I am not asking about why or how gravity should be quantized, or what the problem with renormalization is, or what the discrepancy is between QM and GR is. Those are beautifully described in other questions.
Spacetime is widely accepted as being continuous and showing no discreteness.
And it safely rules out all hypotheses that the spacetime may be built out of discrete, LEGO-like or any qualitatively similar building blocks.
Gravity is accepted to be spacetime curvature.
the Einstein-Hilbert action, or "gravity is the curvature of spacetime"
If spacetime doesn't show any discreteness, then naively I would think that it's curvature doesn't either. Its curvature is gravity itself, so that would mean that gravity doesn't show any discreteness either. But if it doesn't show any discreteness, then it can't be quantized?
The only thing I can think about is the EM field but that does show some discreteness. We do know that Em energy can be transferred in quanta, based on experiments (like the photoelectric effect), and we do know that it can be stored in quantized energy levels in atoms. There is just no example like that for gravity.
So basically the question is whether the gravitational field needs to show some discreteness to be quantized or not.
- If gravity is spacetime curvature, and spacetime doesn't show any discreteness, then gravity doesn't show any discreteness and can't be quantized?