The Lemaître coordinate chart is not geodesically complete. This can be seen by tracing outward-moving radial null geodesics backwards in time. The outward-moving geodesics correspond to the plus sign in the above. Selecting a starting point r > r s at τ τ = 0 , the above equation integrates to r → → + ∞ ∞ as τ τ → → + ∞ ∞ . Going backwards in proper time, one has r → → r s as τ τ → → − − ∞ ∞ . Starting at r < r s and integrating forward, one arrives at r = 0 in finite proper time. Going backwards, one has, once again that r → → r s as τ τ → → − − ∞ ∞ . Thus, one concludes that, although the metric is non-singular at r = r s , all outward-traveling geodesics extend to r = r s as τ τ → → − − ∞ ∞ .