Gauge Gravitational Field in a Fractal Space-Time

Abstract

Considering the fractal structure of space-time, the scale relativity theory in the topological dimension D T = 2 is built. In such a conjecture, the geodesics of this space-time imply the hydrodynamic model of the quantum mechanics. Subsequently, the gauge gravitational field on a fractal space-time is given. Then, the gauge group, the gauge-covariant derivative, the strength tensor of the gauge field, the gauge-invariant Lagrangean, the field equations of the gauge potentials and the gauge energy-momentum tensor are determined. Finally, using this model, a Reissner-Nordström type metric is obtained.

New Metrics from a Fractional Gravitational Field

“…This metric is similar to the Reissner-Nordstrom metric in D = 4. [19,37] (c) For D = 5, α = 2, γ = β and small distances it is easy to check that we obtain a metric similar to the Schwarzschild metric in D = 4:…”

“…It is noteworthy that in Ref. [19], Agop et al constructed a gauge gravitational field in a fractal spacetime based on scale relativity framework with topological dimension 2. It was proved that a Reissner-Nordstrom type metric is obtained from this theory.…”

Reissner—Nordström-de—Sitter-type Solution by a Gauge Theory of Gravity

“…Moreover, using the procedure as same as in Ref. [25], a Reissner-Nordströmde Sitter metric on the gauge group of a fractal spacetime can be given.…”