[Paper Review] Black holes and solution generating techniques

Adriano Viganò|arXiv (Cornell University)|Nov 1, 2022

Relativity and Gravitational Theory4 citations

TL;DR

This paper presents a comprehensive framework for generating exact solutions in General Relativity using solution-generating techniques, particularly Ernst equations and inverse scattering methods, to model black holes in complex environments such as external gravitational fields, expanding bubbles of nothing, and swirling universes. The key contribution is a systematic construction of multi-black hole, charged, rotating, and accelerated black hole solutions with explicit thermodynamic and geometric analyses.

ABSTRACT

Multi-black hole solutions play a relevant role both from the theoretical and the phenomenological point of view. In this Thesis, we construct some regular multi-black hole spacetimes in pure Einstein's General Relativity with the aid of solution generating techniques. We begin with a perspective on the history of solution generating techniques, and then we explain in detail the Ernst formalism and the inverse scattering method. These are the techniques that are applied in the rest of the Thesis. Subsequently, we construct multi-black hole solutions embedded in an external gravitational field: it is possible to obtain an equilibrium configuration in many interesting cases, like a collection of collinear static black holes or a chain of accelerating black holes, by choosing appropriately the multipole parameters of the field. Then, we consider the expanding bubbles of nothing as a background for multi-black hole and black ring solutions. The expanding behaviour of the bubbles provides the force necessary to balance the gravitational attraction among the black holes, and hence to reach the equilibrium. Finally, we construct a solution that represents a black hole embedded in a "swirling" universe, which describes a spacetime whirlpool. Moreover, we discuss the possibility of implementing the swirling background in order to enforce the spin-spin configuration, and reach an equilibrium configuration in a double-Kerr spacetime.

Motivation & Objective

• To develop a unified framework for constructing exact black hole solutions in General Relativity using advanced solution-generating techniques.
• To investigate equilibrium configurations of multiple black holes under external gravitational fields, avoiding conical singularities via regularized solutions.
• To explore the behavior of black holes inside expanding bubbles of nothing, including transitions from black hole binaries to bubbles in 4D and 5D spacetimes.
• To analyze black hole dynamics in a swirling universe background, including geodesics, ergoregions, and thermodynamic properties.
• To extend known solution-generating methods—such as Ehlers and Harrison transformations—to new physical scenarios involving acceleration, rotation, and external fields.

Proposed method

• Employment of the Ernst formalism for stationary, axisymmetric Einstein-Maxwell systems, using Ernst potentials to encode gravitational and electromagnetic fields.
• Application of the inverse scattering method to construct $n$-soliton solutions, including the Kerr–NUT and rotating C-metric spacetimes.
• Utilization of the Ehlers transformation to generate new solutions from seed metrics, particularly for rotating and charged black holes.
• Implementation of the Harrison transformation to include electromagnetic fields via dual transformations in the Ernst equation framework.
• Use of Weyl metrics and rod structures to analyze the topology and regularity of multi-black hole systems and their transition to Kaluza–Klein bubbles.
• Embedding of seed metrics like Zipoy–Voorhees and Schwarzschild into swirling spacetimes via double-Wick rotation and background field construction.

Experimental results

Research questions

• RQ1 Can multi-black hole systems be constructed in equilibrium without conical singularities using solution-generating techniques?
• RQ2 How do black holes and black rings behave inside an expanding bubble of nothing, and what are the conditions for equilibrium?
• RQ3 What are the thermodynamic and geometric properties of a Schwarzschild or Kerr black hole immersed in a swirling universe background?
• RQ4 How do solution-generating techniques such as Ehlers and Harrison transformations extend to non-vacuum and higher-dimensional configurations?
• RQ5 What is the role of multipole moments and rod structures in classifying and regularizing multi-black hole and black ring solutions?

Key findings

• The paper constructs explicit solutions for binary black holes and black rings in equilibrium inside expanding bubbles of nothing, showing that such configurations are possible under specific topological and multipole conditions.
• The transition from a binary black hole system to a bubble of nothing is achieved via a limiting procedure in Weyl coordinates, with the rod structure indicating a topological change from two horizons to a single expanding bubble.
• In the swirling universe background, the Schwarzschild black hole exhibits ergoregions and Petrov type D geometry, with geodesics showing complex orbital behavior depending on angular momentum and background rotation.
• The Kerr black hole in a swirling universe displays non-trivial frame-dragging effects and modified thermodynamic quantities due to the background's non-vacuum nature.
• The Zipoy–Voorhees metric embedded in the swirling background yields a non-spherically symmetric solution with adjustable multipole moments, useful for modeling astrophysical bodies with deviations from spherical symmetry.
• The use of the inverse scattering method successfully generates $n$-soliton solutions, including the rotating C-metric and multi-black hole configurations, with explicit expressions for metric functions and conserved charges.