# Reference frame galiléen

In Physical, a reference frame galiléen , or inertial , is a reference frame in which an object isolated (on which no force is exerted or on which the resultant of the forces is null) is either motionless, or moving from translation rectilinear uniform compared to this reference frame. That means that the principle of Inertie, which is stated in the first law of Newton applies.

A reference frame galiléen is thus named in homage to Galileo and more particularly to the Relativité galiléenne.

By using a motionless reference frame compared to the terrestrial ground, all small mechanical experiment of short duration is described perfectly as proceeding in a Galiléen reference frame (the error is tiny and in general negligible).

In a noninertial Référentiel, which is animated of a movement accelerated compared to a reference frame galiléen, it is necessary to utilize the inertias.

## Applications of the reference frames galiléens

The reference frames galiléens are employed in Newtonian Mécanique and restricted Relativité.

• In Mechanical Newtonian, all the reference marks galiléens is supposed to be equivalent: if two observers O and O' are animated of a uniform translatory movement one compared to the other, then same the laws of the movement must apply to each one of them, that O is supposed to be the “motionless” reference (in which case O' is moving) or that one supposes O' motionless and O moving. Within the framework of Newtonian mechanics, time passes, by assumption, at the same rate/rhythm for all the observers meaning that clock gauged in a reference frame will continue to measure exact durations in any other reference frame, that this one is galiléen or not.

• In the restricted Relativity, the equivalence of the reference frames galiléens is also supposed to be valid. However, the valid assumption of absolute time in all the reference frames is replaced by the assumption of invariance of the Speed of light, which is the same one in any reference frame galiléen. That required the installation of a building theoretical, initiated by Henri Poincaré (1900) in relation to the concept of local time worked out by Hendrik Lorentz and designed, apparently independently, by Albert Einstein. This theory makes it possible observers to define times and distances apparent on the basis of assumption of invariance speed of light in each one of their reference frame and to build a wide frame of reference (the Espace-temps) to measure the time and the distance which separate two events. Thus observers located in different reference frames will obtain a different separation in time and space between two same events. The formulas to convert or “to transform” the values between various reference frames galiléens make it possible each observer to calculate how the laws of physics change when they are observed in another reference frame. Thus, according to the point of view chosen, the time and the distance which separate two events can change, but the interval of Espace-temps is unchanged for him: it is independent of the reference frame galiléen chosen, it is a Invariant.

## Change of reference frame

A change of reference frame is the whole of the laws to apply to convert the physical sizes of a reference frame to another. If conversion relates to the distances and the durations, one speaks about Transformation.

### Newtonian mechanics

Within the framework of Newtonian mechanics, one shows that two reference frames galiléens are necessarily actuated by a relative movement of uniform rectilinear translation. Their relative Speed is thus constant in standard, direction and direction. Let us consider for that two reference frames galiléens: one related to an observer O , the other related to an observer O' . Let us consider moreover one point M which is subjected to no force then, under the terms of the first law of Newton, the body M has a uniform rectilinear motion in each of the two reference marks galiléens O and O' , which one can write in vectorial form:

Thus the fact that in Euclidean geometry $\ vec \left\{OM\right\} = \ vec \left\{OO\text{'}\right\} + \ vec \left\{O\text{'} M\right\}$ and that time is an absolute (independent of the reference frame), it is obtained immediately that: $\ frac \left\{D \ vec \left\{OM\right\}\right\} \left\{dt\right\} = \ frac \left\{D \ vec \left\{OO\text{'}\right\}\right\} \left\{dt\right\} + \ frac \left\{D \ vec \left\{O\text{'} M\right\}\right\} \left\{dt\right\}$ is the law of addition speeds $\ vec \left\{V\right\} = \ vec \left\{v\text{'}\right\} + \ vec \left\{V\text{'}\right\}$

$\ frac \left\{\ overrightarrow \left\{dOM\right\}\right\} \left\{dt\right\} = \ vec \left\{V\right\}$ and $\ frac \left\{\ overrightarrow \left\{dO\text{'} M\right\}\right\} \left\{dt\right\} = \ vec \left\{V\text{'}\right\}$ where $\ vec \left\{V\right\}$ and $\ vec \left\{V\text{'}\right\}$ is constants.

The distance from the observer O to the O' observer can be expressed as well in a reference frame as in the other. Traditional mechanics poses as assumption that $\ vec \left\{OO\text{'}\right\}$ does not depend on the reference frame in which one measures it. That is also true for $\ vec \left\{OM\right\}$ and $\ vec \left\{O\text{'} M\right\}$ and so the following vectorial relation is true in any reference frame (that it is galiléen or not):

$\ vec \left\{OO\text{'}\right\} = \ vec \left\{OM\right\} - \ vec \left\{O\text{'} M\right\}$

In addition, in traditional mechanics, time is absolute , him also independent of the selected reference frame. Consequently the derivative of the vectors OM and O' M are calculated compared to the same temporal variable T and one can thus deduce the relative movement from O compared to O' :

$\ frac \left\{\ overrightarrow \left\{dOO\text{'}\right\}\right\} \left\{dt\right\} = \ frac \left\{\ overrightarrow \left\{dOM\right\}\right\} \left\{dt\right\} - \ frac \left\{\ overrightarrow \left\{dO\text{'} M\right\}\right\} \left\{dt\right\} = \ vec \left\{V\right\} - \ vec \left\{V\text{'}\right\}$

Who means, by definition, which the reference frames galiléens O and O' are animated of a rectilinear motion uniform one compared to the other.

