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The reduced Hamiltonian for next-to-leading-order
spin-squared dynamics of general compact binaries
Steven Hergt, Jan Steinhoff, Gerhard Schäfer
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Steven Hergt, Jan Steinhoff, Gerhard Schäfer. The reduced Hamiltonian for next-to-leadingorder spin-squared dynamics of general compact binaries. Classical and Quantum Gravity, IOP
Publishing, 2010, 27 (13), pp.135007. .
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Confidential: not for distribution. Submitted to IOP Publishing for peer review 10 February 2010
Reduced Hamiltonian for next-to-leading order
Spin-Squared Dynamics of General Compact
Binaries
Steven Hergt, Jan Steinhoff and Gerhard Schäfer
Theoretisch–Physikalisches Institut, Friedrich–Schiller–Universität,
Max–Wien–Platz 1, 07743 Jena, Germany, EU
PACS numbers: 04.25.Nx, 04.20.Fy, 04.70.Bw, 97.60.Jd
Abstract. Within the post Newtonian framework the fully reduced Hamiltonian
(i.e., with eliminated spin supplementary condition) for the next-to-leading order
spin-squared dynamics of general compact binaries is presented. The Hamiltonian
is applicable to the spin dynamics of all kinds of binaries with self-gravitating
components like black holes and/or neutron stars taking into account spin-induced
quadrupolar deformation effects in second post-Newtonian order perturbation
theory of Einstein’s field equations. The corresponding equations of motion for
spin, position and momentum variables are given in terms of canonical Poisson
brackets. Comparison with a nonreduced potential calculated within the Effective
Field Theory approach is made.
NLO Spin-Squared Hamiltonian for Compact Binaries
2
1. Introduction
A crucial prediction of Einstein’s theory of General Relativity is the existence of
gravitational waves (GWs), e.g. resulting from the inspiralling and merging process of
two compact objects. Up till now those waves are purely theoretical predictions with
lack of direct experimental verification, but their direct detection is under preparation
by gravitational wave observatories on Earth, e.g., LIGO, VIRGO, GEO, or LISA, a
future space mission [1].
A first indirect evidence for the existence of GWs was the observation of energy
loss in the orbital motion of the Hulse-Taylor binary pulsar PSR B1913+16 being
in full agreement with the predictions of Einstein’s theory using the quadrupole
radiation formula. This discovery was awarded the Nobel Prize in 1993. In the
meantime another strong indirect evidence occurred with the double pulsar system
PSR J0737-3039A and B [2]. For analysis of the measured GW patterns one has
to provide very accurate templates following from theory. This can be achieved by
numerical calculations with the matching of functions to the results or directly using
analytic tools. In the latter case, waveforms can be obtained only perturbatively
due to missing analytic solutions of the Einstein field equations for two or more
(spinning) compact objects (black holes, neutron stars). One of the most successful
approximation methods is the post-Newtonian one in which the metric keeps close to
the flat spacetime relying on the assumption that the typical velocity v in a system
divided by the speed of light c is always small, v/c ∼ 1. The deviations from the
flat metric can be characterized by the Newtonian potential Φ; so for a binary system,
Φ/c2 ∼ v 2 /c2 ∼ 2 . In an appropriate limit (as → 0), the post-Newtonian (PN)
approximation yields Newton’s equations.
The merging process of two compact objects is divided into three time scale
sectors: inspiral, merger, and ring-down. Each sector delivers characteristic theoretical
wave patterns, that are hoped to be matched against measured signals in the future.
The PN approximation provides an excellent analytic handling for the inspiral phase.
If the PN calculations are very accurate and thus high in order one can make very
sensible predictions when comparing with measured signals. In this article we focus on
calculations of the next-to-leading order (NLO) dynamics of spin-induced quadrupolar
deformation effects. Surely, the most compact dynamical object is the Hamiltonian,
generating the equations of motion, so we calculate in section 2 the NLO spin-squared
one including a constant CQ parameterizing spin-induced quadrupolar deformation
effects. CQ can be given definite values describing black holes (BHs) or neutron stars
(NSs). For neutron stars CQ also depends on the model or equation of state (EoS).
