Virial inequalities for steady states in relativistic galactic dynamics
aa r X i v : . [ m a t h - ph ] D ec Virial inequalities for steady statesin relativistic galactic dynamics ∗ Simone Calogero, Juan Calvo,´Oscar S´anchez & Juan Soler † Abstract
It is well known that steady states of the Vlasov-Poisson system, a widely used model innon-relativistic galactic dynamics, have negative energy. In this paper we derive the analogousproperty for two relativistic generalizations of the Vlasov-Poisson system: The Nordstr¨om-Vlasov system and the Einstein-Vlasov system. In the first case we show that the energy ofsteady states is bounded by their total rest mass; in the second case, where we also assumespherical symmetry, we prove an inequality which involves not only the energy and the restmass, but also the central redshift. In both cases the proof makes use of integral inequalitiessatisfied by time depedent solutions and which are derived using the vector fields multipliersmethod.
AMS classification (2000):
Keywords:
Galactic dynamics, Vlasov-Poisson, Einstein-Vlasov, steady states.
A widely used model in astrophysics for the dynamics of the stars of a galaxy is the Vlasov-Poissonsystem [7]. This model is justified when collisions among the stars and external forces are neglected.Moreover, being a non-relativistic model, the Vlasov-Poisson system ceases to be valid when thestars move with large velocities (of the order of the speed of light) or in the presence of verymassive galaxies, since then relativistic effects become important. Typical relativistic effects arethe redshift of the luminous signals emitted by the galaxy and the formation of black holes. Themodel which is currentely believed to represent the physically correct relativistic generalization ofthe Vlasov-Poisson system is the Einstein-Vlasov system [1], where Poisson’s equation is substitutedby Einstein’s equations of General Relativity. Another relativistic generalization of Vlasov-Poissonis the Nordstr¨om-Vlasov system [9]. This model relies on the same geometric interpretation ofgravity as for the Einstein-Vlasov system. Although it is not physically correct, the Nordstr¨om-Vlasov system is mathematically interesting since it already captures some of the technical andconceptual new difficulties that are encountered when studying a relativistic (Lorentz invariant)system.In this paper we want to investigate the mass-energy bounds (virial inequalities) required for theexistence of steady states to the relativistic models. It is well known in fact that static solutionsof the Vlasov-Poisson system, which correspond to equilibrium configurations of the galaxy, have ∗ The authors have been partially supported by Ministerio de Ciencia e Innovaci´on (Spain), Project MTM2008-05271 and Junta de Andaluc´ıa Project E–792. † Departamento de Matem´atica Aplicada. Facultad de Ciencias, Universidad de Granada. 18071 Granada, Spain.( [email protected], [email protected], [email protected], [email protected] ) Let f = f ( t, x, p ) be the distribution function in phase space for an ensemble of unit mass particles,where f ≥ t ∈ R , x ∈ R and p ∈ R . In the physics of gravitational systems, the particles standfor the stars of a galaxy. In geometric units, i.e., 4 πG = 1, where G is Newton’s gravitational con-stant, the gravitational potential U = U ( t, x ) generated by the galaxy solves the Poisson equation∆ x U = ρ , lim | x |→∞ U = 0 , ∀ t ∈ R , (1a)where ρ = Z R f dp (1b)is the mass density of the galaxy and the boundary condition at infinity means that the galaxy isisolated. The assumption that the stars interact only by gravity leads to the Vlasov equation: ∂ t f + p · ∇ x f − ∇ x U · ∇ p f = 0 . (1c)The system (1) is the Vlasov-Poisson system. The energy H and the mass M of a solution aregiven by H = 12 Z R Z R | p | f dp dx − Z R |∇ x U | dx , M = Z R Z R f dp dx and are conserved quantities. Likewise, the total linear momentum Q and angular momentum L , Q = Z R Z R p f dp dx , L = Z R Z R x ∧ p f dp dx , are conserved quantities. The invariance of Vlasov-Poisson by (time dependent) Galilean transfor-mations is the property that, given u ∈ R and the transformation of coordinates G u : t ′ = t , x ′ = x − ut , p ′ = p − u , then f u ( t, x, p ) = f ( t ′ , x ′ , p ′ ) and U u ( t, x ) = U ( t ′ , x ′ ) solve the system (1) if and only if ( f, U ) does.Note that Q can be made to vanish by a Galilean transformation with velocity u = Q/M ; theresulting reference frame is at rest with respect to the center of mass of the distribution, which isdefined as c ρ ( t ) = M − R R x ρ dx .A galaxy in equilibrium is described by steady states solutions of the Vlasov-Poisson system. Wedistinguish between two types of steady states: Static solutions and traveling steady states. Theformers are defined as time independent solutions of the Vlasov-Poisson system (1) and have totalmomentum Q = 0. A solution f is a traveling steady state (with total momentum Q = 0) if f ◦ G u , where u = Q/M , is a time independent solution of the Vlasov-Poisson system (i.e., astatic solution). Our interest on traveling steady states is motivated by the fact that their energyprovides a lower limit for the energy of totally dispersive solutions, see [11]. Moreover, the non-linear stability theorems proved for the Vlasov-Poisson system consider the traveling steady statesas possible perturbations of a static equilibria, see [10, 19, 24] and references therein.A fundamental property shared by all static solutions of the Vlasov-Poisson system is that of havingnegative energy. The proof goes as follows. Any sufficiently regular solution of the Vlasov-Poissonsystem satisfies the dilation identity: ddt Z R Z R x · p f dp dx = H + E kin , E kin = 12 Z R Z R | p | f dp dx ,
2s it follows by direct computation. If f is a static solution, then the previous identity implies the virial relation H = − E kin , which yields that H < , for static solutions of the Vlasov-Poisson system. (2)For traveling steady states, we just apply to (2) a Galilean transformation with u = ( M ) − Q andwe obtain H < | Q | M , for traveling steady states of the Vlasov-Poisson system. (3)Our purpose in this work is to extend these fundamental inequalities to the relativistic case. Formore information on the Vlasov-Poisson system, we refer to [11, 14, 17, 19].
