# Dynamical invariants for general relativistic two-body systems at the third post-Newtonian approximation

###### Abstract

We extract all the invariants (i.e. all the functions which do not depend on the choice of phase-space coordinates) of the dynamics of two point-masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by showing how a contact transformation can be used to reduce the 3PN higher-order Hamiltonian derived by Jaranowski and Schäfer [2] to an ordinary Hamiltonian. The dynamical invariants for general orbits (considered in the center-of-mass frame) are then extracted by computing the radial action variable as a function of energy and angular momentum. The important case of circular orbits is given special consideration. We discuss in detail the plausible ranges of values of the two quantities , which parametrize the existence of ambiguities in the regularization of some of the divergent integrals making up the Hamiltonian. The physical applications of the invariant functions derived here (e.g. to the determination of the location of the last stable circular orbit) are left to subsequent work.

###### pacs:

04.25Nx, 04.30.Db, 97.60.Jd, 97.60.Lf## I Introduction

Binary sytems made of compact objects (neutron stars or black holes) are the
most promising candidate sources for ground based interferometric
gravitational-wave detectors such as LIGO and VIRGO. In the case of (stellar)
black holes the gravitational waveform will enter the detector bandwidth only at
the last stage of the inspiral motion, just before the inspiral turns into a
plunge. The detection of these gravitational signals will be possible only by
correlating the detector output with sufficiently accurate copies of the
expected
signals (‘‘matched filtering”). It is therefore very important to have the best
possible analytical control of the dynamics of general relativistic two-body
systems. The equations of motion of binary systems have been derived some years
ago at the 5/2 post-Newtonian (2.5PN) approximation^{1}^{1}1We recall that the
“PN approximation” means the obtention of the terms of order
in the equations of motion. [1]. Recently,
it has been possible to derive the third post-Newtonian (3PN) dynamics of
point-mass systems [2], though with some remaining ambiguity due to the
need to regularize the divergent integrals caused by the use of
Dirac-delta-function sources. The knowledge of the conservative part of the
dynamics of binary systems must then be completed by a correspondingly accurate
knowledge of the gravitational-wave luminosity (used, heuristically, to derive
an accurate estimate of the radiation damping effects which drive the inspiral
motion of binary systems). The gravitational-wave luminosity is currently known
to the fractional 2.5PN accuracy [3].
By combining the 2PN-level conservative
dynamics [1] with the 2.5PN gravitational-wave luminosity [3] one
has recently constructed some (improved) filters (P-approximants) [4],
for application to gravitational-wave data analysis problems.

The main purpose of this work is to extract all the invariants, i.e., the functions which do not depend on the choice of coordinates (in space or phase-space), of the 3PN dynamics derived in Ref. [2]. This task is important for three reasons: (i) some of the invariant functions are directly useful for deriving the 3PN “phasing formula” of inspiralling binaries, i.e., for constructing 3PN-accurate gravitational-wave filters; (ii) we shall use, in a companion paper [5], some of the invariants derived below to determine the location of the Last (circular) Stable Orbit which marks the transition between the inspiral and the plunge, and (iii) from a practical point of view, the dynamical invariants will be quite useful for comparing the 3PN ADM Hamiltonian dynamics of [2] with forthcoming derivations of the 3PN equations of motion in harmonic coordinate systems [6].

## Ii Reduction of the 3PN higher-order Hamiltonian

It was shown some years ago [7] that, in most coordinate systems, the conservative part of the PN-expanded equations of motion of two body systems, say , where and where , do not follow from any ordinary Lagrangian . For instance, in harmonic coordinates, and at the 2PN level, one needs to consider an acceleration-dependent Lagrangian [1]. However, it was shown in Ref. [8], and, more generally, in Ref. [7], that any higher-order PN-expanded Lagrangian (where higher derivatives enter only perturbatively) can be reduced to an ordinary Lagrangian by a suitable (higher-order) contact transformation . At the 2PN level the class of coordinates where the dynamics admit an ordinary Lagrangian is rather restricted [7], but it includes in particular the ADM coordinates [9, 10]. At the 3PN level, Jaranowski and Schäfer, who worked within the ADM canonical formalism, have found that the (conservative) ADM dynamics could not be derived from an ordinary Hamiltonian (equivalent to an ordinary Lagrangian ), but that instead it could be derived from a certain higher-order Hamiltonian . [This higher-order matter Hamiltonian is defined by eliminating the field variables , in a certain Routh (i.e., mixed) functional ; see Eq. (33) of Ref. [2].] The meaning of this higher-order matter Hamiltonian is that the equations of motion of the matter can be written (after elimination of the TT variables) as

(1) |

where and denote functional derivatives:

(2) |

It is easily seen that the Hamilton-like equations of motion (1) are equivalent to the Euler-Lagrange equations derived by extremizing the action functional

(3) |

where

(4) |

Indeed, we have

(5) |

To derive the dynamical invariants of the 3PN equations of motion it is convenient to introduce new coordinates (in phase-space ),

(6) |

such that the equations of motion for become some ordinary Hamilton equations

(7) |

This is equivalent to requiring that the equations of motion derive from an action functional of the form

(8) |

where^{2}^{2}2As defined does not depend on , but, by
symmetry, it is convenient to allow for such a dependence (which can be easily
introduced by transforming by parts).

