Is Einstein Still Right?

by Clifford M. Will

  There was one more surprise. The radio pulses from the primary pulsar A were known to be partially eclipsed by the passage of the secondary pulsar B across the line of sight, and the on-off flickering of the signal during the eclipse occurred on a roughly 3 second timescale, the same as the rotation period of pulsar B. This was not because the signal hit the actual neutron star, but because it passed through the highly charged “magnetosphere” of the companion. This is a region of strong magnetic fields and charged particles that wraps around the magnetic equator of the neutron star, and is much larger in diameter than the star itself. It is shaped more like a donut or bagel, with the neutron star residing in the middle of the hole (see Figure 5.5). The pulsar beam from B propagates along the magnetic poles in a direction perpendicular to the plane of the donut. It is essentially the same as the magnetosphere of the Earth, except that the Earth’s magnetosphere is strongly distorted by the solar wind. The dense cloud of charged particles that are trapped within the magnetosphere can absorb and scatter radio waves as effectively as the solid material of the neutron star itself. So, as pulsar A passes behind pulsar B during the orbital motion, its signal can be absorbed or not by the magnetosphere of pulsar B, depending on the orientation of the pulsar at that moment. This model beautifully explained the pattern of eclipses, except for one thing. Over many orbits of the system, the detailed shape of the eclipse pattern drifted with time, contrary to what the model predicted.

  Figure 5.5 Pulsar B rotates about its spin axis with a period of 2.8 seconds, with its magnetic axis (which produces the pulsar beam) and donut-shaped magnetosphere (in gray) not aligned with the spin axis. Left: Pulsar A passes behind pulsar B, but the signal directed out of the page misses the magnetosphere. Right: After half a rotation of pulsar B, the magnetosphere now blocks the signal from pulsar A, causing an “eclipse.”

  General relativity gave the solution. As we saw in Chapter 4, the spin of an object should change direction slowly with time as the object moves through the curved spacetime of another body. This effect, known as the “geodetic precession,” was one of the two effects verified by the Gravity Probe-B experiment. If the spin direction of the pulsar changes slowly with time, then, as can be seen in Figure 5.5, the signals from pulsar A might encounter a fatter or thinner part of the magnetosphere as time goes on. Incorporating this effect into the model gave complete agreement with the eclipse data, including the long-term changes, and the amount of geodetic precession of the spin needed to fit the data agreed with the prediction of general relativity. Sadly, general relativity also killed the “double” pulsar, because by 2008, the spin had precessed so much that the beam of pulsar B no longer passed across the Earth, and the system became a “single” binary pulsar. Pulsar B may reappear in 2035 when the wobbling spin brings its beam back into alignment with the Earth.

  The second stand-out pulsar was discovered in 2014 by Scott Ransom and collaborators, using the Green Bank radio telescope in West Virginia. But instead of being in a binary system, this pulsar was in a triple star system, with two white dwarf companions. This is not as unlikely as it may seem. While the numbers are uncertain, as many as a fifth of the stars in the Milky Way could be in triple star systems. Alpha Centauri, our nearest neighbor, and Polaris, the North Star, are actually in triple systems, with two stars in a fairly close binary orbit and a more distant third star orbiting the pair. This new system is similar in that it consists of a close inner binary and a distant third star. The pulsar has a spin period of 2.73 milliseconds, and a mass of 1.44 solar masses. It is orbited once every 1.6 days by a white dwarf of only one fifth of a solar mass. This pair is orbited by a 0.4 solar mass white dwarf, with an orbital period of 327 days. Both orbits are almost perfect circles, and lie on the same plane (see Figure 5.6).

  Figure 5.6 Pulsar in a triple system. Left: The neutron star is in a 1.6 day orbit with a low-mass white dwarf. Right: This inner binary is in a 327 day orbit with the other white dwarf.

