Relation between quantum effects in General Relativity and embedding theory

TL;DR

This paper explores the connection between quantum effects like Hawking and Unruh phenomena in curved and flat embedded spaces, analyzing conditions for their correspondence and implications for quantum field theory in curved spacetime.

Contribution

It establishes conditions under which Hawking and Unruh effects are equivalent via isometric embeddings, providing a geometric framework for quantum effects in curved spacetime.

Findings

Hawking-Unruh mapping exists for certain hyperbolic embeddings.

Examples where the mapping does not hold are identified.

Restrictions on embedding parameters for correlating two-point functions are derived.

Abstract

We present results relevant to the relation between quantum effects in a Riemannian space and on the surface appearing as a result of its isometric embedding in a flat space of a higher dimension. We discuss the mapping between the Hawking effect fixed by an observer in the Riemannian space with a horizon and the Unruh effect related to an accelerated motion of this observer in the ambient space. We present examples for which this mapping holds and examples for which there is no mapping. We describe the general form of the hyperbolic embedding of the metric with a horizon smoothly covering the horizon and prove that there is a Hawking into Unruh mapping for this embedding. We also discuss the possibility of relating two-point functions in a Riemannian space and the ambient space in which it is embedded. We obtain restrictions on the geometric parameters of the embedding for which such a relation is known.