, consisting of the deflection of light and the time delay
of light.
(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.
from
light deflection and time delay measurements. Its general
relativity value is unity. The arrows denote anomalously large
values from early eclipse expeditions. The Shapiro time-delay
measurements using Viking spacecraft yielded an agreement with GR
to 0.1 percent, and VLBI light deflection measurements have
reached 0.02 percent. Hipparcos denotes the optical astrometry
satellite, which has reached 0.1 percent.
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 
.
|
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 |
