structural:expansion
3D spacetime

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space d s 2 = d x 1 2 + d x 2 2 − c 2 d t 2, {\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2},} we see that the null geodesics lie along a dual-cone (see Fig. 10-2) defined by the equation; d s 2 = 0 = d x 1 2 + d x 2 2 − c 2 d t 2 {\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}} or simply d x 1 2 + d x 2 2 = c 2 d t 2, {\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2},} which is the equation of a circle of radius c dt.