ActionAngle Variables and KAMTheory in General Relativity


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1 ActionAngle Variables and KAMTheory in General Relativity Daniela Kunst, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany Workshop in Oldenburg March 2013
2 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
3 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
4 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
5 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
6 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
7 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
8 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
9 Outline Introduction Schwarzschild spacetime KAMTheorem and nondegeneracyconditions Kerr spacetime KAMTheorem and nondegeneracyconditions Conclusion and Outlook
10 Introduction Integrable dynamical systems Dynamical systems of ndof are described by the Hamiltonian H( q, p) and the equations of motion ṗ i = H(p i, q i ) q i q i = H(p i, q i ) p i The system is integrable if it has n integrals of motion F i, that are in involution. Its phase space is foliated into ndimensional tori.
11 Introduction Integrable dynamical systems Dynamical systems of ndof are described by the Hamiltonian H( q, p) and the equations of motion ṗ i = H(p i, q i ) q i q i = H(p i, q i ) p i The system is integrable if it has n integrals of motion F i, that are in involution. Its phase space is foliated into ndimensional tori.
12 Introduction Integrable dynamical systems
13 Introduction Integrable dynamical systems ActionAngle Variables simplified description of periodic or quasiperiodic motion of integrable systems ActionVariable: specifies a particular torus AngleVariable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )
14 Introduction Integrable dynamical systems ActionAngle Variables simplified description of periodic or quasiperiodic motion of integrable systems ActionVariable: specifies a particular torus AngleVariable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )
15 Introduction Integrable dynamical systems ActionAngle Variables simplified description of periodic or quasiperiodic motion of integrable systems ActionVariable: specifies a particular torus AngleVariable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )
16 Introduction Integrable dynamical systems ActionAngle Variables simplified description of periodic or quasiperiodic motion of integrable systems ActionVariable: specifies a particular torus AngleVariable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )
17 Introduction The KAMTheorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAMTheorem (KolmogorovArnoldMoser)
18 Introduction The KAMTheorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAMTheorem (KolmogorovArnoldMoser) If an unperturbed system is nondegenerate, then for sufficiently small [...] perturbations, most nonresonant invariant tori do not vanish, [...] so that in phase space of the perturbed system, too, there are invariant tori densely filled with phase curves winding around them conditionallyperiodically [...]. V.I. Arnold: Mathematical aspects of classical and celestial mechanics, Springer, Berlin (2006)
19 Introduction The KAMTheorem
20 Introduction The KAMTheorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAMTheorem (KolmogorovArnoldMoser) IsoEnergetic NonDegeneracy Condition (V. Arnold) D A = det ) ω 0 ω 0 ( ω I
21 Schwarzschild Spacetime General Hamiltonian H = p 0 = g0i p i g 00 + [ ( ) ] 1 g ij p i p j +m 2 2 g + 0i 2 p i g 00 g 00
22 Schwarzschild Spacetime... for Schwarzschild geometry (assume θ = π 2 ) [(1 H = r s ) ( r p 2 r + 1 p 2 r 2 θ + ) p2 ϕ sin( π 2 )2 ] (1 + m 2 r s ) r
23 Schwarzschild Spacetime... for Schwarzschild geometry (assume θ = π 2 ) [(1 H = r s ) ( r p 2 r + 1 p 2 r 2 θ + ) p2 ϕ sin( π 2 )2 ] (1 + m 2 r s ) r ActionVariables: I ϕ = p ϕ dϕ = 2πL γ φ I θ = 0 I r = p r dr = 2 γ r ra r p [ ( 1 1 r s ) r 1 ( 1 r s r ] )H L2 r 2 m2 dr
