Turning point principle for relativistic stars

Upon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960's Zel'dovich [50] and Wheeler [22] formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity.


Introduction
In this work we rigorously establish the turning point principle for radial relativistic stars along the so-called mass-radius curve of 1-parameter family of stationary solutions, see Theorem 1.13. This principle was formulated by Zel'dovich [50] and Wheeler, see [22] (pages 60-66), and it is also referred to as the M(R)-method. In the radial setting this is a powerful tool predicting the exact number of unstable eigenmodes for the linearized radial Einstein-Euler system around its dynamic equilibria, based solely on the the location of the equilibrium on the mass-radius curve.
In our previous work [18] jointly with Rein, among other things we introduced the so-called separable Hamiltonian formulation of the linearized Einstein-Euler system, which highlights the symplectic structure in the problem. This proved crucial to a refined understanding of the linearized flow and its decomposition into invariant subspaces, where we used a general framework developed recently by Lin and Zeng [30]. The second main result of this paper is a precise index formula which expresses the number of unstable modes as the difference of the negative Morse index of a certain Schrödinger type operator (1.39) and a quantity we call the winding index which reflects the winding properties of the mass-radius curve, see Definition 1.9 and Theorem 1.11. This result completes a related result from [18] by including equilibria with certain exceptional values of the central redshift parameter.
(1.13) (P3) There exist the inverse of P on [0, ∞) and constants 0 < c 2 s ≤ 1, c 2 > 0 such that |p − c 2 s ρ| ≤ c 2 p 1/2 for all p > 0. (1.14) (P4) For any ρ > 0 we have This is a causality assumption and states that the speed of sound inside the star never exceeds the speed of light.
Assumptions (P1)-(P4), or some qualitatively similar version of those, are quite commonly used in the description of gaseous stars in relativistic astrophysics, see [23,31,18] and references therein. For a detailed study of the equations of states for neutron stars see [20]. Assumption (P2) states that in the region close to vacuum (0 < ρ ≪ 1) the equation of state is effectively described by the classical polytropic power law P (ρ) = kρ γ . On the other hand, in the regime where the density is very large (ρ ≫ 1) assumption (P3) states that to the leading order P (ρ) = c 2 s ρ. Here 0 < c s ≤ 1, which is also a consequence of the causality assumption stated in assumption (P4). We also observe that assumptions (P1) and (P4) imply that P (ρ) ≤ ρ.
We shall refer to the system of equations (1.6)-(1.11) together with assumptions (P1)-(P4) as the spherically symmetric Euler-Einstein system and use the abbreviation EEsystem.
We introduce the unknown y(r) = const. − µ(r), so that ρ can now be expressed through The field equation (1.6) with u = 0 can be rewritten in the form ∂ r r − e −2λ r = 8πr 2 ρ, which immediately yields (1.20) Plugging the above into the field equation (1.7) with u = 0, we finally obtain the fundamental steady state equation satisfied by y: Here p is given in terms of y by the relations p(r) = h(y(r)) = P (g(y(r))). (1.22) The existence of compactly supported steady states follows for example from the work of Ramming and Rein [37], which we state in the following proposition for readers' convenience. there exists a unique smooth solution y = y κ to (1.21), which is defined on [0, ∞) and has a unique zero at some radius R κ > 0. The value R κ is the radius of the star.
Remark 1.2. The existence of compactly supported radial steady star solutions to the Einstein-Euler system is well-known, see [23,34,42,37] and references therein. The assumptions on the equation of state, in particular the lower bound on γ in (P1) can be relaxed, and the finite extent property can also be shown in different ways [34,23,37].
Given y κ , we define ρ κ and λ κ via (1.19) and (1.20) respectively. The metric coefficient µ κ is then obtained through the formula (1.24) For any κ > 0 we refer to the triple (ρ κ , µ κ , λ κ ) as the steady state of the Euler-Einstein system. Therefore κ and z are in a 1-1 relationship and, by slight abuse of terminology, we continue to call κ the central redshift parameter.