  Everything about this system made it pretty useless for testing general relativity in the same manner as either the Hulse–Taylor binary or the double pulsar. The motions were too slow and the orbital separations too large for relativistic effects on the orbits to be very interesting. The orbits were too circular for the periastrons to be even located, let alone for their advances to be measured. The mass of the inner white dwarf was too small for the gravitational redshift of the pulsar signal to be measurable. The inclination of the orbits relative to the plane of the sky was found to be about 40 degrees, so the radio signal from the pulsar would pass nowhere near either white dwarf, making the Shapiro delay negligible. And finally, the emission of gravitational radiation would be far too feeble for the decrease in the orbital periods to be detectable.

  So why was this discovery so fantastic for general relativity? The answer goes back to the founding idea of general relativity, the principle of equivalence. In Chapter 2 we discussed how Einstein’s insight about the equivalence between gravity and acceleration led him to a curved spacetime picture for gravitation. This was based upon the observational fact that bodies fall with the same acceleration, independently of their internal structure or composition. This idea is usually called the “weak equivalence principle,” and was already appreciated in the ancient world. In 400 ce, Ioannes Philiponus wrote “… let fall from the same height two weights, of which one is many times as heavy as the other … the difference in time is a very small one.” Even before Galileo, this principle had been expounded and tested in the 1500s by Giambattista Benedetti and Simon Stevin, and if Galileo actually did drop things from the Leaning Tower of Pisa during his time there between 1589 and 1592, he was probably just demonstrating to his students what was by then a well-known concept. Even Isaac Newton carried out experiments using pendula to test this equality. As we noted in Chapter 2, the challenge of testing the weak equivalence principle to high precision was taken up by Eötvös at the turn of the twentieth century. Because this principle is so crucial for the foundation of Einstein’s theory, the effort to test it to higher and higher precision has continued to the present day. The current state of the art comes from two sources. One is a series of experiments carried out by a group headed by Eric Adelberger at the University of Washington in Seattle, called the “Eöt-Wash” experiments, that show that different materials fall with the same acceleration to a few parts in ten trillion. The other is a space experiment called MICROSCOPE, launched in 2016 by the French Space Agency, that yielded a limit of parts in a hundred trillion.

  One interesting and important conclusion can be drawn from these results. Recall that the mass of an atomic nucleus is made up of the masses of the individual neutrons and protons, but that’s not all. These neutrons and protons are held together by the strong forces that bind the nucleus. Einstein has taught us through special relativity that energy and mass are different manifestations of the same thing. Therefore, the mass of the atomic nucleus is made up of the sum of the masses of the individual neutrons and protons, and the “mass” associated with the binding energy. Now, since different elements have different amounts of nuclear internal energy per unit mass, and since experiments tell us that the different kinds of nuclei fall with the same acceleration, then the energy of the nuclear forces must “fall” with the same acceleration as do the nuclear particles themselves. A similar conclusion applies to the electromagnetic energy associated with the forces between the charged protons and electrons. So it would seem that not only do the fundamental building blocks of matter, such as protons, neutrons and electrons, fall with the same acceleration, but so do the various forms of energy associated with their interactions with each other, such as nuclear, electromagnetic and weak interactions.

  But the standard model of fundamental particles tells us that there is a fourth interaction: gravitation. What about the energy associated with it? Does gravitational energy fall with the same acceleration as matter and the other forms of energy? The experiments we have just described don’t provide a
n answer, because the internal gravitational energy of the laboratory-scale bodies employed in those experiments is utterly negligible. To get a meaningful amount of gravity, you need a large amount of mass, and therefore to test the equivalence principle including gravitational energy you need objects like planets or stars.

  The first person to contemplate this possibility was Kenneth Nordtvedt. Born in Chicago, he received an undergraduate degree from MIT, took a Ph.D. degree at Stanford University, and had post-doctoral research positions back in the Boston area at Harvard and at MIT. But by 1965 he had developed a dislike for the lifestyle and politics of big cities, especially on either of the coasts, and had resolved to head for the heartland of America. When offered an assistant professorship at the then small Montana State University in tranquil and beautiful Bozeman, he accepted readily and headed west to begin his academic career in earnest.

