# Time transfer functions without enhanced terms in stationary spacetimes
Application to an isolated, axisymmetric spinning body

P. Teyssandier<EMAIL_ADDRESS>SYRTE, Observatoire de Paris, PSL
Research University, CNRS, Sorbonne Universités,
UPMC Univ. Paris 06, LNE, 61 avenue de l’Observatoire, F-75014 Paris, France

###### Abstract

We develop a new perturbation method for determining a class of time transfer
functions in a stationary spacetime when its metric is a small deformation of
a background metric for which the time transfer functions are known in a
closed form. The perturbation terms are expressed as line integrals along the
null geodesic paths of the background metric. Unlike what happens with the
other procedures proposed until now, the time transfer functions obtained in
this way are completely free of unbounded terms and do not generate any
enhancement in the light travel time. Our procedure proves to be very
efficient when the background metric is a linearized Schwarzschild-like
metric. Its application to an isolated body slowly rotating about an axis of
symmetry leads to integrals which can be calculated with any symbolic computer
program. Explicit expressions are obtained for the mass dipole and quadrupole
moments and for the leading gravitomagnetic term induced by the spin of the
body. A brief numerical discussion is given for the 2002 Cassini experiment.

PACS numbers

04.25.-g, 04.80.Cc, 95.10.Jk, 95.30.Sf

###### pacs:

04.25.-g, 04.80.Cc, 95.10.Jk, 95.30.Sf

##  I Introduction

Determining the travel time of an electromagnetic signal as a function only
involving the spatial position of the emitter, the spatial position of the
receiver and the instant of arrival of the signal is a problem of crucial
importance in metric theories of gravity. Indeed, knowing such a function –
called a reception time transfer function (TTF) according to a terminology
specified in Linet:2002 ; Leponcin:2004  – makes it possible to model the
relativistic time delay and/or the Doppler observables as measured in space
missions (see, e.g., Bertotti:2008 ; Ashby:2010 ; Hees1:2012 ; Hees2:2014 ;
Hees3:2014 ) or in Very Long Baseline Interferometry observations
Leponcin:2016 ; Soffel:2017 . It also enables one to calculate the propagation
direction of a light ray at the points of emission and observation
Leponcin:2004 ; Leponcin:2008 ; Teyssandier:2012 , thus justifying the
relevancy of the notion of TTF in relativistic astrometry Bertone:2014 ;
Bertone:2017 . Moreover, the TTF formalism turns out to be effective in
studying the phase function associated with a light ray Qin:2017  and the
propagation of light rays in nondispersive isotropic moving media
Bourgoin:2020 ; Bourgoin:2021 ; Bourgoin:2022 .

The explicit calculation of TTFs with the accuracy required by current space
missions and/or relativistic astrometry has given rise to a large body of
works. Two kinds of procedures have been systematically investigated, each
based on the assumption that the TTFs involved in the observations made in the
Solar system can be represented by a general post-Newtonian or post-
Minkowskian expansion. The first kind of procedures consists in determining
the light rays by integration of the null geodesic equations by approximate
asymptotical methods (see, e.g., Klioner:1992 ; Kopeikin:1999 ; Kopeikin:2006
; Klioner:2010 ; Korobkov:2014 ; Zschocke:2015 ; Zschocke:2018 ;
Zschocke1:2022  and Refs. therein). The second kind of approaches avoids the
integration of the geodesic equations either by using Synge’s world function
Linet:2002 ; Leponcin:2004  or by integrating an eikonal equation Ashby:2010 ;
Teyssandier:2008 . These different procedures lead to equivalent formulas
Zschocke:2014 . However, they suffer from two major shortcomings. First, they
allow to determine just one TTF, even in configurations where a gravitational
lensing phenomena may occur Zschocke:2011 . Second, the TTF so far obtained
for a large class of parametrized static spherically symmetric metrics contain
enhanced terms, i.e. terms which become infinite when the emission and
reception points tend to be diametrically opposed Teyssandier:2008 ;
Ashby:2010 ; Klioner:2010 ; Linet:2013 ; Teyssandier:2014 . The same issue
arises when the multipole moments or/and the motion of the source(s) of the
field are taken into account (see, e.g., Kopeikin:1997 ; Kopeikin:2002 ;
Linet:2002 ; Ciufolini:2003 ; Leponcin:2008 ; Korobkov:2014 ; Hees3:2014 ;
Soffel:2015 ; Zschocke:2018 ; Zschocke2:2022 ).

Clearly, we must abandon the assumption that a TTF can be represented by a
post-post-Newtonian or a generalized post-Minkowskian expansion regardless of
the positions of the emitter and the receiver. This crucial point is confirmed
in Linet:2016 , where all the TTFs of a Schwarzschild-like metric are obtained
in an exact closed-form within the linearized, weak-field approximation. These
functions are bounded everywhere, but it is proven that none of them can be
represented by a convergent post-Minkowskian expansion when the emitter and
the receiver are sufficiently close to a superior conjunction.

The aim of the present paper is to extend the approach developed in Linet:2016
to the case where the metric is a weak stationary perturbation of a background
metric whose null geodesics and their associated TTFs are explicitly known in
an analytical closed form. The perturbation terms in each TTF are obtained as
line integrals evaluated along a null geodesic path of the background metric.
The case of a weak stationary perturbation of a linearized Schwarzschild-like
metric is treated in detail. We apply these results to light rays propagating
in the field of an isolated body slowly rotating about an axis of symetry. We
show that the perturbations in the propagation time of light rays due to the
mass and spin multipoles can be calculated with any symbolic computer program.
Explicit expressions are derived for the dipole and quadrupole mass moments,
as well as for the dominant gravitomagnetic term due to the intrinsic angular
momentum of the body. All the expressions are bounded when the positions of
the emitter and the receiver tend to be diametrically opposite.

The applications of our results to the deflection of light and the
gravitational lensing require lengthy calculations which are beyond the scope
of the present work. So we restrict ourselves to comparing the new formulas
with those previously used to determine the post-Newtonian parameter $\gamma$
from data collected during the Cassini mission.

In this paper, $G$ is the Newtonian gravitational constant and $c$ is the
speed of light in a vacuum. Greek indices run from 0 to 3, and Latin indices
run from 1 to 3. Any bold italic letter stands for an ordered triple: we note,
e.g., $\bm{a}=(a^{1},a^{2},a^{3})=(a^{i})$. All the triples are formally
regarded as vectors of a three-dimensional Euclidean space. Given two triples
$\bm{a}$ and $\bm{b}$, $\bm{a}.\bm{b}$ and $\bm{a}\times\bm{b}$ denote their
usual scalar product and exterior product, respectively. The notation
$|\bm{a}|$ stands for the Euclidean norm of $\bm{a}$. The Einstein summation
convention is used for both types of repeated indices, whatever their
respective positions.

##  II Time transfer functions in stationary spacetimes

Throughout this work, the physical metric $g$ is supposed to be stationary.
The signature adopted for $g$ is $(+,-,-,-)$. Spacetime is assumed to be
covered by a global coordinate system $(x^{0}=ct,\bm{x})$, $t$ being a time-
like coordinate and $\bm{x}$ a triple of quasi-Cartesian spatial coordinates:
$\bm{x}=(x^{i})$. This coordinate system is chosen in such a way that the
components of the metric do not depend on $x^{0}$. The light signals are
supposed to propagate through a vacuum, so the light rays are null geodesic
paths of the metric $g$.

Let $\mathcal{G}^{[\sigma]}(\bm{x}_{A},x^{0}_{B},\bm{x}_{B})$ be a generic
member of the family of light rays starting from points-events having a given
spatial position $\bm{x}_{A}$ and arriving at the point-event
$(x_{B}^{0},\bm{x}_{B})$, the brackets around the index $\sigma$ being used to
avoid any confusion with an exponent or a tensorial index. Let us denote by
$(x^{0}_{A})^{[\sigma]}$ the value of $x^{0}$ when the light ray
$\mathcal{G}^{[\sigma]}(\bm{x}_{A},x^{0}_{B},\bm{x}_{B})$ is emitted. Owing to
the stationary character of spacetime, the light travel time
$[x^{0}_{B}-(x^{0}_{A})^{[\sigma]}]/c$ corresponding to this ray depends only
on the spatial positions $\bm{x}_{A}$ and $\bm{x}_{B}$. We can therefore write

$x^{0}_{B}-(x^{0}_{A})^{[\sigma]}=c\mathcal{T}^{[\sigma]}(\bm{x}_{A},\bm{x}_{B}),$
(1)

where $\mathcal{T}^{[\sigma]}$ is the reception time transfer function
associated with the light ray
$\mathcal{G}^{[\sigma]}(\bm{x}_{A},x^{0}_{B},\bm{x}_{B})$. It may be noted
that each path $\mathcal{G}^{[\sigma]}(\bm{x}_{A},x^{0}_{B},\bm{x}_{B})$ can
be deduced from $\mathcal{G}^{[\sigma]}(\bm{x}_{A},0,\bm{x}_{B})$ by a
timelike translation. So, in what follows, any light ray of type $\sigma$
starting from $\bm{x}_{A}$ and arriving at $\bm{x}_{B}$ will be simply denoted
by $\mathcal{G}^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$. Moreover, the term
“reception” will be systematically omitted.

##  III Perturbation approach of the TTF formalism in stationary spacetimes

Let us now enunciate the other general assumptions underlying this paper.

1\. The metric $g$ depends on a family of small dimensionless parameters
$\\{\epsilon_{a},a\in\mathcal{P}\\}$ describing the departure of this metric
from a background stationary metric denoted by $\gamma$; $\mathcal{P}$ may be
an infinite set.

2\. The components $g_{\mu\nu}$ of the metric can be decomposed as

$g_{\mu\nu}(\bm{x};\mathcal{P})=\gamma_{\mu\nu}(\bm{x})+h_{\mu\nu}(\bm{x};\mathcal{P}),$
(2)

where the perturbation terms $h_{\mu\nu}$ are assumed to be represented with a
sufficient accuracy by an expansion as follows

$h_{\mu\nu}(\bm{x};\mathcal{P})=\sum_{a\in\mathcal{P}}h_{\mu\nu}^{a}(\bm{x})+O(\epsilon^{2}),$
(3)

each $h_{\mu\nu}^{a}$ being the first-order perturbation due to $\epsilon_{a}$
and $O(\epsilon^{2})$ standing for all the terms of order
$\epsilon_{a}\epsilon_{b}$, with $(a,b)\in\mathcal{P}\times\mathcal{P}$. Then,
the contravariant components of the metric tensor may be written as

$g^{\mu\nu}(\bm{x};\mathcal{P})=\gamma^{\mu\nu}(\bm{x})+\sum_{a\in\mathcal{P}}k^{\mu\nu}_{a}(\bm{x})+O(\epsilon^{2}),$
(4)

where the $\gamma^{\mu\nu}$ are coefficients satisfying the equations
$\gamma_{\mu\rho}\gamma^{\mu\nu}=\delta_{\rho}^{\nu}$ and the quantities
$k^{\mu\nu}_{a}$ are given by

$k^{\mu\nu}_{a}(\bm{x})=-\gamma^{\mu\alpha}(\bm{x})\gamma^{\nu\beta}(\bm{x})h_{\alpha\beta}^{a}(\bm{x}).$
(5)

We systematically use greek letters to denote the mathematical objects which
only depend on the background metric $\gamma$. We also use the zeroth-order
appellation for these objects. No confusion is possible between the background
metric and the post-Newtonian parameter $\gamma$.

3\. In agreement with a notation already specified in Sect. II, let us denote
by $\Gamma^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$ a generic member of the family
of null geodesic paths of the background metric $\gamma$ starting from the
spatial position $\bm{x}_{A}$ and arriving at the point-event
$(x^{0}_{B},\bm{x}_{B})$, and by $T^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$ the
corresponding zeroth-order time transfer function. It is assumed that whatever
the value of the parameter $\sigma$, there exists a null geodesic
$\mathcal{G}^{[\sigma]}(\bm{x}_{A},\bm{x}_{B};\mathcal{P})$ of the metric $g$
such that the corresponding TTF
$\mathcal{T}^{[\sigma]}(\bm{x}_{A},\bm{x}_{B};\mathcal{P})$ can be written as

$\displaystyle\mathcal{T}^{[\sigma]}(\bm{x}_{A},\bm{x}_{B};\mathcal{P})=T^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})+\sum_{a\in\mathcal{P}}\Delta\mathcal{T}^{[\sigma]}_{a}(\bm{x}_{A},\bm{x}_{B})$
$\displaystyle\qquad\qquad\qquad\quad\quad+O(\epsilon^{2}),$ (6)

where each $\Delta\mathcal{T}^{[\sigma]}_{a}$ denotes the first-order
perturbation term due to the small parameter $\epsilon_{a}$.

To determine the perturbation functions $\Delta\mathcal{T}^{[\sigma]}_{a}$, we
use the property of each function $\mathcal{T}^{[\sigma]}$ to obey an eikonal
equation which can be written as Bel:1994 ; Teyssandier:2008

$g^{00}(\bm{x};\mathcal{P})-2g^{0i}(\bm{x};\mathcal{P})\frac{c\partial\mathcal{T}^{[\sigma]}(\bm{x}_{A},\bm{x};\mathcal{P})}{\partial
x^{i}}+g^{ij}(\bm{x};\mathcal{P})\frac{c\partial\mathcal{T}^{[\sigma]}(\bm{x}_{A},\bm{x};\mathcal{P})}{\partial
x^{i}}\frac{c\partial\mathcal{T}^{[\sigma]}(\bm{x}_{A},\bm{x};\mathcal{P})}{\partial
x^{j}}=0.$ (7)

Substituting the expansions (4) and (6) into Eq. (7), and then retaining only
the terms of orders 0 and 1 with respect to the parameters $\epsilon_{a}$, we
get the system of equations

$\displaystyle\gamma^{00}(\bm{x})-2\gamma^{0i}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}+\gamma^{ij}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial x^{i}}\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial x^{j}}=0,$ (8) $\displaystyle
k_{a}^{00}(\bm{x})-2k_{a}^{0i}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}+k_{a}^{ij}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial x^{i}}\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial x^{j}}$
$\displaystyle+2\left[-\gamma^{0j}(\bm{x})+\gamma^{ij}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}\right]\frac{c\partial\Delta\mathcal{T}^{[\sigma]}_{a}(\bm{x}_{A},\bm{x})}{\partial
x^{j}}=0.$ (9)

Of course, Eq. (8) is the eikonal equation associated with the background
metric $\gamma$.

