The Renaissance of General Relativity
Clifford M. Will
McDonnell Center for the Space Sciences, Washington University
During the two decades from 1960 to 1980, the subject of general relativity experienced a rebirth. Despite its enormous influence on scientific thought in its early years, by the late 1950s general relativity had become a sterile, formalistic, cut off from the mainstream of physics. It was thought to have very little observational contact, and was believed to be an extremely difficult subject to learn and comprehend.
Yet by 1970, it had become one of the most active and exciting branches of physics. It took on new roles both as a theoretical tool of the astrophysicist and as a playground for the elementary-particle physicist. New experiments verified its predictions in unheard-of ways and to remarkable levels of precision.
Fields of study were created, such as "black-hole physics" and "gravitational-wave astronomy," that brought together the efforts of theorists and experimentalists. One of the most remarkable and important aspects of this renaissance was the degree to which experiment and observation motivated and complemented theoretical advances.
This was not always the case. In deriving general relativity during the final months of 1915, Einstein himself was not particularly motivated by a desire to account for observational results. Instead, he was driven by purely theoretical ideas of elegance and simplicity. His goal was to produce a theory of gravitation that incorporated in a natural way both the special theory of relativity that dealt with physics in inertial frames, and the principle of equivalence, the proposal that physics in a frame falling freely in a gravitational field is in some sense equivalent to physics in an inertial frame.
Once the theory was formulated, however, he did try to confront it with experiment by proposing three tests. One of these tests was an immediate success--the explanation of the anomalous advance in the perihelion of Mercury of 43 arcseconds per century, a problem that had bedeviled celestial mechanicians of the latter part of the 19th century. The next test, the deflection of light by the Sun, was such a success that it produced what today would be called a "media event." The measurements of the deflection, amounting to 1.75 arcseconds for a ray that grazes the Sun, by two teams of British astronomers in 1919, made Einstein an instant international celebrity. However, these measurements were not all that accurate, and subsequent measurements weren't much better. The third test, actually proposed by Einstein in 1907, was the gravitational redshift of light, remained unfulfilled until 1960, by which time it was no longer viewed as a true test of general relativity.
Cosmology was the other area where general relativity was believed to have observational relevance. Although the general relativistic picture of the expansion of the universe from a "big bang" was compatible with observations, there were problems. As late as the middle 1950s the measured values of the universal expansion rate implied that the universe was younger than the Earth! Even though this "age" problem had been resolved by 1960, cosmological observations were still in their infancy, and could not distinguish between various alternative models.
The turning point for general relativity came in the early 1960s, when discoveries of unusual astronomical objects such as quasars demonstrated that the theory would have important applications in astrophysical situations. Theorists found new ways to understand the theory and its observable consequences. Finally, the technological revolution of the last quarter century, together with the development of the interplanetary space program, provided new high-precision tools to carry out experimental tests of general relativity.
After 1960, the pace of research in general relativity and in an emerging field called "relativistic astrophysics" began to accelerate. New advances, both theoretical and observational, came at an ever-increasing rate. They included the discovery of the cosmic background radiation; the analysis of the synthesis of helium from hydrogen in the big bang; observations of pulsars and of black-hole candidates; the developments of the theory of relativistic stars and black holes; the theoretical study of gravitational radiation and the beginning of an experimental program to detect it; improved versions of old tests of general relativity, and brand new tests, discovered after 1960; the discovery of the binary pulsar, which provided evidence for gravity waves; the discovery of a gravitational lens; and the beginnings of a unification of gravitation theory with other interactions and with quantum mechanics.
During the two decades following 1960, general relativity rejoined the world of physics and astronomy. Research in relativity took on an increasingly interdisciplinary flavor, spanning such subjects as celestial mechanics, pure mathematics, experimental physics, quantum mechanics, observational astronomy, particle physics, and theoretical astrophysics.
Einstein's Equivalence Principle
The foundations of general relativity are actually quite old, dating back to the equivalence principle of Galileo and Newton. In Newton's view, the principle of equivalence stated that all objects accelerate at the same rate in a gravitational field regardless of their mass or composition. This equality has been verified abundantly over the years, including classic experiments by the Hungarian physicist Baron Lorand von EÖtvÖs around 100 years ago and recent experiments at Princeton and Moscow State Universities. The accuracy of these tests is better than one part in 1011. Einstein's insight was the recognition that, to an observer inside a freely falling laboratory, not only should objects float as if gravity were absent as a consequence of this equality, bus also all laws of nongravitational physics, such as electromagnetism and quantum mechanic, should behave as if gravity were truly absent.
