4 Strong Gravity and Gravitational 3 Tests of Post-Newtonian Gravity3.6 Tests of the strong

3.7 Other tests of post-Newtonian gravity 

3.7.1 Tests of post-Newtonian conservation laws 

Of the five ``conservation law'' PPN parameters tex2html_wrap_inline4367, tex2html_wrap_inline4369, tex2html_wrap_inline4371, tex2html_wrap_inline4373, and tex2html_wrap_inline3765, only three, tex2html_wrap_inline4369, tex2html_wrap_inline4371 and tex2html_wrap_inline3765, have been constrained directly with any precision; tex2html_wrap_inline4367 is constrained indirectly through its appearance in the Nordtvedt effect parameter tex2html_wrap_inline3749, Eq. (37Popup Equation). There is strong theoretical evidence that tex2html_wrap_inline4373, which is related to the gravity generated by fluid pressure, is not really an independent parameter - in any reasonable theory of gravity there should be a connection between the gravity produced by kinetic energy (tex2html_wrap_inline4877), internal energy (tex2html_wrap_inline4879), and pressure (p). From such considerations, there follows [141] the additional theoretical constraint


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 tex2html_wrap_inline4197 .

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 tex2html_wrap_inline4885 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. (39Popup Equation), the electrostatic binding energy tex2html_wrap_inline4887 of an atomic nucleus could make a contribution to the ratio of active to passive mass of the form


The resulting limit on tex2html_wrap_inline4371 from the lunar experiment is tex2html_wrap_inline4891 (TEGP 9.2, 14.3 (d) [147Jump To The Next Citation Point In The Article]).

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 tex2html_wrap_inline4893, a is the semi-major axis, and tex2html_wrap_inline4897 is a unit vector directed from the center of mass to the point of periastron of tex2html_wrap_inline4899 (TEGP 9.3 [147Jump To The Next Citation Point In The Article]). A consequence of this acceleration would be non-vanishing values for tex2html_wrap_inline4901, where P denotes the period of any intrinsic process in the system (orbit, spectra, pulsar periods). The observed upper limit on tex2html_wrap_inline4905 of the binary pulsar PSR 1913+16 places a strong constraint on such an effect, resulting in the bound tex2html_wrap_inline4907 . Since tex2html_wrap_inline3765 has already been constrained to be much less than this (Table  4), we obtain a strong bound on tex2html_wrap_inline4369 alone [146].

3.7.2 Geodetic precession 

A gyroscope moving through curved spacetime suffers a precession of its axis given by


where tex2html_wrap_inline4917 is the velocity of the gyroscope, and U is the Newtonian gravitational potential of the source (TEGP 9.1 [147Jump To The Next Citation Point In The Article]). 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 tex2html_wrap_inline4921 (see below); if achieved, this would substantially improve the accuracy of the parameter tex2html_wrap_inline4351 .

3.7.3 Search for gravitomagnetism 

According to GR, moving or rotating matter should produce a contribution to the gravitational field that is the analogue of the magnetic field of a moving charge or a magnetic dipole. Although gravitomagnetism plays a role in a variety of measured relativistic effects, it has not been seen to date, isolated from other post-Newtonian effects (for a discussion of the evidence for gravitomagnetism in solar system measurements and the binary pulsar, see [100, 99]). The Relativity Gyroscope Experiment (Gravity Probe B or GP-B) at Stanford University, in collaboration with NASA and Lockheed-Martin Corporation, is in the advanced stage of developing a space mission to detect this phenomenon directly [71]. A set of four superconducting-niobium-coated, spherical quartz gyroscopes will be flown in a low polar Earth orbit, and the precession of the gyroscopes relative to the distant stars will be measured. In the PPN formalism, the predicted effect of gravitomagnetism is a precession (also known as the Lense-Thirring effect, or the dragging of inertial frames), given by


where tex2html_wrap_inline4929 is the angular momentum of the Earth, tex2html_wrap_inline4177 is a unit radial vector, and r is the distance from the center of the Earth (TEGP 9.1 [147Jump To The Next Citation Point In The Article]). For a polar orbit at about 650 km altitude, this leads to a secular angular precession at a rate tex2html_wrap_inline4935 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].

3.7.4 Improved PPN parameter values 

A number of advanced space missions have been proposed in which spacecraft orbiters or landers and improved tracking capabilities could lead to significant improvements in values of the PPN parameters, of tex2html_wrap_inline4725 of the Sun, and of tex2html_wrap_inline4807 . Doppler tracking of the Cassini spacecraft (launched to orbit and study Saturn in 1997) during its 2003 superior conjunction could measure tex2html_wrap_inline4351 to a few parts in tex2html_wrap_inline4197, by measuring the time variation of the Shapiro delay [77]. A Mercury orbiter, in a two-year experiment, with 3 cm range capability, could yield improvements in the perihelion shift to a part in tex2html_wrap_inline4187, in tex2html_wrap_inline4351 to tex2html_wrap_inline4949, in tex2html_wrap_inline4807 to tex2html_wrap_inline4953, and in tex2html_wrap_inline4725 to a few parts in tex2html_wrap_inline4193 . Proposals are being developed, primarily in Europe, for advanced space missions which will have tests of PPN parameters as key components, including GAIA, a high-precision astrometric telescope (successor to Hipparcos), which could measure light-deflection and tex2html_wrap_inline4351 to the tex2html_wrap_inline4191 level [9]. Nordtvedt [103] has argued that ``grand fits'' of large solar system range data sets, including ranging to Mercury, Mars and the Moon, could yield substantially improved measurements of PPN parameters.

4 Strong Gravity and Gravitational 3 Tests of Post-Newtonian Gravity3.6 Tests of the strong

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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