One elementary equivalence principle is the kind Newton had in mind when he stated that the property of a body called ``mass'' is proportional to the ``weight'', and is known as the weak equivalence principle (WEP). An alternative statement of WEP is that the trajectory of a freely falling body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition. In the simplest case of dropping two different bodies in a gravitational field, WEP states that the bodies fall with the same acceleration (this is often termed the Universality of Free Fall, or UFF).
A more powerful and far-reaching equivalence principle is known as the Einstein equivalence principle (EEP). It states that:
The second piece of EEP is called local Lorentz invariance (LLI), and the third piece is called local position invariance (LPI).
For example, a measurement of the electric force between two charged bodies is a local non-gravitational experiment; a measurement of the gravitational force between two bodies (Cavendish experiment) is not.
The Einstein equivalence principle is the heart and soul of gravitational theory, for it is possible to argue convincingly that if EEP is valid, then gravitation must be a ``curved spacetime'' phenomenon, in other words, the effects of gravity must be equivalent to the effects of living in a curved spacetime. As a consequence of this argument, the only theories of gravity that can embody EEP are those that satisfy the postulates of ``metric theories of gravity'', which are:
The argument that leads to this conclusion simply notes that,
if EEP is valid, then in local freely falling frames, the laws
governing experiments must be independent of the velocity of the
frame (local Lorentz invariance), with constant values for the
various atomic constants (in order to be independent of
location). The only laws we know of that fulfill this are those
that are compatible with special relativity, such as Maxwell's
equations of electromagnetism. Furthermore, in local freely
falling frames, test bodies appear to be unaccelerated, in other
words they move on straight lines; but such ``locally straight''
lines simply correspond to ``geodesics'' in a curved spacetime
(TEGP 2.3 [147]).
General relativity is a metric theory of gravity, but then so are many others, including the Brans-Dicke theory. The nonsymmetric gravitation theory (NGT) of Moffat is not a metric theory. Neither, in this narrow sense, is superstring theory (see Sec. 2.3), which, while based fundamentally on a spacetime metric, introduces additional fields (dilatons, moduli) that can couple to material stress-energy in a way that can lead to violations, say, of WEP. So the notion of curved spacetime is a very general and fundamental one, and therefore it is important to test the various aspects of the Einstein Equivalence Principle thoroughly.
A direct test of WEP is the comparison of the acceleration of
two laboratory-sized bodies of different composition in an
external gravitational field. If the principle were violated,
then the accelerations of different bodies would differ. The
simplest way to quantify such possible violations of WEP in a
form suitable for comparison with experiment is to suppose that
for a body with inertial mass
, the passive gravitational mass
is no longer equal to
, so that in a gravitational field
g, the acceleration is given by
. Now the inertial mass of a typical laboratory body is made up
of several types of mass-energy: rest energy, electromagnetic
energy, weak-interaction energy, and so on. If one of these forms
of energy contributes to
differently than it does to
, a violation of WEP would result. One could then write
where
is the internal energy of the body generated by interaction
A, and
is a dimensionless parameter that measures the strength of the
violation of WEP induced by that interaction, and
c
is the speed of light. A measurement or limit on the fractional
difference in acceleration between two bodies then yields a
quantity called the ``Eötvös ratio'' given by
where we drop the subscript ``I'' from the inertial masses.
Thus, experimental limits on
place limits on the WEP-violation parameters
.
Many high-precision Eötvös-type experiments have been
performed, from the pendulum experiments of Newton, Bessel and
Potter, to the classic torsion-balance measurements of
Eötvös [58], Dicke [55], Braginsky [31] and their collaborators. In the modern torsion-balance
experiments, two objects of different composition are connected
by a rod or placed on a tray and suspended in a horizontal
orientation by a fine wire. If the gravitational acceleration of
the bodies differs, there will be a torque induced on the
suspension wire, related to the angle between the wire and the
direction of the gravitational acceleration
g
. If the entire apparatus is rotated about some direction with
angular velocity
, the torque will be modulated with period
. In the experiments of Eötvös and his collaborators, the wire
and
g
were not quite parallel because of the centripetal acceleration
on the apparatus due to the Earth's rotation; the apparatus was
rotated about the direction of the wire. In the Dicke and
Braginsky experiments,
g
was that of the Sun, and the rotation of the Earth provided the
modulation of the torque at a period of 24 hr (TEGP
2.4 (a) [147
]). Beginning in the late 1980s, numerous experiments were
carried out primarily to search for a ``fifth force'' (see
Sec.