This idea is now known as the Einstein equivalence principle, and it was a key step, because it implied the converse: that in a reference frame where gravity is felt, such as in a laboratory on the Earth's surface, the effects of gravitation on physical laws can be obtained simply by mathematically transforming the laws from a freely falling frame to the laboratory frame. According to the branch of mathematics known as differential geometry, this is the same as saying that space-time is curved; in other words, that the effects of gravity are indistinguishable from the effects of being in curved space-time.
Gravitational Redshift
An immediate consequence of Einstein's insight is the gravitational redshift effect. It is not a consequence of general relativity itself, although Einstein believed that it was. One version of a gravitational redshift experiment measures the frequency shift between two identical clocks (meaning any device that produces a signal at a well-defined, steady frequency), placed at rest at different heights in a gravitational field. To derive the frequency shift from the equivalence principle, one imagines a freely falling frame that is momentarily at rest with respect to one clock at the moment the clock emits its signal. Because special relativity is valid in that frame, the frequency of the signal is unaffected by gravity as it travels from one clock to the other. However, by the time the signal reaches the second clock, the frame has picked up a downward velocity because it is in free fall in the gravitational field, and therefore, as seen by the falling frame, the second clock is moving upward. Thus the frequency seen by the second clock will appear to be shifted from the standard value by the Doppler effect. For small differences in height h between clocks, the shift in the frequency ∆f is given by
where g is the local gravitational acceleration and c is the speed of light. If the receiver is at a lower height than the emitter, the received signal is shifted to higher frequencies ("blueshift"); if the receiver is higher, the signal is shifted to lower frequencies ("redshift"). The generic name for the effect is "gravitational redshift."
The first and most famous high-precision redshift measurements was the Pound-Rebka experiment of 1960, which measured the frequency shift of gamma-ray photons from the decay of iron-57 (57Fe) as they ascended or descended the Jefferson Physical Laboratory tower at Harvard University.
The most precise gravitational redshift experiment performed to date was a rocket experiment carried out in June 1976. A "hydrogen maser" atomic clock was flown on a Scout D rocket to an altitude of 10 000 km, and its frequency compared to a similar clock on the ground using radio signals. After the effects of the rocket's motion were taken into account, the observations confirmed the gravitational redshift to 0.02%.
Deflection of Light
The first of these is a test that made Einstein's name a household word; the deflection of light. According to general relativity, a light ray which passes the Sun at a distance d is deflected by an angle ∆q = 1."75/d, where d is measured in units of the solar radius, and the notation '' denotes seconds of arc (See Fig. 2). Half of the deflection can be calculated by appealing only to the principle of equivalence. The idea is to determine what observers in a sequence of free-falling frames all along the trajectory of the photon would find for the deflection from one frame to the next. It turns out that the net deflection over all the frames is 0." 875/d. The same result can be obtained by calculating the deflection of a particle by a massive body using ordinary Newtonian theory, and then taking the limit in which the particle's velocity becomes that of light. But the result of both versions is the deflection of light only relative to local straight lines, as defined for example by rigid rods laid end-to-end. However, because of the curvature of space, a local straight line defined by rods passing close to the Sun is itself bent relative to straight lines far from the Sun by an amount that yields the remaining half of the deflection.
The prediction of the bending light by the Sun was one of the great successes of general relativity. Confirmation by the British astronomers Eddington and Crommelin of the bending of optical starlight observed during a total solar eclipse in the first months following World War I helped make Einstein a celebrity. However, those measurements had only 30% accuracy, and succeeding eclipse experiments weren't much better; the results were scattered between one half and twice the Einstein value, and the accuracies were low.
