[Paper Review] Spacetime near Kerr isolated horizon

Jerzy Lewandowski, Carmen Li|arXiv (Cornell University)|Sep 13, 2018

Black Holes and Theoretical Physics9 references4 citations

TL;DR

This paper presents a systematic method to compute the spacetime metric near a Kerr isolated horizon in Bondi-like coordinates, using intrinsic horizon data—specifically the connection and Newman-Penrose spin coefficient π—under the assumption that horizon symmetry extends into the bulk. The key contribution is the explicit derivation of the Kerr-dS horizon metric up to first order in radial expansion, with the pure Kerr case computed to first order, providing a foundational tool for quasi-local black hole studies in general relativity.

ABSTRACT

The theory of isolated horizon provides a quasi-local framework to study the spacetime geometry in the neighbourhood of the horizon of a black hole in equilibrium without any reference to structures far away from the horizon. While the geometric properties of the Kerr-(A)dS and more general algebraically special solutions have drawn substantial interest recently in the isolated horizon formalism, their horizon metrics have never been written down explicitly in the adapted Bondi-like coordinate system. Following the approach by Krishnan and assuming that the horizon symmetry extends to certain order in the bulk, we present in this note a general method to compute the metric functions order by order radially in Bondi-like coordinates in 4-dimensions from a small set of intrinsic data -- the connection and the Newman-Penrose spin coefficient $π$ specified on the horizon cross-section. Applying this general method, we then present the horizon metric of non-extremal Kerr-dS in Bondi-like coordinates. For the pure Kerr case without a cosmological constant, we also show explicitly the metric functions to the first order.

Motivation & Objective

• To develop a systematic, order-by-order method for computing the spacetime metric near a Kerr isolated horizon in Bondi-like coordinates.
• To reconstruct the near-horizon geometry of non-extremal Kerr-dS and pure Kerr black holes using only intrinsic horizon data: the connection and Newman-Penrose spin coefficient π.
• To provide explicit expressions for the metric functions up to first order in radial distance from the horizon, enabling local analysis without global spacetime assumptions.
• To extend the isolated horizon framework to dynamical and cosmological settings, such as Kerr-dS, by adapting the Newman-Penrose formalism to Gaussian null coordinates.

Proposed method

• The method uses the Newman-Penrose formalism in 4-dimensional spacetime with signature (−, +, +, +), leveraging the characteristic initial value problem to reconstruct the spacetime geometry from data on the horizon cross-section.
• It assumes that the horizon symmetry (generated by a null vector field) extends into the bulk up to a certain order in the radial coordinate r, enabling recursive computation of metric functions.
• The initial data consist of the induced metric on the 2-sphere cross-section and the Newman-Penrose spin coefficient π, both specified on the horizon.
• The method applies the Newman-Penrose equations and Bianchi identities order by order in r to determine the metric components gvv, gvθ, gvφ, gθθ, gφφ, and gθφ in Bondi-like coordinates.
• The analysis is restricted to non-extremal black holes due to singularities in the extremal limit; the method breaks down at extremality.
• For Kerr-dS, the method incorporates the cosmological constant via the parameter g = √(3/Λ), with small rotation and cosmological constant assumed (a ≪ r+, g ≪ r+).

Experimental results

Research questions

• RQ1 Can the spacetime metric near a Kerr isolated horizon be systematically reconstructed in Bondi-like coordinates using only intrinsic horizon data?
• RQ2 How can the Newman-Penrose formalism be adapted to compute metric functions order by order in the radial direction from horizon data?
• RQ3 What is the explicit form of the horizon metric for non-extremal Kerr-dS in Bondi-like coordinates?
• RQ4 To what extent can the isolated horizon framework be extended to include cosmological constants and non-vacuum solutions?
• RQ5 Why does the method fail for extremal black holes, and how does this relate to the near-horizon geometry?

Key findings

• The paper derives the first explicit expression for the Kerr-dS horizon metric in Bondi-like coordinates, valid up to first order in radial distance r from the horizon.
• For the pure Kerr case, the metric functions gvv, gvθ, gvφ, gθθ, gφφ, and gθφ are computed explicitly to first order in r, with all components expressed in terms of horizon data: r+, a, and θ.
• The O(r) term in gvv vanishes in the extremal limit (a → r+), consistent with the vanishing surface gravity, but the full extremal limit is singular due to a 1/(a² − r²₊) divergence.
• The metric components depend explicitly on the Newman-Penrose spin coefficient π, which is derived from the rotation 1-form ω(ℓ) on the horizon, confirming its role as a fundamental geometric input.
• The method successfully reconstructs the horizon geometry using only the horizon cross-section data: the induced metric and π, without requiring knowledge of the full spacetime or asymptotic structure.
• The results demonstrate that the isolated horizon framework, when combined with the Newman-Penrose formalism and Bondi-like coordinates, enables a quasi-local reconstruction of black hole spacetimes in a neighborhood of the horizon.