A Bisognano-Wichmann-like Theorem in a Certain Case of a Non Bifurcate Event Horizon related to an Extreme Reissner-Nordstr\"om Black Hole

TL;DR

This paper demonstrates a Bisognano-Wichmann-like theorem for a specific extremal Reissner-Nordström black hole model, showing that the vacuum state has properties similar to Minkowski space and has zero entropy despite horizons.

Contribution

It establishes a Bisognano-Wichmann-like theorem for a non-bifurcate horizon in an extremal R-N black hole model, linking vacuum states and thermal properties.

Findings

Only the Bertotti-Robinson vacuum state satisfies the extendibility conditions.

The vacuum state extends to the whole manifold as the global Carter-like vacuum.

The Carter-like vacuum restricted to a region has zero entropy despite horizons.

Abstract

Thermal Wightman functions of a massless scalar field are studied within the framework of a ``near horizon'' static background model of an extremal R-N black hole. This model is built up by using global Carter-like coordinates over an infinite set of Bertotti-Robinson submanifolds glued together. The analytical extendibility beyond the horizon is imposed as constraints on (thermal) Wightman's functions defined on a Bertotti-Robinson sub manifold. It turns out that only the Bertotti-Robinson vacuum state, i.e. $T=0$, satisfies the above requirement. Furthermore the extension of this state onto the whole manifold is proved to coincide exactly with the vacuum state in the global Carter-like coordinates. Hence a theorem similar to Bisognano-Wichmann theorem for the Minkowski space-time in terms of Wightman functions holds with vanishing ``Unruh-Rindler temperature''. Furtermore, the Carter-like vacuum restricted to a Bertotti-Robinson region, resulting a pure state there, has vanishing entropy despite of the presence of event horizons. Some comments on the real extreme R-N black hole are given.