[Paper Review] Hessling's Quantum Equivalence Principle and the Temperature of an Extremal Reissner-Nordström Black Hole

Valter Moretti|ArXiv.org|Oct 10, 1995

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper investigates the application of Hessling's quantum equivalence principle to a massless scalar field outside an extremal Reissner-Nordström black hole. By refining the Haag-Narnhofer-Stein principle, the study shows that only the Reissner-Nordström vacuum state satisfies the quantum equivalence principle, uniquely selecting a temperature of zero, thus resolving the ambiguity in thermal state selection for extremal black holes.

ABSTRACT

The Hessling improvement of the Haag, Narnhofer and Stein principle is analysed in the case of a massless scalar field propagating outside of an extremal R-N black hole. It is found that this sort of ``Quantum (Einstein's) Equivalence Principle'' selects only the R-N vacuum as a physically sensible state, i.e., it selects the temperature $T=0$ only.

Motivation & Objective

• To resolve the ambiguity in selecting a physically sensible quantum state for a massless scalar field near an extremal Reissner-Nordström black hole.
• To investigate whether the quantum equivalence principle, as refined by Hessling, can uniquely select a preferred vacuum state.
• To determine the resulting temperature of the black hole under this selection criterion.
• To clarify the physical consistency of the Reissner-Nordström vacuum in the context of quantum field theory in curved spacetime.

Proposed method

• Application of Hessling's improvement of the Haag-Narnhofer-Stein principle to quantum field theory in the background of an extremal Reissner-Nordström black hole.
• Analysis of the behavior of a massless scalar field propagating in the exterior region of the extremal black hole.
• Use of the quantum equivalence principle to constrain the set of physically acceptable quantum states.
• Identification of the Reissner-Nordström vacuum as the only state satisfying the improved principle.
• Mathematical derivation showing that only the R-N vacuum state is compatible with the quantum equivalence principle under the given conditions.
• Evaluation of the resulting temperature of the black hole as a consequence of the selected vacuum state.

Experimental results

Research questions

• RQ1 Does Hessling's quantum equivalence principle uniquely select a vacuum state for a massless scalar field in the exterior of an extremal Reissner-Nordström black hole?
• RQ2 Which quantum state(s) satisfy the improved quantum equivalence principle in this spacetime?
• RQ3 What is the resulting temperature of the extremal black hole when only the Reissner-Nordström vacuum is physically allowed?
• RQ4 Can the quantum equivalence principle resolve the ambiguity in thermal state selection for extremal black holes?
• RQ5 Is the Reissner-Nordström vacuum the only physically sensible state under the quantum equivalence principle in this context?

Key findings

• The quantum equivalence principle, as improved by Hessling, uniquely selects the Reissner-Nordström vacuum as the only physically sensible quantum state for a massless scalar field outside an extremal Reissner-Nordström black hole.
• All other candidate vacuum states are ruled out by the requirement of consistency with the quantum equivalence principle.
• The selected vacuum state corresponds to a temperature of zero, implying no thermal radiation from the extremal black hole.
• The result resolves the long-standing ambiguity in assigning a temperature to extremal Reissner-Nordström black holes within quantum field theory.
• The analysis confirms that the extremal black hole cannot emit Hawking radiation if only the R-N vacuum is physically allowed.
• The conclusion is robust under the given framework, as the selection is based on a physically motivated principle rather than arbitrary choice.