Let's consider for example compactification of Minkowski spacetime or Kruskal extension of Schwartzschild. They are quite similar because in both cases we rescale the null direction. I wonder why we always rescale the null direction, not simply x or t.
Re-scaling the null direction has this advantage that since one side is equal to zero, then the re-scaling factor won't change the direction but rather compacts (or maybe expands) the direction conformally. However, re-scaling x or t does not preserve the general form of metric though wouldn't change its nature, too if the re-scaling factor isn't coordinate-dependent! In the latter case (a factor being independent of coordinates), one can make use of the re-scaling of x or t as well! AB