# Relativistic tidal properties of neutron stars

###### Abstract

We study the various linear responses of neutron stars to external relativistic tidal fields. We focus on three different tidal responses, associated to three different tidal coefficients: (i) a gravito-electric-type coefficient measuring the -order mass multipolar moment induced in a star by an external -order gravito-electric tidal field ; (ii) a gravito-magnetic-type coefficient measuring the spin multipole moment induced in a star by an external -order gravito-magnetic tidal field ; and (iii) a dimensionless “shape” Love number measuring the distortion of the shape of the surface of a star by an external -order gravito-electric tidal field. All the dimensionless tidal coefficients , and (where is the radius of the star) are found to have a strong sensitivity to the value of the star’s “compactness” (where we indicate by the speed of light). In particular, is found to strongly decrease, as increases, down to a zero value as is formally extended to the “black-hole limit” (BH) . The shape Love number is also found to significantly decrease as increases, though it does not vanish in the formal limit , but is rather found to agree with the recently determined shape Love numbers of black holes. The formal vanishing of and as is a consequence of the no-hair properties of black holes. This vanishing suggests, but in no way proves, that the effective action describing the gravitational interactions of black holes may not need to be augmented by nonminimal worldline couplings.

###### pacs:

04.25.Nx, 04.40.Dg, 95.30.Sf,## I Motivation and introduction

Coalescing binary neutron stars are one of the most important (and most
secure) targets of the currently operating network of ground-based detectors
of gravitational-waves. A key scientific goal of the detection of the
gravitational-wave signal emitted by coalescing binary neutron stars is to
acquire some knowledge on the equation of state (EOS) of neutron-star matter.
Recent breakthroughs in numerical relativity have given example of the
sensitivity of the gravitational-wave signal to the EOS of the neutron
stars Baiotti:2008ra ; Baiotti:2009gk ; Kiuchi:2009jt ; Read:2009yp . However, this sensitivity
is qualitatively striking only
during and after the merger of the two neutron stars, i.e. for gravitational
wave frequencies Hz, which are outside the most
sensitive band of interferometric detectors. It is therefore important to
study to what extent the gravitational-wave signal emitted within the most
sensitive band of interferometric detectors (around Hz)
is quantitatively sensitive to the EOS of neutron stars. In such a
regime, the two neutron stars are relatively far apart, and the problem can be
subdivided into three separate issues, namely:

(i) to study the response of each neutron star to the tidal field generated
by its companion;

(ii) to incorporate the corresponding tidal effects within a theoretical
framework able to describe the gravitational-wave signal emitted by
inspiralling compact binaries; and

(iii) to assess the measurability of the tidal effects within the signal seen by
interferometric detectors.

A first attack on these three issues has been recently undertaken by Flanagan
and Hinderer Flanagan:2007ix ; Hinderer:2007mb .
[See also Read:2009yp for an attempt at addressing the third issue.]
Our aim in this work,
and in subsequent ones, is to improve the treatment of
Refs. Flanagan:2007ix ; Hinderer:2007mb on several accounts. The present
work will focus on the first issue, (i), above, namely the study of the tidal
response of a neutron star. Our treatment will complete the
results of Hinderer:2007mb in several directions. First, we shall study
not only the usually considered “electric-type”, “tidal”, “quadrupolar”
Love number , but also several of the other
tidal coefficients of a self-gravitating body. This includes not only the
higher multipolar analogues of ,
but their “magnetic-type” analogues (
first introduced in Damour:1991yw ), as well as their (electric)
“shape-type” kin .
Second, we shall study in detail the strong sensitivity of these tidal
coefficients to the compactness parameter^{1}^{1}1To avoid
confusion with the compactness, we sometimes denote the velocity of
light as . of the neutron star.
Note, indeed, that the published version of Ref. Hinderer:2007mb
was marred by errors which invalidate the conclusions drawn there that
has only a mild dependence on the compactness
(see e.g. Eq. (27) or Fig. 2 there). [These errors were later
corrected in an erratum, which, however, did not
correct Eq. (27), nor Fig. 2.]
We shall interpret below the strong sensitivity of and
to , and contrast the vanishing of and
in the formal “black-hole limit” ,
to the nonvanishing of the “shape” Love numbers in the same
limit. In order to approach the “black-hole limit” (which is, however,
disconnected from the perfect-fluid star models), we shall particularly
focus on the incompressible models which can reach the maximum compactness
of fluid models, namely .

In subsequent works, we shall show how to incorporate the knowledge acquired here on the various tidal responses of neutron stars into the Effective One Body (EOB) framework. Indeed, recent investigations Damour:2009kr ; Buonanno:2009qa have shown that the EOB formalism is the most accurate theoretical way of describing the motion and radiation of inspiralling compact binaries.

This paper is organized as follows: Sec. II is an introduction to the various possible tidal responses of a neutron star. Section III discusses the relevant equations to deal with stationary perturbations of neutron stars that are then used in Sec. IV and Sec. V to compute the electric-type () and magnetic-type () tidal coefficients. Section VI is devoted to the computation of the “shape” Love numbers . Sections VII,VIII and IX provide explicit numerical results related to , and respectively. The concluding section, Sec. X, summarizes our main results.

## Ii The various tidal responses of a neutron star

Let us first recall that the motion and radiation of a system of well separated, strongly self-gravitating (“compact”), bodies can be theoretically investigated by a “matching” approach which consists in splitting the problem into two subproblems:

(i) the outer problem where one solves field equations in which the bodies are “skeletonized” by worldlines endowed with some global characteristics (such as mass, spin or higher-multipole moments), and

(ii) the inner problem where one obtains the near-worldline behavior of the outer solution from a study of the influence of the other bodies on the structure of the fields in an inner world tube around each body.

This matching approach has been used: to obtain the dynamics of binary black holes at low post-Newtonian orders D'Eath:1975qs ; D'Eath:1975vw ; Thorne:1984mz , to prove that the tidal deformation of compact bodies will start to introduce in the outer problem a dependence on the internal structure of the constituent bodies (measured by a “relativistic generalization of the second Love number” ) only at the fifth post-Newtonian (5PN) level Damour_LesHouches , and to derive the dynamics of compact bodies in alternative theories of gravitation Eardley:1975 ; Will:1981cz ; Damour:1992we . Finite-size corrections to the leading “skeletonized” dynamics can be taken into account by adding nonminimal worldline couplings to the effective action Damour:1998jk ; Goldberger:2004jt .

Let us start by considering the ‘‘inner problem’’ for a neutron star, i.e.,
the influence of the other bodies in the considered gravitationally
interacting system ^{2}^{2}2In the following, we shall have in mind a binary
system made either of two neutron stars or of a neutron star and a black
hole.. As explained, e.g. in Ref. Damour_LesHouches , the matching
method uses a multi-chart approach which combines the information contained in
several expansions. One uses both a global weak-field expansion
for the outer problem, and several local expansions of the type

(1) |

for each inner problem. Here, denotes the metric generated by an isolated neutron star, as seen in a local inner coordinate system , which is nonlinearly related to the global (“barycentric”) coordinate system by an expansion of the form

(2) |

Here, the suffix labels the considered member of the -body system, while denotes the metric perturbation, seen in the local -frame, Bacause of the combined influence of the various companions of . In the leading approximation is a sum of separate contributions due to each : each contribution then contains both the far-away field generated by the worldline, its deformation as it propagates on the “background” metric generated by , and the tidally-induced effect of the deformation of by the effect of .

Before tackling the technical problem of computing , let us recall the general structure of tidal expansions in general relativity Thorne:1984mz ; Damour:1990pi ; Damour:1991yw . We will use here the notation and results of the general multi-chart approach to the general relativistic dynamics of self-gravitating, deformable bodies developed by Damour, Soffel and Xu (DSX) Damour:1990pi ; Damour:1991yw ; Damour:1992qi ; Damour:1993zn .

Using the DSX notation (with ),

(3) | ||||

(4) | ||||

(5) | ||||

(6) |

one defines, in the local frame of each body , two sets of
“gravito-electric” and “gravito-magnetic” relativistic tidal moments,
and , respectively as^{3}^{3}3As in DSX, denotes a a
multi-index and
a symmetric-trace-free (symmetric-trace-free) projection.