At the heart of our analysis is the formulation of the linearized flow as a separable Hamiltonian system derived in [18]. The natural function spaces contain weights that for each κ > 0 depend on the solution (ρ κ , µ κ , λ κ ). An important role is played by the quantity It is easy to check using property (P1) that the function It is in fact slightly better -a simple consequence of the Hopf lemma is that close to the star boundary ρ ∼ (R κ − r) 1 γ−1 and therefore where we have used (1.27) and the property (P2). Note that we abused the notation slightly by denoting Ψ −1 κ the extension of the reciprocal of Ψ κ on [0, R κ ) to [0, ∞).
(a) The Hilbert space X κ is the space of all spherically symmetric functions in the weighted L 2 space on the set B κ = B Rκ (the ball with radius R κ which is the support of ρ κ ) with weight e 2µκ+λκ Ψ κ and the corresponding inner product, Y κ is the space of radial functions in L 2 (B κ ), and the phase space for the linearized Einstein-Euler system is X κ × Y κ . where ρ is extended by 0 to the region r > R κ .
As proved in [18] (Section 5.2) the formal linearization of the spherically symmetric Einstein-Euler system takes the separable Hamiltonian form where (ρ, v) ∈ X, and κ are anti-self-dual and self-dual respectively. Here the operators A κ : Y * κ → X κ and its dual A ′ κ : X * κ → Y κ are given by Operators A κ and A ′ κ are densely defined and closed, see Section 5.2 of [18]. The conserved energy associated with (1.33) is given by It is important to note that the first order formulation (1.33) can be equivalently replaced by a second order formulation is positive definite. In fact, the number of negative eigenvalues corresponds to the negative Morse index associated with the operator L κ R(Aκ) . The space R(A κ ) ⊂ X κ corresponds to the set of all dynamically accessible perturbations. It is not hard to see (Section 5.3 of [18]) that Remark 1.5. A simple consequence of the above discussion is the Chandrasekhar stability critetrion [8,18]: steady state (ρ κ , µ κ , λ κ ) is spectrally stable if and only if L κ ρ, ρ ≥ 0 for all ρ ∈ X κ with mean 0. A crucial tool in our proof of the turning point principle is the so-called reduced operator discovered in [18]: (1.40) The operator Σ κ :Ḣ 1 r → (Ḣ 1 r ) * is selfdual. From (1.29) and (1.30) it is clear that ∆ κμρ = e µκ+λκ ρ for any ρ ∈ X κ and therefore (1.41) One of the key properties of the reduced operator is described in the following lemma.
The Newtonian limit of the Einstein-Euler system is the well-known gravitational Euler-Poisson system:ρ + div(ρu) = 0 (1.43) Here ρ is the fluid density, u the Newtonian 3-velocity, and φ the gravitational potential satisfying the Poisson equation (1.45). Upon specifying an equation of state p = P (ρ) one finds 1-parameter family of radial equilibria. The most famous among them are the compact Lane-Emden stars, associated with the so-called polytropic equation of state The linear stability of Lane-Emden stars is a classical topic in astrophysics [7] and they also play an important role in our work as suitably rescaled limiting objects in the Newtonian limit κ → 0, see Lemma 2.2. For general (non-polytropic) equations of state, the stability analysis is considerably more complicated due to the absence of exact scaling invariance. In a recent work Lin and Zeng [30] showed that for a very general class of equations of state allowing for compact equilibria, essentially the same turning point principle as proposed by Wheeler applies. In fact, our strategy in this paper is based on analogous steps to [30]. Of central importance in the proof of the turning point principle for the Euler-Poisson system is the the Newtonian limit of the operator Σ κ given by The subscript + in f + refers to the positive part of the function f . Since 1 < γ < 2 we have α > 1 and therefore g 0 is a C 1 -function. It is in particular shown in [30] that The operator Σ 0 can indeed be viewed as the Newtonian limit (κ → 0 + ) of the sequence of operators (Σ κ ) κ>0 . This is a consequence of Lemma 2.2 and Corollary 2.3.
