Dynamics of spinning particles in Schwarzschild spacetime

Transcription

1 Dynamics of spinning particles in Schwarzschild spacetime, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany RTG Workshop, Bielefeld

2 Outline Introduction Dynamics in Schwarzschild spacetime In general Constant Spin Components Isofrequency Pairing Conclusion and Outlook

3 Introduction Why Spinning Particles? Most of astrophysical objects have a non-vanishing spin, which is important for the detection and interpretation of signals from gravitational waves emitted from systems containing spinning objects. Since the fully realistic system is way to complicated to be treated in a reasonable amount of time, we first make a few assumptions: mass monopole and spin dipole non-radiating mass of particle small compared to mass of central object

4 Introduction Why Spinning Particles? Most of astrophysical objects have a non-vanishing spin, which is important for the detection and interpretation of signals from gravitational waves emitted from systems containing spinning objects. Since the fully realistic system is way to complicated to be treated in a reasonable amount of time, we first make a few assumptions: mass monopole and spin dipole non-radiating mass of particle small compared to mass of central object

5 Introduction Hamiltonian Formalism (Barausse et al. 2009) Derivation of Hamiltonian et ei I ee T point particle with attached tetrad eaµ representing spin motion ee I I I Spacetime split into 3+1 by choosing worldline parameter to be coordinate time Find momenta conjugate to coordinates Pµ = pµ + Eµαβ Sαβ I Hamiltonian function H = Pt = pt + Etαβ Sαβ

6 Introduction Hamiltonian Formalism (Barausse et al. 2009) Constraints system of MPD-equations is not closed; thus: Spin Supplementary Condition has to be chosen S µν ω ν = 0 Newton-Wigner: ω µ = p µ me T µ Poisson brackets are replaced by Dirac brackets considering the constraint

7 Introduction Hamiltonian Formalism (Barausse et al. 2009) Canonical Coordinates at linear order in Spin { x i, S J} = O(S 2 ) { Pi, S J} = O(S 2 ) { S I, S J} = ɛ IJK S K + O(S 2 ) The Hamiltonian Replacing kinematical momenta by canonical ones ( ) αγ ij H(x i, P i, S I ) = H NS β i F K i + F K P i F K j t + S K m2 + γ ij P i P j with H NS = β i P i + α m 2 + γ ij P i P j

8 Dynamics in Schwarzschild ST In general Hamiltonian and equations of motion Metric ds 2 = f(r)dt 2 + h(r)dr 2 + r 2 (dθ 2 + sin(θ) 2 dφ 2 ) Hamiltonian H(x i, P j, S K ) =H ns (r, θ, φ, P r, P θ, P φ ) + A(r, θ, φ, P r, P θ, P φ )(P θ S 3 Equations of Motion ṙ = H P r Ṗ r = H r B(r, θ, φ, P r, P θ, P φ )S 1 θ = H P θ Ṗ θ = H θ φ = H P φ Ṗ φ = H φ Ṡ 1 = {S 1, H} Ṡ 2 = {S 2, H} Ṡ 3 = {S 3, H} P φ sin(θ) S 2)

9 Dynamics in Schwarzschild ST In general Constants of Motion S S S 2 3 = S H(x i, P j, S K) = E ( ) J x = P θ sin(φ) + cos(φ) P φ cot(θ) + S1 sin(θ) ( ) J y = P θ cos(φ) + sin(φ) P φ cot(θ) + S1 sin(θ) J z = P φ Check whether the Dirac brackets of the conserved quantities with he Hamiltonian do vanish at linear order in Spin. {S, H} = 0 {H, H} = 0 {J x, H} = O(S 2 ) {J y, H} = O(S 2 ) {J z, H} = O(S 2 )

10 Dynamics in Schwarzschild ST Constant Spin Components Definition of cartesian Spin Components S x = cos(φ) (S 1 sin(θ) + S 2 cos(θ)) S 3 sin(φ) S y = sin(φ) (S 1 sin(θ) + S 2 cos(θ)) S 3 cos(φ) S z = S 1 cos(θ) S 2 sin(θ) Choose the spin vector to point into z-direction S x (θ, φ, S 1, S 2, S 3 ) S y (θ, φ, S 1, S 2, S 3 ) S z (θ, φ, S 1, S 2, S 3 ) = 0 0 S

11 Dynamics in Schwarzschild ST Constant Spin Components Time derivatives of Spin components Since we consider the system at linear order in spin, we require the time derivatives to vanish also at linear order in spin. leads to S Ṡ x Ṡ y Ṡ z = (A + B) cos(φ) (C + D) sin(φ) (A + B) cos(φ) + (C + D) sin(φ) 0 O(S 2 ) O(S 2 ) O(S 2 ) where A, B, C, D are functions of r, P r, θ, P θ. + O(S 2 ) = O(S 2 ) O(S 2 ) O(S 2 )

