A spinning test particle, as described by the Papapetrou equations, appears to be chaotic in Kerr
spacetime, with maximum e-folding timescales of a few×102_{M}_{. The applicability of this result}
is limited by three main factors: (1) chaos appears only for physically unrealistic values of the
spin parameter; (2) other effects, such as tidal coupling, may be important for some astrophysical
systems, violating the pole-dipole approximation implicit in the Papapetrou equations; and (3) we
neglect gravitational radiation. The third limitation is not fatal, since the radiation timescales can
be long enough that chaos, if present in the conservative limit, would have time to manifest itself in
the gravitational radiation of extreme mass-ratio systems.

Figure 2.17: The natural logarithm of the largest ellipsoid axis vs.τ for the Euclidean norm (top) and the projected norm from Sec. 2.4.1 (bottom).

In the unphysical S = 1 limit, the Lyapunov exponents exhibit characteristics expected of a
Hamiltonian system, appearing in pairs±λ(Sec. 2.6.2). There are zero Lyapunov exponents which
correspond to the constants of the motion, but the other exponents are in general nonzero. (For
the Kerr orbits considered in this chapter, we find that two of the three independent exponents
are nonzero, as illustrated in Fig. 2.8.) Typical orders of magnitude for the largest Lyapunov
exponents are a few×10−3M−1for unphysical spins (S= 1). For physically realistic spin parameters
(Sec. 2.7.3), we find thatλmax.few×10−7M−1, corresponding toe-folding timescales of a few×
106_{M}_{. Even this bound appears to be limited only by the total integration time; in all physically}
realistic cases considered,λmaxis indistinguishable from zero (usingS= 0 integrations as a baseline).
From the perspective of gravitational radiation detection, our most important conclusion is that
chaos seems to disappear for physically realistic values of S, i.e., values of S for which the test-
particle approximation and hence the Papapetrou equations are valid. We are unable to comment
on the dynamics of comparable mass-ratio binaries, since such systems are not accurately modeled
by the Papapetrou equations, but forextreme mass-ratio binaries it appears unlikely that chaos will
present a problem for the calculation of theoretical templates for use in matched filters. A more
thorough exploration of parameter space is needed to reach a firmer conclusion (Chapter 3).

### Acknowledgements

I would like to thank Sterl Phinney for his support, encouragement, and excellent suggestions. Thanks also to Scott Hughes for contributing through his ideas and enthusiasm for this project. I would especially like to thank Janna Levin for her careful reading of the paper on which this chapter

is based, and for insightful comments. Finally, I would like to thank Kip Thorne for teaching me general relativity and Jim Yorke for teaching me dynamical systems theory.

### Appendix: Full Jacobian

For reference, we list the derivatives needed to calculate the full Jacobian matrix. From Sec. 2.5.2, we have the following:

∂x˙µ
∂xν =N
pαgαµ,ν+w
µ
,ν+ v
µ_{(}_{v}
αwα,ν+12N w
α_{w}β_{g}
αβ,ν)
(2.57)
∂x˙µ
∂pν
=N(gµν+Wµν+N vµwν) +N vµvαWαν (2.58)
with
Wµν =−∗R∗µανβSαSβ (2.59)
∂x˙µ
∂Sν
=N Vµν+N vµvαVαν (2.60)
with
Vµν =−Sαpβ(∗R∗µαβν−∗R∗µναβ). (2.61)

Now we simply apply the product rule many times:

∂p˙µ
∂xν =−pαSβ R
∗ αβ
µγ ,νvγ+R∗µγαβvγ,ν
+pα Γαβµ,νvβ+ Γαβµvβ,ν
(2.62)
∂p˙µ
∂pν
=−Sβ
R∗_{µγ}νβvγ+R∗_{µγ}αβpα
∂vγ
∂pν
+ Γν_{βµ}vβ+ Γα_{βµ}pα
∂vβ
∂pν
(2.63)
∂p˙µ
∂Sν
=−R∗_{µγ}ανvγpα−R∗µγαβ
∂vγ
∂Sν
pαSβ+ Γαβµpα
∂vβ
∂Sν
∂S˙µ
∂xν =−pµSαpγSδ
R∗α γδ_{β} _{,ν}vβ+R∗α γδ_{β} vβ_{,ν}+ Sα Γαβµ,νv
β_{+ Γ}α
βµv
β
,ν
(2.64)
∂S˙µ
∂pν
=−SαSδvβ
δ_{µ}νR∗α γδ_{β} pγ +pµR∗α νδβ
− pµR∗α γδ_{β} Sα
∂vβ
∂pν
pγSδ+ Γαβµ
∂vβ
∂pν
Sα (2.65)