Thus our result, as it seems to be necessary to accurately measure CQ within future
GW astronomy, could help to find the right EoS. The Hamiltonian in the present
paper is calculated within the canonical formalism of Arnowitt, Deser, and Misner
(ADM) [3]. It should be noted that our Hamiltonian is fully reduced in the sense that
the spin supplementary condition (SSC) is eliminated on the level of the Hamiltonian.
Further we make a formal counting of the spin as c0 and do not distinguish between
fast and slowly spinning objects (see, e.g., [4] and also Appendix A of [5]).
By now there are a lot of results regarding spin effects at the conservative orders
in the PN approximation. The leading order (LO) spin effects are well-known for black
holes, see, e.g., [6, 7, 8, 9, 10]. The LO CQ -dependence is given in [8, 11]. The NLO
spin effects were only tackled recently. The first derivation of the NLO spin-orbit (SO)
equations of motion (EoM) is given in [12] which became further developed in [13],
NLO Spin-Squared Hamiltonian for Compact Binaries
3
both in harmonic gauge. Later, within the ADM canonical formalism, a Hamiltonian
presentation was achieved [14] (see also [15]). The NLO spin(1)-spin(2) dynamics was
found in [16, 15] and confirmed by [17, 18]. Higher PN orders linear in spin were
tackled recently in [19, 20, 5]. In particular, Ref. [19] extended the point-mass ADM
formalism to spinning objects, valid to any order linear in spin. Even Hamiltonians
of cubic and higher order in spin were obtained for binary black holes (BBHs) [21, 4].
Besides quadrupolar deformations induced by proper rotation (spin) and treated in
the present paper, tidal deformations induced through the gravitational field of the
other object were also treated, see, e.g., [22, 23, 24].
A nonreduced potential (i.e., with SSC not eliminated on the level of the potential)
corresponding to the result of the present paper was already calculated in [25], however,
a term relevant for the center-of-mass motion was missing and only found recently [26].
A comparison with the result of the present paper is not trivial, if one wants to avoid
comparing all (rather long) equations of motion; instead it is more efficient to stay on
the level of the (relatively short) potential. In section 3 we sketch how to transform the
potential from [25, 26] into a reduced Hamiltonian where we will find full agreement
with our result of the present paper. Considering the special case of black holes (or
CQ = 1), we already succeeded in calculating the Hamiltonian of the present paper in
[27, 4], providing for the first time both the spin and correct center-of-mass dynamics
in this case. There we were only able to find agreement with [25] in the spin precession
equation, see [28] (after identifying a sign typo in [25]). With the correction in [26] a
full comparison can now be attempted. It will be provided in section 3.
More work needs to be done for an application of the result of the present paper
to GW astronomy. In order to obtain the NLO radiation field (for the SO case see
[29, 30]) the stress-energy tensor has to include spin-squared corrections. This stressenergy tensor arises from the one with a general quadrupole [31] by a spin-squared
ansatz for the mass-quadrupole, see [27]. Moreover, the NLO spin contribution should
be of importance for data analysis. In a recent publication [32] it has been shown that
for maximal spins (aligned with the total orbital angular momentum), the event rates
are roughly thirty times larger than of those matter systems with anti-aligned spins to
orbital angular momentum and eight times as large as for non-spinning binaries. So
especially considering such sources the event rate will increase with the inclusion of
spin effects. Further, for the creation of templates, it is useful to find a parametrization
of the orbits by solving the EoM. It is common to describe the conservative dynamics
in terms of certain orbital elements. Spin precession and dissipative effects can then
be formulated as secular EoM of the orbital elements. For explicit solutions including
spin at LO SO see, e.g., [33, 34].
