A light ray (or photon) which passes the Sun at a distance is deflected by an angle
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 (see Figure 5), and the accuracies were
low.
In recent years, transcontinental and intercontinental VLBI observations of quasars and radio galaxies
have been made primarily to monitor the Earth’s rotation (“VLBI” in Figure 5). These measurements are
sensitive to the deflection of light over almost the entire celestial sphere (at
from the Sun, the
deflection is still 4 milliarcseconds). A 2004 analysis of almost 2 million VLBI observations of 541
radio sources, made by 87 VLBI sites yielded
, or equivalently,
[240].
Analysis of observations made by the Hipparcos optical astrometry satellite yielded a test at the level of
0.3 percent [115]. 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 [257]. The results of
light-deflection measurements are summarized in Figure 5.
A radar signal sent across the solar system past the Sun to a planet or satellite and returned to the Earth
suffers an additional non-Newtonian delay in its round-trip travel time, given by (see Figure 4)
In the two decades following Irwin Shapiro’s 1964 discovery of this effect as a theoretical consequence of
GR, 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 Equation (50). 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 Equation (50
). 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, the Viking Mars landers and orbiters, and the Cassini spacecraft to Saturn, 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 [281
] for discussion and references). The 1976 Viking experiment
resulted in a 0.1 percent measurement [222].
A significant improvement was reported in 2003 from Doppler tracking of the Cassini spacecraft while it
was on its way to Saturn [29], with a result . This was made possible by the
ability to do Doppler measurements using both X-band (
) and Ka-band (
) radar,
thereby significantly reducing the dispersive effects of the solar corona. In addition, the 2002 superior
conjunction of Cassini was particularly favorable: With the spacecraft at 8.43 astronomical
units from the Sun, the distance of closest approach of the radar signals to the Sun was only
.
From the results of the Cassini experiment, we can conclude that the coefficient must be
within at most 0.0012 percent of unity. Scalar-tensor theories must have
to be compatible with
this constraint.
In 2001, Kopeikin [147] suggested that a measurement of the time delay of light from a quasar as the light
passed by the planet Jupiter could be used to measure the speed of the gravitational interaction. He argued
that, since Jupiter is moving relative to the solar system, and since gravity propagates with a finite speed,
the gravitational field experienced by the light ray should be affected by gravity’s speed, since the field
experienced at one time depends on the location of the source a short time earlier, depending on how
fast gravity propagates. According to his calculations, there should be a post-Newtonian
correction to the normal Shapiro time-delay formula (49
) which depends on the velocity of
Jupiter and on the velocity of gravity. On September 8, 2002, Jupiter passed almost in front of a
quasar, and Kopeikin and Fomalont made precise measurements of the Shapiro delay with
picosecond timing accuracy, and claimed to have measured the correction term to about 20
percent [112, 153, 148, 149
].
However, several authors pointed out that this 1.5PN effect does not depend on the speed of
propagation of gravity, but rather only depends on the speed of light [14, 288, 232, 49, 233]. Intuitively, if
one is working to only first order in
, then all that counts is the uniform motion of the planet, Jupiter
(its acceleration about the Sun contributes a higher-order, unmeasurably small effect). But if that is the
case, then the principle of relativity says that one can view things from the rest frame of Jupiter. In this
frame, Jupiter’s gravitational field is static, and the speed of propagation of gravity is irrelevant. A detailed
post-Newtonian calculation of the effect was done using a variant of the PPN framework, in
a class of theories in which the speed of gravity could be different from that of light [288
],
and found explicitly that, at first order in
, the effect depends on the speed of light, not
the speed of gravity, in line with intuition. Effects dependent upon the speed of gravity show
up only at higher order in
. Kopeikin gave a number of arguments in opposition to this
interpretation [149, 151, 150, 152]. On the other hand, the
correction term does show a
dependence on the PPN parameter
, which could be non-zero in theories of gravity with a
differing speed
of gravity (see Equation (7) of [288]). But existing tight bounds on
from other experiments (see Table 4) already far exceed the capability of the Jupiter VLBI
experiment.
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