Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes

TL;DR

This paper develops a fractional gravity framework using non-integer dimensional spacetimes with Caputo derivatives, enabling the construction of exact solutions including fractional black holes and gravitational configurations.

Contribution

It introduces a geometric formalism for fractional gravity with a generalized tensor calculus, extending Einstein gravity to non-integer dimensions and providing methods for exact solution construction.

Findings

Fractional gravitational field equations can be integrated explicitly.

Constructed examples include fractional black holes and ellipsoid configurations.

The approach links fractional calculus with gravitational theories and solutions.

Abstract

We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various Lagrange-Finsler and Hamilton-Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, fractional ellipsoid gravitational configurations and imbedding of such objects in fractional solitonic backgrounds.