# Time delay in the gravitational field of an axisymmetric body at rest with
full mass and spin multipole structure

Sven Zschocke  Institute of Planetary Geodesy - Lohrmann Observatory, TUD
Dresden University of Technology, Helmholtzstrasse 10, D-01069 Dresden,
Germany

###### Abstract

The time delay of a light signal which propagates in the gravitational field
of an isolated body is considered. The body can be of arbitrary but time-
independent shape and inner structure and can be in uniform rotational motion,
while the center of mass of the body is assumed to be at rest. The
gravitational field is given in the post-Newtonian scheme and in terms of the
full set of mass-multipoles and spin-multipoles of the body. The asymptotic
configuration is considered, where source and observer are located at spatial
infinity from the massive body. It is found that in this asymptotic limit the
higher multipole terms of time delay are related to the higher multipole terms
of total light deflection. Furthermore, it is shown that the gauge terms
vanish in this asymptotic configuration. In case of an axisymmetric body in
uniform rotational motion, the higher multipole terms of time delay can be
expressed in terms of Chebyshev polynomials. This fact allows one to determine
the upper limits of the time delay for higher multipoles. These upper limits
represent a criterion to identify those multipoles which contribute
significantly to the time delay for a given accuracy of time measurements. It
is found that the first mass-multipoles with $l\leq 8$ and the first spin-
multipoles with $l\leq 3$ are sufficient for an accuracy on the femto-second
scale of accuracy in time measurements.

###### pacs:

95.10.Jk, 95.10.Ce, 95.30.Sf, 04.25.Nx, 04.80.Cc

##  I Introduction

In the classical Shapiro time delay Shapiro1  one considers the propagation of
an electromagnetic signal, for instance a radar signal or a signal of visible
light, in the gravitational field of a spherically symmetric body. Assume the
space-time is covered by harmonic four-coordinates
$x^{\mu}=\left(x^{0},x^{1},x^{2},x^{3}\right)$, with time component $x^{0}$
and spatial components $x^{1},x^{2},x^{3}$ and the origin of the spatial
coordinates is located at the center of mass of the body. Then, the light
travel time of a signal, emitted at
$\left(t_{0},\mbox{\boldmath$x$}_{0}\right)$ and received at
$\left(t_{1},\mbox{\boldmath$x$}_{1}\right)$, is in the post-Newtonian (PN)
scheme given by MTW ; Book_Clifford_Will

$\displaystyle\left(t_{1}-t_{0}\right)$ $\displaystyle=$
$\displaystyle\frac{R}{c}+\frac{2GM}{c^{3}}\ln\frac{x_{1}+\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{1}}{x_{0}+\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{0}}+{\cal
O}\left(c^{-3}\right)\;,$ (1)

where $M$ is the mass of the body and $\sigma$ is the unit tangent vector
along the light ray at minus infinity. The difference between the light travel
time, $\left(t_{1}-t_{0}\right)$, and Euclidean distance,
$R=\left|\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{0}\right|$, divided by
the speed of light, is the Shapiro time delay and belongs to the four
classical tests of general relativity: perihelion precession of Mercury, light
deflection at the Sun, gravitational redshift of light, and light-travel time
delay. The effect of time delay (1) has been detected in $1968$ Shapiro2  and
$1971$ Shapiro3 , which yields, for the round-trip Earth-Sun-Venus and Venus-
Sun-Earth path, up to $251$ micro-seconds for radar signals grazing the Sun.
The most accurate experiments of time delay measurements have been performed
in $2003$ by using the Saturn orbiter Cassini as reflector, which amounts, for
the round-trip Earth-Sun-Saturn and Saturn-Sun-Earth, up to $288$ micro-
seconds, where a precision on the nano-second scale of accuracy has been
achieved Shapiro6 .

Technological developments in time measurements by means of atomic clocks both
on the ground as well as in space have made giant progress during recent
decades. Some remarks about todays atomic clock standards have been given in
the introductory section in Zschocke_Total_Light_Deflection_15PN . In
particular, the accuracy of the standard deviation of up-to-date optical
atomic clocks is $\Delta t/t=10^{-19}$ Atomic_Clock1 . Such an accuracy
corresponds to a precision of $0.001$ pico-seconds for a light signal which
travels, for instance, from a giant planet of the Solar System towards an
observer located nearby the Earth. In fact, there are several mission
proposals of the European Space Agency (ESA) Astrod1 ; Lator1 ; Odyssey ;
Sagas ; TIPO ; EGE , aiming at time-delay measurements at the pico-second and
sub-pico-second level of accuracy. Furthermore, the accuracy of time
measurements improves by an order of magnitude every seven years Allan_Ashby ,
and in near future the frequency standards will arrive a level of $\Delta
t/t=10^{-20}-10^{-21}$, which corresponds to a precision of $0.01$ femto-
seconds for a light signal from a giant planet of the Solar System towards an
observer located nearby the Earth.

In view of such rapid developments in time measurements, it becomes apparent
that the mass-monopole approximation for Solar System bodies is not sufficient
and a more realistic description of gravitational fields of Solar System
bodies is necessary. The determination of the gravitational fields of Solar
System bodies is achieved by decomposing the metric in terms of mass-
multipoles $M_{L}$ (describe shape and inner structure of the massive body)
and spin-multipoles $S_{L}$ (describe rotational motions and inner currents of
the massive body) of these bodies. Then, the analytical formula for the time-
delay becomes a complicated function of these multipoles Kopeikin1997 ,

$\displaystyle\left(t_{1}-t_{0}\right)$ $\displaystyle=$
$\displaystyle\frac{R}{c}+\sum\limits_{l=0}^{\infty}\Delta\tau_{\rm
1PN}^{M_{L}}\left(t_{1},t_{0}\right)+\sum\limits_{l=1}^{\infty}\Delta\tau_{\rm
1.5PN}^{S_{L}}\left(t_{1},t_{0}\right)$ (2) $\displaystyle+{\cal
O}\left(c^{-4}\right),$

where the first term in the sum $\left(l=0\right)$ is the mass-monopole term
given by (1). In order to decide which multipoles are relevant for a given
accuracy in time measurements, one has to determine the upper limits of the
individual terms of the time delay formula in (2), that means the maximal
absolute value of these terms.

In a recent investigation it has been demonstrated that the effect of total
light deflection is related to Chebyshev polynomials
Zschocke_Total_Light_Deflection_15PN . This fact has allowed for determining
the upper limits of the total light deflection. As in case of total light
deflection it is, therefore, the aim of this investigation to obtain an
analytical formula for the multipole terms in (2) for the asymptotic
configurations, where the source as well as the observer are located at
spatial infinity from the massive body. It it found that in the asymptotic
limit the multipole terms in (2) are related to Chebyshev polynomials, a fact
that allows one to quantify the upper limits of absolute values of these terms
to arbitrary multipole order. Using these criterions, one may easily decide,
which terms in (2) are really relevant for a given goal accuracy in time delay
measurements.

The manuscript is organized as follows: In Section II the metric and the
geodesic equation are considered. The Shapiro time delay is considered in
Section III. In Section IV it is shown that in the asymptotic case, where
source and observer are located infinitely far from the massive body, the
total effect of time delay is related to the total light deflection. These
results are applied in Section V for the case of an axisymmetric body. A
comparison with the literature and some numerical results of the time delay in
the gravitational fields of Solar System bodies are given in Sections VI and
VII. A summary is given in Section VIII. The used notations are explained in
Appendix A, while Appendices B and C contain some further details of the
calculations.

##  II Metric tensor and geodesic equation

###  II.1 Metric tensor

The curved space-time is described by a pair $\left({\cal
M},g_{\mu\nu}\right)$, where ${\cal M}$ is a four-dimensional differentiable
manifold and $g_{\mu\nu}$ is the metric tensor of the manifold, and each point
${\cal P}\in{\cal M}$ represents a possible space-time event; we assume for
the metric signature $\left(-,+,+,+\right)$. These ten components of the
metric tensor $g_{\mu\nu}$ are determined by the ten field equations of
gravity, which are valid in any coordinate system. The Bianchi identities
reduce these field equations to only six independent equations. Therefore,
four gauge conditions are imposed to fix the coordinates, which cover the
physical manifold ${\cal M}$. We adopt harmonic four-coordinates,
$z^{\mu}=\left(z^{0},z^{1},z^{2},z^{3}\right)$, which are imposed by four
harmonic gauge conditions Kopeikin_Efroimsky_Kaplan

$\displaystyle\square_{g}\,z^{\mu}=0\;,$ (3)

where
$\square_{g}=(-g)^{-1/2}\,\partial_{\mu}\,(-g)^{1/2}\,g^{\mu\nu}\,\partial_{\nu}$
is the general-covariant d’Alembert operator. If the gravitational fields are
weak, then it is useful to decompose the metric tensor of the physical space-
time MTW ; Book_Clifford_Will ; Kopeikin_Efroimsky_Kaplan ; Carroll ,

$\displaystyle g_{\mu\nu}\left(t,\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\eta_{\mu\nu}+h_{\mu\nu}\left(t,\mbox{\boldmath$x$}\right),$ (4)

where $\eta_{\mu\nu}=\left(-1,+1,+1,+1\right)$ are the components of
Minkowskian metric, while $h_{\mu\nu}$ are the metric perturbations which are
small corrections to the Minkowskian metric: $\left|h_{\mu\nu}\right|\ll 1$.
The decomposition (4) implies that the metric perturbations can be thought of
as symmetric tensorial fields which propagate in the flat background space-
time MTW ; Kopeikin_Efroimsky_Kaplan ; Carroll . The flat space-time is
described by a pair $\left({\cal M}_{0},\eta_{\mu\nu}\right)$, where ${\cal
M}_{0}$ is the flat background manifold. The curved physical manifold ${\cal
M}$ and the flat background manifold ${\cal M}_{0}$ are diffeomorphic to each
other, which implicates a one-to-one correspondence between the points ${\cal
P}\in{\cal M}$ and the points ${\cal Q}\in{\cal M}_{0}$.

The flat background manifold ${\cal M}_{0}$ is assumed to be covered by
harmonic four-coordinates $x^{\mu}=\left(x^{0},x^{1},x^{2},x^{3}\right)$
which, according to (3), are imposed by the harmonic gauge condition,

$\displaystyle\square\,x^{\mu}=0\;,$ (5)

where $\square=\eta^{\mu\nu}\,\partial_{\mu}\partial_{\nu}$ is the Lorentz-
covariant d’Alembert operator. Then, by inserting the decomposition (4) into
the exact field equations of gravity and keeping only terms linear in the
metric perturbations, one obtains the linearized field equations of gravity,
which in harmonic gauge read

$\displaystyle\square\,\overline{h}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{16\,\pi\,G}{c^{4}}\,T^{\mu\nu}\;,$ (6)

where $T^{\mu\nu}$ is the stress-energy tensor of the source of matter. The
following relations allow one to deduce the metric perturbations $h_{\mu\nu}$
from the metric density perturbations $\overline{h}_{\mu\nu}$ as soon as the
solution of the linearized field equations (6) is found,

$\displaystyle\overline{h}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
h_{\mu\nu}-\frac{1}{2}\;h\;\eta_{\mu\nu}\;,$ (7) $\displaystyle h_{\mu\nu}$
$\displaystyle=$
$\displaystyle\overline{h}_{\mu\nu}-\frac{1}{2}\;\overline{h}\;\eta_{\mu\nu}\;,$
(8)

with $h=h^{\alpha\beta}\,\eta_{\alpha\beta}$ and
$\overline{h}=\overline{h}^{\alpha\beta}\,\eta_{\alpha\beta}$. These relations
are valid at linear order in $h$. Furthermore, in linearized gravity the
indices of tensors are lowered and raised by the Minkowskian metric.

By imposing the Fock-Sommerfeld boundary condition Blanchet_Kopeikin_Schafer ;
KlionerKopeikin1992 , which implies the no-incoming radiation condition as
well as the asymptotic flatness of space-time (isolated source of matter), the
solution of the differential equation (6) is uniquely given by

$\displaystyle\overline{h}^{\mu\nu}=\frac{4\,\pi\,G}{c^{4}}\,\int_{V}d^{3}x^{\prime}\,\frac{T^{\mu\nu}\left(t^{\prime},\mbox{\boldmath$x$}^{\prime}\right)}{\left|\mbox{\boldmath$x$}-\mbox{\boldmath$x$}^{\prime}\right|}\;,$
(9)

where the integral runs over the volume of the compact source of matter (body)
and
$t^{\prime}=t-c^{-1}\,\left|\mbox{\boldmath$x$}-\mbox{\boldmath$x$}^{\prime}\right|$.

