Historically three functions were attributed to time: simultaneity, synchronization and duration. Time became important in physics even before Galileo analyzed how gravity could be diluted by using a slope. Middle Age mathematicians made the first differential calculus computations using time, two centuries before Fermat established calculus.
Newton used calculus for his detailed theory of gravitation. However Isaac thought his own theory made no sense. The problem was that gravity was supposed to act instantaneously at a distance. Isaac thought that “it is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.”
— Isaac Newton, Letters to Bentley, 1692/3
[The picture actually alludes to a completely different work of Poincaré, his discovery that qualitative methods in non solvable differential equations produced results where exact differential equations a la Newton did not: in particular, Poincare’s recurrence theorem… Useful in astronomy.]
Newton’s theory depended crucially on an absolute, universal time: thus the gravity force vector could always point to the center of (the) mass (exerting the gravitational force).
However the wrapping up of the electromagnetic equations by Maxwell showed that light was electromagnetic field travelling at speed c. C was universal. And independent of any “rest frame”. After thinking about the problem for twenty years, Lorentz discovered that, for electromagnetic phenomena to stay the same in a moving frame, one had to introduce what Poincaré called a “Local Time”. Poincaré then pointed out that there was no absolute rest relative to an “ether”, all one could do was to analyze the motion of matter relative to matter.
Then Poincaré thought some more for five years, and published in 1900, in the major Dutch physics Journal, that electromagnetic field retardation and its violation of Newton’s Third Law (Action equals reaction) could be resolved by attributing the inertial mass E/cc to the electromagnetic field.
(Mass = energy was attributed to a number of second order German physicists for Francophobic and nationalistic reasons, and the notion is repeated to this day by ignorant parrots; that would be sort of funny, if it did not distort not just the history of physics, but even the understanding of physics, as the parrots tend to not have as deep an understanding the underlying concepts).
“The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion… From all these results, if they were to be confirmed, would issue a wholly new mechanics which would be characterized above all by this fact, that there could be no velocity greater than that of light, any more than a temperature below that of absolute zero. For an observer, participating himself in a motion of translation of which he has no suspicion, no apparent velocity could surpass that of light, and this would be a contradiction, unless one recalls the fact that this observer does not use the same sort of timepiece as that used by a stationary observer, but rather a watch giving the “local time.[..] Perhaps, too, we shall have to construct an entirely new mechanics that we only succeed in catching a glimpse of, where, inertia increasing with the velocity, the velocity of light would become an impassable limit. The ordinary mechanics, more simple, would remain a first approximation, since it would be true for velocities not too great, so that the old dynamics would still be found under the new” [Poincaré, 1904.]
So after Poincaré’s work, what was the situation? Time is local (yet clocks could be synchronized at a distance), Galilean relativity could be extended to electromagnetism as long as mass = energy.
Are we further along today?
Poincaré kept a distinction between “apparent time” and “ether” given time. Einstein’s variation of the theory does not preserve this distinction (and that makes it false, ha ha ha). I will not go into the details here, as it would be pure research of the sort that 99% of theoretical physicists are unwilling to consider (some other day, in simple words). I am not trying to spite Einstein, long my preferred physicist (no more, though, he has exhausted my patience with vindictive plagiarism, in particular against Poincaré and Karl Popper, let alone abandoning his little daughter). Actually Einstein admitted there was some sort of ether: …”we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable.” [Einstein, 1920.]
But there is much worse: we now know that Quantum Physics ignores Local Time. Quantum Physics brings back the instantaneous interaction at a distance which repulsed Newton. (At least, it appears instantaneous experimentally, so far, and it is certainly instantaneous in the existing Quantum formalism, which, amusingly, is in the same exact situation as Newtonian Physics: the Quantum as we know it today, cannot function without that instantaneous Quantum Interaction.
Whatever happens next, only one thing is clear; those who claim physics has been figured out, know very little, and should be advised to shut up, lest their egregious statements confuse the public about the scientific method.