Gravitational Instantons from Minimal Surfaces

TL;DR

This paper explores gravitational instantons derived from minimal surfaces, focusing on the helicoid, revealing symmetries, separability of equations, and enabling calculation of scalar field fluctuations.

Contribution

It provides a detailed analysis of a specific gravitational instanton from a minimal surface, highlighting its symmetries and solvable equations, which was not previously established.

Findings

The instanton corresponds to a Bianchi Type VII_0 metric.

The metric admits a quadratic Killing tensor, indicating hidden symmetry.

Closed-form scalar Green function allows vacuum fluctuation calculations.

Abstract

Physical properties of gravitational instantons which are derivable from minimal surfaces in 3-dimensional Euclidean space are examined using the Newman-Penrose formalism for Euclidean signature. The gravitational instanton that corresponds to the helicoid minimal surface is investigated in detail. This is a metric of Bianchi Type $VII_0$, or E(2) which admits a hidden symmetry due to the existence of a quadratic Killing tensor. It leads to a complete separation of variables in the Hamilton-Jacobi equation for geodesics, as well as in Laplace's equation for a massless scalar field. The scalar Green function can be obtained in closed form which enables us to calculate the vacuum fluctuations of a massless scalar field in the background of this instanton.