Symmetry and special relativity in Finsler spacetime with constant curvature

Within the framework of projective geometry, we investigate kinematics and symmetry in (α,β)(α,β) spacetime—one special types of Finsler spacetime. The projectively flat (α,β)(α,β) spacetime with constant flag curvature is divided into four types. The symmetry in type A—Riemann spacetime with constant sectional curvature—is just the one in de Sitter special relativity. The symmetry in type B—locally Minkowski spacetime—is just the one in very special relativity. It is found that type C—Funk spacetime and type D—scaled Berwaldʼs metric spacetime both possess the Lorentz group as its isometric group. The geodesic equation, algebra and dispersion relation in the (α,β)(α,β) spacetime are given. The corresponding invariant special relativity in the four types of (α,β)(α,β) spacetime contain two parameters, the speed of light and a geometrical parameter, which may relate to the new physical scale. They all reduce to Einsteinʼs special relativity while the geometrical parameter vanishes.