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. In particular, one can
view the part of the PPN metric (see Box 2) as an analogue of the vector potential of
electrodynamics. In a suitable gauge, and dropping the preferred-frame terms, it can be written
Gravitomagnetism plays a role in a variety of measured relativistic effects involving moving material
sources, such as the Earth-Moon system and binary pulsar systems. Nordtvedt [199, 198] has argued that,
if the gravitomagnetic potential (55) were turned off, then there would be anomalous orbital effects in LLR
and binary pulsar data.
Rotation also produces a gravitomagnetic effect, since for a rotating body, , where
is the angular momentum of the body. The result is a “dragging of inertial frames” around the body,
also called the Lense-Thirring effect. A consequence is a precession of a gyroscope’s spin
according to
The Relativity Gyroscope Experiment (Gravity Probe B or GPB) at Stanford University, in
collaboration with NASA and Lockheed-Martin Corporation [246], recently completed a space mission to
detect this frame-dragging or Lense-Thirring precession, along with the “geodetic” precession (see
Section 3.7.2). A set of four superconducting-niobium-coated, spherical quartz gyroscopes were flown in a
polar Earth orbit ( mean altitude, 0.0014 eccentricity), and the precessions of the gyroscopes
relative to a distant guide star (HR 8703, IM Pegasi) were measured. For the given orbit, the
predicted secular angular precession of the gyroscopes is in a direction perpendicular to the
orbital plane at a rate
. The accuracy goal of the
experiment is about 0.5 milliarcseconds per year. The spacecraft was launched on April 20, 2004, and
the mission ended in September 2005, as scheduled, when the remaining liquid helium boiled
off.
It is too early to know whether the relativistic precessions were measured in the amount predicted by GR, because an important calibration of the instrument exploits the effect of the aberration of starlight on the pointing of the on-board telescope toward the guide star, and completing this calibration required the full mission data set. In addition, part of the measured effect includes the motion of the guide star relative to distant inertial frames. This was measured before, during and after the mission separately by radio astronomers at Harvard/SAO and elsewhere using VLBI, and the results of those measurements were to be strictly embargoed until the GPB team has completed its analysis of the gyro data. Final results from the experiment are expected in 2006.
Another way to look for frame-dragging is to measure the precession of orbital planes of bodies circling a rotating body. One implementation of this idea is to measure the relative precession, at about 31 milliarcseconds per year, 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 (126 degrees per year) 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 concrete plans exist at present to launch a third satellite in a supplementary orbit. Nevertheless, Ciufolini and Pavlis [56] combined nodal precession data from LAGEOS I and II with improved models for the Earth’s multipole moments provided by two recent orbiting geodesy satellites, Europe’s CHAMP (Challenging Minisatellite Payload) and NASA’s GRACE (Gravity Recovery and Climate Experiment), and reported a 5 - 10 percent confirmation of GR. In earlier reports, Ciufolini et al. had reported tests at the the 20 - 30 percent level, without the benefit of the GRACE/CHAMP data [55, 57, 54]. Some authors stressed the importance of adequately assessing systematic errors in the LAGEOS data [226, 133].
A gyroscope moving through curved spacetime suffers a precession of its spin axis given by
where For the GPB gyroscopes orbiting the Earth, the precession is 6.6 arcseconds per year. A goal of GPB is
to measure this effect to ; if achieved, this could bound the parameter
to a part in
, not
competitive with the Cassini bound.
Of the five “conservation law” PPN parameters ,
,
,
, and
, only three,
,
,
and
, have been constrained directly with any precision;
is constrained indirectly
through its appearance in the Nordtvedt effect parameter
, Equation (53
). There is strong
theoretical evidence that
, 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 (
), internal energy (
), and
pressure (
). From such considerations, there follows [275] 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
.
A remarkable planetary test was reported by Bartlett and van Buren [22]. 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 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
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 LLR, 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
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
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