Relativistic Fractional-Dimension Gravity
  This paper presents a relativistic version of Newtonian Fractional-Dimension
Gravity (NFDG), an alternative gravitational model recently introduced and
based on the theory of fractional-dimension spaces. This extended version -
Relativistic Fractional-Dimension Gravity (RFDG) - is based on other existing
theories in the literature and might be useful for astrophysical and
cosmological applications.
  In particular, we review the mathematical theory for spaces with non-integer
dimensions and its connections with the non-relativistic NFDG. The
Euler-Lagrange equations for scalar fields can also be extended to spaces with
fractional dimensions, by adding an appropriate weight factor, and then can be
used to generalize the Laplacian operator for rectangular, spherical, and
cylindrical coordinates. In addition, the same weight factor can be added to
the standard Hilbert action in order to obtain the field equations, following
methods used for scalar-tensor models of gravity, multi-scale spacetimes, and
fractional gravity theories.
  We then apply the field equations to standard cosmology and to the
Friedmann-Lema\^itre-Robertson-Walker metric. Using a suitable weight
$v_{t}\left(t\right)$, depending on the synchronous time $t$ and on a single
time-dimension parameter $\alpha _{t}$, we extend the Friedmann equations to
the RFDG case. This allows for the computation of the scale factor
$a\left(t\right)$ for different values of the fractional time-dimension $\alpha
_{t}$ and the comparison with standard cosmology results. Future additional
work on the subject, including studies of the cosmological late-time
acceleration, type Ia supernovae data, and related dark energy theory will be
needed to establish this model as a relativistic alternative theory of gravity.
