As a physicist, I am among the first to admit that we scientists do not know everything, but some things we do know. The process of science- in the strict sense of the word- is the systematic description of observed phenomena in terms of observed and reproducible processes. The privilege of a working theoretical scientist is to attempt to try and explain things which are not understood or to better explain things that are not understood as well as could be. Similarly, the privilege of an experimental scientist is to attempt to find situations in which the currently accepted explanations do not lead to correct or entirely correct results. Science advances more from what scientists do not understand than by what we do understand.
“Einsteinian relativity” can refer either to special or to general relativity. Both subjects have advanced well beyond the form initially set forth by Einstein, more than a century ago (from 1905) for the special theory of relativity and nearly a century (from 1915) for general relativity.
The essence of special relativity is the fact that the speed of light remains constant in any frame of reference. No matter who observes light and from where or in what situation, the speed of that light will always be identically the same, approximately three hundred million meters per second. This amazing fact was established experimentally by a pair of researchers named Michaelson and Morley before the publication of Einstein paper “On the electrodynamics of moving bodies” [written in German] in 1905 which established special relativity. Special relativity simply develops the physical consequences of this fact systematically. One of those physical consequences is that the speed of light in vacuum becomes a physical limit to speed for any object in the universe. To be precise, phase velocity can exceed the speed of light in vacuum under some limited circumstances but no coherent information or physical object is able to exceed the speed of light in vacuum. Ironically, the way that this has been confirmed is most commonly in the sub-atomic regime. If the speed of light is constant for any object, then the physical passage of time must slow down as an object is accelerated to speeds near that of light. Why must this be so? If the passage of time did not slow down, then light would be seem when measured against the accelerated object to travel at a speed different than the constant light-speed. This slowing down of the passage of time has been measured to incredible precision countless times in particle accelerator experiments by measuring the rates of decay of various radioactive particle accelerated to near light-speed. The time-dilation effect, as it is called, predicted by special relativity exactly describes how decay rates change for accelerated particles.Other examples could be cited as well, but this type of observation serves by itself to establish that special relativity physically works.
General relativity is a modification of Newton’s description of gravity in order to accommodate special relativity. One of the questions which puzzled 19th century scientists was the deviation of the orbit of the planet Mercury, the planet by far closest to the sun, from the orbit predicted by Newton’s law of universal gravitation. One of the less obvious consequences of special relativity is that mass and energy cannot be conserved separately. Energy and mass become under certain circumstances interchangeable according to the famous formula E=mc^2. General relativity takes this into account and so includes a relatively small correction to the sun’s effective mass due to the sun’s energy. For all the planets farther away from the sun, the correction is so small that it can be ignored, but in the case of Mercury the effect is not negligible. In fact, the orbit of Mercury is exactly explained by general relativity. Again, other cases could be cited and a copious description is given in works such as Stephen Hawking and Werner Israel’s landmark book “300 Years of Gravitation” written to commemorate the 300th anniversary of the publication of Newton’s book “Principia” in which Newton presented his theory of gravity, but the case of explaining the orbit of Mercury shows that general relativity physically works.
Any critique or refutation of relativity must stand up to the criterion of Occam’s razor; the simplest explanation consistent with all known facts is probably correct. Hands down, relativity is the simplest explanation for these kinds of observations. Moreover, no known case exists where the physical prediction from relativity in a given physical situation has been shown to be incorrect.