where s is the proper time of this Minkowski observer. Note that only a small portion of his history is covered by the Rindler chart. This shows explicitly why the Rindler chart is not geodesically complete ; timelike geodesics run outside the region covered by the chart in finite proper time. Of course, we already knew that the Rindler chart cannot be geodesically complete, because it covers only a portion of the original Cartesian chart, which is a geodesically complete chart.