When $\bm{x}$ is constrained to vary along the zeroth-order light ray
$\Gamma^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$, Eq. (9) can be written in the form
of an ordinary first-order differential equation governing the perturbation
term $\Delta\mathcal{T}^{[\sigma]}_{a}$. Indeed, let

$x^{0}=\xi^{0}_{[\sigma]}(v),\quad\bm{x}=\bm{\xi}_{[\sigma]}(v)=(\xi^{i}_{[\sigma]}(v))$
(10)

be a system of parametric equations of
$\Gamma^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$, $v$ being an arbitrarily chosen
parameter. Put

$\lambda_{\alpha}^{[\sigma]}(v)=\gamma_{\alpha\mu}(\bm{\xi}_{[\sigma]}(v))\dot{\xi}_{[\sigma]}^{\mu}(v),$
(11)

where $\dot{\xi}_{[\sigma]}^{\mu}\\!\\!=\\!\\!d\xi_{[\sigma]}^{\mu}/dv$. Since
the quantities $\lambda_{\alpha}^{[\sigma]}(v)$ are the covariant components
of the vector tangent to $\Gamma^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$ determined
by the background metric $\gamma_{\mu\nu}$, the partial derivatives of
$T^{[\sigma]}(\bm{x}_{A},\bm{x})$ with respect to $x^{i}$ at point
$\bm{\xi}_{[\sigma]}(v)$ satisfy the relations (see Eq. (40) in Leponcin:2004
)

$\left[\frac{c\partial T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}\right]_{\bm{x}=\bm{\xi}_{[\sigma]}(v)}=-\frac{\lambda^{[\sigma]}_{i}(v)}{\lambda^{[\sigma]}_{0}(v)}.$
(12)

Contracting Eq. (12) by $\gamma^{ij}$ and using the relation

$\gamma^{\alpha j}\lambda_{\alpha}^{[\sigma]}(v)=\dot{\xi}_{[\sigma]}^{j}(v),$
(13)

it is easily seen that

$\left[-\gamma^{0j}+\gamma^{ij}\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}\right]_{\bm{x}=\bm{\xi}_{[\sigma]}(v)}=-\frac{\dot{\xi}_{[\sigma]}^{j}(v)}{\lambda_{0}^{[\sigma]}(v)}.$
(14)

Then, using Eq. (14) and noting that

$\displaystyle\dot{\xi}_{[\sigma]}^{j}(v)\frac{c\partial\Delta\mathcal{T}^{[\sigma]}_{a}(\bm{x}_{A},\bm{\xi}_{[\sigma]}(v))}{\partial
x^{j}}$
$\displaystyle\qquad\qquad\qquad=\frac{cd\Delta\mathcal{T}^{[\sigma]}_{a}(\bm{x}_{A},\bm{\xi}_{[\sigma]}(v))}{dv},$
(15)

it appears that Eq. (9) turns into an ordinary differential equation whose
solution is

$c\Delta\mathcal{T}^{[\sigma]}_{a}(\bm{x}_{A},\bm{x}_{B})=\int_{v_{A}}^{v_{B}}\\!\lambda_{0}^{[\sigma]}(v)\mathcal{H}^{[\sigma]}_{a}(\bm{x}_{A},\bm{\xi}_{[\sigma]}(v))dv,$
(16)

where $v_{A}$ and $v_{B}$ are respectively the values of $v$ at points
$\bm{x}_{A}$ and $\bm{x}_{B}$, and
$\mathcal{H}^{[\sigma]}_{a}(\bm{x}_{A},\bm{x})$ is given by

$\mathcal{H}^{[\sigma]}_{a}(\bm{x}_{A},\bm{x})=\frac{1}{2}k_{a}^{00}(\bm{x})-k_{a}^{0i}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}+\frac{1}{2}k_{a}^{ij}(\bm{x})\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial x^{i}}\frac{c\partial
T^{[\sigma]}(\bm{x}_{A},\bm{x})}{\partial x^{j}}.$ (17)

It follows from Eqs. (11), (12) and (17) that determining the perturbation
term $c\Delta\mathcal{T}^{[\sigma]}_{a}$ comes down to calculating a line
integral along $\Gamma^{\sigma}(\bm{x}_{A},\bm{x}_{B})$, the integrand being
expressed as a functional only involving the metric components
$\gamma_{\mu\nu}$, the functions $\xi^{\mu}(v)$ and their derivatives
$\dot{\xi}_{[\sigma]}^{\mu}(v)$.

Omitting the argument $v$ for the sake of brevity, we have

$\lambda_{0}^{[\sigma]}=\gamma_{00}\dot{\xi}_{[\sigma]}^{0}+\gamma_{0k}\dot{\xi}_{[\sigma]}^{k}.$
(18)

Squaring each side of Eq. (18) and taking into account that
$\gamma_{\mu\nu}\dot{\xi}^{\mu}\dot{\xi}^{\nu}=0$ along
$\Gamma^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$, it may be seen that

$\lambda_{0}^{[\sigma]}=\pm\sqrt{\gamma_{00}\gamma^{\,\ast}_{ij}\dot{\xi}_{[\sigma]}^{i}\dot{\xi}_{[\sigma]}^{j}},$
(19)

the quantities $\gamma^{\,\ast}_{ij}$ being defined as

$\gamma^{\,\ast}_{ij}=-\gamma_{ij}+\frac{\gamma_{0i}\gamma_{0j}}{\gamma_{00}}.$
(20)

Choosing the parameter $v$ so as to have the positive determination of the
right-hand side of Eq. (19), it is immediately seen that Eq. (16) can be
written in the form

$\displaystyle c\Delta\mathcal{T}^{[\sigma]}_{a}(\bm{x}_{A},\bm{x}_{B})$
$\displaystyle=\int_{v_{A}}^{v_{B}}\sqrt{\gamma_{00}(\bm{\xi}_{[\sigma]}(v))}\mathcal{H}^{[\sigma]}_{a}(\bm{x}_{A},\bm{\xi}_{[\sigma]}(v))d\ell_{[\sigma]},$
(21)

where $d\ell_{[\sigma]}$ is the infinitesimal distance along
$\Gamma^{[\sigma]}(\bm{x}_{A},\bm{x}_{B})$ defined as

$d\ell_{[\sigma]}=\sqrt{\gamma^{\,\ast}_{ij}(\bm{\xi}_{[\sigma]}(v))\dot{\xi}_{[\sigma]}^{i}\dot{\xi}_{[\sigma]}^{j}}dv.$
(22)

Since the null geodesics of two conformal metrics are coincident (see, e.g.,
Joshi:2007 ), our problem can be reduced to the calculation of the TTFs for a
metric such that

$g_{00}=1$ (23)

holds at any point. Accordingly, the corresponding background metric can
always be chosen so that

$\gamma_{00}=1$ (24)

and the perturbation terms $h^{a}_{00}$ can be assumed to satisfy the
relations

$h^{a}_{00}=0$ (25)

for any $a$. This remark will be systematically used in the next sections.

##  IV Application to perturbed Schwarzschild-like spacetimes

The procedure outlined in the previous section can be applied when the
physical metric is a weak stationary perturbation of a static, spherically
symmetric metric corresponding to the field of a central mass $M$. We put
$m=GM/c^{2}$. Choosing appropriate quasi-Cartesian coordinates
$(x^{0},\bm{x})$ and putting $r=|\bm{x}|$, the physical $ds^{2}$ can be
considered as conformally related to a $d\bar{s}^{2}$ written in the form

$\displaystyle
d\bar{s}^{2}=(dx^{0})^{2}+2\bar{h}_{0i}(\bm{x};\mathcal{P})dx^{0}dx^{i}$
$\displaystyle\qquad\;\;\;-\left[\mathcal{U}(r)\delta_{ij}-\bar{h}_{ij}(\bm{x};\mathcal{P})\right]dx^{i}dx^{j},$
(26)

where

$\mathcal{U}(r)=1+2\kappa_{1}\frac{m}{r}+2\sum_{n=2}^{\infty}\kappa_{n}\frac{m^{n}}{r^{n}}$
(27)

and the quantities $\bar{h}_{\mu i}(\bm{x};\mathcal{P})$ are assumed to be
small perturbations developable into series analogous to (3), the parameters
$\epsilon_{a}$ characterizing here the sphericity defect of the central body.
The constants $\kappa_{n}$ are some post-Newtonian parameters. In particular,
$\kappa_{1}$ is linked to the Eddington parameter $\gamma$ by the relation

$\kappa_{1}=1+\gamma.$ (28)

In this paper, we are exclusively concerned with light rays propagating in
regions of spacetime such that $r\gg m$. The metric (26) can therefore be
considered as a small perturbation of the background metric defined by

$\gamma_{00}=1,\quad\gamma_{0i}=0,\quad\gamma_{ij}=-\left(1+\frac{2\kappa_{1}m}{r}\right)\delta_{ij},$
(29)

the perturbation potentials being the first-order terms $\bar{h}^{a}_{\mu i}$
involved in the expansion of $\bar{h}_{\mu i}$ and the quantities
$\bar{h}^{\kappa_{n}}_{\mu\nu}$ given by

$\bar{h}^{\kappa_{n}}_{00}=\bar{h}^{\kappa_{n}}_{0i}=0,\quad\bar{h}^{\kappa_{n}}_{ij}=-\frac{2\kappa_{n}m^{n}}{r^{n}}\delta_{ij}$
(30)

for $n\geq 2$.

It is easily inferred from Eqs. (5) that the $\bar{k}^{\mu\nu}_{a}$
corresponding to the $\bar{h}^{a}_{\mu\nu}$ are given by

$\displaystyle\bar{k}_{a}^{00}$ $\displaystyle=-\bar{h}_{00}^{a}=0,$ (31a)
$\displaystyle\bar{k}_{a}^{0i}$
$\displaystyle=\frac{r}{r+2\kappa_{1}m}\bar{h}_{0i}^{a},$ (31b)
$\displaystyle\bar{k}_{a}^{ij}$
$\displaystyle=-\frac{r^{2}}{(r+2\kappa_{1}m)^{2}}\bar{h}_{ij}^{a}.$ (31c)

In the same way, the $\bar{k}^{\mu\nu}_{\kappa_{n}}$ corresponding to the
$\bar{h}^{\kappa_{n}}_{\mu\nu}$ are

$\bar{k}^{00}_{\kappa_{n}}=\bar{k}^{0i}_{\kappa_{n}}=0,\quad\bar{k}_{\kappa_{n}}^{ij}=\frac{2\kappa_{n}m^{n}}{r^{n-2}(r+2\kappa_{1}m)^{2}}\delta^{ij}.$
(32)

It follows from Eqs. (30) that each parameter $\kappa_{n}$ generates a
perturbation of order $m^{n}$. This property implies that only the
perturbations due to $\kappa_{1},\kappa_{2}$ and $\kappa_{3}$ can be
consistently treated in the present work since the terms of order $m^{4}$
which are proportional to $(\kappa_{2})^{2}$ are not taken into account by our
procedure.

The TTFs corresponding to the background metric (29) are obtained in an exact
closed form in Linet:2016 . We are led to distinguish three kinds of
configurations according to the value of the angle $\psi_{AB}$ between the
vectors $\bm{x}_{A}$ and $\bm{x}_{B}$. In what follows, $\psi_{AB}$ is
unambiguously determined by the relations

$\cos\psi_{AB}=\bm{n}_{A}.\bm{n}_{B},\qquad 0\leq\psi_{AB}\leq\pi,$ (33)

where $\bm{n}_{A}$ and $\bm{n}_{B}$ are defined as

$\bm{n}_{A}=\frac{\bm{x}_{A}}{r_{A}},\qquad\bm{n}_{B}=\frac{\bm{x}_{B}}{r_{B}}.$
(34)

###  IV.1 Case where $\psi_{AB}=0$

Since $\bm{n}_{B}=\bm{n}_{A}$ in this case, there exists one and only one
zeroth-order light ray passing through $\bm{x}_{A}$ and $\bm{x}_{B}$, namely
the radial null geodesic of the metric (29) tangent to $\bm{n}_{A}$. We have
$x^{0}_{B}-x^{0}_{A}=|\int_{r_{A}}^{r_{B}}\sqrt{1+2\kappa_{1}m/r}\,dr|$ along
this path. Hence the expression of the corresponding zeroth-order radial TTF:

$cT^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=\left|\sqrt{r_{B}(r_{B}+2\kappa_{1}m)}-\sqrt{r_{A}(r_{A}+2\kappa_{1}m)}\right|+2\kappa_{1}m\left|\ln\left(\frac{\sqrt{r_{B}}+\sqrt{r_{B}+2\kappa_{1}m}}{\sqrt{r_{A}}+\sqrt{r_{A}+2\kappa_{1}m}}\right)\right|,$
(35)

from which it is inferred that

$\left[\frac{c\partial T^{[rad]}}{\partial
x^{i}}\right]_{\bm{x}=r\bm{n}_{A}}=\text{sgn}(r_{B}-r_{A})\sqrt{1+\frac{2\kappa_{1}m}{r}}n^{i}_{A}.$
(36)

Denote by $c\Delta\mathcal{T}_{a}^{[rad]}$ the perturbation due to
$\epsilon_{a}$ and by $c\Delta\mathcal{T}^{[rad]}_{\kappa_{n}}$ the
perturbation due to $\kappa_{n}$ for $n\geq 2$. Substituting for
$\bar{k}_{a}^{\mu\nu}$ from Eqs. (31) into Eq. (17), using Eq. (36) and noting
that Eq. (22) reads

