[Paper Review] The characteristic initial-boundary value problem for the Einstein--massless Vlasov system in spherical symmetry

Georgios Moschidis|arXiv (Cornell University)|Dec 11, 2018

Cosmology and Gravitation Theories6 references20 citations

TL;DR

This paper establishes the well-posedness of the characteristic initial-boundary value problem for the spherically symmetric Einstein–massless Vlasov system in asymptotically Anti-de Sitter spacetime with a reflecting boundary condition at conformal infinity. It proves existence, uniqueness, and continuation criteria for smooth solutions under smallness of $2m/r$, and demonstrates Cauchy stability of AdS spacetime under small perturbations in a low regularity, scale-invariant norm, implying long-time existence for such data.

ABSTRACT

In this paper, we initiate the study of the asymptotically AdS initial-boundary value problem for the Einstein-massless Vlasov system with $Λ<0$ in spherical symmetry. We will establish the existence and uniqueness of a maximal future development for the characteristic initial-boundary value problem in the case when smooth initial data are prescribed on a future light cone $\mathcal{C}^{+}$ emanating from a point at $\{r=0\}$ and a reflecting boundary condition is imposed on conformal infinity $\mathcal{I}$. We will then prove a number of continuation criteria for smooth solutions of the spherically symmetric Einstein-massless Vlasov system, under the condition that the ratio $2m/r$ remains small in a neighborhood of $\{r=0\}$. Finally, we will establish a Cauchy stability statement for Anti-de Sitter spacetime as a solution of the spherically symmetric Einstein-massless Vlasov system under initial perturbations which are small only with respect to a low regularity, scale invariant norm $||\cdot||$. This result will imply, in particular, a long time of existence statement for $||\cdot||$-small initial data. This paper provides the necessary tools for addressing the AdS instability conjecture in the setting of the spherically symmetric Einstein--massless Vlasov system, a task which is carried out in our companion paper. However, the results of this paper are also of independent interest.

Motivation & Objective

• To establish the existence and uniqueness of a maximal future development for the characteristic initial-boundary value problem in spherically symmetric, asymptotically AdS spacetimes with $\Lambda < 0$.
• To derive continuation criteria for smooth solutions under the assumption that $2m/r \ll 1$ near $r=0$, ensuring regularity and extendability.
• To prove a Cauchy stability statement for Anti-de Sitter spacetime under small initial perturbations measured in a low regularity, scale-invariant norm $||\cdot||$, implying long-time existence.
• To provide foundational tools for addressing the AdS instability conjecture in the context of the spherically symmetric Einstein–massless Vlasov system.
• To analyze the behavior of solutions near $r=0$ and at conformal infinity $\mathcal{I}$, particularly under reflecting boundary conditions.

Proposed method

• The study employs double null coordinates to describe spherically symmetric spacetimes and formulates the Einstein–massless Vlasov system in this gauge, enabling a systematic analysis of the evolution equations.
• Smooth initial data are prescribed on a future light cone $\mathcal{C}^+$ emanating from $r=0$, and a reflecting boundary condition is imposed on $\mathcal{I}$, the conformal boundary at infinity.
• The existence and uniqueness of the maximal future development are established via local well-posedness results for characteristic initial data sets, leveraging constraint propagation and gauge normalization.
• A general extension principle is developed for solutions in the domain of outer communication, relying on bounds on the mass-to-radius ratio $2m/r$ and the behavior of the metric components.
• The paper introduces a low regularity, scale-invariant norm $||\cdot||$ to measure initial data, enabling the proof of Cauchy stability for AdS spacetime under such perturbations.
• The analysis includes detailed estimates on the geodesic flow and the behavior of $\partial_v r$, $\partial_u r$, and the mass function $m$, particularly near $\mathcal{I}$ and $r=0$, using integral inequalities and asymptotic expansions.

Experimental results

Research questions

• RQ1 Under what conditions does the characteristic initial-boundary value problem for the spherically symmetric Einstein–massless Vlasov system admit a unique maximal future development in asymptotically AdS spacetime with $\Lambda < 0$?
• RQ2 What continuation criteria ensure the smooth extendability of solutions when $2m/r$ remains small near $r=0$?
• RQ3 Can Cauchy stability of Anti-de Sitter spacetime be established under initial perturbations that are small only in a low regularity, scale-invariant norm $||\cdot||$?
• RQ4 How does the reflecting boundary condition on conformal infinity $\mathcal{I}$ affect the global existence and regularity of solutions?
• RQ5 What is the behavior of the ratio $2m/r$ near the boundary $\mathcal{I}$, and how does it constrain the maximal development of solutions?

Key findings

• The characteristic initial-boundary value problem for the spherically symmetric Einstein–massless Vlasov system is well-posed in asymptotically AdS spacetime with a reflecting boundary condition at $\mathcal{I}$, ensuring existence and uniqueness of a maximal future development.
• Solutions can be smoothly extended across $r=0$ when $2m/r \ll 1$ in a neighborhood of the center, provided the initial data are smooth and satisfy the required compatibility conditions.
• A general extension principle is established for solutions in the domain of outer communication, relying on the boundedness and monotonicity of $\partial_v r$ and the behavior of $m/r$.
• The paper proves that $\lim_{n\to\infty} \frac{2m}{r}(u_n,v_n) = 1$ along a sequence approaching $\mathcal{I}$, indicating the formation of a trapped surface or curvature singularity at the boundary.
• Cauchy stability of AdS spacetime is established under $||\cdot||$-small initial perturbations, implying a long-time existence statement for such solutions.
• The analysis confirms that $\partial_v r < 0$ along $u = u_*$ for $v \geq v_*$, which implies that $\{u = u_*\} \cap \mathcal{I} = \emptyset$, ensuring that $u_* < u_{\mathcal{I}}$, and thus the solution remains regular up to $\mathcal{I}$ as long as $2m/r < 1$.