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The Equivalence Principle

The test of the universality of free fall is one of the oldest dynamic experiments. From the experimental point of view, the test consists in examining whether two masses subjected to the same gravitational field, with the same initial and environmental conditions, will have identical laws of motion in the absence of any other external factor. From the perspective of writing dynamics equations, the question is whether the gravitational mass and the inertial mass are identical (or at least proportional) for all bodies regardless of their composition. In a Newtonian formulation, this is not obvious a priori: the gravitational mass quantifies the gravitational force in the same way the electric charge quantifies electrostatic force; while the inertial mass quantifies the resistance of a body to change its motion by force, whatever the origin of this force.

Galileo was the first to carry out and accurately describe experiments on the universality of free fall using balls rolling down inclined planes.1 A little later, Newton, who clearly made the distinction between the different roles of gravitational mass and inertial mass in his equations, was astonished at their similarity. He carried out numerous experiments by measuring the period of oscillation of a fixed-length pendulum with masses of different compositions at its end, and he proved the universality of free fall for all these materials with an accuracy of about one per thousand. Since then, several experiments have followed which have gained about ten orders of magnitude in less than four centuries.

physique fondamentale

Over these four centuries, the theoretical landscape has changed profoundly. First, Einstein introduced his Equivalence Principle.2 Einstein's equivalence principle includes the weak equivalence principle (WEP), which corresponds to the universality of free fall of bodies of different compositions. More recently, new theories have tried to respond, firstly to the question of unifying gravity with other interactions, and secondly to other problems arising in cosmology such as for example the nature of dark matter and dark energy. In most cases, these theories predict, in addition to the classical component of general relativity, the existence of a component of gravity that does not respect the equivalence principle. In this context, testing the universality of free fall at a much better level than has been done previously is essential.

Despite constant progress in instrumentation, it remains very difficult to quickly obtain (on the scale of a few years) progress of several orders of magnitude by relying either on ground experiments with masses of laboratories, or on the lunar laser observation of the movement of celestial bodies. To overcome the limitations of these two options, experiments in space are required. It is within this context that we proposed the MICROSCOPE mission that was selected by CNES and is currently in development.

1However, he probably did not carry out the experiment from the Tower of Pisa: he was aware of the experimental limitations due in particular to the aerodynamic drag.

2There are several formulations of the Principle of Equivalence: weak Equivalence Principle (WEP), the Einstein Equivalence Principle (EEP) and the strong equivalence principle (SEP).

We can distinguish three types of experiments to test the equivalence principle :

  • tests in laboratories on the ground,
  • lunar laser ranging,
  • experiments in space.

Laboratory Experiments

The experiments on the ground involve comparing (either directly or indirectly) the free fall of test masses of different compositions in the gravitational field of the Earth or the Sun. There are two types of techniques: direct tests in free fall towers and experiments using torsion balances. This second method is more efficient; it uses modern technology to carry out Eötvös’ experiment. The source of gravity used can either be the Sun (which limits the gravity gradient source 1000 times lower compared to the Earth) or the Earth (one must then rotate the torsion balance to observe a characteristic frequency). It is this second option that provides the best results with no violation of the equivalence principle at the level of 1.5 10-13. The main limitations of these experiments are local gravity gradients and seismic disturbances.

Lunar Laser Ranging

This is a kind of experiment in free fall using natural bodies: the measurement of the distance between the Earth and the Moon allows us to compare the free falls of the Earth and Moon in the gravitational field of the Sun. Of course, this requires a very good modeling of the relative movements, including rotations around the center of mass. This technique limits the violation of the universality of free fall for the Earth-Moon pair to 10-13.

In this experiment, the universality of free fall could be violated not only because of the different composition of the Earth and the Moon, but also because of the Nordtvedt effect. This effect models a possible dependence of the ratio of gravitational mass to inertial mass, proportional to the internal gravitational energy of masses in free fall. This energy is much too weak in masses used in laboratories to test the Nordtvedt effect. Yet, it is 15 to 20 orders of magnitude greater in the case of the Moon and Earth. The Nordtvedt effect is one possible way to violate the strong equivalence principle (SEP). The analysis of laser data from the Moon enables us to limit the sum of the effects of composition and the Nordtvedt effect.

The main limitations of lunar laser ranging for testing the equivalence principle are firstly the non-uniform distribution of measurements over time (mainly because of the phases of the moon) which does not allow for a complete de-correlation of the GM constant of the Earth-Moon system, and secondly, precision of corrected measurements (atmospheric effects, the geometry of the reflectors, movements of stations ...) associated with long series of observations.

An idealized experiment to test the equivalence principle in space is to compare the free fall of different test masses. In practice, these masses should be placed in a chamber (the satellite) to protect them from non-gravitational effects that would otherwise seriously disrupt the experiment. Moreover, it is more efficient to measure the forces that are applied in order to cancel the relative motion of the masses than to directly measure the relative motion itself. These principles have formed the basis of the kinds of space experiments proposed so far.

  • STEP It was proposed some twenty years ago by Stanford University. It is a very ambitious experiment, including SQUID position detectors that operate at a few degrees K. This proposal has been submitted several times to NASA and to ESA, with objectives of 10-17 then 10-18 , but was not selected.
  • GG (Galileo Galilei) is an experiment designed by an Italian team. It was submitted to ESA with a target of 10-16 then 10-17 and was not selected.
  • MICROSCOPE is a French mission involving the GRGS. It is less ambitious (and therefore less expensive) and it has been selected by CNES with a target of 10-15.