Euclidean horizon period under a regular radial conformal factor ($\beta =4\pi /|f^{\prime }(r_H)|$)

For a Euclidean radial-time metric $C(r)(f(r)d{t}_E^2+d{r}^2/f(r))$, assume $f$ has a simple horizon zero and $C$ is smooth, finite and positive there. A proper radial coordinate $\rho =2\sqrt{C(r_H)(r-r_H)/f^{\prime }(r_H)}$ puts the leading metric in the form $d{\rho }^2+f^{\prime }(r_H{)}^2{\rho }^{2}d{t}_E^2/4$. Smoothness fixes the displayed period independently of $C(r_H)$. When the Lorentzian time is normalized at infinity, the period is the inverse Hawking temperature in natural units. A conformal factor singular at the horizon need not preserve this argument.

// Target: general-relativity.bigb