$d\ell=\mbox{sgn}(r_{B}-r_{A})\sqrt{1+2\kappa_{1}m/r}dr$ (37)

when $r$ is the choosen parameter, it may be seen that Eq. (21) takes the form

$c\Delta\mathcal{T}^{[rad]}_{a}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=-\int_{r_{A}}^{r_{B}}\left[\bar{h}^{a}_{i0}(r\bm{n}_{A})n^{i}_{A}+\frac{1}{2}\text{sgn}(r_{B}-r_{A})\sqrt{\frac{r}{r+2\kappa_{1}m}}\bar{h}_{ij}^{a}(r\bm{n}_{A})n^{i}_{A}n^{j}_{A}\right]dr.$
(38)

Substituting $\bar{h}^{\kappa_{n}}_{i\alpha}$ for $\bar{h}^{a}_{i\alpha}$ into
Eq. (38) and taking Eqs. (30) into account yields for $n\geq 2$

$c\Delta\mathcal{T}^{[rad]}_{\kappa_{n}}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=\kappa_{n}m^{n}\left|I_{n-1}(r_{A},r_{B})\right|,$
(39)

where $I_{n}(r_{A},r_{B})$ is given by

$I_{n}(r_{A},r_{B})=\int_{r_{A}}^{r_{B}}\frac{dr}{r^{n}\sqrt{r(r+2\kappa_{1}m)}}.$
(40)

We have for $n=2$ and $n=3$:

$\displaystyle
c\Delta\mathcal{T}^{[rad]}_{\kappa_{2}}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})$
$\displaystyle=$
$\displaystyle\frac{2\kappa_{2}m^{2}|r_{B}-r_{A}|}{r_{A}\sqrt{r_{B}(r_{B}+2\kappa_{1}m)}+r_{B}\sqrt{r_{A}(r_{A}+2\kappa_{1}m)}},$
(41) $\displaystyle
c\Delta\mathcal{T}^{[rad]}_{\kappa_{3}}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})$
$\displaystyle=$
$\displaystyle\frac{\kappa_{3}m^{3}|r_{B}^{2}-r_{A}^{2}|}{r_{A}^{2}(r_{B}-\kappa_{1}m)\sqrt{r_{B}(r_{B}+2\kappa_{1}m)}+r_{B}^{2}(r_{A}-\kappa_{1}m)\sqrt{r_{A}(r_{A}+2\kappa_{1}m)}}$
(42)
$\displaystyle\times\left[1-\frac{2\kappa_{1}m}{3(r_{A}+r_{B})}\left(1+\frac{r_{A}}{r_{B}}+\frac{r_{B}}{r_{A}}\right)\right].$

These exact expressions will be useful to prove the continuity of the
perturbation terms proportional to $\kappa_{2}$ and to $\kappa_{3}$ (see the
next subsection). In practice, however, owing to the fact that the quantities
$I_{n}(r_{A},r_{B})$ can be expanded in series in powers of $m/r_{A}$ and
$m/r_{B}$ which are convergent when $\text{inf}(r_{A},r_{B})>2\kappa_{1}m$, we
may content ourselves with approximate expressions as follow:

$c\Delta\mathcal{T}^{[rad]}_{\kappa_{2}}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=\frac{\kappa_{2}m^{2}|r_{B}-r_{A}|}{r_{A}r_{B}}\left[1-\frac{\kappa_{1}m(r_{A}+r_{B})}{2r_{A}r_{B}}+O\left(\frac{m^{2}}{r_{AB}^{2}}\right)\right]$
(43)

and

$c\Delta\mathcal{T}^{[rad]}_{\kappa_{3}}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=\frac{\kappa_{3}m^{3}|r_{B}^{2}-r_{A}^{2}|}{2r_{A}^{2}r_{B}^{2}}\left[1+O\left(\frac{m}{r_{AB}}\right)\right],$
(44)

where

$r_{AB}=\inf(r_{A},r_{B}).$ (45)

###  IV.2 Case where $0<\psi_{AB}<\pi$

Denote by $\bm{k}_{AB}$ the unit vector defined as

$\bm{k}_{AB}=\frac{\bm{n}_{A}\times\bm{n}_{B}}{\sin\psi_{AB}}.$ (46)

0.75 0.75 (10.850454, -4.020682)-1.4169099[linecolor=black, linewidth=0.014,
dimen=outer](10.811061,-3.14)(10.811061,5.2) (9.33553,
-6.8449244)39.466568$\bm{x}_{A}$$\bm{x}_{B}$$\bm{n}_{A}$$\bm{n}_{B}$$\bm{s}$$\varphi$$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$$\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$$O$$\bm{\xi}_{+}(\varphi)$$\Pi_{O}(\bm{k}_{AB})$

Figure 1:  Representation of the zeroth-order light rays joining two given
points in the generic case where the vector positions $\bm{x}_{A}$ and
$\bm{x}_{B}$ are not collinear; these paths are confined in the plane
$\Pi_{O}(\bm{k}_{AB})$ passing through the origin $O$ and containing
$\bm{x}_{A}$ and $\bm{x}_{B}$.

Since the background metric (29) is spherically symmetric, its null geodesics
paths joining $\bm{x}_{A}$ and $\bm{x}_{B}$ are confined in the plane
$\Pi_{O}(\bm{k}_{AB})$ passing through the origin $O$ of the spatial
coordinates and orthogonal to $\bm{k}_{AB}$. Denote by $\varphi$ the angle
between $\bm{n}_{A}$ and the position vector $\bm{x}$ in the plane
$\Pi_{O}(\bm{k}_{AB})$. Then, the first integrals of a null geodesic of the
metric (29) contained in $\Pi_{O}(\bm{k}_{AB})$ may be written as

$\displaystyle\frac{dx^{0}}{d\lambda}=E,$ (47) $\displaystyle
r(r+2\kappa_{1}m)\frac{d\varphi}{d\lambda}=J,$ (48)
$\displaystyle\left(1+\frac{2\kappa_{1}m}{r}\right)\left[\left(\frac{dr}{d\lambda}\right)^{2}+r^{2}\left(\frac{d\varphi}{d\lambda}\right)^{2}\right]=E^{2},$
(49)

where $E$ and $J$ are constants of the motion, $\lambda$ being an affine
parameter.

The absolute value of the ratio $b$ defined by

$b=\frac{J}{E}$ (50)

is the intrinsic impact parameter of the corresponding zeroth-order light ray
Chandrasekhar:1983 . Equation (49) implies that the element $dl$ defined by
Eq. (22) is related to the affine parameter $\lambda$ by the equation
$dl=Ed\lambda$ ($\lambda$ may be chosen so that $E>0$). Hence

$dl=\frac{J}{b}d\lambda.$ (51)

Eliminating $J$ between Eq. (48) and Eq. (51) yields

$dl=\frac{r(r+2\kappa_{1}m)}{b}d\varphi.$ (52)

It is shown in Woude:1922  that there exist two and only two solutions to Eqs.
(47)-(49) passing through $\bm{x}_{A}$ and $\bm{x}_{B}$ when
$0<\psi_{AB}<\pi$. These solutions, denoted by
$\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ and $\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$,
are Keplerian hyperbolas with a focus coinciding with the origin $O$. By
convention, $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ is the path along which
$\varphi$ takes the values $0$ to $\psi_{AB}$, whereas
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$ is the path along which $\varphi$ takes
the values 0 to $\psi_{AB}-2\pi$. For the sake of brevity, we put

$\psi_{AB}^{+}=\psi_{AB},\qquad\psi_{AB}^{-}=\psi_{AB}-2\pi.$ (53)

The TTFs corresponding to $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ and to
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$ will be denoted by
$T^{+}(\bm{x}_{A},\bm{x}_{B})$ and $T^{-}(\bm{x}_{A},\bm{x}_{B})$,
respectively. It will be relevant in some calculations to note that these
functions satisfy the eikonal equation

$\delta^{ij}\frac{c\partial T^{\pm}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}\frac{c\partial T^{\pm}(\bm{x}_{A},\bm{x})}{\partial
x^{j}}=1+\frac{2\kappa_{1}m}{r},$ (54)

which is straiforwardly inferred from Eq. (8) when Eqs. (29) are taken into
account.

Let us denote the parametric equations of $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$
and $\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$ as functions of $\varphi$ by
$\bm{\xi}_{+}(\varphi)$ and $\bm{\xi}_{-}(\varphi)$, respectively.
Substituting for $\bar{k}_{a}^{\mu\nu}$ from Eqs. (31) in Eq. (17), using Eq.
(21) and taking Eq. (52) into account yields an expression as follows for
$c\Delta\mathcal{T}_{a}^{\pm}(\bm{x}_{A},\bm{x}_{B})$:

$c\Delta\mathcal{T}_{a}^{\pm}(\bm{x}_{A},\bm{x}_{B})=-\frac{1}{b_{\pm}}\int_{0}^{\psi_{AB}^{\pm}}\\!r^{2}\left[\bar{h}^{a}_{0i}(\bm{x})\frac{c\partial
T^{\,\pm}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}+\frac{r\bar{h}^{a}_{ij}(\bm{x})}{2(r+2\kappa_{1}m)}\frac{c\partial
T^{\,\pm}(\bm{x}_{A},\bm{x})}{\partial x^{i}}\frac{c\partial
T^{\,\pm}(\bm{x}_{A},\bm{x})}{\partial
x^{j}}\right]_{\bm{x}=\bm{\xi}_{\pm}(\varphi)}\\!d\varphi,$ (55)

where $b_{+}$ (resp. $b_{-}$) is the algebraic value of the impact parameter
of $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ (resp.
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$). We have (see Eq. (31) in Linet:2016 )

$b_{\pm}=\frac{r_{A}r_{B}\displaystyle\sqrt{1-\cos\psi_{AB}}}{2R_{AB}}\left[\sqrt{1+\cos\psi_{AB}+\frac{2\kappa_{1}m(r_{A}+r_{B}-R_{AB})}{r_{A}r_{B}}}\pm\sqrt{1+\cos\psi_{AB}+\frac{2\kappa_{1}m(r_{A}+r_{B}+R_{AB})}{r_{A}r_{B}}}\right],$
(56)

where $R_{AB}$ is defined as

$R_{AB}=|\bm{x}_{B}-\bm{x}_{A}|=\sqrt{r_{A}^{2}+r_{B}^{2}-2r_{A}r_{B}\cos\psi_{AB}}.$
(57)

Note that $b_{+}>0$ and $b_{-}<0$.

The zeroth-order time transfer functions $T^{+}$ and $T^{-}$ corresponding to
the background metric (29) are given by Eq. (39) in Linet:2016 :

$\displaystyle cT^{\,\pm}(\bm{x}_{A},\bm{x}_{B})=$
$\displaystyle\frac{1}{2}\left(\sqrt{r_{A}+r_{B}+R_{AB}}\sqrt{r_{A}+r_{B}+R_{AB}+4\kappa_{1}m}\mp\sqrt{r_{A}+r_{B}-R_{AB}}\sqrt{r_{A}+r_{B}-R_{AB}+4\kappa_{1}m}\right)$
(58)
$\displaystyle\mbox{}+2\kappa_{1}m\ln\left(\frac{\displaystyle\sqrt{r_{A}+r_{B}+R_{AB}+4\kappa_{1}m}+\sqrt{r_{A}+r_{B}+R_{AB}}}{\displaystyle\sqrt{r_{A}+r_{B}-R_{AB}+4\kappa_{1}m}\pm\sqrt{r_{A}+r_{B}-R_{AB}}}\right).\quad$

Substituting $\bm{x}$ for $\bm{x}_{B}$ into Eq. (58) and differentiating with
respect to $x^{i}$ yields

$\frac{c\partial T^{\pm}(\bm{x}_{A},\bm{x})}{\partial
x^{i}}=\frac{1}{2}\left\\{\left[X^{+}(\bm{x}_{A},\bm{x})\pm
X^{-}(\bm{x}_{A},\bm{x})\right]\frac{x^{i}-x_{A}^{i}}{|\bm{x}-\bm{x}_{A}|}+\left[X^{+}(\bm{x}_{A},\bm{x})\mp
X^{-}(\bm{x}_{A},\bm{x})\right]\frac{x^{i}}{r}\right\\},$ (59)

where $X^{+}(\bm{x}_{A},\bm{x})$ and $X^{-}(\bm{x}_{A},\bm{x})$ are given by

$X^{\pm}(\bm{x}_{A},\bm{x})=\sqrt{1+\frac{4\kappa_{1}m}{r_{A}+r\pm|\bm{x}-\bm{x}_{A}|}}.$
(60)

The parametric functions $\bm{\xi}_{\pm}(\varphi)$ describing the hyperbolas
$\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ and $\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$
can be written in the form

$\bm{\xi}_{\pm}(\varphi)=r_{\pm}(\varphi)[\cos\varphi\,\bm{n}_{A}+\sin\varphi\,(\bm{k}_{AB}\times\bm{n}_{A})],$
(61)

where

$r_{\pm}(\varphi)=\frac{p_{\pm}}{1+e_{\pm}\cos(\varphi-\varphi^{\pm}_{P})},$
(62)

the parameters $p_{\pm}$ and the eccentricities $e_{\pm}$ being given by
relations as follow (see Linet:2016 ):

$p_{\pm}=\frac{b^{2}_{\pm}}{\kappa_{1}m},\quad\quad
e_{\pm}=\frac{|b_{\pm}|}{\kappa_{1}m}\sqrt{1+\frac{\kappa_{1}^{2}m^{2}}{b^{2}_{\pm}}},$
(63)

and $\varphi^{+}_{P}$ (resp. $\varphi^{-}_{P}$) denoting the value of
$\varphi$ at the pericenter of $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ (resp.
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$). Given that $\varphi^{+}_{P}$ and
$\varphi^{-}_{P}$ are determined by the equations which express that
$\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ and $\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$
pass through $\bm{x}_{A}$ and $\bm{x}_{B}$, namely

