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General Relativity

Brahim LAMINE

IRAP, CNRS, CNES, Université de Toulouse, France

1.1. The fundamentals of general relativity

1.1.1. The equivalence principle

The equivalence principle is the fundamental element in the construction of the theory of general relativity. It goes far beyond a simple equality between inertial mass and gravitational mass, as we shall see throughout this book. This principle makes it possible to give a geometric meaning to gravitational interaction and therefore allows us to describe the physics in the presence of gravitation, if we understand the physics in the absence of gravitation.

1.1.1.1. Universality of free fall

The fact that all bodies fall in the same way in a gravitational field, independent of their mass, is one of the most universal observations in nature. However, as Einstein noted in 1911 (Einstein 1911), this principle was not at the heart of any fundamental laws describing our representation of the world. In Einstein's own words (translated by Michael D. Godfrey):

This experience, of the equal falling of all bodies in the gravitational field, is one of the most universal which the observation of nature has yielded to us; but in spite of this, this law has found no place in the foundations of our world view (Weltbildes) of the physical universe.

The equivalence principle, on which the theory of general relativity is based, remedies this deficit.

1.1.1.1.1. Gravitational mass versus inertial mass

Inertial mass mi is a concept used for kinematic quantities, for example, linear momentum, and therefore kinetic energy or angular momentum . According to Newton's second law of motion, it is therefore related to the resistance to change (inertia):

where represents the resultant of the external forces. The larger the value of mi, the smaller the variation in the velocity, which is a good representation of the concept of inertia.

The gravitational mass mg is, a priori, another concept, related to gravitational interaction, in which the gravitational mass is interpreted as a gravitational charge. In Newton's description, this interaction is described by a force, , such that its norm is written as:

where g is the gravitational field created by the object of gravitational mass Mg. This latter formula is what makes it possible to interpret gravitational mass1 as a charge, analogous to F = qE for the Coulomb force.

Experiments carried out over centuries have shown that the ratio mg/mi is constant and independent of the material of the object. We can take this constant to be equal to 1, which is the same as redefining the units of gravitational mass and inertial mass. In other words, gravitational mass is equal to inertial mass, up to the accuracy of our measurements, which will be described further in the chapter.

1.1.1.1.2. A history of the measurement of

The experimental measurement of the ratio mg/mi for different materials began almost 500 years ago and is still carried out today with increasingly spectacular accuracy. We will briefly mention a few landmark historical experiments.

Galileo (1564-1642)

The idea was to carry out experiments of free fall in a uniform gravitational field. Because of friction due to air, Galileo carried out these experiments using marbles rolling down inclined planes. This was a brilliant idea, as the solid friction acting on the marble from the plane allowed the marble to roll without sliding and thus eliminate energy loss due to friction2. By marking the position of the marble as a function of time, we can go back to the ratio mg/mi. Indeed:

where is the effective gravitational field3, dependent on the angle, ?, of the inclined plane. By marking the position of the marble as a function of time, we go back to the ratio mg/mi. These clever experiments were, however, limited by the time of the fall. Indeed, a long fall time necessitates a very low inclined plane and a very flat plane. Galileo's experiments were refined thanks to Atwood's machine in 1784. The inclined plane was abandoned in favor of a pulley system, which was much better as it allows small uniform acceleration and therefore a long fall time, in a much simpler way than Galileo's inclined plane (especially because of the question of the flatness of the inclined plane). In Figure 1.1, the ball B has a uniform acceleration given by . It is sufficient to choose a mass mA which is a fraction a of mB, so that . Consequently, the acceleration of the ball B is finally written as:

Once again, the ratio can be measured by monitoring the motion of ball B.

Newton (1642-1727) then Bessel (1784-1846)

Following Galileo's experiments on free fall, Newton introduced the idea of using a simple pendulum, which could oscillate for quite a long time. Furthermore, if the frictional forces are low, the oscillation period is not greatly affected by their presence (although the amplitude is damped):

Figure 1.1. Experiments with free fall with constant acceleration. (a) Galileo's experiment on an inclined plane. (b) Atwood's experiment with a pulley

It is then enough to measure the period of the oscillation T = 2p/?0 for several pendulums made of different materials but of same length4. Although even cleverer than Galileo's inclined plane, these experiments were still limited by the oscillation time.

Eötvös (late 19th, early 20th century)

Eötvös invented a truly ingenious system, this time based on a torsion pendulum. The idea was to make use of the non-Galilean nature of the Earth reference frame, especially the effects of the inertial forces, which depend on the inertial mass. Let us consider a balance with two objects that are oriented in the east-west direction. We suspend the balance at a vertical torsion wire. Due to the inertial drag force in the terrestrial reference frame, if the vertical is not the same for both objects (A and B) the pendulum will rotate around the vertical axis and form an angle with respect to the east-west axis. We will use ae,h/z to denote the horizontal and vertical components of acceleration and drag, and it can be noted that .

Figure 1.2. Eötvös' experiment using a torsion pendulum

In order for the pendulum to be balanced, the moment of the forces (torque) must be balanced with respect to the north-south axis, that is:

Following this, the equilibrium of the moments with respect to the vertical axis yields another equation:

By replacing lB, we finally find at the lowest order:

Therefore, if the ratio mi/mg is not the same for the objects A and B, the pendulum will then begin to rotate around the vertical axis and find the equilibrium when the torque of the restoring force of the torsion wire becomes equal to the previous expression. In this experiment, Eötvös obtained the following constraint on the parameter ? (which is called the Eötvös parameter5):

These experiments based on a torsion pendulum were then refined over time, up to the most advanced experiments currently being carried out by "Eöt-Wash" at Washington University, achieving a precision on ? of 10-13 in 2012 (Wagner et al. 2012).

Lunar laser ranging (LLR)

The concept behind this experiment was to verify that the Earth and the Moon, which have different compositions, fall in the same manner in the Sun's gravitational field. The advantage of this measurement is that it is not limited by the fall time and can thus be integrated over a very long time (over 30 years since the first measurement). The second advantage is that thanks to the development of lasers, it was possible to determine the distance to the Moon up to a millimeter, by measuring the time taken by a beam of light to get reflected. The precision over the Eötvös parameter is 10-13 (Williams et al. 2012), roughly equal to the precision obtained by the research group at Washington university.

MICROSCOPE satellite (launched in 2016)

The final precision over ?, published in 2022 (Touboul et al. 2022), is 2.7 × 10-15, in line with the mission's scientific objectives. The experiment consists of launching into orbit two objects in free fall and then comparing their free fall over duration of 2 h. This is a come back to free fall experiments, but over a long time and with controlled disturbances, thanks to the drag compensation system allowing for true free fall.

Conceptual consequences

The universal nature of bodies falling is therefore one of the...