An Uncertainty Relation of Space-Time

TL;DR

This paper introduces a new space-time uncertainty relation that helps estimate black hole and universe entropies, reproduces the holographic principle, and discusses implications for cosmology and maximal entropy bounds.

Contribution

It proposes a novel space-time uncertainty relation and applies it to derive qualitative entropy estimates, connecting to holographic principles and cosmological entropy bounds.

Findings

Reproduces the holographic principle of 't Hooft and Susskind.

Provides qualitative entropy estimates for black holes and the universe.

Suggests a new formula for maximal entropy differing from Bekenstein's bound.

Abstract

We propose an uncertainty relation of space-time. This relation is characterized by GhT \lesssim \delta V, where T and \delta V denote a characteristic time scale and a spatial volume, respectively. Using this uncertainty relation, we give qualitative estimations for the entropies of a black hole and our universe. We obtain qualitative agreements with the known results. The holographic principle of 't Hooft and Susskind is reproduced. We also discuss cosmology and give a relation to the cosmic holographic principle of Fischler and Susskind. However, as for the maximal entropy of a system with an energy E, we obtain the formula \sqrt{EV/Gh^2}, with V denoting the volume of the system, which is distinct from the Bekenstein entropy formula ER/h with R denoting the length scale of the system.