부산대학교 도서관

Under certain conditions, a (1+1)-dimensional slice g⁁ 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=[lowercase_phi_synonym](r)-1 dr2+[lowercase_phi_synonym](r)dθ2, then Kg=-1/2[lowercase_phi_synonym]″(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. [ABSTRACT FROM AUTHOR]