GRAVITATIONAL THEORIES

Jacob D. Bekenstein

A theory of gravitation is a description of the long range forces that electrically neutral bodies exert on one another because of their matter content. Until the 1910s Sir Isaac Newton's law of universal gravitation, two particles attract each other with a central force proportional to the product of their masses and inversely proportional to the square of the distance between them, was accepted as the correct and complete theory of gravitation: The proportionality constant here is Newton's constant G = 6.67 x 10^-8 dyn cm² g^-2, also called the gravitational constant. This theory is highly accurate in its predictions regarding everyday phenomena. However, high precision measurements of motions in the solar system and in binary pulsars, the structure of black holes, and the expansion of the universe can only be fully understood in terms of a relativistic theory of gravitation. Best known of these is Albert Einstein's general theory of relativity, which reduces to Newton's theory in a certain limit. Of the scores of rivals to general relativity formulated over the last half century, many have failed various experimental tests, but the verdict is not yet in on which extant relativistic gravitation theory is closest to the truth.

NEWTONIAN GRAVITATIONAL THEORY

In alternative language, newtonian gravitational theory states that the acceleration a (the rate of change of the velocity v) imparted by gravitation on a test particle is determined by the gravitational potential ,

a = -dv / dt = -,

and the potential is determined by the surrounding mass distribution by Poisson's partial differential equation

· = 4G.

This formulation is entirely equivalent to Newton's law of gravitation. Because a test particle's acceleration depends only on the potential generated by matter in the surroundings, the theory respects the weak equivalence principle: the motion of a particle is independent of its internal structure or composition. As the subject of Galileo Galilei's apocryphal experiment at the tower of Pisa, this principle is supported by a series of high precision experiments culminating in those directed by Baron Lorand von Eötvos in Budapest in 1922, Robert Dicke at Princeton in 1964, and Vladimir Braginsky in Moscow in 1972.

Highly successful in everyday applications, newtonian gravitation has also proved accurate in describing motions in the solar system (except for tiny relativistic effects), the internal structure of planets, the sun and other stars, orbits in binary and multiple stellar systems, the structure of molecular clouds, and, in a rough way, the structure of galaxies and clusters of galaxies (but see below).

THE GENERAL THEORY OF RELATIVITY

According to newtonian theory, gravitational effects propagate from place to place instantaneously. With the advent of Einstein's special theory of relativity in 1905, a theory uniting the concepts of space and time into that of four dimensional flat space-time (named Minkowski space-time after the mathematician Hermann Minkowski), a problem became discernible with newtonian theory. According to special relativity, which is the current guideline to the form of all physical theory, the speed of light, c = 3 x 10¹⁰ cm s^-1, is the top speed allowed to physical particles or forces: There can be no instantaneous propagation. After a decade of search for new concepts to make gravitational theory compatible with the spirit of special relativity, Einstein came up with the theory of general relativity (1915), the prototype of all modern gravitational theories. Its crucial ingredient, involving a colossal intellectual jump, is the concept of gravitation, not as a force, but as a manifestation of the curvature of space-time, an idea first mentioned in rudimentary form by the mathematician Ceorg Bernhard Riemann in 1854. In Einstein's hands gravitation theory was thus transformed from a theory of forces into the first dynamical theory of geometry, the geometry of four dimensional curved space-time.

Why talk of curvature? One of Einstein's first predictions was the gravitational redshift: As any wave, such as light, propagates away from a gravitating mass, all frequencies in it are reduced by an amount proportional to the change in gravitational potential experienced by the wave. This redshift has been measured in the laboratory, in solar observations, and by means of high precision clocks flown in airplanes. However, imagine for a moment that general relativity had not yet been invented, but the redshift has already been measured. According to a simple argument owing to Alfred Schild, wave propagation under stationary circumstances can display a redshift only if the usual geometric relations implicit in Minkowski space-time are violated: The space-time must be curved. The observations of the redshift thus show that space-time must be curved in the vicinity of masses, regardless of the precise form of the gravitational theory.

Einstein provided 10 equations relating the metric (a tensor with 10 independent components describing the geometry of space-time) to the material energy momentum tensor (also composed of 10 components, one of which corresponds to our previous ). These Einstein field equations, in which both of the previously mentioned constants G and c figure as parameters, replace Poisson's equation. Einstein also replaced the newtonian law of motion by the statement that free test particles move along geodesics, the shortest curves in the space-time geometry. The influential gravitation theorist John Archibald Wheeler has encapsulated general relativity in the aphorism ``curvature tells matter how to move, and matter tells space-time how to curve.'' The Eötvos-Dicke-Braginsky experiments demonstrate with high precision that free test particles all travel along the same trajectories in space-time, whereas the gravitational redshift shows (with more modest precision) these universal trajectories to be identical with geodesics.

