Constructing noncommutative black holes

Abstract

We present a self-contained and consistent formulation of noncommutative (NC) gauge theory of gravity, focusing on spherically symmetric black hole geometries. Our construction starts from the gauge-theoretic viewpoint of Poincaré (or de Sitter) gravity and introduces noncommutativity through the Moyal star product and the Seiberg-Witten map, retaining NC gauge invariance at each order in the deformation parameter $Θ$. Working systematically to second order in $Θ$, we obtain explicit NC corrections to the spin connection, the vierbein, and various geometric objects such as the metric and curvature scalars. Using these results, we compute NC modifications of four-dimensional Schwarzschild and Reissner-Nordström solutions, including scenarios with a cosmological constant, as well as three-dimensional BTZ-type black holes (both uncharged and charged). For each black hole solution, we explore various possible Moyal twists, each of which generally breaks some symmetries and modifies the horizon structure, surface gravity, and curvature invariants. In particular, we show that while the radial location of horizons in Schwarzschild-like solutions remains unchanged for some twists, other twists introduce important but finite deformations in curvature scalars and can decouple the Killing horizon from the causal horizon. Similar patterns arise in the charged and lower-dimensional cases. Beyond constructing explicit examples, our approach provides a blueprint for systematically incorporating short-distance quantum corrections through noncommutativity in gravitational settings. The methods and expansions we present can be extended to more general geometries including rotating black holes and additional matter fields, offering a broad framework for future studies of NC effects in classical solutions of general relativity.