In a pioneering calculation using his early form of the PPN formalism, Nordtvedt  showed that many metric theories of gravity predict that massive bodies violate the weak equivalence principle – that is, fall with different accelerations depending on their gravitational self-energy. Dicke  argued that such an effect would occur in theories with a spatially varying gravitational constant, such as scalar-tensor gravity. For a spherically symmetric body, the acceleration from rest in an external gravitational potential U has the formEg is the negative of the gravitational self-energy of the body (Eg 0). This violation of the massive-body equivalence principle is known as the “Nordtvedt effect”. The effect is absent in GR () but present in scalar-tensor theory (). The existence of the Nordtvedt effect does not violate the results of laboratory Eötvös experiments, since for laboratory-sized objects Eg / m 10–27, far below the sensitivity of current or future experiments. However, for astronomical bodies, Eg / m may be significant (3.6 10–6 for the Sun, 10–8 for Jupiter, 4.6 10–10 for the Earth, 0.2 10–10 for the Moon). If the Nordtvedt effect is present () then the Earth should fall toward the Sun with a slightly different acceleration than the Moon. This perturbation in the Earth-Moon orbit leads to a polarization of the orbit that is directed toward the Sun as it moves around the Earth-Moon system, as seen from Earth. This polarization represents a perturbation in the Earth-Moon distance of the form  for detailed derivations and references; for improved calculations of the numerical coefficient, see [201, 89]).
Since August 1969, when the first successful acquisition was made of a laser signal reflected from the Apollo 11 retroreflector on the Moon, the LLR experiment has made regular measurements of the round-trip travel times of laser pulses between a network of observatories and the lunar retroreflectors, with accuracies that are at the 50 ps (1 cm) level, and that may soon approach 5 ps (1 mm). These measurements are fit using the method of least-squares to a theoretical model for the lunar motion that takes into account perturbations due to the Sun and the other planets, tidal interactions, and post-Newtonian gravitational effects. The predicted round-trip travel times between retroreflector and telescope also take into account the librations of the Moon, the orientation of the Earth, the location of the observatories, and atmospheric effects on the signal propagation. The “Nordtvedt” parameter along with several other important parameters of the model are then estimated in the least-squares method.
Numerous ongoing analyses of the data find no evidence, within experimental uncertainty, for the Nordtvedt effect [295, 296] (for earlier results see [95, 294, 192]). These results represent a limit on a possible violation of WEP for massive bodies of about 1.4 parts in 1013 (compare Figure 1).
However, at this level of precision, one cannot regard the results of LLR as a “clean” test of SEP until one eliminates the possibility of a compensating violation of WEP for the two bodies, because the chemical compositions of the Earth and Moon differ in the relative fractions of iron and silicates. To this end, the Eöt-Wash group carried out an improved test of WEP for laboratory bodies whose chemical compositions mimic that of the Earth and Moon. The resulting bound of 1.4 parts in 1013 [19, 2] from composition effects reduces the ambiguity in the LLR bound, and establishes the firm SEP test at the level of about 2 parts in 1013. These results can be summarized by the Nordtvedt parameter bound = (4.4 4.5) 10–4.
In the future, the Apache Point Observatory for Lunar Laser ranging Operation (APOLLO) project, a joint effort by researchers from the Universities of Washington, Seattle, and California, San Diego, plans to use enhanced laser and telescope technology, together with a good, high-altitude site in New Mexico, to improve the LLR bound by as much as an order of magnitude .
In GR, the Nordtvedt effect vanishes; at the level of several centimeters and below, a number of non-null general relativistic effects should be present .
Tests of the Nordtvedt effect for neutron stars have also been carried out using a class of systems known as wide-orbit binary millisecond pulsars (WBMSP), which are pulsar–white-dwarf binary systems with small orbital eccentricities. In the gravitational field of the galaxy, a non-zero Nordtvedt effect can induce an apparent anomalous eccentricity pointed toward the galactic center , which can be bounded using statistical methods, given enough WBMSPs (see  for a review and references). Using data from 21 WBMSPs, including recently discovered highly circular systems, Stairs et al.  obtained the bound , where . Because for typical neutron stars, this bound does not compete with the bound on from LLR; on the other hand, it does test SEP in the strong-field regime because of the presence of the neutron stars.
