Fractal–cantorian geometry of space-time

Abstract

This contribution is concerned with the extension of fractal theory used for the description of elementary stationary physical fields (gravitational, electric fields, fields of weak and strong interactions) as well as stationary fields of other physical quantities (thermal and acoustic) defined in the authors’ previous contributions to space-time area. This theory, defined generally in E-dimensional Euclidean space, was applied for description of stationary effects in one-, two- and three-dimensional space, respectively ( r = x i + y j + z k , where i , j , k are orthogonal unitary vectors of Euclidean space). The agreement of laws formulated in various science disciplines with presented theory was proven for Euclidean objects (e.g. Newton gravitation law, Coulomb law, Planck’s radiation law, and 1st Fick’s law). In addition, the presented theory enables extension of validity of given laws for objects having fractal character. In this contribution, another extension of fractal theory is presented in the area of so-called pseudo-Euclidean coordinates, where E-dimensional space consists of p Euclidean and q pseudo- Euclidean dimensions ( E = p + q ) . Special case of this space is the space-time ( s = x i + y j + z k + i ct l ) , where number of Euclidean dimensions is p = 3 and number of pseudo-Euclidean dimensions is q = 1 , which are only preserved (i is imaginary unit, c is speed of light and i , j , k , l are orthogonal unitary vectors of Minkowski space). Physical quantities of this four-dimensional orthogonal space are very often transformed into three-dimensional curved space by means of parametric formulation of quantities r ′ ( t ′ ) = x ′ ( t ′ ) i + y ′ ( t ′ ) j + z ′ ( t ′ ) k , ( x ′ = β ( x - vt ) , y ′ = y , z ′ = z , where t ′ = β ( t - ( v / c 2 ) x ) and β = ( 1 - v 2 / c 2 ) - 1 / 2 , respectively). This forms the basis for the formulation of laws of special and general theory of relativity. The time dilatation and the length contraction or relativist transformation of the mass result from these transformations. In many other cases physical laws are formulated in reality in four-dimensional space-time (i.e. by means of independent coordinates x, y, z, t). It is concerned with 2nd Fick’s law, 2nd Fourier’s law, Schrödinger’s equation, continuity equation, etc. However, it is possible to eliminate one coordinate (e.g. time t) from equations by implementation of suitable quantity independent of this coordinate (e.g. on time t) in special cases. It is the case of steady flows of physical quantities (e.g. steady state electric current, steady state heat flow). In this case it is possible to formulate physical laws formally like in stationary cases (i.e. in three-dimensional space with coordinates x, y, z but by means of dynamic physical quantities). In this way the fractal theory of so-called space charge limited currents (SCLC) was solved (give express citation).