$\displaystyle\frac{p_{\pm}}{e_{\pm}r_{A}}-\frac{1}{e_{\pm}}=\cos\varphi^{\pm}_{P},$
(64a)
$\displaystyle\frac{p_{\pm}}{e_{\pm}r_{B}}-\frac{1}{e_{\pm}}=\cos(\psi_{AB}-\varphi^{\pm}_{P}),$
(64b)

it is easily seen that $1/r_{\pm}(\varphi)$ can be written as

$\frac{1}{r_{\pm}(\varphi)}=\frac{\kappa_{1}m}{b^{2}_{\pm}}+\frac{1}{\sin\psi_{AB}}\left[\left(\frac{1}{r_{A}}-\frac{\kappa_{1}m}{b^{2}_{\pm}}\right)\sin(\psi_{AB}-\varphi)+\left(\frac{1}{r_{B}}-\frac{\kappa_{1}m}{b^{2}_{\pm}}\right)\sin\varphi\right].$
(65)

Equations (56), (59)-(60), (63) and (65) constitute the ingredients allowing
to write the explicit expression of the integral appearing in the right hand
side of Eq. (55) when the perturbation terms $\bar{h}^{a}_{\mu\nu}$ are known.
The computation of $c\Delta\mathcal{T}^{\pm}_{a}$ will be greatly simplified
if $\bar{h}^{a}_{ij}=\bar{h}^{a}\delta_{ij}$, since it results from Eq. (54)
that

$\bar{h}^{a}_{ij}(\bm{x})\frac{c\partial T^{\pm}}{\partial
x^{i}}\frac{c\partial T^{\pm}}{\partial
x^{j}}=\bar{h}^{a}(\bm{x})\left(1+\frac{2\kappa_{1}m}{r}\right).$ (66)

In particular, substituting for $\bar{h}_{\mu\nu}^{\kappa_{n}}$ from Eqs. (30)
into Eq. (55) straightforwardly leads to

$c\Delta\mathcal{T}^{\pm}_{\kappa_{2}}(\bm{x}_{A},\bm{x}_{B})=\frac{\kappa_{2}m^{2}}{b_{\pm}}\psi_{AB}^{\pm}$
(67)

and

$\displaystyle\\!\\!\\!c\Delta\mathcal{T}^{\pm}_{\kappa_{3}}(\bm{x}_{A},\bm{x}_{B})$
$\displaystyle=$
$\displaystyle\frac{\kappa_{3}m^{3}}{b_{\pm}^{2}}\bigg{[}\frac{\sin\psi_{AB}^{\pm}}{1+\cos\psi_{AB}^{\pm}}\left(\frac{b_{\pm}}{r_{A}}+\frac{b_{\pm}}{r_{B}}\right)$
$\displaystyle+\frac{\kappa_{1}m}{b_{\pm}}\left(\psi_{AB}^{\pm}-\frac{2\sin\psi_{AB}^{\pm}}{1+\cos\psi_{AB}^{\pm}}\right)\bigg{]}.$

###  IV.3 Case where $\psi_{AB}=\pi$

Let $\bm{k}$ be any unit vector orthogonal to the common direction of
$\bm{n}_{A}$ and $\bm{n}_{B}$. The plane $\Pi_{O}(\bm{k})$ passing through $O$
and orthogonal to $\bm{k}$ contains two zeroth-order null geodesics joining
$\bm{x}_{A}$ and $\bm{x}_{B}$. The arbitrariness in the choice of
$\Pi_{O}(\bm{k})$ implies that our procedure leads in this case to a double
infinity of TTFs whose perturbed part depends on $\bm{k}$. Explicit examples
of this dependence are given in Subsect. V.3. The zeroth-order terms
$cT^{\pm}$ are simply obtained by setting $R_{AB}=r_{A}+r_{B}$ in Eq. (58),
which gives

$\displaystyle cT^{\pm}(r_{A}\bm{n}_{A},-r_{B}\bm{n}_{A})$ $\displaystyle=$
$\displaystyle(r_{A}+r_{B})\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}$
$\displaystyle-\kappa_{1}m\ln\frac{2\kappa_{1}m}{r_{A}+r_{B}}$
$\displaystyle+2\kappa_{1}m\ln\left(1+\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}\right)$

Equation (56) reduces to

$b_{\pm}=\pm\sqrt{\frac{2\kappa_{1}mr_{A}r_{B}}{r_{A}+r_{B}}}.$ (70)

Substituting for $b_{\pm}$ from Eq. (70) into Eqs. (63) yields expressions as
follow for $p_{\pm}$ and $e_{\pm}$:

$\displaystyle p_{\pm}$ $\displaystyle=$
$\displaystyle\frac{2r_{A}r_{B}}{r_{A}+r_{B}},$ (71) $\displaystyle e_{\pm}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2r_{A}r_{B}}{\kappa_{1}m(r_{A}+r_{B})}+1}.$ (72)

Using Eq. (71), it is easily deduced from Eqs. (64) that

$e_{\pm}\cos\varphi^{\pm}_{P}=\frac{r_{B}-r_{A}}{r_{A}+r_{B}}.$ (73)

This relation determines the angular position of the respective pericenters of
$\Gamma^{+}(\bm{x}_{A},\bm{x}_{B};\bm{k})$ and
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B};\bm{k})$ when the geometrically obvious
conditions $0<\varphi^{+}_{P}<\pi$ and $\varphi^{-}_{P}=-\varphi^{+}_{P}$ are
taken into account. Equations (72) and (73) imply that

$e_{\pm}\sin\varphi^{\pm}_{P}=\pm\sqrt{\frac{2r_{A}r_{B}}{\kappa_{1}m(r_{A}+r_{B})}\left(1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}\right)}.$
(74)

To finish, Eqs. (62), (71), (73) and (74) lead to

$\displaystyle\\!\\!\\!\\!\frac{1}{r_{\pm}(\varphi)}$ $\displaystyle=$
$\displaystyle\frac{r_{A}+r_{B}}{2r_{A}r_{B}}+\frac{r_{B}-r_{A}}{2r_{A}r_{B}}\cos\varphi$
(75)
$\displaystyle\pm\sqrt{\frac{r_{A}+r_{B}}{2\kappa_{1}mr_{A}r_{B}}\left(1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}\right)}\sin\varphi.$

Equations (58)-(60), (70), (71) and (75) constitute the whole set of relations
enabling one to get the explicit expressions of the integrals involved in Eq.
(55) when the emitter and the receiver are located in diametrically opposite
directions. The contributions due to $\kappa_{2}$ and $\kappa_{3}$ are given
by formulas as follow:

$\displaystyle c\Delta\mathcal{T}^{\pm}_{\kappa_{2}}(\bm{x}_{A},\bm{x}_{B})$
$\displaystyle=$ $\displaystyle\frac{\kappa_{2}\pi
m}{\sqrt{2\kappa_{1}}}\sqrt{\frac{m(r_{A}+r_{B})}{r_{A}r_{B}}},$ (76)
$\displaystyle c\Delta\mathcal{T}^{\pm}_{\kappa_{3}}(\bm{x}_{A},\bm{x}_{B})$
$\displaystyle=$
$\displaystyle\frac{\kappa_{3}m^{2}}{\kappa_{1}}\frac{r_{A}+r_{B}}{r_{A}r_{B}}\Bigg{[}\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}$
(77)
$\displaystyle+\frac{\pi}{2}\sqrt{\frac{\kappa_{1}m(r_{A}+r_{B})}{2r_{A}r_{B}}}\Bigg{]}.$

For $\psi_{AB}$ close to $\pi$, Eq. (56) implies

$\displaystyle b_{\pm}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{2\kappa_{1}mr_{A}r_{B}}{r_{A}+r_{B}}}$ (78)
$\displaystyle+\frac{r_{A}r_{B}}{2(r_{A}+r_{B})}\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}(\pi-\psi_{AB})$
$\displaystyle+O\left[(\pi-\psi_{AB})^{2}\right].$

Using Eq. (78), it may be seen that Eq. (76) and Eq. (77) are respectively the
limit of Eq. (67) and Eq. (LABEL:Tka3) when $\psi_{AB}\rightarrow\pi$. It is
thus explicitly checked that the functions
$c\Delta\mathcal{T}^{\pm}_{\kappa_{2}}$ and
$c\Delta\mathcal{T}^{\pm}_{\kappa_{3}}$ calculated above in the different
possible configurations are continuous whatever $\bm{x}_{A}$ and $\bm{x}_{B}$.

###  IV.4 Existence of two regimes

Our method is valid only if the zeroth-order light ray along which the
perturbation terms are calculated is entirely located in a region of weak
field, which amounts to supposing that $r\gg m$ at any point of the path.
Since it is assumed in this paper that $\text{inf}(r_{A},r_{B})\gg m$, the
weak-field condition can be violated only if the null geodesic path joining
$\bm{x}_{A}$ and $\bm{x}_{B}$ passes through a pericenter whose radial
coordinate is close to $m$. Denote by $r_{P}^{+}$ (resp. $r_{P}^{-}$) the
value of $r$ at the pericenter of $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ (resp.
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$). It follows from Eqs. (62) and (63) that

$r_{P}^{\pm}=\sqrt{b_{\pm}^{2}+\kappa_{1}^{2}m^{2}}-\kappa_{1}m.$ (79)

Using Eqs. (56) and (79), a discussion which can be summarized as follows
leads to distinguish at least two regimes (see Ref. Linet:2016  for the
details).

1\. First regime.– If $\psi_{AB}$ is such that arccos

$\arccos\left[\frac{\kappa_{1}m(r_{A}+r_{B})}{r_{A}r_{B}}-1\right]\lesssim\psi_{AB}\leq\pi,$
(80)

then $\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ and
$\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$ are located in the region where the weak
field approximation is valid. This regime is typically encountered in a
configuration of gravitational lensing.

2\. Second regime.– If $\psi_{AB}$ is such that

$0<\psi_{AB}\ll\arccos\left[\frac{\kappa_{1}m(r_{A}+r_{B})}{r_{A}r_{B}}-1\right],$
(81)

it may be seen that $r_{P}^{-}\thicksim\kappa_{1}m$, a value which is not
compatible with the weak-field approximation. As a consequence, only the
perturbation terms $c\Delta\mathcal{T}_{a}^{+}$ and
$c\Delta\mathcal{T}_{\kappa_{n}}^{+}$ can be considered as legitimate in the
context of our procedure when condition (81) is met.

The second regime is typically encountered when we make $\psi_{AB}$ tend to
$0$. The computation of the limits of $c\Delta\mathcal{T}^{+}_{\kappa_{2}}$
and $c\Delta\mathcal{T}^{+}_{\kappa_{3}}$ in this case is easy if we note that

$\displaystyle b_{+}$ $\displaystyle=$
$\displaystyle\frac{r_{A}\sqrt{r_{B}(r_{B}+2\kappa_{1}m)}+r_{B}\sqrt{r_{A}(r_{A}+2\kappa_{1}m)}}{2|r_{B}-r_{A}|}\psi_{AB}$
(82) $\displaystyle\times[1+O(\psi_{AB}^{2})]$

when $\psi_{AB}$ is close to 0. Substituting for $b_{+}$ from Eq. (82) into
Eq. (67), it appears that $\lim_{\psi_{AB}\rightarrow
0}c\Delta\mathcal{T}^{+}_{\kappa_{2}}(\bm{x}_{A},\bm{x}_{B})$ coincides with
the right-hand side of Eq. (41). The continuity of the perturbation due to
$\kappa_{2}$ as $\psi_{AB}\rightarrow 0$ is thus directly proved. A similar
conclusion can be drawn for the perturbation due to $\kappa_{3}$. Indeed, Eqs.
(LABEL:Tka3) and (82) lead to an expression of $\lim_{\psi_{AB}\rightarrow
0}c\Delta\mathcal{T}^{+}_{\kappa_{3}}(\bm{x}_{A},\bm{x}_{B})$ which can be
checked with some calculations to be identical to the right-hand side of Eq.
(42).

###  IV.5 TTFs in static, spherically symmetric spacetimes

When the source of the gravitational field is a non-rotating spherically
symmetric body, the perturbation potentials $\bar{h}_{i\alpha}$ vanish. The
TTFs provided by our procedure then reduce to the functions
$\mathcal{T}^{\pm}_{sph}$ given by

$c\mathcal{T}^{\pm}_{sph}=cT^{\pm}+c\Delta\mathcal{T}_{\kappa_{2}}^{\pm}+c\Delta\mathcal{T}_{\kappa_{3}}^{\pm}.$
(83)

It is of interest to compare the expansion of $\mathcal{T}^{+}_{sph}$ in
ascending powers of $m$ with the expression of the TTF obtained in Linet:2013
. It follows from Eq. (58) that $cT^{+}$ can be expanded into a convergent
series in powers of $m$ if and only if

$r_{A}+r_{B}-R_{AB}\geq 4\kappa_{1}m.$ (84)

Since we assume $\inf(r_{A},r_{B})\gg m$, it is easily seen that condition
(84) is equivalent to

$0\leq\psi_{AB}\leq\psi_{m}(r_{A},r_{B}),$ (85)

where $\psi_{m}(r_{A},r_{B})$ is defined as

$\psi_{m}(r_{A},r_{B})=\arccos\left[\frac{4\kappa_{1}m(r_{A}+r_{B}-2\kappa_{1}m)}{r_{A}r_{B}}-1\right].$
(86)

On the other hand, it is straightforwardly inferred from Eq. (56) that $b_{+}$
can be developed into a convergent series in powers of $m$ if and only if

$\frac{2\kappa_{1}m(r_{A}+r_{B}+R_{AB})}{r_{A}r_{B}(1+\cos\psi_{AB})}\leq 1,$
(87)

a condition which can be checked as being satisfied when
$\psi_{AB}\leq\psi_{m}(r_{A},r_{B})$. So, using the expansion of $cT^{+}$ in
powers of $m$ given in Linet:2016  and substituting for $1/b_{+}$ from

$\displaystyle\frac{1}{b_{+}}$ $\displaystyle=$
$\displaystyle\frac{R_{AB}}{r_{A}r_{B}\sin\psi_{AB}}$ (88)
$\displaystyle\times\left[1-\frac{\kappa_{1}m}{1+\cos\psi_{AB}}\frac{r_{A}+r_{B}}{r_{A}r_{B}}+O\left(\frac{m^{2}}{r_{AB}^{2}}\right)\right]$

into Eqs. (67) and (LABEL:Tka3), it is rigorously shown that
$c\mathcal{T}_{sph}^{+}$ can be expanded as

$c\mathcal{T}_{sph}^{+}(\bm{x}_{A},\bm{x}_{B})=c\mathcal{T}_{sph}^{\ast}(\bm{x}_{A},\bm{x}_{B})+O\left(\frac{m^{4}}{r_{AB}^{4}}\right)$
(89)

with

$\displaystyle c\mathcal{T}_{sph}^{\ast}(\bm{x}_{A},\bm{x}_{B})$
$\displaystyle=$ $\displaystyle
R_{AB}+\kappa_{1}m\ln\left(\frac{r_{A}+r_{B}+R_{AB}}{r_{A}+r_{B}-R_{AB}}\right)$
$\displaystyle+\frac{m^{2}R_{AB}}{r_{A}r_{B}}\left(\frac{\kappa_{2}\psi_{AB}}{\sin\psi_{AB}}-\frac{\kappa_{1}^{2}}{1+\cos\psi_{AB}}\right)$
$\displaystyle+\frac{m^{3}R_{AB}(r_{A}+r_{B})}{r_{A}^{2}r_{B}^{2}(1+\cos\psi_{AB})}$
$\displaystyle\;\;\;\times\left(\kappa_{3}-\frac{\kappa_{1}\kappa_{2}\psi_{AB}}{\sin\psi_{AB}}+\frac{\kappa_{1}^{3}}{1+\cos\psi_{AB}}\right)$

if and only if condition (85) is satisfied.