Without limiting generality, one may assume the harmonic four-coordinates
$x^{\mu}=\left(x^{0},x^{1},x^{2},x^{3}\right)$, which cover the flat
background manifold ${\cal M}_{0}$, to be Cartesian. However, from the
harmonic gauge condition (5) one concludes, that the harmonic gauge does not
fix these harmonic four-coordinates $x^{\mu}$ uniquely, but allows for smooth
deformations MTW ; Kopeikin_Efroimsky_Kaplan

$x^{\alpha}_{{\rm can}}=x^{\alpha}+w^{\alpha}(x^{\beta}),$ (10)

if these gauge vector fields $w^{\alpha}$ satisfy $\square\,w^{\alpha}=0$. The
label of these new coordinates $\\{x_{\rm can}\\}$ abbreviates the term
canonical. It is emphasized that the gauge transformations in (10) are tending
to $0$ at spatial infinity and they are small, $|w^{\alpha}|\ll|x^{\alpha}|$,
and in the sense that the derivatives of the gauge functions $w^{\alpha}$ with
respect to space and time are of the same order as the metric perturbations,
$w^{\alpha}_{\,,\,\mu}={\cal O}\left(h^{\alpha}_{\mu}\right)$, hence
$|w^{\alpha}_{\,,\,\mu}|\ll 1$. The residual gauge transformation (10) implies
a residual gauge transformation of the metric tensor,

$g_{\alpha\beta}\left(t,\mbox{\boldmath$x$}\right)=\frac{\partial x^{\mu}_{\rm
can}}{\partial x^{\alpha}}\,\frac{\partial x^{\nu}_{\rm can}}{\partial
x^{\beta}}\,g_{\mu\nu}^{\rm can}\left(t_{\rm can},\mbox{\boldmath$x$}_{\rm
can}\right).$ (11)

By inserting (10) into (11) and performing a series expansion of the metric
tensor on the right-hand side around the old (Cartesian) coordinates
$\\{x\\}$, one obtains ($\partial_{\alpha}\,f\equiv
f_{\,,\,\alpha}\equiv\partial f/\partial x^{\alpha}$):

$g_{\alpha\beta}\left(t,\mbox{\boldmath$x$}\right)=g_{\alpha\beta}^{\rm
can}\left(t,\mbox{\boldmath$x$}\right)+\partial_{\alpha}w_{\beta}\left(t,\mbox{\boldmath$x$}\right)+\partial_{\beta}w_{\alpha}\left(t,\mbox{\boldmath$x$}\right).$
(12)

The unique solution in (9) can be expressed in terms of six Cartesian
symmetric and trace-free (STF) multipoles $\hat{F}_{L}$ (Eq. (8.4) in Thorne
or Eqs. (5.3a) - (5.3c) in Multipole_Damour_2 ),

$\overline{h}^{\mu\nu}\left(t,\mbox{\boldmath$x$}\right)=\frac{4\,G}{c^{4}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\hat{\partial}_{L}\left[\frac{\hat{F}_{L}^{\mu\nu}\left(u\right)}{r}\right]\,,$
(13)

where $u=t-c^{-1}\,r$ is the retarded time,
$r=\left|\mbox{\boldmath$x$}\right|$, the STF multipoles $\hat{F}_{L}$ are
given by Eqs. (5.4a) - (5.4c) in Multipole_Damour_2 , and

$\hat{\partial}_{L}={\rm STF}_{i_{1}\dots i_{l}}\,\frac{\partial}{\partial
x^{i_{1}}}\dots\frac{\partial}{\partial x^{i_{l}}}\,,$ (14)

where the hat in $\hat{F}_{L}$ and in $\hat{\partial}_{L}$ indicates STF
operation with respect to the spatial components of the multi-index
$L=i_{1}\dots i_{l}$. A detailed proof of this theorem in (13) has been
presented in Zschocke_Theorem .

The solution in (13) is unique and represents the most general solution of the
linearized gravity for an isolated compact source of matter. By extensive use
of STF Cartesian tensor techniques, it has been demonstrated in Thorne ;
Blanchet_Damour1 ; Multipole_Damour_2  that the general solution in (13) can
be written in terms of six STF multipoles
$\\{\hat{M}_{L},\hat{S}_{L},\hat{W}_{L},\hat{X}_{L},\hat{Y}_{L},\hat{Z}_{L}\\}$
Thorne ; Blanchet_Damour1 ; Multipole_Damour_2 . Accordingly, the metric
tensor can be written in the following form,

$\displaystyle h_{\alpha\beta}\left(t,\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle h^{{\rm
can}}_{\alpha\beta}\left[\hat{M}_{L},\hat{S}_{L}\right]+\partial_{\alpha}w_{\beta}\left[\hat{W}_{L},\hat{X}_{L},\hat{Y}_{L},\hat{Z}_{L}\right]$
(15)
$\displaystyle+\partial_{\beta}w_{\alpha}\left[\hat{W}_{L},\hat{X}_{L},\hat{Y}_{L},\hat{Z}_{L}\right].$

The canonical part of the metric perturbations in (15) depends on two
multipoles only, namely mass-multipoles and spin-multipoles
$\\{\hat{M}_{L},\hat{S}_{L}\\}$, while the gauge terms of the metric
perturbation in (15) depends on four multipoles,
$\\{\hat{W}_{L},\hat{X}_{L},\hat{Y}_{L},\hat{Z}_{L}\\}$. The form of the
metric perturbations in (15) depends on the chosen coordinate system. For
instance, in canonical coordinates there are no gauge terms at all and only
the canonical term $h^{{\rm can}}_{\alpha\beta}$ would remain. Here, we will
use this most general form in (15) in order to demonstrate that these gauge
terms vanish at spatial infinity.

We will consider the metric of bodies with time-independent multipoles and
where the center of mass of the body is assumed to be at rest with respect to
the harmonic coordinate system. Then, the canonical metric perturbations in
(15) are separated into two pieces, $h^{\rm
can}_{\alpha\beta}=h_{\alpha\beta}^{\left(2\right)\,{\rm
can}}+h_{\alpha\beta}^{\left(3\right)\,{\rm can}}$,

$\displaystyle h_{00}^{\left(2\right)\,{\rm can}}$ $\displaystyle=$
$\displaystyle-\frac{2G}{c^{2}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\hat{\partial}_{L}\frac{\hat{M}_{L}}{r}\;,$
(16) $\displaystyle h_{0i}^{\left(3\right)\,{\rm can}}$ $\displaystyle=$
$\displaystyle+\frac{4G}{c^{3}}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}\,l}{\left(l+1\right)!}\,\epsilon_{iab}\,\hat{\partial}_{aL-1}\frac{\hat{S}_{bL-1}}{r}\;,$
(17)

while $h_{ij}^{\left(2\right)\,{\rm can}}=h_{00}^{\left(2\right)\,{\rm
can}}\,\delta_{ij}$. The derivatives $\hat{\partial}_{L}$ are not acting on
the multipoles, because they are independent of space and time here. These
mass-multipoles and spin-multipoles in (16) and (17) in the stationary case,
that means in case of a time-independent source of matter, are given by

$\displaystyle\hat{M}_{L}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,\hat{x}_{L}\,\Sigma\;,$ (18) $\displaystyle\hat{S}_{L}$
$\displaystyle=$ $\displaystyle\int
d^{3}x\,\epsilon_{jk<i_{l}}\,\hat{x}_{L-1>}\,x^{j}\,\Sigma^{k}\;,$ (19)

where $\Sigma=\left(T^{00}+T^{kk}\right)/c^{2}$ and $\Sigma^{k}=T^{0k}/c$ with
the stress-energy tensor $T^{\mu\nu}$ of the source of matter (cf. Eq. (6)),
and the integration runs over the volume of the body. The time-independence of
the multipoles implies time-independence of the metric.

The gauge functions in (15) have been determined in Thorne ; Blanchet_Damour1
; Multipole_Damour_2 ; Multipole_Blanchet_1  and read:

$\displaystyle w^{0}$ $\displaystyle=$
$\displaystyle+\frac{4G}{c^{3}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\hat{\partial}_{L}\frac{\hat{W}_{L}}{r}\;,$
(20) $\displaystyle w^{i}$ $\displaystyle=$
$\displaystyle-\frac{4G}{c^{2}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\hat{\partial}_{iL}\frac{\hat{X}_{L}}{r}-\frac{4G}{c^{2}}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\hat{\partial}_{L-1}\frac{\hat{Y}_{iL-1}}{r}$
(21)
$\displaystyle-\frac{4G}{c^{2}}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\frac{l}{l+1}\,\epsilon_{iab}\,\hat{\partial}_{aL-1}\frac{\hat{Z}_{bL-1}}{r}\,.$

As mentioned above, these gauge vectors in (20) - (21) represent the most
general form of the gauge terms in the metric tensor. The multipoles
$\hat{W}_{L},\hat{X}_{L},\hat{Y}_{L},\hat{Z}_{L}$ of the gauge functions (20)
and (21) have been determined in Multipole_Damour_2 ; Multipole_Blanchet_1
and read in the stationary case as follows,

$\displaystyle\hat{W}_{L}$ $\displaystyle=$
$\displaystyle+\frac{2l+1}{\left(l+1\right)\left(2l+3\right)}\int
d^{3}x\,\hat{x}_{Lj}\,\Sigma^{j}\;,$ (22) $\displaystyle\hat{X}_{L}$
$\displaystyle=$
$\displaystyle+\frac{2l+1}{\left(2l+2\right)\left(l+2\right)\left(2l+5\right)}\int
d^{3}x\,\hat{x}_{jkL}\,\Sigma^{jk}\;,$ (23) $\displaystyle\hat{Y}_{L}$
$\displaystyle=$ $\displaystyle-\int d^{3}x\,\hat{x}_{L}\,\Sigma^{kk}\;,$ (24)
$\displaystyle\hat{Z}_{L}$ $\displaystyle=$
$\displaystyle-\frac{2l+1}{\left(l+2\right)\left(2l+3\right)}\\!\int
d^{3}x\,\epsilon_{jk<i_{l}}\hat{x}_{L-1>km}\Sigma^{jm},$ (25)

where $\Sigma^{jk}=T^{jk}/c^{2}$, with $T^{jk}$ being the spatial components
of the stress-energy tensor of the body, and the integration runs over the
volume of the body.

We will show that the gauge terms in (15) have no impact on the Shapiro time
delay in case of infinite distance of source and observer from the massive
body (see Appendix B). This specific case reflects the general fact that
$g_{\alpha\beta}$ and $g_{\alpha\beta}^{\rm can}$ in (12) are physically
equivalent, that means they lead to same observables. This statement is valid
in case the gauge transformations (10) are small and are tending to $0$ at
spatial infinity, as emphasized in the text below Eq. (10).

###  II.2 Geodesic equation

In flat Minkowskian space-time, assumed to be covered by Cartesian four-
coordinates, a light signal, emitted by a source at
$\left(t_{0},\mbox{\boldmath$x$}_{0}\right)$ in some direction specified by a
unit vector $\sigma$, propagates along a straight line, given by

$\displaystyle\mbox{\boldmath$x$}_{\rm N}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{0}+c\left(t-t_{0}\right)\mbox{\boldmath$\sigma$}\;,$
(26)

and its absolute value is

$\displaystyle r_{\rm
N}=\sqrt{\left(x_{0}\right)^{2}+2\,\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{0}\;c\left(t-t_{0}\right)+c^{2}\left(t-t_{0}\right)^{2}}\,,$
(27)

where the index N in (26) and (27) stands for Newtonian. In what follows we
also need the so-called impact vector of the unperturbed light ray,

$\displaystyle\mbox{\boldmath$d$}_{\sigma}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\times\left(\mbox{\boldmath$x$}_{\rm
N}\times\mbox{\boldmath$\sigma$}\right),$ (28)

with its absolute value, the impact parameter of the unperturbed light ray,

$\displaystyle
d_{\sigma}=\left|\mbox{\boldmath$\sigma$}\times\mbox{\boldmath$x$}_{\rm
N}\right|.$ (29)

The impact vector (28) points from the center of mass of the body towards the
unperturbed light ray at its closest encounter; for a graphical elucidation
see Fig. 1.