Despite the great contrast between General Relativity and Newtonian theory, predictions of the former approach the latter for systems in which velocities are small compared to c and gravitational potentials are weak enough that they cannot cause larger velocities. This is why we can discuss with newtonian theory the structure of the earth and planets, stars and stellar clusters, and the gross features of motions in the solar system without fear of error.

Einstein noted two other predictions of General Relativity. First, light beams passing near a gravitating body must suffer a slight deflection proportional to that body's mass. First verified by observations of stellar images during the 1919 total solar eclipse, this effect also causes deflection of quasar radio images by the sun, is the likely cause of the phenomenon of ``double quasars'' with identical redshift and of the recently discovered giant arcs in clusters of galaxies (both probably effects of gravitational lensing), and is part and parcel of the black hole phenomenon. In a closely related effect first noted by Irwin Shapiro, radiation passing near a gravitating body is delayed in its flight in proportion to the body's mass, a time delay verified by means of radar waves deflected by the sun on their way from Earth to Mercury and back.

The second effect is the precession of the periastron of a binary system. According to newtonian gravitation, the orbit of each member of a binary is a coplanar ellipse with orientation fixed in space. General relativity predicts a slow rotation of the ellipse's major axis in the plane of the orbit (precession of the periastron). Originally verified in the motion of Mercury, the precession has of late also been detected in the orbits of binary pulsars.

All three effects mentioned depend on features of General Relativity beyond the weak equivalence principle. Indeed, Einstein built into general relativity the much more encompassing ``strong equivalence principle'': the local forms of all nongravitational physical laws and the numerical values of all dimensionless physical constants arc the same in the presence of a gravitational field as in its absence. In practice this implies that within any region in a gravitational field, sufficiently small that space-time curvature may be ignored, all physical laws, when expressed in terms of the space-time metric, have the same forms as required by special relativity in terms of the metric of Minkowski space-time. Thus in a small region in the neighborhood of a black hole (the source of a strong gravitational field) we would describe electromagnetism and optics with the same Maxwell equations used in earthly laboratories where the gravitational field is weak, and we would employ the laboratory values of the electrical permittivity and magnetic susceptibility of the vacuum.

SCALAR TENSOR THEORIES

The strong equivalence principle effectively forces gravitational theory to be General Relativity. Less well tested than the weak version of the principle mentioned earlier, the strong version requires Newton's constant expressed in atomic units to be the same number everywhere, in strong or weak gravitational fields. Stressing that there is very little experimental evidence bearing on this assertion, Dicke and his student Carl Brans proposed in 1961 a modification of general relativity akin to a theory considered earlier by Pascual Jordan. In the Brans-Dicke theory the reciprocal of the gravitational constant is itself a one-component field, the scalar field , that is generated by matter in accordance with an additional equation. Then as well as matter has a say in determining the metric via a modified version of Einstein's equations. Because it involves both metric and scalar fields, the Brans-Dicke theory is dubbed scalar-tensor. Although not complying with the strong equivalence principle, the theory does respect a milder version of it, the Einstein equivalence principle, which asserts that only nongravitational laws and dimensionless constants have their special relativistic forms and values everywhere. Gravitation theorists call theories obeying the Einstein equivalence principle metric theories.

The Brans-Dicke theory also reduces to Newtonian theory for systems with small velocities and weak potentials: It has a newtonian limit. In fact, Brans-Dicke theory is distinguishable from general relativity only by the value of its single dimensionless parameter which determines the effectiveness of matter in producing . The larger , the closer the Brans-Dicke theory predictions are to general relativity. Both theories predict the same gravitational redshift effect, although they predict slightly different light deflection and periastron precession effects; the differences vanish in the limit of infinite . Measurements of Mercury's perihelion precession, radar flight time delay, and radio wave deflection by the sun indicate that is at least several hundred.

Initially a popular alternative to General Relativity, the Brans-Dicke theory lost favor as it became clear that must be very large-an artificial requirement in some views. Nevertheless, the theory has remained a paradigm for the introduction of scalar fields into gravitational theory, and as such has enjoyed a renaissance in connection with theories of higher dimensional space-time.

