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ilasus

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- TL;DR Summary
- About Galilean Transformations

Galilei transformations are a set of equations that describe how the coordinates of a point P change between two reference systems R, R' moving at constant speed v relative to each other. For example, when moving from the reference system R to R', the Galilei transformations are given by the equations:

(g) x' = x – vt, y' = y, z' = z

and when moving from the reference system R' to R, the Galilei transformations are given by the equations:

(g') x = x' + vt', y = y', z = z'

where x, y, z are the coordinates of the point P in the reference system R, and x', y', z' are the coordinates of the point P in the reference system R'. To these equations, classical physics also added the equality t' = t, which states that the time measured in the reference system R' is identical to the time measured in the reference system R. So according to the point of view of the observers in the reference system R, (g), the origin O' of the reference system R' approaches with speed v the point P, and according to the point of view of the observers in the reference system R', (g'), the origin O of the reference system R is moving away from point P with velocity -v. But if we assume that point P is actually an observer, then in the first case, the observer P is at rest at a distance x in the reference frame R, and in the second case, the observer P is at rest at distance x' in the reference frame R'. In other words, the observer P would be an "exception observer", because unlike the other observers, who are at rest in only one of the reference systems R, R', the observer P is at rest in both reference systems. Obviously, the same observer P can be at rest in two different reference frames, but not at the same time. It follows that the time t in which the origin O' approaches the point P at rest in the reference system R, is not identified with the time t' in which the origin O moves away from the point P at rest in the reference system R'. But if t ≠ t', then the positions of the point P calculated in relation to the origins of the reference systems R, R' cannot be identical either: x' ≠ x - vt and x ≠ x' + vt'. This results in a contradiction between the equalities:

(G) x' = x - vt, x = x' + vt, t' = t

and inequalities:

(G*) x' ≠ x – vt, x ≠ x' + vt', t' ≠ t

How is it explained? Or, if you think my logic is flawed, where is the flaw? Thanks.

(g) x' = x – vt, y' = y, z' = z

and when moving from the reference system R' to R, the Galilei transformations are given by the equations:

(g') x = x' + vt', y = y', z = z'

where x, y, z are the coordinates of the point P in the reference system R, and x', y', z' are the coordinates of the point P in the reference system R'. To these equations, classical physics also added the equality t' = t, which states that the time measured in the reference system R' is identical to the time measured in the reference system R. So according to the point of view of the observers in the reference system R, (g), the origin O' of the reference system R' approaches with speed v the point P, and according to the point of view of the observers in the reference system R', (g'), the origin O of the reference system R is moving away from point P with velocity -v. But if we assume that point P is actually an observer, then in the first case, the observer P is at rest at a distance x in the reference frame R, and in the second case, the observer P is at rest at distance x' in the reference frame R'. In other words, the observer P would be an "exception observer", because unlike the other observers, who are at rest in only one of the reference systems R, R', the observer P is at rest in both reference systems. Obviously, the same observer P can be at rest in two different reference frames, but not at the same time. It follows that the time t in which the origin O' approaches the point P at rest in the reference system R, is not identified with the time t' in which the origin O moves away from the point P at rest in the reference system R'. But if t ≠ t', then the positions of the point P calculated in relation to the origins of the reference systems R, R' cannot be identical either: x' ≠ x - vt and x ≠ x' + vt'. This results in a contradiction between the equalities:

(G) x' = x - vt, x = x' + vt, t' = t

and inequalities:

(G*) x' ≠ x – vt, x ≠ x' + vt', t' ≠ t

How is it explained? Or, if you think my logic is flawed, where is the flaw? Thanks.