordinate r is equivalent tostate that r ≥ a ( φ o − . This means that a real, positive DF, determining thestress-energy tensor could be well defined only for r ≥ a ( φ o − φ o that permits aDF well defined at the entire configuration space, r ≥ φ o = 1.The bounding value for φ o can be diminished by tak-ing into account that the DF of eq. (32) is negative for E < m/ E min = m √ A . Therefore, situationswhere √ A < /
2, which in this case equals to state that r < a (3 φ / − r ≥ a (cid:18) φ − (cid:19) , which means that, φ o = 2 / φ o such that F is positive and well defined for r ≥ φ o we guarantee that E ≥ m/ φ o are given by0 ≤ φ o ≤ / , (36)in order to obtain a self-consistent relativistic Hernquistmodel, charaterized by a DF well defined at the entireconfiguration space. V. DISTRIBUTION FUNCTIONS FOR AGENERAL-RELATIVISTIC VERSION OF THEHYPERVIRIAL FAMILY
The formalism used in the preceding sections can alsobe applied in the case of the hypervirial family, to whichHernquist model belongs. In Newtonian gravity, thehipervirial potentials are given by φ n ( r ) = − φ no [1 + ( r/a ) n ] n , (37)where n is a positive integer and φ no , a positive realconstants. Each member is characterized by a DF pro-portional to E (3 n +1) / L n − .As in the case of Hernquist model (the particular case n = 1 of (37)), a physically reasonable relativistic ex-tension, introduced previously in [45], is performed bydefining f = − φ n / T rr = 2 n − f n +2 πa n φ no n Gr − n (1 − f )(1 + f ) = 2 r n T θθ = f (1 − f )( n + 1)(1 + f ) T tt , (38)and T µν = 0 for µ = ν . From (38) is easy to see that P θ = ( n/ P r , which, by using Proposition 1, impliesthat the corresponding DF can be written as F = ξ ( E ) L n − , n = 1 , , ... By introducing the above expression into (19)-(21) weobtain m Z m √ A ξ ∗ ( E ) E (cid:18) E A − m (cid:19) n − d E = 2 A / (cid:16) − √ A (cid:17) n +1 , m Z m √ A ξ ∗ ( E ) (cid:18) E A − m (cid:19) n +12 d E = (cid:16) − √ A (cid:17) n +2 , where ξ ( E ) = ξ ∗ ( E ) ( n + 1) Γ (cid:0) n +12 (cid:1) π a n φ no n GΓ (cid:0) n (cid:1) m . These two relations are essentially the same: the firstone can be obtained by taking the derivative of the sec-ond one with respect to √ A . So, in this case, we canchoose the second one relation (the simpler one) as theintegral equation to be solved, in order to find an explicitexpression for function ξ . Here, for simplicity, we define √ A = x , which leads to m Z mx ξ ∗ ( E ) "(cid:18) Em (cid:19) − x n +12 d E = (cid:16) xm (cid:17) n +1 (1 − x ) n +2 . (39)In order to solve the above relation it is convenient toconsider, separately, two cases: (i) n = 1 , , , ... and (ii) n = 0 , , , ... . Each of these options will lead to twokinds of distribution functions.(i) By choosing n = 2 p + 1, for p = 0 , , , . . . , in eq.(39), we find that ξ p +1 ( E ) = p +4 X k =1 a p +1 (cid:18) Em (cid:19) k − , where the a p +1 are constants that will be specifiedlater (see eq. (41)). Note that the DF correspond-ing to the relativistic extension of Hernquis modelis obtained for p = 0 (or n = 1). The next case, p = 1 (or n = 3) is described by a function ξ ( E ) ∝ (cid:18) − Em (cid:19) " (cid:18) Em (cid:19) − Em + 5 , which must be restricted to a domain given (ap-proximately) by 0 ≤ E/m ≤ . . ≤ E/m ≤
