Chris,
You ask very provocative questions!
When I started my essay on time (about a year ago), I was very perplexed. Well into it, I at least thought that I had made a start. Now, after reading a good share of the essays on this site, I am as perplexed as ever.
There are many good insights here; many ways of looking at time. As you mention, one of the main threads relates time to motion is space. This, I think is a valid and fundamental approach. Every one of our clocks seems to embody this mechanism. There is the ephemeral time of the stars, the pendulum, a weight on a spring. Energy flow in the form of electromagnetic waves gives us the oscillation of an LC circuit at its characteristic frequency and in the quartz crystal of a digital watch. Deeper yet is the interplay of the permittivity and permeability of free space with the speed of light. Lights¡¯ smooth and regular transitions as it cycles between electric and magnetic field oscillating in phase space sets a speed limit for all motion. This gets us into the structure of that ¡®free space,¡¯ the mysterious vacuum. Presumably, the structure of the vacuum relates to the structure of space. From Newtonian mechanics through GR we have this cyclical sine curve; smooth and differentiable; deterministic and understandable, except for one thing; the mystery of what mechanism sets this speed of light. My first guess is that the structure of space sets it. But how? Abstract relations have no speed limit. It must be physical.
Wild conjecture and a hunger for an answer leads me to brew a hearty stew from the various available ingredients: superstrings vibrating in compactified orbafolds at the Planck scale being T-Dual with the curvature of space at the cosmic scale. Could something be oscillating within this specific structure to set the speed of light?
At an even more fundamental level, I think, time is dependent on physical being. The notion that time started at the big bang permeates our thinking. What seem to be the four fundamental attributes of being are mass, energy, space and time. Pairing them in one way, mass-energy and space-time, gives rise to General Relativity. The alternate pairing, mass(* velocity)-space and energy-time, gives rise to Quantum Mechanics. One pairing is commutative, the other is not. That a pair of virtual particles could discontinuously spring into being for a maximum time inversely proportional to it energy can also be taken as a fundamental clock. It also begs the question of that mysterious act of becoming.
As all roads lead to Modesto, I am constantly being led back to the notion of duality (my favorite tool for understanding anything and everything), and particularly, the primary duality of non-being and being.
Thinking works by comparison. For any attribute that can be assigned to the universe as a whole (or anything in it) the dual (opposite or conjugate) attribute must also be assigned as a background against which it is compared. Both halves of the duality must be present to unify the whole. Here is an example: Is the universe symmetric or asymmetric? It is both. With respect to the three mirrors, charge, parity and time (CPT), the universe is assumed to be perfectly symmetric. But with respect to the three taken in pairs or singly, there is an ever so slight asymmetry. Then on the macroscopic scale, entropy renders time almost totally asymmetric. And there seems to be an even more fundamental asymmetry. Through duality being implies non-being but non-being does not imply anything.
At one point, I had myself convinced that the continuity problem (and Zeno¡¯s paradox) was a non issue. GR space is smooth and differentiable; there is a derivative at every point. And even on the quantum scale, I thought, surely even if space is quantized at the Planck scale, uncertainty at that same scale would smooth out the granularity and make it continuous.
Duality throws the continuity of the universe into question. At its most fundamental level it is a unity. However, no physicality accrues to this undifferentiated whole. The act of becoming, non-being into being, is, I think, irreducibly discontinuous and discrete. How does nature jump this gap? By a quantum leap.
So, final answer on the nature of time: Fine tuned into the physical structure of space there is a characteristic frequency of becoming, ¥Äe ¥Ät ¡Ã ©¤/2. This may very well be the fundamental clock.
But there are many types of clocks: The oscillation of a photon, the electron pulsing around its atom, chemical clocks, geologic clocks, biological clocks; each clock relative to the process it is embedded in. Cyclical clocks tick sequential time but they do not explain every aspect of being or becoming. For a more complete view, I think we need a dualistic time, not in the sense of opposites but in the conjugate sense. A complex time.
To your one-or-the-other argument I would add this third possibility that it is both. There are many clocks of both types. At the speed of light, time becomes simultaneous, just like space. Dr. E¡¯s MDT is a fascinating explanation of why that would be. As time is relative, age and lifespan is also relative. For an eye-opener, do the time dilation calculation for the farthest galaxy out for the two cases where the earth is approaching and then receding as we go around the sun. The formula is in Brian Greens¡¯ ¡®The Fabric of the Cosmos.¡¯ Look at a muon hitting our atmosphere.
I agree with you that if you take away the particles and fields there would be no physical time, or at least the real (sequential) part of it. But the imaginary (simultaneous) part would remain.
The real component, which we sense so easily in an intuitive way, where each element of position and momentum collapses irreversibly into the past where it projects, or casts its shadow, into the future by limiting what can happen next. But there are many possibilities which still remain in superposition.
To speak metaphorically, the collapse of the wave function of the universe is ongoing. The ¡®now¡¯ rides the avalanche of discontinuous ticks of collapsing sub-wave functions forming the ordered and unique sequence of the past. The future is the sum over histories of all the multiverses yet in superposition; the un-collapsed portion of the wave function. The few remaining photons at the end of time, still entangled from the big bang, report back what has gone before. From the fecundity of infinite possibilities, the eigenstate that is holographically fruitful is chosen. These metaphors might capture this discontinuous, simultaneous and spontaneous side of nature that no differential equation can.
All the best.
Jim