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 “test” 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).
The Einstein equivalence principle (EEP) is a more powerful and far-reaching concept; 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 fully 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 [281]).
General relativity is a metric theory of gravity, but then so are many others, including the Brans-Dicke theory and its generalizations. Theories in which varying non-gravitational constants are associated with dynamical fields that couple to matter directly are not metric theories. Neither, in this narrow sense, is superstring theory (see Section 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. It is important to point out, however, that there is some ambiguity in whether one treats such fields as EEP-violating gravitational fields, or simply as additional matter fields, like those that carry electromagnetism or the weak interactions. Still, 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. We first survey the experimental tests, and describe some of the theoretical formalisms that have been developed to interpret them. For other reviews of EEP and its experimental and theoretical significance, see [126, 162].
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
, 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
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 [100], Dicke [94],
Braginsky [43], 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, and this difference has a component
perpendicular to the suspension wire, there will be a torque induced on the wire, related to the angle
between the wire and the direction of the gravitational acceleration . 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
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,
was that
of the Sun, and the rotation of the Earth provided the modulation of the torque at a period
of
(TEGP 2.4 (a) [281
]). Beginning in the late 1980s, numerous experiments were
carried out primarily to search for a “fifth force” (see Section 2.3.1), 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 [249, 19
], and have reached levels of
[2
]. The resulting upper limits on
are summarized in Figure 1
(TEGP 14.1 [281
]; for a bibliography of experiments up to 1991,
see [107
]).
Another, known as Satellite Test of the Equivalence Principle (STEP) [247], is under consideration as
a possible joint effort of NASA and the European Space Agency (ESA), with the goal of a
test. STEP would improve upon MICROSCOPE by using cryogenic techniques to reduce
thermal noise, among other effects. At present, STEP (along with a number of variants, called
MiniSTEP and QuickSTEP) has not been approved by any agency beyond the level of basic design
studies or supporting research and development. An alternative concept for a space test of
WEP is Galileo-Galilei [261], which uses a rapidly rotating differential accelerometer as its
basic element. Its goal is a bound on
at the
level on the ground and
in
space.
Although special relativity itself never benefited from the kind of “crucial” experiments, such as the perihelion advance of Mercury and the deflection of light, that contributed so much to the initial acceptance of GR and to the fame of Einstein, the steady accumulation of experimental support, together with the successful merger of special relativity with quantum mechanics, led to its being accepted by mainstream physicists by the late 1920s, ultimately to become part of the standard toolkit of every working physicist. This accumulation included
In addition to these direct experiments, there was the Dirac equation of quantum mechanics and its prediction of anti-particles and spin; later would come the stunningly successful relativistic theory of quantum electrodynamics.
In 2005, on the 100th anniversary of the introduction of special relativity, one might ask “what is there
to test?”. Special relativity has been so thoroughly integrated into the fabric of modern physics that its
validity is rarely challenged, except by cranks and crackpots. It is ironic then, that during the past several
years, a vigorous theoretical and experimental effort has been launched, on an international
scale, to find violations of special relativity. The motivation for this effort is not a desire to
repudiate Einstein, but to look for evidence of new physics “beyond” Einstein, such as apparent
violations of Lorentz invariance that might result from certain models of quantum gravity.
Quantum gravity asserts that there is a fundamental length scale given by the Planck length,
, but since length is not an invariant quantity (Lorentz-FitzGerald
contraction), then there could be a violation of Lorentz invariance at some level in quantum gravity. In
brane world scenarios, while physics may be locally Lorentz invariant in the higher dimensional world,
the confinement of the interactions of normal physics to our four-dimensional “brane” could
induce apparent Lorentz violating effects. And in models such as string theory, the presence of
additional scalar, vector, and tensor long-range fields that couple to matter of the standard model
could induce effective violations of Lorentz symmetry. These and other ideas have motivated
a serious reconsideration of how to test Lorentz invariance with better precision and in new
ways.
A simple and useful way of interpreting some of these modern experiments, called the -formalism, is
to suppose that the electromagnetic interactions suffer a slight violation of Lorentz invariance, through a
change in the speed of electromagnetic radiation
relative to the limiting speed of material test particles
(
, made to take the value unity via a choice of units), in other words,
(see Section 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
[167]. 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 Zeeman-split
ground states of a nucleus of total spin
in a magnetic field;
another is to compare the levels of a complex nucleus with the atomic hyperfine levels of a
hydrogen maser clock. The magnitude of these “clock anisotropies” would be proportional to
.
The earliest clock anisotropy experiments were the Hughes-Drever experiments, performed in the period
1959 - 60 independently by Hughes and collaborators at Yale University, and by Drever at Glasgow
University, although their original motivation was somewhat different [131, 96]. The Hughes-Drever
experiments yielded extremely accurate results, quoted as limits on the parameter in
Figure 2
. Dramatic improvements were made in the 1980s using laser-cooled trapped atoms and
ions [215, 163, 53]. This technique made it possible to reduce the broading of resonance lines caused by
collisions, leading to improved bounds on
shown in Figure 2
(experiments labelled NIST,
U. Washington and Harvard, respectively).
Also included for comparison is the corresponding limit obtained from Michelson-Morley type
experiments (for a review, see [127]). In those experiments, when viewed from the preferred frame, the
speed of light down the two arms of the moving interferometer is , while it can be shown using the
electrodynamics of the
formalism, that the compensating Lorentz-FitzGerald contraction of the
parallel arm is governed by the speed
. Thus the Michelson-Morley experiment and its descendants
also measure the coefficient
. One of these is the Brillet-Hall experiment [46], which used a
Fabry-Perot laser interferometer. In a recent series of experiments, the frequencies of electromagnetic cavity
oscillators in various orientations were compared with each other or with atomic clocks as a function of the
orientation of the laboratory [297
, 168
, 190
, 12, 248]. These placed bounds on
at the level of
better than a part in
. Haugan and Lämmerzahl [125] have considered the bounds that
Michelson-Morley type experiments could place on a modified electrodynamics involving a “vector-valued”
effective photon mass.
