PHY2061 R. D. Field Department of Physics relativity_4a.doc University of Florida Analogy with RotationsConsider two frames of reference the O-frame(label points according to x,y) and the O'-frame(label points according to x',y'). Let the origins of the two frames coincide and rotate the O'-frameabout the z-axis by an angle θ. The two frames are related by the following transformation (i.e.by a rotation). θcossinsincosyxyyxx′+′=′−′=cossinsincosyxyyxx+−=′+=′Vector Notation: ′′−=yxyxcossinsincosrRr′=rr−=cossinsincosR=yxrr′′=′yxrrRotational Invariant: 222222rrryxyxrrr′=′⋅′=′+′=+=⋅=rrrrLorentz Transformation: Let coshθ= γand sinhθ= βγβγthen′′
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