User:Zoltan the Temporal Mechanic/sandbox

Derivation of the Galilean Transformation
The Galilean transformation is the basis from which the equations of special relativity, the Lorentz transformation equations, are derived. The Galilean transformation consists of two inertial systems that are in relative motion along their x-axes. An event occurs in one of the systems. It becomes desirable to measure the event simultaneously from both systems. The simultaneous measurement is enabled by the transfer of an observer from one system to the other, resulting in a transformation of coordinates. Two transformation equations are generated, one for each transfer direction.

Systems S and S’
The Galilean transformation is comprised of two independent Cartesian systems S (x,y,z) and S’ (x’,y’,z’), each is a system of space with its own distinct origin O and O’.

The Motion
Motion is constrained to the x-axis of system S and to the x’-axis of system S’. There is no absolute rest or motion, only relative rest and relative motion. The motion between systems S and S’ is relatively at rest or relatively in motion. The rest or motion can be quantified by the relative velocity v, which is a constant because the systems are inertial, but there’s something else to know. Relative motion isn’t just the opposite of relative rest, relative motion occurs through phases.

Approaching Phase
The earliest and most general phase of relative motion consists of two systems approaching each other. The approaching phase precedes the localized phase.

$$S \longrightarrow S'$$

Localized Phase
The localized phase of relative motion occurs when the origins O and O’ occupy the same point of space. This occurs after the approaching phase, but before the receding phase. This phase is a transient state.

$$S \odot S'$$

Receding Phase
The receding phase of relative motion consists of two systems moving away from each other. The receding phase occurs immediately after the localized phase, it’s the final phase of relative motion.

$$S \leftrightarrow S'$$

The Strobe
A strobe light is finitely displaced from the Origin of system S at a distance $$x$$, relatively at rest. The strobe can be activated to emit a burst of visible light.

Activating the Strobe
The distance from each system to the strobe is a parameter that needs to be measured simultaneously by both systems. The phases of relative motion suggest a strategy for activating the strobe, in a manner that promotes simultaneity. The strategy is to reset the clocks and activate the strobe when the systems localize. This has the effect of initiating receding motion and synchronizing the clocks in systems S and S’.

$$t'=0$$ and  $$t=0$$ when $$S \odot S'$$

The Observer in S
An observer is placed at the origin O of system S. Systems S and S’ localize, clock t is reset, and the strobe is activated. The observer measures the distance $$x$$ to the strobe. When the light signal arrives the observer finds that it has taken an increment of time $$\mathbf \Delta \mathbf  t$$ for the light to travel at the speed of light c from the strobe to O.  The distance $$x$$ to the strobe and the distance $$c \Delta t$$ traveled by the light signal are one and the same distance.

$$x = c \Delta t$$.

The Observer in S’
An observer is placed at the origin O’ of system S’. Systems S and S’ localize, clock t’ is reset, and the strobe is activated. When the light signal arrives the observer measures the distance $$x'$$ to the strobe and finds that it has taken an increment of time $$\mathbf \Delta \mathbf  {t'}$$ for the light to travel at the speed of light c from the strobe to O’. The distance $$x'$$ to the strobe and the distance $$c \Delta {t'}$$ traveled by the light signal are one and the same distance.

$$x' = c \Delta {t'}$$.

The Transfer
The strobe is relatively at rest in system S, displaced from the Origin by a constant distance $$x$$. The observer in system S will measure $$x$$ in all phases of relative motion. The observer in S’ sees the distance $$x'$$ to the strobe first as decreasing, then as increasing. That’s because the initial phase of relative motion between the strobe and system S’ is the approaching phase. The phase difference is explained by the strobe’s displacement from the origin of system S. This seems to complicate the problem of simultaneous measurement, here’s one way to solve that problem. Make the measurement in one system, instantly transfer the observer to the other system, and immediately make the measurement in the other system. This is an idealized scenario, but that would be a strategy for executing simultaneous measurements in systems S and S’.

The Transfer from S to S’
The observer is in system S. The systems localize, the strobe is activated, the clock is reset to $$t=0$$. In the $$\Delta t$$ time it takes for the light signal to travel the distance $$x$$ the two systems, moving with relative velocity v, have separated by $$\mathbf  {v \Delta t}$$due to receding motion. The observer transfers to system S’ and measures the strobe distance as $$x'$$.

$$\mathbf {x' = x - v \Delta t}$$

The Transfer from S’ to S
The observer is in system S’. The systems localize, the strobe is activated, the clock is reset to $$t'=0$$. In the $$\Delta t'$$ time it takes for the light signal to travel the distance $$x'$$ the two systems, moving with relative velocity v, have separated by $$\mathbf {v \Delta t'}$$due to the receding motion. The observer transfers to system S and measures the strobe distance as $$x$$.

$$x'=x-v \Delta t'$$.

Separating the systems on each side of the equation,

$$\mathbf {x=x'+v \Delta t'}$$

Summary
No assumptions have been made about system S being at rest and system S’ being in motion. All considerations of motion have been made using the principle that motion is relative. The Galilean transformation equations for the transfers between systems S and S’ are, by transfer direction:

$$\mathbf {S \rightarrow S'}$$       $$x' = x - v \Delta t$$

$$\mathbf {S' \rightarrow S}$$       $$x=x'+v \Delta t'$$