Gauge-eliminated derivation of the black-hole negative mode

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 nondynamic fields. A broad-range generalization to perturbations around geometries with one nonhomogeneous dimension is discussed.