 Euclidean horizon period under a regular radial conformal factor

ID: euclidean-horizon-period-under-a-regular-radial-conformal-factor

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

ρ = 2 C (r_{H}) (r - r_{H}) / f^{'} (r_{H})

puts the leading metric in the form

d ρ^{2} + f^{'} (r_{H})^{2} ρ^{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.

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