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.

Figure 1

A graph of the perturbation potential $V$ as a function of the radial coordinate $r$ (29) in $r_0=1$ units. Drawn at $d=4$—in higher dimensions $V$ becomes even more concentrated around $r/r_0=1$.