In curved space-time, $\left({\cal M},g_{\mu\nu}\right)$, the trajectory of a
light signal is determined by the geodesic equation, which in terms of
coordinate time reads as follows Brumberg1991 ; Kopeikin_Efroimsky_Kaplan ;
MTW

$\displaystyle\frac{\ddot{x}^{i}\left(t\right)}{c^{2}}+\Gamma^{i}_{\mu\nu}\frac{\dot{x}^{\mu}\left(t\right)}{c}\frac{\dot{x}^{\nu}\left(t\right)}{c}-\Gamma^{0}_{\mu\nu}\frac{\dot{x}^{\mu}\left(t\right)}{c}\frac{\dot{x}^{\nu}\left(t\right)}{c}\frac{\dot{x}^{i}\left(t\right)}{c}=0\;,$
(30)

where a dot denotes total derivative with respect to coordinate time, and
$\Gamma^{\alpha}_{\mu\nu}$ are the Christoffel symbols, given by
Kopeikin_Efroimsky_Kaplan ; Brumberg1991 ; MTW

$\displaystyle\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}\,g^{\alpha\beta}\left(\frac{\partial
g_{\beta\mu}}{\partial x^{\nu}}+\frac{\partial g_{\beta\nu}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\beta}}\right).$ (31)

The geodesic equation is a differential equation of second order of one
variable, $t$, thus a unique solution of (30) necessitates two initial-
boundary conditions: the spatial position of light source
$\mbox{\boldmath$x$}_{0}$ and the unit-direction $\sigma$ of the light signal
at minus infinity Brumberg1991 ; Kopeikin1997 ;
KopeikinSchaefer1999_Gwinn_Eubanks ; KlionerKopeikin1992 ; Zschocke_1PN ;
Zschocke_15PN :

$\sigma$ $\displaystyle=$
$\displaystyle\frac{\dot{\mbox{\boldmath$x$}}\left(t\right)}{c}\,\bigg{|}_{t=-\infty}\quad{\rm
with}\quad\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$\sigma$}=1\;,$ (32)
$\displaystyle\mbox{\boldmath$x$}_{0}\;$ $\displaystyle=$
$\displaystyle\;\mbox{\boldmath$x$}\left(t\right)\bigg{|}_{t=t_{0}}\;.$ (33)

By using the initial-boundary conditions (32) and by inserting the
decomposition (4) into (30), which implies weak gravitational fields, the
solution of the second integration of geodesic equation (30) is given by

$\displaystyle\mbox{\boldmath$x$}\left(t\right)$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{0}+c\left(t-t_{0}\right)\mbox{\boldmath$\sigma$}+\Delta\mbox{\boldmath$x$}\left(t,t_{0}\right),$
(34)

where $\Delta\mbox{\boldmath$x$}$ are small corrections to the trajectory of
the unperturbed light ray (26). Eq. (34) represents the trajectory of a light
signal propagating in the flat background space-time $\left({\cal
M}_{0},\eta_{\mu\nu}\right)$; for a graphical elucidation see Fig. 1.

Figure 1: A geometrical representation of the propagation of a light signal
through the gravitational field of a massive Solar System body at rest. The
axes of inertia are denoted by
$\mbox{\boldmath$e$}_{1},\mbox{\boldmath$e$}_{2},\mbox{\boldmath$e$}_{3}$. In
stationary case, the body is of arbitrary but time-independent shape and inner
structure and the vector of angular velocity $\Omega$ and its absolute value
$\Omega$ are time-independent. The light signal is emitted by the light source
at $\mbox{\boldmath$x$}_{0}$ and propagates along the exact light trajectory
$\mbox{\boldmath$x$}\left(t\right)$. The unit tangent vectors along the light
trajectory at minus and plus infinity are $\sigma$ and $\nu$. The unperturbed
light ray $\mbox{\boldmath$x$}_{\rm N}\left(t\right)$ is given by Eq. (26) and
propagates in the direction of $\sigma$ along a straight line through the
position of the light source at $\mbox{\boldmath$x$}_{0}$. The impact vector
$\mbox{\boldmath$d$}_{\sigma}$ of the unperturbed light ray is given by Eq.
(28).

The solution of the initial value problem (34) implies the following limit,

$\displaystyle\lim_{t\rightarrow
t_{0}}\Delta\mbox{\boldmath$x$}\left(t,t_{0}\right)$ $\displaystyle=$
$\displaystyle 0\;,$ (35)

in order to be consistent with the condition (33).

The geodesic equation in 1.5PN approximation can be deduced from the exact
geodesic equation (30) and is given by Brumberg1991

$\displaystyle\frac{\ddot{x}^{i}\left(t\right)}{c^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,h_{00,i}-h_{00,j}\,\sigma^{i}\sigma^{j}-h_{ij,k}\,\sigma^{j}\sigma^{k}+\frac{1}{2}\,h_{jk,i}\,\sigma^{j}\sigma^{k}$
(36) $\displaystyle-
h_{0i,j}\,\sigma^{j}+h_{0j,i}\,\sigma^{j}-h_{0j,k}\,\sigma^{i}\,\sigma^{j}\sigma^{k}+{\cal
O}\left(c^{-4}\right)\;,$

where we have omitted all those terms which contain a derivative of the metric
perturbations with respect to time, because we consider the stationary case,
that is the case of time-independent metric. Note, that in stationary case the
geodesic equation in 1.5PN approximation of the post-Newtonian (PN) scheme and
the geodesic equation in 1PM approximation of the post-Minkowskian (PM) scheme
agree with each other Klioner_Peip . By inserting the metric perturbation (15)
into the geodesic equation (36), one may separate the geodesic equation into a
canonical term, $\ddot{\mbox{\boldmath$x$}}_{\rm can}$, plus a gauge term,
$\ddot{\mbox{\boldmath$x$}}_{\rm gauge}$, as follows:

$\frac{\ddot{\mbox{\boldmath$x$}}\left(t\right)}{c^{2}}=\frac{\ddot{\mbox{\boldmath$x$}}_{\rm
can}\left(t\right)}{c^{2}}+\frac{\ddot{\mbox{\boldmath$x$}}_{\rm
gauge}\left(t\right)}{c^{2}}+{\cal O}\left(c^{-4}\right),$ (37)

where the spatial components of these terms are

$\displaystyle\frac{\ddot{x}_{\rm can}^{i}\left(t\right)}{c^{2}}$
$\displaystyle=$ $\displaystyle h^{\left(2\right)\,{\rm
can}}_{00,i}-2\,h^{\left(2\right)\,{\rm
can}}_{00,j}\,\sigma^{i}\sigma^{j}-h^{\left(3\right)\,{\rm
can}}_{0i,j}\,\sigma^{j}$ (38) $\displaystyle+h^{\left(3\right)\,{\rm
can}}_{0j,i}\,\sigma^{j}-h^{\left(3\right)\,{\rm
can}}_{0j,k}\,\sigma^{i}\,\sigma^{j}\sigma^{k}\;,$
$\displaystyle\frac{\ddot{x}_{\rm gauge}^{i}\left(t\right)}{c^{2}}$
$\displaystyle=$
$\displaystyle\partial_{j}\,w^{0}_{\,,\,k}\,\sigma^{i}\sigma^{j}\sigma^{k}-\partial_{j}\,w^{i}_{\,,\,k}\,\sigma^{j}\sigma^{k}\;.$
(39)

The metric perturbations in (38) are given by (16) and (17), while the gauge
functions in (39) are given by (20) and (21). The metric tensor with the
metric perturbations (15) is valid in the entire space-time, while in the
geodesic equation (37) with (38) and (39), the arguments of the metric
perturbations are taken along the light trajectory, that means they have to be
replaced by $\mbox{\boldmath$x$}=\mbox{\boldmath$x$}_{\rm N}+{\cal O}(c^{-2})$
and $r=|\mbox{\boldmath$x$}_{\rm N}|+{\cal O}(c^{-2})$.

The integration of (37) yields the light trajectory in 1.5PN approximation,
formally given by

$\mbox{\boldmath$x$}\left(t\right)=\mbox{\boldmath$x$}_{0}+c\left(t-t_{0}\right)\mbox{\boldmath$\sigma$}+\Delta\mbox{\boldmath$x$}_{\rm
can}\left(t,t_{0}\right)+\Delta\mbox{\boldmath$x$}_{\rm
gauge}\left(t,t_{0}\right).$ (40)

In particular, one has to insert the multipole decomposition of the metric
perturbations (15) into the geodesic equation (36) and then one has to apply
advanced integration methods developed in Kopeikin1997 , which yields

$\displaystyle\mbox{\boldmath$x$}\left(t\right)$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{0}+c\left(t-t_{0}\right)\mbox{\boldmath$\sigma$}$
(41) $\displaystyle+\sum\limits_{l=0}^{\infty}\Delta\mbox{\boldmath$x$}_{\rm
1PN}^{M_{L}}\left(t,t_{0}\right)+\sum\limits_{l=1}^{\infty}\Delta\mbox{\boldmath$x$}_{\rm
1.5PN}^{S_{L}}\left(t,t_{0}\right)$
$\displaystyle+\,\Delta\mbox{\boldmath$x$}_{\rm gauge}\left(t,t_{0}\right).$

The first line describes the straight trajectory of the unperturbed light ray,
the second line represents the perturbations to the unperturbed light ray
caused by the canonical terms of the metric tensor, and the third line are the
perturbations to the unperturbed light ray caused by the gauge terms of the
metric tensor. The canonical terms $\Delta\mbox{\boldmath$x$}_{\rm
1PN}^{M_{L}}$ and $\Delta\mbox{\boldmath$x$}_{\rm 1.5PN}^{S_{L}}$ have been
obtained at the very first time by advanced integration methods in
Kopeikin1997 , while the gauge term $\Delta\mbox{\boldmath$x$}_{\rm gauge}$
has been calculated in Appendix B by the same approach developed in
Kopeikin1997 .

##  III Time delay in field of an arbitrary body

In Kopeikin1997  advanced integration methods have been introduced that one
allow to integrate (36) exactly. The basic ideas of the conception has
originally been introduced in Kopeikin1997  for bodies with time-independent
multipoles and where the center of mass of the body is assumed to be at rest
with respect to the harmonic coordinate system. This approach has further been
developed for the case of light propagation in the gravitational field of a
time-dependent source of matter, where the center of mass of the body was also
assumed to be at rest with respect to the harmonic coordinate system
KopeikinSchaefer1999_Gwinn_Eubanks ; KopeikinKorobkovPolnarev2006 ;
KopeikinKorobkov2005 . Later, these mathematical tools developed in
Kopeikin1997 ; KopeikinSchaefer1999_Gwinn_Eubanks ;
KopeikinKorobkovPolnarev2006 ; KopeikinKorobkov2005  have been applied to the
case of light propagation in the gravitational fields of $N$ slowly moving
bodies with time-dependent multipoles Zschocke_1PN ; Zschocke_15PN .

In the approach in Kopeikin1997  two new parameters were introduced,

$\displaystyle c\tau$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm N}\;,$
(42) $\displaystyle\xi^{i}$ $\displaystyle=$ $\displaystyle P^{i}_{j}\,x_{\rm
N}^{j}\;,$ (43)

where $P^{ij}=\delta^{ij}-\sigma^{i}\sigma^{j}$ is a projection operator onto
the plane perpendicular to vector $\sigma$. The unperturbed light ray (26)
expressed in terms of these new variables takes the form

$\displaystyle\mbox{\boldmath$x$}_{\rm N}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\xi$}+c\tau\,\mbox{\boldmath$\sigma$}\;,$ (44)

with its absolute value

$\displaystyle r_{\rm N}=\sqrt{\xi^{2}+c^{2}\tau^{2}}\;.$ (45)

In favor of a simpler notation, we will not introduce new notations for the
unperturbed light ray (26) given in terms of the standard variables
$\left(ct,\mbox{\boldmath$x$}\right)$ and for the unperturbed light ray (44)
given in terms of the auxiliary variables
$\left(c\tau,\mbox{\boldmath$\xi$}\right)$. Similarly, the same notation will
be used for their absolute values in (27) and (45).