2. The NLO Spin-Squared Hamiltonian
We start with giving a short overview concerning the calculation of the Hamiltonian
in question. This calculation is done within the ADM canonical formalism [3]. We use
units in which 16πG = c = 1, where G is the Newtonian gravitational constant and c
the velocity of light. Greek indices will run over 0, 1, 2, 3, Latin over 1, 2, 3. For the
signature of spacetime we choose +2. We employ the following notations: x = xi
(i = 1, 2, 3) denotes a point in the 3-dimensional Euclidean space R3 endowed with a
standard Euclidean metric and a scalar product (denoted by a dot). Letters a and b
are body labels (usually they are set to 1 or 2), so xa ∈ R3 denotes the position of
the ath point mass. We also define ra := x − xa , ra := |ra |, na := ra /ra ; and for
NLO Spin-Squared Hamiltonian for Compact Binaries
4
a 6= b, rab := xa − xb , rab := |rab |, nab := rab /rab ; | · | stands here for the length
of a vector. The linear momentum vector of the ath body is denoted by pa = (pai ),
and ma denotes its mass parameter. The usual flat space spin vector of the ath
body (in local coordinates) is denoted by Sa = Sa(i) in correspondence with our
paper [27] and its associated antisymmetric tensor by Sa(i)(j) = ijk Sa(k) with the
total antisymmetric -symbol defined as 123 = 1. We abbreviate δ (x − xa ) by δa .
The partial differentiation with respect to xi is denoted by ∂i or by a comma, i.e.,
∂i φ ≡ φ,i ; the partial differentiation with respect to xia we denote by ∂ai .
Following the ADM canonical formalism, the independent degress of freedom
of the gravitational field are described by hTijT , the transverse-traceless part of
T
= 0), and by conjugate momenta π ij T T . The needed
hij = gij − δij (hTiiT = 0, hTij,j
energy and linear momentum density expressions are given by
1
γ 2 T µν nµ nν = Hm(atter) ,
1
2
µ
−γ Ti nµ =
m(atter)
Hi
,
(1)
(2)
where γ = det(gij ), γ ij is inverse to gij , nν is a unit timelike normal to hypersurface
x0 = const, and T µν is the energy-momentum tensor of the matter system. Hereafter,
we call its constituents the “particles”, but they may well represent neutron stars or
black holes. This is substantiated by “general relativity’s adherence to the strong
equivalence principle”: black holes and other compact bodies, to some approximation,
obey the same laws of motion as test bodies; see, e.g., [35]. Also, the analysis of the
initial-value solutions for black holes shows that as in electromagnetism, where image
charges are described by delta functions, black holes in full general relativity can be
represented by “image masses” with delta functions support [36]. It is convenient to
choose the following four coordinate conditions
1
(3)
π ii = 0 , gij = ψ 4 δij + hTijT , ψ = 1 + φ .
8
The standard ADM Hamiltonian (cf. [3])
I
H = dSi (gij,i − gjj,i ),
(4)
then becomes, using the Gauss theorem,
Z
H = − d3 x∆φ .
(5)
The integrand ∆φ = ∂i ∂i φ can be expressed in terms of xa , pa , Sa , hTijT and π ij T T
using the constraint equations. By expansions of the field equations in powers of G
and after adopting suitable regularization procedures of integrals involved (see, e.g.,
Ref. [37] and the Appendix in [38]), one can determine the Hamiltonian.
