Zoom-whirl orbits of spinning bodies

in rotating space-times

Balázs Mikóczi(HAS Wigner RCP)

&

Zoltán Keresztes(University of Szeged)

Gravity@Malta 2018

NKFIH PD 116892

UNKP-17-4

Outline

Introducing of the equations of motion for the spinning particles, theMathisson-Papapetrou-Dixon equations (MPD eqs.)

Spin supplementary conditions (SSCs). SSC dependence in compactbinary system

Zoom-whirl orbits and spinning bodies on Schwarzschild and Kerrbackgrounds

Spinning bodies on general, axially symmetric, rotating space-times

Equations and numerical setup

Numerical results; zoom-whirl orbits on Kerr space-time

Numerical results; orbits on rotating space-times (regular black holes,e.g. Hayward and Bardeen classes)

https://www.innovia.com/blog/combinations-defaults-and-priority-taking-nav-accuracy-to-another-dimension

Motion of spinning particles in

gravitational fields The MPD equations can be derived by the

properties of the stress-energy-momentum

tensor (symmetric and divergence-free).

These equations beyond the point particle

equations (EIH eqs.) are the pole-dipole eqs.

in the multipole expansion.

The MPTD eqs. are not closed! We have to

impose spin supplementary conditions

(SSC).

There are two different masses:

.

m=M if the tangent vector uα is parallel to

momentum pα :

We can introduce the magnitude of the spin:

Quantities M, m and S are not conserved

in general case!

where

MPD equations:

m pu"" D/D u

u dX

d

uu 1

S konst

T0x g d3x

p mu uS

the tangent vector

of the wordline

2S2 SS

uα

x(τ)

pα

Uα

Σ(τ,U)spacelike

hypersurface

(SαβUα=0)

p 12

R Su

S pu

#

#

M papa

u p

M

Spin supplementary conditions (SSCs)

III. Corinaldesi-Papapetrou (SSC III):

I. Frenkel-Mathisson-Pirani (SSC I):

IV. Tulczyjew-Dixon (SSC IV):

Tulczyjew, W. M. Acta Phys. Polon., 18, 393. (1959).

Dixon, W. Nuovo Cim., 34, 317. (1964).

Frenkel J. Z. Phys. 37, 243 (1926).

Mathisson M. , Acta. Phys. Polon. 6, 167 (1937).

Pirani, F.A.E. Acta Phys. Polon., 15, 389 (1956).

Corinaldesi, E. & Papapetrou, A.. Proc. Roy. Soc.,A/09, 259

(1951).

II. Newton-Wigner-Pryce (SSC II):

Pryce, M.H.L. Pme R. Soc. A 195 62 (1948).

Newton, T.D. & Wigner, E.P. Rev. Mod. Phys. 21 400 (1949).

Su 0

2S0 uS 0

S0 0

Sp 0

V. Ohashi-Kyrian-Semerák (SSC V):

Ohashi A, Phys. Rev. D 68, 044009 (2003)

Kyrian K & Semerák O, Mon. Not. R. Astron. S. 382, 1922

(2007)

Sw 0

ww 1

w 0M pp

m pu

S SS/2

Conserved quant.

m, S

M, S

* conserved up to linear order in the particle’s spin

M*

m=M

L 2

v2 Gmr G

c2r3v r 2S k 1 2k1

2c2mv a

hijSO

4m

Dr3m SNir j

TT

hij1.5SO

4Dr3

3r ir j

r2r v2S 1 k

r iv 4S 3 2kj

2kvir j 6rr r ir S j

3rr r 2v N r 2rN v

i NjTT

#

#

The Lagrangian (acceleration) of the leading-order spin-orbit interaction is

not unique, it depends on SSC (Kidder 1995):

where

S1

S2

SL

m1

m2

σ

m m1 m2

m1m2

m1 m2

S S1 S2

m2m1

S1 m1m2

S2

k=1 Pirani/Tulczyjew-Dixon SSC I/IVk=½ Pryce-Newton-Wigner SSC IIk=0 Corinaldesi-Papapetrou SSC III

Spin-orbit interaction for compact binaries

The SSC-dependent quantities:- The energy, the orbital angular momentum, the LRL-vector

- The energy and the angular momentum losses

- Waveform, e.g.

Mikóczi B., Phys. Rev. D 95, 064023 (2017)

E,L,A, dE

dt, dL

dt,h

Some milestones for zoom whirl orbits and spinning bodies on Kerr background

Highly eccentric orbits, separatrix (Cutler, Kennefick, Poisson 1994)

Homoclinic orbits in Schwarzschild space-time (Bombelli, Calzetta 1992,Levin, O’Reilly, Copeland 2000)

Numerical investigation of spinning bodies in Kerr geometry (Semerák1999, Hartl 2003)

Zoom-whirl* orbits in Kerr geometry (Glampedakis, Kennefick 2002)

Closed zoom-whirl orbits of test particles in Kerr field (Levin, Perez-Giz2008), e.g.