Remark 1.8. In the context of the Euler-Poisson system, operator Σ 0 is the reduced operator associated with the Lane-Emden steady states with equation of state p = kρ γ . Definition 1.9 (Winding index). An important quantity in our analysis is the winding index i κ : to the radially symmetric Einstein-Euler system.
(i) The number of growing modes n u (κ) of the linearized EE-system around a steady state (ρ κ , λ κ , µ κ ) is given by the formula where i κ is given in Definition 1.9 and n − (Σ κ ) is the negative Morse index of the operator Σ κ .
(ii) The eigenvalues of the linearized system are discrete with finite multiplicity.
Remark 1.12. The discreteness of the spectrum can also be obtained using Sturm-Liouville type methods. The formulation can be essentially read off from Chandrasekhar's pioneering work [8], for mathematically rigorous treatment see for example the work of Makino [33]. Our proof of discreteness in Theorem 1.11 proceeds by a different method and capitalizes crucially on the separable Hamiltonian structure of the linearized operator. The same strategy has been used in the Euler-Poisson case [30] and it is a generally applicable procedure to other systems enjoying the separable Hamiltonian structure.
to the radially symmetric Einstein-Euler system.

1.
Turning Point Principle. The number of growing modes n u (κ) can only change at the extrema of the mass function κ → M κ . At an extremum of κ → M κ , n u (κ) increases by 1 if the sign of d dκ M κ d dκ R κ changes from − to + as κ increases, and similarly it decreases by 1 if the sign of d dκ M κ d dκ R κ changes from + to − as κ increases. Geometrically this implies that we "gain" a growing mode if the mass-radius curve bends counter-clockwise at the extremum of κ → M κ , and we "lose" a growing mode if the mass-radius curve bends clockwise at the extremum of κ → M κ . Here the horizontal axis corresponds to the star radius.
2. The number of growing modes goes to infinity as κ goes to infinity, i.e. (1.53) Remark 1.14. In the physics literature the onset of the "higher and higher order instabilites" [47] as κ → ∞ for static stars with extremely dense cores was first pointed out by Dimitriev and Holin [9] in 1963, as well as Harrison [21] and Wheeler [22].
13 is a (considerable) strengthening of a result in [18], where it was shown that for κ ≫ 1 sufficiently large we have n u (κ) ≥ 1.
One of the central outcomes in the analysis of the Euler-Einstein system in [18] is Theorem 5.20 which proves the existence and the associated exponential trichotomy decomposition of the phase space for the linearized flow (1.33). Theorem 5.20 in [18] is Figure 1: Schematic sketch of a possible mass-radius curve based on the physics literature, see e.g. [22,49,48]. Blue portions correspond to spectrally stable equilibria. The first three local extrema of κ → M κ are labelled by A, B, C. Starting with κ ≪ 1 small to the very right of the curve, equilibria are stable until we reach the first maximum of M κ at the point A (the "white dwarf" region). By Theorem 1.13 equilibria between A and B have 1 growing mode, the branch between B and C is again stable (the "neutron star" region), and all the equilibria pass the point C are unstable. The far left region of the graph features the infinite mass-radius spiral which leads to ever-increasing number of growing modes as it bends counterclockwise.
however not complete, as it does not address the steady states whose central redshifts satisfy the nongeneric condition For readers' convenience we state the complete version of the theorem, however, we only briefly sketch the proof in Section 3 as it follows closely the arguments in [18,30].
Theorem 1.16 (Exponential trichotomy). Let the equation of state ρ → P (ρ) satisfy assumptions (P1)-(P4) and let (ρ κ , µ κ , λ κ ) be the 1-parameter family of steady states given by Proposition 1.1. Then for any κ > 0 the operator J κ L κ generates a C 0 group e tJ κ L κ of bounded linear operators on X κ × Y κ and there exists a decomposition with the following properties: where i κ is defined by (1.51).
