*Jacob D. Bekenstein*

A theory of gravitation is a description of the long range forces thatelectrically neutral bodies exert on one another because of theirmatter content. Until the 1910s Sir Isaac Newton's law of universalgravitation, *two particles attract each other with a central forceproportional to the product of their masses and inversely proportionalto the square of the distance between them*, was accepted as thecorrect and complete theory of gravitation: The proportionalityconstant here is Newton's constant *G* = 6.67 x 10^{-8} dyncm^{2} g^{-2}, alsocalled the gravitational constant. This theory is highly accurate inits predictions regarding everyday phenomena. However, high precisionmeasurements of motions in the solar system and in binary pulsars, thestructure of black holes, and the expansion of the universe can onlybe fully understood in terms of a relativistic theory ofgravitation. Best known of these is Albert Einstein's general theoryof relativity, which reduces to Newton's theory in a certain limit. Ofthe scores of rivals to general relativity formulated over the lasthalf century, many have failed various experimental tests, but theverdict is not yet in on which extant relativistic gravitation theoryis closest to the truth.

In alternative language, newtonian gravitational theory states thatthe acceleration **a** (the rate of change of the velocity **v**)imparted bygravitation on a test particle is determined by the gravitationalpotential ,

**a** = -*d***v** / *dt* = -,

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

**·** = 4*G*.

This formulation is entirely equivalent to Newton's law ofgravitation. Because a test particle's acceleration depends only onthe potential generated by matter in the surroundings, the theoryrespects the weak equivalence principle: *the motion of a particle isindependent of its internal structure or composition*. As the subjectof Galileo Galilei's apocryphal experiment at the tower of Pisa, thisprinciple is supported by a series of high precision experimentsculminating in those directed by Baron Lorand von Eötvos in Budapestin 1922, Robert Dicke at Princeton in 1964, and Vladimir Braginsky inMoscow in 1972.

Highly successful in everyday applications, newtonian gravitationhas also proved accurate in describing motions in the solar system(except for tiny relativistic effects), the internal structure ofplanets, the sun and other stars, orbits in binary and multiplestellar systems, the structure of molecular clouds, and, in a roughway, the structure of galaxies and clusters of galaxies (but seebelow).

### THE GENERAL THEORY OF RELATIVITY

According to newtonian theory, gravitational effects propagate fromplace to place instantaneously. With the advent of Einstein's specialtheory of relativity in 1905, a theory uniting the concepts of spaceand time into that of four dimensional flat space-time (namedMinkowski space-time after the mathematician Hermann Minkowski), aproblem became discernible with newtonian theory. According to specialrelativity, which is the current guideline to the form of all physicaltheory, the speed of light, *c* = 3 x 10^{10} cms^{-1}, is the top speed allowedto physical particles or forces: There can be no instantaneouspropagation. After a decade of search for new concepts to makegravitational theory compatible with the spirit of special relativity,Einstein came up with the theory of general relativity (1915), theprototype of all modern gravitational theories. Its crucialingredient, involving a colossal intellectual jump, is the concept ofgravitation, not as a force, but as a manifestation of the curvatureof space-time, an idea first mentioned in rudimentary form by themathematician Ceorg Bernhard Riemann in 1854. In Einstein's handsgravitation theory was thus transformed from a theory of forces intothe first dynamical theory of geometry, the geometry of fourdimensional curved space-time.

Why talk of curvature? One of Einstein's first predictions was thegravitational redshift: As any wave, such as light, propagates awayfrom a gravitating mass, all frequencies in it are reduced by anamount proportional to the change in gravitational potentialexperienced by the wave. This redshift has been measured in thelaboratory, in solar observations, and by means of high precisionclocks flown in airplanes. However, imagine for a moment that generalrelativity had not yet been invented, but the redshift has alreadybeen measured. According to a simple argument owing to Alfred Schild,wave propagation under stationary circumstances can display a redshiftonly if the usual geometric relations implicit in Minkowski space-timeare violated: The space-time must be curved. The observations of theredshift thus show that space-time must be curved in the vicinity ofmasses, regardless of the precise form of the gravitational theory.

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

Despite the great contrast between General Relativity and Newtoniantheory, predictions of the former approach the latter for systems inwhich velocities are small compared to *c* and gravitational potentialsare weak enough that they cannot cause larger velocities. This is whywe can discuss with newtonian theory the structure of the earth andplanets, stars and stellar clusters, and the gross features of motionsin 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 slightdeflection proportional to that body's mass. First verified byobservations 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'' withidentical redshift and of the recently discovered giant arcs inclusters of galaxies (both probably effects of gravitational lensing),and is part and parcel of the black hole phenomenon. In a closelyrelated effect first noted by Irwin Shapiro, radiation passing near agravitating body is delayed in its flight in proportion to the body'smass, a time delay verified by means of radar waves deflected by thesun on their way from Earth to Mercury and back.

The second effect is the precession of the periastron of a binarysystem. According to newtonian gravitation, the orbit of each memberof a binary is a coplanar ellipse with orientation fixed inspace. General relativity predicts a slow rotation of the ellipse'smajor axis in the plane of the orbit (precession of theperiastron). Originally verified in the motion of Mercury, theprecession has of late also been detected in the orbits of binarypulsars.

