Do Fractals Confirm the General Theory of Relativity?

Abstract

The relatively high abundance of fractal properties of complex systems on Earth and in space is considered an argument in support of the general relativity of the geometric theory of gravity. The fractality may be called the fractal symmetry of physical interactions providing self-similarities of complex systems. Fractal symmetry is discrete. A class of geometric solutions of the general relativity equations for a complex scalar field is offered. This class allows analogy to spatial fractals in large-scale structures of the universe due to its invariance with respect to the discrete scale transformation of the interval d s ↔ q d s ˜ . The method of constructing such solutions is described. As an application, the treatment of spatial variations of the Hubble constant H 0 H S T (Riess et al., 2016) is considered. It is noted that the values H 0 H S T form an almost fractal set. It has been shown that: a) the variation H 0 H S T may be connected with the local gravitational perturbations of the space-time metrics in the vicinity of the galaxies containing Cepheids and supernovae selected for measurements; b) the value of the variation H 0 H S T can be a consequence of variations in the space-time metric on the outskirts of the local supercluster, and their self-similarity indicates the fractal distribution of matter in this region.

General Relativity Fractal for Cosmic Web

“…Thus, we construct the exact solution to the GR-equations: the connection coefficients (2), the Ricci tensor ( 8), the metric tensor (10) and the energy density (11) in terms of the velocities i u and i b . We see from formulae ( 2) and ( 8) - (11), that these solutions change under the discrete scaling transformation (3) as follows:…”

“…, 1.67 2.09 α < < ; and (ii) the geometrical progression consisting of the spatial variations of the Hubble constant [10]. Power laws are characteristic of fractals.…”

“…Power laws are characteristic of fractals. Fractals are often found in astrophysical objects that differ both in the processes occurring in them and in spatial scales (examples are given in article [10]). So far, the fractal properties of the cosmic web are not taken into account in the LCDM model.…”

“…This leads, in particular, to the convergence of the world lines of particles in the direction of speed k W and to the formation of local matter flows and elongated structures. Thus, knowing the distribution of velocities, we can detect the deformation of space-time in a certain region of space-time.Now, we substitute in formula(10) the metric tensor…”