In his "Discorsi e dimostrazioni matematiche ..." published in Leiden in 1638 while he was blind and under house arrest in Italy (Galilei, 1638) Galileo reported results of experiments carried out almost forty years earlier with pendula and the inclined plane. He formulated with astonishing neatness what lately became known as the Principle of Equivalence, according to which all bodies fall in the same way regardless of their mass and composition: ...veduto, dico questo, cascai in opinione che se si levasse totalmente la resistenza del mezzo, tutte le materie descenderebbero con eguali velocitŕ (...having observed this I came to the conclusion that, if one could totally remove the resistance of the medium, all substances would fall at equal speeds). About 80 years after Galileo's first experiments Newton went further, actually recognizing the equivalence of mass and weight. Newton regarded this equivalence as so important that he devoted to it the opening paragraph of the Principia, where (Definition I) he stated: "this quantity that I mean hereafter under the name of ...mass ...is known by the weight ...for it is proportional to the weight as I have found by experiments on pendulums, very accurately made..." (Cajori, 1934).
At the beginning of the 
century, almost 300 years since Galileo's work, Einstein realized that
because of the equivalence between the gravitational (passive) mass 
and the inertial mass 
(i.e. the weak equivalence
principle), the effect of gravitation is locally equivalent to the effect of an
accelerated frame and can be locally cancelled. In a freely falling system all masses fall
equally fast, hence gravitational acceleration has no local dynamical effects. Einstein
then generalized the weak equivalence principle to the strong equivalence
principle, on which he based his theory of General Relativity. The strong
equivalence principle states that in an electromagnetically shielded laboratory,
freely falling and nonrotating, the laws of Physics - including their numerical content -
are independent of the location of the laboratory. In such a laboratory all particles free
of nongravitational forces move with the same acceleration. That is to say, the effects
of gravity, according to General Relativity, are equivalent to the effects of living in a
curved spacetime. In this sense the weak equivalence principle expresses the very
essence of General Relativity and as such it deserves to be tested as accurately as
possible. In the last 30 years since the advent of the space age General Relativity has
been subject to extensive experimental testing as never before in its first 50 years of
existence, and so far it has come out having no real competitors (e.g. Will, 1992); the crucial area where experimental
gravitation is likely to play an important rôle is in the verification of the weak
equivalence principle itself, since it is tantamount to testing whether gravitation can be
ascribed to a metric structure of spacetime.
The total mass-energy of a body is the sum of many terms corresponding
to the energy of all the conceivable interactions and components: 
. For two bodies A and B
of different composition the Eötvös parameter 
can be generalized into
such that a non-zero value of 
would define the violation of equivalence between the inertial and
gravitational mass-energy of the k-th kind. For instance, the rest mass would
contribute (as a fraction of the total) for 
; the nuclear binding energy for 
, the mass difference between neutron and proton for 
(A being the number
of protons plus neutrons and Z the number of protons in the nucleus), the
electrostatic energy of repulsion in the nuclei for 
, the mass of electrons for 
, the antiparticles for 
, the weak interactions responsible of 
decay for 
. From the point of view of conventional field theory, the
verification of all these separate equivalence principles corresponds to a very peculiar
coupling of each field to gravity; whether and why it should be so in all cases is a
mystery. Let us consider the case of antiparticles. A peculiarity of gravity, strictly
related to the Equivalence Principle (EP), is that there is so far no evidence for antigravity,
namely for the possibility that matter is gravitationally repelled by antimatter. A
negative ratio of inertial to gravitational mass would obviously violate the equivalence
principle and forbid any metric theory of gravity. Yet, there are theoretical formulations
which would naturally lead to antigravity (Scherk,
1979), and experiments have been proposed to directly explore the relation between
gravity and antimatter. The idea is to make a Galileo-type mass dropping experiment using
a proton and an antiproton in order to check whether they both fall like ordinary matter
or not. The experiment was proposed to CERN by an international team of scientists (Beverini et al., 1986). Unfortunately, while
experiments concerning the inertial mass of antiparticles have been highly successful, and
these are very accurately known, gravitational experiments (i.e. involving the
gravitational mass of antiparticles) are extremely difficult because of the far larger
electric effects, such as those due to stray electric fields in the drift tube. Indeed,
the latter have so far hindered the experiment mentioned above. In absence of such direct
tests, an improvement by several orders of magnitude of current tests of the weak
equivalence principle with ordinary matter would also be an important constraint as far as
the relation between gravity and antimatter is concerned. Several models of elementary
particles have been proposed in which there are new long range forces between neutral
particles. Generally they lead to forces between two bodies proportional to the product of
two quantum numbers - e.g. their barion numbers - and as such they violate the equivalence
principle. However, their state of development is uncertain and at present experiments on
the weak equivalence principle do not have a precise theory to test and a corresponding
target accuracy.
The best ground experiments to test the weak equivalence principle
have employed one of the most sensitive devices in the history of Physics: the torsion
balance (Eötvös et al., 1922; Roll et al., 1964; Braginsky & Panov, 1972). More recently,
thanks to the debate on the so-called fifth-force raised by Fishbach et al. (1986), a series of revised
torsion balance experiments has been carried out reaching an accuracy of about 1 part
in 
(Adelberger et al., 1990; Su et al., 1994). The main novelty is that the
torsion balance is rotating in order to modulate the signal at higher frequency.