2.3), but their null results also constituted tests of WEP. In the
``free-fall Galileo experiment'' performed at the University of
Colorado, the relative free-fall acceleration of two bodies made
of uranium and copper was measured using a laser interferometric
technique. The ``Eöt-Wash'' experiments carried out at the
University of Washington used a sophisticated torsion balance
tray to compare the accelerations of various materials toward
local topography on Earth, movable laboratory masses, the Sun and
the galaxy [121,
10
], and have recently reached levels of
. The resulting upper limits on
are summarized in Figure
1
(TEGP 14.1 [147
]; for a bibliography of experiments, see [61
]).
The second ingredient of EEP, local Lorentz invariance, has
been tested to high-precision in the ``mass anisotropy''
experiments: The classic versions are the Hughes-Drever
experiments, performed in the period 1959-60 independently by
Hughes and collaborators at Yale University, and by Drever at
Glasgow University (TEGP 2.4 (b) [147]). Dramatically improved versions were carried out during the
late 1980s using laser-cooled trapped atom techniques (TEGP
14.1 [147
]). A simple and useful way of interpreting these experiments is
to suppose that the electromagnetic interactions suffer a slight
violation of Lorentz invariance, through a change in the speed of
electromagnetic radiation
c
relative to the limiting speed of material test particles (
, chosen to be unity via a choice of units), in other words,
(see Sec.
2.2.3). Such a violation necessarily selects a preferred universal
rest frame, presumably that of the cosmic background radiation,
through which we are moving at about 300 km/s. Such a
Lorentz-non-invariant electromagnetic interaction would cause
shifts in the energy levels of atoms and nuclei that depend on
the orientation of the quantization axis of the state relative to
our universal velocity vector, and on the quantum numbers of the
state. The presence or absence of such energy shifts can be
examined by measuring the energy of one such state relative to
another state that is either unaffected or is affected
differently by the supposed violation. One way is to look for a
shifting of the energy levels of states that are ordinarily
equally spaced, such as the four
J
=3/2 ground states of the
Li nucleus in a magnetic field (Drever experiment); another is
to compare the levels of a complex nucleus with the atomic
hyperfine levels of a hydrogen maser clock. These experiments
have all yielded extremely accurate results, quoted as limits on
the parameter
in Figure
2
. Also included for comparison is the corresponding limit
obtained from Michelson-Morley type experiments (for a review,
see [75]).
Recent advances in atomic spectroscopy and atomic timekeeping
have made it possible to test LLI by checking the isotropy of the
speed of light using one-way propagation (as opposed to
round-trip propagation, as in the Michelson-Morley experiment).
In one experiment, for example, the relative phases of two
hydrogen maser clocks at two stations of NASA's Deep Space
Tracking Network were compared over five rotations of the Earth
by propagating a light signal one-way along an ultrastable
fiberoptic link connecting them (see Sec.
2.2.3). Although the bounds from these experiments are not as tight as
those from mass-anisotropy experiments, they probe directly the
fundamental postulates of special relativity, and thereby of LLI
(TEGP 14.1 [147], [144
]).
The principle of local position invariance, the third part of
EEP, can be tested by the gravitational redshift experiment, the
first experimental test of gravitation proposed by Einstein.
Despite the fact that Einstein regarded this as a crucial test of
GR, we now realize that it does not distinguish between GR and
any other metric theory of gravity, but is only a test of EEP. A
typical gravitational redshift experiment measures the frequency
or wavelength shift
between two identical frequency standards (clocks) placed at
rest at different heights in a static gravitational field. If the
frequency of a given type of atomic clock is the same when
measured in a local, momentarily comoving freely falling frame
(Lorentz frame), independent of the location or velocity of that
frame, then the comparison of frequencies of two clocks at rest
at different locations boils down to a comparison of the
velocities of two local Lorentz frames, one at rest with respect
to one clock at the moment of emission of its signal, the other
at rest with respect to the other clock at the moment of
reception of the signal. The frequency shift is then a
consequence of the first-order Doppler shift between the frames.
The structure of the clock plays no role whatsoever. The result
is a shift
where
is the difference in the Newtonian gravitational potential
between the receiver and the emitter. If LPI is not valid, then
it turns out that the shift can be written
where the parameter
may depend upon the nature of the clock whose shift is being
measured (see TEGP 2.4 (c) [147
] for details).
