Relativistic Fractional-Dimension Gravity

Abstract

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, in this work, 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-Lemaître-Robertson-Walker metric. Using a suitable weight vtt, depending on the synchronous time t and on a single time-dimension parameter αt, we extend the Friedmann equations to the RFDG case. This allows for the computation of the scale factor at for different values of the fractional time-dimension αt 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.

Newtonian Fractional-Dimension Gravity and the External Field Effect

“…In this work, we continue the analysis started in previous publications ( [1][2][3][4], hereafter papers I-IV, respectively) of Newtonian Fractional-Dimension Gravity (NFDG), an alternative gravitational model based on the assumption that Newtonian gravity can be extended to astrophysical objects characterized by a fractional dimension D < 3, i.e., different from the standard value of the spatial dimension D = 3.…”

“…In this section we will summarize the fundamental ideas of NFDG, together with its applications to galactic dynamics. More details can be found in papers I-IV [1][2][3][4] and references therein.…”

“…In paper I [1], we generalized the gravitational Gauss's law and used it to introduce the NFDG potential and field for sphericallysymmetric structures; in paper II [2], we extended our analysis to axially-symmetric structures in order to analyze disk components of galaxies, and produced our first rotation curve fit for NGC 6503. In paper III [3], we extended our analysis and included two more rotationally supported galaxies (NGC 7814 and NGC 3741), while in our paper IV [4], we presented a relativistic version of our model (Relativistic Fractional-Dimension Gravity -RFDG).…”

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

“…However, an inspection of the approximated potential led to fixing the fractional exponent α to the value (4.30). It would be interesting to make a more sophisticated analysis to see if this value eventually gives a good fit of data, as suggested by the results of [24].…”

“…Recently, the idea of invoking anomalous dimensions to realize dark-matter galactic observations has been retaken, implicitly, in Newtonian dynamics with a fractional Laplacian [19,20] and, explicitly, in Newtonian fractional-dimension gravity (NFDG) [21][22][23]. A covariant version of NFDG was shown to essentially match with the multi-fractional theory T 1 [24], so that one may wonder whether multi-fractional theories can directly say anything interesting about the dark-matter problem.…”