(TEGP 7.1 [147]), where
is the mass of the Sun and
is the angle between the Earth-Sun line and the incoming
direction of the photon (Figure
4). For a grazing ray,
,
, and
independent of the frequency of light. Another, more useful expression gives the change in the relative angular separation between an observed source of light and a nearby reference source as both rays pass near the Sun:
where
d
and
are the distances of closest approach of the source and
reference rays respectively,
is the angular separation between the Sun and the reference
source, and
is the angle between the Sun-source and the Sun-reference
directions, projected on the plane of the sky (Figure
4). Thus, for example, the relative angular separation between the
two sources may vary if the line of sight of one of them passes
near the Sun (
,
,
varying with time).
It is interesting to note that the classic derivations of the
deflection of light that use only the principle of equivalence or
the corpuscular theory of light yield only the ``1/2'' part of
the coefficient in front of the expression in Eq. (30). But the result of these calculations is the deflection of
light relative to local straight lines, as established for
example by rigid rods; however, because of space curvature around
the Sun, determined by the PPN parameter
, local straight lines are bent relative to asymptotic straight
lines far from the Sun by just enough to yield the remaining
factor ``
''. The first factor ``1/2'' holds in any metric theory, the
second ``
'' varies from theory to theory. Thus, calculations that purport
to derive the full deflection using the equivalence principle
alone are incorrect.
The prediction of the full bending of light by the Sun was one of the great successes of Einstein's GR. Eddington's confirmation of the bending of optical starlight observed during a solar eclipse in the first days following World War I helped make Einstein famous. However, the experiments of Eddington and his co-workers had only 30 percent accuracy, and succeeding experiments were not much better: The results were scattered between one half and twice the Einstein value (Figure 5), and the accuracies were low.
However, the development of VLBI, very-long-baseline radio
interferometry, produced greatly improved determinations of the
deflection of light. These techniques now have the capability of
measuring angular separations and changes in angles as small as
100 microarcseconds. Early measurements took advantage of a
series of heavenly coincidences: Each year, groups of strong
quasistellar radio sources pass very close to the Sun (as seen
from the Earth), including the group 3C273, 3C279, and 3C48, and
the group 0111+02, 0119+11 and 0116+08. As the Earth moves in its
orbit, changing the lines of sight of the quasars relative to the
Sun, the angular separation
between pairs of quasars varies (Eq. (32
)). The time variation in the quantities
d,
,
and
in Eq. (32
) is determined using an accurate ephemeris for the Earth and
initial directions for the quasars, and the resulting prediction
for
as a function of time is used as a basis for a least-squares fit
of the measured
, with one of the fitted parameters being the coefficient
. A number of measurements of this kind over the period 1969-1975
yielded an accurate determination of the coefficient
which has the value unity in GR. A 1995 VLBI measurement using
3C273 and 3C279 yielded
[85].
A recent series of transcontinental and intercontinental VLBI
quasar and radio galaxy observations made primarily to monitor
the Earth's rotation (``VLBI'' in Figure
5) was sensitive to the deflection of light over almost the entire
celestial sphere (at
from the Sun, the deflection is still 4 milliarcseconds). A
recent analysis of over 2 million VLBI observations yielded
[59]. Analysis of observations made by the Hipparcos optical
astrometry satellite yielded a test at the level of 0.3
percent [66]. A VLBI measurement of the deflection of light by Jupiter was
reported; the predicted deflection of about 300 microarcseconds
was seen with about 50 percent accuracy [129]. The results of light-deflection measurements are summarized in
Figure
5
.
where
(
) are the vectors, and
(
) are the distances from the Sun to the source (Earth),
respectively (TEGP 7.2 [147
]). For a ray which passes close to the Sun,
where d is the distance of closest approach of the ray in solar radii, and r is the distance of the planet or satellite from the Sun, in astronomical units.
In the two decades following Irwin Shapiro's 1964 discovery of
this effect as a theoretical consequence of general relativity,
several high-precision measurements were made using radar ranging
to targets passing through superior conjunction. Since one does
not have access to a ``Newtonian'' signal against which to
compare the round-trip travel time of the observed signal, it is
necessary to do a differential measurement of the variations in
round-trip travel times as the target passes through superior
conjunction, and to look for the logarithmic behavior of
Eq. (34). In order to do this accurately however, one must take into
account the variations in round-trip travel time due to the
orbital motion of the target relative to the Earth. This is done
by using radar-ranging (and possibly other) data on the target
taken when it is far from superior conjunction (i.e. when the
time-delay term is negligible) to determine an accurate ephemeris
for the target, using the ephemeris to predict the PPN coordinate
trajectory
near superior conjunction, then combining that trajectory with
the trajectory of the Earth
to determine the Newtonian round-trip time and the logarithmic
term in Eq. (34
). The resulting predicted round-trip travel times in terms of
the unknown coefficient
are then fit to the measured travel times using the method of
least-squares, and an estimate obtained for
.
The targets employed included planets, such as Mercury or Venus, used as passive reflectors of the radar signals (``passive radar''), and artificial satellites, such as Mariners 6 and 7, Voyager 2, and the Viking Mars landers and orbiters, used as active retransmitters of the radar signals (``active radar'').
The results for the coefficient
of all radar time-delay measurements performed to date
(including a measurement of the one-way time delay of signals
from the millisecond pulsar PSR 1937+21) are shown in
Figure
5
(see TEGP 7.2 [147
] for discussion and references). The Viking experiment resulted
in a 0.1 percent measurement [111].
From the results of VLBI light-deflection experiments, we can
conclude that the coefficient
must be within at most 0.014 percent of unity.
Scalar-tensor theories must have
to be compatible with this constraint.
Table 4:
Current limits on the PPN parameters. Here
is a combination of other parameters given by
.
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The Confrontation between General Relativity and
Experiment
Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |