Gravitational fields and the theory of general relativity
- Also called:
- gravitation
News •
In Einstein’s theory of general relativity, the physical consequences of gravitational fields are stated in the following way. Space-time is a four-dimensional non-Euclidean continuum, and the curvature of the Riemannian geometry of space-time is produced by or related to the distribution of matter in the world. Particles and light rays travel along the geodesics (shortest paths) of this four-dimensional geometric world.
There are two principal consequences of the geometric view of gravitation: (1) the accelerations of bodies depend only on their masses and not on their chemical or nuclear constitution, and (2) the path of a body or of light in the neighborhood of a massive body (the Sun, for example) is slightly different from that predicted by Newton’s theory. The first is the weak principle of equivalence. Newton himself performed experiments with pendulums that demonstrated the principle to better than one part in 1,000 for a variety of materials, and, at the beginning of the 20th century, the Hungarian physicist Roland, Baron von Eötvös, showed that different materials accelerate in Earth’s field at the same rate to within one part in 109. More-recent experiments have shown the equality of accelerations in the field of the Sun to within one part in 1011. Newtonian theory is in accord with these results because of the postulate that gravitational force is proportional to a body’s mass.
Inertial mass is a mass parameter giving the inertial resistance to acceleration of the body when responding to all types of force. Gravitational mass is determined by the strength of the gravitational force experienced by the body when in the gravitational field g. The Eötvös experiments therefore show that the ratio of gravitational and inertial mass is the same for different substances.
In Einstein’s theory of special relativity, inertial mass is a manifestation of all the forms of energy in a body, according to his fundamental relationship E = mc2, E being the total energy content of a body, m the inertial mass of the body, and c the speed of light. Dealing with gravitation, then, as a field phenomenon, the weak principle of equivalence indicates that all forms of nongravitational energy must identically couple to or interact with the gravitational field, because the various materials in nature possess different fractional amounts of nuclear, electrical, magnetic, and kinetic energies, yet they accelerate at identical rates.
In the theory of general relativity, the gravitational field also interacts with gravitational energy in the same manner as with other forms of energy, an example of that theory’s universality not possessed by most other theories of gravitation.
The Sun has an appreciable fraction of internal gravitational energy, and the repetitions of the Eötvös experiments during the 1970s, with the Sun instead of Earth as the attracting mass, revealed that bodies accelerate at identical rates in the Sun’s field as well as in that of Earth. Extremely accurate laser measurements of the distance of the Moon from Earth have made possible a further test of the weak principle of equivalence. The chemical constitutions of Earth and the Moon are not the same, and so, if the principle did not hold, they might accelerate at different rates under the Sun’s attraction. No such effect has been detected.
Newton’s third law of dynamics states that every force implies an equal and opposite reaction force. Modern field theories of force contain this principle by requiring every entity that is acted upon by a field to be also a source of the field. An experiment by the American physicist Lloyd Kreuzer established to within 1 part in 20,000 that different materials produce gravitational fields with a strength the same as that of gravitational fields acting upon them. In this experiment a sphere of solid material was moved through a liquid of identical weight density. The absence of a gravitational effect on a nearby Cavendish balance instrument during the sphere’s motion is interpreted as showing that the two materials had equal potency in producing a local gravitational-field anomaly.
Other experiments have confirmed Einstein’s predictions to within a few percent. Using the Mössbauer effect to monitor the nuclear reabsorption of resonant gamma radiation, a shift of wavelength of the radiation that traveled vertically tens of meters in Earth’s gravitational field was measured, and the slowing of clocks (in this case the nuclear vibrations are clocks) as predicted by Einstein was confirmed to 1 percent precision. If ν and Δν are clock frequency and change of frequency, respectively, h is the height difference between clocks in the gravitational field g. Then 
The paths of particles and light
The idea that light should be deflected by passing close to a massive body had been suggested by the British astronomer and geologist John Michell in the 18th century. However, Einstein’s general relativity theory predicted twice as much deflection as Newtonian physics. Quick confirmation of Einstein’s result came from measuring the direction of a star close to the Sun’s direction during an expedition led by the British astronomer Sir Arthur Stanley Eddington to observe the solar eclipse of 1919. Optical determinations of the change of direction of a star are subject to many systematic errors, and far better results have been obtained of the directions of spacecraft with radio interferometers of very long baselines. The effect comes from the decrease in the speed of light near a massive object (the Sun). That decrease has also been found directly from the round-trip travel times for radar pulses between Earth and other inner planets or artificial satellites passing behind the Sun and has confirmed to about 4 percent the prediction of an additional time delay Δt given by the following formula, in which MS is the Sun’s mass, R1 and R2 are the distances from the Sun to Earth and to the other reflecting body, and D is the distance of closest approach of the radar pulses to the Sun (ln stands for natural logarithm): 
The additional precession of the orbit of Mercury of 43 arc seconds per century was known before the development of the theory of general relativity. With radar measurements of the distances to the planets, similar anomalous precessions have been estimated for Venus and Earth and have been found to agree with general relativity.
Gravitational radiation
According to general relativity, the curvature of space-time is determined by the distribution of masses, while the motion of masses is determined by the curvature. In consequence, variations of the gravitational field should be transmitted from place to place as waves, just as variations of an electromagnetic field travel as waves. If the masses that are the source of a field change with time, they should radiate energy as waves of curvature of the field. There are strong grounds for believing that such radiation exists. One particular double-star system has a pulsar as one of its components, and, from measurements of the shift of the pulsar frequency due to the Doppler effect, precise estimates of the period of the orbit show that the period is changing, corresponding to a decrease in the energy of the orbital motion. Gravitational radiation is the only known means by which that could happen.
Double stars in their regular motions (such as that for which a change in period has been detected) and massive stars collapsing as supernovas have been suggested as sources of gravitational radiation, and considerable theoretical effort has gone into calculating the signals to be expected from those and other sources.
Gravitational radiation is very weak. The changes of curvature would correspond to a dilation in one direction and a contraction at right angles to that direction. One scheme, first tried out about 1960, employed a massive cylinder that might be set in mechanical oscillation by a gravitational signal. The authors of this apparatus argued that signals had been detected, but their claim was not substantiated. In a more fruitful scheme an optical interferometer is set up with freely suspended reflectors at the ends of long paths that are at right angles to each other. Shifts of interference fringes corresponding to an increase in length of one arm and a decrease in the other would indicate the passage of gravitational waves. Such an interferometer, the Laser Interferometer Gravitational-Wave Observatory (LIGO), made the first detection of gravitational radiation in 2015. Two black holes about 1.3 billion light-years away spiraled into each other. The black holes were 36 and 29 times the mass of the Sun and formed a new black hole 62 times the mass of the Sun. In the merger, three solar masses were converted to energy in gravitational waves; the amount of power radiated was 50 times more than that of all the stars shining in the universe in that moment. As of 2024, LIGO has made 90 detections of gravitational radiation. Eighty-three were from mergers of a black hole binary, two were the merger of a neutron star binary, and three were the possible merger of a black hole with a neutron star. Two detections were from the merger of a black hole with an object that has a mass between those known for black holes and neutron stars.
Kenneth L. Nordtvedt Alan H. CookSome astronomical aspects of gravitation
As stated above, studies of gravity allow the masses and densities of celestial bodies to be estimated and thereby make it possible to investigate the physical constitutions of stars and planets. Because gravitation is a very weak force, however, its distinctive effects appear only when masses are extremely large. The idea that light might be attracted gravitationally had been suggested by Michell and examined by the French mathematician and astronomer Pierre-Simon Laplace. Predictions by classical physics and general relativity that light passing close to the Sun might be deflected are described above. There are two further consequences for astronomy. Light from a distant object may pass close to objects other than the Sun and be deflected by them. In particular, they may be deflected by a massive galaxy. If some object is behind a massive galaxy, as seen from Earth, deflected light may reach Earth by more than one path. Operating like a lens that focuses light along different paths, the gravity of the galaxy may make the object appear multiple; examples of such apparently double objects have been found.
Both Michell and Laplace pointed out that the attraction of a very dense object upon light might be so great that the light could never escape from the object, rendering it invisible. Such a phenomenon is a black hole. The relativistic theory of black holes has been thoroughly developed in recent years, and astronomers have conducted extensive observations of them. One possible class of black holes comprises very large stars that have used up all of their nuclear energy so that they are no longer held up by radiation pressure and have collapsed into black holes (less-massive stars may collapse into neutron stars). Supermassive black holes with masses millions to billions times that of the Sun are thought to exist at the centers of most galaxies.
Black holes, from which no radiation is able to escape, cannot be seen by their own light, but there are observable secondary effects. If a black hole were one component of a double star, the orbital motion of the pair and the mass of the invisible member might be derived from the oscillatory motion of a visible companion. Because black holes attract matter, any gas in the vicinity of an object of this kind would fall into it and acquire, before vanishing into the hole, a high velocity and consequently a high temperature. The gas may become hot enough to produce X-rays and gamma rays from around the hole. Such a mechanism is the origin of at least some powerful X-ray and radio astronomical sources, including those at the centers of galaxies and quasars. In the case of the massive galaxy M87, the supermassive black hole at its center, which has a mass 6.5 billion times that of the Sun has been directly observed.
Experimental study of gravitation
The essence of Newton’s theory of gravitation is that the force between two bodies is proportional to the product of their masses and the inverse square of their separation and that the force depends on nothing else. With a small modification, the same is true in general relativity. Newton himself tested his assumptions by experiment and observation. He made pendulum experiments to confirm the principle of equivalence and checked the inverse square law as applied to the periods and diameters of the orbits of the satellites of Jupiter and Saturn.
During the latter part of the 19th century, many experiments showed the force of gravity to be independent of temperature, electromagnetic fields, shielding by other matter, orientation of crystal axes, and other factors. The revival of such experiments during the 1970s was the result of theoretical attempts to relate gravitation to other forces of nature by showing that general relativity was an incomplete description of gravity. New experiments on the equivalence principle were performed, and experimental tests of the inverse square law were made both in the laboratory and in the field.
There also has been a continuing interest in the determination of the constant of gravitation, although it must be pointed out that G occupies a rather anomalous position among the other constants of physics. In the first place, the mass M of any celestial object cannot be determined independently of the gravitational attraction that it exerts. Thus, the combination GM, not the separate value of M, is the only meaningful property of a star, planet, or galaxy. Second, according to general relativity and the principle of equivalence, G does not depend on material properties but is in a sense a geometric factor. Hence, the determination of the constant of gravitation does not seem as essential as the measurement of quantities like the electronic charge or Planck’s constant. It is also much less well determined experimentally than any of the other constants of physics.
Experiments on gravitation are in fact very difficult, as a comparison of experiments on the inverse square law of electrostatics with those on gravitation will show. The electrostatic law has been established to within one part in 1016 by using the fact that the field inside a closed conductor is zero when the inverse square law holds. Experiments with very sensitive electronic devices have failed to detect any residual fields in such a closed cavity. Gravitational forces have to be detected by mechanical means, most often the torsion balance, and, although the sensitivities of mechanical devices have been greatly improved, they are still far below those of electronic detectors. Mechanical arrangements also preclude the use of a complete gravitational enclosure. Last, extraneous disturbances are relatively large because gravitational forces are very small (something that Newton first pointed out). Thus, the inverse square law is established over laboratory distances to no better than one part in 104.
The inverse square law
Recent interest in the inverse square law arose from two suggestions. First, the gravitational field itself might have a mass, in which case the constant of gravitation would change in an exponential manner from one value for small distances to a different one for large distances over a characteristic distance related to the mass of the field. Second, the observed field might be the superposition of two or more fields of different origin and different strengths, one of which might depend on chemical or nuclear constitution. Deviations from the inverse square law have been sought in three ways:
- The law has been checked in the laboratory over distances up to about 1 meter.
- The effective value of G for distances between 100 meters and 1 km has been estimated from geophysical studies.
- There have been careful comparisons of the value of the attraction of Earth as measured on the surface and as experienced by artificial satellites.
Early in the 1970s an experiment by the American physicist Daniel R. Long seemed to show a deviation from the inverse square law at a range of about 0.1 meter. Long compared the maximum attractions of two rings upon a test mass hung from the arm of a torsion balance. The maximum attraction of a ring occurs at a particular point on the axis and is determined by the mass and dimensions of the ring. If the ring is moved until the force on the test mass is greatest, the distance between the test mass and the ring is not needed. Two later experiments over the same range showed no deviation from the inverse square law. In one, conducted by the American physicist Riley Newman and his colleagues, a test mass hung on a torsion balance was moved around in a long hollow cylinder. The cylinder approximates a complete gravitational enclosure and, allowing for a small correction because it is open at the ends, the force on the test mass should not depend on its location within the cylinder. No deviation from the inverse square law was found. In the other experiment, performed in Cambridge, Eng., by Y.T. Chen and associates, the attractions of two solid cylinders of different mass were balanced against a third cylinder so that only the separations of the cylinders had to be known; it was not necessary to know the distances of any from a test mass. Again no deviation of more than one part in 104 from the inverse square law was found. Other, somewhat less-sensitive experiments at ranges up to one meter or so also have failed to establish any greater deviation.
The geophysical tests go back to a method for the determination of the constant of gravitation that had been used in the 19th century, especially by the British astronomer Sir George Airy. Suppose the value of gravity g is measured at the top and bottom of a horizontal slab of rock of thickness t and density d. The values for the top and bottom will be different for two reasons. First, the top of the slab is t farther from the center of Earth, and so the measured value of gravity will be less by 2(t/R)g, where R is the radius of Earth. Second, the slab itself attracts objects above and below it toward its center; the difference between the downward and upward attractions of the slab is 4πGtd. Thus, a value of G may be estimated. Frank D. Stacey and his colleagues in Australia made such measurements at the top and bottom of deep mine shafts and claimed that there may be a real difference between their value of G and the best value from laboratory experiments. The difficulties lie in obtaining reliable samples of the density and in taking account of varying densities at greater depths. Similar uncertainties appear to have afflicted measurements in a deep bore hole in the Greenland ice sheet.
New measurements have failed to detect any deviation from the inverse square law. The most thorough investigation was carried out from a high tower in Colorado. Measurements were made with a gravimeter at different heights and coupled with an extensive survey of gravity around the base of the tower. Any variations of gravity over the surface that would give rise to variations up the height of the tower were estimated with great care. Allowance was also made for deflections of the tower and for the accelerations of its motions. The final result was that no deviation from the inverse square law could be found.
A further test of the inverse square law depends on the theorem that the divergence of the gravity vector should vanish in a space that is free of additional gravitational sources. An experiment to test this was performed by M.V. Moody and H.J. Paik of the University of Maryland with a three-axis superconducting gravity gradiometer that measured the gradients of gravity in three perpendicular directions. The sum of the three gradients was zero within the accuracy of the measurements, about one part in 104.
The absolute measurements of gravity described earlier, together with the comprehensive gravity surveys made over the surface of Earth, allow the mean value of gravity over Earth to be estimated to about one part in 106. The techniques of space research also have given the mean value of the radius of Earth and the distances of artificial satellites to the same precision. Thus, it has been possible to compare the value of gravity on Earth with that acting on an artificial satellite. Agreement to about one part in 106 shows that, over distances from the surface of Earth to close satellite orbits, the inverse square law is followed.
Thus far, all of the most reliable experiments and observations reveal no deviation from the inverse square law.
The principle of equivalence
Experiments with ordinary pendulums test the principle of equivalence to no better than about one part in 105. Eötvös obtained much better discrimination with a torsion balance. His tests depended on comparing gravitational forces with inertial forces for masses of different composition. Eötvös set up a torsion balance to compare, for each of two masses, the gravitational attraction of Earth with the inertial forces due to the rotation of Earth about its polar axis. His arrangement of the masses was not optimal, and he did not have the sensitive electronic means of control and reading that are now available. Nonetheless, Eötvös found that the weak equivalence principle (see above Gravitational fields and the theory of general relativity) was satisfied to within one part in 109 for a number of very different chemicals, some of which were quite exotic. His results were later confirmed by the Hungarian physicist János Renner. Renner’s work has been analyzed recently in great detail because of the suggestion that it could provide evidence for a new force. It seems that the uncertainties of the experiments hardly allow such analyses.
Eötvös also suggested that the attraction of the Sun upon test masses could be compared with the inertial forces of Earth’s orbital motion about the Sun. He performed some experiments, verifying equivalence with an accuracy similar to that which he had obtained with his terrestrial experiments. The solar scheme has substantial experimental advantages, and the American physicist Robert H. Dicke and his colleagues, in a careful series of observations in the 1960s (employing up-to-date methods of servo control and observation), found that the weak equivalence principle held to about one part in 1011 for the attraction of the Sun on gold and aluminum. A later experiment by the Russian researcher Vladimir Braginski, with very different experimental arrangements, gave a limit of about one part in 1012 for platinum and aluminum.
Galileo’s supposed experiment of dropping objects from the Leaning Tower of Pisa has been reproduced in the laboratory with apparatuses used to determine the absolute value of gravity by timing a falling body. Two objects, one of uranium, the other of copper, were timed as they fell. No difference was detected.
Laser-ranging observations of the Moon in the LAGEOS (laser geodynamic satellites) experiment have also failed to detect deviations from the principle of equivalence. Earth and the Moon have different compositions, the Moon lacking the iron found in Earth’s core. Thus, if the principle of equivalence were not valid, the accelerations of Earth and the Moon toward the Sun might be different. The very precise measurements of the motion of the Moon relative to Earth could detect no such difference.
By the start of the 21st century, all observations and experiments on gravitation had detected that there are no deviations from the deductions of general relativity, that the weak principle of equivalence is valid, and that the inverse square law holds over distances from a few centimeters to thousands of kilometers. Coupled with observations of electromagnetic signals passing close to the Sun and of images formed by gravitational lenses, those observations and experiments make it very clear that general relativity provides the only acceptable description of gravitation at the present time.
The constant of gravitation
The constant of gravitation has been measured in three ways:
- The comparison of the pull of a large natural mass with that of Earth
- The measurement with a laboratory balance of the attraction of Earth upon a test mass
- The direct measurement of the force between two masses in the laboratory
The first approach was suggested by Newton; the earliest observations were made in 1774 by the British astronomer Nevil Maskelyne on the mountain of Schiehallion in Scotland. The subsequent work of Airy and more-recent developments are noted above. The laboratory balance method was developed in large part by the British physicist John Henry Poynting during the late 1800s, but all the most recent work has involved the use of the torsion balance in some form or other for the direct laboratory measurement of the force between two bodies. The torsion balance was devised by Michell, who died before he could use it to measure G. Cavendish adapted Michell’s design to make the first reliable measurement of G in 1798; only in comparatively recent times have clearly better results been obtained. Cavendish measured the change in deflection of the balance when attracting masses were moved from one side to the other of the torsion beam. The method of deflection was analyzed most thoroughly in the late 1800s by Sir Charles Vernon Boys, an English physicist, who carried it to its highest development, using a delicate suspension fibre of fused silica for the pendulum. In a variant of that method, the deflection of the balance is maintained constant by a servo control.
The second scheme involves the changes in the period of oscillation of a torsion balance when attracting masses are placed close to it such that the period is shortened in one position and lengthened in another. Measurements of period can be made much more precisely than those of deflection, and the method, introduced by Carl Braun of Austria in 1897, has been used in many subsequent determinations. In a third scheme the acceleration of the suspended masses is measured as they are moved relative to the large attracting masses.
In another arrangement a balance with heavy attracting masses is set up near a free test balance and adjusted so that it oscillates with the same period as the test balance. The latter is then driven into resonant oscillations with an amplitude that is a measure of the constant of gravitation. The technique was first employed by J. Zahradnicek of Czechoslovakia during the 1930s and was effectively used again by C. Pontikis of France some 40 years later.