(ii) The quadratic form (L κ ·, ·) X vanishes on E u,s , but is non-degenerate on E u ⊕ E s , and In the generic case d dκ M κ = 0, we have k 0 = 0 and therefore the flow is Lyapunov stable on the center space E c . Remark 1.17. Invariant subspaces and the exponential trichotomy are important for a refined description of the dynamics in the vicinity of the equilibria. Our result is closely related to the criticality picture emerging in the description of contrasting dynamics near nontrivial steady states, which is largely based on numerical and heuristic arguments [14]. In the context of neutron stars, Noble and Choptuik [35] numerically probed the dynamics near the unstable equilibria of the Einstein-Euler system, using the initial velocity and the central density (or equivalently κ) to parametrize their perturbations. The resulting dynamic picture is very rich, and leads to collapsing, dispersive, and time-periodic solutions with data starting out close to unstable equilibria.
As already explained, the statement of Theorem 1.13 goes back to Zel'dovich [50] and Wheeler [22], see also Section 10.11 of the book by Zel'dovich and Novikov [52]. It is also referred to as the static criterion or the static approach [5,46] which, as formulated in the original work of Zel'dovich [50], asserts that a growing mode is gained or lost at the extrema of the κ → M (κ) curve -specifically only at the maxima and minima and at no other extrema [46]. The word "static" is used, as the stability can be read off from the location of the equilibrium on the mass-radius curve, which are natural astrophysical observables; in the process we avoid potentially cumbersome eigenvalue computations [47]. Our formulation of this principle follows closely the one in [22]. In 1965 Thorne [47] gave a more precise version of Wheeler's Turning Point Principle, and provided heuristic arguments for the main conclusion of part (ii) of Theorem 1.13. In 1970 Calamai [6] similarly gave a more refined argument for the static approach to stability. Various heuristic treatments of the "static approach" can be found in the textbook by Shapiro and Teukolsky [41] and Straumann [45].
The first comprehensive treatments of the (linear) stability study of the isentropic relativistic dynamic equilibria (stars) started with the seminal contribution of Chandrasekhar [8], which after the pioneering work of Oppenheimer and Volkov from 1939 [36] gave a big boost to the study of dynamic stability properties of stars. For a historical overview we refer the reader to the summer school notes of Thorne [47] and the review paper of Bisnovatyi-Kogan [5]. Chandrasekhar [8] linearized the problem in the co-moving coordinates and formulated the spectral stability problem in terms of a suitable Rayleigh-Ritz minimization principle for the eigenvalues of the linearized operator. An alternative, purely "Eulerean" characterization of spectral stability was derived by Thorne in [22] in terms of the second variation of the ADM-mass M under the constraint of constant total particle number N . For more details we point the reader to [18] and references therein. At the same time as Wheeler's work on turning point principle [22] Bardeen [3] proposed a slightly different turning point principle for so-called hot stars (where the thermodynamic temperature is not zero), also relying on the M (R)-diagram. A nice overview is given by Bardeen,Thorne,and Meltzer [4], where both the spectral stability of a single star, as well as their behaviour along the mass-radius curve is discussed.
When studying the stability of self-gravitating systems, a distinction is made between the dynamic stability/instability -studied in this paper -and the thermodynamic stability/instability, see the work of Green, Schiffrin, and Wald [15] for an extensive discussion. The latter instability sets in when an energy-like quantity -typically the entropy -can be infinitesimally increased with perturbations that keep other relevant conserved quantities infinitesimally zero. This notion of stability is in general not equivalent to dynamic stability, but one can often formulate turning point principles along 1-parameter family of equilibria where entropy, or a binding energy is plotted against some other relevant conserved quantity. A general criterion for determining turning point instabilities in this context was given by Sorkin [43,44], which was later applied to the study of thermodynamic (in)stability of axisymmetric stars by Friedman, Ipser, and Sorkin [12]. More recently, thermodynamic stability of radial and axisymmetric equilibria of the Einstein-Euler system was investigated by Schiffrin and Wald [40], Roupas [39], both works containing a number of references on the topic.