We thus recover Eqs. (17), (62), (86) and (87) obtained in Linet:2013 , while
bringing a very important precision since we now specify the condition of
convergence of the expansion provided by Eqs. (89) and (LABEL:dTsp).

As a final remark, it may be noted that the limit of the right-hand side of
Eq. (LABEL:dTsp) when $\psi_{AB}\rightarrow 0$ is the development of
$c\mathcal{T}_{sph}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{B})$ obtained by
expanding $\left|\int_{r_{A}}^{r_{B}}\sqrt{\mathcal{U}(r)}dr\right|$ into a
series of powers of $m$ up to and including the third order in $m$ (see Eqs.
(19)-(22) in Linet:2013 ). This conclusion is directly inferred from
$\lim_{\psi_{AB}\rightarrow 0}R_{AB}=|r_{B}-r_{A}|$.

##  V Application to a spinning axisymmetric body

The formulas obtained in Sec. IV can be applied when the gravitational field
is generated by an isolated, axisymmetric body of mass $M$ slowly rotating
about its axis of symmetry with a constant angular velocity $\bm{\omega}$. The
problem is treated here within the post-Newtonian framework for fully
conservative theories. Modeling the light propagation for the current
experiments performed in the Solar System or the astrometry at a level of 1
microarcsecond requires to retain only the terms of order $c^{-2}$ and
$c^{-3}$ in the metric Klioner:1992 ; Klioner:2003 . So we will content
ourselves with a physical metric having the form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2W}{c^{2}}\right)(dx^{0})^{2}+\frac{4\kappa_{1}}{c^{3}}(\bm{W}.\bm{d}\bm{x})dx^{0}$
(91) $\displaystyle\mbox{}-\left(1+\frac{2\gamma
W}{c^{2}}\right)\delta_{ij}dx^{i}dx^{j},$

where $W$ is the Newtonian-like potential of the body, namely

$W(\bm{x})=G\int_{\mathcal{D}}\frac{\rho^{\ast}(\bm{x}^{\prime})}{|\bm{x}-\bm{x}^{\prime}|}d^{3}\bm{x}^{\prime},$
(92)

and $\bm{W}$ is the gravitomagnetic vector defined as

$\bm{W}(\bm{x})=G\int_{\mathcal{D}}\frac{\rho^{\ast}(\bm{x}^{\prime})(\bm{\omega}\times\bm{x}^{\prime})}{|\bm{x}-\bm{x}^{\prime}|}d^{3}\bm{x}^{\prime},$
(93)

$\rho^{\ast}$ being the conserved mass density of the body (see, e.g.,
Will:2018 ) and $\mathcal{D}$ the volume occupied by the matter.

The origin $O$ of the spatial coordinates $x^{i}$ is assumed to be situated on
the axis of symmetry. Then, outside any sphere of radius $r_{0}$ centered on
$O$ and encompassing the central body, $W/c^{2}$ may be represented by the
well-known multipole expansion

$\frac{W(\bm{x})}{c^{2}}=\frac{m}{r}\left[1-\sum_{n=1}^{\infty}J_{n}\left(\frac{r_{0}}{r}\right)^{n}P_{n}(\bm{s}.\bm{n})\right],$
(94)

where $\bm{s}$ is a unit vector parallel to the axis of symmetry, $\bm{n}$ is
defined as

$\bm{n}=\frac{\bm{x}}{r},$ (95)

$P_{n}$ denotes the Legendre polynomial of degree $n$ and $J_{n}$ is the
corresponding mass multipole moment of the body. In most cases, $r_{0}$ can be
chosen as the equatorial radius of the body. Of course, we have $J_{1}=0$ if
the origin $O$ coincides with the center of mass.

Let $\bm{S}$ be the intrinsic angular momentum of the body rotating about its
axis of symmetry. We have

$\bm{S}=S\bm{s},$ (96)

where $S=I\omega$, $I$ being the moment of inertia of the body about its axis
of symmetry and $\omega$ being defined by $\bm{\omega}=\omega\bm{s}$. By
analogy with the Kerr metric, it is convenient to introduce the quantity $a$
having the dimension of a length defined as

$a=\frac{S}{Mc}.$ (97)

With this notation, the gravitomagnetic vector $\bm{W}/c^{3}$ may be written
in the form (see, e.g., Linet:2002  and references therein)

$\frac{\bm{W}(\bm{x})}{c^{3}}=\frac{ma(\bm{s}\times\bm{x})}{2r^{3}}\left[1-\sum_{n=1}^{\infty}K_{n}\left(\frac{r_{0}}{r}\right)^{n}\\!P^{\prime}_{n+1}(\bm{s}.\bm{n})\right],$
(98)

where $P^{\prime}_{n+1}(x)$ denotes the derivative of the Legendre polynomial
$P_{n+1}(x)$ with respect to $x$ and each $K_{n}$ is a constant depending on
the matter distribution inside the body.

For light rays, the conformal metric obtained by dividing the right-hand side
of Eq. (91) by $1-2W/c^{2}$ can be used in place of the initial metric. Since
the terms of order $c^{-4}$ are neglected, the conformal metric to be
considered here reduces to

$\displaystyle d\bar{s}^{2}$ $\displaystyle=$
$\displaystyle(dx^{0})^{2}+\frac{4\kappa_{1}}{c^{3}}\left({\bm{W}}.{\bm{d}}\bm{x}\right)dx^{0}$
(99)
$\displaystyle\mbox{}-\left(1+\frac{2\kappa_{1}W}{c^{2}}\right){\bm{d}}\bm{x}^{2}.$

In accordance with the methodology developed in Section IV, the right-hand
side of Eq. (99) is regarded as a stationary perturbation of the background
metric defined by Eq. (29). The perturbation parameters $\epsilon_{a}$ are
then identified with the dimensionless quantities $J_{n}$, $a/m$ and $K_{n}$.
The non-zero perturbations of the background metric may be written as follows:

$\bar{h}_{ij}^{J_{n}}(\bm{x})=2\kappa_{1}mJ_{n}\frac{r_{0}^{n}}{r^{n+1}}P_{n}(\bm{s}.\bm{n})\delta_{ij}$
(100)

for the contribution due to the multipole $J_{n}$,

$\bar{\bm{h}}^{S}(\bm{x})=\frac{\kappa_{1}ma(\bm{s}\times\bm{n})}{r^{2}}$
(101)

for the dominant contribution due to $\bm{S}$ and

$\bar{\bm{h}}^{K_{n}}(\bm{x})=-\kappa_{1}maK_{n}(\bm{s}\times\bm{n})\frac{r_{0}^{n}}{r^{n+2}}P^{\prime}_{n+1}(\bm{s}.\bm{n})$
(102)

for the contribution of the spin multipole $K_{n}$, where we use the vector
notation

$\bar{\bm{h}}^{a}=(\bar{h}^{a}_{0i}).$ (103)

###  V.1 Case where $\psi_{AB}=0$

Substituting $\bar{h}_{ij}^{J_{n}}$ for $\bar{h}^{a}_{ij}$ into Eq. (38) gives

$\displaystyle
c\Delta\mathcal{T}_{J_{n}}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})$
$\displaystyle=$ $\displaystyle-\kappa_{1}mJ_{n}r_{0}^{n}|I_{n}(r_{A},r_{B})|$
(104) $\displaystyle\times P_{n}(\bm{s}\cdot\bm{n}_{A}),$

where the quantities $I_{n}(r_{A},r_{B})$ are defined by Eq. (40).

Equation (59) implies that the gradient of $cT^{+}(\bm{x}_{A},\bm{x})$ and
$\bm{x}$ are collinear when $\bm{x}=r\bm{n}_{A}$. As a consequence, it follows
from Eqs. (38), (101) and (102) that

$c\Delta\mathcal{T}_{S}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=0$ (105)

and

$c\Delta\mathcal{T}_{K_{n}}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=0.$ (106)

Taking into account the relation

$I_{n}(r_{A},r_{B})=\frac{1}{n}\left(\frac{1}{r_{A}^{n}}-\frac{1}{r_{B}^{n}}\right)\left[1+O\left(\frac{m}{r_{AB}}\right)\right]$
(107)

and Eqs. (105)-(106), it may be seen that Eqs. (35) and (104) lead to an
expansion as follows:

$c\mathcal{T}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})=|r_{B}-r_{A}|+\kappa_{1}m\left\\{\left|\ln\frac{r_{B}}{r_{A}}\right|-\sum_{n=1}^{\infty}\frac{1}{n}J_{n}\left|\frac{r_{0}^{n}}{r_{A}^{n}}-\frac{r_{0}^{n}}{r_{B}^{n}}\right|P_{n}(\bm{s}.\bm{n}_{A})\right\\}+O\left(\frac{m^{2}}{r_{AB}^{2}}\right),$
(108)

where $r_{AB}$ is defined by Eq. (45).

According to the assumption made at the beginning of this section, we may
content ourselves with the approximation provided by Eq. (108). However, exact
expressions or full expansions are necessary for checking the continuity of
the perturbation functions associated with the mass and spin multipoles. In
the next subsection, explicit expressions are obtained for the perturbation
terms due to $J_{1}$, $J_{2}$ and $\bm{S}$ when $\bm{n}_{A}$ and $\bm{n}_{B}$
are not collinear. So, it will be essential to know the exact expressions of
$c\Delta\mathcal{T}_{J_{1}}^{[rad]}$ and $c\Delta\mathcal{T}_{J_{2}}^{[rad]}$
which are directly inferred from (104), namely:

$\displaystyle
c\Delta\mathcal{T}_{J_{1}}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})$
$\displaystyle=$
$\displaystyle-2\kappa_{1}mJ_{1}r_{0}\frac{|r_{B}-r_{A}|(\bm{s}.\bm{n}_{A})}{r_{A}\sqrt{r_{B}(r_{B}+2\kappa_{1}m)}+r_{B}\sqrt{r_{A}(r_{A}+2\kappa_{1}m)}},$
(109) $\displaystyle
c\Delta\mathcal{T}_{J_{2}}^{[rad]}(r_{A}\bm{n}_{A},r_{B}\bm{n}_{A})$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa_{1}mJ_{2}r_{0}^{2}\frac{|r_{B}^{2}-r_{A}^{2}|[3(\bm{s}.\bm{n}_{A})^{2}-1]}{r_{A}^{2}(r_{B}-\kappa_{1}m)\sqrt{r_{B}(r_{B}+2\kappa_{1}m)}+r_{B}^{2}(r_{A}-\kappa_{1}m)\sqrt{r_{A}(r_{A}+2\kappa_{1}m)}}$
(110)
$\displaystyle\mbox{}\times\left[1-\frac{2\kappa_{1}m}{3(r_{A}+r_{B})}\left(1+\frac{r_{A}}{r_{B}}+\frac{r_{B}}{r_{A}}\right)\right].$

###  V.2 Case where $0<\psi_{AB}<\pi$

Substituting $\bar{h}^{J_{n}}_{ij}$ for $\bar{h}^{a}_{ij}$ into Eq. (55) and
taking into account Eq. (66) yields an expression as follows for
$c\Delta\mathcal{T}^{\pm}_{J_{n}}$:

$c\Delta\mathcal{T}_{J_{n}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=-\frac{\kappa_{1}m}{b_{\pm}}J_{n}r_{0}^{n}\mathcal{J}^{\pm}_{n}(\bm{x}_{A},\bm{x}_{B}),$
(111)

where

$\mathcal{J}^{\pm}_{n}(\bm{x}_{A},\bm{x}_{B})=\int_{0}^{\psi_{AB}^{\pm}}P_{n}\left[\frac{\bm{s}\cdot\bm{\xi}_{\pm}(\varphi)}{r_{\pm}(\varphi)}\right]\frac{d\varphi}{r_{\pm}^{n-1}(\varphi)},$
(112)

$\bm{\xi}_{\pm}(\varphi)/r_{\pm}(\varphi)$ being directly inferred from Eq.
(61) and $1/r_{\pm}(\varphi)$ being given by Eq. (65).