The three-vector $\xi$ in (43) actually coincides with the impact vector
defined by (28). The use of two different notations for the same vector is
appropriate for the following reason: the three-vector $\xi$ is laying in the
two-dimensional plane perpendicular to $\sigma$, hence only two components are
independent, which implies $\partial\xi^{i}/\partial\xi^{j}=P^{i}_{j}$.
However, in practical calculations it is convenient to treat the spatial
components of this vector as formally independent, which implies
$\partial\xi^{i}/\partial\xi^{j}=\delta^{i}_{j}$. Therefore, a subsequent
projection onto this two-dimensional plane by means of $P^{ij}$ is necessary
KopeikinSchaefer1999_Gwinn_Eubanks ; Book_Soffel_Han . That is why two
different notations for $\xi$ and $\mbox{\boldmath$d$}_{\sigma}$ are in use
here. Then, for a spatial derivative expressed in terms of these new
variables, one obtains

$\displaystyle\frac{\partial}{\partial x^{i}}$ $\displaystyle=$ $\displaystyle
P_{i}^{j}\,\frac{\partial}{\partial\xi^{j}}+\sigma_{i}\,\frac{\partial}{\partial
c\tau}\;.$ (46)

Using (46) and the binomial theorem, one finds the differential operator in
(14) expressed in terms of these new variables,

$\displaystyle\widehat{\partial}_{L}$ $\displaystyle=$ $\displaystyle{\rm
STF}_{i_{1}\dots
i_{l}}\;\sum\limits_{p=0}^{l}\frac{l!}{\left(l-p\right)!\;p!}\;\sigma_{i_{1}}\,...\,\sigma_{i_{p}}\,P_{i_{p+1}}^{j_{p+1}}\;...\;P_{i_{l}}^{j_{l}}$
(47)
$\displaystyle\times\frac{\partial}{\partial\xi^{j_{p+1}}}\;...\;\frac{\partial}{\partial\xi^{j_{l}}}\;\left(\frac{\partial}{\partial
c\tau}\right)^{p}\,,$

where the symbol widehat indicates STF operation with respect to the spatial
indices $i_{1}\dots i_{l}$. Here we prefer to use the operator as given by Eq.
(47) where $\partial\xi^{i}/\partial\xi^{j}=\delta^{i}_{j}$, while if one
applies the operator as given by Eq. (24) in Kopeikin1997  then
$\partial\xi^{i}/\partial\xi^{j}=P^{i}_{j}$. The final results of either these
operations are identical. Then, using the basic integral (25) in Kopeikin1997
, one finds for the second integration the formula given by Eq. (27) in
Kopeikin1997 , which leads to the solution for the second integration of
geodesic equation (36).

From the solution for the light trajectory in (41), one obtains the time-of-
flight in the gravitational field of a body at rest with full mass-multipole
and spin-multipole structure, given by the following formula Kopeikin1997 ,

$\displaystyle\left(t_{1}-t_{0}\right)$ $\displaystyle=$
$\displaystyle\frac{R}{c}+\sum\limits_{l=0}^{\infty}\Delta\tau_{\rm
1PN}^{M_{L}}\\!\left(t_{1},t_{0}\right)+\sum\limits_{l=1}^{\infty}\Delta\tau_{\rm
1.5PN}^{S_{L}}\\!\left(t_{1},t_{0}\right)$ (48)
$\displaystyle+\,\Delta\tau_{\rm gauge}\left(t_{1},t_{0}\right)$

up to terms of the order ${\cal O}(c^{-4})$, where

$\displaystyle\Delta\tau_{\rm 1PN}^{M_{L}}\left(t_{1},t_{0}\right)$
$\displaystyle=$
$\displaystyle-\,\frac{1}{c}\,\mbox{\boldmath$\sigma$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}^{M_{L}}\left(t_{1},t_{0}\right),$ (49) $\displaystyle\Delta\tau_{\rm
1.5PN}^{S_{L}}\left(t_{1},t_{0}\right)$ $\displaystyle=$
$\displaystyle-\,\frac{1}{c}\,\mbox{\boldmath$\sigma$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1.5PN}^{S_{L}}\left(t_{1},t_{0}\right),$ (50) $\displaystyle\Delta\tau_{\rm
gauge}\left(t_{1},t_{0}\right)$ $\displaystyle=$
$\displaystyle-\,\frac{1}{c}\,\mbox{\boldmath$\sigma$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
gauge}\left(t_{1},t_{0}\right).$ (51)

The explicit expressions for the mass-multipole (gravitoelectric) term (49)
reads Kopeikin1997

$\displaystyle\Delta\tau_{\rm 1PN}^{M_{L}}\left(t_{1},t_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{2G}{c^{3}}\frac{\left(-1\right)^{l}}{l!}\hat{M}_{L}$
$\displaystyle\times\left(\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)\bigg{|}_{\tau=\tau_{1}}-\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)\bigg{|}_{\tau=\tau_{0}}\right),$

and the spin-multipole (gravitomagnetic) term (50) reads Kopeikin1997  (see
also footnote $3$ in Ciufolini )

$\displaystyle\Delta\tau_{\rm 1.5PN}^{S_{L}}\left(t_{1},t_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4G}{c^{4}}\frac{\left(-1\right)^{l}\,l}{\left(l+1\right)!}\,\epsilon_{abc}\,\sigma^{c}\,\hat{S}_{bL-1}$
$\displaystyle\times\\!\left(\\!\widehat{\partial}_{aL-1}\ln\left(r_{\rm
N}+c\tau\right)\bigg{|}_{\tau=\tau_{1}}-\widehat{\partial}_{aL-1}\ln\left(r_{\rm
N}+c\tau\right)\bigg{|}_{\tau=\tau_{0}}\right).$

The gauge term (51) is determined in Appendix B, where it is shown that this
term vanishes at minus and plus infinity,

$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty\atop\tau=\tau_{1}\rightarrow+\infty}\Delta\tau_{\rm
gauge}\left(t_{1},t_{0}\right)=0\;.$ (54)

That means, the gauge terms have no impact on the Shapiro time delay when
source and observer are located at spatial infinity, where the space-time is
the flat Minkowski space. Thus, relation (54) is just a specific example of
the general fact, that observables are independent of the chosen gauge. A
similar conclusion is valid for the total light deflection, that is the
bending of light in case the source and observer are located at spatial
infinity. In Zschocke_Total_Light_Deflection_Proceeding  it has been shown
that the gauge terms have no impact on the total light deflection, which is
also an observable.

Thus, Eqs. (LABEL:Shapiro_Mass_Multipole) and (LABEL:Shapiro_Spin_Multipole)
represent the effect of time delay and were also given in the textbook
Book_Soffel_Han . In (LABEL:Shapiro_Mass_Multipole) and
(LABEL:Shapiro_Spin_Multipole) the differentiations have to be performed.
Afterwards one has to substitute the unperturbed light ray (44) and its
absolute value (45) by the standard expressions as given by (26) and (27). At
the very end of these differentiations, the sub-labels in
(LABEL:Shapiro_Mass_Multipole) and (LABEL:Shapiro_Spin_Multipole) are replaced
by

$\displaystyle c\tau_{0}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\left(t_{0}\right),$ (55) $\displaystyle c\tau_{1}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\left(t_{1}\right),$ (56)

and $P^{ij}\xi_{j}$ by the spatial components of the impact vector
$d^{i}_{\sigma}$. Further details about this approach can be found in
Kopeikin1997 ; Book_Soffel_Han ; Zschocke_Time_Delay_2PN .

The time delay in 1.5PN approximation, given by Eq. (48), is valid for finite
distances of source and observer from the gravitating body. According to (40),
the spatial coordinates of source, $\mbox{\boldmath$x$}_{0}$, and observer,
$\mbox{\boldmath$x$}_{1}$, are related to the spatial coordinates of the
unperturbed light signal at time of emission, $t_{0}$ and time of observation,
$t_{1}$, as follows,

$\displaystyle\mbox{\boldmath$x$}_{0}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{\rm N}\left(t_{0}\right),$ (57)
$\displaystyle\mbox{\boldmath$x$}_{1}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{\rm N}\left(t_{1}\right)+{\cal
O}\left(c^{-2}\right).$ (58)

Therefore, after performing the differentiations, one may replace these
awkward terms $\mbox{\boldmath$x$}_{\rm N}\left(t_{0}\right)$ and
$\mbox{\boldmath$x$}_{\rm N}\left(t_{1}\right)$ in the arguments of
(LABEL:Shapiro_Mass_Multipole) and (LABEL:Shapiro_Spin_Multipole) just by the
exact positions of source and observer.

Like in case of total light deflection Kopeikin1997 ;
Zschocke_Total_Light_Deflection_15PN , we will determine the total effect of
time delay (LABEL:Shapiro_Mass_Multipole) and (LABEL:Shapiro_Spin_Multipole).
That means, we will consider astrometric configurations, where both the source
and the observer are located at spatial infinity from the gravitating body. In
particular, we will determine the time delay for asymptotic configurations of
source and observer, where the limits are

$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{0}$
$\displaystyle\rightarrow$ $\displaystyle-\infty\;,$ (59)
$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{1}$
$\displaystyle\rightarrow$ $\displaystyle+\infty\;.$ (60)

Roughly to speak, these conditions represent configurations, where the massive
body is located somewhere „between“ source and observer, as shown in Fig. 2.
In view of (55) - (60), these asymptotic limits (59) and (60) in terms of the
arguments of the time delay in (LABEL:Shapiro_Mass_Multipole) and
(LABEL:Shapiro_Spin_Multipole) reads

$\displaystyle c\tau_{0}\rightarrow-\infty\,,$ (61) $\displaystyle
c\tau_{1}\rightarrow+\infty\,.$ (62)

The time delay (LABEL:Shapiro_Mass_Multipole) and
(LABEL:Shapiro_Spin_Multipole) depends on the impact vector of the unperturbed
light ray, which is constant for a given light ray. Therefore, these limits in
(61) and (62) have to be taken along the unperturbed light trajectory with
constant impact vector, as displayed by Fig. 2. In particular, one finds the
following limits,

$\displaystyle\lim_{\tau_{0}\rightarrow-\infty}\,\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\left(t_{0}\right)}{r_{\rm N}\left(t_{0}\right)}$ $\displaystyle=$
$\displaystyle-1\,,$ (63)
$\displaystyle\lim_{\tau_{1}\rightarrow+\infty}\,\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\left(t_{1}\right)}{r_{\rm N}\left(t_{1}\right)}$ $\displaystyle=$
$\displaystyle+1\,,$ (64)

where $r_{\rm N}\left(t_{0}\right)=|\mbox{\boldmath$x$}_{\rm
N}\left(t_{0}\right)|$ and $r_{\rm
N}\left(t_{1}\right)=|\mbox{\boldmath$x$}_{\rm N}\left(t_{1}\right)|$. Thus,
the angles are
$\delta(\mbox{\boldmath$\sigma$},\mbox{\boldmath$x$}_{0})\rightarrow\pi$ and
$\delta(\mbox{\boldmath$\sigma$},\mbox{\boldmath$x$}_{\rm
N}\left(t_{1}\right))\rightarrow 0$ in these limits, a result which is
elucidated by the graphical representation in Fig. 2.

Figure 2: This diagram elucidates the limits in (61) and (62). The angles are
$\alpha_{0}=\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$x$}_{0}\right)$
and $\alpha_{1}=\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$x$}_{\rm
N}\left(t_{1}\right)\right)$. The dashed arrows to the right and left show the
directions to which the spatial positions of source,
$\mbox{\boldmath$x$}_{0}=\mbox{\boldmath$x$}_{\rm N}\left(t_{0}\right)$, and
the spatial position of the unperturbed light signal $\mbox{\boldmath$x$}_{\rm
N}\left(t_{1}\right)$ at time of observation, are shifted up to infinity. The
limits in (61) and (62) implicitly assume that the impact vector of the
unperturbed light ray remains constant: $\mbox{\boldmath$d$}_{\sigma}={\rm
const}$.

The mass-monopole term ($l=0$ in (LABEL:Shapiro_Mass_Multipole)) has already
been given by Eq. (1), which is logarithmically divergent. Therefore, we
consider the asymptotic limit only for mass-multipole terms with $l\geq 1$ and
for spin-multipole terms with $l\geq 1$. We use the following notation for
these asymptotic limits,

$\displaystyle\Delta\tau_{\rm 1PN}^{M_{L}}$ $\displaystyle=$
$\displaystyle\lim_{\tau_{0}\rightarrow-\infty\atop\tau_{1}\rightarrow+\infty}\Delta\tau_{\rm
1PN}^{M_{L}}\left(t_{1},t_{0}\right),$ (65) $\displaystyle\Delta\tau_{\rm
1.5PN}^{S_{L}}$ $\displaystyle=$
$\displaystyle\lim_{\tau_{0}\rightarrow-\infty\atop\tau_{1}\rightarrow+\infty}\Delta\tau_{\rm
1.5PN}^{S_{L}}\left(t_{1},t_{0}\right),$ (66)

where $\Delta c\tau_{\rm 1PN}^{M_{L}}\left(t_{1},t_{0}\right)$ and $\Delta
c\tau_{\rm 1.5PN}^{S_{L}}\left(t_{1},t_{0}\right)$ were given by Eqs.
(LABEL:Shapiro_Mass_Multipole) and (LABEL:Shapiro_Spin_Multipole). In Appendix
C the following results for these limits are shown

$\displaystyle\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)\bigg{|}_{\tau=\tau_{0}\rightarrow-\infty}$ $\displaystyle=$
$\displaystyle
2\,\widehat{\partial}_{L}\,\ln\left|\mbox{\boldmath$\xi$}\right|,$ (67)
$\displaystyle\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)\bigg{|}_{\tau=\tau_{1}\rightarrow+\infty}$ $\displaystyle=$
$\displaystyle 0\;,$ (68)

which are valid for $l\geq 1$. These limits in (67) and (68) can also nicely
be verified for the first few orders in $l$ just by explicit computation. Let
us notice, that on the left-hand side of these relations one has, first of
all, to perform the differentiations by using the differential operator in
(47) and afterwards one has to calculate the limits. Then, on the right-hand
side of these relations only the term with $p=0$ in the differential operator
(47) contributes, which is given by (124) in Appendix C; see also comment in
the text below Eq. (133).

