= Schwarzschild time-translation Killing field in Kruskal coordinates
{c}
{title2=$K=(-U\partial_U+V\partial_V)/(4m)$}
For null coordinates normalized by $UV=(2m-r)e^{r/(2m)}$, the displayed field agrees with exterior Schwarzschild time translation. Its flow rescales $U$ and $V$ inversely, preserving their product, the <radius> and $dU\,dV$. It therefore preserves the extended <metric> across both horizons. Its squared norm is $1-2m/r$: it is timelike in both exterior regions, spacelike in the black-hole and white-hole interiors, and nonzero null on the horizons away from the bifurcation two-<sphere>. At that <sphere> it vanishes. The <zero vector> should not be mistaken for a nonzero null generator. This gives an explicit symmetry that survives where the original time coordinate ceases to be regular.
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