1, in order to have a positive DF. For theother cases, p = 2 , , .. the function ξ also can bewritten in the form ξ p +1 ∝ (1 − ε ) p +2 g ( ε ), where g is a polynomial of degree p + 1 in ε .(ii) The case in which n is even, i.e. n = 2 p for p =0 , , , . . . in equation (39), demands a little moreattention. By computing the derivative in x of (39), p + 1 times, we have m Z mx ξ ( E )d E q ( E/m ) − x ∝ (cid:18) − x dd x (cid:19) p +1 " x p +1 (1 − x ) p +2 m p +1 (2 p + 1)!! . Note that the right side has the form of an Abel in-tegral, so the function ξ can be determined explic-itly by performing the Abel transformation. Thus,after some calculations we find ξ p ( E ) = 2 Em p +2 X k =0 b pk Z Em x k − q x − (cid:0) Em (cid:1) d x. where b pk are constants given by relations (43).For example, the case p = 1 (or n = 2), for whichthe L -dependence is dropped, lead us to the DFcorresponding to the relativistic extension of thePlummer model: ξ ∝ E − q − E m + π q − E m (cid:16) E m + Em (cid:17) − π (cid:16) E m + E m + Em (cid:17) ln (cid:18) mE q − E m + mE (cid:19) We can summarize our results through the followingrelations F (odd) n = l n − n +2 X k =1 a nk ǫ k − , n = 1 , , , ... (40) Table I: Upper bound value of φ no for different n .n φ no where a nk = (cid:18) n + 2 k (cid:19) ( − k + n − ( k + n + 1)!! k π a n φ no n Gm Γ (cid:0) n (cid:1) k !!( n + 1)!! × ( n + 1) √ πΓ (cid:18) n + 12 (cid:19) , n = 1 , , , .., (41)for DFs with odd index, and F (even) n = l n − ǫ n +2 X k =0 b nk Z ǫ x k − d x √ x − ǫ , n = 0 , , , ..., (42)where b n = ( − − n ( n + 1) Γ (cid:0) n +12 (cid:1) π a n φ no n Gm √ πΓ (cid:0) n (cid:1) , b n = 0 ,b nk = (cid:18) n + 2 k (cid:19) ( k + n + 1)!!( − k − n ( k − π a n φ no n Gm ( k − n + 1)!! × ( n + 1) Γ (cid:0) n +12 (cid:1) √ πΓ (cid:0) n (cid:1) , k ≥ , n = 0 , , , ... (43)As done in Sec. IV, we can choose the values of φ no so that F n be positive everywhere in configuration space, r >
0. Figure 5 suggests that the upper bound for φ no decreases with n , as confirmed by the values of Table I. VI. CONCLUSION
We derived an analytic expression for the DF cor-responding to the general relativistic extension of theHernquist model presented in [45]. In the derivationwe considered the self-gravitating equations for asymp-totically flat static isotropic space-times, from which weestablished that anisotropic models so that P θ = kP r ,with k constant, are characterized by a DF of the form F = ξ ( E ) L k − (proposition 1). For the Hernquistcase, corresponding to k = 1 /
2, we find F ( E, L ) ∝ L − (2 E/m −
1) (1 − E/m ) , from which we establishedthat the upper bound of free parameter φ o is 2 / r > - E (cid:144) m F Ž E (cid:144) m - - - - E (cid:144) m Figure 3: Dimensionless DF corresponding to the hypervirial model n = 3, for different values of L/m : 0.2 (blue), 0.5 (violet),1 (yellow), 1.5 (green). This DF is positive for 0 ≤ E/m ≤ . . ≤ E/m ≤ E/m >
1, this DF has negative values (right panel). E (cid:144) m F Ž - - - - E (cid:144) m E (cid:144) m Figure 4: Dimensionless DF corresponding to the hypervirial model n = 2, which is a relativistic extension of Plummermodel. The DF is positive for 0 ≤ E/m ≤ . . ≤ E/m < . < E/m < . E (cid:144) m F Ž n Figure 5: Dimensionless DF, ˜ F n = 2 m π Gφ nno a n F n , forthe general relativistic extension of the Hypervirial family as afunction of E/m with
L/m = 2, for different values of n : n=1(red), n=2 (Blue), n=3(Green), n=5 (Orange), n=7 (Purple),n=9 (Brown). Hypervirial family, which satisfies P θ = ( n/ P r for the n th member (Hernquist model is the first member, n =1). Proposition 1 implies that the DF corresponding tothe n th member is of the form F n = ξ n ( E ) L − n , wherewe have to distinguish between odd and even values of n ,in order to encompass in a simple fashion all cases (eqs.(40) and (42)). Thus we find two subfamilies in the setof hypervirial models, which now can be regarded as aself-consistent family of models in the context of generalrelativity.We note that the free parameter φ no , corresponding to the n th member of the hypervirial family, has an up-per bound which diminishes by increasing n . Such upperbound, as in the case of Hernquist model, was chosen insuch a way that the DF was positive for r >