  Although his Ph.D. thesis was in solid-state physics, around 1967 he turned his attention to gravity and asked whether a massive body with its own internal gravity, such as the Earth, would fall in an external gravitational field with the same acceleration as, say, a ball of lead. To try to answer this question, Nordtvedt devised a way of treating the motion of planetary-size bodies that would be valid in any curved spacetime theory of gravity, or at least in a broad class of such theories. The equations he developed could encompass general relativity, the Brans-Dicke theory, then the leading alternative theory to Einstein’s, and many others, in one fell swoop. To find the actual prediction of a chosen theory, such as general relativity, all one had to do was to specialize the equations by fixing the numerical values of certain coefficients that appeared in them. The calculations were complicated, with many, many terms in the final equation describing the acceleration of a massive body, but when all was said and done, two remarkable results emerged.

  First, when the equations were specialized to general relativity there was a tremendous cancelation of terms, and the result was that different massive bodies would have exactly the same acceleration, regardless of how much internal gravity they possessed. Therefore, in general relativity the acceleration of gravitationally bound bodies was predicted to be the same as that of laboratory-size bodies. This beautiful prediction of general relativity, the equivalence of acceleration of bodies from the smallest to the largest sizes, is sometimes called the strong equivalence principle. Later research would show that this equivalence also applies to neutron stars and even black holes.

  There was another remarkable result of Nordtvedt’s calculations. In most other theories of gravity, including that of Brans and Dicke, the complete cancelation did not occur and a small difference in acceleration remained, depending on how strongly bound by internal gravity the bodies were. Therefore, even though these theories guaranteed that laboratory-size bodies fall with the same acceleration, satisfying the weak equivalence principle, as soon as one considered bodies with significant amounts of self-gravitational binding, the bodies would fall differently. In other words, in such theories, gravitational energy falls at a slightly different rate than mass and other forms of energy, such as nuclear energy, electromagnetic energy and so on. Thus, theories such as the Brans–Dicke theory were compatible with the weak equivalence principle, but not compatible with the strong equivalence principle. Today this is called the Nordtvedt effect.

  Nordtvedt then proposed to search for this effect in the motion of the Moon. Consider the acceleration of the Earth and the Moon in the field of the Sun (see Figure 5.7). The gravitational energy per unit mass of the Moon is about one twenty-fifth that of the Earth, so they could in principle fall with different accelerations because the Earth is more tightly bound by its own gravity than is the Moon. Suppose, for the sake of argument, that the Moon falls with a slightly larger acceleration than the Earth (whether it is larger or smaller depends on the theory of gravity). The Moon orbits the Earth, but is being accelerated toward the Sun slightly more strongly than the Earth is; therefore, on each succeeding orbit the Moon is pulled a little closer to the Sun. What started out as a nearly circular orbit becomes elliptical, and on each orbit the ellipse becomes more and more elongated toward the Sun, until the Moon is pulled catastrophically from the hold of the Earth and plunges with a great splash into the Sun. Is the Nordtvedt effect a lunar calamity? Actually not, because we have forgotten an important fact: the Sun is in orbit about the Earth (as seen from the Earth’s frame, of course). Thus, just as the Moon’s orbit is elongated toward the Sun on one revolution of the Moon, by the next lunar revolution, 27 days later, the Sun has moved by about 27 degrees in its orbit (the Sun’s rate of revolution about the Earth is 360 degrees in 365 days, or about 1 degree per day), so on the next revolution, the elongation must occur in a direction toward the new position of the Sun. On the following revolution of the Moon, the elongation must be directed toward a still newer position, and so on. Therefore, instead of building up to a disastrous size, the elongation of the Moon’s orbit that would be caused by the Nordtvedt effect maintains a fixed size, but is always oriented with its long axis toward the Sun. If the Moon were predicted to fall with a slightly smaller acceleration than the Earth, then the elongation would be in the opposite direction, with its long axis directed away from the Sun. If, as in general relativity, the two fall with the same acceleration, there would be no predicted elongation of this sort.

