A non-zero value for any of these parameters would result in a violation of conservation of momentum, or of Newton's third law in gravitating systems. An alternative statement of Newton's third law for gravitating systems is that the ``active gravitational mass'', that is the mass that determines the gravitational potential exhibited by a body, should equal the ``passive gravitational mass'', the mass that determines the force on a body in a gravitational field. Such an equality guarantees the equality of action and reaction and of conservation of momentum, at least in the Newtonian limit.

A classic test of Newton's third law for gravitating systems was carried out in 1968 by Kreuzer, in which the gravitational attraction of fluorine and bromine were compared to a precision of 5 parts in .

A remarkable planetary test was reported by Bartlett and van Buren [11]. They noted that current understanding of the structure of the Moon involves an iron-rich, aluminum-poor mantle whose center of mass is offset about 10 km from the center of mass of an aluminum-rich, iron-poor crust. The direction of offset is toward the Earth, about to the east of the Earth-Moon line. Such a model accounts for the basaltic maria which face the Earth, and the aluminum-rich highlands on the Moon's far side, and for a 2 km offset between the observed center of mass and center of figure for the Moon. Because of this asymmetry, a violation of Newton's third law for aluminum and iron would result in a momentum non-conserving self-force on the Moon, whose component along the orbital direction would contribute to the secular acceleration of the lunar orbit. Improved knowledge of the lunar orbit through lunar laser ranging, and a better understanding of tidal effects in the Earth-Moon system (which also contribute to the secular acceleration) through satellite data, severely limit any anomalous secular acceleration, with the resulting limit

According to the PPN formalism, in a theory of gravity that violates conservation of momentum, but that obeys the constraint of Eq. (39), the electrostatic binding energy of an atomic nucleus could make a contribution to the ratio of active to passive mass of the form

The resulting limit on from the lunar experiment is (TEGP 9.2, 14.3 (d) [147]).

Another consequence of a violation of conservation of momentum is a self-acceleration of the center of mass of a binary stellar system, given by

where
,
*a*
is the semi-major axis, and
is a unit vector directed from the center of mass to the point
of periastron of
(TEGP 9.3 [147]). A consequence of this acceleration would be non-vanishing
values for
, where
*P*
denotes the period of any intrinsic process in the system
(orbit, spectra, pulsar periods). The observed upper limit on
of the binary pulsar PSR 1913+16 places a strong constraint on
such an effect, resulting in the bound
. Since
has already been constrained to be much less than this
(Table
4), we obtain a strong bound on
alone [146].

where
is the velocity of the gyroscope, and
*U*
is the Newtonian gravitational potential of the source (TEGP
9.1 [147]). The Earth-Moon system can be considered as a ``gyroscope'',
with its axis perpendicular to the orbital plane. The predicted
precession is about 2 arcseconds per century, an effect first
calculated by de Sitter. This effect has been measured to about
0.7 percent using Lunar laser ranging data [56,
154].

For a gyroscope orbiting the Earth, the precession is about 8 arcseconds per year. The Stanford Gyroscope Experiment has as one of its goals the measurement of this effect to (see below); if achieved, this would substantially improve the accuracy of the parameter .

where
is the angular momentum of the Earth,
is a unit radial vector, and
*r*
is the distance from the center of the Earth (TEGP 9.1 [147]). For a polar orbit at about 650 km altitude, this leads to a
secular angular precession at a rate
arcsec/yr. The accuracy goal of the experiment is about 0.5
milliarcseconds per year. The science instrument package and the
spacecraft are in the final phases of assembly, with launch
scheduled for 2002.

Another proposal to look for an effect of gravitomagnetism is to measure the relative precession of the line of nodes of a pair of laser-ranged geodynamics satellites (LAGEOS), ideally with supplementary inclination angles; the inclinations must be supplementary in order to cancel the dominant nodal precession caused by the Earth's Newtonian gravitational multipole moments. Unfortunately, the two existing LAGEOS satellites are not in appropriately inclined orbits, and no plans exist at present to launch a third satellite in a supplementary orbit. Nevertheless, by combing nodal precession data from LAGEOS I and II with perigee advance data from the slightly eccentric orbit of LAGEOS II, Ciufolini et al. reported a partial cancellation of multipole effects, and a resulting 20 percent confirmation of GR [34].

The Confrontation between General Relativity and
Experiment
Clifford M. Will
http://www.livingreviews.org/lrr-2001-4
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