∂S˙µ
∂Sν
=−pµpγvβ
R∗ν γδ_{β} Sδ+R∗α γνβ Sα
−pµR∗α γδβ Sα
∂vβ
∂Sν
pγSδ+ Γνβµvβ+ Γαβµ
∂vβ
∂Sν
Sα (2.66)

Accidentally leaving off the final term in ∂ ˙

Sµ

∂Sν

led to the robust but spurious chaotic behavior mentioned in Sec. 2.5.1.

## Bibliography

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[3] S. Suzuki and K. Maeda, Phys. Rev. D55, 4848 (1997).

[4] J. D. Schnittman and F. A. Rasio, Phys. Rev. Lett.87, 121101 (2001). [5] N. J. Cornish, Phys. Rev. D 64, 084011 (2001).

[6] N. J. Cornish and J. Levin, gr-qc/0207016.

[7] N. J. Cornish and J. Levin, Phys. Rev. Lett.89, 179001 (2002). [8] S. A. Hughes, Class. Quantum Grav.18, 4067 (2001).

[9] http://lisa.jpl.nasa.gov.

[10] K. T. Alligood, T. D. Sauer, and J. A. Yorke,Chaos: An Introduction to Dynamical Systems (Springer, New York, 1997).

[11] N. Cornish and J. Levin, Phys. Rev. Lett.78, 998 (1997).

[12] E. Ott,Chaos in Dynamical Systems(Cambridge University Press, Cambridge, England, 1993). [13] C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973). [14] A. Papapetrou, Proc. R. Soc. LondonA209, 248 (1951).

[15] W. G. Dixon, Proc. R. Soc. LondonA314, 499 (1970).

[16] C. Møller,Theory of Relativity (Oxford University Press, London, 1972). [17] B. M. Barker and R. F. O’Connell, Gen. Relativ. Gravit.5, 539 (1974). [18] L. E. Kidder, Phys. Rev. D52, 821 (1995).

[20] G. B. Cook, S. L. Shapiro, and S. A. Teukolsky, Astrophys. J.424, 823 (1994). [21] D. Lai, F. A. Rasio, and S. L. Shapiro, Astrophys. J. Suppl. Ser.88, 205 (1993). [22] M. Nauenberg, Astrophys. J.175, 417 (1972).

[23] V. S. Geroyannis and P. J. Papasotiriou, Astrophys. J.534, 359 (2000). [24] P. Wiggins and D. Lai, Astrophys. J.532, 530 (2000).

[25] T. Tanaka, Y. Mino, M. Sasaki, and M. Shibata, Phys. Rev. D54, 3762 (1996). [26] J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys.57, 617 (1985).

[27] V. Karas and D. Vokrouhlick´y, Gen. Relativ. and Grav.24, 729 (1992).

[28] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, England, 1992).

[29] J. Binney and S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, NJ, 1987).

[30] S. Hojman, Ph.D. thesis, Princeton University (1975).

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[32] A. J. Hanson and T. Regge, Ann. Phys. (N.Y.)87, 498 (1974).

[33] S. Hughes,http://www.tapir.caltech.edu/listwg1/EMRI/hughes geodesic.html. [34] L. S. Finn and K. S. Thorne, Phys. Rev. D62, 124021 (2000).

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### Chapter 3

## A survey of spinning test particle

## orbits in Kerr spacetime

This chapter has been accepted for publication in Phys. Rev. D; available as M. D. Hartl, gr- qc/0302103.

### Abstract

We investigate the dynamics of the Papapetrou equations in Kerr spacetime. These equations provide a model for the motion of a relativistic spinning test particle orbiting a rotating (Kerr) black hole. We perform a thorough parameter space search for signs of chaotic dynamics by calculating the Lyapunov exponents for a large variety of initial conditions. We find that the Papapetrou equations admit many chaotic solutions, with the strongest chaos occurring in the case of eccentric orbits with pericenters close to the limit of stability against plunge into a maximally spinning Kerr black hole. Despite the presence of these chaotic solutions, we show that physically realistic solutions to the Papapetrou equations are not chaotic; in all cases, the chaotic solutions either do not correspond to realistic astrophysical systems, or involve a breakdown of the test-particle approximation leading to the Papapetrou equations (or both). As a result, the gravitational radiation from bodies spiraling into much more massive black holes (as detectable, for example, by LISA, the Laser Interferometer Space Antenna) should not exhibit any signs of chaos.