Suspensions for two-arm balances for the comparison of masses and for torsion balances have been studied intensively by T.J. Quinn and his colleagues at the International Bureau of Weights and Measures, near Paris, and they have found that suspensions with thin ribbons of metal rather than wires provide the most stable systems. They have used balances with such suspensions to look for deviations from the predictions of general relativity and have most recently used a torsion balance with ribbon suspension in two new determinations of the constant of gravitation.
Many new determinations of G were made in the five years from 1996 to 2001 and are summarized in the table. However, despite the great attention given to systematic errors in those experiments, it is clear from the range of the results that serious discrepancies, much greater than the apparent random errors, still afflict determinations of G. In 2001 the best estimate of G was taken to be 6.67553 × 10−11 m3 s−2 kg−1. Results before 1982 indicate a lower value, perhaps 6.670, but those from 1996 onward suggest a higher value.
| Values of the constant of gravitation | |||
| author | year | method | G (in units of 10–11 m3s–2kg–1) |
| H. Cavendish | 1798 | torsion balance (deflection) | 6.754 |
| J.H. Poynting | 1891 | common balance | 6.698 |
| C.V. Boys | 1895 | torsion balance (deflection) | 6.658 |
| C. Braun | 1897 | torsion balance (deflection) | 6.658 |
| C. Braun | 1897 | torsion balance (period) | 6.658 |
| P.R. Heyl | 1930 | torsion balance (period) | 6.669 |
| J. Zahradnicek | 1932 | torsion balance (resonance) | 6.659 |
| P.R. Heyl, P. Chrzanowski | 1942 | torsion balance (period) | 6.672 |
| C. Pontikis | 1972 | torsion balance (resonance) | 6.6714 |
| G.G. Luther and W.R. Towler | 1982 | torsion balance (period) | 6.6726 |
| H. de Boer | 1987 | mercury flotation (deflection) | 6.667 |
| W. Michaelis et al. | 1996 | flotation (null deflection) | 6.7164 |
| C.H. Bagley and G.G. Luther | 1997 | torsion balance (period) | 6.6740 |
| O.V. Karagioz et al. | 1998 | torsion balance (period) | 6.6729 |
| J. Luo et al. | 1999 | torsion balance (period) | 6.6699 |
| M.P. Fitzgerald, T.R. Armstrong | 1999 | torsion balance (null deflection) | 6.6742 |
| F. Nolting et al. | 1999 | common balance | 6.6754 |
| U. Kleinvoss et al. | 1999 | pendulum deflection | 6.6735 |
| J.H. Gundlach, S.M. Merkowitz | 2000 | torsion balance (acceleration) | 6.67422 |
| T.J. Quinn et al. | 2001 | torsion balance (servo) | 6.67553 |
| T.J. Quinn et al. | 2001 | torsion balance (deflection) | 6.67565 |
The variation of the constant of gravitation with time
The 20th-century English physicist P.A.M. Dirac, among others, suggested that the value of the constant of gravitation might be proportional to the age of the universe; other rates of change over time also have been proposed. The rates of change would be extremely small, one part in 1011 per year if the age of the universe is taken to be 1011 years; such a rate is entirely beyond experimental capabilities at present. There is, however, the possibility of looking for the effects of any variation upon the orbit of a celestial body, in particular the Moon. It has been claimed from time to time that such effects may have been detected. As yet, there is no certainty.
Fundamental character of G
The constant of gravitation is plainly a fundamental quantity, since it appears to determine the large-scale structure of the entire universe. Gravity is a fundamental quantity, whether it is an essentially geometric parameter, as in general relativity, or the strength of a field, as in one aspect of a more-general field of unified forces. The fact that, so far as is known, gravitation depends on no other physical factors makes it likely that the value of G reflects a basic restriction on the possibilities of physical measurement, just as special relativity is a consequence of the fact that, beyond the shortest distances, it is impossible to make separate measurements of length and time.
Alan H. Cook