In the case of the Nordstr¨om-Vlasov system, which will be presented in Section 2, the generalizationof (2) is that the energy of regular steady states is bounded by their mass, i.e. H ≤ M . (4)Furthermore, the counterpart to (3) for traveling steady states is p H − | Q | ≤ M . (Of course, the invariants H , M , Q have to be redefined in an appropriate way for this new system,see Section 2). Moreover the equality sign could only hold for steady states with unboundedsupport. We remark that in the case of the Vlasov-Poisson system the supremum of the steadystates energy coincide with the infimum energy of totally dispersive time dependent solutions,see [11]. The analogous statement for the Nordstr¨om-Vlasov system is currently not known, dueto the difficulties in defining a Lorentz invariant concept of total dispersion. We also remark thatthe bound H < M , which holds for all regular and compactly supported static solutions of theNordstr¨om-Vlasov system, is crucial in the proof of orbit stability of the polytropic steady statesestablished in [10].For the spherically symmetric Einstein-Vlasov system, analyzed in Section 3, we derive an inequal-ity that involves not only the energy (ADM mass, H ) and the mass (rest mass, M ) of the steadystate, but also the central redshift Z c : Z c ≥ (cid:12)(cid:12)(cid:12)(cid:12) MH − (cid:12)(cid:12)(cid:12)(cid:12) . (5)The metric of the space-time for spherically symmetric static solutions of the Einstein-Vlasovsystem is determined, following the notation in Section 3, by two functions λ ( r ) ≥ µ ( r ) ≤ Z c := e − µ (0) −
1. It is the redshift of a photon emitted from the center of the galaxy. Theestimate (5) can thus be seen as an upper bound for µ (0). Similarly, the celebrated Buchdahl’sinequality in General Relativity [27] can be seen as an upper bound on the metric component λ ( r )for spherically symmetric steady states of the Einstein-matter equations. A quite general versionof the Buchdahl inequality was proved recently in [2] and readssup r ≥ (cid:16) − e − λ ( r ) (cid:17) ≤ , or equivalently sup r ≥ λ ( r ) ≤ ln 3 . (6)For static shells the Buchdahl inequality is equivalent to a lower bound for the external radius.We will show that estimate (5) leads to an upper bound on the internal radius. We refer to [5] for3n analitical/numerical investigation of the Buchdahl inequality in the context of the sphericallysymmetric Einstein-Vlasov system.We do not know whether, as for the Vlasov-Poisson and the Nordstr¨om-Vlasov system, the inequal-ity (5) could also be related to the problem of stability of spherically symmetric static solutions.This is a difficult question to answer, since the stability problem for the Einstein-Vlasov sys-tem is still poorly understood. However it is worth noticing that heuristic and numerical studies[29, 30, 26] indicate that the regime of stability of compact galaxies is indeed characterized bythe central redshift and the fractional binding energy (defined as 1 − H/M ). Moreover it wasconjectured that the binding energy maximum along a steady state sequence signals the onsetof instability. There are several numerical studies on the problem of stability for the sphericallyEinstein-Vlasov system; we refer to [4, 5, 23].A last basic comment on (5) is that, as opposed to the inequalities that hold for steady statesof the Vlasov-Poisson and the Nordstr¨om-Vlasov system, the bound (5) contains a quantity, thecentral redshift, which is not preserved along time dependent solutions. It is therefore not clearwhether one can interpret (5) as the exact analog of the mass-energy inequalities for the steadystates of the Vlasov-Poisson and Nordstr¨om-Vlasov system.We conclude this Introduction with a brief explanation on how we prove our main results. Asa first step we employ the vector fields multipliers method to the local conservation laws for theNordstr¨om-Vlasov and the Einstein-Vlasov system to establish a virial identity which has to besatisfied by all time dependent solutions. These identities are of independent interest and couldbe useful to derive space-time (Morawetz type) estimates for the evolution problem. The virialidentities restricted to time independent solutions give rise, after applying some simple bounds onthe moments of the distribution f , to the virial inequalities (4)-(5). We write the Nordstr¨om-Vlasov system in the formulation used in [10]: ∂ t f + p p e φ + | p | · ∇ x f − ∇ x (cid:18)q e φ + | p | (cid:19) · ∇ p f = 0 , (7a) ∂ t φ − ∆ x φ = − e φ Z R f dp p e φ + | p | . (7b)Here f = f ( t, x, p ) ≥ φ = φ ( t, x ). The physical interpretation of a solution ( f, φ ) is thefollowing: The space-time is the Lorentzian manifold ( R , g = e φ η ), where η is the Minkowskimetric, whereas f is the kinetic distribution function of particles (the stars of a galaxy) movingalong the geodesic curves of the metric g . The motion along geodesics reflects the condition thatgravity is the only interaction among the particles. The system has been written in units suchthat 4 πG = c = 1, where G is Newton’s gravitational constant and c the speed of light. If thelatter is restored in the equations, in the limit c → ∞ one recovers the Vlasov-Poisson systemin the gravitational case. For a proof of the latter