What Dimensions Do the Time and Space Have: Integer or Fractional?

Abstract

A theory of time and space with fractional dimensions (FD) of time and space ($d_α, α=t,{\bf r})$ defined on multifractal sets is proposed. The FD is determined (using principle of minimum the functionals of FD) by the energy densities of Lagrangians of known physical fields. To describe behaviour of functions defined on multifractal sets the generalizations of the fractional Riemann-Liouville derivatives $D_{t}^{d(t)}$ are introduced with the order of differentiation (depending on time and coordinate) being equal the value of fractional dimension. For $d_{t}=const$ the generalized fractional derivatives (GFD) reduce to ordinary Riemann-Liouville integral functionals, and when $d_{t}$ is close to integer, GFD can be represented by means of derivatives of integer order. For time and space with fractional dimensions a method to investigate the generalized equations of theoretical physics by means of GFD is proposed. The Euler equations defined on multifractal sets of time and space are obtained using the principle of the minimum of FD functionals. As an example, a generalized Newton equation is considered and it is shown that this equation coincide with the equation of classical limit of general theory of relativity for $d_{t} \to 1$. Several remarks concerning existence of repulsive gravitation are discussed. The possibility of geometrization all the known physical fields and forces in the frames of the fractal theory of time and space is demonstrated.