The Connection between Noneuclidean Geometry and Special Relativity in an Expanding Universe

The homogeneous Lorentz group is also the isometry group of noneuclidean geometry in hyperbolic space, but the connection has not been fully exploited in special relativity. In a 1907 lecture Minkowski recognized that the velocity \textbf{v} in special relativity generates a noneuclidean manifold. He soon showed this to be part of a covariant 4-vector $\textsf {w}=\left( {1-v^2/c^2} \right)^{-1/2}\left( {v_x ,v_y ,v_z ,ic} \right)$, the vector to the 3-surface of a 4-sphere of imaginary radius $ic$ in velocity space. Unable to identify a comparable geometry in position space, he omitted all mention of this velocity symmetry in later publications. Had the Hubble expansion (1927) been known, he could have used the Hubble time $t_H$, a cosmic time variable $t={t_H} +\delta t$, and a position 4-vector $\textsf {s}=\left( {t/t_H} \right)\left( {x,y,z,i\left[ {c^2{t_H} ^2+r^2} \right]^{1/2}} \right)$, an expanding hypersphere of imaginary radius $R\left( t \right)=ict$. The interval between two local events is, to first order, $\Delta r=\left( {\Delta x,\Delta y,\Delta z,ic\Delta t} \right)$. This is the Minkowski 4-vector in differential form, but the source of its imaginary time is identified as the cosmological expansion. An extended Lorentz group follows if the 4-vectors are replaced by tensors of position and velocity.