Relativity Lite

96 | Appendix Since c and A are invariant constants, one may integrate * to obtain V c A c V c x x = + ( )   sinh / τ arcsinh 0 , (A6) where we have set τ 0 = 0 , and V x (τ = τ 0 = 0) = V x0 . We integrate again † to obtain X X c A A c V c V c x x x x − = + ( ) [ ] − +     0 2 0 0 2 2 1 cosh / τ arcsinh , (A7) where we have used arcsinh arccosh y y ( ) = + ( ) 1 2 . Using (A6) in (A5a), A A A c V c x x = + ( )   cosh / τ arcsinh 0 . (A8) Dividing (A6) by (A8) and using (A4c) and (A4b), we have v c A c V c x = + ( )   tanh / τ arcsinh 0 . (A9a) Now using (A2e) and (A4b) in (A8) and integrating dt d d A c V c d x = = = + ( )   γ τ τ τ τ sec cosh / Θ arcsinh 0 (A10) yields t c A A c V c V c x x = + ( )   − ( ) sinh / τ arcsinh 0 0 , (A11) where we have synchronized clocks at t 0 = τ 0 = 0 . Note that this is not γ τ : it is only for differentials ( dt = γ dτ ) where this is true for accelerated systems. For extended durations, one must use (A11) and its inverse relation, τ = +     −     c A At V c c A V c x x arcsinh arcsinh 0 0 . (A11a) Also note that the limit is t V c A x → + → 0 0 2 2 1 τ . (A11b) Inverting (A9a) gives V c A c v c x 0 = − +         sinh τ arctanh (A9b) * I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 99, No. 2.261(b). † I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 155, No. 2.248.1.