statement and general information on theNordstr¨om-Vlasov system we refer to [12]. The global regularity of solutions is studied in [8, 13].The local energy, momentum and stress tensor of a solution ( f, φ ) of (7) are defined respectively4s ( i, j = 1 , , h ( t, x ) = Z R q e φ + | p | f dp + 12 ( ∂ t φ ) + 12 |∇ x φ | ,q i ( t, x ) = Z R p i f dp − ∂ t φ ∂ i φ ,τ ij ( t, x ) = Z R p i p j p e φ + | p | f dp + ∂ i φ ∂ j φ + 12 δ ij (cid:2) ( ∂ t φ ) − |∇ x φ | (cid:3) , where ∂ i denotes the partial derivative along x i . These quantities are related by the conservationlaws ∂ t h + ∇ x · q = 0 , ∂ t q i + ∂ j τ ij = 0 , (9)the sum over repeated indexes being understood. Upon integration, the previous identities lead tothe conservation of the total energy and of the total momentum: H ( t ) = Z R h ( t, x ) dx = constant , Q ( t ) = Z R q ( t, x ) dx = constant . Moreover, solutions of the Nordstr¨om-Vlasov system satisfy the conservation of the total rest mass: M ( t ) = Z R ρ ( t, x ) dx = constant , which is obtained by integrating the local rest mass conservation law ∂ t ρ + ∇ x · j = 0 , ρ = Z R f dp , j = Z R p p e φ + | p | f dp . (10)The system (7) satisfies the fundamental property of Lorentz invariance. Precisely, let ( t ′ , x ′ ) be asystem of coordinates in Minkowski space obtained from ( t, x ) by a Lorentz boost, that is t ′ = u t − u · x , x ′ = x − u t + u − | u | ( u · x ) u , where u is a fixed vector in R and u = p | u | . The inverse Lorentz transformation is obtainedby exchanging u with − u , that is t = u t ′ + u · x ′ , x = x ′ + u t ′ + u − | u | ( u · x ′ ) u , (11)which we shorten by ( t, x ) = L u ( t ′ , x ′ ). Define the field φ u in the new coordinates by φ u ( t ′ , x ′ ) = φ ◦ L u ( t ′ , x ′ ) . Introduce the new momentum variable p ′ = p − u q e φ ( t,x ) + | p | + u − | u | ( u · p ) u or, inverting, p = p ′ + u q e φ ( t,x ) + | p ′ | + u − | u | ( u · p ′ ) u . (12)We shall write ( t, x, p ) = L u ( t ′ , x ′ , p ′ ) to shorten the set of transformations (11)-(12). Finally,define the distribution function in the new variables as f u ( t ′ , x ′ , p ′ ) = f ◦ L u ( t ′ , x ′ , p ′ ) .
5n the language of special relativity, one says that f and φ transform like scalar functions underLorentz transformations. The Lorentz invariance of the Nordstr¨om-Vlasov system means that thepair ( f, φ ) solves the system (7) in the coordinates ( t, x, p ) if and only if ( f u , φ u ) satisfies thesame system in the coordinates ( t ′ , x ′ , p ′ ). Thus, in particular, also the mass-energy-momentum of( f u , φ u ) is conserved along the time evolution, M [ f u ] = constant , H [ f u , φ u ] = constant , Q [ f u , φ u ] = constant . We shall need the relation between the mass-energy-momentum of ( f, φ ) and of ( f u , φ u ), which isderived the following lemma . Lemma 1.
For all u ∈ R , M [ f u ] = M [ f ] , (13a) H [ f u , φ u ] = p | u | H [ f, φ ] − Q [ f, φ ] · u , (13b) Q [ f u , φ u ] = Q [ f, φ ] − H [ f, φ ] u + u − | u | ( u · Q [ f, φ ]) u . (13c) Proof.
Since the mass-energy-momentum of both pairs ( f, φ ) and ( f u , φ u ) is conserved, it is suf-ficient to prove the relations (13) for the initial value of M ( u ) := M [ f u ], H ( u ) := H [ f u , φ u ] and Q ( u ) := Q [ f u , φ u ]. We restrict ourselves to prove the invariance of the total mass, the proof forthe other transformations being similar. We shall need that, by (12), q e φ ( t,x ) + | p ′ | = u q e φ ( t,x ) + | p | − u · p or, inverting, q e φ ( t,x ) + | p | = u q e φ ( t,x ) + | p ′ | + u · p ′ . (14)In order to prove (13a) we write M ( u ) = Z R Z R f u (0 , x ′ , p ′ ) dx ′ dp ′ = Z R Z R f ◦ L u (0 , x ′ , p ′ ) dx ′ dp ′ = Z R Z R f (cid:18) u · x ′ , x ′ + u − | u | ( u · x ′ ) u, p ′ + u q e φ u (0 ,x ′ ) + | p ′ | + u − | u | ( u · p ′ ) u (cid:19) dx ′ dp ′ . Next we make the change of variable x = x ′ + u − | u | ( u · x ′ ) u , p = p ′ + u q e φ u (0 ,x ′ ) + | p ′ | + u − | u | ( u · p ′ ) u . The Jacobian of this transformation is given by J = b u · p ′ + p e φ u (0 ,x ′ ) + | p ′ | p e φ u (0 ,x ′ ) + | p ′ | , where b u = u/u . Using (12) and (14) we obtain J = − b u · p p e φ u (0 ,x ′ ) + | p | ! − . In the language of special relativity, the lemma establishes that M transforms like a scalar function, whereasthe quadruple ( H, Q ) transforms like a four-vector under Lorentz transformations. dx ′ dp ′ = J − dx dp , we obtain M ( u ) = Z R Z R f ( b u · x, x, p ) − b u · p q e φ ( b u · x,x ) + | p | dx dp = Z R ( ρ ( b u · x, x ) − b u · j ( b u · x, x )) dx , where ρ and j are defined by (10). Taking the partial derivative ∂ u i of the previous expression weget ∂ u i M ( u ) = Z R ( ∂ t ρ ∂ u i ( b u · x ) − ( ∂ u i b u k ) j k − b u k ∂ t j k ∂ u i ( b u · x )) ( b u · x, x ) dx = Z R ( − ∂ u i ( b u · x )( ∂ x k j k + b u k ∂ t j k ) − ( ∂ u i b u k ) j k ) ( b u · x, x ) dx = − Z R ( ∂ u i ( b u · x ) ∂ x k [ j k ( b u · x, x )] − ( ∂ u i b u k ) j k ( b u · x, x )) dx = 0 , where we used the continuity equation (10) to pass from the first to the second line and integrationby parts to pass from the third to the last line. Thus we obtained that ∇ u M ( u ) = 0, i.e., M ( u ) = M (0), which yields the claim on the invariance of the total rest mass. Remark 1.