Following the procedure outlined in our previous papers [4, 27], the Hamiltonian
and the other generators are constructed as to fulfill the Poincaré algebra up to 2PN
order depending on standard canonical variables
{xia , pbj }
= δji δab ,
{Sa(i) , Sa(j) } = ijk Sa(k) ,
(6)
(7)
with all other brackets being zero. The coefficient equations resulting from this
procedure will change due to a modified Hamiltonian and CoM vector entering the
NLO Spin-Squared Hamiltonian for Compact Binaries
5
crucial relation {Gi , H} = Pi see Equation (2.4) in [4]. The modification leading to
spin quadrupolar deformation effects of a general compact object has to be made in the
leading order spin-squared Hamiltonian labeled as HS12 in equation (2.8) in [4], which
now has to include a general spin-quadrupole constant CQ see [11]; additionally an
appropriate S12 -CoM vector has to be found. Both can be accomplished by adopting
the static source expression for Hm from our paper [27] equation (4) and incorporating
the CQ constant reading
c1 ij 1
I δ1
+
gmn γ pj γ ql γ mi,p γ nk,q Ŝ1ij Ŝ1kl δ1
HSmatter
2 ,static =
1
m1 1
8m1
;ij
1 ij mn kl
+
,
(8)
γ γ γ ,m Ŝ1ln Ŝ1jk δ1
4m1
,i
1
c1 = − CQ ,
(9)
2
meaning CQ = 1 for BH. Symbolic abbreviations in this formula are taken unaltered
from the original paper thus denoting the same mathematical objects. This means
that S(i) being given in an Euclidean basis can be related to a spin tensor Ŝij in a
coordinate basis with the help of a triad (dreibein) ei(j) by Ŝij = ei(k) ej(l) klm S(m) .
The dreibein as a function of the metric is just ei(j) = ψ 2 δij because the metric can
be taken as conformally flat, gij = ψ 4 δij , in our approximation. The mass-quadrupole
tensor of object 1, I1ij , is given by
2
I1ij ≡ γ ik γ jl γ mn Ŝ1km Ŝ1nl + S21 γ ij ,
3
2S21 = γ ik γ jl Ŝ1ij Ŝ1kl = const .
(10)
(11)
(i)(j)
2
2
2 2
The relation to Qij
= m21 Qij
1 and a1 is I1
1 and S1 = m1 a1 , so in leading order the
ij
related quadrupole-moment tensor Q1 is just given by
1 2 ij
(i) (j)
Qij
(12)
1 = a1 a1 − a1 δ
3
This static source alone is also enough to determine all the G2 terms (static,
free of linear momenta) of the Hamiltonian in question. The LO spin-squared
R
HamiltonianRand the S12 -CoM vector are calculated via the formulae H = − d3 x∆φ
and Gi = − d3 x xi ∆φ, respectively, with a post-Newtonian perturbatively expanded
∆φ and HSmatter
according to equations (4.14) - (4.16) in [4]. The results are
2
1 ,static
G m1 m2
S21
(S1 · n12 )2
CQ
HS 2 =
CQ 3
− 2 ,
(13)
3
1
2 r12
m21
m1
and
GS12
"
#
2
(S1 · n12 ) S1
m2
(S1 · n12 )
S21
ν1
=G
+
(ν2 x1 + ν3 x2 ) + 3 (ν4 x1 + ν5 x2 )
(14)
2
3
m1
r12
r12
r12
with coefficients
1 3
ν1 = − − CQ ,
2 2
1 3
ν5 = − − CQ .
2 4
ν2 =
3
CQ ,
4
ν3 =
3
CQ ,
4
ν4 =
1 1
+ CQ ,
2 4
(15)
NLO Spin-Squared Hamiltonian for Compact Binaries
6
The non-static (momenta based) terms of the Hamiltonian will be determined via the
same ansatzes for the source terms in Hm and Him as in [4] with the LO quadrupole
moment (12) reading
"
2
X
1
mb
p2
ij
2 1/2
+ λ1 b Qij
∂i ∂j
CQ Qij
Hm =
−
b ∂i ∂j − pb · (ab × ∂) + γ pbi pbj + mb
2
2
2mb b
b=1
#
λ2
λ3 2
ij
ij
2
+
(pb · ∂)Qb pbi ∂j +
a (pb · ∂) − λ8 pb · (ab × ∂) Qb ∂i ∂j δb ,
(16)
mb
mb b
Him
= −2
2
X
"
Qkl
p
∂
∂
+
λ
p
∂
∂
+
λ
(p
·
∂)
δ
∂
λ
5 bk l i
6 bi k l
7
b
li k
b
b=1
+
λ4 a2b (pb
#
1 kl
1
mb
(ab × ∂)i 1 − Qb ∂k ∂l − pbi δb .
