Alberto said:
> If two events A and B are *not* timelike, are they
> spacelike?
No. There's a third possibility - the interval between them might be
null (or lightlike). In special relativity, two events at timelike
separation lie inside each other's light cones, two events at null
separation lie on each other's light cones and two events at spacelike
separation lie outside each other's light cones. In general relativity
things are more subtle because the light cones only exist in the
tangent spaces at each event and so we have to talk about timelike
curves (whose tangent vectors are everywhere timelike) or causal curves
(whose tangents are everywhere either timelike or null), and
chronological and causal futures and pasts and suchlike. The same sorts
of conclusions still hold though - if you can join two events by a
chronological curve then you can make a coordinate system in which they
happen at the same place at different times (in fact you can make an
infinite number of such coordinate systems) and if you can't join two
points by a causal curve then you can make coordinate systems in which
they happen at the same time but different places. But you shouldn't be
too attached to coordinate systems really.
> or [I guess this is equivalent]
>
> If there are two events A and B such that each line
> joining A to B must have at least one arc that is
> spacelike or two timelike arcs that are reversed in
> time relative to each other, then is there any reference
> system for which A and B are simultaneous?
Yes, I think that's true.
> What I am thinking is about abnormal situations, like
> one event A inside a Black Hole and the other event B
> outside it.
Yes, the same things apply then. In fact, it's possible to make the
causal structure of black holes nice and clear by drawing things called
Penrose diagrams. Here's a page I found showing the Schwarzschild black
hole in Kruskal coordinates and the corresponding Penrose diagram:
http://www.phy.syr.edu/courses/PHY312.98Spring/projects/jebornak/html/penrose.html
(the basic idea is to take the Kruskal coordinates diagram and make a
conformal transformation that brings the points at infinity in to a
finite distance).
Rich, catching up on email.
