Pnin said:
We see from the picture that there are more ways to “go to infinity” in a null direction than in a timelike or spacelike direction.
I would not put it this way. If you look at how the null curves go, you will see that the different points of past null infinity are each the "source" of a different ingoing null curve, and the different points of future null infinity are each the "destination" of a different outgoing null curve. Furthermore, each ingoing null curve is matched at ##r = 0## to an outgoing null curve, so we can speak of a family of null curves that cover the entire spacetime, each one of which (a) starts at a distinct point on past null infinity, (b) covers a distinct set of points within the spacetime, and (c) ends at a distinct point on future null infinity.
So at any given
point in the spacetime, you have only
two null curves passing through it--one coming from past null infinity (which will go through ##r = 0## and then head out to future null infinity) and one going to future null infinity (because it has already gone through ##r = 0##). But you have
multiple timelike and spacelike curves passing through that same point, because those curves can go in any direction that is within the appropriate limits of the light cones (whereas a null curve can only go
on the light cones). So actually, your statement is backwards: there are many more timelike or spacelike directions at any given point in the spacetime than there are null directions, and since each such direction is a different "way to go to infinity", there are many more ways to go to infinity in a timelike or spacelike direction than in a null direction.
Your second way of phrasing your question is better; see below.
Pnin said:
Why do massive particles have all just one common point in infinity in space, future and space respectively, but for massless there are (infinitely) many infinities?
This is a better way of phrasing the question because it focuses on the
number of points at infinity instead of the
number of ways to go to infinity from a particular point in the spacetime. As my response above shows, the two are not the same.
I don't know that there is a good single well-accepted answer to this question. What follows is just my personal opinion.
A key property of null geodesics that is not shared by timelike and spacelike geodesics is that they are causal boundaries. In other words, null geodesics show the causal structure of the spacetime. So if we imagine moving along some chosen spacelike or timelike geodesic, we will encounter the whole family of null geodesics, and each one will be the causal boundary of a different causal region. That causal boundary property has to extend all the way to infinity, since the causal structure of the spacetime is preserved by the conformal transformations that are used to construct Penrose diagrams. That means each null geodesic has to have its own distinct endpoint at infinity.
However, the same is not true for timelike and spacelike geodesics because they are not causal boundaries. If we imagine moving along some chosen null geodesic, we will encounter the whole family of timelike and spacelike geodesics, but there is nothing in the causal structure of the spacetime that distinguishes anyone of them from any other. There is nothing that requires them to have distinct endpoints at infinity, because they are not causal boundaries of different causal regions.
