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Updated December 12, 2023

Equivalence principle

Equivalence principle

Image: This train carcass abandoned in the sun in the hostile desert of Patagonia (Argentina), shows the inscription of an equation. This austere equation Rμν -½ gμν R = -8πG/c4 Tμν is one of the modified equations of the theory of General Relativity by Albert Einstein. The left part represents the curvature of space-time and the right part represents the mass/energy content of space-time.
During the second half of the 19th century, the question of synchronizing clocks for train departures and arrivals arose.

What is the equivalence principle?

The equivalence principle is a fundamental concept in Albert Einstein's theory of general relativity (1979-1955).

This principle states that locally, in a small volume of spacetime, gravitational effects are indistinguishable from constant acceleration. In other words, an observer cannot tell the difference between the gravity he feels and an equivalent constant acceleration.
During a parabolic flight, the sensation of weightlessness felt by the occupants inside the plane is an illusion of "zero gravity" or "zero g" but expresses this equivalence well.

More precisely, the principle of equivalence is expressed by the equality between inert mass and gravitational mass.

Inert mass measures the resistance of an object to being accelerated when a force is applied, while gravitational mass measures the gravitational pull experienced by an object.
The inert mass is the mass as it appears in Newton's second law (1642-1727), F = ma
The gravitational mass is the mass that appears in Newton's gravitational equation, which determines the magnitude of the gravitational attraction between two objects, F = G m1 m2/ r2

The equivalence principle states that these two masses, inert mass and gravitational mass, are equivalent, meaning that the trajectory of an object under the influence of gravity depends only on its inert mass, and not on its internal composition. So, according to the equivalence principle, there is an equality between inert mass and gravitational mass, meaning that how an object responds to gravity is entirely determined by its inert mass, as if gravity were not than constant acceleration.

This has profound implications for understanding gravity. For example, in a vacuum, all objects, regardless of their mass, fall with the same acceleration under the influence of gravity.
If the Moon and a rock were placed in the same orbit around Earth, they would fall toward Earth at the same speed. This is because, in a stable orbit, all orbiting objects fall freely under the influence of gravity.
In the context of free fall in orbit, the mass of the object does not influence the speed of fall. It is a fundamental principle of Galilean relativity and special relativity. Mass does not play a role in the free fall time of an object in a gravitational field, as long as we neglect the effects of atmospheric friction.

The equivalence principle is at the heart of the mathematical formulation of general relativity, where gravity is interpreted as a curvature of spacetime caused by the presence of mass and energy.

By applying this principle, Einstein was able to develop a unified theory of gravitation, replacing the classical Newtonian conception of gravity.

“Is it conceivable that the principle of relativity also applies to systems that are accelerated relative to each other?”
This question was asked by Albert Einstein in his 1907 paper entitled "On the Relativity of Gravitation and the Influence of Gravitation on the Propagation of Light."
In this sentence, Einstein assumes that the principle of relativity concerns not only movements at uniform speed, but also movements accelerated relative to each other. In other words, there is no reason why the gravitational field should escape the principle of relativity.
The question of whether this principle could also be applied to accelerated systems eventually led Einstein to develop the theory of general relativity, where gravity is interpreted as a curvature of space-time due to the presence of mass and of energy.
“We will hypothesize the complete physical equivalence between a gravitational field and the corresponding acceleration of the reference system.”

Einstein's general relativity equation has the following form: \[ G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \] Gμν is the spacetime curvature tensor, gμν is the metric tensor, Λ is the cosmological constant, G is the gravitational constant, c is the speed of light, and Tμν is the energy-momentum tensor which describes the distribution of matter and energy.

N.B.: The most remarkable thing is that Einstein deduced even before the development of his theory of general relativity, as early as 1907, that clocks must be influenced by the gravitational field.
“The process involved in the clock takes place all the more quickly as the gravitational potential is greater.”
The frequencies of atoms being clocks (the atomic clock did not exist at that time), Einstein assumed that the frequencies of atoms were modified by the gravitational potential and he deduced: "Light coming from the solar surface has a wavelength approximately 2x10-6 longer than that of light emitted on Earth by identical substances."

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