12 Dynamics in Schwarzschild ST Constant Spin Components Time derivatives of Spin components S (A + B) cos(φ) (C + D) sin(φ) (A + B) cos(φ) + (C + D) sin(φ) 0 + O(S 2 ) = O(S 2 ) O(S 2 ) O(S 2 ) Therewith we obtain conditions on A, B, C, D: 1. A + B = 0 and C + D = 0 or C + D = O(S) 2. A + B = O(S) and C + D = 0 or C + D = O(S)

13 Dynamics in Schwarzschild ST Constant Spin Components Equatorial Motion and Radial Infall Known solutions to this problem are two types of motion: Equatorial Motion Valid solution to 1. and 2.: θ = π 2 and P θ = 0 yield motion in equatorial plane Radial infall Valid solution to 1. and 2.: P φ = S and P θ = 0 lead to L x = 0, L y = 0, L z = 0

14 Dynamics in Schwarzschild ST Constant Spin Components Motion on a cone New solution: Motion of a spinning particle on a cone P φ = 0 and P θ = 0 satisfying the two conditions for constant spin components: θ = 0 ṗ θ = 0 φ 0 But how does it look like?

15 Dynamics in Schwarzschild ST Constant Spin Components Motion on a cone

16 Dynamics in Schwarzschild ST Constant Spin Components Characterise structure of orbits with P φ = 0 and P θ = 0 bound orbits ṙ needs at least two zeros (turning points of radial motion) - Non-existent! scattering orbits ṙ needs at least one zero (one turning point) - Existent, but not real! infalling orbits ṙ has no zero at all - Existent and allowed!

17 Dynamics in Schwarzschild ST Constant Spin Components Integration In order to obtain a solution to ṙ, φ we first substitute P r by terms of H. Then divide φ by ṙ in order to obtain: dφ = SH ( dr ± r( (1 H 2 )r+2)r r 2(H r+ r 2) ( cot 2 (θ)+ r 2 r ) dr ± r( (1 H 2 )r+2)hr ) Integration results in an expression containing hypergeometric Appell functions.

18 Dynamics in Schwarzschild ST Isofrequency pairing of orbits Warburton, Barack and Sago, 2013 Bound orbits cannot be uniquely characterised by their frequencies. How does the spin affect this characteristic?

19 Dynamics in Schwarzschild ST Isofrequency pairing of orbits Bound orbits in equatorial plane We set θ = π 2, P θ = 0 and thus the Hamiltonian looks like r 2 H = r r ( P 2 r (r 2) + r ) + P 2 φ r 2 + P φ S ( ) r 3 r(p 2 r (r 2)+r)+P2 φ + r 2 1 P φ (r 2)S r 3 r(p 2 r (r 2)+r)+P2 φ r 2 This relation can be used to eliminate P r by solving a cubic equation. P 2 r = P2 φ + r2 ω 2 r 2 ω = with k = 0, 1, 2. r 2 2r 4p ( r, H, P φ, S ) 3 sin ( α ( r, H, P φ, ) S + 2πk ) ( a r, H, P φ, ) S 3 3

20 Dynamics in Schwarzschild ST Isofrequency pairing of orbits Calculation of frequencies In order to calculate Ω r and Ω φ we have to find a relation between H, P φ and r a, r p. turning points: P r = 0 Energy value at r a and r p : H = ri P2 φ + P φ(r i 2)S r i r 2 i r 3 i 1+ P2 φ r 2 i P φ S r 3 i 1+ P2 φ r 2 i +1 Expand: H H 0 + H 1 S and P φ P φ0 + P φ1 S Taylor expansion of the two equations about S = 0 up to first order Solve for H 0 (r a, r p ), H 1 (r a, r p ) and P φ0 (r a, r p ), P φ1 (r a, r p )

21 Dynamics in Schwarzschild ST Isofrequency pairing of orbits Calculation of frequencies The frequencies can be determined by the following relations: T r = 2 Ω r = 2π T r ra Ω φ = 1 T r Tr r p 1 ṙ dr 0 dφ dt dt Using the relations between r a, r p and p, e it is possible to obtain Ω r (p, e) and Ω φ (p, e) so that we can analyse possible isofrequent orbits in a similar way to Warburton 2013.

22 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

23 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

24 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

25 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

26 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

27 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

28 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

29 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

30 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?

31 Conclusion and Outlook Hamiltonian Formalism spinning testparticle in curved spacetime Newton Wigner SSC Canonical Coordinates Dynamics in Schwarzschild Spacetime constant spin components: equatorial motion, radial infall and motion on a cone influence of spin on the isofrequency pairing of orbits Future work Are there any other kinds of motion with constant spin components? How does it look in Kerr ST? How does the spin parameter change the characteristics of the isorfrequency pairing? What about Kerr ST? Another aspect is the investigation of chaos?