Space–time slices and surfaces of revolution

Under certain conditions, a $(1+1)$-dimensional slice $ĝ$ of a spherically symmetric black hole space–time 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 three-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 $g=φ(r)−1 dr2+φ(r)dθ2$⁠, then $Kg=−12φ″(r)$⁠. This note shows that metrics $g$ and $ĝ$ 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 $ĝ$ is given by a simple formula that seems not to be widely known.