Lorentz Group Action on Ellips Space

Abstract

The ellips space E has been constructed as cartesian product R+ × R+ × [ π 2 , π 2 ]. Its elements, (a, b, θ), is called as an ellipse with eccentricity is = p1 − b2/a2 if b < a and is = p1 − a2/b2 if a > b. The points (a, b, π/2) is equal to (b, a, 0). The action of subgrup SOoz(3, 1) of Lorentz group SOo(3, 1), containing Lorentz transformations on x−y plane and rotations about z axes, on E is defined as Lorentz transformation or rotation transformation of points in an ellipse. The action is effective since there are no rigid points in E. The action is also not free and transitive. These properties means that Lorentz transformations change any ellips into another ellips. Although mathematically we can move from an ellipse to another one with the bigger eccentrity but it is imposible physically. This is occured because we donot include the speed parameter into the definition of an ellipse in E.