Hawking into Unruh mapping for embeddings of hyperbolic type

TL;DR

This paper investigates the conditions under which Hawking into Unruh mapping occurs for hyperbolic embeddings of spacetime metrics with a time-like Killing vector, establishing criteria for its existence and explaining known exceptions.

Contribution

It proves that Hawking into Unruh mapping exists for hyperbolic embeddings of any metric with a time-like Killing vector and smooth horizon coverage, without requiring additional symmetries.

Findings

Hawking into Unruh mapping occurs under specific hyperbolic embedding conditions.

Examples lacking the mapping do not meet these conditions.

The results generalize previous knowledge beyond symmetric cases.

Abstract

We study the conditions of the existence of Hawking into Unruh mapping for hyperbolic (Fronsdal-type) embeddings of metric into the Minkowski space, for which timelines are hyperbolas. Many examples are known for global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics with some symmetry. However the examples of embeddings, both smooth and hyperbolic, for which there is no mapping, were also given. In the present work we prove that Hawking into Unruh mapping takes place for a hyperbolic embedding of an arbitrary metric with a time-like Killing vector and a Killing horizon if the embedding of such type exists and smoothly covers the horizon. At the same time we do not assume any symmetry (spherical for example), except the time translational invariance which corresponds to the existence of a time-like Killing vector. We show that the known examples of the absence of mapping do not satisfy the formulated conditions of its existence.