In a pioneering calculation using his early form of the PPN formalism, Nordtvedt [196] 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 [228] 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 has
the form
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 (
) level, and that may soon approach
(
). 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
(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 [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
. These results can be summarized by the Nordtvedt parameter bound
.
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 [296].
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 [201].
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 [86], which can be bounded using
statistical methods, given enough WBMSPs (see [243] for a review and references). Using data
from 21 WBMSPs, including recently discovered highly circular systems, Stairs et al. [244
]
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) [281]). An bound on
of
from the period derivatives
of 21 millisecond pulsars was reported in [26, 244]; improved bounds on
were achieved using LLR
data [191], and using observations of the circular binary orbit of the pulsar J2317+1439 [25].
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
does change with cosmic evolution, its rate of variation should be of the order of the expansion rate of the
universe, i.e.
, where
is the Hubble expansion parameter and is given by
, where current observations of the expansion of the
universe give
.
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) [281
]), 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
per year. For an initial analysis
along these lines, see [212
]. It has been suggested that radar observations of the planned 2012
Bepi-Colombo Mercury orbiter mission over a two-year integration with
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
.
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
, given
by
[200]. Thus, the bounds quoted in Table 5 for the binary pulsar PSR
1913+16 and others [143] (see also [87
]) 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 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
then was within 20 percent of
today.
Assuming a power-law variation of
then yields a bound on
today shown in
Table 5.
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