The contributions due to the gravitomagnetic potentials are given by

$c\Delta\mathcal{T}_{S}^{\pm}(\bm{x}_{A},\bm{x}_{B})=-\kappa_{1}ma(\bm{s}.\bm{k}_{AB})\int_{0}^{\psi^{\pm}_{AB}}\frac{d\varphi}{r_{\pm}(\varphi)}$
(113)

and

$c\Delta\mathcal{T}_{K_{n}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=\kappa_{1}ma(\bm{s}.\bm{k}_{AB})K_{n}r_{0}^{n}\mathcal{K}_{n}^{\pm}(\bm{x}_{A},\bm{x}_{B}),$
(114)

where

$\mathcal{K}^{\pm}_{n}(\bm{x}_{A},\bm{x}_{B})=\int_{0}^{\psi^{\pm}_{AB}}P^{\prime}_{n+1}\left[\frac{\bm{s}\cdot\bm{\xi}_{\pm}(\varphi)}{r_{\pm}(\varphi)}\right]\frac{d\varphi}{r_{\pm}^{n+1}(\varphi)}.$
(115)

The integrals involved in Eqs. (112), (113) and (115) can be calculated with
any symbolic computer program. In this work, we only calculate
$c\Delta\mathcal{T}_{J_{1}}^{\pm}$, $c\Delta\mathcal{T}_{J_{2}}^{\pm}$ and
$c\Delta\mathcal{T}_{S}^{\pm}$.

Using Eq. (46), it is easily seen that
$\mathcal{J}_{1}^{\pm}(\bm{x}_{A},\bm{x}_{B})$ can be written as

$\mathcal{J}^{\pm}_{1}(\bm{x}_{A},\bm{x}_{B})=\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}(\bm{s}.\bm{n}_{A}+\bm{s}.\bm{n}_{B}).$
(116)

Obtaining $\mathcal{J}_{2}^{\pm}$ in a convenient form for discussing the
behavior of $c\Delta\mathcal{T}_{J_{2}}^{\pm}(\bm{x}_{A},\bm{x}_{B})$ when
$\psi_{AB}\rightarrow 0$ or $\psi_{AB}\rightarrow\pi$ requires some rather
tricky transformations detailed in the Appendix. Getting
$c\Delta\mathcal{T}_{S}^{\pm}$ is elementary.

We find:

$\displaystyle c\Delta\mathcal{T}_{J_{1}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=$
$\displaystyle-\kappa_{1}mJ_{1}\frac{r_{0}}{b_{\pm}}\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}(\bm{s}.\bm{n}_{A}+\bm{s}.\bm{n}_{B}),$
(117) $\displaystyle c\Delta\mathcal{T}^{\pm}_{J_{2}}(\bm{x}_{A},\bm{x}_{B})=$
$\displaystyle\frac{1}{2}\kappa_{1}mJ_{2}\left(\frac{r_{0}}{b_{\pm}}\right)^{2}\\!\Bigg{\\{}\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}\left(\frac{b_{\pm}}{r_{A}}+\frac{b_{\pm}}{r_{B}}-\frac{2\kappa_{1}m}{b_{\pm}}\right)$
(118)
$\displaystyle\times\left[1-2(\bm{s}.\bm{n}_{A}).(\bm{s}.\bm{n}_{B})-(\bm{s}\times\bm{k}_{AB})^{2}(1-\cos\psi_{AB})\right]$
$\displaystyle-\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}\left[\left(\frac{b_{\pm}}{r_{A}}-\frac{\kappa_{1}m}{b_{\pm}}\right)(\bm{s}.\bm{n}_{A})^{2}+\left(\frac{b_{\pm}}{r_{B}}-\frac{\kappa_{1}m}{b_{\pm}}\right)(\bm{s}.\bm{n}_{B})^{2}\right]$
$\displaystyle+\frac{\kappa_{1}m}{2b_{\pm}}\Big{[}\left[3(\bm{s}.\bm{k}_{AB})^{2}-1\right]\psi_{AB}^{\pm}+3\left[(\bm{s}\times\bm{k}_{AB})^{2}\cos\psi_{AB}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\right]\sin\psi_{AB}\Big{]}\Bigg{\\}},$
$\displaystyle c\Delta\mathcal{T}_{S}^{\pm}(\bm{x}_{A},\bm{x}_{B})=$
$\displaystyle-\kappa_{1}m\frac{a(\bm{s}.\bm{k}_{AB})}{b_{\pm}}\left[\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}\left(\frac{b_{\pm}}{r_{A}}+\frac{b_{\pm}}{r_{B}}-\frac{2\kappa_{1}m}{b_{\pm}}\right)+\frac{\kappa_{1}m}{b_{\pm}}\psi_{AB}^{\pm}\right].$
(119)

It follows from a result established in Subsect. IV.5 that the perturbation
terms given by Eqs. (117)-(119) can be expanded in convergent series in powers
of $m$ when condition (85) is satisfied. Considering only the second regime
and substituting for $1/b_{+}$ from Eq. (88), we get for the contributions due
to $J_{1}$ and $J_{2}$:

$\displaystyle c\Delta\mathcal{T}_{J_{1}}^{+}(\bm{x}_{A},\bm{x}_{B})=$
$\displaystyle-\kappa_{1}mJ_{1}\frac{r_{0}R_{AB}}{r_{A}r_{B}}\frac{\bm{s}.\bm{n}_{A}+\bm{s}.\bm{n}_{B}}{1+\cos\psi_{AB}}+O\left(\frac{m^{2}}{r_{AB}^{2}}\right),$
(120) $\displaystyle c\Delta\mathcal{T}_{J_{2}}^{+}(\bm{x}_{A},\bm{x}_{B})=$
$\displaystyle\frac{1}{2}\kappa_{1}mJ_{2}\frac{r_{0}^{2}}{r_{A}r_{B}}\frac{R_{AB}}{1+\cos\psi_{AB}}\bigg{\\{}\frac{1-(\bm{s}.\bm{n}_{A})^{2}}{r_{A}}+\frac{1-(\bm{s}.\bm{n}_{B})^{2}}{r_{B}}$
(121)
$\displaystyle-\left[(\bm{s}\times\bm{k}_{AB})^{2}(1-\cos\psi_{AB})+2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\right]\left(\\!\frac{1}{r_{A}}+\frac{1}{r_{B}}\\!\right)\\!\bigg{\\}}+O\left(\frac{m^{2}}{r_{AB}^{2}}\right),$

and for the dominant contribution of the angular momentum

$c\Delta\mathcal{T}_{S}^{+}(\bm{x}_{A},\bm{x}_{B})=-\kappa_{1}ma(\bm{s}.\bm{k}_{AB})\frac{r_{A}+r_{B}}{r_{A}r_{B}}\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}+O\left(\frac{m^{2}}{r_{AB}^{2}}\right).$
(122)

Equations (121) and (122) are equivalent to formulas previously obtained by
other procedures within the first post-Newtonian approximation (see, e.g.,
Kopeikin:1997 ; Kopeikin:2002 ; Linet:2002 ; Ciufolini:2003 ; Leponcin:2008 ).

Case where $\bm{s}.\bm{n}_{A}=\bm{s}.\bm{n}_{B}=0$.– The zeroth-order paths
$\Gamma^{+}(\bm{x}_{A},\bm{x}_{B})$ and $\Gamma^{-}(\bm{x}_{A},\bm{x}_{B})$
are then confined in the plane passing through $O$ and orthogonal to the axis
of rotation of the central body (this plane is the equatorial plane if $O$ is
chosen so that $J_{1}=0$). Equation (117) reduces to
$c\Delta\mathcal{T}_{J_{1}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=0$ and Eqs.
(118)-(119) are significantly simplified, since
$\bm{s}.\bm{n}_{A}=\bm{s}.\bm{n}_{B}=0$ implies that
$\bm{s}\times\bm{k}_{AB}=0$ and $(\bm{s}.\bm{k}_{AB})^{2}=1$. More generally,
the calculation of the integrals $\mathcal{J}_{n}$ and $\mathcal{K}_{n}$ is
then greatly facilitated since $\bm{s}.\bm{n}(\varphi)=0$ for any $\varphi$,
which means that the terms $P_{n}[\bm{s}.\bm{n}(\varphi)]$ and
$P^{\prime}_{n+1}[\bm{s}.\bm{n}(\varphi)]$ reduce to the constants $P_{n}(0)$
and $P^{\prime}_{n+1}(0)$, respectively. In particular, it follows from the
relations $P_{2k+1}(0)=P^{\prime}_{2k}(0)=0$ that

$c\Delta\mathcal{T}_{J_{2k+1}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=c\Delta\mathcal{T}_{K_{2k+1}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=0$
(123)

for $k=0,1,2,\ldots$

###  V.3 Case where $\psi_{AB}=\pi$

According to the considerations developed in Subsec. IV.3, the perturbation
terms depend now on an arbitrary unit vector $\bm{k}$ orthogonal to
$\bm{n}_{A}$. In this case, $\psi_{AB}^{\pm}=\pm\pi$ and the function
$1/r_{\pm}(\varphi)$ occurring in the integrals involved in Eqs. (111), (113)
and (114) is given by Eq. (75). Taking into account the expression of
$b_{\pm}$ supplied by Eq. (70), we get expressions as follow for the terms
which must be added to the zeroth-order TTFs given by Eq. (LABEL:T0pi):

$\displaystyle
c\Delta\mathcal{T}^{\pm}_{J_{1}}(r_{A}\bm{n}_{A},-r_{B}\bm{n}_{A};\bm{k})=$
$\displaystyle\mp
J_{1}r_{0}\sqrt{\frac{2\kappa_{1}m(r_{A}+r_{B})}{r_{A}r_{B}}}[(\bm{s}\times\bm{k}).\bm{n}_{A}],$
(124) $\displaystyle
c\Delta\mathcal{T}^{\pm}_{J_{2}}(r_{A}\bm{n}_{A},-r_{B}\bm{n}_{A};\bm{k})=$
$\displaystyle\frac{1}{2}J_{2}\frac{r_{0}^{2}(r_{A}+r_{B})}{r_{A}r_{B}}\Bigg{\\{}\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}\left[2(\bm{s}.\bm{k})^{2}+(\bm{s}.\bm{n}_{A})^{2}-1\right]$
(125)
$\displaystyle+\sqrt{\frac{\kappa_{1}m(r_{A}+r_{B})}{2r_{A}r_{B}}}\left[\pi\frac{3(\bm{s}.\bm{k})^{2}-1}{4}\mp
2\frac{r_{B}-r_{A}}{r_{A}+r_{B}}[(\bm{s}\times\bm{k}).\bm{n}_{A}](\bm{s}.\bm{n}_{A})\right]\Bigg{\\}}$

and

$\displaystyle
c\Delta\mathcal{T}^{\pm}_{S}(r_{A}\bm{n}_{A},-r_{B}\bm{n}_{A};\bm{k})=$
$\displaystyle\mp
a(\bm{s}.\bm{k})\sqrt{\frac{2\kappa_{1}m(r_{A}+r_{B})}{r_{A}r_{B}}}\left[\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}+\frac{\pi}{2}\sqrt{\frac{\kappa_{1}m(r_{A}+r_{B})}{2r_{A}r_{B}}}\right].$
(126)

It follows from these formulas that the perturbation terms obtained with our
procedure are continuous functions when the emitter and the receiver are
located in diametrically opposite directions. Indeed, return to Eqs.
(117)-(119) and let $\bm{n}_{B}$ tend to $-\bm{n}_{A}$ in such a way that
$\bm{k}_{AB}\rightarrow\bm{k}$ when $\psi_{AB}\rightarrow\pi$. Using the
expansion

$b_{\pm}=\pm\sqrt{\frac{2\kappa_{1}mr_{A}r_{B}}{r_{A}+r_{B}}}+\frac{r_{A}r_{B}}{2(r_{A}+r_{B})}\sqrt{1+\frac{2\kappa_{1}m}{r_{A}+r_{B}}}(\pi-\psi_{AB})+O\left[(\pi-\psi_{AB})^{2}\right],$
(127)

it can be checked that Eqs. (124)-(126) are recovered.

##  VI Application to the Cassini experiment

Equation (LABEL:dTsp) has been used to discuss in depth the numerical
determination of the post-Newtonian parameter $\gamma$ with an accuracy
$\sigma_{\gamma}\approx 2.3\times 10^{-5}$ from the data collected during the
Cassini mission (see Ashby:2010  and Refs. therein). For this determination,
the relevant configurations were light rays almost grazing the surface of the
Sun and corresponding to an emitter and a receiver located in almost
diametrically opposed positions. So it is of interest to check if the
condition of convergence (85) was satisfied or not, and then to compare the
numerical values of the travel time of light provided by Eq. (83) and by Eq.
(LABEL:dTsp), respectively.

For the discussion, it is convenient to introduce the dimensionless parameters

$l_{A}=\frac{r_{c}}{r_{A}},\quad
l_{B}=\frac{r_{c}}{r_{B}},\quad\epsilon_{c}=\frac{m}{r_{c}},$ (128)

where $r_{c}$ is the Euclidean distance between the origin $O$ and the
straight line passing through $\bm{x}_{A}$ and $\bm{x}_{B}$, given by

$r_{c}=\frac{r_{A}r_{B}\sin\psi_{AB}}{R_{AB}}.$ (129)

It is clear that $l_{A}\ll 1$ and $l_{B}\ll 1$ in the Cassini mission.
Moreover, it may be assumed that $r_{c}\geq r_{\odot}$, with $r_{\odot}$ being
the radius of the Sun ($r_{\odot}$ = 696,000 km), which means that
$\epsilon_{c}\leq 2.15\times 10^{-6}$.

Since we are concerned here with configurations such that
$\pi/2<\psi_{AB}<\pi$, it is easily inferred from Eqs. (57) and (129) that

$\cos\psi_{AB}=l_{A}l_{B}-\sqrt{(1-l_{A}^{2})(1-l_{B}^{2})}.$ (130)

Consequently, the necessary and sufficient condition of convergence expressed
by Eq. (85) can be written in the form

$\displaystyle\sqrt{(1-l_{A}^{2})(1-l_{B}^{2})}\leq$ $\displaystyle
1+l_{A}l_{B}-4\kappa_{1}\epsilon_{c}$ (131)
$\displaystyle\times(l_{A}+l_{B}-2\kappa_{1}\epsilon_{c}l_{A}l_{B}).\quad$

Squaring each side of Eq. (131) and then simplifying, it is easily checked
that a condition like

$l_{A}+l_{B}-8\kappa_{1}\epsilon_{c}(1+l_{A}l_{B})\geq 0$ (132)

is sufficient to insure the convergence of the expansion given by Eqs.
(89)-(LABEL:dTsp). For the Cassini mission, we have

$r_{A}\geq 1\text{au},\quad r_{B}=1\text{au},$ (133)

where au denotes the astronomical unit. Hence

$l_{A}+l_{B}>4.64\times 10^{-3}$ (134)

since $r_{c}\geq r_{\odot}$. On the other hand, the inequality
$1+l_{A}l_{B}\leq 2$ is always satisfied since
$r_{c}\leq\text{inf}(r_{A},r_{B})$. As a consequence, assuming that
$\kappa_{1}=2$ leads to

$8\kappa_{1}\epsilon_{c}(1+l_{A}l_{B})<6.86\times 10^{-5}$ (135)

for any light ray travelling in the Solar System. It follows from Eqs. (134)
and (135) that condition (132) was fully satisfied in the part of the Cassini
mission exploited for determining $\gamma$. The TTF expansion whose first
terms are given by Eq. (LABEL:dTsp) was therefore rightfully used.