However, constancy of is not conceptually required. In the generic scalar-tensor theory studied by Peter Bergmann, Robert Wagoner, and Kenneth Nordtvedt, is itself a general function of (). It remains true that in regions of space-time where () is numerically large, the theory's predictions approach those of general relativity. It is even possible for () to evolve systematically in the favored direction. Thus in the variable mass theory (VMT, see Table 1), a scalar-tensor theory devised to test the necessity for the strong equivalence principle, the expansion of the universe forces evolution of toward a particular value at which () diverges. Thus, late in the history of the universe (and today is late), localized gravitational systems are accurately described by general relativity although the assumed gravitational theory is scalar-tensor.

Table 1. Comparison of Selected Gravitational Theories

Theory	Metric	Other Fields	Free Elements	Status
Newtonian (1687)1	Nonmetric	Potential	None	Nonrelativistic theory
Nordstrom (1913)1,2	Minkowski	Scalar	None	Fails to predict observed light detection
Einstein's General Relativity (1915)1, 2	Dynamic	None	None	Viable
Belifante-Swihart (1957)2	Nonmetric	Tensor	K parameter	Contradicted by Dicke-Braginsky experiments
Brans-Dicke (1961)1-3 Generic Scalar	Dynamic	Scalar	parameter	Viable for > 500
Tensor (1970)2	Dynamic	Scalar	2 free functions	Viable
Ni (1970)1, 2	Minkowski	Tensor, Vector, and Scalar	One parameter, 3 functions	Predicts unobserved preferred-frame effects
Will-Nordtvedt (1972)2	Dynamic	Vector	None	Viable
Rosen (1973)2	Fixed	Tensor	None	Contradicted by binary pulsar data
Rastall (1976)2	Minkowski	Tensor, vector	None	Viable
VMT (1977)2	Dynamic	Scalar	2 parameters	Viable for a wide range of the parameters
MOND (1983)4	Nonmetric	Potential	Free function	Nonrelativistic theory
1 Misner, Thorne, and Wheeler (1973)
2 Will (1981)
3 Dicke (1965)
4 Milgrom (1989)

More than two score relativistic theories of gravitation have been proposed. Some have no metric; others take the metric as fixed, not dynamic. These have usually fared badly in light of experiment. Among metric theories those involving a vector field or a tensor field additional to the metric can display a preferred frame of reference or spatial anisotropy effects (phenomena that depend on direction in space). Both effects may contradict a variety of modern experiments. Table 1 gives a sample of theories of gravitation, summarizing the main ingredients of each theory and its experimental status.

All relativistic gravitational theories mentioned so far have a newtonian limit, a tacit requirement of candidate relativistic gravitational theories until very recently. Now, if the correct gravitational theory is general relativity or any of its traditional imitations, then newtonian theory should satisfactorily describe galaxies and clusters of galaxies, astrophysical systems involving small velocities and weak potentials. But there is mounting observational evidence that this can be the case only if galaxies and clusters of galaxies are postulated to contain large amounts of dark matter. Thus far this dark matter has not been detected independently of the preceding argument.

Might not this missing mass puzzle signal instead the break-down of the newtonian limit of gravitational theory for very large systems? In this connection several schemes alternative to Newtonian theory have been proposed. A well developed one is the modified newtonian dynamics or MOND (see Table 1), in which the relation between newtonian potential and the resulting acceleration is regarded as departing from newtonian form for gravitational fields with magnitude of below 10^-8 cm s^-2. In galaxies and clusters of galaxies (with no dark matter assumed) the gravitational fields are weaker than this, and a breakdown of newtonian predictions having nothing to do with dark matter is expected. With its one postulated relation, MOND ties together a number of empirical relations in extragalactic astronomy. A nonrelativistic gravitational theory containing the MOND relation has been set forth, and relativistic generalizations of these ideas are currently under study.

1. Dicke, R. (1965). The Theoretical Significance of Experimental Relativity. Gordon and Breach, New York.
2. Milgrom, M. (1989). Alternatives to dark matter. Comments Astrophysics 13 215.
3. Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Graviation. W.H. Freeman, San Francisco.
4. Will, C. (1981). Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge.
5. Will, C. (1986). Was Einstein Right? Basic Books, New York.
6. See also Black Holes, Stellar, Observational Evidence; Black Holes, Theory; Dark Matter, Cosmological; Gravitational Lenses; Missing Mass, Galactic; Pulsars, Binary; Stars, Neutron, Physical Properties and Models.