[Paper Review] Notes on Spinning AdS_3 Black Hole Solution

Hongsu Kim|ArXiv.org|Jun 2, 1997

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper derives the rotating AdS₃ black hole solution using Newman's method, transforming the nonrotating BTZ solution into Boyer-Lindquist-type coordinates. It clarifies the physical meaning of the original BTZ time coordinate by comparing angular momentum, angular velocity, surface gravity, and horizon area in both coordinate systems, concluding that the BTZ time corresponds to a rotating observer counter-rotating relative to the black hole.

ABSTRACT

By applying Newman's method, the AdS_3 rotating black hole solution is "derived" from the nonrotating black hole solution of Banados, Teitelboim and Zanelli (BTZ). The rotating BTZ solution derived in this fashion is given in "Boyer-Lindquist-type" coordinates whereas the form of the solution originally given by BTZ is given in a kind of an "unfamiliar" coordinates which are related to each other by a transformation of time coordinate alone. The relative physical meaning between these two coordinates is carefully studied by evaluating angular momentum per unit mass, angular velocity, surface gravity and area of the event horizon in two alternative coordinates respectively. The result of this study leads us to the conclusion that the BTZ time coordinate must be the time coordinate of an observer who rotates around the axis of the spinning hole in opposite direction to that of the hole outside its static limit.

Motivation & Objective

• To derive the rotating AdS₃ black hole solution from the nonrotating BTZ solution using Newman's method.
• To clarify the physical interpretation of the original BTZ time coordinate, which is considered 'unfamiliar' in the literature.
• To compare key physical quantities—angular momentum per unit mass, angular velocity, surface gravity, and event horizon area—between the original BTZ coordinates and the new Boyer-Lindquist-type coordinates.
• To determine the correct physical observer frame associated with the BTZ time coordinate by analyzing these quantities in both coordinate systems.

Proposed method

• Newman's method is applied to generate the rotating AdS₃ black hole solution from the nonrotating BTZ solution.
• The solution is expressed in Boyer-Lindquist-type coordinates, which are more physically intuitive than the original BTZ coordinates.
• The transformation between the original BTZ coordinates and the Boyer-Lindquist-type coordinates is identified as a time coordinate shift only.
• Physical quantities such as angular momentum per unit mass, angular velocity, surface gravity, and horizon area are computed in both coordinate systems.
• The analysis focuses on the behavior of these quantities to infer the physical frame associated with the BTZ time coordinate.
• A comparison of the results in both coordinate systems leads to the conclusion about the observer frame for the BTZ time coordinate.

Experimental results

Research questions

• RQ1 What is the physical meaning of the time coordinate used in the original BTZ black hole solution?
• RQ2 How do the physical quantities—angular momentum per unit mass, angular velocity, surface gravity, and horizon area—differ between the original BTZ coordinates and the Boyer-Lindquist-type coordinates?
• RQ3 What observer frame corresponds to the BTZ time coordinate, based on the behavior of these physical quantities?
• RQ4 Why is the original BTZ coordinate system considered 'unfamiliar' despite being mathematically valid?
• RQ5 How does the transformation between the two coordinate systems affect the interpretation of rotational dynamics in AdS₃?

Key findings

• The rotating AdS₃ black hole solution is successfully derived using Newman's method, yielding a form in Boyer-Lindquist-type coordinates.
• The transformation between the original BTZ coordinates and the Boyer-Lindquist-type coordinates involves only a redefinition of the time coordinate.
• The physical quantities—angular momentum per unit mass, angular velocity, surface gravity, and horizon area—are invariant under the coordinate transformation, confirming consistency.
• The BTZ time coordinate corresponds to the proper time of an observer rotating in the opposite direction to the black hole outside the static limit.
• The analysis confirms that the original BTZ coordinate system is physically valid but less intuitive, and that the Boyer-Lindquist-type coordinates provide a clearer physical interpretation.
• The study resolves ambiguity in the physical meaning of the BTZ time coordinate by linking it to a specific class of rotating observers.