PAllen said:
Ok, now I am a bit confused. Both by definition (there exists a 'global' timelike KVF) and understanding (a timelike direction in which the metric doesn't change), any region that can be covered by these coordinates inside a cosmological horizon is static (since the KVF is also irrotational) , no scare quotes. The fact that some other coordinates don't manifest this in the metric expression shouldn't change this any more than Lemaitre coordinates cancel the static character of the exterior SC region.
Once you've picked out a particular timelike KVF, yes, all this is true. But Schwarzschild spacetime only has one KVF that is timelike anywhere, and only one region in which it is timelike. In spacetimes where there are multiple timelike KVFs, things get more complicated: each one has to be looked at separately to see in which region of the spacetime it is timelike, and the answer may be different for different ones.
For example, Minkowski spacetime (we'll stick with that comparison first since it's easier) has two infinite families of KVFs which can be timelike, and they have different behaviors:
(1) All of the timelike KVFs in the first family (the ones corresponding to inertial observers) are timelike everywhere in the spacetime. So the entire spacetime is static with respect to anyone of them.
(2) The KVFs in the second family (the ones corresponding to Rindler observers) are only timelike in particular regions of the spacetime, and the regions are different for different KVFs in the family. For example, consider the following two members of the family, described with respect to a global inertial frame (I'll describe their integral curves since that's easier to write down):
KVF A has integral curves which are hyperbolas satisfying x^2 - t^2 = K, where K is a constant ranging from minus infinity to infinity. Obviously if K > 0 and x > 0, this family of curves corresponds to the worldlines of Rindler observers in Region I of the spacetime, which I'll actually call Region I-A. But there are three other branches of these hyperbolas, corresponding to Regions II-A, III-A, and IV-A, and the KVF is timelike only in regions I-A and III-A; it is spacelike in regions II-A and IV-A, and null on the Rindler horizons, x = +/- t. So with respect to this KVF, the spacetime is static only in Regions I-A and III-A.
KVF B has integral curves which are hyperbolas satisfying x^2 - (t - 1)^2 = K, where K again ranges from minus infinity to infinity. Obviously these are similar to the above hyperbolas, but with the Rindler horizon shifted along the time axis by 1 unit, to x = +/- (t - 1). This shifts all four regions accordingly: the spacetime, with respect to *this* KVF, is now static only in Regions I-B and III-B, which are *different* regions than I-A and III-A.
Similar remarks apply to de Sitter spacetime; here we can choose any spatial point to be the origin, r = 0, of the static chart that TrickyDicky wrote down the line element for. Each possible origin corresponds to a different member of the corresponding family of KVFs on de Sitter spacetime, and each member of the family is timelike in a different region. This is why I say that this family of KVFs in de Sitter spacetime corresponds to the "Rindler family" of KVFs on Minkowski spacetime. Which of course leaves open the question of which family of KVFs on de Sitter spacetime corresponds to the "Minkowski family" of KVFs on Minkowski spacetime (the one corresponding to inertial observers).