[Paper Review] On the metric bundles of axially symmetric spacetimes

D. Pugliese, Hernando Quevedo|arXiv (Cornell University)|Oct 11, 2019

Astrophysical Phenomena and Observations15 references6 citations

TL;DR

This paper introduces metric bundles—families of axially symmetric spacetimes parameterized by a characteristic photon orbital frequency—as a geometric framework to unify black hole and naked singularity physics. By analyzing the extended plane of metric parameters, the authors show that metric bundles are tangent to the horizon curve, revealing connections between black holes and naked singularities through limiting frequencies, with Killing throats and bottlenecks emerging as key structures that signal horizon remnants and pre-horizon regimes.

ABSTRACT

We present the definition of metric bundles in axially symmetric geometries and give explicit examples for solutions of Einstein equations. These structures have been introduced in Pugliese and Quevedo (2019) to explain some properties of black holes (BHs) and naked singularities (NSs), investigated through the analysis of the limiting frequencies of stationary observers, which are at the base of a Killing horizon definition for these black hole spacetimes. In Pugliese and Quevedo (2019), we introduced the concept of NS Killing throats and bottlenecks associated to, and explained by, the metric bundles. In particular, we proved that the horizon frequency can point out a connection between BHs and NSs. We detail this definition in general and review some essential properties of metric bundles as seen in different frames and exact solutions.

Motivation & Objective

• To define and formalize the concept of metric bundles in axially symmetric spacetimes, particularly for solutions of Einstein's equations.
• To explore the geometric and physical connections between black holes and naked singularities through the lens of characteristic photon frequencies.
• To establish a link between metric bundles and Killing horizons, including the emergence of Killing throats and bottlenecks as horizon remnants.
• To analyze the role of metric bundles in black hole thermodynamics and the pre-horizon regime, particularly in Kerr, Kerr-Newman, Reissner-Nordström, and Kerr-de Sitter spacetimes.
• To generalize the framework to include cosmological constants and extend the analysis beyond the equatorial plane.

Proposed method

• Define metric bundles as one-parameter families of spacetimes tied to a fixed characteristic photon orbital frequency $\omega$, with parameters related through a specific functional relation.
• Construct the extended plane as a 2D (or 3D) parameter space of metric parameters (e.g., $a/M$, $r/M$, $\Lambda$) to visualize families of solutions.
• Use the Boyer-Lindquist coordinates to express the Kerr, Kerr-Newman, Reissner-Nordström, and Kerr-de Sitter metrics and derive their metric bundle forms.
• Identify the horizon curve in the extended plane as the envelope of all metric bundles, with bundles tangent to it at specific points.
• Analyze the radii $r_s^{\pm}(\omega, a)$ of light surfaces to define Killing throats and bottlenecks in the $r$-$\omega$ plane.
• Introduce the concept of horizon replicas—points on a bundle where the bundle frequency matches the horizon frequency at $r_{\pm}(a_p)$—to link black hole and naked singularity geometries.

Experimental results

Research questions

• RQ1 How can metric bundles be defined in axially symmetric spacetimes to unify the description of black holes and naked singularities?
• RQ2 What is the geometric relationship between metric bundles and the horizon curve in the extended parameter plane?
• RQ3 How do Killing throats and bottlenecks emerge from the analysis of light surface radii, and what do they signify in terms of horizon remnants?
• RQ4 In what way do metric bundles reveal connections between extreme black holes and weak naked singularities through frequency matching?
• RQ5 How do the properties of metric bundles in the Kerr-de Sitter spacetime differ due to the presence of a positive cosmological constant?

Key findings

• Metric bundles are tangent to the horizon curve in the extended plane, with the horizon curve emerging as the envelope of all such bundles.
• For weak naked singularities (WNSs), the spin-mass ratio $a/M$ close to the extreme black hole value $a/M=1$ corresponds to a portion of the inner horizon, while strong naked singularities (SNSs) with $a>2M$ are associated with the outer horizon.
• Killing bottlenecks—localized narrowing of Killing throats—occur in the extreme Kerr spacetime, where the throat closes on the horizon, indicating a remnant of the horizon structure.
• The concept of horizon replicas arises when a bundle's frequency $\omega_b(a)$ matches the horizon frequency $\omega_H^\pm(a_p)$ at $r_{\pm}(a_p)$, indicating a geometric correspondence between black hole and naked singularity solutions.
• In the Kerr-de Sitter spacetime, the metric bundle is explicitly derived in terms of $\Lambda$, $\omega$, $a$, and $r$, showing complex horizon structures influenced by the cosmological constant.
• The analysis reveals that characteristic frequencies in metric bundles influence exterior properties of black holes and provide a new interpretation of Killing horizons as collective features of solution families.