Figure 3.8 Excess travel time of light. An emitter on an enormous rigid ring sends light signals to a receiver on the opposite side. (a) The Sun is far to the left, so the travel time is the diameter of the ring divided by the usual speed of light. (b) The Sun is about to pass through the center of the ring, so the signal suffers an excess delay caused by curved spacetime near the Sun. (c) The Sun is far to the right, so the travel time returns to normal. (d) Plot of the travel time vs. time.
Whether or not we use the words “light slows down near the Sun” is purely a question of semantics. Because the observer on the ring who receives the light ray never goes near the Sun to make the measurement, she can’t really make such a judgment; and if she had made such a measurement in a freely falling laboratory near the Sun, she would have found the same value for the speed of light as in a freely falling laboratory far from the Sun, and might have thoroughly confused herself. All the observer can say with no fear of contradiction is that she observed an excess time of travel that depended on how close the light ray came to the Sun. The only sense in which it can be said that the light slowed down is mathematical: in a particular mathematical representation of the equations that describe the motion of the light ray, what general relativists call a particular coordinate system, the light appears to have a variable speed. But in a different mathematical representation (a different coordinate system), this statement might be false. Nevertheless, the observable quantities, such as the net time of travel, are the same no matter what representation is used. This is one of those cases in relativity where the careless use of words or phrases that are not based on observable quantities can lead to confusion or contradiction. We have already seen an example of this in our discussion in Chapter 2 of what “really” changed in the gravitational redshift, the clock rates or the signal frequency.
Given this discussion, you might be tempted to assume that Einstein derived this delay effect and proposed that it be measured. But he didn’t. The effect was derived by the radio astronomer Irwin I. Shapiro in 1964.
After receiving a Ph.D. in physics from Harvard University in 1955, Shapiro had been at MIT’s Lincoln Laboratory working on the problem of “radar ranging.” The idea was to bounce radar signals off planets such as Venus and Mercury and measure the round-trip travel times. This new technique had already led to improved determinations of the astronomical unit, the mean radius of the Earth’s orbit, and promised significant improvements in determining the orbits of the planets.
Shapiro had only a passing acquaintance with general relativity, and might not ever have considered it relevant to radar ranging had it not been for a lecture he attended in 1961 on measurements of the speed of light. Purely in passing, the speaker mentioned that according to general relativity, the speed of light is not constant. This statement puzzled Shapiro, because he had always thought that according to relativity, the speed of light should be constant. He knew, of course, that general relativity predicts that light should be deflected by a gravitating body, and following the same logic as we presented for the fish in water, he asked if its speed would also be affected.
To be fair, Einstein had considered the possibility of speed variation. Once he understood, from the principle of equivalence, that gravity could have an effect on light (the gravitational redshift), he attempted to construct a theory of gravity in which the speed of light would vary in the vicinity of a gravitating body. It was the equations from this specific theory that he used in 1911 to calculate the (one-half) bending of light. As we discussed on page 42, we’re talking here about the speed in a specific coordinate system.
However, for some reason Einstein did not take the next step, the one that Shapiro took. Shapiro consulted the classic general relativity textbook by Eddington and found that, according to the equations of the full general theory, the “effective” speed of light should indeed vary, just as it did in Einstein’s earlier model (in the full theory of general relativity the effect was doubled, just as it was for the deflection of light). Shapiro then applied these equations to the problem of the round trip of a radar signal to a distant object and found, in agreement with our qualitative argument, that the radar signal should take slightly longer to make the round trip than one would have expected on the basis of Newtonian theory and a constant speed of light. The additional delay would increase if the signal passed closer to the Sun.
In the solar system, the effect would be most noticeable when the target was on the opposite side of the Sun from the Earth, so that the signal would pass very near the Sun on its trip, as in panel (b) of Figure 3.8. Such a configuration is called superior conjunction (when both planets are on the same side of the Sun, it is called inferior conjunction). For example, Shapiro found that a radar signal sent from Earth to Mars at superior conjunction that just grazes the surface of the Sun suffers a round-trip delay of 250 millionths of a second (250 microseconds). Don’t forget that the total round-trip travel time for such a signal is about 42 minutes! So the idea here would be to detect an additional delay in the round trip of 250 microseconds on a total travel time of three-quarters of an hour. This might seem to be a hopeless proposition until you realize that the distance that light travels in 250 microseconds is 75 kilometers. So the delay represents an apparent shift in the distance to the target of half of this, or about 38 kilometers. Since Shapiro saw that radar ranging could potentially achieve a precision in distance between Earth and planets corresponding to a few kilometers, then perhaps this effect could be observed. The problem was that no radio telescopes at the time had the capability of sending a powerful enough radar signal to any planet at superior conjunction and detecting the extremely weak return signal. So Shapiro’s calculation lay in his desk for two years.
