Dynamical axisymmetric compact objects in General Relativity

Technical Summary: Dynamical Axisymmetric Compact Objects in General Relativity

Problem Statement
The search for exact solutions describing asymptotically Friedmann-Lemaître-Robertson-Walker (FLRW) compact objects within General Relativity remains a significant challenge. While progress has been made in the spherically symmetric case (e.g., McVittie, Thakurta, and Fonarev solutions), the field is limited by the scarcity of exact solutions for axisymmetric compact objects embedded in cosmological backgrounds. A primary difficulty lies in the inherent dynamism of these spacetimes, which lack a timelike Killing vector, rendering standard horizon definitions (based on Killing horizons) inapplicable. Furthermore, existing methods to generate such solutions often rely on conformal rescaling of vacuum solutions, which can introduce pathologies in the matter sector (e.g., violations of energy conditions) or fail to solve the dynamics self-consistently. Additionally, analyzing the causal structure of these dynamical geometries is nontrivial without a generalized tool to identify trapping horizons beyond spherical symmetry.

Methodology
The authors employ a two-pronged approach combining solution-generating techniques with geometric analysis tools:

1. Solution-Generating Technique: The paper develops a new method to construct exact, non-stationary, axisymmetric solutions for the self-interacting Einstein-scalar system. This technique synthesizes two existing frameworks:

   • Buchdahl's Theorem: Used to promote a static vacuum solution to a static, axisymmetric solution with a scalar field (hair).
   • Fonarev's Method: Used to introduce time-dependence and a self-interacting Liouville potential ($V(\phi) \propto e^{\xi_3 \phi}$) to a static seed.
   • Extension: The authors prove an Extended Fonarev Theorem, demonstrating that these methods can be combined. Starting from a static, axisymmetric vacuum seed (satisfying specific ignorable coordinate conditions), one can generate a time-dependent, axisymmetric solution with a scalar field and a specific potential. The construction involves a conformal rescaling of the metric by a function of the ignorable coordinate and a corresponding shift in the scalar field profile.

2. Geometric Analysis (Mean Curvature Vector): To analyze the causal structure and locate horizons in the absence of spherical symmetry, the authors utilize the Mean Curvature Vector (MCV) (and its dual), originally introduced by Anco.

   • The MCV generalizes the Kodama vector (which exists only in spherical symmetry) to arbitrary spacetimes.
   • The norm of the MCV (or its dual) provides a foliation-independent (though embedding-dependent) quantity. Its vanishing identifies marginally trapped or anti-trapped surfaces.
   • The authors outline how the sign of the MCV norm distinguishes between untrapped, trapped, and anti-trapped regions, and how Lie derivatives of expansions along null directions characterize the nature of the horizon (e.g., black hole vs. cosmological).

Key Contributions and Results

1. First Exact Axisymmetric Dynamical Solution:
   The authors construct the first exact, non-vacuum, asymptotically FLRW, axisymmetric solution of the Einstein-scalar system with a compact source.

   • Seed: The static seed is the Zipoy-Voorhees (ZV) metric (also known as the $\gamma$-metric), a vacuum solution representing a deformed Schwarzschild geometry with a quadrupole moment.
   • Resulting Solution: Applying the Extended Fonarev theorem yields a time-dependent metric (Eq. 4.4) and a scalar field (Eq. 4.5). The solution is characterized by mass $M$, scalar charge parameters ($\beta, \delta$), and a time-scaling parameter $C$.
   • Asymptotics: At large radii, the solution reduces to a flat FLRW metric filled with a homogeneous, time-dependent scalar field, confirming its cosmological embedding.

2. Horizon Analysis via MCV:
   The paper applies the MCV framework to the new ZV-FJNW solution to identify trapping horizons.

   • Embedding Dependence: Unlike in spherical symmetry, where the horizon is naturally defined by $r = \text{const}$, the axisymmetric nature of the ZV metric causes the norm of the MCV to depend on the polar angle $\theta$. Consequently, the condition for a marginally trapped surface ($H^2 = 0$) cannot be satisfied globally by a simple $r = \text{const}$ embedding.
   • Analytic Limitations: The authors demonstrate that while the MCV successfully identifies the presence of marginally trapped behavior locally, a full characterization of the horizon structure requires a deformed embedding $r = r(\theta)$. This necessitates numerical methods for a complete analytic description, which the authors note is beyond the scope of the current work but outlines the necessary procedure.
   • Special Cases: The paper notes that for certain geometries where the angular dependence factorizes (e.g., specific magnetized extensions), the trivial embedding remains valid, allowing for fully analytic horizon analysis.

Significance and Claims
The paper claims to advance the description of dynamical compact objects in cosmology in two specific ways:

• Framework for Axisymmetric Dynamics: It provides a rigorous solution-generating framework that moves beyond spherical symmetry, enabling the construction of exact dynamical black (and white) hole solutions with scalar hair in an FLRW background. This addresses the limitation of previous works that were largely confined to spherical seeds or pathologically constructed conformal rescalings.
• Generalized Horizon Tools: It advocates for the use of the Mean Curvature Vector as the natural extension of the Kodama vector for axisymmetric spacetimes. The work clarifies that while the MCV offers a coordinate-independent method to identify trapped regions, its application to generic axisymmetric dynamical spacetimes introduces an embedding dependence that must be handled carefully, often requiring numerical intervention to resolve the horizon geometry fully.

The authors conclude that their results offer a framework for further investigating dynamical axisymmetric compact objects, including scenarios involving dynamical scalar accretion and the study of primordial black holes (PBHs) in cosmological settings. They explicitly state that the physical relevance of their specific solution (regarding the formation of a black hole versus a transient trapped region) requires further numerical investigation to determine the precise horizon trajectories and their evolution over time.