# Principle of relativity

In its modern expression, the principle of relativity affirms that the physical laws are the same ones for all the observers. That does not mean that the physically measurable events in an experiment are the same ones for the various observers, but that the measurements made by the various observers check the same equations. However, for two prepared experiments in an identical way in two distinct reference frames subjected to the same gravitational constraints (all the two inertial ones for example) the laws are rigorously identical and give identical measurements in their respective reference frames. It is said that the laws are “invariant by change of reference frame”, or which they are “covariantes”.

## Formulations

Definition: A reference frame galiléen (or inertial ) is a reference frame in which any free body (not influenced by outside) which is at rest remains there indefinitely, and any free body moving remains with constant vector Speed (and thus also with Angular momentum constant).

Principle of relativity of Galileo : all the laws of mechanics are identical in all the reference frames galiléens.

Property : that is to say ($R$) is a reference frame galiléen, one a: if ($R_*$) is a reference frame moving by translation at constant speed V compared to ($R$), then ($R_*$) is him also galiléen.
Remark : one will take guard with the fact that the reciprocal one of the property is not true, contrary to what seemed obvious with all until Albert Einstein works out the Principe of equivalence.

Comment : the principle has two significances here.

That the same experiment seen since the two different reference frames galiléens, ($R$) and ($R_*$), follows a law which is expressed same manner when she is formulated in the coordinates of one or other of the reference frames.
And also that an experiment made with identical in two unspecified reference frames galiléens follows, in each one, the same law and gives the same observations exactly.

mathematical Structure used : space closely connected and vectorial of dimension 3, time paramètrisant the trajectories and the states of the system étudié.
Transformations of Galileo :
If $\ vec r$ is the vector punctual coordinate in ($R$) and $\ vec r_*$ is the vector coordinated of the same point in ($R_*$), then one a:

$\ vec R = \ vec r_* + T. \ vec V$ and $\ T = t_*$

The principle of relativity of Galileo is expressed as well like the need for the invariance of the equations of the movement compared to the transformations of Galileo.

the second equality means that time is the same one in both référentiels.

the first equality is equivalent to the law of composition speeds: $\ vec v = \ vec v_* + \ vec V \ Longleftrightarrow \ frac \left\{D \ vec R\right\} \left\{dt\right\} = \ frac \left\{D \ vec r_*\right\} \left\{dt\right\} + \ vec V \ Longleftrightarrow D \ vec R = D \ vec r_* + \ vec V .dt \ Longleftrightarrow \ vec R = \ vec r_* + \ vec V .t$ (with a constant vector near)
It is also equivalent to the independence of acceleration (and thus of the force $\ vec F = m \ ddot \ vec r$ being exerted on the body) compared to the inertial reference frame of the observer: $\ ddot \ vec R = \ ddot \ vec r_* \ Longleftrightarrow \ frac \left\{d^2 \ vec R\right\} \left\{dt^2\right\} = \ frac \left\{d^2 \ vec r_*\right\} \left\{dt^2\right\} \ Longleftrightarrow \ frac \left\{D \ vec R\right\} \left\{dt\right\} = \ frac \left\{D \ vec r_*\right\} \left\{dt\right\} + \ vec V \ Longleftrightarrow \ vec R = \ vec r_* + \ vec V .t$ (except for a constant vector)

### In restricted Relativity

The definition of a reference frame galiléen is the same one as in traditional mechanics.

The principle of relativity undergoes a small change:
Principle of relativity : all the laws of physics are identical in all the reference frames galiléens.

One adds to it a second principle in conformity with the electromagnetism of Maxwell: “speed of light in the vacuum does not depend on the speed of its source”, which one can also express “the value speed of light in the vacuum is the same one in all the reference frames galiléens”.

The property is always true:
Property : that is to say ($R$) is a reference frame galiléen, one a: if ($R_*$) is a reference frame moving by translation at constant speed V compared to ($R$), then ($R_*$) is him also galiléen.
Remark : the reciprocal one of the property is implicitly allowed. In restricted relativity the studied reference frames are those which are inertial and which are supposed in translations at constant speed the ones compared to the others. The Gravitation is not treated by this theory.

Comment : for the principle of relativity, idem with the comment made in the paragraph above of traditional mechanics. For the second principle: one can include/understand the need of it if it is considered that speed of light is a measurement of two identical experiments (emission of light) made in two different reference frames galiléens: its measurement must be the same one in both (but to admit that it is necessary to be convinced that the ether does not have its place in physics). Moreover, mathematics proposes, with the only principle of relativity, to have a speed indépassable and unchanged of a reference frame galiléen to the other (this speed, with the choice, being finished or infinite).

mathematical Structure used : for each reference frame galiléen a vectorial space closely connected and of dimension 3, and a time paramètrisant the trajectories and the states of the system étudié.

Consequences : speed of light in the vacuum is a speed indépassable in any reference frame; two simultaneous events in the reference frame ($R$) can not the being in ($R_*$); measurements of the time intervals, lengths, speeds and accelerations change from one reference frame to another; etc…

Transformations of Lorentz : these transformations express the changes of measurements of the time intervals, lengths and speeds from one inertial reference frame to another; the principle of relativity, in restricted relativity, is also expressed like the need for the invariance of the equations of physics by these transformations.

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