By inserting (67) and (68) into (LABEL:Shapiro_Mass_Multipole) and
(LABEL:Shapiro_Spin_Multipole) one obtains for the time delay in the
asymptotic limit where source and observer are located at spatial infinity,
the following expressions:

$\displaystyle\Delta\tau_{\rm 1PN}^{M_{L}}$ $\displaystyle=$
$\displaystyle-\frac{4G}{c^{3}}\frac{\left(-1\right)^{l}}{l!}\hat{M}_{L}\;\widehat{\partial}_{L}\,\ln\left|\mbox{\boldmath$\xi$}\right|,$
(69) $\displaystyle\Delta\tau_{\rm 1.5PN}^{S_{L}}$ $\displaystyle=$
$\displaystyle-\frac{8G}{c^{4}}\frac{\left(-1\right)^{l}\,l}{\left(l+1\right)!}\,\epsilon_{abc}\,\sigma^{c}\,\hat{S}_{bL-1}\;\widehat{\partial}_{aL-1}\,\ln\left|\mbox{\boldmath$\xi$}\right|,$

which are valid for $l\geq 1$. In order to determine the time delay by means
of the expressions (69) and (LABEL:Total_Shapiro_Spin_Multipole), one has to
calculate the term

$\displaystyle\widehat{\partial}_{L}\,\ln\left|\mbox{\boldmath$\xi$}\right|$
$\displaystyle=$ $\displaystyle{\rm STF}_{i_{1}\dots
i_{l}}\;P_{i_{1}}^{j_{1}}\;...\;P_{i_{l}}^{j_{l}}\,\frac{\partial}{\partial\xi^{j_{1}}}\;...\;\frac{\partial}{\partial\xi^{j_{l}}}\;\ln\left|\mbox{\boldmath$\xi$}\right|.$

In our investigation Zschocke_Total_Light_Deflection_15PN  it has been shown
that this term is given by the following expression

$\displaystyle\widehat{\partial}_{L}\,\ln\left|\mbox{\boldmath$\xi$}\right|$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{l+1}}{\left|\mbox{\boldmath$\xi$}\right|^{l}}\;{\rm
STF}_{i_{1}\dots
i_{l}}\;\sum\limits_{n=0}^{[l/2]}G_{n}^{l}\;P_{i_{1}i_{2}}\,\dots\,P_{i_{2n-1}i_{2n}}$
(72)
$\displaystyle\times\frac{\xi_{i_{2n+1}}\,\dots\,\xi_{i_{l}}}{\left|\mbox{\boldmath$\xi$}\right|^{l-2n}}\;,$

which is valid for any natural number $l\geq 1$, and the coefficients are
given by Zschocke_Total_Light_Deflection_15PN

$\displaystyle G^{l}_{n}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n}\,2^{l-2n-1}\,\frac{l!}{n!}\,\frac{\left(l-n-1\right)!}{\left(l-2n\right)!}\;.$
(73)

Inserting (72) into (69) and (LABEL:Total_Shapiro_Spin_Multipole) completes
the calculation of the time delay in the gravitational field of a massive body
with full (time-independent) mass-multipole and spin-multipole structure, if
the source and the observer are at infinite distance from the massive body.

##  IV Relation between time delay and total light deflection

The tangent vector of the light trajectory at minus infinity, $\sigma$, has
been defined by Eq. (32), and the tangent vector of the light trajectory at
plus infinity, $\nu$, is defined by

$\nu$ $\displaystyle=$
$\displaystyle\frac{\dot{\mbox{\boldmath$x$}}\left(t\right)}{c}\,\bigg{|}_{t=+\infty}\quad{\rm
with}\quad\mbox{\boldmath$\nu$}\cdot\mbox{\boldmath$\nu$}=1\;.$ (74)

A graphical representation of the three-vector in (74) is given in Fig. 1. The
angle of total light deflection is defined as angle
$\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}\right)$ between
these tangent vectors,

$\displaystyle\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}\right)$
$\displaystyle=$
$\displaystyle\arcsin\left|\mbox{\boldmath$\sigma$}\times\mbox{\boldmath$\nu$}\right|.$
(75)

The tangent vector $\nu$ can be expanded in terms of mass-multipoles
$\hat{M}_{L}$ and spin-multipoles $\hat{S}_{L}$ Klioner1991 ;
Zschocke_Total_Light_Deflection_15PN

$\nu$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}+\sum\limits_{l=0}^{\infty}\mbox{\boldmath$\nu$}^{M_{L}}_{\rm
1PN}+\sum\limits_{l=1}^{\infty}\mbox{\boldmath$\nu$}^{S_{L}}_{\rm 1.5PN}+{\cal
O}\left(c^{-4}\right),$ (76)

where the individual terms are given by Eqs. (48) and (49) in
Zschocke_Total_Light_Deflection_15PN . By inserting (76) into (75) one finds
that the angle of total light deflection is also expanded in terms of mass-
multipoles and spin-multipoles,

$\displaystyle\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{l=0}^{\infty}\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}^{M_{L}}_{\rm
1PN}\right)+\sum\limits_{l=1}^{\infty}\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}^{S_{L}}_{\rm
1.5PN}\right)+{\cal O}\left(c^{-4}\right).$

The individual terms are given by Kopeikin1997 ;
Zschocke_Total_Light_Deflection_15PN

$\displaystyle\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}_{\rm
1PN}^{M_{L}}\right)$ $\displaystyle=$
$\displaystyle-\frac{4G}{c^{2}}\,\frac{1}{\left|\mbox{\boldmath$\xi$}\right|}\frac{\left(-1\right)^{l}}{\left(l-1\right)!}\,\hat{M}_{L}\,\widehat{\partial}_{L}\;\ln\left|\mbox{\boldmath$\xi$}\right|,$
(78)
$\displaystyle\delta\left(\\!\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}_{\rm
1.5PN}^{S_{L}}\\!\right)$ $\displaystyle=$
$\displaystyle-\frac{8G}{c^{3}}\frac{1}{\left|\mbox{\boldmath$\xi$}\right|}\,\frac{\left(-1\right)^{l}l^{2}}{\left(l+1\right)!}\,\epsilon_{abc}\,\sigma^{c}\,\hat{S}_{bL-1}\widehat{\partial}_{aL-1}\ln\left|\mbox{\boldmath$\xi$}\right|,$

which are valid for $l\geq 1$. By comparing (78) and
(LABEL:light_deflection_spin) with Eqs. (69) and
(LABEL:Total_Shapiro_Spin_Multipole) one recovers the following remarkable
relations between the multipole terms of the total effect of time delay and
the multipole terms of the angle of total light deflection:

$\displaystyle\Delta\tau_{\rm 1PN}^{M_{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{l}\,\frac{\left|\mbox{\boldmath$\xi$}\right|}{c}\,\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}^{M_{L}}_{\rm
1PN}\right),$ (80) $\displaystyle\Delta\tau_{\rm 1.5PN}^{S_{L}}$
$\displaystyle=$
$\displaystyle\frac{1}{l}\,\frac{\left|\mbox{\boldmath$\xi$}\right|}{c}\,\delta\left(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\nu$}^{S_{L}}_{\rm
1.5PN}\right),$ (81)

which are valid for multipoles of order $l\geq 1$ and
$|\mbox{\boldmath$\xi$}|=d_{\sigma}$ is the impact parameter of the
unperturbed light ray (29). These relations (80) and (81) are strictly valid
for light signals which propagate in the gravitational fields of a body at
rest, having the full set of mass-multipoles and spin-multipoles. That means,
the body can be of arbitrary shape, inner structure and can be in arbitrary
but uniform rotational motions and stationary inner currents.

##  V Time delay in field of an axisymmetric body

The largest effect of Shapiro effect is expected from the Sun and the giant
planets of the Solar System. In order to determine the Shapiro time delay one
needs the explicit form for mass-multipoles (18) and for spin-multipoles (19).
For an estimation of the individual terms in (69) and
(LABEL:Total_Shapiro_Spin_Multipole), one may approximate the Sun and the
giant planets by a rigid axisymmetric body with radial dependent mass
distribution and in uniform rotational motion with angular velocity $\Omega$
around the symmetry axis of the body, which is aligned with the $x^{3}$-axis
of the coordinate system. A graphical representation of this configuration is
given by Fig. 3. Then, the higher mass-multipoles and spin-multipoles for such
a body are given by Zschocke_Time_Delay_2PN ;
Zschocke_Total_Light_Deflection_15PN

$\displaystyle\hat{M}_{L}$ $\displaystyle=$
$\displaystyle-M\left(P\right)^{l}\,J_{l}\;\delta^{3}_{<{i_{1}}}\;\dots\;\delta^{3}_{{i_{l}}>}\;,$
(82) $\displaystyle\hat{S}_{a}$ $\displaystyle=$
$\displaystyle+\kappa^{2}\,M\,P^{2}\,\Omega\,\delta_{3a}\;,$ (83)
$\displaystyle\hat{S}_{L}$ $\displaystyle=$
$\displaystyle-M\left(P\right)^{l+1}\,\Omega\,J_{l-1}\,\frac{l+1}{l+4}\;\delta^{3}_{<{i_{1}}}\;\dots\;\delta^{3}_{{i_{l}}>}\;,$
(84)

where (82) is valid for any natural number of $l\geq 2$, while (84) is valid
for any natural number of $l\geq 3$. The mass-dipole term, that is $l=1$ in
(82) vanishes in case the origin of coordinate system is located at the center
of mass of the body (i.e. $J_{1}=0$) and will, therefore, not be considered in
what follows. Here, $M$ is the Newtonian mass of the body, $P$ its equatorial
radius, $J_{l}$ are the actual zonal harmonic coefficients of index $l$,
$\kappa^{2}$ is the dimensionless moment of inertia, $\Omega$ is the angular
velocity of the rotating body and
$\delta^{3}_{<{i_{1}}}\;\dots\;\delta^{3}_{{i_{l}}>}={\rm STF}_{i_{1}\dots
i_{l}}\;\delta_{3i_{1}}\dots\delta_{3i_{l}}$ denotes products of Kronecker
symbols which are symmetric and traceless with respect to indices $i_{1}\dots
i_{l}$. It is noticed that the definition of the angular velocity $\Omega$ has
an unambiguous meaning only at linear order around flat space-time.

Figure 3: A geometrical representation of the propagation of a light signal
through the gravitational field of an axisymmetric Solar System body at rest.
The axes of inertia are denoted by
$\mbox{\boldmath$e$}_{1},\mbox{\boldmath$e$}_{2},\mbox{\boldmath$e$}_{3}$. The
body is in uniform rotational motion with angular velocity $\Omega$ around the
axis of symmetry $\mbox{\boldmath$e$}_{3}$. The light signal is emitted by the
light source at $\mbox{\boldmath$x$}_{0}$ and propagates along the exact light
trajectory $\mbox{\boldmath$x$}\left(t\right)$. The unit tangent vectors along
the light trajectory at minus and plus infinity are $\sigma$ and $\nu$. The
unperturbed light ray $\mbox{\boldmath$x$}_{\rm N}\left(t\right)$ is given by
Eq. (26) and propagates in the direction of $\sigma$ along a straight line
through the position of the light source at $\mbox{\boldmath$x$}_{0}$. The
impact vector $\mbox{\boldmath$d$}_{\sigma}$ of the unperturbed light ray is
given by Eq. (28).