Turning point principles play an important role in the study of other relativistic selfgravitating systems. An important open problem in this context is the stability of radially symmetric galaxies, which are equilibria of the asymptotically flat Einstein-Vlasov system. Going back to Zel'dovich and Podurets [51], it is conjectured and numerically verified (see also Zel'dovich and Novikov [52], and more recent numerical investiagtion by Andréasson and Rein [1]) that the stability of suitable 1-parameter families exhibits a single exchange of stability to instability at some critical value of central redshift κ = κ max . At κ max the so-called fractional binding energy has a maximum and for κ > κ max the equilibria are dynamically unstable. This stability scenario is very different from the mass-radius turning point principle that we prove in Theorem 1.13, as in the case of stars stability can in principle be exchanged to instability, and then back to stability [22,28], see Figure 1. The works [19,18] show that the steady states are spectrally stable for small values of κ and spectrally unstable for large values of κ respectively, which is consistent with the Zeldovitch-Podurets stability picture. The "large central redshift" instability is driven by the existence of an exponentially growing mode. To prove the existence of the growing mode and understand the invariant subspaces requires the full power of the separable Hamilton formulation of the Einsten-Vlasov system [18,29], as variational principles are inadequate for this purpose in the context of the Vlasov theory. We also mention that related binding energy criteria play a role in the study of the stability of so-called boson stars [25] as well as black holes/black rings in higher dimensions [11,2,40].
In the Newtonian context, we already mentioned that gaseous stars radial equilibria are embedded in 1-parameter families of the gravitational Euler-Poisson system (1.43)-(1.45). On the other hand, the Newtonian limit of the Einstein-Vlasov system is the gravitational Vlasov-Poisson system and also admits 1-parameter families of radially symmetric equilibria, i.e. steady galaxies, for a given microscopic equation of state. While the Zel'dovich/Wheeler turning point principle was shown to be true in the macroscopic Euler-Poisson case [30], such a principle is wrong for the Vlasov-Poisson case. To illustrate this, the well-known 1-parameter family of King solutions of the Vlasov-Poisson system possesses a mass-radius graph which spirals in to some asymptotic value (M ∞ , R ∞ ) with infinitely many winding points [38], but it is nonlinearly dynamically stable for any value of the central macroscopic density ρ 0 > 0 [16,17,26,27]. The inadequacy of the mass-radius diagram to predict the offset of (linear) instability for the kinetic models such as Einstein-Vlasov and Vlasov-Poisson is intimately related to the more complicated Hamiltonian structure by comparison to their macroscopic (gaseous) counterparts. In particular, the space of dynamically accessible perturbations in the kinetic setting is infinite-codimensional, which is one of the reasons why an extension of our analysis in the present work to the Einstein-Vlasov system is difficult.
We can however prove the radial equilibria of the Einstein-Vlasov system possess no growing modes for all κ < κ max , see Theorem 4.1. This is a consequence of Theorem 1.13 and the so-called macro-micro stability principle proved in Theorem 5.26 in [18]. The latter shows that, in a certain precise sense, the steady states of the Einstein-Vlasov system are "more stable" than the steady states of the Einstein-Euler system.
