In one simple example [92], one can write the Lagrangian for the low-energy limit of a string-inspired theory in the so-called “Einstein frame”, in which the gravitational Lagrangian is purely general relativistic:

where is the non-physical metric, is the Ricci tensor derived from it, is a dilaton field, and , and are functions of . The Lagrangian includes that for the electromagnetic field , and that for particles, written in terms of Dirac spinors . This is not a metric representation because of the coupling of to matter via and . A conformal transformation , , puts the Lagrangian in the form (“Jordan” frame) One may choose so that the particle Lagrangian takes the metric form (no explicit coupling to ), but the electromagnetic Lagrangian will still couple non-metrically to . The gravitational Lagrangian here takes the form of a scalar-tensor theory (see Section 3.3.2). But the non-metric electromagnetic term will, in general, produce violations of EEP. For examples of specific models, see [254, 85]. Another class of non-metric theories are included in the “varying speed of light (VSL)” theories; for a detailed review, see [178].On the other hand, whether one views such effects as a violation of EEP or as effects arising from additional “matter” fields whose interactions, like those of the electromagnetic field, do not fully embody EEP, is to some degree a matter of semantics. Unlike the fields of the standard model of electromagnetic, weak and strong interactions, which couple to properties other than mass-energy and are either short range or are strongly screened, the fields inspired by string theory could be long range (if they remain massless by virtue of a symmetry, or at best, acquire a very small mass), and can couple to mass-energy, and thus can mimic gravitational fields. Still, there appears to be no way to make this precise.

As a result, EEP and related tests are now viewed as ways to discover or place constraints on new physical interactions, or as a branch of “non-accelerator particle physics”, searching for the possible imprints of high-energy particle effects in the low-energy realm of gravity. Whether current or proposed experiments can actually probe these phenomena meaningfully is an open question at the moment, largely because of a dearth of firm theoretical predictions.

On the phenomenological side, the idea of using EEP tests in this way may have originated in the middle
1980s, with the search for a “fifth” force. In 1986, as a result of a detailed reanalysis of Eötvös’ original
data, Fischbach et al. [108] suggested the existence of a fifth force of nature, with a strength of about a
percent that of gravity, but with a range (as defined by the range of a Yukawa potential, )
of a few hundred meters. This proposal dovetailed with earlier hints of a deviation from the
inverse-square law of Newtonian gravitation derived from measurements of the gravity profile
down deep mines in Australia, and with emerging ideas from particle physics suggesting the
possible presence of very low-mass particles with gravitational-strength couplings. During the
next four years numerous experiments looked for evidence of the fifth force by searching for
composition-dependent differences in acceleration, with variants of the Eötvös experiment or with
free-fall Galileo-type experiments. Although two early experiments reported positive evidence, the
others all yielded null results. Over the range between one and 10^{4} meters, the null experiments
produced upper limits on the strength of a postulated fifth force between 10^{–3} and 10^{–6} of the
strength of gravity. Interpreted as tests of WEP (corresponding to the limit of infinite-range
forces), the results of two representative experiments from this period, the free-fall Galileo
experiment and the early Eöt-Wash experiment, are shown in Figure 1. At the same time,
tests of the inverse-square law of gravity were carried out by comparing variations in gravity
measurements up tall towers or down mines or boreholes with gravity variations predicted using
the inverse square law together with Earth models and surface gravity data mathematically
“continued” up the tower or down the hole. Despite early reports of anomalies, independent
tower, borehole, and seawater measurements ultimately showed no evidence of a deviation.
Analyses of orbital data from planetary range measurements, lunar laser ranging (LLR), and laser
tracking of the LAGEOS satellite verified the inverse-square law to parts in 10^{8} over scales of 10^{3}
to 10^{5} km, and to parts in 10^{9} over planetary scales of several astronomical units [250]. A
consensus emerged that there was no credible experimental evidence for a fifth force of nature, of a
type and range proposed by Fischbach et al. For reviews and bibliographies of this episode,
see [107, 109, 110, 4, 278].

Although the idea of an intermediate-range violation of Newton’s gravitational law was dropped, new ideas
emerged to suggest the possibility that the inverse-square law could be violated at very short ranges, below
the centimeter range of existing laboratory verifications of the behavior. One set of
ideas [13, 11, 221, 220] posited that some of the extra spatial dimensions that come with string theory
could extend over macroscopic scales, rather than being rolled up at the Planck scale of 10^{–33} cm, which
was then the conventional viewpoint. On laboratory distances large compared to the relevant scale of the
extra dimension, gravity would fall off as the inverse square, whereas on short scales, gravity would fall off
as 1 / R^{2+n}, where n is the number of large extra dimensions. Many models favored n = 1 or n = 2.
Other possibilities for effective modifications of gravity at short range involved the exchange of light scalar
particles.

Following these proposals, many of the high-precision, low-noise methods that were developed for tests of WEP were adapted to carry out laboratory tests of the inverse square law of Newtonian gravitation at millimeter scales and below. The challenge of these experiments has been to distinguish gravitation-like interactions from electromagnetic and quantum mechanical (Casimir) effects. No deviations from the inverse square law have been found to date at distances between 10 m and 10 mm [171, 130, 129, 52, 170]. For a comprehensive review of both the theory and the experiments, see [3].

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