Under certain conditions, a (1+1)-dimensional slice g^​ of a
spherically symmetric black hole spacetime can be equivariantly embedded in
(2+1)-dimensional Minkowski space. The embedding depends on a real parameter
that corresponds physically to the surface gravity κ of the black hole
horizon.
Under conditions that turn out to be closely related, a real surface that
possesses rotational symmetry can be equivariantly embedded in 3-dimensional
Euclidean space. The embedding does not obviously depend on a parameter.
However, the Gaussian curvature is given by a simple formula: If the metric is
written g=ϕ(r)−1dr2+ϕ(r)dθ2, then
\K_g=-{1/2}\phi''(r).
This note shows that metrics g and g^​ occur in dual pairs, and that
the embeddings described above are orthogonal facets of a single phenomenon. In
particular, the metrics and their respective embeddings differ by a Wick
rotation that preserves the ambient symmetry.
Consequently, the embedding of g depends on a real parameter. The ambient
space is not smooth, and κ is inversely proportional to the cone angle
at the axis of rotation. Further, the Gaussian curvature of g^​ is given
by a simple formula that seems not to be widely known.Comment: 15 pages, added reference