Plan of the paper. In Section 2 we prove a number of spectral properties of the linearized operator as it changes with the parameter κ. Lemma 2.2 (unsurprisingly) shows that in the κ → 0 limit we recover the corresponding Newtonian problem, which is then used in conjunction with Lemma 2.6 to compute both the kernel and the negative Morse index of Σ κ at small values of the parameter κ > 0. This is used as a starting point for the continuity argument. Next, Lemma 2.5 gives a sharp characterization of the kernel of Σ κ for any value of κ > 0 in terms of the critical points of the map κ → Mκ Rκ . The third most relevant result of Section 2 is the "jump-lemma" formulated in Lemma 2.8, showing that the negative Morse index n − (Σ κ ) can jump only at the critical points of the map κ → Mκ Rκ and that this jump is equal to the jump of the winding index i κ , see Definition 1.9. Section 3 is devoted to the proofs of Theorems 1.11 and 1.13, building on the preparatory results from Section 2. In Section 4 we state and provide the proof of a sufficient stability condition for the radial equilibria of the asymptotically flat Einstein-Vlasov system.
Recall the definitions of g (1.19) and g 0 (1.48). It follows from (2.71) that there exists a sufficiently small κ 0 such that (2.72) and the claim follows.
Since y κ (r) = µ κ (R κ ) − µ κ (r), in light of (2.59) it is natural to defineμ κ through the relationshipμ A simple corollary of Lemma 2.2 are the following a priori bounds.

Lemma 2.2 and Corollary 2.3 imply that for all
where in the last bound we have used that the supports of g κ •ȳ κ and g 0 • y 0 are both contained in [0, S 0 ]. Finally, since φ L 6 (R 3 ) ∇φ L 2 (R 3 ) for any φ ∈Ḣ 1 r it follows from Hölder's inequality that φ L 2 (B S 0 (0)) ≤ φ L 6 (R 3 ) ≤ ∇φ L 2 (R 3 ) . Therefore, using Cauchy-Schwarz and (2.97) we conclude By (1.50) the operator Σ 0 is nondegenerate and therefore by Proposition 2.3 in [30] it follows that n − (Σ κ ) = n − (Σ 0 ) = 1 for sufficiently small κ. By the same proposition, the value of n − (Σ κ ) can only change for those κ > 0 where the kernel of Σ κ is nontrivial, i.e. only at the critical points of κ → Mκ Rκ . Strictly speaking, to apply Proposition 2.3 from [30] we must show that the operators Σ κ , Σ 0 satisfy the assumption (G3) from [30]. By definition an operator L : X → X * satisfies the property (G3) if it is bounded and self-dual and the Hilbert space X can be decomposed into the direct sum of three closed subspaces and moreover 1) Lu, u < 0 for all u ∈ X − \ {0} and 2) there exists a constant δ > 0 such that Lu, u ≥ δ u 2 X for all u ∈ X + . The self-duality and boundedness of Σ κ is clear. To see that the decomposition (2.98) holds we consider the operatorΣ κ defined above. We note that for any µ ∈Ḣ 1 r we have Σ κ µ, µ = Σ κ φ, φ , where φ = e µκ+λκ µ. It suffices to show that Σ κ φ, φ ≥ δ 1 ∇φ 2 with some δ 1 > 0 and for φ in a finite co-dimensional subspace ofḢ 1 r . This will in particular imply that the kernel and the space corresponding to the negative part of the spectrum of the operator Σ κ are at most finite-dimensional. To prove (2.99) we writẽ Since there exists c 0 > c 1 > 0 such that the operator S 1 2 0 :Ḣ 1,r → L 2 r is an isomorphism. For φ ∈Ḣ 1,r we define ψ = S 1 2 0 φ ∈ L 2 r . Since (Σ κ φ, φ) L 2 r = (S κ ψ, ψ) L 2 r , the proof of (2.99) is reduced to check that (S κ ψ, ψ) L 2 r is uniformly positive for ψ in a finite co-dimensional subspace of L 2 r . We shall show that the operatorS is compact. Then it follows that the operatorS κ has finite dimensional eigenspaces for negative and zero eigenvalues, andS κ is uniformly positive on the complement space. To show the compactness ofS κ − id, we take a sequence (ψ n ) ⊂ L 2 r such that ψ n ⇀ 0 weakly in L 2 and show that S κ − id ψ n L 2 → 0, as n → ∞. Indeed, by Hardy's inequality in Fourier space, 0 ψ n is bounded inḢ 1 r and V κ has compact support. This shows that Σ κ satisfies the property (G3) for κ > 0. The same proof works for Σ 0 . = 0 for some κ > 0. By Lemma 2.5, we have Σ κ ( d dκ µ κ ) = 0 which is equivalent to Integrating the above relation over B κ and using g ′ (y κ ) d dκ µ κ = − d dκ ρ κ , we obtain where the last equality follows by our assumption. Therefore d dr e µκ+λκ d dκ µ κ r=Rκ = 0, and thus from (2.100) we conclude d dr e µκ+λκ d dκ µ κ = 0 for all r ≥ R κ . Since d dκ µ κ vanishes at asymptotic infinity, we conclude d dκ µ κ = 0 for all r ≥ R κ . Again by (2.100) we have d dκ µ κ = 0 for all r ≥ 0, which is clearly a contradiction, since d dκ µ κ r=0 = − d dκ y κ r=0 = −1.