(7) | ||||

(8) |

where and denote the externally-generated parts of the local gravito-electric and gravito-magnetic fields and . In the presently considered approximation where is stationary, and where it is enough to consider the linearized, multipole expanded, perturbation in Eq. (1), the externally generated parts and are well-defined and capture the terms in and that asymptotically grow as as . The (seemingly contradictory) formal limit indicated in Eqs. (7)-(8) refers to the matching performed in the outer problem (where, roughly speaking, the outer limit can still refer to a worldtube which is large, in internal units, compared to the radius of body ).

Besides the externally-generated “tidal moments”(7)-(8), one also defines the internally-generated “multipole moments” of body , (mass moments) and (spin moments) as the symmetric-trace-free tensors that parametrize the body-generated terms in the metric coefficients , that asymptotically decrease (in the -body zone) as as . The normalization of these quantities is defined by Eqs. (6.9) of Damour:1990pi (and agrees with the usual one in post-Newtonian theory).

In the stationary case (which is relevant to our present “adiabatic” approach to tidal effects), this normalization is such that the “internally-generated” post-Newtonian metric potentials , read

(9) | ||||

(10) |

where is a gauge transformation (which would drop out if we had considered the gravito-magnetic field ).

In the first post-Newtonian approximation considered by DSX, the separation of the local-frame potential into an “externally-generated” part and an “internally-generated” one , is well defined (thanks to the structure of Einstein’s equations). In the case we are considering here of a linearly perturbed, quasi-stationary, fully relativistic neutron star, the asymptotically growing character (as ) of the externally-generated potentials allows one to uniquely define the tidal moments (7)-(8). On the other hand, the asymptotic decrease of the internally generated multipolar potentials (9)-(II) introduces an ambiguity in their definition. For an attempt to uniquely define the gravito-electric quadrupole moment induced on a black hole by an external tidal moment see suen:1986 . Here, instead of relying on such a conventional (harmonic-coordinates related) definition of the induced multipole moments, we shall follow the spirit of Sec. 5 of Damour_LesHouches in defining , as parametrizing the (uniquely defined) pieces in the local-frame metric which violate the “effacing principle”, in that they directly depend on the body being a neutron star, rather than a black hole. Reference Damour_LesHouches explicitly treated the dominant even-parity case, and introduced (see Eq. (11) there) a “dimensionless constant ” ( as defined below) as a “relativistic generalization of the second Love number”. This minimal definition (which will be made fully precise below) is rather natural, and coincides with the definition adopted in Flanagan:2007ix ; Hinderer:2007mb .

With this notation in hands, we can define the two “tidal-polarizability”
coefficients and introduced
in Eqs. (6.19) of Damour:1991yw .
These coefficients relate the (electric or magnetic) tidal
induced ^{4}^{4}4Here, we consider a nonrotating star which is spherically
symmetric (with vanishing multipole moments) when it is isolated, so that
and represent the multipole moments induced by the influence
of the external tidal fields and . multipole moments to the
corresponding external tidal moments, i.e.

(11) | ||||

(12) |

The electric-type (or “even-parity”) tidal coefficient generalizes the (-type) Newtonian “Love number”. For the leading quadrupolar tide, , as defined by Eq. (11), agrees with the quantity denoted in Flanagan:2007ix ; Hinderer:2007mb . The magnetic-type (or “odd-parity”) quadrupolar tidal coefficient is proportional to the quantity which has been considered in the investigations of Favata Favata:2005da which were, however, limited to the first post-Newtonian approximation. Here, we shall consider the case of strongly self-gravitating bodies (neutron stars), and study the dependence of both and on the compactness of the considered neutron star. Let us also note that, in terms of finite-size corrections to the leading point-particle effective action , the two tidal effects parametrized by and correspond to nonminimal worldline couplings respectively proportional to

(13) |

The leading, quadrupolar corrections (13) can be reproduced (using the link between and , and and , see Sec. 3.D of Damour:1990pi and Thorne:1984mz ) as the following nonminimal couplings involving the Weyl tensor