For reason of completeness, we notice the upper limit of the Shapiro time
delay caused by the mass-monopole (1), which reads Book_Clifford_Will

$\displaystyle\left|\Delta\tau_{\rm 1PN}^{M_{0}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{2GM}{c^{3}}\,\ln\frac{4\,x_{0}\,x_{1}}{\left(d_{\sigma}\right)^{2}}\;.$
(85)

Now we consider the upper limits of the higher mass-multipoles and spin-
multipoles, that means the upper limits of the absolute values in the
asymptotic limits (65) and (66). In order to get the total effect of time
delay in the gravitational field of an axisymmetric body in uniform rotational
motion, one has to insert the multipoles (82) - (84) into Eqs. (69) and
(LABEL:Total_Shapiro_Spin_Multipole) and taking account of relation (72).
However, in view of relations (80) and (81), which are valid for bodies of
arbitrary shape, inner structure and uniform rotational motions, it is easier
to use the results for the total light deflection angle which have been
determined for axisymmetric bodies in Zschocke_Total_Light_Deflection_15PN .
In particular, in our investigation Zschocke_Total_Light_Deflection_15PN  it
has been found that the angle of total light deflection is related to
Chebyshev polynomials. This relation has been established for mass-multipoles
by Eq. (114) in Zschocke_Total_Light_Deflection_15PN  and for spin-multipoles
by Eq. (121) in Zschocke_Total_Light_Deflection_15PN . Hence, one obtains for
the time delay of light signals in the field of an axisymmetric body the
following expressions in the asymptotic limit,

$\displaystyle\Delta\tau_{\rm 1PN}^{M_{L}}$ $\displaystyle=$
$\displaystyle-\frac{4GM}{c^{3}}\frac{J_{l}}{l}\left(\frac{P}{d_{\sigma}}\right)^{l}$
(86)
$\displaystyle\times\left[1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}\right]^{[l/2]}T_{l}\left(x\right),$

for the mass-multipole terms, which is valid for $l\geq 2$. For the effect of
time delay caused by the spin-dipole one obtains, by means of Eq. (120) in
Zschocke_Total_Light_Deflection_15PN , the following expression in the
asymptotic limit,

$\displaystyle\Delta\tau_{\rm 1.5PN}^{S_{1}}$ $\displaystyle=$
$\displaystyle-\frac{4GM}{c^{4}}\,P\,\Omega\,\kappa^{2}\,J_{0}\left(\frac{P}{d_{\sigma}}\right)\,\frac{\left(\mbox{\boldmath$\sigma$}\times\mbox{\boldmath$d$}_{\sigma}\right)\cdot\mbox{\boldmath$e$}_{3}}{d_{\sigma}}\;,$

with $J_{0}=-1$, and for the effect of time delay caused by higher spin-
multipoles,

$\displaystyle\Delta\tau_{\rm 1.5PN}^{S_{L}}$ $\displaystyle=$
$\displaystyle-\frac{8GM}{c^{4}}\,P\,\Omega\,J_{l-1}\left(\frac{P}{d_{\sigma}}\right)^{l}\,\frac{\left(\mbox{\boldmath$\sigma$}\times\mbox{\boldmath$d$}_{\sigma}\right)\cdot\mbox{\boldmath$e$}_{3}}{d_{\sigma}}\frac{1}{l+4}$
(88)
$\displaystyle\times\left[1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}\right]^{[l/2]}\;U_{l-1}\left(x\right),$

which is valid for $l\geq 3$. The variable in (86) and (88) reads
Zschocke_Total_Light_Deflection_15PN

$\displaystyle x$ $\displaystyle=$
$\displaystyle\left(1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}\right)^{-1/2}\;\left(\frac{\mbox{\boldmath$d$}_{\sigma}\cdot\mbox{\boldmath$e$}_{3}}{d_{\sigma}}\right),$
(89)

which is a real number. It has already been shown in
Zschocke_Total_Light_Deflection_15PN  that the interval of the argument is, in
fact, given by $-1\leq x\leq+1\;$. The power representation of Chebyshev
polynomials of first kind reads Arfken_Weber

$\displaystyle T_{l}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{l}{2}\sum\limits_{n=0}^{[l/2]}\frac{\left(-1\right)^{n}}{n!}\,\frac{\left(l-n-1\right)!}{\left(l-2n\right)!}\,\left(2x\right)^{l-2n}\,,$
(90)

where $l\geq 1$. The power representation of Chebyshev polynomials of second
kind reads Arfken_Weber

$\displaystyle U_{l}\left(x\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{[l/2]}\frac{\left(-1\right)^{n}}{n!}\,\frac{\left(l-n\right)!}{\left(l-2n\right)!}\,\left(2x\right)^{l-2n}\;,$
(91)

where $l\geq 0$. The remarkable feature that the total effect of time delay is
given in terms of Chebyshev polynomials allows for a straightforward
determination of the upper limit of the total effect of time delay, because
the upper limits of Chebyshev polynomials are given by

$\displaystyle\left|T_{l}\right|\leq 1\quad{\rm
and}\quad\left|U_{l-1}\right|\leq l\;.$ (92)

By inserting (92) into (86) and (88) yields for the absolute values of the
Shapiro time delay induced by the mass-multipoles and spin-multipoles the
following inequalities: for the mass-multipole terms ($l\geq 2$) one obtains,

$\displaystyle\left|\Delta\tau_{\rm 1PN}^{M_{L}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{4GM}{c^{3}}\,\frac{\left|J_{l}\right|}{l}\left(1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}\right)^{[l/2]}\left(\frac{P}{d_{\sigma}}\right)^{l},$
(93)

for the spin-dipole term ($l=1$) one finds,

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{1}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{4GM}{c^{4}}P\,\Omega\,\kappa^{2}\left(1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}\right)^{[l/2]}\left(\frac{P}{d_{\sigma}}\right),$

and for the spin-multipole terms ($l\geq 3$) one obtains,

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{L}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{8GM}{c^{4}}P\,\Omega\frac{\left|J_{l-1}\right|l}{l+4}\left(1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}\right)^{[l/2]}\\!\left(\frac{P}{d_{\sigma}}\right)^{l}.$

These upper limits (93) and (LABEL:upper_limit_S_L) represent a criterion to
identify those multipoles which contribute significantly to the time delay for
a given accuracy of time measurements. By taking into account that
$|\,1-\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$e$}_{3}\right)^{2}|\leq
1$ one may derive simpler expressions for these upper limits, namely for the
mass-multipoles ($l\geq 2$),

$\displaystyle\left|\Delta\tau_{\rm 1PN}^{M_{L}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{4GM}{c^{3}}\,\frac{\left|J_{l}\right|}{l}\,\left(\frac{P}{d_{\sigma}}\right)^{l},$
(96)

for the spin-dipole ($l=1$),

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{1}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{4GM}{c^{4}}P\,\Omega\,\kappa^{2}\left(\frac{P}{d_{\sigma}}\right),$
(97)

and for the spin-multipoles ($l\geq 3$),

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{L}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{8GM}{c^{4}}\,P\,\Omega\,\frac{l}{l+4}\,\left|J_{l-1}\right|\,\left(\frac{P}{d_{\sigma}}\right)^{l}\;.$
(98)

These upper limits have also been presented by Eqs. (39) and (41) in
Zschocke_Time_Delay_2PN . However, the coefficients in front of Eqs. (39) and
(41) in Zschocke_Time_Delay_2PN  were only given for the very few first
multipoles for ellipsoidal bodies, while here these upper limits in (96) and
(98) are valid for any multipole order and for the more general case of
axisymmetric bodies.

Finally, we notice the upper limits for grazing rays, for the mass-multipole
terms ($l\geq 2$) given by

$\displaystyle\left|\Delta\tau_{\rm 1PN}^{M_{L}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{4GM}{c^{3}}\,\frac{\left|J_{l}\right|}{l}\,,$ (99)

for the spin-dipole ($l=1$) given by

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{1}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{4GM}{c^{4}}P\,\Omega\,\kappa^{2}\;,$ (100)

and for the spin-multipole terms ($l\geq 3$) one obtains

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{L}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{8GM}{c^{4}}\,P\,\Omega\,\frac{l}{l+4}\,\left|J_{l-1}\right|\,.$
(101)

Numerical values of these limits (99) - (101) are given by Table 2 for Solar
System bodies.

##  VI Comparison with the literature

The upper limits of Shapiro time delay for mass-monopole (85) and spin-dipole
(100) are well-known and agree, for instance, with Eq. (40.13) in MTW  and Eq.
(75) in Klioner1991 , respectively.

Upper limits for the effect of time delay caused by higher mass-multipoles and
spin-multipoles of the massive body are rare and have, thus far, only been
presented in our previous investigation in Zschocke_Time_Delay_2PN . In order
to compare the upper limits in (96) and (98) with those upper limits presented
by Eqs. (39) and (41) in Zschocke_Time_Delay_2PN , we rewrite (96) and (98) in
the following form:

$\displaystyle\left|\Delta\tau_{\rm 1PN}^{M_{L}}\right|$ $\displaystyle\leq$
$\displaystyle
A_{l}\,\frac{GM}{c^{3}}\,\left|J_{l}\right|\,\left(\frac{P}{d_{\sigma}}\right)^{l}\;{\rm
with}\;A_{l}=\frac{4}{l}\;,$ (102)

which is valid for $l\geq 2$, and for the spin-multipole terms one obtains

$\displaystyle\left|\Delta\tau_{\rm 1.5PN}^{S_{L}}\right|$ $\displaystyle\leq$
$\displaystyle
B_{l}\,\frac{GM}{c^{4}}\,P\,\Omega\,\left|J_{l-1}\right|\,\left(\frac{P}{d_{\sigma}}\right)^{l}\;{\rm
with}\;B_{l}=\frac{8\,l}{l+4}\;,$

which is valid for $l\geq 3$. These coefficients, for the first few mass-
multipoles and spin-multipoles read

$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle 2\,,\quad
A_{4}=1\,,\quad A_{6}=\frac{2}{3}\,,\quad A_{8}=\frac{1}{2}\,,\quad
A_{10}=\frac{2}{5}\,,$ $\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle\frac{24}{7}\,.$ (104)

In Eqs. (42) and (43) in Zschocke_Time_Delay_2PN  the following coefficients
were presented:

$\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle\frac{11}{5}\,,\quad
A_{4}=\frac{7}{6}\,,\quad A_{6}=\frac{3}{5}\,,\quad A_{8}=\frac{3}{10}\,,\quad
A_{10}=\frac{3}{20}\,,$ $\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle\frac{7}{6}\,.$ (105)

The coefficients in (105) have been obtained for the case of finite spatial
distances of source and observer from the massive body, while the coefficients
in (104) are valid for infinite spatial distances of source and observer from
the massive body. Since the general case of finite distances contains also the
specific case of infinite distances, one would expect that the coefficients in
(105) are slightly larger than the coefficients in (104). In fact, this is the
case for the multipole orders $l=2$ and $l=4$, while the coefficients for
$l=6$ in (105) and (104) are almost equal. These facts are also reflected by
the numerical values presented here in Table II and in Table II in our
previous investigation in Zschocke_Time_Delay_2PN . On the other side, the
upper limits of mass-multipole terms of order $l=8$ and $l=10$ and the upper
limit of the spin-multipole term of order $l=3$, presented in our previous
investigation in Zschocke_Time_Delay_2PN , are too small. These deviations
have carefully been analyzed and were caused by an inaccuracy in the
analytical calculations in Zschocke_Time_Delay_2PN , which were assisted by
computer algebra systems because of the involved algebraic structure of the
expressions for the Shapiro time delay, especially in case of higher
multipoles and for finite distances.

##  VII Numerical values of time delay in the Solar System

For the numerical values of time delay we take the parameter of Solar System
bodies as given by Table 1.

