Is Spacetime Countable?

TL;DR

This paper explores whether spacetime is fundamentally discrete or continuous, proposing a scale-invariant view where only the shapes of configurations matter, challenging traditional notions of spacetime structure in quantum gravity.

Contribution

It introduces a scale-invariant framework where spacetime's discreteness is rejected, reinterpreting relativity as a shape-based theory without a minimum length.

Findings

Spacetime may be fundamentally continuous, not discrete.

Scale invariance implies no minimum length in spacetime.

Relativity can be reformulated as a shape-based theory.

Abstract

Is there a number for every bit of spacetime, or is spacetime smooth like the real line? The ultimate fate of a quantum theory of gravity might depend on it. The troublesome infinities of quantum gravity can be cured by assuming that spacetime comes in countable, discrete pieces which one could simulate on a computer. But, perhaps there is another way? In this essay, we propose a picture where scale is meaningless so that there can be no minimum length and, hence, no fundamental discreteness. In this picture, Einstein's Special Relativity, suitably modified to accommodate an expanding Universe, can be reinterpreted as a theory where only the instantaneous shapes of configurations count.