• DOI:10.1063/1.1808487
• Corpus ID: 119085754

Space–time slices and surfaces of revolution

@article{Giblin2004SpacetimeSA,
  title={Space–time slices and surfaces of revolution},
  author={John T. Giblin and Andrew D. Hwang},
  journal={Journal of Mathematical Physics},
  year={2004},
  volume={45},
  pages={4551-4559}
}

• J. Giblin, Andrew D. Hwang
• Published 3 June 2004
• Mathematics
• Journal of Mathematical Physics

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 in...