Proof. Analogously to [30] we define the operator Locally around κ =κ there exists a curve ℓ(κ) of eigenvalues ofL κ such that ℓ(κ) = 0. The associated eigenvalues are normalized so that Letting r → ∞, we obtain In particular,

Proofs of the main theorems
Proofs of Theorems 1.11 and 1.13 follow closely the structure of proofs of Theorems 1.1 and 1.2 in [30].
Since for any α, β ∈ R we have it is clear that n − (L κ S 0 ) = 1. It thus follows that where we have used Lemma 1.7 and Definition 1.9.

Sufficient stability condition for the Einstein-Vlasov equilibria
The unknowns in the EV-system are the Lorentzian manifold (M, g) and the phase-space distribution function f which is supported on the mass-shell submanifold of the tangent bundle and solves the Vlasov equation. To find radially symmetric isotropic steady states, one prescribes a microscopic equation of state where E is the local particle energy and E 0 some cut-off energy. Following [37,18] we assume that Φ satisfies the assumption (Φ1) (see (3.3) in [18]) which we repeat here for completeness; we assume that Φ ∈ L ∞ loc ([0, ∞)) is a non-negative function, such that Φ(η) = 0 for all η ≤ 0 and there exists a − 1 2 < k < 3 2 and constants c 1 , c 2 such that c 1 η k ≤ Φ(η) ≤ c 2 η k , for all η ≥ 0 sufficiently small.
For a fixed Φ satisfying these assumptions, by analogy to the Einstein-Euler system one obtains a 1-parameter family of steady states κ → (f κ , µ κ , λ κ ) of the asymptotically flat radial Einstein-Vlasov system (see Section 3  For such a family there is a canonical mapping Φ → P Φ (Section 3.2 of [18]) which yields a macroscopic equation of state ρ → P Φ (ρ) satisfying assumptions (P1)-(P4). For instance, using equations (3.5)-(3.6) in [18] it is easy to see that in the small 0 ≤ ρ ≪ 1 region the Taylor expansion of P Φ about ρ = 0 takes the form Our assumptions on the range of k ensure that 4 3 < γ k < 2 and therefore assumption (P2) is satisfied. It is easy to see that the remaining assumptions (P1),(P3)-(P4) also hold. The resulting 1-parameter family of steady states κ → (ρ κ , µ κ , λ κ ) of the Einstein-Euler system given by Proposition 1.1 has the identical mass-radius curve as the family κ → (f κ , µ κ , λ κ ). A simple corollary of Theorem 1.13 is then Theorem 4.1 (Sufficient stability condition for the Einstein-Vlasov equilibria). Let Φ satisfy the above assumptions. The 1-parameter family of steady states κ → (f κ , µ κ , λ κ ) associated with the microscopic state function Φ is spectrally stable for all values of κ ∈ (0, κ max ), where κ max > 0 is the first maximum of the ADM-mass κ → M κ .