ASTR 498: Special Relativity Practice Problems Part 1: Basics A four-vector is written with a greek superscript: x" or xfi are examples. This has four components, one for each of the four spacetime coordinates you have chosen. For example, in Cartesian coordinates, the difierential separation vector is dx" = (dt;dx;dy;dz). The invariant interval ds (an interval between events that is the same as measured by any observer) is related to the difierential separation vector dx" through the metric tensor g"": ds2 = g""dx"dx" (1) where we use the Einstein summation convention that for a repeated up and down index, we sum over all spacetime components. For example, in Cartesian coordinates, " = t;x;y;z and the same with ", so ds2 = gttdt2 + gtxdtdx + gtydtdy + gtzdtdz + ::: (2) In this problem set we will opt for simplicity and always use Cartesian coordinates. Note, though, that in general we could use other coordinate systems (e.g., spherical polar coordi- nates). We are also focusing on at spacetim...