24 Schwarzschild Spacetime KAMTheorem and the isoenergetic nondegeneracy condition ActionVariables: I ϕ = p ϕ dϕ = 2πL γ φ I θ = 0 I r = p r dr = 2 γ r ra r p [ ( 1 1 r s ) r 1 ( 1 r s r ] )H L2 r 2 m2 dr The Hamiltonian is implicitly given as: H(I r, I ϕ )
25 Schwarzschild Spacetime KAMTheorem and the isoenergetic nondegeneracy condition I ϕ = p ϕdϕ = 2πL γ φ I r = p rdr = 2 γ r ra r p = H(I r, I ϕ) implicitly [ ( r s ) ( r 1 r s r ) H L2 r 2 m2 ] dr Fundamental Frequencies defined as ω i = H by the equations of motion and applying the I i implicit function theorem yields: ω r = ( Ir H ) 1 ω ϕ = 1 2π I r L ω r (calculation based on W. Schmidt 2002)
26 Schwarzschild Spacetime KAMTheorem and the isoenergetic nondegeneracy condition Fundamental Frequencies ω r = ( Ir H ) 1 ω ϕ = 1 I r 2π L ωr isoenergetic nondegeneracy condition D A = det ( ω I = ω3 r (2π) 2 ) ω = ω3 r 2 I r ω 0 (2π) 2 L 2 (( ) L c c L 1 K(k) L k k L ( )) E(k) 1 k K(k) 0 2 which means, that the map (I) ( ωϕ = 1 ) I r ω r 2π L bijective at fixed H must be
27 Schwarzschild Spacetime KAMTheorem and the isoenergetic nondegeneracy condition D A ( L c c L 1)K(k) L k k L ( E(k) 1 k 2 K(k)) > 0 L c c L 1 < 0 L c c L 1 = 0 H = 1 bdries of bound motion
28 Schwarzschild Spacetime KAMTheorem and the isoenergetic nondegeneracy condition c DA ( Lc L 1)K(k) L c c L L k E(k) ( k L 1 k2 K(k)) > 0 1<0 L c 1=0 c L H =1 bdries of bound motion H = const ωϕ ωr = const bdries of bound motion
29 KerrSpacetime... for Kerr geometry H = ar sp ϕ ρ 2 +r s(r 2 +a 2 )r r + 2 p 2 r + p2 θ + ρ2 rsr sin(θ) 2 p2 ϕ + ρ2 m 2 ρ 2 +r s+(r 2 +a 2 )r ( + with = r 2 + a 2 r sr and ρ 2 = r 2 + a 2 cos(θ) 2 ) 2 ar sp ϕ ρ r 2 +r s(r 2 +a 2 )r
30 KerrSpacetime... for Kerr geometry H = ar sp ϕ ρ 2 +r s(r 2 +a 2 )r r + 2 p 2 r + p2 θ + ρ2 rsr sin(θ) 2 p2 ϕ + ρ2 m 2 ρ 2 +r s+(r 2 +a 2 )r ( + with = r 2 + a 2 r sr and ρ 2 = r 2 + a 2 cos(θ) 2 ) 2 ar sp ϕ ρ r 2 +r s(r 2 +a 2 )r ActionVariables: I ϕ = p ϕ dϕ = 2πL z γ φ 1 2 C + Lz (1 H 2 )a 2 (1 y 2 ) L2 z y I θ = p θ dθ = 2 γ θ θ min (1 y) [(C + L 2 z )y (1 H 2 )a 2 (1 y)y L 2 z] dy ra 1 [ ] I r = p r dr = 2 ((r γ r 2 + a 2 )H al 2 z) 2 (r 2 + (L 2 z ah) 2 + C) dr r p
31 Kerr Spacetime KAMTheorem and the isoenergetic nondegeneracy Condition ActionVariables: I ϕ = p ϕ dϕ = 2πL z γ φ 1 2 C + Lz (1 H 2 )a 2 (1 y 2 ) L2 z y I θ = p θ dθ = 2 γ θ θ min (1 y) [(C + L 2 z )y (1 H 2 )a 2 (1 y)y L 2 z] dy ra 1 [ ] I r = p r dr = 2 ((r γ r 2 + a 2 )H al 2 z) 2 (r 2 + (L 2 z ah) 2 + C) dr r p The Hamiltonian is implicitly given as: C(I θ, I ϕ, H) H(I r, I θ, I ϕ )
32 Kerr Spacetime KAMTheorem and the isoenergetic nondegeneracy Condition I ϕ = p ϕdϕ = 2πL z γ φ 1 I θ = p θ dθ = 2 γ θ θ min I r = p r dr = 2 γ r ra rp = H(I r, I θ, I ϕ) implicitly C + L 2 z (1 H2 )a 2 (1 y 2 ) L2 z y (1 y) [ (C + L 2 z )y (1 H2 )a 2 (1 y)y L 2 z 1 2 [ ((r 2 + a 2 )H al 2 z ] dy ) 2 (r 2 + (L z ah) 2 + C)] dr Fundamental Frequencies ω r = I θ C Q I r C ω θ = Q ω ϕ = 1 [ Ir 2π L ω r + I ] θ L ω θ with Q = I θ I r Ir I θ C H C H (calculation based on W. Schmidt 2002)
33 Kerr Spacetime KAMTheorem and the isoenergetic nondegeneracy Condition Fundamental Frequencies ω r = I θ C Q I r C ω θ = Q ω ϕ = 1 [ Ir 2π L ωr + I ] θ L ω θ isoenergetic nondegeneracy condition ( ω ) ω D A = det I 0 ω 0 which means, that the map (I) must be bijective at fixed H ( ωr, ω ) ( ϕ = 1 [ Iθ ω θ ω θ 2π L + I ] r ω r, ω ) r L ω θ ω θ
34 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the nonresonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?
35 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the nonresonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?
36 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the nonresonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?
37 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the nonresonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?
38 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the nonresonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?
39 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the nonresonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?
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