(TEGP 7.1 ), 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 .
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 . Analysis of observations made by the Hipparcos optical astrometry satellite yielded a test at the level of 0.3 percent . 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 . 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 ). 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  for discussion and references). The Viking experiment resulted in a 0.1 percent measurement .
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
Clifford M. Will
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