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].
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).
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 |