However, the development of long-baseline radio interferometry produced a method for greatly improved determinations of the deflection of light. Radio interferometry is a technique of combining widely separated radio telescopes in such a way that the direction of a source of radio waves can be measured by determining the difference in phase of the signal received at the different telescopes. Modern interferometry has the ability to measure angular separations and changes in angles as small as 10-4 arcseconds. Coupled with this technological advance is a series of heavenly coincidences: each year groups of strong quasars pass near the Sun (as seen from the Earth). The idea is to measure the differential deflection of radio waves from one quasar relative to those from another as they pass near the Sun. A number of measurements of this kind occurred almost annually over the period from 1969 to 1975, yielding a confirmation of the predicted deflection to 1.5%.
The 1979 discovery of the "double" quasar Q0957 + 561 converted the deflection of light from a test of relativity to an important effect in astronomy and cosmology. The "double" quasar was found to be a multiple image of a single quasar caused by the gravitational lensing effect of a galaxy or a cluster of galaxies along the line of sight between us and the quasar. Several other such lenses have been found.
Closely related to light deflection is the "Shapiro time delay," a retardation of light signals that pass near the Sun. For instance, for a signal that grazes the Sun on a round trip from Earth to Mars at superior conjunction (when Mars is on the far side of the Sun), the round trip travel time is increased over what Newtonian theory would give about 250 m s. The effect decreases with increasing distance of the signal from the Sun.
In the two decades following radio-astronomer Irwin Shapiro's 1964 discovery of this effect, several high-precision measurements have been made using the technique of radar-ranging to planets and spacecraft. Three types of targets were employed; planets such as Mercury and Venus, free-flying spacecraft such as Mariners 6 and 7, and combinations of planets and spacecraft, known as "anchored spacecraft," such as the Mariner 9 Mars orbiter and the 1976 Mars landers and orbiters. The Viking experiments produced dramatic results, agreeing with the general relativistic prediction to one part in a thousand. This corresponded to a measurement accuracy in the Earth-Mars distance of 30 meters.
Perihelion Shift
The explanation of the anomalous perihelion shift of Mercury's orbit was an early triumph of general relativity (See Figs. 3 and 4). This had been an unsolved problem in celestial mechanics for over half a century, since the announcement by Le Verrier in 1859 that, after the perturbing effects of the planets on Mercury's orbit had been accounted for, there remained in the data an unexplained advance in the perihelion of Mercury. The modern value for this discrepancy is 42.98 arcseconds per century. Mandy ad hoc proposals were made in an attempt to account for this excess, including the existence of a new planet Vulcan near the Sun, or a deviation from the inverse square law of gravitation. But each was doomed to failure. General relativity accounted for the anomalous shift in a natural way. Radar measurements of the orbit of Mercury since 1966 have led to improved accuracy, so that the relativistic advance is known to about 0.5%.
Other measurements carried out since 1960 have given further support to general relativity. Observations of the motion of the Moon using laser ranging to a collection of specially designed reflectors, deposited on the lunar surface during the Apollo and Luna missions, have shown that the Moon and the Earth accelerate toward the Sun equally to a part in 1011, and important confirmation of the equivalence principle applied to planetary bodies. Other measurements of planetary and lunar orbits have shown that the gravitational constant is a true constant of nature; it does not vary with time as the universe ages. Ambitious experimenters are currently designing an experiment using supercooled quartz gyroscopes to be placed in Earth orbit, to try to detect an important general relativistic effect called the "dragging of inertial frames," caused by the rotation of the Earth.
General relativity has passed every experimental test to which it has been put, and many alternative theories have fallen by the wayside. Most physicists now take the theory for granted, and look to see how it can be used as a practical tool in physics and astronomy.
Gravitational Radiation
One of these new tools is gravitational radiation, a subject almost as old as general relativity itself. By 1916, Einstein had succeeded in showing that the field equations of general relativity admitted wave-like solutions analogous to those of electromagnetic theory. For example, a dumbbell rotating about an axis passing at right angles through its handle will emit gravitational waves that travel at the speed of light. They also carry energy away from the dumbbell, just as electromagnetic waves carry energy away from a light source. That was about all that was heard on the subject for over 40 years, primarily because the effects associated with gravitational waves were extremely tiny, unlikely (it was thought) ever to be of experimental or observational interest.
But in 1968 the idea of gravitational radiation was resurrected by the stunning announcement by Joseph Weber that he had detected gravitational radiation of extraterrestrial origin by using massive aluminum bars as detectors. A passing gravitational wave acts as an oscillating gravitational force field that alternately compresses and extends the bar in the lengthwise direction. However, subsequent observations by other researchers using bars with sensitivity that was claimed to be better than Weber's failed to confirm Weber's results. His results are now generally regarded as a false alarm, although there is still no good explanation for the "events" that he recorded in his bars.