### 3.1

### Introduction

The present chapter continues the investigation initiated in Chapter 2, which considered the dy- namics of spinning test particles (i.e., compact astrophysical objects) orbiting a spinning black hole (Kerr spacetime). The primary objective is to determine whether or not the orbits of spinning com- pact objects spiraling into much more massive black holes are chaotic. Evidence for chaotic orbits in

relativistic binaries has been presented in [1–6] (although only [1] was directly concerned with the ex- treme mass ratio systems we consider here). An astrophysical example of the systems we consider is a solar-mass black hole or neutron star orbiting a supermassive black hole in a galactic nucleus. Such systems are potentially important sources of gravitational radiation in frequency bands detectable by space-based gravitational wave detectors such as the proposed Laser Interferometer Space An- tenna (LISA) mission. The methods of data analysis for signals from such detectors typically rely on matched filters, which use a discrete mesh in parameter space to achieve a given signal-to-noise ratio. The presence of chaos would cause the number of templates needed to fit a given signal to grow exponentially with decreasing mesh size, potentially overwhelming the computers performing the analysis.

Many studies have used the Papapetrou equations to investigate the dynamics of spinning test particles in the background spacetime of a central black hole (including most recently [1, 7–10]; see [9] and [10] for a more thorough literature review). We found in Chapter 2 that the Papapetrou equations formally admit chaotic solutions in Kerr spacetime, but the chaos seemed to disappear for physically realistic spins. This conclusion was tentative, since we investigated only a few values of the many parameters appearing in the equations. In the present study, we undertake a thorough search of parameter space in order to determine the prevalence of chaotic orbits, both for dynamically interesting (but physically unrealistic) orbits with high spin parameter S and for astrophysically relevant systems with smaller spin.

As in Chapter 2, we use Lyapunov exponents to measure the divergence of nearby trajectories and to provide an estimate for the timescale of the divergence. For chaotic systems, initial conditions separated by a small distance0diverge exponentially, with the separation growing as(τ) =0eλτ, whereλis the Lyapunov exponent. For nonchaotic orbits,λ= 0, but whenλis nonzero it provides thee-folding timescale τλ = 1/λ for chaotic behavior to manifest itself. In this chapter we use two

rigorous methods for determining the maximum Lyapunov exponentλmax described in Chapter 2. Further details appear below in Sec. 3.4 and in Sec. III of Chapter 2.

Our parameter space search makes use of a convenient orbital parameterization technique, which allows us to specify desired values of pericenterrp and orbital inclinationι(Sec. 3.3). We vary these

orbital parameters along with the spin magnitude, eccentricity, Kerr spin angular momentuma, and spin inclination, and calculate the largest Lyapunov exponent for each set of initial conditions. A complete description of our methods for searching parameter space appears in Sec. 3.5. Although not exhaustive, the resulting survey of parameter space gives a thorough picture of chaos in the Papapetrou equations with a Kerr background.

We set G=c = 1, use sign conventions as in MTW [11], and use Boyer-Lindquist coordinates (r, θ, φ) for Kerr spacetime. We use vector arrows for 4-vectors and boldface for Euclidean vectors. The symbol log denotes the natural logarithm.

### 3.2

### Spinning test particles

We use the Papapetrou equations of motion [12] to model a spinning test particle. As reformulated by Dixon [13], these equations describe the motion of a “pole-dipole particle,” which neglects effects smaller than those due to the mass monopole and spin dipole (thus neglecting the tidal coupling, which is a mass quadrupole effect). The Papapetrou equations describe a particle with negligible mass compared to the masses responsible for generating the background spacetime. In our case, we writeµfor the test particle’s mass andM for the mass of the central Kerr black hole; the test particle limit then requires thatµM.

### 3.2.1

### Equations of motion

Dixon’s formulation for the equations of motion uses an antisymmetric spin tensorSµν _{to represent}

the spin angular momentum of the particle. The covariant derivative of the 4-momentumpµ_{deviates}

from the nonspinning (geodesic) case by a term representing a coupling of the spin to the background
spacetime curvature. If we writevµ _{for the 4-velocity, the full equations of motion appear as follows:}

dxµ
dτ = v
µ
∇~vpµ = −1_{2}Rµναβv
ν_{S}αβ _{(3.1)}
∇~vSµν = 2p[µvν].