All three effects mentioned depend on features of General Relativitybeyond the weak equivalence principle. Indeed, Einstein built intogeneral relativity the much more encompassing ``strong equivalenceprinciple'': *the local forms of all nongravitational physical laws andthe numerical values of all dimensionless physical constants arc thesame in the presence of a gravitational field as in its absence*. Inpractice this implies that within any region in a gravitational field,sufficiently small that space-time curvature may be ignored, allphysical laws, when expressed in terms of the space-time metric, havethe same forms as required by special relativity in terms of themetric of Minkowski space-time. Thus in a small region in theneighborhood of a black hole (the source of a strong gravitationalfield) we would describe electromagnetism and optics with the sameMaxwell equations used in earthly laboratories where the gravitationalfield is weak, and we would employ the laboratory values of theelectrical permittivity and magnetic susceptibility of the vacuum.

### SCALAR TENSOR THEORIES

The strong equivalence principle effectively forces gravitationaltheory to be General Relativity. Less well tested than the weakversion of the principle mentioned earlier, the strong versionrequires Newton's constant expressed in atomic units to be the samenumber everywhere, in strong or weak gravitational fields. Stressingthat there is very little experimental evidence bearing on thisassertion, Dicke and his student Carl Brans proposed in 1961 amodification of general relativity akin to a theory considered earlierby Pascual Jordan. In the Brans-Dicke theory the reciprocal of thegravitational constant is itself a one-component field, the scalarfield , that is generated bymatter in accordance with an additionalequation. Then as well asmatter has a say in determining the metricvia a modified version of Einstein's equations. Because it involvesboth metric and scalar fields, the Brans-Dicke theory is dubbedscalar-tensor. Although not complying with the strong equivalenceprinciple, the theory does respect a milder version of it, theEinstein equivalence principle, which asserts that onlynongravitational laws and dimensionless constants have their specialrelativistic forms and values everywhere. Gravitation theorists calltheories obeying the Einstein equivalence principle metric theories.

The Brans-Dicke theory also reduces to Newtonian theory for systemswith small velocities and weak potentials: It has a newtonianlimit. In fact, Brans-Dicke theory is distinguishable from generalrelativity only by the value of its single dimensionless parameter which determines the effectiveness of matter in producing . Thelarger , the closer theBrans-Dicke theory predictions are to generalrelativity. Both theories predict the same gravitational redshifteffect, although they predict slightly different light deflection andperiastron precession effects; the differences vanish in the limit ofinfinite . Measurements ofMercury's perihelion precession, radarflight time delay, and radio wave deflection by the sun indicate that is at least several hundred.

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

However, constancy of isnot conceptually required. In the genericscalar-tensor theory studied by Peter Bergmann, Robert Wagoner, andKenneth Nordtvedt, isitself a general function of (). It remainstrue that in regions of space-time where () is numerically large,the theory's predictions approach those of general relativity. It iseven possible for () to evolvesystematically in the favoreddirection. Thus in the variable mass theory (VMT, seeTable 1), ascalar-tensor theory devised to test the necessity for the strongequivalence principle, the expansion of the universe forces evolutionof toward a particular value atwhich () diverges. Thus, late inthe history of the universe (and today is late), localizedgravitational systems are accurately described by general relativityalthough the assumed gravitational theory is scalar-tensor.

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) |

### OTHER THEORIES

More than two score relativistic theories of gravitation have beenproposed. Some have no metric; others take the metric as fixed, notdynamic. These have usually fared badly in light of experiment. Amongmetric theories those involving a vector field or a tensor fieldadditional to the metric can display a preferred frame of reference orspatial anisotropy effects (phenomena that depend on direction inspace). Both effects may contradict a variety of modern experiments.Table 1 gives a sample of theories ofgravitation, summarizing themain ingredients of each theory and its experimental status.

All relativistic gravitational theories mentioned so far have anewtonian limit, a tacit requirement of candidate relativisticgravitational theories until very recently. Now, if the correctgravitational theory is general relativity or any of its traditionalimitations, then newtonian theory should satisfactorily describegalaxies and clusters of galaxies, astrophysical systems involvingsmall velocities and weak potentials. But there is mountingobservational evidence that this can be the case only if galaxies andclusters of galaxies are postulated to contain large amounts of darkmatter. Thus far this dark matter has not been detected independentlyof the preceding argument.

Might not this missing mass puzzle signal instead the break-down ofthe newtonian limit of gravitational theory for very large systems? Inthis connection several schemes alternative to Newtonian theory havebeen proposed. A well developed one is the modified newtonian dynamicsor MOND (see Table 1), in which the relationbetween newtonianpotential and the resulting acceleration is regarded as departing fromnewtonian form for gravitational fields with magnitude of below10^{-8} cm s^{-2}. In galaxies and clusters of galaxies(with no dark matterassumed) the gravitational fields are weaker than this, and abreakdown of newtonian predictions having nothing to do with darkmatter is expected. With its one postulated relation, MOND tiestogether a number of empirical relations in extragalactic astronomy. Anonrelativistic gravitational theory containing the MOND relation hasbeen set forth, and relativistic generalizations of these ideas arecurrently under study.

**Additional Reading**

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