The crucial advantage of an EP space experiment in low Earth orbit is
that the driving signal in the field of the Earth is given by the entire value of
its gravitational acceleration. At 
altitude this amounts to 
(G is the universal constant of gravity, 
the mass of the Earth and R
the orbital radius of the satellite) as opposed to a maximum value of 
for a torsion balance on the ground in the
field of the Earth (at 
latitude)
and 
in the field of the Sun. In
contrast a short range EP experiment has nothing to gain from going into space, since much
bigger source masses are available on the ground. The scientific objective of the
nondrag-free GG mission proposed here is to test the weak equivalence principle in low
Earth orbit to 1 part in 
. This
requires to measure the effects of a very small differential acceleration 
. An accuracy of 
in 
would be a four order of magnitude improvement with
respect to the current best ground tests. Even with a further improvement of ground
results (e.g. to an accuracy of 1 part in 
) space would still allow a great leap forward. In order to avoid complexity,
increase reliability and reduce cost, the whole mission design is focused on - and
optimized for - EP testing only. Therefore, it does not offer a variety of scientific
objectives as it has become the case for STEP (Worden
& Everitt, 1973; Worden, 1976; Worden, 1987; Barlier et al., 1991; Blaser et al., 1993; Blaser et al., 1994; Blaser et al., 1996) and as proposed for the
gravity mission based on the so-called SEE (Satellite Energy Exchange) method (Sanders & Deeds, 1992). In addition the
experiment is noncryogenic, the advantage being that there is no need to operate a
cryostat in space in a very delicate, small-force experiment in which perturbations from
the cryostat itself should be accurately controlled.
The main features of the mission derive from a careful analysis of ground based experiments on one side and the zero-g space environment on the other. EP experiments in the laboratory have demonstrated the advantage of rotating the torsion balance. As for vibrational noise (the analog of seismic noise on the ground), work done within the VIRGO project now under construction in Pisa has led to the development of very efficient attenuators. GG employs spinning test masses as well as mechanical suspensions. In order to reduce tidal (purely classical) gravitational perturbations, the two test masses - in the shape of hollow cylinders - are placed one inside the other. A precursor of GG, designed as a co-experiment in a multipurpose spacecraft, has appeared in 1992 (Bramanti et al., 1992) but indeed the first proposal for an EP space experiment with a spinning apparatus goes back to 1970 (Chapman & Hanson, 1970). It must be realized that an EP experiment with spinning test masses can be severely limited by centrifugal forces. An important feature of GG is that, for the first time, it exploits the effect of self-centering of bodies in supercritical rotation. It is worth stressing that, while an EP violation signal is modulated at the spin frequency, the effects of a large number of perturbing forces (e.g. due to inhomogeneities of the test bodies, spacecraft mass anomalies, nonuniform thermal expansion, parasitic capacitances, etc.) appear as DC because the entire system is spinning.
Small force gravitational experiments in space usually need to be performed inside a spacecraft equipped with thrusters in order to compensate - to some extent and in the appropriate frequency range - for the effects of the residual atmosphere and/or solar radiation acting on its external surface (and not on the test bodies inside it). However, ordinary impulsive thrusters cannot be used because they would induce vibrations of the spacecraft also at the frequency of the signal. Proportional thrusters have therefore been studied, based either on appropriate mechanical valves or on field emission electric propulsion (FEEP), but so far their in-flight performances have not been tested. Since the only scientific objective of GG is to test the equivalence principle, i.e. to detect any tiny difference in the gravitational attraction of the Earth on two bodies of equal mass and different composition, it is by definition a differential experiment. Hence, perturbations which produce the same effect on both test masses (Common Mode Effect) do not compete with the signal. If the masses are suspended (rather than free falling) any external drag effect will appear as an inertial force applied to their centres of mass and differ (for sufficiently equal masses) only because of differences in the suspension mechanisms. We have therefore devoted special care in devising a coupled suspension of the test bodies which allows us an accurate balancing and consequent reduction of the differential effects of inertial forces (Common Mode Rejection, CMR). Such a balancing would allow us to operate the experiment in non drag-free conditions, i.e. to retain entirely the perturbation from air drag (and solar radiation pressure) on the spacecraft while making small only the differential effects of the inertial forces that it generates on the suspended test bodies. Drag effects could be essentially eliminated without inducing vibrational noise were the experimental apparatus totally uncoupled from the spacecraft. This appears to be an interesting possibility in the case of a passive experimental apparatus; in fact it was proposed for measuring the universal constant of gravitation G with a miniature planet-satellite system inside an orbiting spacecraft (Nobili et al., 1990), and in general for measuring the effects of pure gravitational interaction of the test bodies (Sanders & Deeds, 1992). It is however much less attractive when dealing with an active apparatus which needs to get power from solar cells placed on the outer surface of the spacecraft and to be electrically discharged. In GG the thin springs which connect the laboratory to the spacecraft can be used for accommodating the required number of wires, and the fact of not having free floating masses avoids the build up of electrostatic charges.
In Section 2 we describe the
spacecraft and the experiment. Section 3 shows, first in
the framework of a simplified model and then with finite element numerical simulations,
the advantages of supercritical rotation, the effect of self-centering and the concept of
electrostatic active damping. Section 4 is devoted to
tidal effects. The effects of inertial forces and the balancing procedure are discussed in
Section 5 . Thermal effects, thermal noise and the
requirements on thermal stability (to be met by passive insulation) are computed in Section 6 . The capacitance read out system is presented in
Section 7 ; its sensitivity and the required level of
balancing appear to be adequate for the experiment. Section
8 is devoted to showing that coupling with the higher mass moments of the test bodies
from the monopole of the Earth and from nearby mass anomalies is sufficiently small. Section 9 deals with electrostatic and magnetic
perturbations, reaching the conclusion that there is no need to reduce the magnetic field
of the Earth inside the satellite. A procedure for locking the suspended masses during
launch and properly unlocking them in orbit once the nominal spin rate has been reached is
proposed in Section 10 . A preprint of this paper is
available (Nobili et al., 1994).
Copyright © 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.
(Anna Nobili- nobili@dm.unipi.it)
