Quantum deformations of Schwarzschild and Schwarzschild-de Sitter spacetimes

TL;DR

This paper constructs quantum versions of Schwarzschild and Schwarzschild-de Sitter spacetimes using noncommutative geometry, computes their metrics and curvatures, and shows they satisfy a proposed noncommutative Einstein equation up to second order.

Contribution

It introduces quantum deformations of classical spacetimes within a noncommutative geometric framework and proposes a corresponding noncommutative Einstein equation.

Findings

Quantum Schwarzschild and Schwarzschild-de Sitter spacetimes are constructed.

Metrics and curvatures are explicitly computed.

The quantum spacetimes satisfy the noncommutative Einstein equation up to second order.

Abstract

A quantum Schwarzschild spacetime and a quantum Schwarzschild-de Sitter spacetime with cosmological constant $\Lambda$ are constructed within the framework of a noncommutative Riemannian geometry developed in an earlier publication. The metrics and curvatures of the quantum Schwarzschild spacetime and the quantum Schwarzschild-de Sitter spacetime are computed. It is shown that up to the second order in the deformation parameter, the quantum spacetimes are solutions of a noncommutative Einstein equation which is proposed in this paper.