Wightman Functions' Behaviour on the Event Horizon of an Extremal Reissner-Nordstr\"om Black Hole

TL;DR

This paper investigates the behavior of thermal Wightman functions near the event horizon of an extremal Reissner-Nordström black hole, introducing a new method to analyze invariant geodesic lengths and examining quantum state prescriptions.

Contribution

It introduces a novel method using invariant geodesic lengths to analyze Wightman functions on black hole horizons and extends prior prescriptions to extremal Reissner-Nordström black holes.

Findings

Weak HNS prescription holds for all finite temperatures.

The extremal Reissner-Nordström vacuum satisfies weak HNS and Hessling's prescriptions.

Results likely extend beyond massless fields and large black hole masses.

Abstract

A weaker Haag, Narnhofer and Stein prescription as well as a weaker Hessling Quantum Equivalence Principle for the behaviour of thermal Wightman functions on an event horizon are analysed in the case of an extremal Reissner-Nordstr\"{o}m black hole in the limit of a large mass. In order to avoid the degeneracy of the metric in the stationary coordinates on the horizon, a method is introduced which employs the invariant length of geodesics which pass the horizon. First the method is checked for a massless scalar field on the event horizon of the Rindler wedge, extending the original procedure of Haag, Narnhofer and Stein onto the {\em whole horizon} and recovering the same results found by Hessling. Afterwards the HNS prescription and Hessling's prescription for a massless scalar field are analysed on the whole horizon of an extremal Reissner-Nordstr\"{o}m black hole in the limit of a large mass. It is proved that the weak form of the HNS prescription is satisfyed for all the finite values of the temperature of the KMS states, i.e., this principle does not determine any Hawking temperature. It is found that the Reissner-Nordstr\"{o}m vacuum, i.e., $T=0$ does satisfy the weak HNS prescription and it is the only state which satisfies weak Hessling's prescription, too. Finally, it is suggested that all the previously obtained results should be valid dropping the requirements of a massless field and of a large mass black hole.