An uncertainty relation of space-time

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  • 23 March 1999
Abstract
We propose an uncertainty relation of space-time. It is characterized by G \hbar T \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 the cosmology and show 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 a formula \sqrt{E V / G \hbar^2} with V denoting the volume of the system, which is distinct from the Bekenstein entropy formula E R / \hbar with R denoting the length scale of the system.

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