I get my "demonstration" of time dilation from the textbook thought experiment.
A laser is mounted on a cart with a reflective ceiling. At $t=0$ the cart starts moving and the laser is fired. When the laser is reflected back at the starting point the (thought) experiment stops.
Now, two different observers, one sharing the frame of the cart and another standing on the ground perpendicular to the cart will observe two different things. For the first one, the laser bounces back and then down in a straight line. For the second one, the light travels in a triangular pattern which is longer than the path observed by the first guy.
Given that the speed of light is constant, the time has to dilate/contract.
Why is the speed of light held constant here? Could we work out a physics where time is absolute but the maximum speed of light variable?
Best Answer
Keep in mind that several other Einsteinian effects are hard to explain in an absolute-time scenario and are tested. For these purposes I would concentrate on:
The existence of a speed limit for massive particles
Looping accelerator systems based RF cavities only work because once the particles have enough energy their speed is effectively constant (and that speed is within a hair's breadth of $c$). But this constant speed doesn't mean constant kinetic energy or momentum: the accelerator continue to add energy and momentum with measurable consequences in the bends and the experimental halls.
Conversion of other energies to mass and vice versa
We can measure the kinetic energy and mass of reaction products in particle physics experiments and when we produce heavier products than we started with the extra mass is related to lost kinetic energy in keeping with $E = mc^2$. Likewise when particles decay to lighter products the products have extra kinetic energy in keeping with the mass difference.
The twin paradox
It is not just a fanciful notion you find in books, but something that we do with unstable particles in looping accelerators (example, muon-g2).
To fit all of those that into a absolute time framework will require more (and to my mind very arbitrary) assumptions. By contrast the Lorentz symmetry of special relativity explains them all in one go and is motivated by the structure of Maxwell's equations (that is has an experimental basis and is not at all arbitrary).