Is Time Truly Continuous or Discrete?

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If time did come in small lumps, it doesn't follow that all these lumps are synchronized so that they march forward like billions of tiny clocks that all tick together. It seems likely that one "clock" or lump of time might tick (pass by), then another somewhere else, then the first one again. So it could be nothing like a video sequence where all the tiny dots that make up the picture "jump" from one frame to the next at the same moment.

Each dot in the TV analogy (or sub-atomic particle in the real world) might jump between states independently and with some random factor as well. When studying particle radiation, I never encountered any suggestion that there are "allowed times" and "disallowed times" for the radiation to be detected. I suspect you could prove that there is no such thing.

If time was discrete, but asychronized with a random variance at the microscopic level, it could give a macroscopic impression of continuity which would emerge from billions of tiny lumps of time that pass endlessly in no particular order throughout a given volume of space.
 
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The discrete arguments seem like pretty good thinking to me. But there seems to be another aspect too, one which does not negate the quantum way of describing time. Using an analogy, say you blow a balloon up and then connect the opening to a device that pressure causes to open and close (releasing a "quanta" of air). As the air pressure decreases in the balloon, the device takes longer to open up, but it will still function as long as any air pressure is inside the balloon. Looking at the point of the opening, there the process is discrete; but looking at the whole ballon, the process is continuous in the sense that is will continue non-stop until all the air pressure in the balloon is gone.
 
Let's look for a reason for time. The Theory of Relativity makes the equivalence principle obvious, and the primary equivalence in this case is time/space. Although space may be thought of as void, with the ideas of quanta it has become usual to think of it as very particulate. The fabric of space comes to mind. Perhaps it becomes no smaller than quark size, but nonetheless something very tiny represents the fabric of space between stars as well as between the parts of atoms making up the stars. In addition to being the underlying substance, space fabric would also be the vector of energy transfer and propagation. The total energy of the Universe is tied to this space fabric. It is the natural tendency of anything in motion to continue in motion and this is the thing we call energy. Energy is change, and at the quanta level there definitely is a lot of change. Think of the change as a structural pattern, such that the rate of energy transfer can be depended on for everything, from a macro object such as a rock to the vibrations of atoms. However, motion affects this transfer of energy, and clocks, whether windup or atomic, are timed by the transfer of quanta. Atomic behaviour depends on it. So it seems plausible that at light and sublight speeds the rate of transfer would be slowed considerably. Could this be what we are measuring and callling time? Why is the speed of light what it is? Try it this way: Accelerate an object. As the object is pushed faster, what happens? It moves away from the energy source (this is true of rockets, too) and the faster it moves away, the less energy is transfered. This effect is usually equated as an increase in mass requiring more and more energy to accelerate, by I think mathematically if we calculate the difference in transfer rate due to motion, the result would be the same increased energy requirement as given by mass increase calculations. So, anyway, if time is a transfer rate, the rate would be different for things at rest compared to when they are in motion. Continuous or discrete? Look at the quanta.