*This name may have been suggested by Kip Thorne

Closed orbits (z=2,w=0,v=1) and (z=1,w=3,v=0,)Eccentric zoom-whirl orbitHomoclinic orbit

We use the Tulczyjew-Dixon SSC:

Spinning bodies on rotating spacetimes

Sp 0

ub m

M2pb 2SbaRaecdpeScd

4M2RaecdSaeScd

m pu

M papa Dynamical mass (conserved)

Kinematical mass (not conserved)where

Sa 1

2MabcdpbScd

Relation between the 4-velocity and the 4-momentum:

Definition of the 4-spinvector:

where is the antisymmetric 4-dimensional Levi-Civita tensor;

S2 SaSa 1

2ScdScdThe magnitude of the spin is conserved.

g

Generalized rotating spacetimes

The line element of asymptotically flat,

rotating, axially symmetric black hole in

Boyer-Linquist coordinates is:

- The µ and a are the mass and the rotation parameters of black hole

- The qe is related to the electric charge (Qe is the electric charge of

the black hole).

- The α and β are function of r

- The σ is the dimension on length squared

r2 a2 cos2 ,

B r2 a2

A r2 a22 a2 sin2

r2 a2 2 r qe

#

#

#

#

[1] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963).

[2] E. T. Newmann, A. I. Janis, J. Math. Phys. 6, 915 (1965).

[3] B. Toshmatov, Z. Stuchlík, B. Ahmedov, Phys. Rev. D 95,

084037 (2017).

[4] F. Filippini, G Tasinato, , (2017) [arXiv:1709.02147].

[5] J. M. Bardeen, Proc. GR5, Tbilisi USSR, 174 (1968).

[6] S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006).

[7] Z-Y Fan, X. Wang, Phys. Rev. D 94, 124027 (2016).

Kerr Kerr Einstein gravity Vector-Tensor

spacetime Newman coupled to nonlinear gravity model

[1] spacetime [2] electrodynamics [3] in vacuum [4]

0 0 qm3 r/ r qm

/ 0

0 0 0 k2Qe2r2/

qe 0 Qe2 0 Qe

2

Bardeen class Hayward class Maxwell theory

of black holes [5] of black holes [6] is recovered

in the weak field limit [7]

2 1

B qm

3 r

r2 qm2 /2

,

H qm

3 r

r qm

3

DUa

d 1

2R

bcd

a UbScd ,

DSab

d paUb uaUb ,

ub mM

Ub 2SbaRaecdUeScd

4 RaecdSaeScd

#

#

#

Numerical setup

Dimensionless variables for

numerical investigation:

14 first-order differential equations

(the bar notations is ignored):

Finally, 9 independent equations

remain since there are 5

constraints:

UaUa 1 ,

SaUa 0,

SaSa S2,

E Ut 12

Sabagbt ,

Jz U 12

Sabagb

#

#

#

#

#

ds2 ds2

2,

, ūā dx ā

d uā, Ūā p ā

M Uā . .

The coordinate space:

(U0)a, (S0)

a

The initial values are x r2 a2 cos sin ,

y r2 a2 sin sin ,

z rcos #

Numerical resultsEquatorial orbits (θ=π/2)

Kerr BH

Hayward BH,

γ=3, qm=0.5

a=0.99(ur=-0.02644,uφ=0.00406)

Bardeen BH

γ=4, qm=0.5

a=0.6(ur= -0.02644,uφ=0.00606)

≈

Hayward/Bardeen BH

Numerical results IIEquatorial zoom-whirl orbits on regular black holes

Hayward,

γ=3, qm=0.9

(a=0.99,ur=-0.02644,uφ=0.00206)

Bardeen

γ=4, qm=0.9

(a=0.99,ur=-0.02644,uφ=0.00206)

Hayward,

γ=5, qm=0.9

Bardeen

γ=5, qm=0.9

Numerical results III

Outer static limit rE+

Inner event horizon r-

Outer event horizon r+

Kerr geometry

Ergoregion

Distant orbits around a Kerr black hole

r 2 a2 ,

rE 2 a2 cos2

a=0.5

E=0.97

L=1

Spinning particle (S=1)Non-spinning particle (S=0)

a=0.5

E=0.97

L=1

Event horizons (grr=0)

and static limit surfaces (gtt=0)

Numerical results IVClose orbits around a Kerr black hole

(a=0.99,ur=-0.02644, uφ=0.00406)

Non-spinning particle (S=0) Spinning particle (S=1)

(a=0.99,ur=-0.02644, uφ=0.00406)

Summary

We have considered the orbits of test particles and spinning bodies on

Kerr geometry.

The spin may influence the orbit strongly in Kerr field.

We have demonstrated the existence of zoom-whirl orbits on a class

of regular black hole space-times.

The future plan is the detailed discussion of zoom whirls orbits

around regular black hole space-times and the vector-tensor models in

vacuum and we want to investigate the influence of the spin on these

orbits and how the spin evolves along these orbits.

http://aasnova.org/2016/03/23/dance-of-two-monster-black-holes/

Thank you for your

attention!