"This is a Galilean transformation: If a reference frame K' moves with constant velocity v relative to K, then x' = x - vt, and, of course, t' = t"
So why is the source “transformed” into the observer frame? It shouldn't. If it is, then the radiation from it is also “transformed,” and the observers there only detect what observers in the source frame detect of the radiation from the source while at rest with the source.
In high school algebra, one is taught to precisely define the terms. What do we want to discover? How about we discover what observers in a moving frame, at rest with each other, will detect from radiation in the form of a light pulse radiating from a source; where the source and observers are at continuously changing distance? (that is: approaching or receding)
Consider the “(K)” coordinate system:
The source is fixed at the origin. There it will stay.
Since a light pulse travels at about one foot per nanosecond, we let each integer on the x and y axis equal one nanosecond and simultaneously, one foot.
Now, two parameters about the light pulse's position are determined by distances between coordinate pairs, simply by observing the "x" and "y" coordinates, namely:
The time delay of the light pulse and the distance it has traveled,
Or the time delay and distance from the source to the observer/detector.:
Thus: x+d is also x+t, or we might say x+d=x+t, and y+d=y+t.
Consider the “(K')” coordinate system:
The source/emitter is not allowed in this “(K')” system, to avoid confusion.
The origin of this “(K')” system is occupied by a detector/observer. [no source]
This “(K')” detector/observer momentarily occupies [passes over] the source/emitter in the “K” at the very instant the pulse is radiated by said source. This common/simultaneous event associates the “(K')” system to/with the “(K)” system.
All additional observer/detectors in this “(K')” system remain at fixed distances to the origin/observer/detector of the “(K')” system. [Not to the source, in “(K)”]
Assume the “(K')” system origin approaches the source from the right: therefore it is moving to the left.
Do not picture more than the one source, the one pulse of light, and one expanding wave front or you will become confused.
Consider that as the radiated wave front expands in the “(K)” system, the entire “(K')“ system (with all its observer/detectors fixed in relation to its origin) moves parallel to the path of the “(K')” origin as it moves relative to the “(K)” origin/source.
Each detector/observer in the “(K')“ system only has one instance to detect the “light pulse wave front” expanding in the “(K)“ system. [This is because the coordinate systems are changing distance slower than the expansion of the radiation of the light pulse.]
To avoid considering the positions of too many detectors/observers in the “(K')” system, we will, in this discussion, only consider detection of the light pulse wave front as it expands along the x and y axis of the “(K)” system,
At each “foot/nanosecond” along the positive y axis in the “(K)” system, [representing the expansion of the light pulse there,] we place a detector/observer in the “(K')” system, keeping in mind that the whole “(K')” system is moving relative to the “(K)” system axis.
Thus, if we choose an arbitrary change in distance between the two systems of half the speed of light, the x coordinate of each “(K')” detector/observer will increase by .5, and the "y" coordinate will increase by 1, with respect to the “(K')” origin.
These detector/observers will appear as a diagonal line extending up from the origin in this system, as the wave front advances up the "y" axis in the “(K)” system.
Now, considering the observer/detectors along the "x" axis in the “(K')” system, as it approaches the source/origin in the “(K)” system, the “y” coordinate will not change value. The "x" coordinate will increase by one half in relation to the “(K')” origin. Each of these detectors must face the origin, since the wave fronts are approaching the detectors as they approach the source.
You see, so far we have in the “(K')” system, a series of detectors slanting up and to the right from the origin, and a series of closely spaced detectors along the "x" axis, placed in the “(K')” system as each foot/second of the expanding wave front was reached in the “(K)” system. This series of approaching detectors start detecting when the origin passes over the source.
This next step is where I believe Einstein and his followers miss the application/ or perform another slight of hand, regarding their “transform:”
As the origin of the “(K')” system passes over the “(K)” source, the other side of the wave front will not be detected, because it has already expanded before the “(K')” origin arrived.
Thus, to detect this oppositely expanding wave front pulse, additional detectors must be added to the “(K')” system to the left of the origin detector, spaced out relative to the origin of their [the “(K')”] system, by one and one half feet apart. Silly Einsteinians call these “future” detectors. They are simply part of the “(K')” system. There is no future or past involved since all the detection is simultaneous with the conjunction of the wave front pulse. There is only one pulse/wave front, we are just detecting it at various time/distances, as the “(K')” system passes over the “(K)” system.
You should comprehend that a continuous source of light radiating in the "(K)" system by one detector approaching along the “x” axis in the “(K')” system will detect a Doppler blue shifted light, by the close spacing of the detectors in the approach phase of the above scenario. Likewise a red shifted light results from the receding detector by the increased spacing of the "future" detectors to the left of the "(K')" origin in that system.
I sense a disturbance in the farce.