(9) |

When so expressed at the level of action functionals, the problem of reducing the ‘higher-order’ action to an ordinary action is quite similar to the problem of the order-reduction of (perturbative) higher-order actions which was solved in full generality in Ref. [7]. It is then an easy task to adapt the techniques used in [7] to solve our present problem.

First, we note that, taking into account the fact that the Hamiltonian has the perturbative structure, , the identities (5) for the variational derivatives of imply

(10a) | |||

(10b) |

Here and (with
= 2PN or N) denote some explicit functions of and which
are the right-hand-sides of the usual Hamilton equations at the 2PN (or
Newtonian) level. For instance, at the Newtonian level , . Note that in Eqs. (10), and
in the reasoning below, we are working ‘off shell’, i.e., we consider virtual
motions which do not necessarily satisfy the equations of motion.
[As we are using
identities for action functionals, it is essential to work off shell.]
Inserting the perturbative identities (10) in and Taylor expanding yields the
identity^{3}^{3}3In Eq. (11) and elsewhere, we abbreviate the notation
by suppressing the (summed over) indices in , , etc.

(11) |

where denotes the naive ‘order-reduced’ Hamiltonian obtained by using the (lower-order) equations of motion to eliminate the higher-order derivative terms (a wrong procedure in general):

(12) |

[As indicated in Eq. (12), it is sufficient to use the Newtonian equations of motion because and enter only at the level.] The ‘double-zeros’ in Eq. (11) denote all the terms generated by the Taylor expansion which would be at least quadratic in and . As is well known [11, 7] such terms can be perturbatively neglected (even off shell) because they do not contribute to the equations of motion at the 3PN level.

If one inserts the identity (11) in the original action functional (3), one sees that has the desired ordinary form (8) modulo some extra terms which are linear in and . As in Ref. [7] such terms can be eliminated by noticing that (to first order) the effect of the shift of dynamical variables (6) on the Lagrangian reads (by virtue of the definition of functional derivatives)

(13) |

where is some linear form in and . [It is sufficient to work to linear order because and are .]

By combining the two identities (11) and (13) we find that, if we define the ordinary Lagrangian (considered in phase-space) , associated to the ordinary (naively reduced) Hamiltonian , Eq. (12),

(14) |

we have the identity

(15) |

Therefore, if we shift the phase-space coordinates by defining

(16) |

we find (noticing that both total differentials, and double-zeros, are negligible in action functionals) that the original equations of motion (1) are transformed, when rewritten in coordinates, into the Euler-Lagrange equations of the ‘ordinary’ phase-space Lagrangian , i.e. into ordinary Hamilton equations

(17) |

Summarizing: it is licit to naively reduce the higher-order original Hamiltonian by replacing the lower-order equations of motion in the offending derivatives , (thereby defining the reduced Hamiltonian , Eq. (12)), if one adds the correcting information that the ordinary Hamilton equations defined by the reduced hold in the new phase-space coordinates defined by Eq. (16).

Let us note for completness that the conserved energy defined by the autonomous Hamiltonian becomes, when expressed in the original variables

(18) |

where

(19) |

is, indeed, easily checked to be a conserved quantity associated with the higher-order dynamics (1).

Let us now consider the explicit application of the previous results to the
case
at hand. Following [2] it is sufficient to consider the dynamics of
the
relative motion of a two-body system, considered in the center-of-mass frame
(). It is convenient to work with the following reduced
variables^{4}^{4}4Note that [2] use units where .

(20) |

where

(21) |

In Eq. (20) the superscript NR denotes a ‘non-relativistic’ (higher-order) Hamiltonian, i.e. the Hamiltonian without the rest-mass contribution . is, to start with, a function of , , , and . Here the dot denotes the reduced-time derivative:

(22) |

From Eq. (16) above and Eq. (68) of [2] one finds that the
coordinate shift needed to transform into an ordinary Hamiltonian reads (on shell^{5}^{5}5Working
on shell here is equivalent to neglecting some double-zero terms, see, e.g. Ref. [7]., i.e. by using the (Newtonian) equations of motion after
the differentiations exhibited in Eq. (16))

(23a) | |||||

(23b) | |||||

Note again that and are of 3PN order.

