2 Equivalence Principle

The gravitational acceleration of massive bodies toward other bodies is dependent on the nonlinear properties of gravity within metric theories of gravity [48Jump To The Next Citation Point]. Tracking this acceleration provides a measurement of how gravity pulls on the gravitational binding energy and how gravitational binding energy affects inertia. This probe of nonlinear gravity is explicitly singled out in measurements of the Parameterized Post-Newtonian parameter β discussed below, but it is also implicitly contained within the Einstein Equivalence Principle.

The Equivalence Principle (EP), which states the equality of gravitational and inertial mass, is central to the theory of GR. The EP comes in two flavors: the weak (WEP) and the strong (SEP). The WEP pertains to nongravitational contributions to mass: namely, Standard Model contributions of nuclear and electromagnetic energy, gluons, plus quark masses and their kinetic energies. Nucleons of differing fractional electro-weak and nuclear binding energies might exhibit different couplings to gravity in the case of a WEP violation. The SEP extends the WEP to include gravitational self-energy of a body, addressing the question of how gravity pulls on itself and, therefore, accessing the nonlinear aspect of gravity.

While the EP must hold true in GR, nearly all alternative theories of gravity predict a violation of the EP at some level. Efforts to formulate a quantum description of gravity generally introduce new scalar or vector fields that violate the EP [13, 14Jump To The Next Citation Point]. These violations manifest themselves in the equations of motion for massive self-gravitating bodies, as well as preferred frame and preferred-location effects on the gravitational constant. GR may be the only metric theory of gravity that is dependent on the SEP holding true [72], distinguishing it from all other theories of gravity. Therefore, probing the validity of the EP is one of the strongest ways of testing GR. This test is often considered one of the most powerful ways to search for new physics beyond the standard model [12].

Precision tests of the EP generally test the Universality of Free Fall (UFF): all test bodies have the same gravitational acceleration in a uniform gravitational field. Tests of the UFF are performed by comparing the gravitational accelerations a1 and a2 of different test bodies:

Δa-- --a1 −-a2-- ( MG--) (MG--) a = 1(a + a ) = M − M , (1 ) 2 1 2 I 1 I 2
where MG is the gravitational mass and MI is the inertial mass of each test body. Laboratory masses lack measurable gravitational self-energy, so the classical Eötvös type experiments, which compare the acceleration of test bodies with different compositions only probe the WEP [60Jump To The Next Citation Point].

In the late 1960s, Nordtvedt recognized that the SEP could be tested by comparing the gravitational acceleration of two massive bodies [48, 47]. For each body, the gravitational to inertial mass ratio can be written as:

MG-- --U-- MI = 1 + ηM c2, (2 )
where U is the gravitational self-energy of the test body:
∫ ′ U = G- ρ-(r)ρ(r-)d3rd3r′, (3 ) 2 |r − r′|
M c2 is the body’s total mass energy, and η is a dimensionless constant that is identically zero if the EP holds true.

For a uniform sphere of radius R, U∕M c2 = − 3 GM ∕5Rc2. However, due to their complex interior structure, the gravitational self-energy for astronomical bodies is generally computed numerically. An Earth model based on the model described in [25] yields a self-energy of [78]:

( U ) ---2- = − 4.64 × 10−10, (4 ) M c Earth
while a Moon with a 20% iron core yields [75]:
( ) -U--- −11 M c2 Moon = − 1.90 × 10 . (5 )
Nordtvedt realized that a violation of the EP would cause the Earth and Moon to fall at different rates toward the Sun resulting in a polarization of the lunar orbit [49]. This polarization shows up in LLR as a displacement along the Earth-Sun line with a 29.53 day synodic period. Detailed solutions to the equations of motion for the Earth-Moon-Sun system [54, 15] find that the radial perturbation of the Earth-Moon distance due to an EP violation is
[(M ) (M ) ] δr = − 2.9427 × 1010 ---G − --G- cosD [m ], (6 ) MI Earth MI Moon
where D is the angle between the mean longitude of the Moon and the mean longitude of the Sun as observed from the Earth (synodic period). Combining Equation (2View Equation) with the estimated self-energy for the Earth and Moon, we find that Equation (6View Equation) becomes:
δr = 13.1η cosD [m ]. (7 )
Recent solutions using LLR data yield an EP test numerically comparable with present laboratory limits, at a part in 1013 [79Jump To The Next Citation Point, 60]. Since the Earth and Moon not only have different gravitational self-energy, but also have different compositions the LLR measurements alone do not provide a pure test of SEP [52]. To separate the WEP and SEP effects and eliminate the possibility of a conspiratorial cancellation, the Eöt-Wash group at the University of Washington performed a torsion balance experiment using test masses of similar composition to the Earth and Moon [4]. Combining the torsion balance results with the latest LLR analyses produced the best test of the SEP to date [79Jump To The Next Citation Point, 80Jump To The Next Citation Point]:
( ) Δ MG-- = (− 2.0 ± 2.0) × 10−13, (8 ) MI SEP
η = (4.4 ± 4.5) × 10− 4. (9 )
Because Earth’s self-energy contributes 4.5 × 10–10 of its total mass, this translates to a SEP test of 0.04%.

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