In the fall of 1964, two events caused Shapiro to retrieve his superior conjunction calculation and take it more seriously. The first was the completion of the Haystack radar antenna in Westford, Massachusetts. The second was the birth of his son on 30 October. As often happens in creative endeavors, the event in his personal life may have elevated him to a higher level of awareness or of mental activity, for soon thereafter, while describing the time delay idea to a colleague at a party, he suddenly realized that Haystack might be able to range to Mercury at superior conjunction and provide a means to test the time delay prediction (Mars would be too far away at superior conjunction for Haystack to record a measurable signal). Shapiro decided then to write up his superior conjunction calculation for Physical Review Letters. The paper was submitted in the middle of November, and published under the title “Fourth Test of General Relativity” in late December, 1964. (The first three tests were the gravitational redshift, the light deflection and the perihelion advance of Mercury, the three proposed by Einstein.) In time, the effect would come to be called the Shapiro time delay.
The principle behind the measurement of the time delay is very much the same as the principle behind the measurement of the deflection of light. Just as we could not measure the deflection of a single star, we cannot detect the time delay in a single radar shot. The reason, of course, is that we cannot “turn off” the gravitational field of the Sun in order to see what the star’s “true” position is or to see what the “flat spacetime” round-trip travel time would have been. To get at the deflection, we had to compare the position of a star or quasar relative to other stars or quasars both when its light passed far from the Sun, and when its light passed very near the Sun. By the same token, to see the time delay, we must compare the round-trip travel time of a radar signal to the planet when the signal passes far from the Sun with that when the signal passes close to the Sun.
When the signal to the planet passes far from the Sun, the Shapiro time delay is relatively small, and the round-trip travel time is closer to being a measure of the “true” distance. This corresponds to the situation in Figure 3.4 where the signal traverses a portion of space that is virtually flat. As the planet moves into superior conjunction, however, and the signal passes closer a
nd closer to the Sun, the Shapiro time delay becomes a larger contribution to the round-trip travel time.
However, even though the radar signal may go near the Sun, the planet itself never does. Its orbit is well away from the Sun, on the order of 230 million kilometers for Mars or 58 million kilometers for Mercury, for instance. Because of this, the planet always moves through a region of low spacetime warpage, and maintains a relatively low velocity; therefore, the relativistic effects on its orbit are small. To the accuracy desired for a time delay measurement, its orbit can be described quite adequately by standard Newtonian gravitational theory. Therefore, even though the planet moves during the experiment, its motion can be predicted accurately. Because of this circumstance, the time delay can be measured in four steps: (1) by ranging to the planet for a period of time when the signal stays far from the Sun, determine the parameters that describe its orbit at that time; (2) using the orbit equations of Newtonian theory, including the perturbations from all the other planets, make a prediction of its future orbit and that of the Earth, including especially the period of superior conjunction where the action will occur; (3) using the predicted orbit, calculate the round-trip travel times of signals to the planet assuming no Shapiro time delay; and (4) compare these predicted round-trip travel times with those actually observed during superior conjunction, attribute the difference to the Shapiro time delay, and see how well it agrees with the prediction of general relativity.
Within about a month of submitting his paper on the time-delay effect to Physical Review Letters, Shapiro’s colleagues at Lincoln Laboratory set out to upgrade the Laboratory’s Haystack radar by increasing its power fivefold and by making other electronic improvements. This would give them the capability to get a decent echo from Mercury and also Venus at superior conjunction, and to measure the round-trip travel times to within 10 microseconds. By late 1966 the improved system was ready, just in time for the 9 November superior conjunction of Venus. Unfortunately, Venus goes through superior conjunction only about once every year and a half, so after observing Venus they then turned the radar sights on Mercury. Because Mercury orbits the Sun almost three times faster than Venus, it has a superior conjunction more often, about three times per year, giving more opportunities to measure the time delay. Measurements were made during the 18 January, 11 May and 24 August 1967 conjunctions of Mercury. All told, over four hundred radar “observations” were used. Most of these measurements (the ones not taken near superior conjunction of either of the planets) were combined with existing optical observations of Mercury and Venus available through the US Naval Observatory to accomplish the first step of the method, namely, to establish accurate orbits for the two planets. The remaining radar measurements centered around the superior conjunctions were then used to compare the predicted time delays with the observed time delays (because of large amounts of noise, the Venus data turned out not to be very useful). The results using Mercury data agreed with general relativity to within 20 percent. The first new test of Einstein’s theory since 1915 was a reality.
But the story does not end there. During the summer of 1965, while Shapiro and his colleagues were busy working on the Haystack radar, a US spacecraft hurtled past Mars, the first man-made object to encounter the “red planet.” The spacecraft was Mariner 4, and on its way by the planet it took twenty-one pictures and examined the Martian atmosphere using radio waves. Buoyed by the success of Mariner 4, NASA in December 1965 authorized two more missions to Mars, Mariner 6 and 7 in 1969 (Mariner 5 was a Venus mission) and Mariner 8 and 9 in 1971, and planners began to think seriously about Martian landers. While these missions would bring planetary exploration to a zenith, at least temporarily, they would also have crucial consequences for general relativity.