· ∂)∂i +
4
6
2
(17)
Notice that the static term in Hm involves the CQ constant which follows from the
expansion of (8) being the only modification of Eq. (4.11) in [4]. These sources
allow the calculation of the NLO spin-squared Hamiltonian (with yet undetermined
coefficients) leading to the same coefficient equations (4.50) - (4.62) in [4] except for
the test particle terms β2 and β4 which are just multiplied by the quadrupole constant
CQ . These equations have to be matched to the ones resulting from the requirement of
fulfilling the Poincaré algebra which now include the CQ constant and for that reason
will slighty differ from the equations (3.8) - (3.22) in [4]. The matching procedure then
fixes all the coefficients left attributing to the source term coefficients the values
7 3
5 3
1
λ1 = − CQ , λ2 = − + CQ , λ3 = − , λ4 = 0,
(18)
4 2
4 2
24
CQ
1
1 CQ
1
λ5 =
−
, λ6 = − +
, λ7 = ,
(19)
12
6
8
4
8
which agrees with our results obtained in [4] when setting CQ = 1. In view of section 3
we label from now on standard canonical variables with a ‘hat’ specifying its affiliation
to the Newton-Wigner (NW) SSC except for the momentum which is chosen to be the
same for NW and covariant SSC. The ‘hated’ variables are then called NW variables
in the sense that they are standard canonical, meaning
{x̂ia , p̂bj } = δji δab ,
{Ŝa(i) , Ŝa(j) } = ijk Ŝa(k) ,
p̂ai = pai ,
(20)
(21)
(22)
all other brackets being zero. Subtleties arising from that definition of NW variables
are discussed in detail in our Comment [28]. The resulting NLO spin-squared
Hamiltonian for general compact binaries reads
ADMcan
HNLO
S12
"
G m2
= 3
r̂12 m31
15
−
+
4
21 9
− + CQ p21 (Ŝ1 · n̂12 )2
8
4
9
5 3
CQ (p1 · n̂12 )(Ŝ1 · n̂12 )(Ŝ1 · p1 ) + − + CQ (Ŝ1 · p1 )2
2
4 2
NLO Spin-Squared Hamiltonian for Compact Binaries
7
!
9 3
5 5
2 2
2 2
− CQ p1 Ŝ1
+ − + CQ (p1 · n̂12 ) Ŝ1 +
8 2
4 4
1
15
− CQ (p1 · n̂12 )(p2 · n̂12 )(Ŝ1 · n̂12 )2
m21
4
21
+ 3 − CQ (p1 · p2 )(Ŝ1 · n̂12 )2
4
3 9
+ − + CQ (p2 · n̂12 )(Ŝ1 · n̂12 )(Ŝ1 · p1 )
2 2
3
+ −3 + CQ (p1 · n̂12 )(Ŝ1 · n̂12 )(Ŝ1 · p2 )
2
3 3
− CQ (Ŝ1 · p1 )(Ŝ1 · p2 )
+
2 2
3 3
− CQ (p1 · n̂12 )(p2 · n̂12 )Ŝ21
+
2 4
!
#
9
C
3 9
3
Q
p2 (Ŝ1 · n̂12 )2 − p22 Ŝ21
+ − + CQ (p1 · p2 )Ŝ21 +
2 4
m1 m2 4 2
4
"
m2 G2 m2
1
Ŝ21
+ 4
2 + CQ +
1 + 2CQ
r̂12
2
m1
#
3
m2 (23)
1 + 6CQ
(Ŝ1 · n̂12 )2 ,
+ −3 − CQ −
2
m1
+
for CQ = 1 being in full agreement with the result for BH presented for the first time
in [27].