From this relation one deduces the Transformations from Galileo, which make it possible to define a Relation of equivalence between the reference frames.

### Restricted relativity

Einstein affirmed that if the light is propagated at the speed C in a single absolute reference frame (as that were postulated in physique pre-relativist where one supposed the existence of a ether), then the transformations of space and time are done using the Transformations of Galileo. On the other hand, in the case of restricted relativity, in fact the Transformations of Lorentz apply, which coincide with the transformations of Galileo for relatively low speeds compared to speed of light.

## Case of the general theory of relativity

The theory of the General relativity of Einstein treats accelerated reference frames by replacing the Euclidean Espace “flat” of the relativity restricted by a not-Euclidean Géométrie “curved”. In general relativity, the principle of inertia is replaced by the Principe of equivalence which postulates that the inertial mass and the mass engrave are identical. A consequence of this principle is that the objects move according to Géodésique S and that it is more acquired only isolated objects will preserve a uniform rectilinear relative movement indefinitely. This phenomenon of deviation means that the reference frames galiléens are local and either total as it is the case in Newtonian mechanics or restricted relativity.

However, general relativity can coincide with the relativity restricted on small areas of the space time where the effects of curve are less important, in which case the reasoning related to the reference frames galiléens can again be applied. Following what, restricted relativity is sometimes described as a “local theory” of general relativity (that relates to however only the application of the theory and not an established relation).

## Analysis criticizes concept of reference frame galiléen

The reference to the Lois of the movement of Newton does not establish with it only coherence. One starts by choosing an isolated system: the resultant of the force S which are applied to him is null. One defines then a reference frame galiléen: the Speed of the system isolated compared to such a reference frame is constant. All that extremely well, but how does one define functions a force? How can one decide if a system is subjected to a whole of forces whose resultant is null? The only possible answer consists in checking that its movement is rectilinear and uniform compared to a reference frame galiléen (!) It is thus necessary to give itself at the beginning an arbitrarily selected reference frame galiléen.

Newton is left there rather well seemingly while affirming the existence an absolute space (and thus implicitly galiléen). All reference frames known as " galiléens" are thus in rectilinear motion and uniform compared to a reference frame related to absolute space, and thus the laws of Newton can be expressed within an absolute formal framework. They are not used any more to qualify a reference frame like " galiléen" , but are used on the contrary to decide if a system is isolated or not. One can then tolerate the expression " reference frame galiléen" although this one concerns an abuse language. The immediate consequence of this abuse is however rather underhand because it authorizes the assignment of a " state of mouvement" like intrinsic property of a system: for example it will be affirmed that a system is " moving accéléré" , which concerns a catégorielle error.

All the building collapses as soon as one abstains from postulating the existence of a space and an absolute time. The theory of the restricted Relativité does not escape a double circularity from its postulate founder. On the one hand the triplet of properties associated with a system " who is subjected to no force" , " isolé" , " moving galiléen" is defined circularly like above exposed, and in addition, one must refer to physical concepts such as the " force" to pose the formal space-time framework on the base of which one intends to set up a physical theory, and in particular the concept of " force". It is then perfectly clear that the expression " state of mouvement" cannot indicate any more one intrinsic property of a system, but only one relation between the system and a certain reference frame. The abuse language " reference frame galiléen" raise here still of a catégorielle error.

The solution of this imbroglio is outlined higher: " it is necessary to give oneself at the beginning a reference frame galiléen arbitrarily choisi". To make " like si" it was about the absolute space of Newton. But for being satisfied of this point of view located , it is necessary to build " the point of view of nowhere " , i.e. to extract the Invariant from all the points of view located, of which each one reflects a particular choice of the reference frame " pseudo-absolu" of departure. One can then create the concept of " class of equivalence galiléenne" stipulating that two reference frames are equivalent to the direction " galiléen" term if and only if their relative movement is rectilinear and uniform. The Accélération becomes a relative concept then making it possible to differentiate or compare the classes of equivalence. A theory " relativiste" would require whereas the " laws of the physique" expressed in various reference frames belonging to the same class can only differ by the value the relative speed of these reference frames. And if they were expressed in reference frames belonging to different classes, they could differ only by the value from relative acceleration between the relative classes and speeds between the reference frames.

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