## Iii Order-reduced 3PN Hamiltonian

As explained above the phase-space coordinate transformation (23) maps the original higher-order 3PN ADM dynamics onto ordinary Hamilton equations, with an Hamiltonian defined by the ‘naive’ procedure (12), i.e. by using in the following replacement rules (in our reduced units where , with )

(24) |

To simplify the writing we shall henceforth drop all the primes (but the reader should remember that we henceforth work in the shifted coordinate system ). We also drop the overbar, on the Hamiltonian, indicating that we have used the replacement rules (24). Finally, using Eqs. (68) and (71) of [2], we obtain the following ordinary Hamiltonian (with )

(25) |

where

(26) | |||

(27) | |||

(28) | |||

(29) |

The parameters and appearing in the 3PN Hamiltonian parametrize the existence of ambiguities in the regularization (as it is presently performed) of some of the divergent integrals making up the Hamiltonian.

It was shown by Damour [12] that the internal structure of the compact bodies making up a binary system (e.g. a neutron star versus a black hole) start influencing the dynamics only at the 5PN level (; see Eq. (19) in Sec. 5 of Ref. [12]). Therefore, one expects that the present 3PN ambiguities are of technical nature, and not linked to real physical ambiguities. The best way to resolve these ambiguities would be to work out in detail, at the 3PN level, the general matching method outlined in Ref. [12], and applied there at the 2PN level. However, the technical difficultities of working at the 3PN level are such that this route has not yet been attempted.

The ‘kinetic ambiguity’ (proportional to
the kinetic energy ) was explicitly introduced in Ref. [2] (see
Eq. (69) there; with corresponding to
modulo the shift of normalization discussed below). The
‘static ambiguity’ was discussed in Ref. [13]. In the
present paper, we normalize the static ambiguity such that the value
corresponds to the static interaction^{6}^{6}6By ‘static
interaction’ we mean the part of the Hamiltonian which remains when setting to
zero both the momenta , and the independent
gravitational-wave degrees of freedom , (time-symmetric
conformally flat data).
between two ‘Brill-Lindquist black holes’ (considered instantaneously at rest;
see [13]). The normalization of the kinetic ambiguity used in Eq.
(III) of this paper also differs from that used in Eq. (71) of
[2].
The motivation for this new normalization is given in Appendix A of the present
paper, where we calculate, following the route of Ref. [2], a new
‘reference’ Hamiltonian which corresponds to this normalization. In terms of
these two normalizations, the Hamiltonian explicitly written in [2]
corresponds to the values and ; this means that
(after having applied the replacement rule (24) and dropped the primes)

(30) |

Though the two parameters and appear separately in Eq. (III), all the dynamical invariants of the 3PN Hamiltonian involve only the combination

(31) |

Our results below will verify explicitly that only the combination appears, but it is, in fact, easy to understand why. Indeed, as we shall calculate the dynamical invariants of , our results would be the same if we transformed by an arbitrary canonical transformation. If we consider ‘a small’ canonical transformation of order , it is enough to work at linear order. At linear order a canonical transformation is parametrized by an arbitrary generating function , and its effect on the Hamiltonian is to change it by the Poisson bracket :

(32) |

If we choose we find

(33) |

Therefore by choosing (for instance) , we see that we can eliminate at the cost of replacing by .

An important issue, when discussing the consequences of the 3PN Hamiltonian (25), is the influence of the lack of precise knowledge of the combination , Eq. (31), on physical observables. First, let us emphasize that, among the hundreds of contributions to the 3PN Hamiltonian which have been computed in Ref. [2], most of them, though they are in general given by a divergent integral, seem to be regularizable in an unambigous manner. Indeed, if one considers separately the divergences near one particle (say in a volume near particle 1), and regularizes them ‘à la Riesz’, i.e. by introducing a regularizing factor in the integrand, the analytic continuation in of the regularized integral does not contain a pole for most terms. This suggests that the well-defined analytic continuation of down to defines, without apparent ambiguity, the regularized value of . [In the notation of Appendix B of [2] we are considering that (i.e. ), in a -restricted integral which behaves as , before taking the limit .]

However, a limited class of ‘dangereous’ integrals involve a simple pole as : (where the ‘infrared’ lengthscale is linked to the size of the volume ). A remarkable fact (emphasized in Section IV of [2]) is, however, that the complete combination of dangerous integrals appearing in is such that all the pole terms cancel: . This also implies that the logarithms depending on the ‘ultraviolet’ regularizing length cancel in the Hamiltonian [2]. In other words, the combination of dangerous integrals appearing in is either ‘finite’ (convergent) or globally of a ‘non dangerous’ type. This is good news, but this still leaves an ambiguity in the finite value of because it was noticed in [2] that the final regularized result depends on the way one writes the ‘bare’ integrand (when transforming it using standard rules: integration by parts, and Leibniz’s rule). In addition to this ambiguity linked to a subset of divergent integrals, there are also ambiguities in the ‘contact’ terms in the Hamiltonian (of the type ). One needs to regularize the (in general singular) limit . In [2] one used an Hadamard-like ‘partie-finie’ prescription () for doing that. This prescription is, however, ambiguous at 3PN, notably because [13], where denotes the Newtonian potential. As said above, the final ambiguities in the Hamiltonian concern only two quite specific types of terms, parametrized by and in Eq. (III).

