Quantum Uncertainties of Static Spherically Symmetric Spacetimes

TL;DR

The authors develop a canonical quantization framework for static spherically symmetric spacetimes using a Static Equal Radius (SER) prescription, showing that classical Schwarzschild–(A)dS solutions emerge in the semiclassical limit via an Ehrenfest-type relation. They derive quantum uncertainty relations between geometric operators, notably between the metric component $\hat g$ and its conjugate momentum, and between noncommuting Weyl-ordered combinations that connect to black hole thermodynamics and generalized uncertainty concepts. The analysis yields both lower and upper bounds on the central mass parameter $M_0$ as functions of the cosmological constant $\Lambda$, with striking numerical echoes to observed cosmic and astrophysical scales when $\Lambda$ is set to its observed value. These results suggest that quantum gravitational effects can imprint measurable, large-scale features and offer new angles on the cosmological constant problem, black hole interiors, and possible observational implications in astrophysics and cosmology.

Abstract

We present a canonical quantization framework for static spherically symmetric spacetimes described by the Einstein-Hilbert action with a cosmological constant. In addition to recovering the classical Schwarzschild-(Anti)-de Sitter solutions via the Ehrenfest theorem, we investigate the quantum uncertainty relations that arise among the geometric operators in this setup. Our analysis uncovers an intriguing relation to black hole thermodynamics and opens a new angle towards generalized uncertainty relations. We further obtain an upper and a lower limit of the mass that is allowed in our model, for a given value of the cosmological constant. Both limits, when evaluated for the known value of the cosmological constant, have a stunning relation to observed bounds. These findings open a promising avenue for deeper insights into how quantum effects manifest in spacetime geometry and gravitational systems.