Action-Angle Variables and KAM-Theory in General Relativity

Action-Angle Variables and KAM-Theory in General Relativity Daniela Kunst, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany Workshop in Oldenburg 11-13 March 2013

Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

Introduction Integrable dynamical systems Dynamical systems of n-dof 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 n-dimensional tori.

Introduction Integrable dynamical systems

Introduction Integrable dynamical systems Action-Angle Variables simplified description of periodic or quasi-periodic motion of integrable systems Action-Variable: specifies a particular torus Angle-Variable: 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 )

Introduction The KAM-Theorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAM-Theorem (Kolmogorov-Arnold-Moser)

Introduction The KAM-Theorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAM-Theorem (Kolmogorov-Arnold-Moser) If an unperturbed system is non-degenerate, then for sufficiently small [...] perturbations, most non-resonant 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 conditionally-periodically [...]. V.I. Arnold: Mathematical aspects of classical and celestial mechanics, Springer, Berlin (2006)

Introduction The KAM-Theorem

Introduction The KAM-Theorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAM-Theorem (Kolmogorov-Arnold-Moser) Iso-Energetic Non-Degeneracy Condition (V. Arnold) D A = det ) ω 0 ω 0 ( ω I

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

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

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 Action-Variables: 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

Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition Action-Variables: 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 )

Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition I ϕ = p ϕdϕ = 2πL γ φ I r = p rdr = 2 γ r ra r p = H(I r, I ϕ) implicitly [ ( 1 1 1 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)

Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition Fundamental Frequencies ω r = ( Ir H ) 1 ω ϕ = 1 I r 2π L ωr iso-energetic non-degeneracy 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

Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy 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

Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy 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

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 Action-Variables: 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

Kerr Spacetime KAM-Theorem and the iso-energetic non-degeneracy Condition Action-Variables: 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 )

Kerr Spacetime KAM-Theorem and the iso-energetic non-degeneracy 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)

Kerr Spacetime KAM-Theorem and the iso-energetic non-degeneracy 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 iso-energetic non-degeneracy 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

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 non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?