To numerically compare the results provided by the functions
$c\mathcal{T}_{sph}^{+}$ and $c\mathcal{T}_{sph}^{\ast}$, it is necessary to
know the expressions of $\sin\psi_{AB}$ and $R_{AB}$. Equation (130) yields

$\sin\psi_{AB}=l_{A}\sqrt{1-l_{B}^{2}}+l_{B}\sqrt{1-l_{A}^{2}}.$ (136)

The expression of $R_{AB}$ is then straightforwardly obtained by substituting
for $\sin\psi_{AB}$ from Eq. (136) into Eq. (129):

$R_{AB}=r_{c}\frac{l_{A}\sqrt{1-l_{B}^{2}}+l_{B}\sqrt{1-l_{A}^{2}}}{l_{A}l_{B}}.$
(137)

Furthermore, it is convenient to use the ratio $m/b_{+}$ written in the
following form, directly deduced from Eq. (56):

$\frac{m}{b_{+}}=\frac{\sqrt{1+\cos\psi_{AB}+2\kappa_{1}\epsilon_{c}\left(l_{A}+l_{A}+\sin\psi_{AB}\right)}-\sqrt{1+\cos\psi_{AB}+2\kappa_{1}\epsilon_{c}\left(l_{A}+l_{A}-\sin\psi_{AB}\right)}}{2\kappa_{1}\sqrt{1-\cos\psi_{AB}}},$
(138)

where $\cos\psi_{AB}$ and $\sin\psi_{AB}$ have to be replaced by their
expressions yielded by Eq. (130) and Eq. (136), respectively.

The quantities $c\mathcal{T}^{+}_{sph}$ and $c\mathcal{T}^{\ast}_{sph}$ can be
expressed as functions of
$l_{A},l_{B},r_{c},m,\kappa_{1},\kappa_{2},\kappa_{3}$. For the comparison, we
assume that the parameters $\kappa_{1},\kappa_{2}$ and $\kappa_{3}$ take their
values predicted by general relativity, namely

$\kappa_{1}=2,\quad\kappa_{2}=\frac{15}{4},\quad\kappa_{3}=\frac{9}{2}.$ (139)

Given $r_{A}$ and $r_{B}$,
$|c\mathcal{T}^{+}_{sph}-c\mathcal{T}^{\ast}_{sph}|$ is a decreasing function
of $r_{c}$. So the most favorable estimate will be obtained for
$r_{c}=r_{\odot}$. Then, with $r_{B}=1$ au, it is easily checked that
$|c\mathcal{T}^{+}_{sph}-c\mathcal{T}^{\ast}_{sph}|$ increases up to 27.35
$\mu$m as $r_{A}$ increases up to 40 au. Our correction is thus much smaller
than the uncertainty on the position of the Cassini spacecraft, which was
typically about 30 cm.

##  VII Conclusion

In this paper we develop a perturbation method allowing to determine a class
of TTFs for a stationary metric which is a weak perturbation of a background
metric being itself stationary. This method works when the light rays of the
background metric are explicitly known as parametric functions only involving
the spatial positions of the emitter and the receiver. Equations (17), (21)
and (22) provide the basic results as simple integrals taken along the zeroth-
order light rays.

When the metric is a stationary perturbation of a Schwarzschild-like metric,
the expressions of the TTFs in the generic case are given by Eqs. (55) and
(58), the zeroth-order light rays being Keplerian hyperbolas.

For an isolated, slowly rotating axisymmetric body, the integrals giving the
perturbation terms due to the mass and spin multipoles can be calculated with
any symbolic computer program. We get explicit expressions of these integrals
for the mass multipoles $J_{1}$, $J_{2}$ and for the intrinsic angular
momentum $\bm{S}$ of the rotating body.

The procedure set in the present work carries out a noticeable improvement on
the previous attempts to determine the time transfer functions since the
expressions obtained for the TTFs are totally devoid of enhanced terms
whatever the positions of the emitter and the receiver. Furthermore, a
criterion of convergence of the expansion of the TTFs in series of powers of
$m$ is supplied by Eqs. (85) and (86).

It can be underlined that our calculations remain valid when the emitter and
the receiver are in diametrically opposite positions. So the present
methodology will certainly be useful for modeling the difference in light
travel time between two different images occuring in a gravitational lensing
configuration.

###### Acknowledgements.

This paper is dedicated to the memory of Bernard Linet, with whom we had very
stimulating discussions which are at the origin of this work. We warmly thank
Jean Teyssandier for his careful reading of the manuscript.

*

##  Appendix A Calculation of $c\Delta\mathcal{T}_{J_{2}}^{\pm}$

Let us put

$A^{\pm}(\bm{x}_{A},\bm{x}_{B})=\int_{0}^{\psi_{AB}^{\pm}}\left\\{\sin^{2}\psi_{AB}-3\left[(\bm{s}.\bm{n}_{A})\sin(\psi_{AB}-\varphi)+(\bm{s}.\bm{n}_{B})\sin\varphi\right]^{2}\right\\}d\varphi$
(140)

and

$\displaystyle B^{\pm}(\bm{x}_{A},\bm{x}_{B};p)$ $\displaystyle=$
$\displaystyle\int_{0}^{\psi_{AB}^{\pm}}\left\\{\sin^{2}\psi_{AB}-3\left[(\bm{s}.\bm{n}_{A})\sin(\psi_{AB}-\varphi)+(\bm{s}.\bm{n}_{B})\sin\varphi\right]^{2}\right\\}$
(141)
$\displaystyle\times\left[\left(\frac{p}{r_{A}}-1\right)\sin(\psi_{AB}-\varphi)+\left(\frac{p}{r_{B}}-1\right)\sin\varphi\right]d\varphi.$

Since $P_{2}(x)=(3x^{2}-1)/2$, Eq. (111) may be written as follows for $n=2$ :

$c\Delta\mathcal{T}_{J_{2}}^{\pm}(\bm{x}_{A},\bm{x}_{B})=\frac{1}{2}\kappa_{1}mJ_{2}\left(\frac{r_{0}}{b_{\pm}}\right)^{2}\frac{\kappa_{1}m}{b_{\pm}}\frac{A^{\pm}(\bm{x}_{A},\bm{x}_{B})\sin\psi_{AB}+B^{\pm}(\bm{x}_{A},\bm{x}_{B},p_{\pm})}{\sin^{3}\psi_{AB}}.$
(142)

The problem is reduced to the calculation of $A^{+}(\bm{x}_{A},\bm{x}_{B})$
and $B^{+}(\bm{x}_{A},\bm{x}_{B};p)$ since it is easily checked that

$\frac{A^{-}(\bm{x}_{A},\bm{x}_{B})\sin\psi_{AB}+B^{-}(\bm{x}_{A},\bm{x}_{B};p)}{\sin^{3}\psi_{AB}}=\frac{A^{+}(\bm{x}_{A},\bm{x}_{B})\sin\psi_{AB}+B^{+}(\bm{x}_{A},\bm{x}_{B};p)}{\sin^{3}\psi_{AB}}-2\pi
P_{2}(\bm{s}.\bm{k}_{AB}).$ (143)

Using a symbolic computer program and grouping the terms involving the
parameter $p$ give

$\displaystyle\frac{A^{+}(\bm{x}_{A},\bm{x}_{B})\sin\psi_{AB}+B^{+}(\bm{x}_{A},\bm{x}_{B};p)}{\sin^{3}\psi_{AB}}$
$\displaystyle=$
$\displaystyle\left[1-3\frac{(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\cos\psi_{AB}}{2\sin^{2}\psi_{AB}}\right]\psi_{AB}$
(144)
$\displaystyle+3\frac{[(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}]\cos\psi_{AB}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})}{2\sin\psi_{AB}}$
$\displaystyle+\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}p\bigg{[}\frac{1-(\bm{s}.\bm{n}_{A})^{2}}{r_{A}}+\frac{1-(\bm{s}.\bm{n}_{B})^{2}}{r_{B}}$
$\displaystyle-\frac{(\bm{s}.\bm{n}_{A}+\bm{s}.\bm{n}_{B})^{2}}{1+\cos\psi_{AB}}\left(\\!\frac{1}{r_{A}}+\frac{1}{r_{B}}\\!\right)\bigg{]}$
$\displaystyle+\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}\left[(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}-2+2\frac{(\bm{s}.\bm{n}_{A}+\bm{s}.\bm{n}_{B})^{2}}{1+\cos\psi_{AB}}\right].$

Several terms involved in the right-hand side of Eq. (144) can be transformed
in such a way that their apparent divergence for $\psi_{AB}\rightarrow 0$ or
$\psi_{AB}\rightarrow\pi$ is eliminated, as it is shown in what follows.

1.– Using the identity

$(\bm{s}\times(\bm{n}_{A}\times\bm{n}_{B}))^{2}=(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\cos\psi_{AB},$
(145)

and then taking into account Eq. (46), it is easily seen that

$\frac{(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\cos\psi_{AB}}{\sin^{2}\psi_{AB}}=(\bm{s}\times\bm{k}_{AB})^{2}=1-(\bm{s}.\bm{k}_{AB})^{2}.$
(146)

As a consequence

$1-3\frac{(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\cos\psi_{AB}}{2\sin^{2}\psi_{AB}}=\frac{3(\bm{s}.\bm{k}_{AB})^{2}-1}{2}=P_{2}(\bm{s}.\bm{k}_{AB}).$
(147)

2.– From Eq. (146), it is inferred that

$\frac{[(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}]\cos\psi_{AB}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})(1-\sin^{2}\psi_{AB})}{\sin\psi_{AB}}=[1-(\bm{s}.\bm{k}_{AB})^{2}]\sin\psi_{AB}\cos\psi_{AB}.$

Therefore

$\frac{[(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}]\cos\psi_{AB}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})}{\sin\psi_{AB}}=\left\\{[1-(\bm{s}.\bm{k}_{AB})^{2}]\cos\psi_{AB}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\right\\}\sin\psi_{AB}.$
(148)

3.– Equation (145) may be rewritten in the form

$(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}=(\bm{s}\times\bm{k}_{AB})^{2}\sin^{2}\psi_{AB}+2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\cos\psi_{AB}.$
(149)

This relation implies

$[(\bm{s}.\bm{n}_{A})+(\bm{s}.\bm{n}_{B})]^{2}=\left[(\bm{s}\times\bm{k}_{AB})^{2}(1-\cos^{2}\psi_{AB})+2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\right](1+\cos\psi_{AB}).$
(150)

As a consequence

$\frac{[(\bm{s}.\bm{n}_{A})+(\bm{s}.\bm{n}_{B})]^{2}}{1+\cos\psi_{AB}}=\left[1-(\bm{s}.\bm{k}_{AB})^{2}\right](1-\cos\psi_{AB})+2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B}).$
(151)

Inserting Eqs. (147), (148) and (151) into Eq. (144), and then taking Eq.
(143) into account, we get

$\displaystyle\frac{A^{+}(\bm{x}_{A},\bm{x}_{B})\sin\psi_{AB}+B^{+}(\bm{x}_{A},\bm{x}_{B};p)}{\sin^{3}\psi_{AB}}$
$\displaystyle=$ $\displaystyle P_{2}(\bm{s}.\bm{k}_{AB})\psi_{AB}$ (152)
$\displaystyle+\frac{3}{2}\left\\{\left[1-(\bm{s}.\bm{k}_{AB})^{2}\right]\cos\psi_{AB}-2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\right\\}\sin\psi_{AB}$
$\displaystyle+\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}\big{[}(\bm{s}.\bm{n}_{A})^{2}+(\bm{s}.\bm{n}_{B})^{2}+4(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})$
$\displaystyle-2\cos\psi_{AB}-2(\bm{s}.\bm{k}_{AB})^{2}(1-\cos\psi_{AB})\big{]}$
$\displaystyle+\frac{\sin\psi_{AB}}{1+\cos\psi_{AB}}p\left[\frac{1-(\bm{s}.\bm{n}_{A})^{2}}{r_{A}}+\frac{1-(\bm{s}.\bm{n}_{B})^{2}}{r_{B}}\right.$
$\displaystyle\left.-\left\\{[1-(\bm{s}.\bm{k}_{AB})^{2}](1-\cos\psi_{AB})+2(\bm{s}.\bm{n}_{A})(\bm{s}.\bm{n}_{B})\right\\}\frac{r_{A}+r_{B}}{r_{A}r_{B}}\right].$

Hence the formula (118) when Eq. (63) is taken into account and the terms are
suitably grouped together.

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# The Importance of Accurate Alignments in End-to-End Speech Synthesis

###### Abstract

Unit selection synthesis systems required accurate segmentation and labeling
of the speech signal owing to the concatenative nature. Hidden Markov model-
based speech synthesis accommodates some transcription errors, but it was
later shown that accurate transcriptions yield highly intelligible speech with
smaller amounts of training data. With the arrival of end-to-end (E2E)
systems, it was observed that very good quality speech could be synthesised
with large amounts of data. As end-to-end synthesis progressed from Tacotron
to FastSpeech2, it has become imminent that features that represent prosody
are important for good-quality synthesis. In particular, durations of the sub-
word units are important. Variants of FastSpeech use a teacher model or forced
alignments to obtain good-quality synthesis. In this paper, we focus on
duration prediction, using signal processing cues in tandem with forced
alignment to produce accurate phone durations during training.

