Fractional Einstein field equations in $2+1$ dimensional spacetime

TL;DR

This paper introduces a new fractional derivative to formulate Einstein's field equations in 2+1 dimensions, addressing divergence issues and exploring solutions like the BTZ black hole within a fractional calculus framework.

Contribution

A novel fractional derivative operator is proposed that preserves constants, enabling the formulation of fractional Einstein equations in 2+1 dimensions and analysis of black hole solutions.

Findings

The new derivative resolves divergence issues in fractional gravity.

BTZ black hole solutions are recovered for fractional parameters near one.

Effective matter resembles a charged BTZ with anisotropic cosmological constant.

Abstract

In this work, we introduce a new fractional derivative that modifies the conventional Riemann-Liouville operator to obtain a set of fractional Einstein field equations within a 2+1 dimensional spacetime by assuming a static and circularly symmetric metric. The main reason for introducing this new derivative stems from addressing the divergence encountered during the construction of Christoffel symbols when using the Caputo operator and the appearance of unwanted terms when using the Riemann-Liouville derivative because of the well-known fact that its action on constants does not vanish, as expected. The key innovation of the new operator ensures that the derivative of a constant is zero. As a particular application, we explore whether the Ba\~nados-Teitelboim-Zanelli black hole metric is a solution to fractional Einstein equations. Our results reveal that for values of the fractional parameter close to one, the effective matter sector corresponds to a charged BTZ solution with an anisotropic cosmological constant.