Primordial axisymmetric compact objects in General Relativity

TL;DR

This work develops axisymmetric, dynamical compact objects embedded in a cosmological background within General Relativity. It introduces a novel solution-generating framework that combines Buchdahl’s static-axisymmetric construction with Fonarev’s time-dependent extension to produce exact axisymmetric Einstein–Scalar spacetimes with a Liouville potential, yielding the first dynamical axisymmetric black/white hole solutions in an asymptotically FLRW universe. It further advocates the mean curvature vector as a natural, foliation-independent generalization of the Kodama vector for locating (anti-)trapping horizons beyond spherical symmetry. By applying the method to the Zipoy–Voorhees seed, the paper provides an explicit axisymmetric, time-dependent Zipoy–Voorhees solution, analyzes its asymptotic cosmology and horizon dynamics, and discusses implications for primordial black hole phenomenology and dynamical horizon thermodynamics in non-spherical spacetimes.

Abstract

The search for exact solutions describing asymptotically FLRW compact objects in General Relativity (GR) remains a notoriously challenging problem. To a large extent, progress has been restricted to the spherically symmetric sector, with the exception of the Kerr-de Sitter and Thakhurta solutions. In this work, we present two new results that advance the description of axisymmetric compact objects embedded in a cosmological background. We first introduce a new solution-generating technique allowing one to construct non-stationary and axisymmetric solutions of the self-interacting Einstein-Scalar system. Using this method, we present the first exact solution which describes a dynamical axisymmetric black (or white) hole embedded in an expanding or contracting cosmology. We provide a detailed investigation of its properties, and in particular its dynamical trapping (or anti-trapping) horizons. To that end, we use the mean curvature vector (MCV) which stands as a natural generalization of the Kodama vector beyond spherical symmetry. The norm of this vector provides a foliation-independent quantity to locate the trapped/anti-trapped and untrapped regions and characterize the causal nature of a given geometry without specific symmetry requirement. The solution-generating method and the techniques to analyze the new solutions provide new powerful tools to further explore the description and the phenomenology of dynamical compact objects embedded in cosmology, in particular those of primordial black holes.