Fractional Einstein field equations in \(2+1\) dimensional spacetime

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 Bañados-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.