Conformal symmetries and MOTS stability

Abstract: Let ${Σ_t}$ be a spacelike foliation of a spacetime $\mathcal{M}$, and suppose each $Σ_t$ is foliated by 2-surfaces $\mathcal{S}$ with spacelike unit normal in $Σ_t$. We show that under mild energy conditions, a MOTS $\mathcal{S}$ that intersects integral curves of past-pointing conformal Killing vector field lying in the normal space of $\mathcal{S}$ is strictly stable and evolves smoothly to a spacelike horizon. We also show that if the restriction of the divergence of the vector field to $\mathcal{S}$ is non-negative, $\mathcal{S}$ is unstable, and if negative and $\mathcal{S}$ is a 2-sphere, $\mathcal{S}$ must be strictly stable.

• The paper establishes rigorous theorems linking conformal Killing vectors with the stability of marginally outer trapped surfaces under the null energy condition.
• It derives conditions using elliptic operators and principal eigenvalues to classify MOTS as strictly stable, marginally stable, or unstable.
• The study introduces a projection formalism for CKVs onto the MOTS normal bundle, elucidating the transition from trapped surfaces to smooth horizons.

Conformal Symmetries and the Stability of Marginally Outer Trapped Surfaces (MOTS)

Overview

This paper provides a comprehensive analysis of the interplay between conformal symmetries and the stability of marginally outer trapped surfaces (MOTS) in a general relativistic spacetime. The work refines and extends the theory of MOTS by incorporating the presence of conformal Killing vector fields (CKVs) and explores conditions under which MOTS evolve smoothly into horizons or exhibit instability, with a focus on the behavior dictated by the associated conformal structure.

MOTS Geometry and Stability

A MOTS is defined by the vanishing of the outgoing null expansion $\theta_k$ on a spacelike 2-surface $\mathcal{S}$, embedded within a spacetime foliation. The stability of a MOTS is governed by a second-order elliptic operator acting on functions on $\mathcal{S}$, introduced originally by Andersson, Mars, and Simon. The principal eigenvalue $\lambda_0$ of this operator determines whether the surface is strictly stable ($\lambda_0 > 0$), marginally stable ($\lambda_0 = 0$), or unstable ($\lambda_0 < 0$). Strict stability ensures that the MOTS evolves into a smooth, generically spacelike, marginally outer trapped tube (MOTT) under suitable energy conditions, particularly the null energy condition (NEC).

Conformal Symmetries and Their Induced Constraints

The interaction between conformal symmetries and MOTS stability is formalized via the conformal Killing equation (CKE)

$\mathcal{L}_\eta g_{ab} = 2\varphi g_{ab}$

where $\eta^a$ is a CKV and $\varphi$ is the conformal divergence, determined by the divergence of $\mathcal{S}$0. A key insight is that when the CKV reduces to a true Killing vector for an associated conformal metric in a spacetime neighborhood, the variations of trap surfaces inherit remarkable properties.

The paper develops a formalism for projecting CKVs onto the normal bundle of $\mathcal{S}$1:

$\mathcal{S}$2

and demonstrates that under certain differential conditions, the CKV generates a true symmetry of the conformal metric, simplifying the analysis of its effects on MOTS.

Main Theoretical Results

Several new theorems are established regarding the existence, stability, and evolution of MOTS in the presence of CKVs:

• Theorem 1: In a spacetime admitting a past-pointing CKV in the normal space of $\mathcal{S}$3, which is a Killing vector for the conformal metric in a neighborhood, any MOTS intersecting integral lines of $\mathcal{S}$4 is strictly stable and evolves smoothly toward a spacelike horizon given the NEC holds strictly somewhere on $\mathcal{S}$5.
• Theorem 2: For a stable MOTS under the above assumptions, the CKV cannot be timelike at any point of intersection; i.e., for a MOTS to remain stable, the CKV must not lie within the interior of the light cone at any point on $\mathcal{S}$6.
• Corollary: If, conversely, the CKV is timelike at any point on $\mathcal{S}$7, the MOTS is necessarily unstable.

Moreover, the analysis draws sharp connections between the sign of the divergence of the CKV restricted to $\mathcal{S}$8 and the surface’s stability: non-negative divergence (i.e., $\mathcal{S}$9) guarantees instability, while negative divergence ensures strict stability for spherical MOTS.

The work further provides a class of explicit spacetimes (e.g., LRS spacetimes with particular metric forms) satisfying the necessary integrability conditions for the reduction of the CKV to a Killing vector for the conformal metric, and explores (with explicit counterexamples) when this structure fails, such as in the de Sitter spacetime with umbilical slicing.

Implications and Extensions

The principal implication is the identification of conformal symmetries as powerful geometric constraints on the existence, stability, and evolution of MOTS. The results synthesize and substantially generalize prior no-go theorems for the presence of MOTS in stationary or symmetric spacetimes by invoking conformal, not merely isometric, symmetries.

The explicit eigenfunction calculation associated with the stability operator in the presence of a CKV illuminates how the null energy condition mediates the transition from instability to stability, and provides a practical mechanism for stability analysis in applications, such as numerical relativity and the analysis of horizon formation in dynamical settings.

This framework may inform future studies of horizon formation, cosmic censorship, and the quasi-local characterization of black hole boundaries, particularly in spacetimes with rich symmetry groups. In addition, by clarifying the connection between local surface geometry and the ambient conformal structure, the results suggest lines of inquiry into the structure of dynamical horizons and the role of almost-symmetric (e.g., almost Killing) fields in near-equilibrium settings.

Conclusion

This work establishes rigorous connections between conformal symmetries and MOTS stability, providing necessary and sufficient conditions for the strict stability, instability, and local evolution of MOTS in terms of the orientation and divergence of CKVs. The results deepen understanding of quasi-local horizons in general relativity, delineate explicit geometric and energy conditions under which MOTS evolve into smooth horizons, and identify configurations precluding stable trapping. These advances offer new tools for analyzing black hole boundary dynamics in highly symmetric or conformally structured spacetimes and set the stage for further investigations into the interplay between conformal geometry and spacetime singularity formation.