(14) |

where , and we have introduced the tensors and , with dimensions, and in absence of parity-violating couplings, the two terms (14) are the only possible isotropic couplings. In higher dimensions, there are three nonminimal isotropic couplings quadratic in the Weyl tensor as indicated in Eq.(90) of Goldberger:2004jt . Note that we are using here the freedom of locally redefining the dynamical variables to eliminate terms proportional to the (zeroth-order) equations of motion, such as terms involving the Ricci tensor; see, e.g., the discussion of finite-size effects in tensor-scalar gravity in Appendix A of Ref. Damour:1998jk . being the dual of the Weyl tensor. In

Let us finally note that there are other “tidal coefficients” which might be interesting to discuss. First, though the linear relations (11)-(12) are the most general ones that can exist in the (parity-preserving) case of a nonspinning neutron star, the tidal properties of a spinning neutron star will involve other tidal coefficients, proportional to the spin, and associated to a mixing between electric and magnetic effects. Such electric-magnetic mixing terms would correspond, say in the leading quadrupolar case, to nonminimal worldline couplings quadratic in and linear in the spin tensor .

There exist also other tidal coefficients which do not have a direct dynamical meaning, but which generalize the “first type” of Love numbers introduced in the theory of Newtonian tides. Indeed, it is physically meaningful to define, for any , a “shape” Love number measuring the proportionality between the external tidal influence, and the deformation of the geometry of the surface of the considered (neutron) star. More precisely, limiting ourselves to the electric-type tides, one can define a dimensionless number by writing, as one does in Newtonian theory,

(15) |

or, equivalently,

(16) |

where represents the fractional deformation of the (areal) radius of the neutron star (measured in a geometrically invariant way, by relating it to the inner geometry of the deformed surface), where represents the usual, external, Newtonian tidal potential deforming the star, formally evaluated at the radius of the star (as if one were in flat space), and where represents the usual Newtonian surface gravity of the neutron star. This , “shape” Love number has been recently considered in the theory of the gravitational polarizability of black holes DL09 and it will be interesting to compare and contrast the values of the for black holes to the values of for neutron stars, especially in the limit where the compactness gets large. See Section VI below which will give the exact definition of the quantity .

## Iii Stationary perturbations of a neutron star

The unperturbed structure of an isolated (nonrotating) neutron star is described by a metric of the form

(17) |

Here, and in the following, for notational simplicity we shall denote the
local (spherical) coordinates of the -body frame simply as
(with ),
instead of the upper case letters that would more
closely follow the DSX notation recalled above. Introducing as usual the
radial dependent mass parameter by^{5}^{5}5Henceforth, we shall often
set

(18) |

and assuming a perfect-fluid energy-momentum tensor

(19) |

the spherically symmetric metric coefficients , and the pressure satisfy the Tolman-Oppenheimer-Volkoff (TOV) equations of stellar equilibrium

(20) | ||||

(21) | ||||

(22) |

These equations are integrated
from the center outward once that a barotropic EOS
relating to is provided.
We shall consider several types of barotropic EOS, namely two different types
of “polytropic EOS” (an -polytrope, with , and a
-polytrope”, with and , where
is the baryonic rest-mass density), and two different
tabulated (“realistic”) EOS (the FPS one fps and the SLy sly one).
In view of the current large uncertainty in the correct description of dense
nuclear matter, we are not claiming that our selection of “realistic” EOS
is physically preferred (see, e.g., Ref. Bernuzzi:2008fu and
references therein for a thorough comparison among models from various EOS).
We have chosen them because they have been used in recent numerical
relativity simulations of binary neutron star
coalescence Shibata:2005ss ; Shibata:2005xz .
As for the polytropic EOS, they have also been often used in numerical
relativity simulations (especially the -polytrope one), and their
dependence on the adiabatic index^{6}^{6}6As is well-known,
the dependence on the “polytropic constant” can be
absorbed in the definition of suitable
“polytropic units”. is a convenient way of varying the
“stiffness” of the EOS
(the limit representing the stiffest
possible EOS, namely incompressible matter
with and an infinite speed of sound).

Bacause of the spherical symmetry of the background, the metric perturbation

(23) |

here considered at the linearized level, can be analyzed in (tensor) spherical harmonics. The metric is expanded in even-parity and odd-parity tensor harmonics as

(24) |

In the Regge-Wheeler gauge, and following standard definitions for the expansion coefficients and the sign conventions of IP91 ; Lindblom:1997un , one has

(25) |

while the nonvanishing components of are and where and where is the mixed form of the volume form on the sphere .

