Mathematics – Logic
Scientific paper
Feb 2010
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010aps..apr.w1031s&link_type=abstract
American Physical Society, APS April Meeting 2010, February 13-16, 2010, abstract #W1.031
Mathematics
Logic
Scientific paper
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 v in special relativity generates a noneuclidean manifold. He soon showed this to be part of a covariant 4-vector w=( 1-v^2/c^2 )-1/2( vx,vy,vz,ic ), 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 tH, a cosmic time variable t=tH +δt, and a position 4-vector s=( t/tH )( x,y,z,i[ c^2tH ^2+r^2 ]^1/2 ), an expanding hypersphere of imaginary radius R( t )=ict. The interval between two local events is, to first order, δr=( δx,δy,δz,icδt ). 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. )
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