Nevertheless, Weber's experiments did initiate the program of gravitational-wave detection, and inspired other groups to build better detectors. Currently a dozen laboratories around the world are engaged in building and improving upon the basic "Weber bar" detector, some using bigger bars, some using smaller bars, some working at room temperature, and some working near absolute zero. The goal in all cases is to reduce noise from thermal, electrical, and environmental sources in order to detect the very weak oscillations produced by a gravitational wave. For a bar of one meter length, the challenge is to detect a variation in length smaller than 10-20 meters, or 10-5 of the radius of a proton.
Another important type of detector is the laser interferometer, currently under development at several laboratories. This device operates on the same principle as the interferometer used by Michelson and Morley, but uses a laser as the light source. A passing gravitational wave will change the length of one arm of the apparatus relative to the other, and cause the interference pattern to vary. One ambitious proposal is to build two independent interferometers, each with arms lengths of 4 km, and separated by 1000 km.
It is hoped that these detectors will eventually be sensitive enough to detect gravitational waves from many sources, both in our galaxy and in distant galaxies, such as collapsing stars, double star systems, colliding black holes, and possibly even gravitational waves left over from the big bang. When this happens, a new field of "gravitational-wave astronomy" will be born.
Although gravitational radiation has not been detected directly, we know that it exists, through a remarkable system known as the binary pulsar. Discovered in 1974 by radio astronomers Russell Hulse and Joseph Taylor, it consists of a pulsar (which is a rapidly spinning neutron star) and a companion star in orbit around each other. Although the companion has not been seen directly, it is also believed to be a neutron star. The two bodies are circling so closely÷their orbit is about the size of the Sun, and the period of the orbit is eight hours÷that the effects of general relativity are very large. For example, the periastron shift (the analogue of the perihelion shift for Mercury) is more than 4 degrees per year.
Furthermore, the pulsar acts as an extremely stable clock, its pulse period of approximately 59 milliseconds drifting by only a quarter of a nanosecond per year. By measuring the arrival times of radio pulses at Earth, observers were able to determine the motion of the pulsar about its companion with amazing accuracy. For example, the accurate value for the orbital period is 27906.981163 seconds, and the orbital eccentricity is 0.617 127.
Like a rotating dumbbell, an orbiting binary system should emit gravitational radiation, and in the process lose some of its orbital energy. This energy loss will cause the pulsar and its companion to spiral in toward each other, and the orbital period to shorten. According to general relativity, the predicted decrease in the orbital period is 75 microseconds per year. The observed decrease rate is in agreement with the prediction to about 4%. This confirms the existence of gravitational radiation and the general relativistic equations that describe it
Black Holes
One of the most important and exciting aspects of the relativity renaissance is the study and search for black holes. The subject began in the early 1960s as astrophysicists looked for explanations of the quasars, warmed up in the early 1970s following the possible detection of a black hole in an X-ray source, became white hot in the middle 1970s when black holes were found theoretically to evaporate, and continues today as one of the most active branches of general relativity.
However, the first glimmerings of the black hole idea date back to the 18th century, in the writings of a British amateur astronomer, the Reverend John Michell. Reasoning on the basis of the corpuscular theory that light would be attracted by gravity in the same way that ordinary matter is attracted, he noted that light emitted from the surface of a massive body such as the Earth or the Sun would be reduced in velocity by the time it reached great distances. (Michell of course did not know special relativity.) How large would a body of the same density as the Sun have to be in order that light emitted from it would be stopped and pulled back before reaching infinity? The answer he obtained was 500 times the diameter of the Sun. Light could therefore never escape from such a body. In today's language, such an object would be a supermassive black hole of about 100 million solar masses.
Although the general relativistic solution for a nonrotating black hole was discovered by Karl Schwarzschild in 1916, and a calculation of gravitational collapse to a black hole state was performed by J. Robert Oppenheimer and Hartland Snyder in 1939, black hole physics didn't really begin until the middle 1960s, when astronomers confronted the problem of the energy output of the quasars, and the mathematician Roy Kerr discovered the rotating black-hole solution.
