When does the Hawking into Unruh mapping for global embeddings work?

Abstract

Abstract:We discuss for which smooth global embeddings of a metric into a Minkowskian spacetime the Hawking into Unruh mapping takes place. There is a series of examples of global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics. These examples use Fronsdal-type embeddings for which timelines are hyperbolas. In the present work we show that for some new embeddings (non Fronsdal-type) of the Schwarzschild and Reissner-Nordström metrics there is no mapping. We give also the examples of hyperbolic and non hyperbolic type embeddings for the de Sitter metric for which there is no mapping. For the Minkowski metric where there is no Hawking radiation we consider a non trivial embedding with hyperbolic timelines, hence in the ambient space the Unruh effect takes place, and it follows that there is no mapping too. The considered examples show that the meaning of the Hawking into Unruh mapping for global embeddings remains still insufficiently clear and requires further investigations.

Algebra of implicitly defined constraints for gravity as the general form of embedding theory

“…Let us calculate the Poisson brackets of the existing four constraints (16), (17) with each other. We will show that, despite the implicit form of the constraint (17), it is still possible to calculate the Poisson brackets of this constraint with other quantities.…”

“…A fundamental difference between the embedding theory from rather similar approach of brane theory [14] is the lack of gravity in the bulk because it is postulated that it is flat. In the framework of the embedding ideas, there were also attempts to associate quantum effects in the spacetime and in the bulk [15][16][17] (see also references in [17]), for which purpose constructed were explicit embeddings of physically interesting solutions of the Einstein equations, see, for example, [18][19][20][21] and references in [21].…”

“…In [23] it has been verified that this property is also preserved when one partially fixes the gauge in the action by identifying the internal and external times, y 0 (x µ ) = x 0 . Let us return to the "full" embedding theory where the Einstein constraints are not imposed and the fourth primary constraint has the form (17). In this case it cannot be written as an explicit function of the canonical variables since (14), being a multidimensional cubic equation with respect to n a , has no explicit solution.…”

Global Embeddings of BTZ and Schwarzschild-ADS Type Black Holes in a Flat Space

“…smoothness at all r > 0) of embeddings usually is not required; it is necessary only in the neiborhood of the outer horizon. It should be also noted that Hawking into Unruh mapping works only for embeddings of hyperbolic type [27,37]. Since the new embeddings of static BTZ and Schwarzschild-AdS spacetimes which were constructed here do not belong to this type (they are closer to the elliptic type), their usage would not lead to the appearance of this mapping.…”

“…radiation encountered by observer who moves along the embedding surface of this spacetime in a flat ambient space. The approach has been applied to many spacetimes with horizons (see, e.g., references in [27]), for which many explicit embeddings have been constructed. This paper is organized as follows.…”