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

Valter Moretti·Stefan Steidl
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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.

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Valter Moretti, Stefan Steidl. 1996-04-12. A Bisognano-Wichmann-like Theorem in a Certain Case of a Non Bifurcate Event Horizon related to an Extreme Reissner-Nordström Black Hole. https://doi.org/10.1088/0264-9381%2F13%2F8%2F009

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