STRACTWhy Do We Always Rescale the Null Direction in Conformal Compactification?

[paweld]

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.

 

[Altabeh]

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