Relativity Lite

98 | Appendix For small A , this simplifies to γ τ → − + − − −           → A x x v c A X X c v c v c 0 2 2 0 2 2 2 2 1 1 1 1 ( ) , (A14c) which reduces to the nonaccelerated value for A = 0 . Upon inverting (A13) and expanding the denominator in a series, one obtains d A X X c V c A X X c V x x x x x x τ ( ) = − − ( ) − + − ( ) + 2 0 2 0 2 2 0 2 0 2 1 2 3 1 2 c V c dt x 2 2 0 4 4 2 1 4     + +         ( ) ... , (A15a) of which (A1) is the first approximation if V x0 = 0 . Note that with A = 0 , one regains the special-relativistic equation for time dilation at constant velocity V x0 (which equals v 0 in lowest order). Finally, we note that the uniform acceleration of the rocket in the same direction as the change in position, as in figure 2a of chapter 4, feels like (is equivalent to) gravitational acceleration in the opposite direction to the change in position in figure 2b of chapter 4. So we replace A x A r F m r V r m GM r thrusters gravity g ∆ ∆ ∆ ⇒ − = − = − = ( ) ( ) 1 2 (A16) to give * the Schwarzschild solution of chapter 4, t r GM c 2 2 2 1 1 1 2 = − × τ . (A15b) Alternatively, one can prove that the arc in figure 10 of chapter 4 really has length c t A x c c ∆ ∆ ∆ = +    1 2 τ . (A17) The relations derived above are necessary for this second proof. The arc-length is defined parametrically by L c dx cd dt d d c A = ′     + ′     ′ = + ∫ τ τ τ τ τ 2 2 2 0 1 2sinh c d         ∫ ′ 0 τ τ . (A18) * To obtain this from the above expression, we first redefine our distance coordinate as decreasing downward rather increas- ing downward as above, by changing the sign of all r ’s. Next we note that m A r is just the gravitational potential energy −G m M/r .