Does Schwarzschild metric in Kruskal-Szekeres coordinates admit asymptotic ($r \to +\infty$) timelike observers? I thank in advance whoever will answer my question.
Schwarzschild metric in Schwarzschild coordinates in $\mathbb{R}^{1,3}$ is [1]:
$$ds^2=-\bigg(1-\displaystyle\frac{2M}{r}\bigg)dt^2+\bigg(1-\frac{2M}{r}\bigg)^{-1}dr^2+r^2d\Omega^2.$$
When $r\to+\infty$ then $g_{\mu\nu} \to \eta_{\mu\nu}$, with $\eta_{\mu\nu}$ Minkowski metric (which obviously admits timelike observers e.g. $(1,0,0,0)$). By operating the change of coordinate of Kruskal-Szekeres we have [1]:
$$ ds^2=\frac{32M^3}{r}e^{-r/2M}(-dT^2+dX^2)+r^2d\Omega^2$$
with $r=2M(1+W_0[(X^2-T^2)/e])$ and $W_0$ Lambert function. In this case, when $r \to +\infty$ we get $ds^2 \to r^2d\Omega^2>0$ i.e. any vector becomes spacelike there. How can this discrepancy be explained?
Thanks a lot.

References
[1] http://bascom.brynmawr.edu/physics/courses/325/Problem_Sets/Review_PS_3.pdf
 A: You're getting tripped up in two different ways here.
(1) When you reason about the sizes of the terms in the K-S metric, you're ignoring the fact that the expression you're using is written using a mixture of variables from the two coordinate systems. This point is easier to understand with a simpler example. Suppose that $ds^2=e^{-r}dX^2$, where $r=\ln X$. (As pointed out in a comment by A.V.S., the log is the approximate behavior of the W function.) Implicit differentiation then gives $dX=e^r dr$, so that the line element can be rewritten as $ds^2=e^{r}dr^2$, and the coefficient clearly doesn't approach zero exponentially.
(Of course every point has a complete light-cone, including the timelike portions, so we knew in advance that there had to be a mistake in your conclusion that all vectors are spacelike in K-S coordinates at large $r$.)
(2) If we want a curve to approach spacelike infinity, it's not enough that it have $r\rightarrow\infty$. For example, a null geodesic can have $r\rightarrow\infty$, but it's going to approach null infinity, not spacelike infinity. The difference between the different idealized boundaries has to do with how fast $r$ approaches infinity compared to how fast $t$ approaches infinity.
If you look at a Penrose diagram for the Schwarzschild spacetime, it's clear that you can't have a timelike world-line that approaches spacelike infinity.
A: I don't think there are any $r\to\infty$ asymptotic timelike observers. There seems to be a limit of that sort in some coordinates, like Eddington-Finkelstein $(u,r)$, but it's an illusion. Conformal spatial infinity is a single point with no time dimension, and its causal past only contains past infinity.
The $r\to\infty$ limit of stationary worldlines on the Penrose diagram covers past and future null infinity, with a crossover at spatial infinity. This is the same thing that happens at $r=2M$, and the meaning is the same: the observer can't be timelike in the limit, and has to be null instead.
In Minkowski space, you could put the (future) observer at $(t{-}r)(s)=s$, with $t{+}r$ constant and tending to $+\infty$. The "redshift" seen by this observer is the same as that seen by an observer at finite $r$, if you take $s$ to be its "proper time". In the K-S case, a stationary worldline at finite $r$ is given by
$$(T{\pm}X)(t) = \pm R \exp(\pm t/4M)$$
where $t$ is the Schwarzschild time and $R=\sqrt{X^2-T^2}$, and the observer at infinity can be at
$$(T{-}X)(s) = -R_\infty \exp(-s/4M)$$
where $R_\infty$ is any positive value (adjustable by shifting the origin of $s$). The "redshift" this observer sees is the limit of redshifts seen by stationary timelike observers.
A: Thank you very much for your kind and detailed answers.
I had thought about both the problem of the change of coordinates and mixed coordinates; however, what is not clear to me is the reason why $r = 2M$ still remains the same region in K-S coordinates while $r \to +\infty$ does not. Is this due to the fact that the K-S coordinates are constructed ad-hoc to do that (i.e. to keep the event horizon at $r = 2M$)?
Moreover, when I write $r \to +\infty$ I mean "in a region very far from the black hole" (i.e. $r >> 2M\;|\;g_{\mu\nu} \sim \eta_{\mu\nu}$). This is very useful for calculating time dilation and redshift between a near-horizon observer and a static Earth observer [2].
To "obtain" this region (where $g_{\mu\nu} \sim \eta_{\mu\nu}$) in Schwarzschild coordinates it suffices to perform $r \to +\infty$; but how can I identify the same region using Kruskal-Szekeres coordinates? How can I go far away from the black hole and put a static timelike observer there?
Thanks again for your help and concern.

References
[2] http://www.roma1.infn.it/teongrav/leonardo/bh/bhcap3.pdf
