5 The Nature of Spacetime

The recent and unexpected measurement of the accelerating expansion of the universe has provided new motivation for exploring the nature of spacetime. Models that predict modification of gravity at large distances, such as brane-world models, have recently become of interest [18]. These theories exhibit a strong coupling phenomenon that makes the gravitational force source-dependent. These theories become testable at shorter distances, where the coupling sets in for lighter sources [19]. The Earth-Moon system provides a testbed for investigating the nature of spacetime at solar-system scales. For example, GR predicts that a gyroscope moving through curved spacetime will precess with respect to a rest frame. This is referred to as geodetic or de Sitter precession. The Earth-Moon system behaves as a gyroscope with a predicted geodetic precession of 19.2 msec/year. This is observed using LLR by measuring the lunar perigee precession [8]. The current limit on the deviation of the geodetic procession from the GR prediction is: −3 Kgp = (1.9 ± 6.4) × 10 [79Jump To The Next Citation Point]. This measurement can also be used to set a limit on a possible cosmological constant: − 2 Λ < 10 −26 km [61], which has implications for our understanding of dark energy.

It is also useful to look at violations of GR in the context of metric theories of gravity. Parameterized Post-Newtonian (PPN) formalism provides a convenient way to describe a class of deviations from GR [50Jump To The Next Citation Point]. The most often considered PPN parameters are γ and β: γ indicates how much space curvature is produced per unit mass, while β indicates how nonlinear gravity is (self-interaction). γ and β are identically one in GR. Also of interest are the preferred-frame parameters α1 and α2, which are identically zero in GR [73, 57].

Limits on γ can be set from geodetic precession measurements [50], but the best limits presently come from measurements of the gravitational time delay of light, i.e., the Shapiro effect [62]. Doppler measurements to the Cassini spacecraft set the current limit on γ: (γ − 1) = (2.1 ± 2.3) × 10−5 [9].

The Equivalence Principle parameter η depends on the PPN parameters β and γ:

η = 4β − γ − 3. (14 )
Combining the Cassini value for γ with the LLR value for η provides the best limit on β: −4 (β − 1) = (1.2 ± 1.1) × 10 [79]. Scalar tensor theories with ‘attractors’ for the cosmic background scalar-field dynamics predict a residual γ − 1 and perhaps β − 1 of order 10−7– 10−5 today [14], within reach of advanced LLR and spacecraft time-delay measurements.

A nonzero preferred frame would show up as an oscillation of the lunar range at the sum and the difference of the anomalistic frequency and the annual period [16, 37]. Recent analysis of LLR data sets the current limit on the PPN parameter − 5 α1 = (− 7 ± 9 ) × 10 [38Jump To The Next Citation Point]. LLR has also been used to set a limit on α2 = (1.8 ± 2.5 ) × 10− 5 [38Jump To The Next Citation Point]. However, the close solar spin axis alignment with the total solar system angular momentum produces a much tighter constraint on α2 of order 10− 7 [51].

Lunar laser ranging also places limits on the gravitomagnetic interaction, the same physical interaction that leads to the Lense–Thirring and Schiff precession phenomena as tested by precession of the Laser Geodynamics Satellites (LAGEOS) orbital plane and by the precession of a gyroscope in Gravity Probe B respectively [41]. In the case of the lunar orbit, rotation is not involved, but rather translation of the Earth and Moon point-masses in the solar system barycenter frame that produce 6 meter amplitude range signatures at both the synodic frequency and twice the synodic frequency. The amplitudes of these signatures are frame-dependent, reflecting the deep connection gravitomagnetism has with the covariant property of relativistic dynamics. Soffel et al. showed the need for the gravitomagnetic term in the LLR equations of motion at the level of 0.15%, whether confined to a PPN context or allowed to vary independently [67]. If another experiment claimed a gravitomagnetic, or “frame-dragging” departure from GR at even the 1% level, LLR data would stand in conflict [39].

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