Global solutions in gravity. Euclidean signature

TL;DR

This paper introduces a method for constructing maximally extended two-dimensional surfaces with a given metric, using an approach analogous to conformal block methods, demonstrated through the Schwarzschild solution.

Contribution

It proposes a novel method for constructing extended surfaces in 2D with specified metrics, applicable to Lorentzian signatures, illustrated by the Schwarzschild example.

Findings

Method successfully constructs maximally extended surfaces

Applicable to a wide class of 2D metrics with one Killing vector

Demonstrated with Schwarzschild solution

Abstract

We consider a wide class of two-dimensional metrics having one Killing vector. The method is proposed for the construction of maximally extended surfaces with the given Riemannian metric which is the analog of the conformal block method for two-dimensional Lorentzian signature metrics. The Schwarzschild solution is considered as an example.