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3.7 Other tests of post-Newtonian gravity

3.7.1 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. In particular, one can view the g0i 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

g = − 1-(4 γ + 4 + α )V . (55 ) 0i 2 1 i
At PN order, this contributes a Lorentz-type acceleration v × B g to the equation of motion, where the gravitomagnetic field Bg is given by i Bg = ∇ × (g0ie ).

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 [199198] has argued that, if the gravitomagnetic potential (55View Equation) 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, V = − 12x × J∕r3, where J 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 S according to

( ) dS 1 1 J − 3n (n ⋅ J) dτ- = ΩLT × S, ΩLT = − 2- 1 + γ + 4-α1 ------r3-----, (56 )
where n is a unit radial vector, and r is the distance from the center of the body (TEGP 9.1 [281Jump To The Next Citation Point]).

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 (642 km 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 1 1 −3 −1 2(1 + γ + 4α1 ) × 41 × 10 arcsec yr. 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 [555754]. Some authors stressed the importance of adequately assessing systematic errors in the LAGEOS data [226133].

3.7.2 Geodetic precession

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

( ) dS- 1- dτ = ΩG × S, ΩG = γ + 2 v × ∇U, (57 )
where v is the velocity of the gyroscope, and U is the Newtonian gravitational potential of the source (TEGP 9.1 [281Jump To The Next Citation Point]). 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 0.6 percent using LLR data [95294295].

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 8 × 10–5; if achieved, this could bound the parameter γ to a part in 104, not competitive with the Cassini bound.

3.7.3 Tests of post-Newtonian conservation laws

Of the five “conservation law” PPN parameters ζ1, ζ2, ζ3, ζ4, and α3, only three, ζ2, ζ3, and α3, have been constrained directly with any precision; ζ1 is constrained indirectly through its appearance in the Nordtvedt effect parameter ηN, Equation (53View Equation). There is strong theoretical evidence that ζ4, 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 (ρv2), internal energy (ρΠ), and pressure (p). From such considerations, there follows [275] the additional theoretical constraint

6ζ4 = 3α3 + 2 ζ1 − 3ζ3. (58 )

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 105.

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 10 km from the center of mass of an aluminum-rich, iron-poor crust. The direction of offset is toward the Earth, about 14°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 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

||(mA ∕mP )Al − (mA ∕mP )Fe|| ||-------------------------|| < 4 × 10−12. (59 ) (mA ∕mP )Fe
According to the PPN formalism, in a theory of gravity that violates conservation of momentum, but that obeys the constraint of Equation (58View Equation), the electrostatic binding energy E e of an atomic nucleus could make a contribution to the ratio of active to passive mass of the form
1 mA = mP + -ζ3Ee. (60 ) 2
The resulting limit on ζ3 from the lunar experiment is ζ3 < 1 × 10−8 (TEGP 9.2, 14.3 (d) [281Jump To The Next Citation Point]). Nordtvedt [203] has examined whether this bound could be improved by considering the asymmetric distribution of ocean water on Earth.

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

1- m-μ-δm------e------ aCM = − 2 (ζ2 + α3 )a2a m (1 − e2)3∕2nP, (61 )
where δm = m1 − m2, a is the semi-major axis, and nP is a unit vector directed from the center of mass to the point of periastron of m1 (TEGP 9.3 [281Jump To The Next Citation Point]). A consequence of this acceleration would be non-vanishing values for d2P∕dt2, where P denotes the period of any intrinsic process in the system (orbit, spectra, pulsar periods). The observed upper limit on 2 2 d Pp ∕dt of the binary pulsar PSR 1913+16 places a strong constraint on such an effect, resulting in the bound −5 |α3 + ζ2| < 4 × 10. Since α3 has already been constrained to be much less than this (see Table 4), we obtain a strong solitary bound on ζ2 < 4 × 10− 5 [280].
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