rajnz00 » January 17th, 2017, 3:34 am wrote: Though the discreteness of space solves Zeno's paradox, there are many other (more satisfactory) solutions.

The discreteness of space is not a very satisfactory solution, if at all. It merely transfers the question from how can Achilles catch the tortoise? or how can the arrow reach its target? to why are space and time discrete?

Aristotle's explanation, that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small, finally solved mathematically by Newton, as a convergent series, with a finite sum, seems to be more satisfying.

In physics it is possible to unite Quantum Mechanics with General Relativity, such that the states of energy and matter are quantized, but the time and space are modeled as continuous in the sense of calculus. This is not impossible, and it does yield results in physics. The branch of physics which commits to this ontology is called Semiclassical Gravity

In Semiclassical Gravity, fundamental particles act according their Schroedinger Wave, but propagate in a smooth spacetime background, which is allowed in some cases to be curved.

https://en.wikipedia.org/wiki/Quantum_field_theory_in_curved_spacetime

https://en.wikipedia.org/wiki/Dirac_equation_in_curved_spacetime

If you get around to following the above links, I would like you to note in particular : the section Further reading: books and relevant papers. See that there are entire books on this subject, and many of them. It is actually a branch of physics with active research. Another anecdote is that the leading researcher for several decades was Stephen Hawking.

In regards to the central question of this thread :

why are space and time discrete?

We can now answer this question in a circuitous manner. We will obtain the answer by asking why is it that Semiclassical Gravity does not match our universe?

Well pragmatically, the theory will calculate observable quantities which will match what we measure in real experiments. But that's not , as you say, "satisfying". There are known approximations in Semiclassical Gravity. It presumes that the thing which is curving the spacetime is (1.) really big and (2.) really far away . Where "far away" is meant to indicate very much larger than inter-atomic distances. The approximation is that the particles which are moving around in this curved spacetime are so tiny that we can ignore their own gravitational contribution. Assume their gravitational mass is zero. Everyone in the room knows this is a fudge away from "true reality" yet accurate enough to still produce sensible answers at the "bottom of the chalkboard".

Classical GR (lets call it) admits solutions which are sensible and consistent, but are singularities. We call the resulting stellar objects Black Holes. Again, Semiclassical Gravity, with its distance approximations will likely be very good at modelling the true behavior of physics around a Black Hole at large distances, particularly when such objects are hundreds of kilometers wide. SCG will not predict what is going on in the very center of a black hole at where the equations predict a singularity. We need not despair. Physics has run into this problem before. Very sensible physics would predict that electrons will spiral into the nucleus immediately, because electrons are negatively charged and the nucleus is positively charged. We look around the room and see "gazillions" of stable atoms where this is electron death spiral is not happening. (While electron capture does happen on occasion,) the vast majority of electrons don't do this. Instead, at best the electron reduces itself to a ground state, and the quantization of the electron's energy forbids it getting lower than that ground state. The electron acts consistent with classical electromagnetism, in a general sense, it will stay very nearby its host atom, but when squeezed with small scales, the DISCRETE NATURE of electrons will appear starkly, and allow for stable atoms.

IF we are work-a-day physicists, the singularity of black holes really forces our hand on this issue. But it's not the only "problem". On a forum like this one, material ontology and philosophical viewpoints carry more weight. We know the fundamental particles themselves must generate a local gravitational effect. Gravity is worse in this regard, since GR demands that any and all particles create gravity, up to and including massless photons. We would be inconsistent with our logical principles if we were to whitewash this gravity with a wave of our hands like SCG does -- and just say "Dont worry about that" , "Just set it to zero and pull the crank."

We are at an impasse now. We have two choices. Either we can say that finding a theory of gravity at the atomic nuclear scales is forever a mystery to science, math , and philosophy, pick up our rucksack and Go home.

OR!

We can try to suck it up and scale across this deep canyon and make progress towards an answer. The most conservative thing, to begin our journey would be to proceed like so: Gravitation field follows SCG on large scales, but on small scales, quantization effects will emerge prominently. (Analogize the electron which acts like a bullet in the tube of an old TV, but acts like a very different beast around an atom). That is to say, to be extra super duper careful about this, we simply state that the Gravitational Field must be quantized. That statement is non-mystical , pragmatic, and non-committal to any crazy ontological commitments about the nature of space and time.

The gravitational field must be quantized.

It is not mysticism. It is not internet guy quackery. It just means, lets find out some equations that might model quantization of gravity at small scales, without embuing the results with ontological flights-of-fancy. We may not understand what the equation "Really means" in the deepest philosophical sense, but it will be incremental, conservative progress across our canyon.

This has already been done. It was first done by John Wheeler and Bryce DeWitt.

The next question that plagues us: Does the procedure of quantization of gravity, necessarily entail that space and time are discrete? The technical answer to that question is "no", "no, it does not entail such."

Of course we are using the word "entail" in the stronger sense that would appear in a peer-reviewed physics paper. On a forum like this, we can toy with being a little more .. 'abusive' as it were. You can review the recent literature on quantum gravity. Be it the ADM formalism, Loop Quantum Gravity, String Theory, M Theory, Causal Dynamical Triangulations, Topos theory, Knot theory --- ya know --- get all the book and articles on those topics and spread them all over a big mahogany table and "party all night". Have at it.

You can decide for yourself how long you can continue to clutch on to the ontology of space and time being infinitely-divisible continuums in the calculus sense. Look in your own heart, and your own mind, and decide whether the material spread all over your desk is commensurate with that -- or whether it is more sensible to say that space at the smallest scales comes in some kind of discretized packets.

I'm not gonna force anyone or make someone uncomfortable by presuming the nature of those packets... (ya know, I'm not gonna get all up in your face with my personal theories and jam them in your face every other thread ... ) but no, generally speaking, progress can be made by letting go of one's pre-baked prejudices and considering, (at the least, entertaining) the supposition that space is in discretized packets. Then upon entertaining such, we must answer how the packets "jump" from one discrete state to the next. Well, perhaps the "jumps" are instantaneous, like the still pictures that compose a movie at 30 frames-per-second. But then that would mean time itself must be "ticking" ... and et cetera.