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, 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 $v_t\left(t\right)$, 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.

arXiv commentMinor changes and references added. 18 pages, including one figure. Final version published in Universe, MDPI

References in corpus (29)

• Modified Gravity and Cosmology
• The Confrontation between General Relativity and Experiment
• The Cosmological Constant
• Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions
• Modified Newtonian Dynamics as an Alternative to Dark Matter
• SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves
• Quintessence: A Review
• The Radial Acceleration Relation in Rotationally Supported Galaxies
• One Law To Rule Them All: The Radial Acceleration Relation of Galaxies
• Fractal universe and quantum gravity
• Modified Gravity and Cosmology: An Update by the CANTATA Network
• Quantum field theory, gravity and cosmology in a fractal universe
• Geometry and field theory in multi-fractional spacetime
• Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies
• Vector Calculus in Non-Integer Dimensional Space and its Applications to Fractal Media
• Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space
• Multifractional theories: an unconventional review
• Multi-scale gravity and cosmology
• MOND-like Fractional Laplacian Theory
• Multiscale spacetimes from first principles
• Multifractional theories: an updated review
• On the Kuzmin model in fractional Newtonian gravity
• Applications of Fractional Calculus to Newtonian Mechanics
• Towards multifractional calculus
• Quantum scalar field theories with fractional operators
• Dark energy in multi-fractional spacetimes
• Newtonian Fractional-Dimension Gravity and Rotationally Supported Galaxies
• Newtonian Fractional-Dimension Gravity and MOND
• Newtonian Fractional-Dimension Gravity and Disk Galaxies

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