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
Dipartimento di Fisica, Università di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento,
Via Sommarive 14 I-38050, Povo (TN), Italy

Stefan Steidl
Institut für Theoretische Physik der Universität Innsbruck, Victor-Franz-Hess-Haus, Technikerstraße 25, A-6020 Innsbruck, Austria

Received 23 October 1995, in final 26 February 1996

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 extremal R-N black hole are given.

PACS numbers. 0462, 0470D, 1110W.

Classical and Quantum Gravity 13, 2121-2143 (1996).