Table 1: Numerical parameter for Schwarzschild radius $GM/c^{2}$, equatorial radius $P$ and actual zonal harmonic coefficients $J_{l}$ of Sun, Jupiter, and Saturn. The values for $GM/c^{2}$ and equatorial $P$ are taken from Ellipticity . The value for $J_{l}$ for the Sun are from J_n_Sun  and references therein. The values $J_{l}$ with $n=2,4,6$ for Jupiter and Saturn are taken from Book_Zonal_Harmonics , while $J_{l}$ with $n=8$ for Jupiter and Saturn come from Zonal_Harmonics_Jupiter  and Zonal_Harmonics_Saturn , respectively. The angular velocities $\Omega=2\pi/T$ (with rotational period $T$) are presented by NASA planetary fact sheets. The dimensionless moment of inertia $\kappa^{2}$ is defined by Eq. (106) and their values have been determined in Ellipticity . Parameter | Sun | Jupiter | Saturn
---|---|---|---
$GM/c^{2}\,[{\rm m}]$ | $1476.8$ | $1.41$ | $0.42$
$P\,[{\rm m}]$ | $696\times 10^{6}$ | $71.5\times 10^{6}$ | $60.3\times 10^{6}$
$J_{2}$ | $1.7\times 10^{-7}$ | $14.696\times 10^{-3}$ | $16.291\times 10^{-3}$
$J_{4}$ | $9.8\times 10^{-7}$ | $-0.587\times 10^{-3}$ | $-0.936\times 10^{-3}$
$J_{6}$ | $4\times 10^{-8}$ | $0.034\times 10^{-3}$ | $0.086\times 10^{-3}$
$J_{8}$ | $-4\times 10^{-9}$ | $-2.5\times 10^{-6}$ | $-10.0\times 10^{-6}$
$\Omega\,[{\rm sec}^{-1}]$ | $2.865\times 10^{-6}$ | $1.758\times 10^{-4}$ | $1.638\times 10^{-4}$
$\kappa^{2}$ | $0.059$ | $0.254$ | $0.210$

The parameter $\kappa^{2}$ in (83) is defined by Ellipticity  (see also Eqs.
(B60) - (B62) in Zschocke_Time_Delay_2PN )

$\displaystyle\kappa^{2}$ $\displaystyle=$
$\displaystyle\frac{I}{M\,P^{2}}\;,$ (106)

where $I$ is the moment of inertia of the real Solar System body under
consideration, which is related to the body’s angular momentum via
$|\mbox{\boldmath$S$}|=I\,\Omega$. For a spherically symmetric body with
uniform density $\kappa^{2}=2/5$ Ellipticity , while for real Solar System
bodies $\kappa^{2}<2/5$ because the mass densities are increasing towards the
center of the massive bodies. The values of $\kappa^{2}$ are given in Table 1
for the Sun and giant planets of the Solar System bodies.

Numerical values of the upper limits in (99) - (101) are presented in Table 2
for the first mass-multipoles and spin-multipoles in case of grazing rays at
the Sun and the giant planets of the Solar System.

Table 2: The effect of (one-way) Shapiro time delay caused by the mass-multipole $\Delta\tau^{M_{l}}_{\rm 1PN}$ and spin-multipole terms $\Delta\tau^{S_{l}}_{\rm 1.5PN}$ in the gravitational field of the Sun and giant planets of the Solar System according to the upper limits (99) - (101). The time delay is given in units of pico-seconds: $1\,{\rm ps}=10^{-12}\,{\rm sec}$. A blank entry means less than $0.001$ pico-seconds. The values are given for grazing rays (impact parameter $d_{\sigma}$ equals body’s equatorial radius $P$). The numerical values should be compared with the assumed goal accuracy of $0.001\,{\rm ps}$ in time-delay measurements. Object | $\Delta\tau^{M_{2}}_{\rm 1PN}$ | $\Delta\tau^{M_{4}}_{\rm 1PN}$ | $\Delta\tau^{M_{6}}_{\rm 1PN}$ | $\Delta\tau^{M_{8}}_{\rm 1PN}$ | $\Delta\tau^{S_{1}}_{\rm 1.5PN}$ | $\Delta\tau^{S_{3}}_{\rm 1.5PN}$
---|---|---|---|---|---|---
Sun | $1.68$ | $4.83$ | $0.13$ | $0.01$ | $7.73$ | $-$
Jupiter | $138.24$ | $2.76$ | $0.11$ | $0.01$ | $0.20$ | $0.010$
Saturn | $45.65$ | $1.31$ | $0.08$ | $0.01$ | $0.04$ | $0.003$

##  VIII Summary

In this investigation, the time delay in the gravitational field of a body at
rest with full multipole structure has been considered. In particular, the
impact of higher mass-multipoles and spin-multipoles on time delay has been
determined. Two main results were found:

  1. (1)

If the source and the observer are located at spatial infinity from the
massive body, then the individual multipole terms of time delay are related to
the individual multipole terms of total light deflection. These relations are
given by (80) and (81), which are valid for multipoles of order $l\geq 1$.

  2. (2)

In case of an axisymmetric massive body, these individual multipole terms of
time delay can be expressed in terms of Chebyshev polynomials: mass-multipole
terms of time delay are related to Chebyshev polynomials of first kind (86)
and spin-multipole terms of time delay are related to Chebyshev polynomials of
second kind (88).

These remarkable facts allow one to determine strict upper limits for the
absolute value of the multipole terms in this asymptotic configuration, where
the source and the observer are located at spatial infinity from the massive
body. These strict upper limits are given by Eqs. (99) and (101), which
represent a criterion to identify those multipoles which contribute
significantly to the time delay for a given accuracy of time measurements. The
coincidence between the total effect of time delay and total light deflection
for higher multipoles, as demonstrated by relations (80) and (81), must have a
deep reason, which represents a problem for subsequent investigations.

Numerical values of the Shapiro time delay, based on these upper limits, have
been calculated and presented in Table 2. These numerical results show that
the impact of the first mass-multipoles with $l\leq 8$ and the first spin-
multipoles with $l\leq 3$ on the effect of time-delay are relevant for an
accuracy on the femto-second scale of accuracy in time measurements, and might
in principle be detected with present-day atomic clocks or, at least, with the
next generation of atomic clocks.

The more realistic astrometric configurations, where source and observer are
located at finite spatial distances from the massive body, will be considered
in a subsequent investigation. It is, however, to expect that the upper limits
of higher multipoles on the effect of time delay for such scenarios will not
much be different from the asymptotic limit which has been considered in this
investigation.

##  IX Acknowledgment

This work was funded by the German Research Foundation (Deutsche
Forschungsgemeinschaft DFG) under grant number 447922800. Sincere gratitude is
expressed to Prof. Sergei A. Klioner, Prof. Michael H. Soffel, Prof. Ralf
Schützhold, Prof. William G. Unruh, Priv.-Doz. Günter Plunien, Dr. Alexey
Butkevich, Dipl-Inf. Robin Geyer, Prof. Burkhard Kämpfer, and Prof. Laszlo
Csernai for inspiring discussions about astrometry and general theory of
relativity.

##  Appendix A Notation

The following notation is in use:

  * •

Newtonian constant of gravitation: $G$.

  * •

vacuum speed of light in flat space-time: $c$.

  * •

Newtonian mass of the body: $M$.

  * •

Equatorial radius of the body: $P$.

  * •

Angular velocity of the body: $\Omega$.

  * •

Zonal harmonic coefficients of the body: $J_{l}$.

  * •

$\eta_{\alpha\beta}=\eta^{\alpha\beta}$ is the metric tensor of flat space-
time.

  * •

$g^{\alpha\beta}$ and $g_{\alpha\beta}$ are the contravariant and covariant
components of the metric.

  * •

$g={\rm det}\left(g_{\mu\nu}\right)$ is the determinant of the metric.

  * •

$n!=n\left(n-1\right)\left(n-2\right)\cdot\cdot\cdot 2\cdot 1$ is the
factorial; by definition: $0!=1$.

  * •

Lower case Greek indices take values 0,1,2,3.

  * •

The contravariant components of four-vectors:
$a^{\mu}=\left(a^{0},a^{1},a^{2},a^{3}\right)$.

  * •

Lower case Latin indices take values 1,2,3.

  * •

The three-dimensional coordinate quantities (three-vectors) referred to the
spatial axes of the reference system are in boldface: $a$.

  * •

The contravariant components of three-vectors:
$a^{i}=\left(a^{1},a^{2},a^{3}\right)$.

  * •

The absolute value of a three-vector:
$a=|\mbox{\boldmath$a$}|=\sqrt{a^{1}\,a^{1}+a^{2}\,a^{2}+a^{3}\,a^{3}}$.

  * •

The scalar product of two three-vectors:
$\mbox{\boldmath$a$}\,\cdot\,\mbox{\boldmath$b$}=\delta_{ij}\,a^{i}\,b^{j}=a^{i}\,b^{i}$
with Kronecker delta $\delta_{ij}$.

  * •

The angle between two three-vectors $a$ and $b$ is designated as
$\delta\left(\mbox{\boldmath$a$},\mbox{\boldmath$b$}\right)$.

##  Appendix B Proof of Eq. (54)

In this appendix we will demonstrate the limit (54). The gauge terms in the
geodesic equation (37) are given by Eq. (39), which consists of two pieces,

$\displaystyle\ddot{\mbox{\boldmath$x$}}_{\rm gauge}$ $\displaystyle=$
$\displaystyle\ddot{\mbox{\boldmath$x$}}_{\rm
g1}+\ddot{\mbox{\boldmath$x$}}_{\rm g2}\;.$ (107)

Their spatial components are given by

$\displaystyle\frac{\ddot{x}_{\rm g1}^{i}\left(t\right)}{c^{2}}$
$\displaystyle=$
$\displaystyle+\partial_{j}\,w^{0}_{\,,\,k}\,\sigma^{i}\sigma^{j}\sigma^{k}\;,$
(108) $\displaystyle\frac{\ddot{x}_{\rm g2}^{i}\left(t\right)}{c^{2}}$
$\displaystyle=$
$\displaystyle-\partial_{j}\,w^{i}_{\,,\,k}\,\sigma^{j}\sigma^{k}\;,$ (109)

where the gauge vectors are given by Eqs. (20) and (21). Let us consider the
first term (108). Using
$\left(r^{-1}\right)_{\,,\,jk}=3x_{j}x_{k}/r^{5}-\delta_{jk}/r^{3}$, one
obtains

$\displaystyle\frac{\ddot{\mbox{\boldmath$x$}}_{\rm g1}\left(t\right)}{c^{2}}$
$\displaystyle=$
$\displaystyle+\frac{8G}{c^{3}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\hat{\partial}_{L}\,\frac{\hat{W}_{L}}{\left(r_{\rm
N}\right)^{3}}\,\mbox{\boldmath$\sigma$}$ (110)
$\displaystyle-\frac{12G}{c^{3}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\hat{\partial}_{L}\,\frac{\hat{W}_{L}}{\left(r_{\rm
N}\right)^{5}}\,\left(d_{\sigma}\right)^{2}\,\mbox{\boldmath$\sigma$}\;.$

In (110) the replacements
$\mbox{\boldmath$x$}\rightarrow\mbox{\boldmath$x$}_{\rm N}$ and $r\rightarrow
r_{\rm N}=|\mbox{\boldmath$x$}_{\rm N}|$ have been performed (cf. text below
Eq. (39)), where the unperturbed light ray $\mbox{\boldmath$x$}_{\rm N}$ is
given by Eq. (26) and its absolute value by Eq. (27). In addition, the
relation $\left(\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\right)^{2}=\left(r_{\rm N}\right)^{2}-\left(d_{\sigma}\right)^{2}$ has been
used.

The expression (110) has to be integrated over the time variable. To apply the
advanced integration methods developed by Kopeikin1997 , we have to transform
(110) from $\left(ct,\mbox{\boldmath$x$}\right)$ into terms of two new
variables, $c\tau=\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm N}$
and $\xi^{i}=P^{ij}\,x_{\rm N}^{j}$, which are independent of each other, and
obtain (note that after differentiation are performed,
$\mbox{\boldmath$\xi$}=\mbox{\boldmath$d$}_{\sigma}$, hence
$\left(d_{\sigma}\right)^{2}=\mbox{\boldmath$\xi$}\cdot\mbox{\boldmath$\xi$}=\xi^{2}$)

$\frac{\ddot{\mbox{\boldmath$x$}}_{\rm
g1}\left(t\right)}{c^{2}}=+\frac{4G}{c^{3}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\hat{W}_{L}\,\widehat{\partial}_{L}\left(\frac{2}{\left(r_{\rm
N}\right)^{3}}-\frac{3\;\left(\xi\right)^{2}}{\left(r_{\rm
N}\right)^{5}}\right)\,\mbox{\boldmath$\sigma$}\;,$ (111)

where the double-dot in (111) means twice the total derivative with respect to
variable $\tau$, and $r_{\rm N}$ is the absolute value of the unperturbed
light ray in terms of these new variables (45). Let us note that the left-hand
side in (111) depends on variable of coordinate time, because in (111) the
differentiations have to be performed, and afterwards one has to replace
$c\tau$ by $\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\left(t\right)$ and $P^{ij}\xi_{j}$ by $d^{i}_{\sigma}$; see text below Eq.
(61). The differential operator (111) has been given by Eq. (47). To get the
coordinate velocity of the light signal, one has to integrate (111) over
variable $c\tau$ and obtains for the spatial components

$\displaystyle\frac{\Delta\dot{x}^{i}_{\rm g1}\left(t\right)}{c}$
$\displaystyle=$
$\displaystyle-\frac{4G}{c^{3}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\hat{W}_{L}\,\widehat{\partial}_{L}\,\frac{c\tau}{\left(r_{\rm
N}\right)^{3}}\;\sigma^{i}$ (112) $\displaystyle=$
$\displaystyle+\frac{4G}{c^{3}}\frac{\partial}{\partial
c\tau}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{L}\,\frac{\hat{W}_{L}}{r_{\rm
N}}\;\sigma^{i}\;.$