3. Comparison with the NLO spin(1)spin(1) potential
In order to transform the Routhian R obtained within the Effective Field Theory
(EFT) approach [25, 26] to a nonreduced Hamiltonian H, we first have to eliminate
the acceleration term [26] with the help of the Newtonian equations of motion
(corresponding to a redefinition of the position variables, see [39]). This generates
corrections terms to the order G2 in the NLO spin-squared potential. Next, one
∂R
must replace velocities by canonical momenta pa = ∂v
to get the Hamiltonian by a
a
Legendre transformation, i.e.,
H = v1 · p 1 + v2 · p 2 − R .
(24)
The canonical momentum p1 necessary to cover all NLO spin effects explicitly reads
1 2
Gm1 m2
[6v1 − 7v2 − (n12 · v2 )n12 ]
p 1 = 1 + v1 m1 v1 +
2
2r12
G
+ 2 [m2 (n12 × S1 ) + 2m1 (n12 × S2 )] ,
(25)
r12
and similarly for particle 2. The Poisson brackets at this stage are
{xia , pbj }
{Sa(i) , Sa(j) }
= δji δab ,
=
ijk Sa(k)
(26)
,
(27)
NLO Spin-Squared Hamiltonian for Compact Binaries
{Sa(i) , Sa(0)(j) }
{Sa(0)(i) , Sa(0)(j) }
8
= ijk Sa(0)(k) ,
(28)
−ijk Sa(k)
(29)
=
,
and zero otherwise.‡ Notice that these are not yet the reduced or standard canonical
(0)(i)
is still an independent degree of freedom and was not eliminated
brackets as Sa
using the covariant SSC Saµν uν = 0.
(0)(i)
It is well known that one has to proceed to Dirac brackets (DBs) if Sa
is going
to be eliminated from the Hamiltonian H using a SSC, see, e.g., [41, 40]. However, it
(j)
is possible to find new variables x̂ia , p̂bj and Ŝa for which the Dirac brackets take on
the standard form,
{x̂ia , p̂bj }DB
{Ŝa(i) , Ŝa(j) }DB
= δji δab ,
=
ijk Ŝa(k)
(30)
,
(31)
and zero otherwise. These new variables can only be unique up to canonical
transformations. This freedom allows us to choose p1i = p̂1i , as for the flat space
case [40]. A possible transition to x̂ia and Ŝaj then reads,
p1[i Ŝ1(j)](k) p1k
3p21
2Gm2
S1(i)(j) = Ŝ1(i)(j) −
1−
− 2 p1[i Ŝ1(j)](k) p1k
m21
4m21
m1 r̂12
3G
G
+
p1[i Ŝ1(j)](k) p1k +
p1[i Ŝ1(j)](k) n̂k12 (n̂12 · p2 )
m1 r̂12
m1 r̂12
2G
2Gm2
k
p
n̂
+ 2 2 p1[i Ŝ1(j)](l) Ŝ1(k)(l) n̂k12 +
(32)
Ŝ
Ŝ
1[i
1(j)](l)
2(k)(l)
12 ,
2
m1 r̂12
m1 r̂12
x1i = x̂1i −
1
p21
m2 p1k Ŝ1(i)(k)
p
1
−
−G 2
Ŝ
1k 1(i)(k)
2m21
4m21
m1
r̂12
3 p2k Ŝ1(i)(k)
G n̂k12 (n̂12 · p2 )Ŝ1(i)(k)
+ G
+
2
m1 r̂12
2
m1 r̂12
k
Ŝ
Ŝ
n̂
n̂k12 Ŝ1(i)(l) Ŝ2(k)(l)
m2 1(k)(l) 1(i)(l) 12
+G 2
+
G
,
2
2
m1
r̂12
m1 r̂12
(33)
where the antisymmetrization of indices pertaining to a tensor Aij is defined as
A[ij] = 1/2(Aij − Aji ). The rather complicated form of these variable transformations
reflects the complicated structure of the DBs for self-interacting spinning objects in the
covariant SSC. We will elaborate on its specific calculation in another paper. Notice
that these results are applicable to all NLO spin effects (for the spin-orbit contributions
the corrected form of the tetrad from [28] has to be inserted into the SSC). To best
of our knowledge this is the first time that DBs are applied to gravitationally selfinteracting spinning objects. In [20] test-spinning objects are considered and [40]
covers the flat space case only.
