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The easy way to see that this has no real physical meaning is to note that ##\partial_t## is a Killing field also in the interior region (although it is no longer time-like). Thus, if you go along a coordinate line of ##t##, you will end up in exactly the same situation as you were before. If I am not mistaken, the transformation generated by ##\partial_t## is just a hyperbolic rotation of the Kruskal-Szekeres coordinates. Since the metric in those coordinates is on the form (with ##T## and ##X## as the KS coordinates)Then, as t goes to infinity, you get ever closer the horizon measured along Kruskal simultaneity line (you must go outside SC coordinates to measure distance to the horizon).

$$

ds^2 = f(r) (dT^2 - dX^2) - r^2 d\Omega^2,

$$

and ##r## is constant along the hyperbolae of constant ##T^2 - X^2##, this is invariant under hyperbolic rotations between ##X## and ##T##.