A similar calculation can be performed for the second gauge term (109), which
yields

$\displaystyle\frac{\Delta\dot{x}^{i}_{\rm g2}\left(t\right)}{c}$
$\displaystyle=$ $\displaystyle-\frac{4G}{c^{2}}\frac{\partial}{\partial
c\tau}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{iL}\,\frac{\hat{X}_{L}}{r_{\rm
N}}$ (113) $\displaystyle-\frac{4G}{c^{2}}\frac{\partial}{\partial
c\tau}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{L-1}\,\frac{\hat{Y}_{iL-1}}{r_{\rm
N}}$ $\displaystyle-\frac{4G}{c^{2}}\frac{\partial}{\partial
c\tau}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\frac{l}{l+1}\,\epsilon_{iab}\,\widehat{\partial}_{aL-1}\,\frac{\hat{Z}_{bL-1}}{r_{\rm
N}}\,.$

The second integration over variable $c\tau$, from lower integration limit
$c\tau_{0}$ to upper integration limit $c\tau_{1}$, can be performed
immediately and yields

$\displaystyle\Delta x^{i}_{\rm g1}\left(t_{1},t_{0}\right)$ $\displaystyle=$
$\displaystyle\Delta x^{i}_{\rm g1}\left(t_{1}\right)-\Delta x^{i}_{\rm
g1}\left(t_{0}\right),$ (114) $\displaystyle\Delta x^{i}_{\rm
g2}\left(t_{1},t_{0}\right)$ $\displaystyle=$ $\displaystyle\Delta x^{i}_{\rm
g2}\left(t_{1}\right)-\Delta x^{i}_{\rm g2}\left(t_{0}\right),$ (115)

with

$\displaystyle\Delta x^{i}_{\rm g1}\left(t\right)$ $\displaystyle=$
$\displaystyle+\frac{4G}{c^{3}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{L}\,\frac{\hat{W}_{L}}{r_{\rm
N}}\;\sigma^{i}\;,$ (116) $\displaystyle\Delta x^{i}_{\rm g2}\left(t\right)$
$\displaystyle=$
$\displaystyle-\frac{4G}{c^{2}}\sum\limits_{l=0}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{iL}\,\frac{\hat{X}_{L}}{r_{\rm
N}}$ (117)
$\displaystyle-\frac{4G}{c^{2}}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{L-1}\,\frac{\hat{Y}_{iL-1}}{r_{\rm
N}}$
$\displaystyle-\frac{4G}{c^{2}}\epsilon_{iab}\sum\limits_{l=1}^{\infty}\frac{\left(-1\right)^{l}}{l!}\,\frac{\left(-1\right)^{l}}{l!}\,\widehat{\partial}_{aL-1}\,\frac{\hat{Z}_{bL-1}}{r_{\rm
N}}\;.$

The time-dependence of the gauge terms (114) and (115) are just via the
spatial positions of the unperturbed light ray at the time of emission and
reception, $\mbox{\boldmath$x$}_{\rm N}\left(t_{0}\right)$ and
$\mbox{\boldmath$x$}_{\rm N}\left(t_{1}\right)$. In line with (107), the terms
(114) and (115) are added together, so we arrive at

$\Delta\mbox{\boldmath$x$}_{\rm
gauge}\left(t_{1},t_{0}\right)=\Delta\mbox{\boldmath$x$}_{\rm
g1}\left(t_{1},t_{0}\right)+\Delta\mbox{\boldmath$x$}_{\rm
g2}\left(t_{1},t_{0}\right).$ (118)

By inserting the differential operator (47) into these solutions (116) and
(117) one finds that these terms in (116) and (117) vanish separately in the
asymptotic limit,

$\displaystyle\lim_{\tau=\pm\infty}\Delta\mbox{\boldmath$x$}_{\rm
g1}\left(\mbox{\boldmath$x$}_{\rm N}\left(t\right)\right)$ $\displaystyle=$
$\displaystyle 0\,,$ (119)
$\displaystyle\lim_{\tau=\pm\infty}\Delta\mbox{\boldmath$x$}_{\rm
g2}\left(\mbox{\boldmath$x$}_{\rm N}\left(t\right)\right)$ $\displaystyle=$
$\displaystyle 0\,,$ (120)

with $c\tau=\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}\left(t\right)$. Eqs. (119) and (120) imply for (118) the limits

$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty\atop\tau=\tau_{1}\rightarrow+\infty}\Delta\mbox{\boldmath$x$}_{\rm
gauge}\left(t_{1},t_{0}\right)=0\;,$ (121)

which is more general that the asserted relation in (54).

##  Appendix C Proof of Eqs. (67) and (68)

In this appendix we will show relations (67) and (68). That means, we consider
the limits

$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty\atop\tau=\tau_{1}\rightarrow+\infty}\,\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right),$ (122)

where $r_{\rm N}=\sqrt{\xi^{2}+c^{2}\tau^{2}}$ with
$\xi^{2}=\mbox{\boldmath$\xi$}\cdot\mbox{\boldmath$\xi$}$, while the
differential operator is given by Eq. (47), and $c\tau_{0}$ and $c\tau_{1}$
are defined by Eqs. (55) and (56), respectively.

First of all, we consider the first derivative of the logarithm in (122) with
respect to variable $c\tau$,

$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty\atop\tau=\tau_{1}\rightarrow+\infty}\,\frac{\partial}{\partial
c\tau}\,\ln\left(r_{\rm N}+c\tau\right)$ $\displaystyle=$
$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty\atop\tau=\tau_{1}\rightarrow+\infty}\,\frac{1}{\sqrt{\xi^{2}+c^{2}\tau^{2}}}=0\;.$

Clearly, any further derivative of this term, with respect to either variable
$\xi$ or $c\tau$, increases the inverse power of $c\tau$ at least by one
order. Therefore, all those terms in the differential operator (47) which
contain at least one derivative with respect to variable $c\tau$ will vanish
in these limits. Hence, one has only to consider the term with $p=0$ of the
differential operator in Eq. (47), given by

$\displaystyle\widehat{\partial}^{\;\;p=0}_{L}$ $\displaystyle=$
$\displaystyle{\rm STF}_{i_{1}\dots
i_{l}}\;P_{i_{1}}^{j_{1}}\;...\;P_{i_{l}}^{j_{l}}\,\frac{\partial}{\partial\xi^{j_{1}}}\;...\;\frac{\partial}{\partial\xi^{j_{l}}}\;,$
(124)

in line with the comment in the text below Eq. (68).

The logarithm in (122) can be written in the form

$\displaystyle\ln\left(r_{\rm N}+c\tau\right)$ $\displaystyle=$
$\displaystyle\ln\left|c\tau\right|+\ln\left(\sqrt{1+\xi^{2}/c^{2}\tau^{2}}\pm
1\right),$ (125)

where the plus sign in the argument of the logarithm is for $c\tau>0$, while
the minus sign in the argument of logarithm is for $c\tau<0$. The first
logarithm on the right-hand side of (125) can be omitted, because of

$\displaystyle\frac{\partial}{\partial\xi^{j_{p+1}}}\;...\;\frac{\partial}{\partial\xi^{j_{l}}}\;\ln\left|c\tau\right|=0\;.$
(126)

Thus, one only needs to consider the second logarithmic term on the right-hand
side of (125). In order to evaluate these limits in (122), it is appropriate
to introduce the dimensionless three-vector,

$\displaystyle\zeta^{a}$ $\displaystyle=$
$\displaystyle\frac{\xi^{a}}{c\tau}\quad\rightarrow\quad\frac{\partial}{\partial\xi^{a}}=\frac{1}{c\tau}\,\frac{\partial}{\partial\zeta^{a}}\;,$
(127)

where the absolute value of this three-vector,
$\zeta=\sqrt{\mbox{\boldmath$\zeta$}\cdot\mbox{\boldmath$\zeta$}}$, is a small
quantity in the limits $c\tau\rightarrow\pm\infty$, that means
$\zeta\rightarrow 0$, because the impact vector, $\xi$, remains constant; see
also Fig. 2. The differential operator in (124) in terms of three-vector
$\zeta^{a}$ transforms into

$\displaystyle\widehat{\partial}^{\;p=0}_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(c\tau\right)^{l}}\,{\rm STF}_{i_{1}\dots
i_{l}}\;P_{i_{1}}^{j_{1}}\;...\;P_{i_{l}}^{j_{l}}\,\frac{\partial}{\partial\zeta^{j_{1}}}\;...\;\frac{\partial}{\partial\zeta^{j_{l}}}\,.$
(128)

Let us consider the term with the plus sign in (122), which in terms of three-
vector (127) reads

$\displaystyle\lim_{\tau=\tau_{1}\rightarrow+\infty}\,\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)=\lim_{\tau=\tau_{1}\rightarrow+\infty}\frac{1}{\left(c\tau\right)^{l}}$
$\displaystyle\times\,{\rm STF}_{i_{1}\dots
i_{l}}P_{i_{1}}^{j_{1}}\;...\;P_{i_{l}}^{j_{l}}\,\frac{\partial}{\partial\zeta^{j_{1}}}\;...\;\frac{\partial}{\partial\zeta^{j_{l}}}\,\ln\left(\sqrt{1+\zeta^{2}}+1\right).$
(129)

The term in the second line of (129) is finite, even for any values of
$\zeta^{2}$. Thus, in view of the prefactor $(c\tau)^{-l}$, the term (129)
vanishes in the limit $c\tau\rightarrow+\infty$ and one obtains

$\displaystyle\lim_{\tau=\tau_{1}\rightarrow+\infty}\,\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)$ $\displaystyle=$ $\displaystyle 0\;.$ (130)

Let us consider the term with the minus sign in (122), which in terms of
three-vector (127) reads

$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty}\,\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)=\lim_{\tau=\tau_{0}\rightarrow-\infty}\frac{1}{\left(c\tau\right)^{l}}$
$\displaystyle\times\,{\rm STF}_{i_{1}\dots
i_{l}}P_{i_{1}}^{j_{1}}\;...\;P_{i_{l}}^{j_{l}}\,\frac{\partial}{\partial\zeta^{j_{1}}}\;...\;\frac{\partial}{\partial\zeta^{j_{l}}}\,\ln\left(\sqrt{1+\zeta^{2}}-1\right).$
(131)

The term in the second line of (131) diverges in the limit $\zeta\rightarrow
0$. To determine this limit, a series expansion of the logarithm in (131) is
performed for $\zeta\ll 1$, which reads

$\displaystyle\ln\left(\sqrt{1+\zeta^{2}}-1\right)$ $\displaystyle=$
$\displaystyle-\ln\left(2\right)+\ln\left(\zeta^{2}\right)-\frac{\zeta^{2}}{4}+{\cal
O}\left(\zeta^{4}\right).$

The constant $\ln(2)$ does not contribute, because a derivative of a constant
with respect to variable $\zeta$ in (131) vanishes. Similarly, the third term
of power $\zeta^{2}$ and also terms of higher powers ${\cal
O}\left(\zeta^{4}\right)$ on the right-hand side in (LABEL:Appendix2_60)
vanish in the limit $\zeta\rightarrow 0$ and, in addition, also in view of the
prefactor $(c\tau)^{-l}$ in (131). Thus, only the logarithmic term remains,
keeping in mind that the differential operator (128) is acting on this
logarithm. By transforming (LABEL:Appendix2_60) from three-vector $\zeta$ back
into three-vector $\xi$ and by accounting for the statements below Eq.
(LABEL:Appendix2_60), one arrives at

$\displaystyle\lim_{\tau=\tau_{0}\rightarrow-\infty}\,\widehat{\partial}_{L}\,\ln\left(r_{\rm
N}+c\tau\right)$ $\displaystyle=$ $\displaystyle
2\,\widehat{\partial}^{\;p=0}_{L}\ln\left|\mbox{\boldmath$\xi$}\right|.$ (133)

By relations (130) and (133) the validity of Eqs. (67) and (68) has been
shown. Let us notice again, that the differential operator
$\hat{\partial}_{L}$ on the left-hand side in (130) and (133) is given by Eq.
(47), while on the right-hand side the differential operator
$\widehat{\partial}^{\;p=0}_{L}$ is given by (124). However, one may keep the
differential operator $\hat{\partial}_{L}$ on the right-hand side in (130) and
(133), because derivatives with respect to variable $c\tau$ in the
differential operator (47) would not contribute anyway.

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