|
Astrophysical observations have also been used to bound Lorentz violations. For example, if photons satisfy the Lorentz violating dispersion relation
where Mattingly [182] gives a thorough and up-to-date review of both the theoretical frameworks and the
experimental results for tests of LLI.
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
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 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) [281
]).
The most precise standard redshift test to date was the Vessot-Levine rocket experiment that took place
in June 1976 [264]. A hydrogen-maser clock was flown on a rocket to an altitude of about 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
[258]. This bound has
been improved using more stable frequency standards, such as atomic fountain clocks [120, 216, 23
]. The
current bound, from comparing a Cesium atomic fountain with a Hydrogen maser for a year, is
[23].
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 ), was measured to about 10 percent [251]. Two measurements of the redshift
using stable oscillator clocks on spacecraft were made at the one percent level: One used the Voyager
spacecraft in Saturn’s gravitational field [158], while another used the Galileo spacecraft in the Sun’s
field [160].
The gravitational redshift could be improved to the level using an array of laser
cooled atomic clocks on board a spacecraft which would travel to within four solar radii of the
Sun [180].
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 positional accuracies of around (even better in its
military mode), and 50 nanoseconds in time transfer accuracy, anywhere on Earth. Yet the difference in rate
between satellite and ground clocks as a result of 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 [15, 16]; for a popular essay,
see [287].)
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) [281] or [97]. For a comprehensive recent review both of experiments and of theoretical
ideas that underly proposals for varying constants, see [262
].
Experimental bounds on varying constants come in two types: bounds on the present rate of variation,
and bounds on the difference between today’s value and a value in the distant past. The main example of
the former type is the clock comparison test, in which highly stable atomic clocks of different fundamental
type are intercompared over periods ranging from months to years (variants of the null redshift experiment).
If the frequencies of the clocks depend differently on the electromagnetic fine structure constant , the
electron-proton mass ratio
, or the gyromagnetic ratio of the proton
, for example, then a limit
on a drift of the fractional frequency difference translates into a limit on a drift of the constant(s). The
dependence of the frequencies on the constants may be quite complex, depending on the atomic
species involved. The most recent experiments have exploited the techniques of laser cooling and
trapping, and of atom fountains, in order to achieve extreme clock stability, and compared the
Rubidium-87 hyperfine transition [181
], the Mercury-199 ion electric quadrupole transition [31
], the
atomic Hydrogen
transition [111
], or an optical transition in Ytterbium-171 [209
],
against the ground-state hyperfine transition in Cesium-133. These experiments show that, today,
.
The second type of bound involves measuring the relics of or signal from a process that occurred in the distant past and comparing the inferred value of the constant with the value measured in the laboratory today. One sub-type uses astronomical measurements of spectral lines at large redshift, while the other uses fossils of nuclear processes on Earth to infer values of constants early in geological history.
|
Earlier comparisons of spectral lines of different atoms or transitions in distant galaxies and quasars
produced bounds or
on the order of a part in 10 per Hubble time [298]. Dramatic
improvements in the precision of astronomical and laboratory spectroscopy, in the ability to model the
complex astronomical environments where emission and absorption lines are produced, and in the ability to
reach large redshift have made it possible to improve the bounds significantly. In fact, in 1999, Webb et
al. [269, 193] announced that measurements of absorption lines in Mg, Al, Si, Cr, Fe, Ni, and Zn in quasars
in the redshift range
indicated a smaller value of
in earlier epochs, namely
, corresponding to
(assuming
a linear drift with time). Measurements by other groups have so far failed to confirm this non-zero
effect [242
, 51, 219]; a recent analysis of Mg absorption systems in quasars at
gave
[242].
Another important set of bounds arises from studies of the “Oklo” phenomenon, a group of natural,
sustained fission reactors that occurred in the Oklo region of Gabon, Africa, around 1.8 billion years
ago. Measurements of ore samples yielded an abnormally low value for the ratio of two isotopes of
Samarium,
. Neither of these isotopes is a fission product, but
can be depleted by a
flux of neutrons. Estimates of the neutron fluence (integrated dose) during the reactors’ “on” phase,
combined with the measured abundance anomaly, yield a value for the neutron cross-section for
1.8
billion years ago that agrees with the modern value. However, the capture cross-section is extremely
sensitive to the energy of a low-lying level (
), so that a variation in the energy of this level of
only
over a billion years would change the capture cross-section from its present value by
more than the observed amount. This was first analyzed in 1976 by Shlyakter [241]. Recent
reanalyses of the Oklo data [72, 116, 210] lead to a bound on
at the level of around
.
In a similar manner, recent reanalyses of decay rates of in ancient meteorites (4.5 billion years
old) gave the bound
[205].
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