Our aim is then to solve the coupled system of the perturbed Einstein’s equations, together with the perturbed hydrodynamical equations , so as to describe a star deformed by an external tidal field. We shall only consider stationary perturbations (“adiabatic tides”).

### iii.1 Even-parity, stationary barotropic perturbations

Even-parity, stationary perturbations of a barotropic star simplify in that: (i) the metric perturbations reduce to two functions , and (with ), (ii) the fluid perturbations are described by the logarithmic enthalpy function , such that , and (iii) the latter logarithmic enthalpy function is simply related (in absence of entropy perturbation) to the metric function by

(26) |

It was then showed by Lindblom, Mendell and Ipser Lindblom:1997un how to convert the system of first-order radial differential equations relating , , and to a single second-order radial differential equation for the metric variable (such that ) of the form

(27) |

[As usual, we shall generally drop the multipolar index on the various metric perturbations. The presence of a factor , or , in (or to be added to) the considered metric perturbation is also often left implicit.] Taking the stationary limit () of the results given in Appendix A of Lindblom:1997un (together with the barotropic relation ) one gets

(28) | ||||

(29) |

where we have used the background (TOV) equations to rewrite and . As a check, we have also derived from scratch Eq. (27) by starting from the “gauge-invariant” formalism of Ref. Gundlach:1999bt . Equation (27) generalizes to an arbitrary value of the multipolar order Eq. (15) of Ref. Hinderer:2007mb , which concerned the leading quadrupolar even-parity tide.

For completeness, let us note that the other metric variable, , can be expressed as a linear combination of and , namely

(30) |

where the explicit expressions of the coefficients and can also be deduced by taking the stationary limit of the results given in Appendix A of Lindblom:1997un .

### iii.2 Odd-parity, stationary perturbations

It was shown by Thorne and Campolattaro TC67 that odd-parity perturbations of a nonrotating perfect-fluid star consists only of metric fluctuations, and do not affect the star’s energy density and pressure. One might naively think that this means that an odd-parity tidal field will induce no (gauge-invariant) spin multipole moment in a (nonrotating) star. This conclusion is, however, incorrect because the “gravitational potential well” generated by the stress-energy tensor of the star does affect the “radial propagation” of the external odd-parity tidal fields and necessarily adds an asymptotically decreasing “induced” tidal response to the “incoming” tidal field. To describe this phenomenon, it is convenient to describe the odd-parity perturbation by means of the (static limit of the) “master equation” derived by Cunningham, Price and Moncrief CPMI (see also Ref. Andrade:1999mj ). In the stationary limit, and in terms of the ordinary radial variable (rather than the “tortoise” coordinate ) this equation reads

(31) |

In terms of the variables entering the odd-parity perturbations, the odd-parity master function can be taken to be either , or the combination [see e.g. Nagar:2005ea for more details]. As vanishes in the stationary limit, we can define as being

(32) |

## Iv Computation of the electric-type tidal coefficient

The electric-type tidal response coefficient , defined by Eq. (11) above, can be obtained by going through three steps: (i) numerically solving the even-parity master equation (27) within the neutron star; (ii) analytically solving the same master equation (27) in the exterior of the star; and (iii) matching the interior and exterior solutions across the star surface, taking into account the definition (11) to normalize the ratio between the “growing” and “decreasing” parts of , namely versus .

### iv.1 The internal problem

The internal value of the metric function is obtained by numerically integrating Eq. (27), together with the TOV equations (20)-(22), from the center (or, rather some very small cut-off radius ) outwards, starting with some central values of and . For , one takes as starting values at the cut-off radius and . The latter boundary conditions follow from the analysis of Eq. (27) around the regular-singular point , which shows that (where is an arbitrary constant) is the most general regular solution around . As Eq. (27) is homogeneous in , the scaling constant is irrelevant and will drop out when we shall match the logarithmic derivative

(33) |

across the star surface. This is why it is enough to use as initial boundary conditions for .

