Is the Idea of a Continuum Always an Approximation to the Physical?

Question: When thinking of continuums the most notable seems to be space-time but they also mark a simplification to reality like in continuum mechanics, often taught when learning the tensor calculus needed for general relativity.

The question is that for general relativity when a geodesic becomes incomplete as can happen in a singularity situation for black holes, what does this say about the idea of a continuum as space-time in general relativity? Does this mark the limit of applicability of the continuum concept? Is space-time continuum truly only a mathematical approximation to something physical? (If this last question is too philosophical, omit it.)

Good question. The problem is to find a viable model of discrete spacetime.

I think that the continuum is always an idealization of physical reality, even in classical mechanics. My DE professor used to say: "Reality is discrete if you look closely enough." Nevertheless, we use DE to describe it successfully. Doesn't the continuum already break down in QM? Nevertheless, we use continuous transformation groups" (old-fashioned for Lie groups) to describe it. Our models break in extreme situations, Newton at high speed, and GR (possibly?) at close ranges.

walkeraj said:

for general relativity when a geodesic becomes incomplete as can happen in a singularity situation for black holes, what does this say about the idea of a continuum as space-time in general relativity?

Nothing. You already posted a separate thread on this, which has now been closed as it is based on an invalid premise.

walkeraj said:

Does this mark the limit of applicability of the continuum concept?

No. See above.

walkeraj said:

Is space-time continuum truly only a mathematical approximation to something physical?

This is a separate question from the above two, and this thread should be limited to discussing it. The short answer is that this is still an open question and is a subject of research in quantum gravity. So far nobody has come up with a model that makes any useful predictions that are testable with our current technology and have passed any such tests.

This question is impossible to answer, and at best is philosophical (see PF Rules) and at worst...um...worse. It boils down to "As we look at smaller and smaller scales, mighy we discover that thinsg we thinka re continuous are really discrete (or for that matter, things we think are discrete are just conglomerations of things we thought were continuous.): Maybe yes, maybe no. No way to tell.

But without comparison to the real world, it ain't science.

fresh_42 said:

I think that the continuum is always an idealization of physical reality, even in classical mechanics.

There are others that also share this opinion, but it is important to recognize that as of today there is no experimental evidence to support that idea. It is in the theoretical physics literature, but without any experimental validation.