Spacetime Slices and Surfaces of Revolution

TL;DR

This paper explores the conditions under which certain 2D slices of black hole spacetimes and rotational surfaces can be embedded into 3D spaces, revealing a duality and a shared phenomenon involving parameters like surface gravity and Gaussian curvature.

Contribution

It demonstrates a duality between embeddings of black hole slices and rotational surfaces, linking parameters such as surface gravity and Gaussian curvature through a unified framework.

Findings

Embeddings depend on parameters like surface gravity and cone angle.

Metrics and embeddings form dual pairs related by Wick rotation.

Gaussian curvature formulas are simple and previously not widely known.

Abstract

Under certain conditions, a $(1+1)$-dimensional slice $\hat{g}$ 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 $\kappa$ 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 $g = \phi(r)^{-1} dr^2 + \phi(r) d\theta^2$, then $\K_g=-{1/2}\phi''(r)$. This note shows that metrics $g$ and $\hat{g}$ 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 $\kappa$ is inversely proportional to the cone angle at the axis of rotation. Further, the Gaussian curvature of $\hat{g}$ is given by a simple formula that seems not to be widely known.