0. How-ever, one could choose different upper bounds for theseparameters when taking into account a reduced configu-ration space, for example given by r ≥ r ∗ , where r ∗ is apositive constant. This can be used to model situationscomposed by two solutions of Einstein equations, one ofthem defined in 0 < r < r ∗ (the solution inside the regionbounded by the shell r = r ∗ ) and the other one, an Hy-pervirial solution, defined in r ≥ r ∗ . In such a case, theDF has to be defined by parts and junction conditionshas to be satisfied in the shell r = r ∗ (see for example[55]). Appendix A: Hernquist potential in NewtonianGravity
The distribution function (DF) for the Hernquist po-tential is given by [45] F ( ε, L ) = Aε β L α (A1)where L is the norm of the specific angular momentumand ε = φ ∗ − E is the relative energy (in this case wehave to set φ ∗ = 0). This is the same distribution func-tion used by Nguyen et al. in order to develop a family ofpotential-density pairs, including the Hernquist model as0a particular case. The mass density can be found by inte-grating the distribution function over the velocity space, ρ = Z F ( ε, L )d ν, which, by introducing (A1) and using spherical coordi-nates, leads to ρ = Z π Z π Z ν e Aε β L α ν sin η d ν d η d κ (A2)where ν e = √− φ is the escape velocity and φ is thegravitational potential. Since in spherical coordinates wecan write L = r ν sin η and ε = − E = − φ − ν /
2, wehave ρ = 2 πA Z π sin α +1 η d η Z ν e (cid:18) − ν − φ (cid:19) β r α ν α +2 d ν The first integral above is basically a constant, so bytaking 2 πA R π sin α +1 η d η = B , we have ρ = Br α Z ν e (cid:18) − ν − φ (cid:19) β ν α +2 d ν (A3)Now, in order to compute the second integral, it can becast as ρ = Br α Z √− φ φ β (cid:18) − ν φ − (cid:19) β ν α +2 d ν, where, by making the substitution x = ν /φ , the integralbecomes ρ = Br α φ β + α +1 α +1 r φ Z − ( − x − β x α +1 / d x, Again, the last integral is a constant. With this in mindand organizing the terms, we have ρ = Cr α φ β + α +3 / (A4)Now it is possible to calculate the potential through thePoisson equation, ∇ φ = 4 πGρ = 4 πCGr α φ β + α +3 / Since α and β are parameters, it is straightforward toprove that φ = − φ o r/a , is a solution of the equation for α = − / β = 2 and4 πCG = − /φ o a , where a is the characteristic radius ofthe system.Now returning to the expression (A4) of the densityand thus using the values α = − / β = 2, we cancompute the constant A : C = 2 πA Z π d η Z − ( − x − d x = − π A , then C = − π A − πGφ o a , which lead us to A = 34 π φ o aG (A5)In summary, we can establish that the distributionfunction, the gravitational potential and the mass densityfor the Hernquist model are given by F ( ε, L ) = 34 π φ o aG ε L − (A6) φ = − φ o r/a (A7) ρ = − πGφ o a r − φ = φ o πGar (cid:18)
11 + r/a (cid:19) (A8) Appendix B: Demonstration of Lemma 1
In this appendix we provide a proof by reductio adabsurdum of lemma 1, used to obtain theorem 1:
Lemma 1 If F is a DF satisfying the self-gravitationequations (19), (20) and (21), then lim L → (cid:0) L F (cid:1) = 0 .Proof . If one supposes thatlim L → (cid:0) L F (cid:1) = 0 , then, from the definition of limit, for every δ > ǫ > L such that 0 < L < δ and F ( E, L ) >ǫL − .On the other hand, since F must be a continuous func-tion, then L F is a continuous function too, so there ex-ists a region centered in L such that F > ǫL − , i.e. F >ǫL − for every L belonging to L − δL < L < L + δL .All of the above holds for every choice of 0 < δ < δL .Then, if we choose δ in such a way that 0 < L < δ and,therefore, L falls inside the interval ( L − δL, L + δL ),then for such δ there exists an ǫ > < L < δ we have F > ǫL − .Now, by choosing L m to be smaller than δ , we canwrite Z L m F ( E, L ) L d L p L m − L ≥ ǫ Z L m d LL p L m − L . Note that the right hand side integral does not convergeand the left hand side integral must converge since ρ ,given by (23), is finite. 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