Tests of the parameter γ

3.4 Tests of the parameter $γ$

With the PPN formalism in hand, we are now ready to confront gravitation theories with the results of solar-system experiments. In this section we focus on tests of the parameter $γ$ , consisting of the deflection of light and the time delay of light.

3.4.1 The deflection of light

A light ray (or photon) which passes the Sun at a distance d is deflected by an angle

(TEGP 7.1 [281

]), where $M ⊙$ is the mass of the Sun and $Φ$ is the angle between the Earth-Sun line and the incoming direction of the photon (see Figure 4

). For a grazing ray, $d ≈ d⊙$ , $Φ ≈ 0$ , 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 d_r are the distances of closest approach of the source and reference rays respectively, $Φr$ is the angular separation between the Sun and the reference source, and $χ$ is the angle between the Sun-source and the Sun-reference directions, projected on the plane of the sky (see 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 ( $d ∼ R ⊙$ , $dr ≫ d$ , $χ$ varying with time).

Figure 4:

Geometry of light deflection measurements.

It is interesting to note that the classic derivations of the deflection of light that use only the corpuscular theory of light (Cavendish 1784, von Soldner 1803 [277 ]), or the principle of equivalence (Einstein 1911), yield only the “1/2” part of the coefficient in front of the expression in Equation (46 ). 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 $γ$ , local straight lines are bent relative to asymptotic straight lines far from the Sun by just enough to yield the remaining factor “ $γ ∕2$ ”. The first factor “1/2” holds in any metric theory, the second “ $γ∕2$ ” 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 (see Figure 5 ), and the accuracies were low.

Figure 5:

Measurements of the coefficient (1 + $γ$ )/2 from light deflection and time delay measurements. Its GR value is unity. The arrows at the top denote anomalously large values from early eclipse expeditions. The Shapiro time-delay measurements using the Cassini spacecraft yielded an agreement with GR to 10^–3 percent, and VLBI light deflection measurements have reached 0.02 percent. Hipparcos denotes the optical astrometry satellite, which reached 0.1 percent.

However, the development of radio-interferometery, and later of very-long-baseline radio interferometry (VLBI), produced greatly improved determinations of the deflection of light. These techniques now have the capability of measuring angular separations and changes in angles to accuracies better than 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 $δθ$ between pairs of quasars varies (see Equation (48 )). The time variation in the quantities $d$ , $d r$ , $χ$ , and $Φ r$ in Equation (48 ) is determined using an accurate ephemeris for the Earth and initial directions for the quasars, and the resulting prediction for $δθ$ as a function of time is used as a basis for a least-squares fit of the measured $δθ$ , with one of the fitted parameters being the coefficient $1 (1 + γ ) 2$ . A number of measurements of this kind over the period 1969 – 1975 yielded an accurate determination of the coefficient $1(1 + γ ) 2$ . A 1995 VLBI measurement using 3C273 and 3C279 yielded $(1 + γ)∕2 = 0.9996 ± 0.0017$ [164 ].

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 90°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 $(1 + γ )∕2 = 0.99992 ± 0.00023$ , or equivalently, $γ − 1 = (− 1.7 ± 4.5) × 10 −4$ [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 .

3.4.2 The time delay of light

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 )

where $xe$ ( $x⊕$ ) are the vectors, and $re$ ( $r⊕$ ) are the distances from the Sun to the source (Earth), respectively (TEGP 7.2 [281

]). 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 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 $xe (t)$ near superior conjunction, then combining that trajectory with the trajectory of the Earth $x (t) ⊕$ 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 $1 2(1 + γ)$ are then fit to the measured travel times using the method of least-squares, and an estimate obtained for $1 (1 + γ ) 2$ .

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 $1(1 + γ) 2$ 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 $γ − 1 = (2.1 ± 2.3) × 10−5$ . This was made possible by the ability to do Doppler measurements using both X-band (7175 MHz) and Ka-band (34316 MHz) 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 $1.6R ⊙$ .

From the results of the Cassini experiment, we can conclude that the coefficient $1(1 + γ ) 2$ must be within at most 0.0012 percent of unity. Scalar-tensor theories must have $ω > 40, 000$ to be compatible with this constraint.

3.4.3 Shapiro time delay and the speed of gravity

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 ^1/2-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 $v ∕c$ , 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 $v∕c$ , 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 $v∕c$ . Kopeikin gave a number of arguments in opposition to this interpretation [149 , 151 , 150 , 152 ]. On the other hand, the $v∕c$ correction term does show a dependence on the PPN parameter $α1$ , which could be non-zero in theories of gravity with a differing speed $cg$ of gravity (see Equation (7) of [288 ]). But existing tight bounds on $α1$ from other experiments (see Table 4) already far exceed the capability of the Jupiter VLBI experiment.

Table 4:

Current limits on the PPN parameters. Here $ηN$ is a combination of other parameters given by $η = 4β − γ − 3 − 10ξ∕3 − α + 2α ∕3 − 2ζ ∕3 − ζ ∕3 N 1 2 1 2$ .

Parameter	Effect	Limit	Remarks
$γ − 1$	time delay	2.3 $×$ 10^–5	Cassini tracking
	light deflection	4 $×$ 10^–4	VLBI
$β − 1$	perihelion shift	3 $×$ 10^–3	J₂ = 10^–7 from helioseismology
	Nordtvedt effect	2.3 $×$ 10^–4	$ηN = 4β − γ − 3$ assumed
$ξ$	Earth tides	10^–3	gravimeter data
$α1$	orbital polarization	10^–4	Lunar laser ranging
		2 $×$ 10^–4	PSR J2317+1439
$α2$	spin precession	4 $×$ 10^–7	solar alignment with ecliptic
$α 3$	pulsar acceleration	4 $×$ 10^–20	pulsar $P˙$ statistics
$ηN$	Nordtvedt effect	9 $×$ 10^–4	lunar laser ranging
$ζ 1$	—	2 $×$ 10^–2	combined PPN bounds
$ζ2$	binary acceleration	4 $×$ 10^–5	$¨Pp$ for PSR 1913+16
$ζ3$	Newton’s 3rd law	10^–8	lunar acceleration
$ζ4$	—	—	not independent (see Equation (58 ))