According to the transformation law of the total momentum Q , the Lorentz transfor-mation that makes Q to vanish, i.e. that moves the reference frame to the center of mass system ,is the transformation L u with u = Q/ p H − | Q | . The energy of the transformed solution is p H − | Q | . The conservation laws (9) can be expressed in a more coincise form as ∂ µ T µν = 0 , µ, ν = 0 , . . . , , x = t , (15)where T µν is the stress-energy tensor, whose components are given by T = − h , T i = − q i , T ij = τ ij . Indexes are raised and lowered with Minkowski’s metric η µν = diag( − , , , ξ µ = ξ µ ( t, x ), integrating on a compact spacetime regionΩ with piecewise differentiable boundary ∂ Ω and applying the divergence theorem we obtain theintegral identity Z ∂ Ω T µν ξ ν n µ dσ = Z Ω T µν ∂ µ ξ ν dtdx , (16)where n µ denotes the exterior normal vector field to the boundary ∂ Ω and dσ the invariant volumemeasure thereon. The identities obtained from (16) upon a specific choice of the vector fieldmultiplier go under the general name of virial identities . We prove here one that applies to regularasymptotically flat solutions . By this we mean that f ∈ C , φ ∈ C ∩ L , the mass and energy arefinite and lim R →∞ Z S ( R ) h ( t, x ) dS R = 0 , ∀ t ∈ R . (17) To be more precise, this is called the center of momentum system . In relativity there is no general acceptanceon the concept of center of mass. ω we shall denote the outward unit normal to S ( R ) = { x : | x | = R } , and dS R stands for theinvariant volume measure on S ( R ). Moreover we denote by χ ( r ), r >
0, a function that satisfies: χ ∈ C , χ ′ ∈ L ∞ , χr ∈ C ∩ L ∞ . (18) Lemma 2.
Let I ( t ) = Z R χ ( r ) (cid:0) q · ω − r − φ ∂ t φ (cid:1) dx , r = | x | . For all regular asymptotically flat solutions of (7) the following identity holds: d I dt = Z R χ ′ h dx + Z R χr e φ ( φ − Z R f p e φ + | p | dp dx + Z R (cid:16) χr − χ ′ (cid:17) " | ω ∧ ∇ x φ | + Z R | ω ∧ p | + e φ p e φ + | p | f dp dx − Z R χ ′′ r φ dx . (19) Proof.
