Like most grad student in physics, I wish I had had Richard Feynman as a teacher. For those who don't know the myth, he was a professor at Caltech, a member of the manhattan project, and a Nobel prize winner for the development of Quantum Electro Dynamics (QED).
Mostly, as an undergrad student, I admired him for his textbooks, the Feynman lectures, a collection of classes he gave at Caltech. It was love at first read. I remember being engrossed with the books sitting on the floor in a bookstore in Paris and just reading through the electromagnetic chapters. I was impressed at how clear and fluent the presentation was and how rigorously mathematical it was. Clearly Feynman was a man who liked to think through things, on his own terms. Only when he had mastered a thought would he teach it. This is part of what makes him so interesting to physics students: the insights he developed. The magic has not changed and 25 years later, I still read Feynman when I want to get to the bottom of some things.
Ironically what he struggled with he put in a little book called "six not so easy pieces". That book is mostly about SR. It is clear he struggled with SR, everyone does. Oh sure, we can calculate, a high school kid can compute a square root, what underlies the axiomatic of SR is still a subject of debate (and scorn in many academic circles). But there he is at his best. Struggling, reaching for explanation.
He actually tried to make sense of dilating time in a very physical way by postulate that time was counted in clicks of a phenomena that would stretch in space as it would move through a real background. The geometric pythagorean relation yielding the proper Lorentz factor. He immediately dismisses the insight as a toy model and that real clocks do not behave that way.
He unfortunately oscillates between 'perceives' and 'is' 2 important concepts which are usually glanced over in most SR treatments. Some people like to interpret Lorentz transformation as 'projections' in our measurement system. This explains why to 2 observers time dilates for each without contradiction. The explanation of the clock above is an example of a "IS" interpretation. That the actual time spent in flight by a photon is proportional to the distance it travels and that that distance stretches by the geometric factor when the 'clock' is in movement. An element of reality is attached to 'time'. The other interpretation the 'relativistic' one is rooted in 'perception' and measurement, not an actual intrinsic property of 'time'. Feynman is not afraid of sharing where he is at, his thoughts for pedagogical purposes even if he doesn't reach a conclusion on the nature of time.
He has another deep insight in the 6 easy pieces about the nature of time. He mentions that the only invariant in the theory is the number of oscillations between respective processes. If it takes a photon 1 phase oscillation and another process n oscillations, then if you define time as that number n (counting in unit of the photon process). That ratio is an invariant even if the number representing the photon 'time' varies.
One is then lead to imagine that time is defined by that light process. A phase oscillation is the unit of time. Proper time is the time a process takes defined as a number of fundamental oscillations. A lot of the SR properties arise from that definition.
GR, acceleration and time.
The third insight very clearly presented is that an acceleration will induce a doppler effect on the time definition above. As the source goes to you, the frequency is increased, as the source leaves the frequency is lowered. Time, this time 'perceived' by the observer at rest "seems" to slow down. The source frequency hasn't changed, just like the siren of the doppler doesn't change, just the pitch we perceive.