Machine Learning For Energy Consumption Forecasting

Machine Learning For Energy Consumption Forecasting

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Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center

Continued Discussion and Extended Analysis (Part 2)

Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center

Continued Discussion and Extended Analysis (Part 3)

Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center

Continued Discussion and Extended Analysis (Part 4)

Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center

Continued Discussion and Extended Analysis (Part 5)

Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center

Continued Discussion and Extended Analysis (Part 6)

Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center

Continued Discussion and Extended Analysis (Part 7)

Stability and instability of self-gravitating relativistic matter

Mahir Hadžić∗, Zhiwu Lin†, Gerhard Rein‡

July 1, 20

Abstract

We consider steady state solutions of the massive, asymptotically flat, spherically

symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clus- ters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift κ > 0. We prove their linear instability when κ is sufficiently large, i.e., when they are strongly relativistic, and that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in

the 1960s by Zel’dovich & Podurets (for the Einstein-Vlasov system) and by Harrison,

Thorne, Wakano, & Wheeler (for the Einstein-Euler system). Our results are in sharp

contrast to the corresponding non-relativistic, Newtonian setting.

We carry out a careful analysis of the linearized dynamics around the above steady

states and prove an exponential trichotomy result and the corresponding index theo- rems for the stable/unstable invariant spaces. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

1 Introduction and main results 2

1.1 The Einstein-Vlasov system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Einstein-Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation and history of the problem . . . . . . . . . . . . . . . . . . . . . 5

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Einstein-Vlasov and Einstein-Euler system in spherical symmetry 1

3 Steady states 1

3.1 The basic set-up; one-parameter families . . . . . . . . . . . . . . . . . . . . 1

3.2 Microscopic and macroscopic equations of state . . . . . . . . . . . . . . . . 1

3.3 The Bisnovatyi-Kogan–Zel’dovich (BKZ) solution . . . . . . . . . . . . . . . 2

3.4 The ultrarelativistic limit κ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Stability analysis for the Einstein-Vlasov system 3

4.1 Dynamic accessibility and the linearized Einstein-Vlasov system . . . . . . . 3

4.2 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 3

4.3 Linear exponential instability . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.4 The Hamiltonian structure of the linearized Einstein-Vlasov system . . . . . 4

4.5 The reduced Einstein-Vlasov operator . . . . . . . . . . . . . . . . . . . . . 5

4.6 Main results for the Einstein-Vlasov system . . . . . . . . . . . . . . . . . . 5

5 Stability analysis for the Einstein-Euler system 6

5.1 The number density n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5.3 Variational approach to stability . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 The reduced Einstein-Euler operator . . . . . . . . . . . . . . . . . . . . . . 7

5.5 Main results for the Einstein-Euler system . . . . . . . . . . . . . . . . . . . 7

5.6 A negative energy direction for κ sufficiently large . . . . . . . . . . . . . . 8

5.7 The micro-macro stability principle . . . . . . . . . . . . . . . . . . . . . . . 8

A Auxiliary results for self-adjoint operators 8

1 Introduction and main results

We consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with

signature (− + + +). The Einstein equations read

Gαβ = 8πTαβ , (1.1)

where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energy-momentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. To obtain a closed system, the field equations (1.1) have to be supplemented by evolution equations for the matter and by the definition of Tαβ in terms of the matter and the metric. We consider two matter models, namely a collisionless gas as described by the

collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein-Vlasov and Einstein-Euler system respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results.

1.1 The Einstein-Vlasov system

In the case of a collisionless gas matter is described by the number density f of the particles on phase space. The world line of a test particle on M obeys the geodesic equation ẋα = pα , ṗα = −Γαβγ pβ pγ , where xα denote general coordinates on M , pα are the corresponding canonical momenta, Γαβγ are the Christoffel symbols induced by the metric gαβ , the dot indicates differentiation with respect to proper time along the world line of the particle, and the Einstein summation

convention is applied. We assume that all the particles have the same rest mass m ≥ 0 and move forward in time, i.e., their number density f is a non-negative function supported on the mass shell n o P M := gαβ pα pβ = −m , pα future pointing ,

a submanifold of the tangent bundle T M of the spacetime manifold M which is invariant under the geodesic flow. Since we are interested in the massive case m > 0, we normalize m = 1. But at a crucial point in our analysis the massless case m = 0 will play an important role. Letting Latin indices range from 1 to 3 we use coordinates (t, xa ) with zero shift which implies that g0a = 0. On the mass shell P M the variable p then becomes a function of the remaining variables (t, xa , pb ):

p p p = −g 1 + gab pa pb . Since the particles move like test particles in the given metric, their number density f = f (t, xa , pb ) is constant along the geodesics and hence satisfies the Vlasov equation pa 1 ∂t f + 0 ∂xa f − 0 Γaβγ pβ pγ ∂pa f = 0. (1.2) p p

The energy-momentum tensor is given by

Tαβ = pα pβ f |g|1/2 , (1.3)

−p where |g| denotes the modulus of the determinant of the metric, and indices are raised and lowered using the metric, i.e., pα = gαβ pβ . The system (1.1), (1.2), (1.3) is the Einstein-

Vlasov system in general coordinates. We want to describe isolated systems, and therefore

we require that the spacetime is asymptotically flat. In Section 3 we will see that steady states of this system can be obtained via an ansatz f = φ(E) (1.4) where E = −p is the local or particle energy and φ is a prescribed ansatz function; we refer to (1.4) as the microscopic equation of state. We can now state our main result for the Einstein-Vlasov system in an informal way; the precise meaning of the parameter κ > 0 is explained in Section 3, in particular (3.10) and (3.11).

Theorem 1. Let (fκ )κ>0 be a one-parameter family of spherically symmetric steady states

to the Einstein-Vlasov system with a fixed, decreasing microscopic equation of state φ and κ the central redshift of the corresponding steady state. (a) For κ sufficiently large, the associated steady state is dynamically unstable in the sense that the linearized Einstein-Vlasov system possesses an exponentially growing mode. (b) For any κ > 0 the phase space of the linearized system splits into three invari- ant subspaces: the stable, unstable, and center