  In this paper we establish a fractional generalization of Einstein field
equations based on the Riemann-Liouville fractional generalization of the
ordinary differential operator $\partial_\mu$. We show some elementary
properties and prove that the field equations correspond to the regular
Einstein field equations for the fractional order $\alpha = 1$. In addition to
this we show that the field theory is inherently non-local in this approach. We
also derive the linear field equations and show that they are a generalized
version of the time fractional diffusion-wave equation. We show that in the
Newtonian limit a fractional version of Poisson's equation for gravity arises.
Finally we conclude open problems such as the relation of the non-locality of
this theory to quantum field theories and the possible relation to fractional
mechanics.