Some theories of gravity violate SEP by predicting that the outcomes of local gravitational experiments may depend on the velocity of the laboratory relative to the mean rest frame of the universe (preferred-frame effects) or on the location of the laboratory relative to a nearby gravitating body (preferred-location effects). In the post-Newtonian limit, preferred-frame effects are governed by the values of the PPN parameters , , and , and some preferred-location effects are governed by (see Table 2).
The most important such effects are variations and anisotropies in the locally-measured value of the gravitational constant which lead to anomalous Earth tides and variations in the Earth’s rotation rate, anomalous contributions to the orbital dynamics of planets and the Moon, self-accelerations of pulsars, and anomalous torques on the Sun that would cause its spin axis to be randomly oriented relative to the ecliptic (see TEGP 8.2, 8.3, 9.3, and 14.3 (c) ). An bound on of 4 10–20 from the period derivatives of 21 millisecond pulsars was reported in [26, 244]; improved bounds on were achieved using LLR data , and using observations of the circular binary orbit of the pulsar J2317+1439 . Negative searches for these effects have produced strong constraints on the PPN parameters (see Table 4).
Most theories of gravity that violate SEP predict that the locally measured Newtonian gravitational constant may vary with time as the universe evolves. For the scalar-tensor theories listed in Table 3, the predictions for can be written in terms of time derivatives of the asymptotic scalar field. Where G does change with cosmic evolution, its rate of variation should be of the order of the expansion rate of the universe, i.e. , where H0 is the Hubble expansion parameter and is given by H0 = 100 h km s–1 Mpc–1 = 1.02 10–10 h yrˆ-1, where current observations of the expansion of the universe give h 0.73 0.03.
Several observational constraints can be placed on , one kind coming from bounding the present rate of variation, another from bounding a difference between the present value and a past value. The first type of bound typically comes from LLR measurements, planetary radar-ranging measurements, and pulsar timing data. The second type comes from studies of the evolution of the Sun, stars and the Earth, big-bang nucleosynthesis, and analyses of ancient eclipse data. Recent results are shown in Table 5.
The best limits on a current come from LLR measurements (for earlier results see [95, 294, 192]). These have largely supplanted earlier bounds from ranging to the 1976 Viking landers (see TEGP, 14.3 (c) ), which were limited by uncertain knowledge of the masses and orbits of asteroids. However, improvements in knowledge of the asteroid belt, combined with continuing radar observations of planets and spacecraft, notably the Mars Global Surveyor (1998 – 2003) and Mars Odyssey (2002 – present), may enable a bound on at the level of a part in 1013 per year. For an initial analysis along these lines, see . It has been suggested that radar observations of the planned 2012 Bepi-Colombo Mercury orbiter mission over a two-year integration with 6 cm rms accuracy in range could yield ; an eight-year mission could improve this by a factor 15 [187, 17].
Although bounds on from solar-system measurements can be correctly obtained in a phenomenological manner through the simple expedient of replacing by in Newton’s equations of motion, the same does not hold true for pulsar and binary pulsar timing measurements. The reason is that, in theories of gravity that violate SEP, such as scalar-tensor theories, the “mass” and moment of inertia of a gravitationally bound body may vary with variation in G. Because neutron stars are highly relativistic, the fractional variation in these quantities can be comparable to , the precise variation depending both on the equation of state of neutron star matter and on the theory of gravity in the strong-field regime. The variation in the moment of inertia affects the spin rate of the pulsar, while the variation in the mass can affect the orbital period in a manner that can subtract from the direct effect of a variation in G, given by . Thus, the bounds quoted in Table 5 for the binary pulsar PSR 1913+16 and others  (see also ) are theory-dependent and must be treated as merely suggestive.
In a similar manner, bounds from helioseismology and big-bang nucleosynthesis (BBN) assume a model for the evolution of G over the multi-billion year time spans involved. For example, the concordance of predictions for light elements produced around 3 minutes after the big bang with the abundances observed indicate that G then was within 20 percent of G today. Assuming a power-law variation of then yields a bound on today shown in Table 5.
© Max Planck Society and the author(s)