  In a series of recent papers we put forward a ``fractional gravity''
framework striking an intermediate course between a modified gravity theory and
an exotic dark matter (DM) scenario, which envisages the DM component in
virialized halos to feel a non-local interaction mediated by gravity. The
remarkable success of this model in reproducing several aspects of DM
phenomenology motivates us to look for a general relativistic extension.
Specifically, we propose a theory, dubbed Relativistic Scalar Fractional
Gravity or RSFG, in which the trace of the DM stress-energy tensor couples to
the scalar curvature via a non-local operator constructed with a fractional
power of the d'Alembertian. We derive the field equations starting from an
action principle, and then we investigate their weak field limit, demonstrating
that in the Newtonian approximation the fractional gravity setup of our
previous works is recovered. We compute the first-order post-Newtonian
parameter $\gamma$ and its relation with weak lensing, showing that although in
RSFG the former deviates from its GR values of unity, the latter is unaffected.
We also perform a standard scalar-vector-tensor-decomposition of RSFG in the
weak field limit, to highlight that gravitational waves propagate at the speed
of light, though also an additional scalar mode becomes dynamical. Finally, we
derive the modified conservation laws of the DM stress energy tensor in RSFG,
showing that a new non-local force emerges, and hence that the DM fluid
deviates from the geodesic solutions of the field equations.