HereRµ_{ναβ} is the Riemann curvature tensor of the spacetime, which in our case corresponds to the
Riemann tensor for the Kerr metric. Note that in the case of vanishing spin (all components ofSµν

equal to zero) or flat spacetime (all components of Rµ_{ναβ} equal to zero), we recover the geodesic
equation∇~vpµ= 0.

For numerical and conceptual purposes, we use a reformulation of the equations of motion in terms of the momentum 1-formpµ and spin 1-formSµ. The spin 1-form can be derived from the

4-momentum and spin tensor using

Sµ=−1_{2}µναβuνSαβ, (3.2)

where uν _{=}_{p}ν_{/µ}_{. The spin 1-form is easier to visualize than the spin tensor, and the Papapetrou}

equations are better-behaved numerically in the S → 0 limit when expressed in terms of the spin 1-form. Details of this formulation appear in Chapter 2 and the appendix.

### 3.2.2

### General constraints

The Papapetrou system of equations (3.1) is highly constrained, even in arbitrary spacetimes. The 4- momentum and spin satisfy normalization conditions that are conserved by the equations of motion:

pµpµ=−µ2 (3.3)
and
1
2S
µν_{S}
µν =S2. (3.4)

A further condition is required to identify a unique worldline for the center of mass:

pµSµν = 0. (3.5)

A more detailed discussion of this “spin supplementary condition” appears in Chapter 2.

### 3.2.3

### Kerr constraints

The Papapetrou equations share an important property with geodesic motion, namely, to each symmetry of the background spacetime (typically represented by a Killing vector) there corresponds a constant of the motion. Kerr spacetime has two such symmetries, which provide two constraints in addition to those described in the previous section. (The Killing tensor that leads to the Carter constant in the geodesic does not correspond to a conserved quantity when the spin is nonzero; see Sec. 3.3.2.3 below.)

To each Killing vector ξ~ there corresponds a constant given by the standard expression for geodesics, plus a contribution due to a coupling with the spin tensor:

Cξ =ξµpµ−_{2}1ξµ;νSµν. (3.6)

Kerr spacetime’s two Killing vectors ~ξt _{=} _{∂/∂t} _{and} ~_{ξ}φ _{=} _{∂/∂φ} _{then give energy and} _{z} _{angular}

momentum conservation:

−E=pt−1_{2}gtµ,νSµν (3.7)

and

Jz=pφ−1_{2}gφµ,νSµν. (3.8)

### 3.2.4

### The spin parameter

### S

The spin magnitudeSthat appears in Eq. (3.4) quantifies the size of the spin and thus plays a crucial role in determining the behavior of spinning test particle systems. As in Chapter 2, we measure

distances and times in terms of M and momenta in terms of µ. In these units, S is measured in units of µM. When measured in traditional geometrized units, the spin of a black hole can be as large as its mass squared, i.e., Sgeom,max = µ2. In this case the spin parameter goes like

S ∼µ2_{/}_{(}_{µM}_{) =}_{µ/M}_{, which is necessarily small for a test particle system. This result applies to}
all astrophysically relevant systems, as shown in Chapter 2. Thus, we have the important physical
constraint that the spin parameter S must satisfy S 1 for all physically realistic systems. A
thorough discussion of various possibilities (including neutron stars and white dwarfs) in Chapter 2
showed that realistic values ofS for LISA sources fall in the range 10−4–10−7.

The smallness of the spin’s effect is not dependent on our choice to measure S in terms of µM.
If instead we measureSin terms ofµ2_{, then the equation of motion for}_{S}µν _{becomes (rewriting}_{∇}

~
v
asD/dτ)
D(Sµν_{/µ}2_{)}
d(τ /M) = 2
1
µp
[µ_{v}ν]_{,} _{(3.9)}

where we have factored out from each variable its corresponding scale factor. This gives

DSµν dτ = µ M 2p[µvν], (3.10)

so that in these units the spin’s effective magnitude is suppressed by a factor ofµ/M. MeasuringSin terms ofµMeffectively absorbs the unavoidable smallness of the spin effect into the spin parameter itself.