The current work aims to highlight the importance of accurate alignments for
good-quality synthesis. An attempt is made to train the E2E systems with
accurately labeled data, and compare the same with approximately labeled data.

Index Terms—  end-to-end speech synthesis, accurate alignments, signal
processing, FastSpeech2

##  1 Introduction

This work explores the importance of accurate alignments in the context of
end-to-end (E2E) text-to-speech synthesis (TTS) systems. Specifically, the
focus is on improving the duration prediction (and thereby the synthesis
quality) of TTS systems by correcting the phoneme alignments of the training
data with the aid of signal processing cues.

The E2E approach is the popular state-of-the-art speech synthesis paradigm due
to its ease in training systems to obtain high-quality speech. The initial E2E
systems were primarily attention-based, such as Tacotron [1], Tacotron2 [2].
The main goal of the attention module in TTS tasks is to learn the alignments
between characters/phones and mel-spectrogram frames. The attention module
learns soft alignments, in comparison to hard alignments used in traditional
TTS approaches such as unit selection synthesis (USS) and hidden Markov model
(HMM) based speech synthesis systems (HTS). The attention network is trained
to act as a duration predictor during synthesis.

One of the main drawbacks of attention-based networks is that the alignments
may not be learnt correctly. Coupled with an auto-regressive decoder, the
synthesis output is prone to errors, such as insertion or deletion of phones.
Hence, the focus of E2E technology has shifted to improving duration
prediction during synthesis. The duration information corresponding to the
training data can be learnt in different ways. FastSpeech [3] uses alignments
predicted by a teacher model. Some architectures such as Fastspeech2 [4] and
DurIAN [5] employ external aligners for this purpose. Other recent works, such
as [6, 7, 8, 9] learn the alignments internally.

In the context of HMM-based systems, [10, 11] have studied the effect of
accurate alignments on synthesis quality and intelligibility, highlighting the
importance of accurate boundaries for training. In this paper, we attempt to
answer the question of “How relevant are accurate alignments in the context of
E2E systems?”. Such a study is very important to produce good quality speech
as duration is an important prosody marker. We employ an external aligner, the
hybrid segmentation (HS) algorithm, which combines signal processing cues in
tandem with deep learning techniques [11], to obtain accurate alignments for
the training data. We use the FastSpeech2 architecture [4] and the HiFi-GAN
vocoder [12] for E2E training111In this paper, we refer to the two-stage
pipeline of generating mel-spectrograms and then reconstructing waveforms as
E2E training.. The synthesis quality of systems trained with different
alignment techniques are compared– using a teacher model, Montreal forced
aligner (MFA) [13] and hybrid segmentation. We also conduct experiments in a
low-resource scenario. Subjective evaluations and qualitative observations
indicate that accurate alignments are crucial for generating good-quality
speech, especially in low-resource scenarios.

The rest of the paper is organised as follows. Section 2 reviews the related
work. The proposed work and related experiments are presented in Sections 3
and 4, respectively. The work is concluded in Section 5.

##  2 Related Work

This section presents recent literature dealing with duration modeling in E2E
training. In FastSpeech [3], duration information is obtained from a
transformer TTS [14], which is considered a teacher model. External aligners
are used in a few papers– MFA [13, 4], HMM-based [15], and connectionist
temporal classification (CTC) based [16]. TTS systems trained in [6, 8, 9, 17,
18] learn duration information internally using HMM-based approaches. In [6,
8, 9], soft and hard alignments are learnt with monotonicity constraint. Glow-
TTS [19] uses normalizing flows and dynamic programming to determine the most
probable monotonic alignment between text and the latent representation of the
audio. In [20], word-level hard alignments are obtained from an external
aligner and soft phone alignments are learnt using a word-to-phoneme attention
network. A recently developed network called SoftSpeech [21] proposes a soft
length regulator for unsupervised duration modeling within the FastSpeech2
network.

Among the presented literature, [18, 21] demonstrate the capability of their
TTS systems in low resource scenarios. In the current work, we use an external
aligner, namely, the hybrid segmentation algorithm [11], which is a signal
processing aided approach. FastSpeech2 is chosen as the mel-spectrogram
generation model as it provides more variance information with pitch and
energy.

##  3 Proposed Work

In this section, we present our proposed approach. We first explain two of the
alignment modules that are widely used– (1) the teacher-student approach and
(2) MFA, which is an external aligner. Then we present the hybrid HMM-GD-DNN
segmentation (HS) approach, the alignment which we propose to be used for
FastSpeech2. The E2E system considered in this work has a 2-stage pipeline:
(1) text to mel-spectrogram conversion (FastSpeech2), and (2) speech
reconstruction using a vocoder (HiFi-GAN). Figure 1 presents the flowchart of
systems trained with different alignments.

Fig. 1: Flowchart illustrating the training and synthesis phases of the TTS
systems. The grey box corresponds to the aligner used. Paths A, B and C refer
to the alignments obtained using the teacher model, MFA and hybrid
segmentation, respectively. Only one of the paths among A, B and C are taken
during training.

###  3.1 Alignment modules

We employ three techniques to obtain the phoneme durations required for
FastSpeech2, as described below.

####  3.1.1 Teacher-student approach (TS)

In the teacher-student approach, phoneme durations from an auto-regressive
Tacotron2 teacher model (or transformer network) are fed to FastSpeech2 model
training. From a trained teacher model, encoder-decoder attention alignments
are extracted for every $<$text, audio$>$ pair as described in [3].

####  3.1.2 Montreal forced aligner (MFA)

MFA [13] is an open-source speech-text aligner that provides phone and word
boundaries. MFA performs triphone modeling and performs speaker adaptation to
model inter-speaker differences. Models are trained in MFA using the Kaldi
speech recognition toolkit [22].

####  3.1.3 Hybrid HMM-GD-DNN segmentation (HS)

Hybrid segmentation is an alignment technique that combines the complementary
features of machine learning and signal processing based approaches to
generate accurate phone boundaries [10, 11]. HMM-based forced alignment does
not accurately model the location of phone boundaries. Hence, in [10], these
boundaries are corrected using signal processing based cues. Specifically, a
group delay (GD) based algorithm is used to obtain accurate syllable
boundaries. However, the drawback of the GD-based technique is that it doesn’t
capture the correct number of syllable boundaries as it is agnostic to the
text. Hence, spurious GD boundaries are estimated, and the GD boundary closest
to an HMM boundary is considered the correct syllable boundary [10]. Then the
phone boundaries are re-estimated within these syllable boundaries instead of
re-estimating across the entire utterance. Additionally, sub-band spectral
flux (SBSF) is used as a cue for correcting boundaries of fricatives,
affricates and nasals [23].

In [11], the accuracy of phone boundaries and synthesis quality are compared
across TTS systems trained with only deep neural network (DNN) alignments and
with DNN alignments employing boundary correction. In the latter, the
alignments obtained by the hybrid HMM-GD technique are considered initial
alignments for DNN segmentation. Experiments show that the synthesis quality
with boundary correction is better than with only DNN alignments. Motivated by
this, we use the hybrid HMM-GD-DNN alignments for FastSpeech2 training, and
compare systems trained with the other machine-learning based alignments
discussed previously.

Fig. 2: An example of a Hindi waveform (bottom panel), its spectrogram (fourth
panel), and phone-level alignments obtained from different techniques (top 3
panels). The highlighted regions indicate the alignments in MFA and the
correct alignments obtained using hybrid segmentation.

Figure 2 shows a sample Hindi waveform, its spectrogram and phone-level
alignments obtained from different techniques. It is clearly seen that the
alignments of TS are not correct. Although MFA has better alignments, the
boundaries are more refined with HS.

###  3.2 Text processing

We first convert the text to its phone-based representation using the unified
parser for Indian languages [24]. The unified parser takes a word as input and
applies relevant language-specific rules to generate the phone-based output in
the common label set (CLS) representation. This output is further processed
such that each phone is represented by a single character, as described in
[25].

Based on the duration information, each phone in the text is assigned a value
equal to the number of frames. A comma is included in the text wherever the
aligner has predicted a short pause (sp). An additional symbol “$” is included
for beginning and end silence regions (SIL) in the audio (if present).
Initially, spaces in the text were removed, as they are linguistic units and
do not carry any meaning in the acoustic space. However, it was observed that
systems trained with spaces still performed better. Hence, spaces were
retained in the training text and correspondingly assigned zero duration. An
example of a sample Hindi text and its processed version is given in Table 1.

Table 1: An example of Hindi text processed for FastSpeech2

###  3.3 FastSpeech2 architecture

We use a transformer-based encoder-decoder FastSpeech2 architecture in our
work [4]. In FastSpeech2, the text is converted to phoneme embeddings which
are then passed through a series of feed-forward transformer (FFT) blocks to
generate the phoneme hidden sequence. The phoneme hidden sequence is then
expanded to match the length of the mel-spectrogram sequence based on the
duration information. Then the expanded phoneme sequence is passed through
another set of FFT blocks at the decoder to generate mel-spectrogram frames.
To provide more variance information, pitch and energy embeddings are also
added to the phoneme hidden state.

During training, the phoneme durations are obtained from a teacher model (such
as Tacotron2) or an external aligner. Pitch and energy values are extracted
from the ground-truth audio files. Duration, pitch and energy predictors are
trained and optimized with mean square error (MSE) loss. During synthesis,
these prosodic features are predicted by the network.

###  3.4 HiFi-GAN vocoder

HiFi-GAN is a GAN-based vocoder capable of producing high-fidelity speech from
mel-spectrograms [12]. It is a non-autoregressive vocoder that models periodic
patterns in speech audio. HiFi-GAN has a smaller footprint size and a higher
synthesis speed compared to most neural vocoders.

##  4 Experiments and Results

Systems are trained on Hindi male dataset (8.5 hours) from the open-source
Indic TTS database [26]. Three systems are trained based on the alignments
used– (1) from Tacotron2 as the teacher model (TS), (2) with MFA, and (3)
using hybrid HMM-GD-DNN alignments (HS). We also compare these models in a
low-resource scenario, i.e., using only 1 hour of training data.

For training Tacotron2 and FastSpeech2 models, the ESPNet toolkit was used
[27], with the default parameters. HiFi-GAN models were trained using an open-
source code222https://github.com/jik876/hifi-gan. The hybrid segmentation
code333www.iitm.ac.in/donlab/tts/hybridSeg.php, implemented using HTK [28] and
Kaldi [22] toolkits, was used. For the low-resource scenario, corresponding
models were trained using only 1 hour of data444Synthesised samples:
www.iitm.ac.in/donlab/preview/ICASSP2023_Anusha_HS/index.html.

We conduct pairwise comparison (PC) tests to assess the performance across
different systems. In PC test, listeners are presented with a pair of audio
files in random order of systems, and asked to give their preference. The test
sentences were selected from the web, ensuring coverage of different domains–
news, sports, entertainment, and technical lectures. 14 native Hindi listeners
participated in each PC test and evaluated 10 pairs of audio files. Results of
the PC test comparing TS vs. HS and MFA vs. HS systems with full data are
presented in Table 2. On average, the system with HS is preferred in more than
56% of the cases, with an equal preference of 34% across the competing
systems.

Table 2: PC test results for full data (preference in $\%$) System pairs | TS/MFA | HS | Equal
---|---|---|---
TS vs. HS | 10.99 | 51.65 | 37.36
MFA vs. HS | 7.69 | 61.54 | 30.77

Further, a PC test is also conducted in the low-resource scenario. The system
with TS is excluded from the experiments as the training of the Tacotron2 (1
hour) teacher model failed due to lack of adequate data. Consequently, the
FastSpeech2 student model also did not train well. 14 native listeners
evaluated a set of 10 audio pairs. Results of the PC test comparing MFA and HS
systems are presented in Table 3. Although the preference for the HS system in
the low resource scenario has reduced, the system still outperforms the MFA (1
hour) model.

Table 3: PC test results with 1 hour data (preference in $\%$) System pairs | MFA | HS | Equal
---|---|---|---
MFA vs. HS | 16.48 | 36.26 | 47.25

To demonstrate that HS provides more accurate boundaries, we manually align 10
ground truth utterances at the phone level. The average of absolute boundary
differences with different alignments (of full data) is given in Table 4. It
is clearly seen that HS provides better alignments compared to MFA.

Table 4: Alignment accuracy Alignment technique | MFA | HS
---|---|---
Duration difference (in ms) | 11.88 | 4.40
Fig. 3: Spectrograms of synthesised utterances of systems (with full data)
using MFA (top) and HS (bottom) corresponding to the text “tarika hai”. Fig.
4: Spectrograms of synthesised utterances of systems (with 1 hour data) using
MFA (top) and HS (bottom) corresponding to the text “sahi dxhang”.

Figures 3 and 4 show spectrograms of utterances in which the MFA systems have
errors (highlighted regions). In Figure 3, the MFA system produces the
affricate “c” instead of voiceless stop consonant “t” in “tarika”. In Figure
4, the aspirated voiced stop consonant “dxh” in “dxhang” is missed by the MFA
system, while uttered correctly by the HS system (as evidenced by the voice
bar).

Our observations from the experiments in this work are summarised here.
Although the alignments from the teacher model are not correct in most places,
the FastSpeech2 student model still learns to generate good-quality speech,
given enough amount of training data. But to further improve the pronunciation
of sounds in the generated output, more accurate alignments are required.
Experiments show that accurate alignments are especially important in low-
resource scenarios.

##  5 Conclusion

In this work, we have studied the role of accurate alignments in the context
of training FastSpeech2 based E2E system. We show that systems trained with
the hybrid HMM-GD-DNN technique with refined boundaries produce better quality
speech compared to the popularly used MFA aligner, even in a low-resource
scenario. This work can be further explored in the context of direct text-to-
wave E2E systems.

##  6 Acknowledgement

We want to thank the Ministry of Electronics and Information Technology
(MeitY), Government of India (GoI), for funding the project “Speech
Technologies in Indian Languages” (SP/21-22/1960/CSMEIT/003119).

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