In Appendix A of this paper, we discuss in more detail these ambiguities, and, we try to estimate what are the corresponding plausible ranges of values of the two ambiguity parameters and . For instance, when transforming (e.g. by integration by parts) the divergent integrands we generate some (rational) values for and . By doing this in various ways, we get an idea of the range of values that these parameters are likely to be in. The final result of Appendix A below is

(34) |

As the symmetric mass ratio ranges between 0 and , the ranges (34) imply that . In the following, we shall therefore consider as fiducial range of variation for the (not yet known) combination the simple range

(35) |

## Iv Dynamical invariants of the 3PN Hamiltonian

In this section we derive a complete (and, in fact, overcomplete) set of invariant functions of the 3PN dynamics. Only such functions enter the principal observables that can be measured from infinity (say, by gravitational wave observations). However, not all the invariants we derive here can be considered as being directly observable. In a subsequent paper, we shall discuss in detail one of the most important observable: the Last Stable (circular) Orbit.

We follow closely a work of Damour and Schäfer [14] in which they derived the invariants of the 2PN dynamics. One first uses the invariance of the Hamiltonian under time translations and spatial rotations. This yields the conserved quantities

(36) |

Here denotes the total ‘non relativistic’ energy (without rest-mass contribution), and the total angular momentum of the binary system in the center of mass frame. Using the Hamilton-Jacobi approach, the motion in the plane of the relative trajectory is obtained, in polar coordinates , by separating the variables and in the reduced planar action

(37) |

Here and is an ‘effective potential’ for the radial motion which is obtained by solving the first equation (36) for after having replaced by

(38) |

Working iteratively in the small parameter , and consistently neglecting all terms , one finds that is given, at 3PN order, by the following seventh-degree polynomial in :

(39) |

The coefficients , , start at Newtonian order, while the extra terms start at the PN order indicated as superscript. All the coefficients , , , are polynomials in and . Their explicit expressions are given in Appendix B.

The Hamilton-Jacobi theory shows that the observables of the 3PN motion are deducible from the (reduced) ‘radial action integral’

(40) |

For instance, the periastron-to-periastron period and the periastron advance per orbit are obtained by differentiating the radial action integral:

(41) | |||

(42) |

To evaluate the invariant function we follow Ref. [14], which was itself following a method devised by Sommerfeld [15] within the context of the ‘old theory of quanta’. This method is explained in detail in the Appendix B of Ref. [14]. The basic idea is the following: define and consider the integral

(43) |

Here is a formal expansion parameter (actually in the final calculations, one takes into account the fact that the higher ’s are multiplied by or , with being in factor of ) and one wishes to compute the expansion of as : . To start with, the limits of integration, and , are the two exact (-dependent) real roots of . [We work in the case where , , and .] The idea is to consider the function as a complex analytic function defined in a suitably cut complex -plane: with, say, a cut along the real segment linking to , and additional cuts from, say, to the other roots. This allows one to rewrite the real integral (43) as a contour integral

(44) |

where one should note the different numerical prefactor, and where denotes any (counterclockwise, simply winding) contour enclosing the two real roots , , and avoiding all the cuts. [The phases of and are defined to be zero when lies on the real axis on the right of , while the phase of is defined as .] If we keep the contour away from the segment , we can now directly expand the integrand in the contour integral (44) in powers of . This generates the successive terms of the expansion , with each term being given by a contour integral made of a sum of contributions of the form

(45) |

Each basic integral appears in the expansion of multiplied by a coefficient which is a polynomial in the ’s. [At any given PN order, there are only a finite number of integrals to compute; see below.]

There are then two ways to compute the ’s. The simplest (the one advocated by Sommerfeld) is to expand out the contour (away from the natural cut associated to the ‘unperturbed’ quadratic form ) until it is deformed into: (i) a clockwise contour around the origin , and (ii) anticlockwise contour at infinity. [Here, one considers that the -integrals are defined in a newly cut plane, with a cut only along the segment .] The value of is then simply given by applying (twice) Cauchy’s residue theorem: it is enough to read off the coefficients of in the two Laurent expansions of the integrand of Eq. (45) for and (keeping track of phases and signs). The other way to compute the ’s consists in shrinking down the contour onto the real axis, so as to get (twice) a real integral from to (with ), plus two circular integrals around and . As shown in Appendix B of [14] one can then prove that the -expansion () of is simply given by