Hawking into Unruh mapping for embeddings of hyperbolic type

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.