Along with everyone else Einsteinhad expected an ether to be discoverable by empirical means. Thus it was a blow when no one could find any fault with the Michelson Morely experimental results, or achieve any different results. This was a blow, because much brilliant mathematical conception had been driven and derived on the hypothesis of an ether.
Einsteins solution came from searching the philosophical musings of the greats, particularly Newton, and the works of the Grassmann's, and other contemporaries like Maxwell. It dawned on him thst the current arguments tearing physics apart were due to differences in point of view. He soon realised that Lagrangian mechanics viewed space as a kind of box in front of the observer. The contents of the box evolve in the observers notion of time. However, those who expected an ether held the Newtonian view. They were inside a boundless space, various media obscuring our view of god of which the last may well have been an ether. Thus things evolved in gods time, an absolute time over which the individual had no relational control.
However, if there was no ether, the only thing left to vibrate and undulate was space itself. This could be achieved by dynamically altering the boundary conditions of the box of the Lagrangian. But then that meant that everything within the box was altered and thus not perceivable by an observer within the box.
It was but a short step to the notion of relativity. If light was an Undulatory wave of this Lagrangian box then the observer had to be outside of it receiving the information. But any colleague had to be inside the box as a separate observer. The transmission of information from inside to outside the box was the problem Einstein solved by using and setting the speed of light as a fundamental constant in space, both inside and outside the box!
Suppose now I am outside a box containing two observers in relative motion to each other, what relationships do I derive? In particular, if the rule is electromagnetic information travels at a constant speed as an undulating wave. Suddenly within the box all distance can be interpreted as time for light to travel and all time as distance light has travelled.
Then relative speed becomes important because if an object moves very slowly relative to another light can pass information, and clocks can be compared, and velocities calculated or distances calculated in proportion to light speed. Moreover, the undulation of the box transmits this information to me as an outside observer. But I observe the evolutions within the box in my own light speed determined time! So every bit of information is relative to the observers space time reference frame, and that reference frame is ultimately dictated by the constant speed of light.
This was a clever obscuring of the notion of ether. To further obscure it, Einstein cloaked it in tensor mathematics.
It was well enough obscured to provide a platform for all competing scientists to unite around. A few Aether Wave theorists were not convinced and continued to promote a marginalised view of the data, little realising that they were saying the same thing as Einstein, but in less mathematical language!
The main " trick" was to do away with the notion of absolute time. Einstein initially felt that this did away with god, but as he worked through his theoretical manipulations he realised this only made a different god out of different predicates of ontology. Not that he was a deep theologian , but he was a deep philosopher.
Doing away with time seems radical, but in fact time is one of the oldest human constructed notions. The word itself derives from the practice of recording positions of astronomical bodies. When one is chosen as a metronome all others are thereby " timed", that is their relative positions plotted relative to the position of the metronome in space. We therefore need a notion of spatial position and change before we can construct a relative position as a notion of proportions of a standard metronome. If now we have an actual standard metronome, namely the constant speed of light we can standardise all records of position relative to that constant as fixed proportions. This then reveals that these positions are relative to the observer, and more disengenuously, that the metronome of time is also relative to the observer! Thus the essence of spacetime is that one cannot assume that time is constant or the same for each observer. The only constant allowed is the speed of light.
when Einstein started with the Galilean principle he started with a relationship that was dynamic, but displayed by a fixed notation. Represented graphically this was odd. The notion of magnitude meant that any label was referring to a continuously varying experience, but clearly some parts of that experience appear fixed relative to an observer. It suddenly became clear that by using Lagrangian mechanics he could notate this experience clearly. It was then that he realised thst what he thought of as just a spatial experience of extension could be conceived of as an experience of space and time extension, where space was fixed unles varied as a function of time. The beauty of his conception was its relativity. Space only varied as a function of time according to each observer, not according to God. Thus the Galilean principle of physical laws remaining the same relative to each observer was maintained not by psychological deception, but by fundamental subjective time distinctions. These must come from a fundamental constant truth, and that was not god's absolute time, as Newton thought, but natures absolute speed, the speed of light..
Trying to do math without making this assumption is where most find difficulty with Einsteins approach,mand indeed Einstein found it difficult too. Certainly in some of his presentations he was not clear at all, and actually used some examples which countered his argument, especially the Galilean transforms. The difficulty is in calling them transforms as per Galileo when in fact they are Lagrangian transforms. The difference is subtle. For Galileo the equations are fixed not morph able. By this he establishes the principle of the uniformity of physical laws in all frames moving uniformly relative to each other. Specifically Galileo specifies a closed room or box. For an observer outside the box everything in the box moves as one. Thus the position of each object in the box can be resolved into two parts: one in the observers local reference frame and another in the boxes reference frame . This is a Galilean transform.
However if time is allowed in the observers reference frame, then the successive positions of objects may be plotted , but how? There is no time dimension in a Galilean transform. We have to introduce a fourth axis and thus we create space time. Having done this theoreticians distort the model by typically leaving out the z and the t axes and talking using the x and y axes, altering the formula somewhat mysteriously. The position on the x axis now encodes t on the time axis also.
Now a swap is done that is Lagrangian and not Galilean. Suddenly a distance becomes a velocity product with time. This is fine in a Lagrangian mechanic, but totally unacceptable in a Galilean. The velocity in a Lagrangian is defined, but in a Galilean it has to be measured, and that is the difference. The Lagrangian establishes a variational calculus from which we may derive structural solutions at maxima or minima. These we can resolve into measurement schemes and thereby test the veracity of the mechanical model. To claim to do that through a Galilean transform is contradictory, for at all times in a Galilean transform one observer can measure the position of the box and compute its velocity making the whole example apparently futile!
The Lagrangian constraint that Einstein wished to establish was that for the situation of a uniform velocity, with no acceleration the constraint resulted in the relative velocity v between the two observers, but as the relative velocity increased to c other relations, principally distance and time altered accordingly. Distance and time dilated!
Mathematically this was straightforward, and always occurs when ratios are compared. Normally, however, commensurability keeps relative scales in order. What makes the difference is the constant the speed of light. As soon as a constant is introduced into a comparison, as being the same on both sides of a duality, the constraint actually restricts the mathematical notation to thst of spherical trigonometry!
Thus we derive results entirel consistent with spherical trigonometry, and in fact we determine thst the Lagrangian box is in fact a Lagrangian sphere!
The question was, does this apply to the real world? Do objects shrink and time shrink the faster they go? Well the answer is an obvious yes. It is obvious because we experience perspective dilation every day and trigonometry accounts very well for it, if inaccurately as measurement approaches the upper and lower limits. Thus measurements taken optically and not corrected for perspective give biased readings in actuality. Objects very far away and thus requiring a long time for light to travel will be overly prallaxed and appear smaller than they actually are close up. Correcting for this enables physics to be compared universally. If this is not corrected, then parallax issues and perspective issues distort reconstructions of distant stars and planets and their orbits.
When Kepler worked out elliptical orbits as the correct form, it was only after trying everything else. It was the assumption of circularity which caused the difficulty. Observations of a circle, inside a circle will necessarily appear elliptical due to perspective transforms. We actually measure what we experience, not what an ideal form determines.
Hence the comparison of times in this interdependent Lagrangian sphere requires so called time dilation equations, but they are just trigonometric ratio constraints due to applying and maintaining Pythagorean principles for the right triangle or spherical trigonometry.