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.