Relativity Lite

Appendix | 99 where we have set V x0 = 0 . As with elliptical integrals, one expands the radical in the small parameter A τ/c and uses (A7) and (A11) to obtain L c A c d c c A = + ′ + ⋅ ⋅ ⋅     ′ = +         ∫ 1 2 2 0 sinh si τ τ τ τ nh cosh A c A c c τ τ τ τ                 − + ⋅ ⋅ ⋅     = 1 2 2 + − +                      c A A c A c 2 2 1 1 sinh cosh τ τ      + ⋅ ⋅ ⋅ = + − +     + ⋅ ⋅ ⋅ c ct A x x c τ 2 2 1 0 2 ( ) , (A19a) where we have set X x = x and X x0 = x 0 for a more conventional notation. Finally, suppose the length of this arc is L = c t to first approximation, as labeled in figure 10 of chapter 4; then (A19a) becomes ct A x x c c 1 1 2 1 2 0 2 − − +     + ⋅ ⋅ ⋅     = ( ) τ (A19b) or t A x x c 1 0 2 − − + ⋅ ⋅ ⋅     = ( ) τ , (A19c) giving (A14a) to first approximation. Thus, the assumption L = c t is a consistent one, show- ing that the gravitational red-shift factor is essentially just the stretched path-length for a beam of light in a gravitational field. Finally, one may want a relation like (A12a) involving just position and velocity, which may be obtained by solving (A7) and (A9) using the relation cosh sinh 2 2 1 Φ Φ − = , v V A x x V c A x x c c V x x x 2 0 2 0 0 2 2 0 2 2 0 2 2 1 = + − + + −     + ( ) ( ) + − + + −     − → → 2 1 2 0 0 2 2 0 2 2 0 0 A x x V c A x x c c A x x x V ( ) ( ) ( 0 0 2 2 0 0 2 2 ) ( ) ( ) ( ) + −     + − + −     A x x c c A x x A x x c c 2 . (A20a) When A is small, this reduces to the nonrelativistic form for constant A : 1 2 2 0 2 0 v v v dv Adx A x x − ( ) = = = − ∫ ∫ ( ) . (A20b) It is generally more convenient to calculate v from A via the relation (A13): dt V c A x x c d d v c d x = + + −         ≡ ≡ − 1 1 1 0 2 2 0 2 2 2 ( ) τ γ τ τ , (A21)