In (16) we use Ω = [0 , T ] × B ( R ), where B ( R ) = { x : | x | ≤ R } and ξ µ : ξ = 0 , ξ i = χ ( r ) ω i . We obtain "Z B ( R ) χ ( r ) q · ω dx T = Z T Z S ( R ) χ ( r ) τ ij ω i ω j dS R dt + Z T Z B ( R ) h(cid:16) χ ′ − χr (cid:17) τ ij ω i ω j + χr δ ij τ ij i dx dt , (20)where for any function g ( t ) we denote [ g ( t )] T = g ( T ) − g (0). Using the bound | τ ij ω i ω j | ≤ h and (17) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S ( R ) χ ( r ) τ ij ω i ω j dS R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k χ k ∞ Z S ( R ) h dS R → , R → ∞ . Then, letting R → ∞ in (20) we obtain (cid:20)Z R χ ( r ) q · ω dx (cid:21) T = Z T Z R h(cid:16) χ ′ − χr (cid:17) τ ij ω i ω j + χr δ ij τ ij i dx dt , whence ddt Z R χ ( r ) q · ω dx = Z R h(cid:16) χ ′ − χr (cid:17) τ ij ω i ω j + χr δ ij τ ij i dx . We compute δ ij τ ij = Z R | p | f dp p e φ + | p | + 32 ( ∂ t φ ) − |∇ x φ | and τ ij ω i ω j = Z R ( ω · p ) f dp p e φ + | p | + ( ω · ∇ x φ ) + 12 [( ∂ t φ ) − |∇ x φ | ] . ddt Z R χ ( r ) q · ω dx = Z R χr "Z R | p | f dp p e φ + | p | + 32 ( ∂ t φ ) − |∇ x φ | dx + Z R (cid:16) χ ′ − χr (cid:17) "Z R ( ω · p ) f dp p e φ + | p | + ( ω · ∇ x φ ) + 12 [( ∂ t φ ) − |∇ x φ | ) dx . Using that | ω ∧ y | = | y | − | ω · y | , for all vectors y ∈ R , we can rewrite the previous equation as ddt Z R χ ( r ) q · ω dx = Z R χr (( ∂ t φ ) − |∇ x φ | ) dx − Z R χ ′ Z R e φ f dp p e φ + | p | dx + Z R χ ′ h dx + Z R (cid:16) χr − χ ′ (cid:17) Z R | ω ∧ p | f dp p e φ + | p | + | ω ∧ ∇ x φ | ! dx . (21)Moreover, using (7b) and integrating by parts twice, we find ddt Z B ( R ) χr φ ∂ t φ dx = Z B ( R ) χr ( ∂ t φ ) − |∇ x φ | − φ Z R e φ f dp p e φ + | p | ! dx + 12 Z B ( R ) ∆ (cid:16) χr (cid:17) φ dx + Z S ( R ) χr φ ω · ∇ x φ dS R − Z S ( R ) ω · ∇ (cid:16) χr (cid:17) φ dS R . (22)Applying the Cauchy-Schwartz inequality, the regularity of the solution and the assumptions on χ , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S ( R ) χr φ ω · ∇ x φ dS R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k L ( S ( R )) k∇ x φ k L ( S ( R )) ≤ C sZ S ( R ) h ( t, x ) dS R → , R → ∞ , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z S ( R ) ω · ∇ (cid:16) χr (cid:17) φ dS R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 R Z S ( R ) (cid:12)(cid:12)(cid:12) χ ′ − χr (cid:12)(cid:12)(cid:12) φ dS R ≤ CR , where C is a constant independent from R . Thus taking the limit R → ∞ in (22) we get − ddt Z R χr φ ∂ t φ dx = − Z R χr ( ∂ t φ ) − |∇ x φ | − φ Z R e φ f dp p e φ + | p | ! dx − Z R ∆ (cid:16) χr (cid:17) φ dx . (23)The quantity ∆ (cid:0) χr (cid:1) is nothing but χ ′′ r . The sum of (21) and (23) yields the desired result.9 .2 Virial inequalities for steady states As in the Vlasov-Poisson case, we distinguish between two types of steady states. Static solutions,which are defined as time independent solutions of the Nordstr¨om-Vlasov system (7), and travelingsteady states, which are defined as solutions f ( t, x, p ) such that f ◦ L u , where u = Q/ p H − | Q | ,is a time independent solution of the Nordstr¨om-Vlasov system (i.e., a static solution). For staticsolutions one has Q = 0, whereas Q = 0 for traveling steady states. Note that for static solutions(that vanish at infinity) the field is determined by f through a non-linear Poisson equation. Thuswhen we refer to a steady state solution we mean simply the distribution function f . The maingoal of this section is to prove the following property of steady states to the system (7). Theorem 1.
Let f be a static regular asymptotically flat solution of (7) . Then H ≤ M . (24)
Traveling steady states satisfy p H − | Q | ≤ M . Moreover, equality in (24) implies that thesupport of the static solution is unbounded.Proof. The statement on traveling steady states follows by applying the Lorentz transformation L u with u = Q/ p H − | Q | to the inequality for static solutions, thus it suffices to prove thelatter. To this purpose consider a function χ that, in addition to (18), satisfies χr − χ ′ ≥ , χ ′′ ≤ . (25)Next we observe the simple inequality y − ≥ − e − y , with equality if and only if y = 0. Using φ − ≥ − e − φ (26)in the identity (19) we obtain d I dt ≥ Z R (cid:16) χ ′ h − χr ρ (cid:17) dx . (27)In particular, for time independent solutions we have Z R (cid:16) χ ′ h − χr ρ (cid:17) dx ≤ . (28)Let R > χ ( r ) = χ R ( r ) given by χ ( r ) = (cid:26) r for r R , R − R r + R r for r > R . This function satisfies the properties (18) and (25). The left hand side of (28) becomes Z R (cid:16) χ ′ h − χr ρ (cid:17) dx = Z B ( R ) ( h − ρ ) dx + Z B ( R ) c (cid:16) χ ′ h − χr ρ (cid:17) ≥ Z B ( R ) ( h − ρ ) dx − C Z B ( R ) c h dx + Z B ( R ) c ρ dx ! = Z B ( R ) ( h − ρ ) dx + ε ( R ) , (29)where ε ( R ) → R → ∞ . Thus, assuming H > M , there exists R > ε ( R ) < ( H − M ) / R B ( R ) ( h − ρ ) dx > ( H − M ) /
2, for all
R > R , whence Z R (cid:16) χ ′ h − χr ρ (cid:17) dx >
14 ( H − M ) > , φ nevervanishes in the support of f (it is strictly negative) and thus the stronger inequality φ − > e − φ holds instead of (26). Thus also the inequality in (28) is strict. Since the last member of (29) goesto zero for R → ∞ when H = M , the claim follows. Remark 2.
Theorem 1 improves a similar result proved in [10] in two aspects. Firstly, in [10] thefact that the strict inequality holds for compactly supported steady states was overlooked. Secondlythe result presented here requires less decay than the inequality proved in [10] and therefore appliesto more general steady states. In particular, this result allows to remove some technical hypothesisin the stability result obtained in [10].
