The previous definition of the ICO is motivated by our comparison with the results of numerical
relativity. Indeed we shall confront the prediction of the standard (Taylor-based) post-Newtonian approach
with a recent result of numerical relativity by Gourgoulhon, Grandclément, and Bonazzola [123, 126
].
These authors computed numerically the energy of binary black holes under the assumptions of conformal
flatness for the spatial metric and of exactly circular orbits. The latter restriction is implemented by
requiring the existence of an “helical” Killing vector, which is time-like inside the light cylinder associated
with the circular motion, and space-like outside. In the numerical approach [123
, 126
] there are no
gravitational waves, the field is periodic in time, and the gravitational potentials tend to zero
at spatial infinity within a restricted model equivalent to solving five out of the ten Einstein
field equations (the so-called Isenberg-Wilson-Mathews approximation; see Ref. [114] for a
discussion). Considering an evolutionary sequence of equilibrium configurations Refs. [123
, 126
]
obtained numerically the circular-orbit energy
and looked for the ICO of binary black
holes (see also Refs. [52, 124, 154] for related calculations of binary neutron and strange quark
stars).
Since the numerical calculation [123, 126
] has been performed in the case of corotating black
holes, which are spinning with the orbital angular velocity
, we must for the comparison
include within our post-Newtonian formalism the effects of spins appropriate to two Kerr black
holes rotating at the orbital rate. The total relativistic mass of the Kerr black hole is given
by34
To take into account the spin effects our first task is to replace all the masses entering the energy
function (194) by their equivalent expressions in terms of
and the two irreducible masses. It is clear
that the leading contribution is that of the spin kinetic energy given by Equation (199
), and it comes from
the replacement of the rest mass-energy
(where
). From Equation (199
) this effect is
of order
in the case of corotating binaries, which means by comparison with Equation (194
) that it
is equivalent to an “orbital” effect at the 2PN order (i.e.
). Higher-order corrections
in Equation (199
), which behave at least like
, will correspond to the orbital 5PN order
at least and are negligible for the present purpose. In addition there will be a subdominant
contribution, of the order of
equivalent to 3PN order, which comes from the replacement
of the masses into the “Newtonian” part, proportional to
, of the energy
(see
Equation (194
)). With the 3PN accuracy we do not need to replace the masses that enter into the
post-Newtonian corrections in
, so in these terms the masses can be considered to be the irreducible
ones.
Our second task is to include the specific relativistic effects due to the spins, namely the spin-orbit (SO)
interaction and the spin-spin (SS) one. In the case of spins and
aligned parallel to the orbital
angular momentum (and right-handed with respect to the sense of motion) the SO energy reads
|
In conclusion, we find that the location of the ICO as computed by numerical relativity, under the
helical-symmetry and conformal-flatness approximations, is in good agreement with the post-Newtonian
prediction. See also Ref. [88] for the results calculated within the effective-one-body approach
method [60, 61
] at the 3PN order, which are close to the ones reported in Figure 1
. This agreement
constitutes an appreciable improvement of the previous situation, because the earlier estimates of
the ICO in post-Newtonian theory [145] and numerical relativity [180, 9] strongly disagreed
with each other, and do not match with the present 3PN results. The numerical calculation of
quasi-equilibrium configurations has been since then redone and refined by a number of groups, for both
corotational and irrotational binaries (see in particular Ref. [74]). These works confirm the previous
findings.
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