**4.4 The extreme mass-ratio case**

**4.4.2 Comparing post-Newtonian and Papapetrou orbits**

We mentioned in the introduction the investigations of the dynamics of a spinning test particle in a background spacetime. ThePapapetrou equations model this system; if we choose the background to be the Kerr metric (as in Chapters 2 and 3), then the Papapetrou equations model the orbit of a small rotating black hole orbiting a supermassive rotating black hole. The Papapetrou system should therefore agree closely with the post-Newtonian equations whenm1m2andr=|X| M. We can compare the PN equations directly with the Papapetrou equations (as reformulated by Dixon [16]) by examining the covariant time-derivative of the 4-momentum:

Dpµ
dτ =−
1
2R
µ
ναβv
ν_{S}αβ_{,} _{(4.23)}

whereRµ_{ναβ} is the Riemann tensor of the spacetime,vν _{is the 4-velocity, and}_{S}αβ _{is the spin tensor.}

For vanishing spin, this is simply the geodesic equation, so the Papapetrou equations include the Newtonian, 1PN, and 2PN terms. Since the Kerr metric is an exact description of a rotating black hole, the geodesic equation also includes theS1part of the spin-orbit coupling in Eq. (4.6), as well as theS1S1quadrupole term [Eq. (4.10)]. [Here we identifyS1/m1with the Kerr parametera, the spin angular momentum per unit mass. This is valid in the extreme mass-ratio limit, but see Eq. (4.17) above for the more general expression.]

For nonvanishing spin, the right-hand term in Eq. (4.23) is known to reproduce the post-
Newtonian spin-orbit coupling to leading order when expanded in v2_{/c}2 _{(that is, 1PN) [17]. This}
means that the spatial part of the expression 1

2R

µ ναβv

ν_{S}αβ_{, expanded to lowest order in}_{v}2_{/c}2_{, is}
equal to the piece ofdP/dtgiven byHSOin Eq. (4.1). (There is an important subtlety here: the def-
initions for the center of mass used for the Papapetrou equations and the Hamiltonian PN equations
differ by a factor of order 1PN: Xcm,PN=Xcm,Papa+_{2}1_{µ}v×S. The correspondence between the

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Figure 4.8: The orbit of two maximally spinning black holes in the extreme mass ratio (m1

m2) limit, using the post-Newtonian equations (dark) and the Papapetrou equations (light). The Papapetrou initial conditions satisfy the conditions of eccentricitye= 0.5, pericenterrp= 30, and

orbital inclinationι= 10◦. The analogous PN orbit starts with the same position and momentum. In both cases, the spin of the small body initially points 45◦ from the vertical. The PN equations of motion include all the PN terms from Sec. 4.2.1 except theS2S2 quadrupole term.

Papapetrou spin-orbit coupling and the PN spin-orbit coupling is only evident once this substitution has been made.) To higher order, the right-hand side of Eq. (4.23) includes products ofS with the Riemann tensor, which is a function of a =S1/m1; since S in the Papapetrou equations is S2 in our treatment of the PN equations, this means that Eq. (4.23) should include the S1S2 coupling term [Eq. (4.9)], although this has not to our knowledge ever been verified explicitly. Finally, the Papapetrou equations model apole-dipole particle, so they do not include theS2S2quadrupole term of the smaller body [Eq. (4.11)].

We can compare Papapetrou and PN orbits more directly by constructing analogous initial conditions. We first select values of the eccentricitye, the pericenterrp, and the orbital inclination

angleι for the Papapetrou equations, as discussed in Chapter 3. We then use the resulting values of position, momentum, and spin as the initial conditions for the PN equations of motion (with the

S2S2term turned off). The orbits agree closely, especially at higher radii, as shown in Fig. 4.8. The quantitative agreement is worse at lower radii due to the violation of the conditionrM required for the validity of the PN equations, but the qualitative behavior—especially the precession of the orbital plane—is essentially the same.

As a more quantitative comparison, we can use the PN equations to investigate the errors in the Papapetrou system caused by deviations from the test particle approximation. As discussed at length in Chapter 2, if we measure distances in terms of the central massMP and momenta in terms

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0Figure 4.9: The error in the Papapetrou radial periodTr,P vs. mass ratio for an eccentric (e= 0.5)

orbit with pericenter rP = 30. Here ∆ is the fractional difference between the Papapetrou radial

period and the post-Newtonian radial period: ∆ =Tr,P/Tr,PN−1. Even in them2→0 limit, we are unable to find initial conditions leading to exactly the same periods, which we correct by subtracting the fractional difference ∆0, which is the extrapolation of ∆ to the case of vanishing mass ratio. (In this case, ∆0= 0.003.) It is evident that the Papapetrou radial period differs significantly from the corresponding PN period whenm2&0.1m1, indicating a failure of the test-particle approximation.

of the particle massµP, then the spinS of the small body is measured in terms ofµPMP. (Here we

use the subscriptP for Papapetrou, to distinguish the Papapetrou masses from the total massM

and reduced massµ used for the PN system.) This means that a small maximally spinning black hole in a maximal Kerr background has a spin parameter of

S= maximum spin µPMP = µ 2 P µPMP = µP MP 1. (4.24)

The last step uses the requirement that the second body be a test particle. Deviations from this approximation are equivalent to increasing the value ofS, so thatS = 1 corresponds toµP =MP,

i.e., an equal mass binary.

Our strategy is to consider initial conditions where both the Papapetrou system and PN system are valid: orbits with extreme mass ratios (Papapetrou valid) and relatively large pericenters (PN valid). Using the average radial periodTr, we can compare the two systems quantitatively as we vary

the mass ratio: from its initial small value (taken to be 10−5) we increase m2 until the mass ratio is 1. Since the PN equations are valid in the equal mass (m1 =m2) limit, the difference between the Papapetrou and PN values ofTr is due to the violation of the test particle approximation. The

results appear in Fig. 4.9 for an eccentric orbit and Fig. 4.10 for a near-circular orbit. In both cases, the Papapetrou period deviates significantly from the PN value whenm2&0.1m1, which indicates a failure of the test-particle approximation.

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0Figure 4.10: The error in the Papapetrou radial period Tr,P vs. mass ratio for a near-circular

(e= 0.05) orbit with pericenterrP = 30. As in Fig. 4.9, ∆ is the fractional difference between the

Papapetrou radial period and the post-Newtonian radial period (∆ =Tr,P/Tr,PN−1), and ∆0is the extrapolation of ∆ to the case of vanishing mass ratio. (In this case, ∆0= 0.001.) The Papapetrou radial period differs significantly from the corresponding PN period when m2&0.1m1, indicating a failure of the test-particle approximation.