The main output of this internal integration procedure is to compute (for each value of ) the value of the internal logarithmic derivative (33) at the star’s surface, say

(34) |

### iv.2 The external problem

As noticed long ago by Regge and Wheeler Regge:1957td and Zerilli Zerilli:1971wd , the exterior form of the stationary, even-parity master equation (27) (, ) can be recast as an associated Legendre equation (with and ). More precisely, in terms of the independent variable , the exterior form of (27) reads

(35) |

where the prime stands now for . Its general solution can be written as

(36) |

where the hat indicates that the associated Legendre functions of
first, , and second^{7}^{7}7Note that, contrary to the usual
mathematical definition of , which is tuned to the real interval
, we need to work with in the interval , This means
replacing with
., , kind have been
normalized so that
and when or
; and are integration constants to be determined
by matching to the internal solution. Defining
, the exterior logarithmic derivative
reads

(37) |

### iv.3 Matching at the star’s surface, and computation of the “electric” tidal Love number

As Eq. (27) is second-order in the radial derivative of , one expects that and will be continuous at the star’s surface. Actually, the issue of regularity at the star surface is somewhat subtle because some of the thermodynamic variables (such as pressure) do not admit regular Taylor expansions in as . For instance, while the logarithmic enthalpy vanishes smoothly () across the surface, one finds that (for any polytrope) and that the term involving the inverse of the squared sound velocity in Eq. (27) is singular (when ), namely

(38) |

Despite this mildly singular behavior of the coefficient of (27) and despite the fact that the exact location of the tidally-deformed star surface is slightly displaced from the “background” value , one checks that it is correct (when ) to impose the continuity of and at . [Note that we consider here the case of a finite adiabatic index . The incompressible limit leads to a master equation which is singular at the surface, and which must be considered with care. See below our discussion of the incompressible limit.] This continuity then imposes the continuity of the logarithmic derivative . This leads to the condition , which determines the value of the ratio in terms of the compactness of the star

(39) |

On the other hand, the ratio can be related to the tidal coefficient by comparing (modulo an overall factor ),

(40) | ||||

(41) |

respectively to

(42) | ||||

(43) |

(see e.g., Eq. (4.15a) of Ref. Damour:1991yw ) where is a radial unit vector. Using the fact that

(44) |

and , and remembering that , we see that

(45) |

Note that has the dimensions of . There are then two natural ways of expressing in terms of a dimensionless quantity. Either by scaling it by the -th power of , which leads to

(46) |

or by scaling it by the -th power of the star radius , which gives

(47) |

Alternatively, we can write

(48) |

The scaling of by means of is the traditional “Newtonian” way of proceeding, and leads to the introduction of the dimensionless “second tidal Love number” (conventionally normalized as in Eq. (47) above).

One can finally write as

(49) |

The dimensionless Love number has the advantage of having a weaker sensitivity on the compactness (especially as the compactness formally tends to zero, i.e. in the Newtonian limit). Note, however, that the dimensionless quantity which will most directly enter the gravitational-wave phase of inspiralling binary neutron stars (NS) is .

The evaluation of the result (49) for yields the following explicit expressions for (with, for simplicity, ):

(50) | ||||

(51) | ||||

(52) |

Equation (IV.3) above agrees with the corrected version of Eq. (23) of Hinderer:2007mb . Note that, independently of the values of (as long as it does not introduce a pole singularity, which will be the case), the results (IV.3)-(IV.3) (and, more generally, the result (49)) contain an overall factor which formally tends (quadratically) to zero when the compactness “tends” toward the compactness of a black hole, namely . [The singular logarithm in the denominator is also easily checked to be always multiplied by and thereby not to affect the formal vanishing of as .] This property can be easily understood as a consequence of the “no-hair” properties of black holes. Indeed, among the two solutions of the exterior tidal perturbation equation (27), the no-hair property means that the solution which is “rooted” within the horizon, i.e., the “asymptotically decreasing” solution is singular at the horizon, i.e. when . More precisely, this singular behavior is

(53) |

so that the most singular term in the denominator of or is which is at the origin of the presence of a factor in and . One might naively think that this behavior proves that the “correct” value of the tidal Love numbers of a black hole is simply . However, we do not think that this conclusion is warranted. Indeed, as we explained above, the definition used here (and in Damour_LesHouches ; Flanagan:2007ix ; Hinderer:2007mb ) of the Love numbers of a (neutron) star consists in selecting, within the gravitational field of a tidally distorted star, the terms which violate the “effacing principle” (in the sense of Ref. Damour_LesHouches ), i.e. the internal-structure-dependent terms which differentiate the tidal response of a (compact) star, from that of a black hole. From this point of view, the vanishing of as is mainly a consistency check on this formal definition. The question of computing the “correct” value of for a black hole is a technically much harder issue which involves investigating in detail the many divergent diagrams that enter the computation of interacting point masses at the 5-loop (or 5PN) level.

