  The negative mode of the Schwarzschild black hole is central to Euclidean
quantum gravity around hot flat space and for the Gregory-Laflamme black string
instability. Numerous gauges were employed in the past to analyze it. Here
_the_ analytic derivation is found, based on postponing the gauge fixing, on
the power of the action and on decoupling of non-dynamic fields. A broad-range
generalization to perturbations around arbitrary co-homogeneity 1 geometries is
discussed.