The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates is given by thefollowing set of equations (in units G = c = 1): ∂ t f + e µ − λ v p | v | · ∇ x f − (cid:16) λ t x · vr + e µ − λ µ r p | v | (cid:17) xr · ∇ v f = 0 , (30) e − λ (2 rλ r −
1) + 1 = 8 πr h , (31a) e − λ (2 rµ r + 1) − πr p rad , (31b) λ t = − πre λ + µ q , (31c) e − λ (cid:18) µ rr + ( µ r − λ r )( µ r + 1 r ) (cid:19) − e − µ ( λ tt + λ t ( λ t − µ t )) = 4 πp tan , (31d)where h ( t, r ) = Z R p | v | f dv , p rad ( t, r ) = Z R (cid:16) x · vr (cid:17) f dv p | v | ,q ( t, r ) = Z R x · vr f dv , p tan ( t, r ) = Z R (cid:12)(cid:12)(cid:12) x ∧ vr (cid:12)(cid:12)(cid:12) f dv p | v | . The functions p rad and p tan are the radial and tangential pressure; h is the energy density and q the local momentum density . As usual, f ≥ t ∈ R , x ∈ R , v ∈ R . The variable v is not the canonicalmomentum of the particles, the latter being denoted by p in the previous sections. The function f is spherically symmetric in the sense that f ( t, x, v ) = f ( t, Ax, Av ), for all A ∈ SO (3). The symbol ∧ denotes the standard vector product in R , and for a function g = g ( t, r ), r = | x | , we denoteby g t and g r the time and radial derivative, respectively. By abuse of notation, g ( t, r ) = g ( t, x )for any spherically symmetric function. The functions λ, µ determine the metric of the space-timeaccording to ds = − e µ dt + e λ dr + r dω , (32)where dω is the standard line element on the unit sphere. The system is supplied with theboundary conditions lim r →∞ λ ( t, r ) = lim r →∞ µ ( t, r ) = λ ( t,
0) = 0 , (33) We adopt the same notation as in the previous sections, although this differs from the standard notation forthe Einstein-Vlasov system. ≤ f (0 , x, v ) = f in ( x, v ) , f in ( Ax, Av ) = f in ( x, v ) , ∀ A ∈ SO (3) . The reader is referred to [18, section 1.1] for a detailed derivation of the system. Throughout thispaper we assume that f is a regular solution of (30)-(31) in the sense defined in [20]. In particular, f ( t, x, v ) is C and has compact support in ( x, v ), for t ∈ [0 , T ], and for any T >
0. For regularsolutions, the metric coefficients are C functions of their arguments. We emphasize that theexistence and uniqueness of global regular solutions to the Cauchy problem for the system (30)-(31) is open for general initial data. We also remark that the equation λ r + µ r = 4 πre λ ( h + p rad ) , (34)follows by (31a)-(31b); by (34) we have λ r + µ r > > λ + µ > µ (0 , t ) . (35)The ADM mass (or energy) H and the total rest mass M of a solution to the spherically symmetricEinstein-Vlasov system are defined by H = Z R Z R p | v | f dv dx , M = Z R Z R e λ f dv dx (36)and are constant for regular solutions . Related to the ADM mass we have the quasi-local mass,defined by m ( t, r ) = 4 π Z r s h ( t, s ) ds = r (cid:0) − e − λ (cid:1) , (37)where we used (31a). Thus lim r →∞ m ( t, r ) = H .For later convenience, we recall that the non-zero Christoffel symbols for the metric (32) are givenby Γ = µ t , Γ a = µ r x a r , Γ ab = e λ − µ ) λ t x a x b r , Γ a = e − λ − µ ) µ r x a r , Γ a b = λ t x a x b r , Γ cab = λ r x c x b x a r + 1 − e − λ r (cid:18) δ cb − x b x c r (cid:19) x a r . Note also that | g | = e λ +2 µ is the determinant of the metric.The stress-energy tensor T µν for Vlasov matter in spherical symmetry is given by T = e − µ h , T a = e − λ − µ q x a r , (38a) T ab = e − λ p rad x a x b r + 12 p tan (cid:18) δ ab − x a x b r (cid:19) (38b)and satisfies the conservation law ∇ µ T µν = 0 . (39) The other two conserved quantities, the linear momentum Q and angular momentum L , considered in theprevious sections are identically zero in the present context by spherical symmetry. The identities (39) are a consequence of the Vlasov equation alone, see [15]. .1 Virial identities for time dependent solutions To begin with we derive an integral identity for the spherically symmetric Einstein-Vlasov systemas we did in Lemma 2 for the Nordstr¨om-Vlasov system, i.e., using the vector fields multipliersmethod. Actually, the identity in Lemma 3 below is valid not only for the Einstein-Vlasov system,but for all matter models in spherical symmetry. This is due to the fact that equation (39), whichis the starting point for deriving the integral identity, must be satisfied by all matter models forcompatibility with the Einstein equations.Multiplying the conservation law (39) by a vector field ξ µ , integrating on a compact spacetimeregion Ω with piecewise differentiable boundary ∂ Ω and applying the divergence theorem we obtainthe integral identity Z ∂ Ω J µ η µ dσ g = Z Ω T µν ∇ µ ξ ν dg , (40)where η µ is the normal covector related to the boundary, J µ = T µν ξ ν = T µν ξ ν is the currentassociated to the vector field ξ µ and ∇ µ ξ ν = ∂ µ ξ ν − Γ σµν ξ σ is the covariant derivative of the vectorfield. Moreover dg is the invariant volume element on the spacetime and dσ g the invariant volumeelement induced on ∂ Ω. Lemma 3.
