Abstract
We live in the 4-D Euclidean space. The 4th dimension is assigned as the absolute time (ct) axis and energy axis (cPt = E0) based on 4-dimensional Euclidean space. This 4th dimension can be indirectly felt through the observable relative time (ctl) and observable total energy (cPtl = E). The space-time distance is d(x1x2x3x4) = ctl. The modified Lorentz transformations are introduced by the time-matching of the absolute times in the 4-D Euclidean space. The size of x’ (or Dx’) of the moving object is expanded to the size of x = gx’ (or Dx = gDx’). These modified Lorentz transformations are approximated to the Lorentz transformations as t à tl when v/c << 1 and to the Galilean transformations as v/c is close to zero. The relative time (tl) and energy (E) are defined as the 4-dimensional distance and 4-dimensional volume, respectively. The geometrical space-time shape has the (x1,x2,x3,ct) coordinate system with the metric signature of (+ + + +) but not the (x1,x2,x3,ctl) coordinate system with the metric signature of (+ - - -). Therefore, d(x1x2x3x4)2 = (ctl)2 = (ct)2 +x2 = x12 + x22 + x32 + x42 and V(x1x2x3x4) = E = mc2 = D(ct)Dx1Dx2Dx3 from (x1,x2,x3,x4) of the geometrical space-time shape. The warped shape can be described as the wave function of the quantum mechanics. The instant force action, twin paradox and possible space travel are explained by the absolute time and wave function collapse of the modified Lorentz transformations and quantum mechanics.
Introduction: space–time and the wave function
