May I suggest we begin with reasons why Special Relativity is an impossibility? The first problems with this work of trickery are the incompleteness of the two postulates. These half statements are used to lead readers to draw wrong conclusions.
The first postulate only says that physics works the same in all inertial frames of reference, moving or not.
This is only half of the correct statement, which is, "within moving or stationary inertial frames, physics provides the same results AS MEASURED BY AN OBSERVER IN THE FRAME OF INTEREST.
This fuller statement about the use of inertial frames gives a different picture of the way we should apply this knowledge. It means that the results of measurements are only going to be identical across any inertial frame, PROVIDING that the measurement is ALSO done within that individual frame.
So a ball moving up and down in a speeding car will be measured as moving up and down in exactly the same way in a stationary car, PROVIDING the measurements are taken by some observer within that car.
IF, and only if an observer outside in the car wants to measure the balls action, AND STILL GET THE SAME RESULT as the cars occupant gets, WOULD BE to take the measurements THEN APPLY a compensation to allow for the fact that the car is in motion relative to his position when he took his measurements.
ONLY then, can the Galilean principal of the equality of inertial frames still work. Both moving observer and stationary observers will find that physics works the same in all inertial frames.
But because EInstein only states HALF of this basic principal of Physics, he is able to trick and fool the reader with this half truth postulate. Einsteins hypothesis is wrong when he allows the stationary observer to measure the motion of the light pulse in the moving vehicle, BUT FAIL TO APPLY THE REQUIRED COMPENSATION. Then Einstein claims that there will be a discrepancy between the two obsevers requiring a Lorentz transformation, when in reality, if physics is done correctly, there is no discrepancy.
The half postulate makes no sense if you think about it, only the fuller, complete statement makes sense.
And armed with a correct understanding of how inertial frames work, the reader of Einstein's paper will come to the place where Einsrtein's weak logical arguments fall down, because of this one deceptively stated postulate.
But it gets worse, as the second postulate is also a half-truth. And also allows the formation of the deception that is called Special Relativity.
We will look at this later.
I'll give a try by writing about the origins of relativity.
I don't know who developed galilean relativity, but it was in times where time was considered as absolute, universal.
The galilean relativity is simple, and explain basic behaviors observed in nature (like sound waves and Doppler effect):
Being (x,t) variables for unidimensional positions (one axis) and absolute time then, moving in an inertial platform at
a constant longitudinal velocity, the observer on the moving platform has coordinates for space and time as (x', t'),
in a way that they relate to static coordinates (x,t) by the formulae:
x' = x - v.t
t' = t
So far so good. But, in January 1897 and prior to the Michelson-Morley experiment, the german physicist Woldemar Voigt
published a paper about sound waves and the Doppler effect, taking into account that information between the resting frame
and the moving frame takes a finite amount of time to reach the moving observer.
He, then, introduced the concept of LOCAL TIME, which is pre-Lorentz's relativity attempts (PROPER TIME), and he wrote
a modified galilean set of equations which accounted for retardation in the time it takes the sound to reach the moving
observer, when it was emitted by the observer at rest (think of a train, a station and two observers at each platform).
Voigt introduced these modifications:
x' = x - v.t
t' = t - x.v/w2
where w was the speed of sound.
With these modifications, the local time t' of the moving observer is different from the time "t" of the observer at rest.
If the fixed observer at station (x = 0) emits a sound, these things happens:
Synchronizing clocks for both observers WHEN t = 0 (this is set in this way) it gives that position x' equals position x
(checked by observing markers at both reference axis x and x').
But sound hasn't reached the moving observer yet. Then, even when it's measured that position x' is equal to position x,
the time t' for the sound wave is delayed by an amount:
t' = -v.x/w2 (which implies that t' is delayed in reference to t).
The moving observer will have to wait a time t' = -v.x/w2 to hear the sound emitted by the fixed observer at x=0.
Remember that the observer at x is not moving. Is standing still at the train station.
The following things can happen:
1) If the speed of sound is infinite, then t' = 0 when t = 0 (what gives a truly galilean relativity with absolute time).
This means that there is no delay, and that time is absolute.
2) If x = x' is too large when t is set to 0 (t = 0) (i.e.: 1 Km far away from the station), the moving observer will have
to wait in order to hear the sound wave emitted at the station (x = 0) a delay given by -v.x/w2.
3) The same thing happens if the velocity v is high, because the sound wave has to catch up the train, and the
delay is still -v.x/w2 .
4) If the velocity of the train is equal to that of the sound, then the delay for the sound emitted at x=0 has a delay -x/w,
For hypersonic values of "v", the formula from 3) still applies, only that the delay becomes larger and larger.
The above concepts were the basis of the paper of Voigt for the propagation of sound. His work is more complex than my
simplification, as he uses a 3D representation for the sound wave and there are no trains or "observers", but is the genesis
of Lorentz-Poincarè-Einstein relativity.
Credits for Voigt are: the introduction of LOCAL TIME and the introduction of the GAMMA FACTOR, even when he doesn't
apply it to the x axis or the time.
Lorentz was looking for "length contraction" only, in order to explain the "fail" of the Michelson-Morley experiment.
However, the expression of "time dilation" appeared as a collateral "damage" in the Lorentz-Poincarè relativity.
When Einstein used all of the above and abolished the ether and any absolute frame of reference, he made profit
of the time-dilation consequences of the Lorentz-Poincarè relativity and ASSERTED that this was a REAL THING.
This happened in the Einstein's 1905 paper about relativity and, for unknown reasons, physics never went back from this.
My advice: start with Voigt's LOCAL TIME concept (which Lorentz inherited).
Once you accept it (or reject it) you can give a try to Einstein's PROPER TIME, which is equal to Gamma Factor x Voigt's Local Time.
And this is the core of the einstenian relativity, being Gamma Factor the root of all evils around special relativity, as it replace the speed
of sound by the speed of light, and then it's declared that "c" is an unbreakable limit for any speed at THIS world.
Hope this can help you a little. At least, I tried to show you how the concepts evolved with time since 1897.