where 
is the negative of the gravitational self-energy
of the body (
). 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,
, far below the sensitivity of
current or future experiments. However, for astronomical bodies,
may be significant (
for the Sun, 
for Jupiter,
for the Earth, 
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
where 
and 
are the angular frequencies
of the orbits of the Moon and Sun around the Earth (see
TEGP 8.1 [147
] for detailed derivations and references; for
improved
calculations of the numerical coefficient, see [102, 53]).
Since August 1969, when the first successful acquisition was made
of a laser signal reflected from the Apollo 11 retroreflector on
the Moon, the lunar laser-ranging experiment (LURE) 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
approaching 50 ps (1 cm). 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 observatory, 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.
Several independent analyses of the data found no evidence, within
experimental uncertainty, for the Nordtvedt effect (for recent results
see [56, 154, 96]). Their results
can be summarized by the bound 
.
These results represent a limit on a possible violation of WEP for
massive bodies of 5 parts in 
(compare Figure 1). For
Brans-Dicke theory, these results force a lower limit on the
coupling constant 
of 1000. Note that,
at this level of precision, one cannot regard the results of lunar laser
ranging 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 four parts in [10] reduces the ambiguity in the Lunar laser ranging
bound, and establishes the firm limit on the universality of
acceleration of gravitational binding energy at the level of 
.
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 [102].
, 
, 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) [147]). An improved bound on of 
from the period derivatives of 20 millisecond pulsars
was reported in [13]; improved bounds on
were achieved using lunar laser ranging data [95],
and using
observations of the circular binary
orbit of the pulsar J2317+1439 [12].
Negative searches for these effects have
produced strong constraints on the PPN parameters (Table 4).
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 
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
using methods that include studies of the evolution of the Sun,
observations of lunar occultations (including analyses of ancient
eclipse data), lunar laser-ranging measurements,
planetary
radar-ranging measurements, and pulsar timing data.
Laboratory experiments may one day lead to interesting limits (for
review and references to past work see TEGP 8.4 and 14.3 (c) [147]). Recent
results are shown in Table 5.
Table 5: Constancy of the gravitational constant. For the
pulsar data, the bounds are dependent upon the theory of gravity in
the strong-field regime and on neutron star equation of state.
The best limits on still
come from ranging measurements to the Viking landers and
Lunar laser ranging measurements [56, 154, 96].
It has
been suggested that radar observations of a Mercury orbiter over a
two-year mission (30 cm accuracy in range) could yield
.
Although bounds on from solar-system measurements can be
correctly
obtained in a phenomenological manner through the simple expedient of
replacing G 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 
[101]. Thus,
the bounds quoted in Table 5
for the binary pulsar PSR 1913+16 [51] and the
pulsar PSR 0655+64 [69]
are theory-dependent and must be treated as merely
suggestive.
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The Confrontation between General Relativity and Experiment Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