At the Jet Propulsion Laboratory (JPL) in Pasadena, California, where the Mariner program was headquartered, the relativistic time delay was also on people’s minds, and they began to wonder if there was any way to make use of Mariner 6 and 7 to measure the time delay. In fact, two JPL scientists, Duane Muhleman and Paul Reichley, had calculated the delay effect of general relativity on radar propagation independently of Shapiro, although they only published the results in internal JPL reports. There was no reason in principle why a measurement of the delay should not be possible. Other than in size, there is no fundamental difference between a planet and a spacecraft. The orbit of the spacecraft can be determined by tracking, and its trajectory during superior conjunction can be predicted, just as for ranging to the planet, and the time delay of the radar ranging signal during superior conjunction can be measured and compared with the prediction of general relativity.
Mariners 6 and 7 were launched on 24 February and 27 March 1969, and reached Mars by the end of July. Both spacecraft performed their primary tasks of observing Mars’ surface and atmosphere beautifully, and then left the planet to go into orbit around the Sun. Between December 1969 and the end of 1970, several hundred range measurements were made to each spacecraft, with the heaviest concentration, involving almost daily measurements, around the time of each superior conjunction—on 29 April 1970 for Mariner 6 and on 10 May for Mariner 7. Neither spacecraft actually went behind the Sun. Because of the tilt of their post-Martian orbits, they both passed by the Sun slightly to the north, Mariner 6 about 1° away, Mariner 7 about 1.5° away, as seen from Earth. For Mariner 6, the distance of closest approach of the radar signal at superior conjunction was about 3.5 solar radii, corresponding to a Shapiro time delay of 200 microseconds out of a total round-trip travel time of 45 minutes. For Mariner 7, the radar signals came no closer than about 5.9 solar radii, giving a slightly smaller time delay of 180 microseconds. After feeding all the observations into the computer, they found that the measured delays agreed with the predictions of general relativity to within 3 percent. This was a dramatic improvement over the 20 percent figure from Venus and Mercury ranging.
Of course, the planetary radar ranging people at Lincoln Laboratory had not been idle since 1967. They had continued to make radar observations of Mercury and Venus using both the Haystack antenna and the Arecibo radio telescope in Puerto Rico. In fact, during late January and early February 1970, while the JPL rangers were busy getting distances to the Mariner spacecraft on their approach to superior conjunction, Venus passed through its own superior conjunction, bombarded almost twice a week by radar signals from Haystack and Arecibo. Data from that Venus conjunction, and from the numerous Mercury conjunctions between 1967 and the end of 1970, once again yielded relativistic time delays in agreement with general relativity, this time at the 5 percent level.
It was soon realized, however, that each method, planetary vs. spacecraft tracking, had advantages and disadvantages. One advantage of planets is that they are massive and therefore are completely unaffected by the constant bombardment of the solar wind and solar radiation pressure. Spacecraft, by contrast, are light and have large antennae and solar panels, and so they tend to get jostled around a lot on their way through the rough neighborhood of interplanetary space. This is important because of the need to predict the orbit accurately during the time of superior conjunction when range measurements are supposed to yield the Shapiro delay.
An advantage of spacecraft is that they receive the radar signal from Earth, pass it through a transponder (the same device we encountered in Chapter 2), which boosts the signal’s power and beams it right back to Earth, leading to very accurate round-trip travel times. By contrast, planets are poor reflectors of radar beams, and also have valleys and mountains that introduce uncertainties in the “true” round-trip travel time.
The way to combine the transponding capabilities of spacecraft with the imperturbable motions of planets was to anchor a spacecraft to a planet, by having the spacecraft orbit the planet, or even better by letting the spacecraft land on the planet.
The first anchored spacecraft was Mariner 9, the orbiter that reached Mars in November 1971 just in time to photograph the raging dust storm that obliterated most of t
he planetary surface for several weeks. The next Martian superior conjunction of 8 September 1972 gave a confirmation of the Einsteinian time delay to 2 percent, only a modest improvement over the previous results, but enough to prove the power of the anchoring idea.
And then came Viking. The Viking landers on Mars were a spectacular achievement for planetary exploration, with their close-up views of the Martian surface, their analyses of the atmosphere, and their search for signs of life in the Martian soil. But to the general relativist they were even more beautiful, for they were the perfect anchored spacecraft for the time delay experiment. After a ten-month voyage, the first Viking spacecraft reached Mars in mid June, 1976. After several weeks studying possible landing sites, Lander 1 was detached from the orbiter and descended to a plain called Chryse on 20 July. Eighteen days later, the second Viking reached Mars, and on 3 September, Lander 2 dropped to the surface in a region called Utopia Planitia.
While much of the world focused its attention on the remarkable photographs and scientific data radioed back to Earth by the landers and orbiters, Shapiro’s group at MIT and the JPL team, now working together, began to prepare for the 26 November superior conjunction. With two landers and two orbiters all providing ranging data, they had an excellent configuration of anchored spacecraft so as to avoid the errors of random orbit perturbations.
Is Einstein Still Right? Page 9