# How do conservation of energy and relativity hold if spacetime is discrete?

I understand the principle of conservation of energy and momentum in classical physics. I am aware of the fact that such conservation laws arise due to fairly deep mathematical symmetries in the natural laws, such as time-invariance.

Some theoretical physicists propose that time might actually be comprised of discrete units as opposed to being continuous. It seems clear to me that discrete units of time would break conservation of energy and relativity.

Consider that, in special relativity, simultaneity depends on your inertial frame. If time is discrete then there will be spacetime rounding errors as the reference frames for multiple observers diverge and later reconcile. This will introduce rounding errors in energies.

Wouldn't this mean that conservation of energy is only an approximation at best?

Wouldn't this mean that conservation of energy is no longer caused by symmetries?

Wouldn't this mean, philosophically speaking, that it's incredibly weird that something very close to conservation of energy still holds, even though it is no longer mathematically necessary?

Is there a mathematically sound way to discretize spacetime without breaking the definition of conservation of energy?

• On the difficulty of Lorentz invariance in a discrete spacetime, see this answer physics.stackexchange.com/a/34436/123208 by Gerard 't Hooft. – PM 2Ring Jul 13 '20 at 18:21
• By Noether's theorem energy and momentum conservation requires continuous symmetry. – my2cts Jul 13 '20 at 18:51
• In models where spacetime is discrete, energy momentum conservation is NOT a true symmetry, but it does arise as an effective symmetry at low energies. – Prahar Mitra Jul 13 '20 at 19:42