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

Hawking into Unruh mapping for embeddings of hyperbolic type