Assume that ( h, q, p rad , p tan ) satisfy the compatibility condition (39) , where T µν isthe stress-energy tensor (38) . In addition, we assume that h ( t, · ) , q ( t, · ) , p rad ( t, · ) , p tan ( t, · ) , havecompact support. Given any smooth function χ ( t, r ) in W , ∞ loc and any solution of (31) define I ( t ) = Z R χ q ( t, r ) dx . Then the following integral identity is verified: d I dt = Z R (cid:20) e µ − λ p rad ∂χ∂r − e µ − λ χ (cid:18) hµ r + p rad λ r − p tan r (cid:19) + q (cid:18) ∂χ∂t − χλ t (cid:19)(cid:21) dx . (41) Proof.
In (40) we use ξ = 0 ,ξ i = χ ( t, r ) x i r . After a long but straightforward computation we obtain T µν ∇ µ ξ ν = e − λ p rad ∂χ∂r + e − λ − µ q ∂χ∂t − χ (cid:20) e − λ hµ r + 2 qλ t e − λ − µ + e − λ p rad λ r − p tan e − λ r (cid:21) . We will choose Ω to be the coordinate image of a cylinder [0 , T ] × B ( R ). In this fashion, we havethat Z Ω T µν ∇ µ ξ ν dg = Z T Z | x |≤ R (cid:20) e µ − λ p rad ∂χ∂r − e µ − λ χ (cid:18) hµ r + p rad λ r − p tan r (cid:19) + q (cid:18) ∂χ∂t − χλ t (cid:19)(cid:21) dx dt. (42)Now we compute the corresponding boundary integral in (40). First, the current reads J = qχe − λ − µ ,J a = e − λ p rad χ ( r ) x a r . Next we write ∂ Ω = A ∪ A ∪ A , where In the case of a perfect fluid, the compatibility condition (39) is the system of Euler equations. A = { t = T, | x | ≤ R } . The outer unit normal is e µ ( r,T ) dt ; the induced metric is e λ ( r,T ) dr + r dω , the volume element e λ ( r,T ) dx . • A = { t = 0 , | x | ≤ R } . The outer unit normal is − e µ ( r, dt ; the induced metric is e λ ( r, dr + r dω , the volume element e λ ( r, dx . • A = { < t < T, | x | = R } . The outer unit normal has the form − e λ ( t,R ) x i R dx i . The metricis ds = − e µ ( t,R ) dt + R dω , the volume element e µ ( t,R ) dS R dt , where dS R is the surfaceelement on the sphere of radius R .Summing up we get Z ∂ Ω J µ η µ dσ g = Z | x |≤ R q ( T, r ) χ ( T, r ) dx − Z | x |≤ R q (0 , r ) χ (0 , r ) dx − Z T Z | x | = R p tan ( t, R ) χ ( t, R ) e µ ( t,R ) − λ ( t,R ) dS R dt . (43)Having assumed that the matter quantities are compactly supported in the variable r , the boundaryintegral vanishes in the limit R → ∞ , whereas the other integrals remain bounded . Thus in thelimit we obtain (cid:20)Z R qχ dx (cid:21) T = Z T Z R (cid:20) e µ − λ p rad ∂χ∂r − e µ − λ χ (cid:18) hµ r + p rad λ r − p tan r (cid:19) + q (cid:18) ∂χ∂t − χλ t (cid:19)(cid:21) dx dt , which is the integral version of (41).We shall now derive two particular cases of the identity (41). First let us choose χ = e λ F ( r ) , for a smooth function F . We have ∂ t χ = 2 λ t e λ F ( r ) and then ∂ t χ − χλ t = 0. In this way equation(41) implies (cid:20)Z R qe λ F dx (cid:21) T = Z T Z R e µ + λ (cid:20) p rad F ′ + F (cid:18) λ r p rad − hµ r + p tan r (cid:19)(cid:21) dx dt . (44)Note now that, using (34), (31b) and (37), λ r p rad − hµ r = p rad ( λ r + µ r ) − µ r ( p rad + h ) = ( λ r + µ r ) (cid:18) p rad − µ r e − λ πr (cid:19) = − m πr ( λ r + µ r ) . Then (44) becomes (cid:20)Z R qe λ F dx (cid:21) T = Z T Z R e µ + λ (cid:20) p rad F ′ + p tan Fr − Fr m ( λ r + µ r )4 πr (cid:21) dx dt . (45)In the integral in the right hand side we use that − Z T Z R e µ + λ Fr ( λ r + µ r ) m πr dxdt = − Z T Z ∞ de λ + µ dr Fr m dr dt Of course the compact support condition can be replaced by a suitable decay assumption. Z T Z ∞ ddr (cid:18) F mr (cid:19) e λ + µ drdt − HT (cid:18) lim r →∞ F ( r ) r (cid:19) . This leads to (cid:20)Z R qe λ F dx (cid:21) T = − HT (cid:18) lim r →∞ F ( r ) r (cid:19) + Z T Z R e λ + µ (cid:20) p rad F ′ + p tan Fr + h Fr + m πr ddr (cid:18) Fr (cid:19)(cid:21) dx dt . (46)Finally for F ( r ) = r we obtain (cid:20)Z R qe λ r dx (cid:21) T = − HT + Z T Z R e λ + µ (cid:0) p rad + p tan + h (cid:1) dx dt . (47) The existence of steady states solutions to the Einstein-Vlasov system is well understood, we referto [5, 16] and the references therein. The identity (47) restricted to steady states imply H = Z R e λ + µ ( p tan + p rad + h ) dx . (48) Remark 3.
