Negative mode of Schwarzschild black hole from the thermodynamic instability

The thermodynamic instability, for example the negative heat capacity, of a black hole implies the existence of off-shell negative mode(s) (unstable perturbation(s)) around the black hole geometry in the Euclidean path integral formalism of quantum gravity. We explicitly construct an off-shell unstable perturbation inspired from the negative heat capacity in the case of Schwarzschild black hole with/without a cosmological constant. We carefully check the boundary conditions, i.e. the regularity at the horizon, the traceless condition, and the normalizability.