A black hole is formed when a star has exhausted the thermonuclear fuel necessary to produce the heat and pressure that support it against gravity. The star begins to collapse, and if it is massive enough, it continues to collapse until the radius of the star approaches a value called the gravitational radius or Schwarzschild radius. In the nonrotating spherical case, this radius is 2GM/c2, where M is the mass of the star. For a body of one solar mass, the gravitational radius is about 3 km; for a body of the mass of the Earth, it is about 9 mm. An observer sitting on the surface of the star sees the collapse continue to smaller and smaller radii, until both star and observer reach the origin r = 0, with consequences too horrible to describe in detail. On the other hand, an observer at great distances observes the collapse to slow down as the radius approaches the gravitational radius, a result of the gravitational redshift of the light signals sent outward. However, in this case, the redshifting or slowing down becomes so extreme that the star appears almost to stop just outside the gravitational radius. The distant observer never sees any signals emitted by the falling observer once the latter is inside the gravitational radius. Any signal emitted inside can never escape the sphere bounded by the gravitational radius, called the "event horizon."
Since the middle 1960s the laws of "black hole physics" have been established and codified. For example, it was discovered that Kerr (rotating) and Schwarzschild (nonrotating) solutions are unique; there are no other black-hole solutions in general relativity. Inside every black hole resides a "singularity," a pathological region of spacetime where the gravitational forces become infinite, where time comes to an end for any observer unfortunate enough to hit the singularity, indeed where all the laws of physics break down. Fortunately the event horizon of the black hole prevents any bizarre phenomena that such singularities might cause from reaching the outside world (this notion has been dubbed "cosmic censorship"). In 1974, Stephen Hawking discovered that the laws of quantum mechanics applied to the outside of a black hole required it to evaporate by the creation of particles with a thermal energy spectrum, and to have an associated temperature and entropy. The temperature of a Schwarzschild black hole is T = hc3/8πkGM, where h is Planck's constant and k is Boltzmann's constant. This discovery demonstrated a remarkable connection between gravity, thermodynamics, and quantum mechanics which helped renew the theoretical quest for a grand synthesis of all the fundamental interactions. For black holes of astronomical masses, however, the evaporation is completely negligible, since for a solar-mass black hole, T 10-6 K.
Although a great deal is known about black holes in theory, rather less is known about them observationally. There are several instances in which the evidence for the existence of black holes is impressive, but in all cases it is indirect. For instance, in the X-ray source Cygnus XI, the source of the X-rays is believed to be a black hole with a mass larger than about six solar masses in orbit around a giant star (See figs. 5 and 6). The object can not be a neutron star, because general relativity predicts that neutron stars must be lighter than about 3 solar masses. Thus the object must be a black hole. The X-rays are emitted by matter pulled from the surface of the companion star and sent into a spiraling orbit around the black hole. Similarly, there is evidence in the centers of certain galaxies, such as M87 and possibly even our own, of collapsed objects between 102 and 108 solar masses. A black hole model is consistent with the observations, although it is not necessarily required. Accretion of matter onto supermassive rotating black holes may produce the jets of outflowing matter that are observed in many quasars and active galactic nuclei. These and other astrophysical processes that might aid in the detection of black holes are being studied by relativists and astrophysicists.
Cosmology
The other area in which there has been a renaissance for general relativity is cosmology. Although Einstein in 1917 first used general relativity to calculate a model for the universe as a hole, the subject was not considered a serious branch of physics until the 1960s, when astronomical observations lent credence to the idea that the universe was expanding from a "big bang." For instance, observations of the rates at which galaxies are receding from us, couples with improved determinations of their distances, implied that the universe was at least 10 billion years old, a value that was consistent with other observations such as the age of the Earth and of old star clusters.
In 1965 came the discovery of the cosmic background radiation by Arno Penzias and Robert Wilson. This radiation is the remains of the hot electromagnetic black-body radiation that once dominated the universe in its earlier phase, now cooled to 3 kelvins by the subsequent expansion of the universe. Next came calculations of the amount of helium that would be synthesized from hydrogen by thermonuclear fusion in the very early Universe, around 1000 seconds after the big bang. The amount, approximately 25% by weight, was in agreement with the abundances of helium observed in stars and in interstellar space. This was an important confirmation of the hot big-bang picture, because the amount of helium believed to be produced by fusion in the interiors of stars is woefully inadequate to explain the observed abundances.
