From minimal-length quantum theory to modified gravity

TL;DR

The paper addresses how generalized uncertainty principle (GUP) deformations implementing a minimal length can be systematically connected to semi-classical gravity by reconstructing effective gravitational actions from entropy corrections. Starting from a generic GUP deformation $Ξ(Δp)$, the authors derive corrections to black hole entropy and use Wald's entropy formula to map these corrections into higher-curvature terms in $f(R)$ and $f(R, R_{\mu\nu}R^{\mu\nu})$ gravity, establishing a concrete dictionary between GUP parameters and curvature coefficients. They show that leading GUP-induced entropy corrections correspond to $R^{3/2}$ and $R^{2}(\ln R^2-1)$ terms, with higher orders mapping to $R^{1+k/2}$, and reveal an undetermined function in the $f(R, P)$ extension that does not affect entropy but can influence dynamics. An explicit astrophysical application yields Yukawa-like corrections to the Newtonian potential and places stringent bounds on the minimal length via solar light-deflection measurements, illustrating how quantum-gravity phenomenology can be probed with astrophysical data. Overall, the work provides a principled framework for embedding quantum gravity signatures into extended gravity theories and demonstrates a viable route for observational tests of minimal-length scenarios.

Abstract

In this work, we consider generalized uncertainty principles (GUPs) that incorporate a minimal length through generic momentum-dependent deformation functions. We thus develop a systematic approach connecting such a framework to effective gravitational actions extending general relativity. By examining quantum gravity-motivated corrections to black hole entropy induced by the GUP and employing Wald's formalism, we reconstruct modifications to Einstein's gravity within the contexts of $f(R)$ and $f(R, R_{μν} R^{μν})$ theories. In this way, we establish a direct mapping between the GUP parameters and the higher-order curvature coefficients in the gravitational Lagrangian. As an illustrative application, we compute corrections to the general relativistic prediction for light deflection, which in turn allows us to infer a stringent upper bound on the minimal measurable length. Our results show that GUP-induced effects can be consistently embedded into extended gravity theories, offering a promising framework for testing quantum gravity phenomenology through astrophysical and cosmological observations.