  Figure 5.7 Lunar catastrophe or orbit perturbation? If the Moon fell with larger acceleration than the Earth toward the Sun, its orbit would become progressively more elongated until it was pulled into the Sun. But because the Earth–Sun orientation is changing because of the Earth’s orbital motion, the elongation never builds up and instead merely produces a shifted orbit that always points toward the Sun (solid curves). In general relativity, the elongation does not occur at all (dashed curves).

  The crucial question is how large this effect might be. When Nordtvedt put in all the numbers, he found that the size of the elongation in the Brans–Dicke theory, for instance, could be as large as 1.3 meters, or about 4 feet. In general relativity, of course, the effect was zero. While this may seem like a ridiculously small effect, it would soon become eminently measurable.

  After Apollo 11 astronaut Neil Armstrong took his first step on the Moon on 21 July 1969, he had a number of tasks to perform, one of which was to walk a few hundred meters from the lander and place on the lunar surface a device called a “retroreflector,” an early version of the retroreflectors used on the LAGEOS satellites (page 75). This was a flat surface embedded with cube corner reflectors that could take a laser beam sent from Earth and reflect it back in the same direction from which it came. One could then send a short laser pulse from Earth, have it bounce off the retroreflector and return to Earth. Measuring the round-trip travel time of the pulse would give a measure of the Earth–Moon distance, with a precision that was expected to be of the order of 100 centimeters, easily sufficient to look for a possible Nordtvedt effect. Within a week and a half of deployment of the Apollo 11 retroreflector, astronomers at Lick Observatory in California had succeeded in bouncing laser pulses off it, and measuring the round-trip travel time to a precision corresponding to several meters. Subsequently, four other retroreflectors were placed on the Moon, two US devices, deployed by astronauts during Apollo 14 and 15, and two French-built reflectors, deposited during the Soviet unmanned missions Luna 17 and 21. By 1975, analyses of the laser ranging data showed absolutely no evidence of the Nordtvedt effect, to a precision of 30 centimeters, to the delight of Nordtvedt and anybody who feels that general relativity is correct. As Nordtvedt was fond of saying, “scientifically, zero can be just as important a number as any other.”

  Today, lunar laser ranging is carried out at over forty observatories worldwide, with ranging precisions in the millimeter regime, yielding important science about the Earth–Moon orbit, the rotation of the Moon, continental drift on Earth, and even whether Newton’s constant of gravitation is constant in time (it is, to the uncertainty in t
he measurement, which is a few parts in ten trillion per year). And recent analyses have continued to show no evidence of the Nordtvedt effect. One way to summarize the results is to state that the Earth and Moon fall toward the Sun with the same acceleration to a few parts in ten trillion, comparable to the limits achieved by tests of the weak equivalence principle, such as the Eöt-Wash and MICROSOPE measurements.

  The pulsar in a triple system carried this test of the Nordtvedt effect into a new realm of extreme gravity. The system is a variant of the Earth–Moon–Sun system, with the inner neutron star / white dwarf binary substituting for the Earth and Moon, and the outer white dwarf substituting for the Sun. The mass relationships are different, with the Sun dominating the masses in the solar system case, and the neutron star dominating in the pulsar case. But the question is the same: do the neutron star and its white dwarf companion fall with the same acceleration toward the distant white dwarf?

  The crucial difference is this. Whereas the internal gravitational binding energy of the Earth and the Moon represent only about a billionth of their total mass, the gravitational energy of a neutron star represents as much as 15 percent of its total mass. In other words, if you could somehow go into the neutron star and count up all the protons, neutrons, electrons and any other exotic particles that you might find, multiply each by its mass and add it all up, you would get something like 1.6 solar masses. The actual measured mass is 1.4 solar masses. The difference of about 0.2 solar masses, multiplied by the square of the speed of light, is the gravitational binding energy, and is in fact a negative number.1 This is analogous to the phenomenon by which the mass of a helium nucleus is slightly smaller than the mass of four hydrogen nuclei (protons), so that when four protons fuse to form helium in the Sun, that mass difference becomes the energy on which we rely (in this example, the binding energy comes from the strong nuclear interactions). The binding energy of a typical white dwarf is parts in ten thousand of its mass, much larger than that of the Earth or Moon, but much smaller than that of a neutron star.