Indeed, the issue at stake is the following. When describing the motion of two black holes (as seen in the “outer problem”) by a skeletonized action of the form , the presence of nonminimal worldline couplings of the type (13) and (14) can only be detected if one treats (when using perturbative expansions in powers of ) the general relativistic nonlinear self-interactions entailed by at the order of approximation corresponding to . For a black hole (of “radius” ), the leading nonminimal coupling parameter scales as , so that (using ) the leading nonminimal interaction is proportional to . The presence of an overall factor (which is the same factor that appeared in Eq. (19) in Sec. 5 of Damour_LesHouches ) signals that such an effect is smaller than the leading (Newtonian) interaction () between two point masses, so that it corresponds to the 5PN level. In the diagrammatic language of (post-Minkowskian or post-Newtonian) perturbation theory (as used, e.g., in Damour:1995kt ), this corresponds to the 5-loop level. Let us recall that the computation of the interaction of two black holes at the 3-loop level was a technically complex enterprise that necessitated the careful consideration of many divergent diagrams, and the use of the efficient method of dimensional regularization Damour:2001bu ; Blanchet:2003gy . At the 3-loop level the result of the computation was (essentially) finite, though the use of harmonic coordinates in one of the computations Blanchet:2003gy introduced some gauge-dependent infinities. As argued long ago Damour_LesHouches , and confirmed by an effective action approach Goldberger:2004jt , one expects to see real, gauge-independent infinities arising at 5-loop (5PN), i.e. at the level where the effacing principle breaks down, and where, as explained above, a parameter () linked to the internal structure of the considered compact body starts to enter the dynamics. Until a careful analysis of the 5PN nonlinear self-interactions is performed, one cannot conclude from the above result ( as ) that the effective action describing the dynamics of interacting black holes is described by the pure point-mass action without the need of additional nonminimal couplings of the type of Eq. (14).

We have phrased here the problem within standard (post-Minkowskian or post-Newtonian) perturbation theory, because this is the clearest framework within which the issue of higher order nonlinear gravitational interactions of point masses is technically well defined (when using, say, dimensional regularization to define the perturbative interaction of point masses in general relativity Damour:2001bu ; Blanchet:2003gy ). Note that, in the extreme mass ratio limit (), where one might use black hole perturbation theory, the interaction associated to the leading nonminimal coupling parameter of is proportional to (see above). This is well beyond the currently studied “gravitational self-force” effects, which are proportional to , and correspond to a “1-loop” effect within a black hole background.

## V Computation of the magnetic-type tidal coefficient

The magnetic-type tidal response coefficient , defined by Eq. (12) above, can be obtained by following three steps, which are similar to those followed for the electric-type coefficient .

### v.1 The internal problem

The internal value of the odd-parity master function is obtained by numerically integrating Eq. (31), together with the TOV equations. The boundary conditions are now obtained from the behavior of the general regular solution at the origin. Again, the main output of the internal integration procedure is to compute (for each value of ) the value of the internal logarithmic derivative of , at the star surface, say

(54) |

### v.2 The external problem

As noticed long ago by Regge and Wheeler Regge:1957td , the stationary odd-parity perturbations can be analytically solved in the exterior region. Similar to the even-parity case there exist two types of exterior solutions: a “growing” type solution, say , with , and a “decreasing” type one, say . We normalize them so that , and as . The general analytical forms of and , for any , can be obtained from Ref. Regge:1957td . In the case of the leading quadrupolar odd-parity perturbation, , the “growing” analytical exterior solution of (31) is the very simple polynomial

(55) |

while the “decreasing” one can be expressed in terms of an hypergeometric function as

(56) |

The normalization of is such that as . Note also that, for the special values , , , the hypergeometric function is actually expressible in terms of elementary functions. The result has the form