Local Well-Posedness for the Bartnik Stationary Extension Problem near Schwarzschild Spheres

TL;DR

This work advances the Bartnik mass program by establishing local well-posedness for the Bartnik stationary metric extension near Schwarzschild spheres, extending prior static analyses to genuinely stationary settings. By working in a double geodesic gauge, the authors reduce the Einstein vacuum equations to a coupled elliptic–transport system and reveal a decoupling at the linearized level between the metric/potential sector and a boundary-value problem for the twist form θ. They formulate a robust Banach-space framework with Bochner-measurable spaces and solve the θ-boundary value problem via a detailed spherical harmonic analysis, obtaining uniform estimates to apply the implicit function theorem. A key outcome is the existence and uniqueness (up to isometry) of a stationary vacuum extension for Bartnik data near Schwarzschild, with a canonical θ-gauge fixing that persists under Lorentz boosts of the boundary. The results pave the way for further study of stationary Bartnik extensions and their implications for quasi-local mass in general relativity.

Abstract

We investigate the Bartnik stationary extension conjecture, which arises from the definition of the spacetime Bartnik mass for a compact region in a general initial data set satisfying the dominant energy condition. This conjecture posits the existence and uniqueness (up to isometry) of an asymptotically flat stationary vacuum spacetime containing an initial data set $(M, \mathfrak{g}, Π)$ that realizes prescribed Bartnik boundary data on $\partial M$, consisting of the induced metric, mean curvature, and appropriate components of the spacetime extrinsic curvature $Π$. Building on the analytic framework developed in arXiv:2411.02801 for the static case, we show that, in a double geodesic gauge, the stationary vacuum Einstein equations reduce to a coupled system comprising elliptic and transport-type equations, with the genuinely stationary contributions encoded in an additional boundary value problem for a $1$-form $θ$. We establish local well-posedness for the Bartnik stationary metric extension problem for Bartnik data sufficiently close to that of any coordinate sphere in any initial data set (possibly non time-symmetric) in Schwarzschild spacetime. This includes spheres arbitrary close to the apparent horizon in the initial data set. A key feature of our framework is that the linearized equations decouple: the equations for the metric and potential reduce to the previously solved static case, while the boundary value problem for $θ$ is treated independently. We prove solvability of this boundary value problem in the Bochner-measurable function spaces adapted to the coupled system developed in arXiv:2411.02801, establishing uniform estimates for the vector spherical harmonic decomposition of $θ$.

Figures (2)

• Figure 1: An illustration of an initial data set $(M,\mathfrak{g}_f, \Pi_f)$ in the spacetime defined by $(M,g,u,\theta)^{(4)}$ is equivalent to the $\{t=0\}$ initial data set in the spacetime defined by $(M,g,u,\theta+df)^{(4)}$.
• Figure 2: Illustration of the gauge freedom for initial data sets in a stationary spacetime defined by $(M, g, u, \theta)^{(4)}$. Each colored curve $M_{f_i}$ represents a different spacelike hypersurface given by the graph $t = f_i(x)$, where the functions $f_i$ satisfy the boundary conditions $f_i|{\partial M} = 0$, $df_i|{\partial M} = 0$ so that all initial data sets realize the same Bartnik boundary data on $\partial M$. Proposition \ref{['uniquef-prop']} shows that there exists a unique choice of hypersurface—depicted in green, $M_{f_2}$—for which the corresponding $1$-form $\theta + df_2$ satisfies the $\theta$-geodesic gauge condition $\Delta_{sc}((\theta + df_2)(\frac{\partial}{\partial r})) = 0$. This means that for any prescribed Bartnik data, there is a unique representative in the gauge where $\theta$ is determined by this elliptic condition and the corresponding initial data slice is then the ${t=0}$ hypersurface in the isometric stationary spacetime defined by $(M,g,u,\theta +df_2)^{(4)}$. Thus, fixing the $\theta$-gauge is equivalent to selecting a unique initial data slice among all those realizing the same boundary data in the stationary spacetime.

Theorems & Definitions (61)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Definition 2.6
• Lemma 2.7
• proof