Relativity Lite

Appendix | 97 and taking the same limit of the square of this, one recovers the special relativistic result, t A c v c v A A → + − +         → + → → 0 2 0 2 1 1 sinh τ τ arctanh c v v c 2 2 2 2 1 1 − ( ) = − τ τ . (A11c) Using (A7) and (A11) in cosh sinh 2 2 1 Φ Φ − = , one obtains A X X c V c At c V c x x x x − ( ) + +         − +       = 0 2 0 2 2 2 0 2 1 1 , (A12a) which is the equation of a hyperbola, reducing to the equation for a parabola, X X At V t x x x − ( ) ≅ + 0 2 0 1 2 , (A12b) when A t c << and V c x 0 << . * Relation (A12a) may also be derived by coordinate time inte- gration, as shown in Sears and Brehme. † Using (A7) in (A12a) gives sinh / A c c At c V c V x x τ + ( )     − +       arcsinh 0 2 0 2 = 0 . (A12c) Moving the second term to the right-hand side and taking the square root and then the derivative of both sides gives two useful differential relations: dt A c V c d At c V c x x = +           = + + cosh τ τ arcsinh 0 0 1       2 d τ . (A13) The first equality can also be found by using (A4c), γ = V x0 /v , which is (A6) divided by (A9). Also (A12a) gives a third form, dt V c A X X c d x x x = + + −         1 0 2 2 0 2 ( ) τ (A14a) and from (A9b) and (A14a) γ τ = + − + ( )   + −     1 2 0 2 sinh / ( ) A c c A X X c x x arctanh v  . (A14b) * The next higher terms in the series expansion of 1 1 0 2 0 2 2 + +       − + At c V c V c x x are − − −     − 3 4 1 2 1 8 1 4 0 2 2 0 2 2 2 2 0 2 2 V c At c V c A t c At V c V x x x x t 0 . † F. W. Sears and R. W. Brehme, Introduction to the Theory of Relativity (Addison-Wesley, Reading, MA, 1968), p. 102 ff.