The Routhian from [25] now leads us to the reduced NLO spin-squared
Hamiltonian in the form
"
G
m2
21 9
EFTcan
− + CQ p21 (Ŝ1 · n̂12 )2
HNLOS 2 = 3
1
r̂12 m31
8
4
‡ Notice that in [17, 25] a different sign convention was used for the Poisson brackets of the spin. As
usual, we are not showing the canonical conjugate of the spin here; see [40].
NLO Spin-Squared Hamiltonian for Compact Binaries
9
21 9
7 3
− CQ (p1 · n12 )(Ŝ1 · n̂12 )(Ŝ1 · p1 ) + − + CQ (Ŝ1 · p1 )2
+
4
2
4 2
!
21 9
7 9
2 2
2 2
+ − + CQ (p1 · n̂12 ) Ŝ1 +
− CQ p1 Ŝ1
8
2
4 4
1
15
− CQ (p1 · n̂12 )(p2 · n̂12 )(Ŝ1 · n̂12 )2
m21
4
21
+ 3 − CQ (p1 · p2 )(Ŝ1 · n̂12 )2
4
9
+ −3 + CQ (p2 · n̂12 )(Ŝ1 · n̂12 )(Ŝ1 · p1 )
2
3
+ −3 + CQ (p1 · n̂12 )(Ŝ1 · n̂12 )(Ŝ1 · p2 )
2
15
3
+ 2 − CQ (Ŝ1 · p1 )(Ŝ1 · p2 ) + 3 − CQ (p1 · n̂12 )(p2 · n̂12 )Ŝ21
2
4
!
#
C
9
13
3
Q
p2 (Ŝ1 · n̂12 )2 − p22 Ŝ21
+ −2 + CQ (p1 · p2 )Ŝ21 +
4
m1 m2 4 2
4
"
m2 1
G2 m2
1
Ŝ21
+ 4
+ 3CQ
2 + CQ +
r̂12
2
m1 2
#
3
m2 1
+ 6CQ
(34)
(Ŝ1 · n̂12 )2 .
+ −3 − CQ −
2
m1 2
+
This Hamiltonian and the one calculated in the last section should differ only up
to a canonical transformation. It should thus be possible to generate the difference
EFTcan
ADMcan
by a canonical transformation of the form
∆HNLOS12 = HNLOS
2 − HNLOS 2
1
1
can
∆HNLOS12 = {HN , gNLOS
2} ,
1
(35)
with HN being the Newtonian Hamiltonian of a two-body system and g being an
appropriate generator. It turns out that with the generator
"
#
G m2
1
1
can
2
(36)
gNLOS 2 = 2
− + CQ (p1 · n̂12 )Ŝ1 + (Ŝ1 · n̂12 )(Ŝ1 · p1 ) ,
1
r̂12 m21
2
2
equation (35) can be fulfilled and so agreement is achieved. This means that the
ADMcan
calculated with the aid of the ADM method, in terms of
Hamiltonian HNLOS
2
1
invariant physical quantities, agrees with the Routhian from above, hence there is
ADMcan
correctly describes a binary consisting of BHs and/or
great confidence that HNLOS
2
1
NSs or other kinds of compact objects in post-Newtonian Einsteinian theory. The
new Hamiltonian may find immediate application in the problem of motion of orbiting
binaries as investigated and solved in e.g., [34, 42].
Acknowledgments
The authors wish to thank M. Tessmer for useful discussions. This work is supported
by the Deutsche Forschungsgemeinschaft (DFG) through SFB/TR7 “Gravitational
Wave Astronomy”.
NLO Spin-Squared Hamiltonian for Compact Binaries
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