The fundamental identity (48) can be proved directly using the Einstein equationsfor static spherically symmetric spacetimes, see [3]. Our derivation has two advantages. Firstly, weobtained (48) as a special case of a more general identity which holds for time dependent solutions,see Lemma 3. Secondly, the technique of the vector fields multipliers, which we used to derive (48),can also be used on spacetimes which are not spherically symmetric and therefore our argumentcould be useful to prove generalizations of (48) for solutions with less symmetry.This identity leads naturally to a bound on the central redshift Z c = e − µ (0) − ∈ [0 , + ∞ )in terms of the mass-energy of the static solution. We consider only static solutions of the spheri-cally symmetric Einstein-Vlasov system. Proposition 1.
Let f be a static solution of the spherically symmetric Einstein-Vlasov systemwith compact support. Then the following inequality holds true e µ (0) ≤ HM if H ≤ M H H − M if H ≥ M i.e Z c ≥ (cid:12)(cid:12)(cid:12)(cid:12) MH − (cid:12)(cid:12)(cid:12)(cid:12) . (49) Proof.
Since µ is increasing, µ ( r ) ≥ µ (0) and so Z R e λ + µ ( p rad + p tan + h ) ≥ e µ (0) Z R e λ h ≥ M e µ (0) . Using this in (48) gives e µ (0) ≤ HM . (50)15oreover p rad + p tan + h = h + Z R (cid:16) x · vr (cid:17) f dv p | v | + Z R (cid:12)(cid:12)(cid:12) x ∧ vr (cid:12)(cid:12)(cid:12) f dv p | v | = 2 h + Z R | v | p | v | − p | v | ! f dv = 2 h − Z R f dv p | v | Thus, since λ + µ > µ (0) and e µ ≤ ≤ e λ , Z R e λ + µ ( p rad + p tan + h ) dx ≥ e µ (0) (2 H − M )and so by (48), e µ (0) ≤ H H − M , when
HM > . (51)The result follows from (50) and (51) and taking into account that HM ≤ H H − M is satisfied in the case < HM ≤ Let R be the radius support of the steady state. Using that µ is negative and increasing and thatthe steady state matches the Schwarzschild solution at r = R we obtain the bound e µ (0) ≤ e µ ( R ) = r − HR . (52)The inequality (52) can be combined with (49) to obtain an upper bound on e µ (0) in terms of R , H and M .Now, we consider briefly an important class of steady states, namely the Jeans type steady states,see [28]. For these steady states the distribution function f has the form f ( x, v ) = ψ ( E, F ) , where E = e µ p | v | , F = | x ∧ v | . (53)Since the particles energy E and the angular momentum F are conserved quantities, the particledensity (53) is automatically a solution of the Vlasov equation. The existence of Jeans type steadystates is then obtained by replacing the ansatz f = ψ ( E, F ) into the (time independent) Einsteinequations and proving existence of global solutions for the resulting system of ODEs. We referto [22] where this procedure is carried out for a large class of profiles ψ ; moreover the Jeans typesteady states constructed in [22] all have compact support and satisfy that ∃ E ∈ (0 ,
1) such that ψ = 0 , for E ≥ E . (54)Thus E is the maximum particle energy in the ensemble. The property (54) is necessary in orderthat the distribution function (53) be asymptotically flat and with finite energy. For Jeans typesteady states one obtains a new estimate on e µ (0) in a straightforward way: H = Z R Z R p | v | f dv dx = Z R Z R e − λ − µ e µ p | v | e λ f dv dx ≤ E e µ (0) M , e µ (0) ≤ E MH . (55)In fact, for Jeans’ type solutions we have [22] E = r − HR and thus, combining (55) with (52) we conclude that for Jeans’ type steady states the inequality e µ (0) ≤ e µ ( R ) = min (cid:26) , MH (cid:27) r − HR hods.Consider now the case of a static shell. Let f be a static shell solution of the spherically symmetricEinstein-Vlasov system with inner radius R and outer radius R . Using (31b) we can write µ (0)as follows µ (0) = − Z ∞ e λ (cid:16) mr + 4 πrp rad (cid:17) dr = − Z ∞ − m/r (cid:16) mr + 4 πrp rad (cid:17) dr . By Buchdahl’s inequality (6), the identity (37) and the bound p rad ≤ h , we obtain µ (0) ≥ − Z ∞ R (cid:16) mr + 4 πrp rad (cid:17) dr ≥ − Z ∞ R (cid:18) Hr + 4 πrh (cid:19) dr ≥ − HR − R Z ∞ R πr h dr = − HR . Now, we use the upper estimates on µ (0) of the Proposition 1 to find µ (0) = ln (cid:18) Z c + 1 (cid:19) ≤ ln (cid:12)(cid:12) MH − (cid:12)(cid:12) + 1 ! . Combining both estimates we obtain R ≤ H ln (cid:0)(cid:12)(cid:12) MH − (cid:12)(cid:12) + 1 (cid:1) , i.e., the inner radius of a static shell with given ADM energy and rest mass cannot be arbitrarilylarge. References [1] H. Andr´easson:
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