In regards to the "Galilean Transform:"
IMHO, nearly everyone is missing the fact that all observers in either frame must remain fixed in relation to the origin of their respective frame/system. Whereas the source may be placed anywhere in the source frame, there is less confusion, as a convenience, if it is placed at the origin of the source frame.
Next IMHO, nearly everyone is missing the fact that there are two entities that are undergoing motion in the "Galilean Transform." One is the expanding pulse wave of light (EMR). The other is the "moving coordinate system/reference frame." I pointed out in the post above, that the source is only in one frame or the other, and either frame may be taken as the "rest frame." In the discourse below, I am taking the source frame as the rest frame.
The expanding pulse wave of light (EMR) is generated by the source. There is only one source under investigation, and the pulse light wave expands away from it in all directions at the same speed. IMHO, this fact generates an expanding sphere with the source at the center. It is/must be depicted as existing in this form only in the source frame/coordinate system.
For emphasis: The source exists, fixed to the origin, only in the source frame/coordinate system.
It cannot be both fixed in the source frame and fixed anywhere in the moving system. [except "instantaneously." Over many "instants," the source will become a series of points in the moving system, forming a line there. (As long as the moving system is moving in a straight line, unvarying in velocity.) ]
In reality, the only way the light/EMR sphere can be known is to be detected by sensors made of matter. I am naming these "sensors" observers
. In the source system, they are fixed to the source's coordinate system origin. The observers in the moving system are at fixed distances from each other and the moving system origin.
In the moving coordinate system each observer has fixed coordinates (x,y,z).
The source however, as depicted in the moving system, (moving along the X axis in the moving system) will have multiple X axis coordinates; one for each instant of time as time passes. So . . . the source coordinates as "transformed to the moving system" are (xt=0
,y,z); (xt+successive instants of time
,y,z): essentially forming a line in the moving system graph.
Therefore, moving system observers will encounter successive wave fronts of the expanding sphere at decreasing radii as they approach the source, and increasing radii wave fronts as they recede from the source. Meanwhile the source itself will appear to approach or recede from the moving observer. The source approaching or receding from the moving observer is what creates the line depiction of the source in the moving frame/system. (It is not seen in reality as a line, it is just seen as increasing or diminishing in size with change in distance from the observer.)
Concurrently in the source frame/system, each observer will encounter fixed radii wave fronts of the expanding sphere. The source appears unchanging in size for each observer in the source frame/system.
In the conventional "Galilean Transform,"
And the transform from the other frame/coordinate system back is x= (x'+vt)
In reality, the transform is non-reversible, their "event" is not defined, and the source is not "in" the opposite frame. If "x' " represents an observer, and (x-vt) represents that same observer in the other system at a particular instant, then, over a duration of time the (x+vt) will depict a line in the said system, just as the source does in that system. Of what use is that?
Logic tells me that the moving system has it's own observers fixed in relation to the moving system origin; likewise the source system has it's own observers fixed in relation to the source system origin. "x" and "x' " are separate entities; and in the moving frame/system, any one observer is both approaching/receding the source and being impinged by wave fronts that have traveled away from the source. The observers do not detect the source directly, they only detect the wave fronts.
Thus, the "transform" is non-reversible.
So . . . what exactly is the event which Einsteinians use as the basis of all their musings???
I sense a disturbance in the farce.