# General Relativity

General relativity, based on the principle of general covariance which extends the principle of relativity to non-inertial reference frames, is a relativistic theory of gravitation, that is to say, it describes the influence on the movement of the stars of the presence of matter and, more generally, of energy, taking into account the principles of special relativity. General relativity encompasses and supplants Isaac Newton’s theory of universal gravitation, which represents its limit at low speeds (compared to the speed of light) and weak gravitational fields.

The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.

General relativity is mainly the work of Albert Einstein, of which it is considered the major achievement, which he developed between 1907 and 1915. The names of Marcel Grossmann and David Hilbert are also associated with it, the first having helped Einstein to familiarize himself with the mathematical tools necessary for understanding the theory (differential geometry), the latter having gone through the final stages together with Einstein leading to the finalization of the theory after the latter had presented it to him during the course of the year 1915 the general ideas of his theory.

General relativity is based on concepts radically different from those of Newtonian gravitation. It states in particular that gravitation is not a force, but is the manifestation of the curvature of space (in fact of space-time), curvature itself produced by the distribution of matter. This relativistic theory of gravitation gives rise to effects that are absent from the Newtonian theory but verified, such as the expansion of the Universe, or potentially verifiable, such as gravitational waves and black holes. None of the many experimental tests carried out to date (2009) have been able to put it at fault, with the possible exception of the Pioneer anomaly which could be the first indication of a discrepancy between the observed phenomena and general relativity, although other interpretations of this phenomenon are possible.

## Need for a relativistic theory of gravitation

The presence of matter modifies the geometry of space-time.

The theory of universal gravitation proposed by Newton at the end of the 17th century is based on the notion of gravitational force acting according to the principle of action at a distance, that is to say the fact that the force exerted by a body (for example the Sun) on another (the Earth) is determined by their relative position at a given moment, whatever the distance separating them. This instantaneous nature is incompatible with the idea of ​​special relativity proposed by Einstein in 1905. Indeed, according to the latter, no information can propagate faster than the speed of light in a vacuum. Moreover, the principle of action at a distance is based on that of the simultaneity of two events: the force that the Sun exerts on the Earth at a given instant is determined by their properties “at this instant”.

Special relativity stipulates that the concept of simultaneity of two events is not defined, the perception of simultaneity being different from one observer to another as long as they are driven by a non-zero relative speed. These contradictions lead Einstein in 1907 to reflect on a theory of gravitation that is compatible with special relativity. The result of his quest is the theory of general relativity.

## From Galilean relativity to special relativity

In the 16th century, Galileo affirmed and explained that the laws of physics are the same in reference frames in rectilinear and uniform translation with respect to each other. This is the principle of relativity (of Galileo).

It will also use the additivity of speeds, according to which any speed can be reached, the whole thing being only a question of means. If a ball is traveling at 10 km/h in a train (and in the direction of travel) which is itself going at 100 km/h relative to the ground, then the ball is traveling at 110 km/h relative to the ground.

In his Mechanics, Isaac Newton presupposed that bodies were endowed with absolute speed, in other words that they were either “really” at rest or “really” in motion. He also noticed that these absolute velocities were not measurable except relative to the velocities of other bodies (in the same way, the position of a body was only measurable relative to that of another body, etc.). Consequently, all the laws of Newtonian mechanics had to operate identically whatever the body considered and whatever its movement.

However, Newton believed that his theory could not make sense without the existence of an absolute fixed reference frame in which the speed of any body could be measured, even if it could not be detected.

In fact, it is possible in practice to build a Newtonian mechanics without this assumption: the resulting theory (named Galilean relativity, moreover) has no particular operational interest and should not be confused with the relativity of ‘Einstein which implies moreover the constancy of the speed of light in all frames of reference and less the Galilean hypothesis that the relative speeds add up (these two postulates are indeed mutually incompatible).

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