The first successful, high-precision redshift measurement was
the series of Pound-Rebka-Snider experiments of 1960-1965 that
measured the frequency shift of gamma-ray photons from
Fe as they ascended or descended the Jefferson Physical
Laboratory tower at Harvard University. The high accuracy
achieved - one percent - was obtained by making use of the
Mössbauer effect to produce a narrow resonance line whose shift
could be accurately determined. Other experiments since 1960
measured the shift of spectral lines in the Sun's gravitational
field and the change in rate of atomic clocks transported aloft
on aircraft, rockets and satellites. Figure
3
summarizes the important redshift experiments that have been
performed since 1960 (TEGP 2.4 (c) [147
]).
The most precise standard redshift test to date was the
Vessot-Levine rocket experiment that took place in June
1976 [131]. A hydrogen-maser clock was flown on a rocket to an altitude of
about 10,000 km and its frequency compared to a similar
clock on the ground. The experiment took advantage of the masers'
frequency stability by monitoring the frequency shift as a
function of altitude. A sophisticated data acquisition scheme
accurately eliminated all effects of the first-order Doppler
shift due to the rocket's motion, while tracking data were used
to determine the payload's location and the velocity (to evaluate
the potential difference
, and the special relativistic time dilation). Analysis of the
data yielded a limit
.
A ``null'' redshift experiment performed in 1978 tested
whether the
relative
rates of two different clocks depended upon position. Two
hydrogen maser clocks and an ensemble of three
superconducting-cavity stabilized oscillator (SCSO) clocks were
compared over a 10-day period. During the period of the
experiment, the solar potential
changed sinusoidally with a 24-hour period by
because of the Earth's rotation, and changed linearly at
per day because the Earth is 90 degrees from perihelion in
April. However, analysis of the data revealed no variations of
either type within experimental errors, leading to a limit on the
LPI violation parameter
[130]. This bound has been improved using more stable frequency
standards [68,
109
]. The varying gravitational redshift of Earth-bound clocks
relative to the highly stable Millisecond Pulsar PSR 1937+21,
caused by the Earth's motion in the solar gravitational field
around the Earth-Moon center of mass (amplitude 4000 km),
has been measured to about 10 percent, and the redshift of stable
oscillator clocks on the Voyager spacecraft caused by Saturn's
gravitational field yielded a one percent test. The solar
gravitational redshift has been tested to about two percent using
infrared oxygen triplet lines at the limb of the Sun, and to one
percent using oscillator clocks on the Galileo spacecraft (TEGP
2.4 (c) [147
] and 14.1 (a) [147
]).
Modern advances in navigation using Earth-orbiting atomic
clocks and accurate time-transfer must routinely take
gravitational redshift and time-dilation effects into account.
For example, the Global Positioning System (GPS) provides
absolute accuracies of around 15 m (even better in its military
mode) anywhere on Earth, which corresponds to 50 nanoseconds in
time accuracy at all times. Yet the difference in rate between
satellite and ground clocks as a result of special and general
relativistic effects is a whopping 39
microseconds
per day (
from the gravitational redshift, and
from time dilation). If these effects were not accurately
accounted for, GPS would fail to function at its stated accuracy.
This represents a welcome practical application of GR! (For the
role of GR in GPS, see [8]; for a popular essay, see [140].)
Local position invariance also refers to position in time. If
LPI is satisfied, the fundamental constants of non-gravitational
physics should be constants in time. Table
1
shows current bounds on cosmological variations in selected
dimensionless constants. For discussion and references to early
work, see TEGP 2.4 (c) [147].
Constant k | Limit on
![]() ![]() |
Method |
|
||
Fine structure constant
![]() |
![]() |
H-maser vs. Hg ion clock [109] |
![]() |
![]() |
|
![]() |
Oklo Natural Reactor [41![]() |
|
![]() |
21-cm vs. molecular absorption at
Z
=0.7 [57![]() |
|
|
||
Weak interaction constant
![]() |
1 |
![]() ![]() |
0.1 | Oklo Natural Reactor [41] | |
0.06 | Big Bang nucleosynthesis [91, 112] | |
|
||
e-p mass ratio | 1 | Mass shift in quasar spectra at
![]() |
|
||
Proton g-factor (![]() |
![]() |
21-cm vs. molecular absorption at Z =0.7 [57] |
![]() |
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 |