Today, the general relativistic hot big-bang model of the universe has broad acceptance, and cosmologists now focus their attention on more detailed issues, such as how galaxies and other large-scale structures formed out of the hot primordial soup, and on what the universe might have been like earlier than 1000 seconds, all the way back to 10-36 s (and some brave cosmologists are going back even further) when the laws of elementary-particle physics may have played a major role in the evolution of the universe.
Acceptance of General Relativity
One of the outgrowths of the renaissance of general relativity that has occurred since 1960 has been a change in attitude about the importance and use of the theory. Its importance as a fundamental theory of the nature of spacetime and gravitation has not been diminished in the least; if anything, it has been enhanced by the flowering of research in the subject that has taken place. Its importance as a foundation for other theories of physics has been strengthened by current searches for unified quantum theories of nature that incorporate gravity along with the other interactions.
But the real change in attitude about general relativity has been in its use as a tool in the real world. In astrophysical situations, general relativity plays a central role in the study of neutron stars, black holes, gravitational lenses, relativistic binary star systems, and the universe as a whole. Gravitational radiation may one day provide a completely new observational tool for exploring and examining the cosmos.
Relativity even plays a role in everyday life. For example, the gravitational redshift effect on clocks must be taken into account in satellite-based navigation systems, such as the US Air Force's Global Positioning System, in order to achieve the required positional accuracy of a few meters or time transfer accuracy of a few nanoseconds.
To general relativists, always eager to find practical consequences of their subject, these have been very welcome developments!
Suggested Readings
Davies, P.C.W. The Search for Gravity Waves, Cambridge, England, Cambridge University Press, 1980.
Greenstein, C. Frozen Star: Of Pulsars, Black Holes and the Fate of Stars, New York, Freundlich books, 1984.
Weinberg, S. The First Three Minutes: A Modern View of the Origin of the Universe, New York, Basic Books, 1977.
Will, C.M. Was Einstein Right? Putting General Relativity to the Test, New York, Basic Books, 1986.
Essay Questions
Figure Legends
Figure 1
The gravitation curvature of space. According to general relativity, gravity curves spacetime. This drawing shows how space is curved around a massive object such as the sun or a star. The red shaded region in the center indicates the location of the star. The greatest curvature is found immediately above the star's surface. Far from the star, where gravity is weak, spacetime is almost perfectly flat.
Figure 2
Deflection of starlight passing near the sun. Because of this effect, the sun and other remote objects can act as a gravitational lens. In his general relativity theory, Einstein calculated that starlight just grazing the sun's surface should be deflected by an angle of 1."75.
Figure 3
Advance of perihelion of Mercury. The elliptical orbit of Mercury about the sun rotates very slowly relative to the system connected with the sun. General relativity successfully explains this small effect, which predicts that the direction of the perihelion should change by only 43 arcseconds per century.
Figure 4
Under the general theory of relativity, the presence of a massive body essentially warps the space nearby. This can account for both the bending of light near the sun and the advance of the perihelion point of Mercury by 43 arcseconds per century more than would otherwise be expected. The diagram shows how a two-dimensional surface warped into three dimensions can change the direction of a "straight" line that is constrained to its surface; the warping of space is analogous, although with a greater number of dimensions to consider. The effect is similar to the golfer putting on a warped green. Though the ball is hit in a straight line, we see it appear to curve. (Adapted from Pasachoff, Jay M., Astronomy, 3rd ed., Saunders College Publishing, 1987).
Figure 5
The overexposed dark object in the center of this negative print is the blue supergiant star HDE 226868, which is thought to be the companion of the first black hole to be discovered, Cygnus X-1. The black hole in Cygnus X-1 that is though to be orbiting the supergiant star is not visible. The image of the supergiant appears so large because it is overexposed on the film. (From Pasachoff, Astronomy.)
Figure 6
The Cygnus X-1 black hole. The stellar wind from HDE 226868 pours matter onto a huge disk around its black hole companion. The infalling gases are heated to enormous temperatures as they spiral toward the black hole. The gases are so hot that they emit vast quantities of x-rays.