Relativity Lite

94 | Appendix We also note that   v dr dt = (A2d) and γ τ = dt d . (A2e) c/ γ c v c V V t θ θ v γ /c 2 r Figure 1. Equations (A3a, b). Graphical depiction of relationships among invariant, spatial, and temporal components of various four-vectors. Now, from the Pythagorean theorem applied to the upper triangle in figure 1, or using the definition of the proper-time interval for a flat metric, η µν = − − −         1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (with the Einstein convention that repeated indices imply a sum over all possible values), c d dX dX dX dX c dt dr c v dt 2 2 2 2 2 2 2 2 τ η µ µ µν µ ν ≡ = − = − ( ) = . (A3a) Then dividing by dτ 2 , c V V t r 2 2 2 = − (A3b)