The whole point of spacetime according to relativity is that it is not fundamentally divided up into “time” and “space.” The light cones, demarcating the accessible past and future of each event, are not added on top of the straightforward Newtonian decomposition of spacetime into time and space; they replace that structure entirely. Time can be measured along each individual world line, but it’s not a built-in feature of the entire spacetime.
