Extensions of spacetime Bartnik data and estimates for the Bartnik mass outside of time-symmetry

1. Problem Statement

The paper addresses the challenge of estimating the Bartnik mass (a quasi-local mass definition) for initial data sets that are not time-symmetric.

• Context: The Bartnik mass of a boundary $\Sigma$ is defined as the infimum of the ADM mass over all admissible asymptotically flat extensions of the boundary data that satisfy the Dominant Energy Condition (DEC) and contain no horizons enclosing $\Sigma$.
• The Gap: While significant progress has been made in the time-symmetric case (where the second fundamental form $K \equiv 0$), the general non-time-symmetric case (where $K \neq 0$) remains largely unexplored. In the general case, the boundary data includes not just the induced metric $\gamma$ and mean curvature $H$, but also the trace of $K$ ($P = \text{tr}_\Sigma K$) and a 1-form $\omega_\perp = K(\nu, \cdot)$.
• Goal: The authors aim to construct explicit admissible extensions for Bartnik data $(\Sigma, \gamma, H, P, \omega_\perp)$ where $P \neq 0$ and $\omega_\perp = 0$, thereby providing upper bounds for the Bartnik mass in the non-time-symmetric regime.

2. Methodology

The authors employ a constructive approach involving the creation of "collar" regions and gluing them to exterior models.

A. The Model Class of Initial Data

They utilize a specific class of initial data sets modeled on spherically symmetric graphs in Schwarzschild spacetimes.

• Structure: The manifold is $M = (a, b) \times S^2$ with a metric $g = A^2 ds^2 + f(s)^2 \gamma(s)$ and a second fundamental form $K$ determined by a smooth function $x(r)$.
• Key Property: This ansatz ensures the momentum constraint is satisfied identically ($J=0$) and allows for explicit control over the energy density $\mu$.
• Dominant Energy Condition (DEC): They derive precise inequalities involving the scalar curvature of the metric path, the function $x(r)$, and its derivatives to ensure $\mu \geq |J|$.

B. The "Collar" Construction

The core of the construction is a "collar" extension on a cylinder $M = [0, 1] \times S^2$ that:

1. Deforms the Metric: Smoothly interpolates the given boundary metric $\gamma_o$ to a round metric (using a path of metrics $\gamma(s)$).
2. Matches Boundary Data: Prescribes the boundary values for $H$ and $P$ at $s=0$.
3. Satisfies DEC: Ensures the strict Dominant Energy Condition holds throughout the collar.
4. Avoids Trapped Surfaces: Ensures no weakly trapped surfaces (MOTS/MITS) exist within the extension.

C. Gluing and Bending Lemmas

To connect the collar to an exterior solution:

• Gluing Lemma: A modification of existing lemmas (Mantoulidis-Schoen, etc.) allows gluing two spherically symmetric initial data sets while preserving the DEC, provided specific conditions on the derivatives of the radial function and the monotonicity of a function $G_x(r) = x^2 + 2xx'r$ are met.
• Bending Lemma: Used to perturb the energy density in a small annular region to ensure strict positivity ($\mu > 0$) where necessary for the gluing process.

D. Two Approaches to Estimation

The paper presents two distinct methods to derive mass estimates:

1. Direct Construction (Theorem 4.12): Constructing a non-time-symmetric extension directly into a Schwarzschild spacetime of mass $m$. This yields an upper bound based on the Hawking mass of the boundary and the geometry of the metric path.
2. Reduction to Time-Symmetry (Theorem 6.1): Constructing a deformation on a cylinder that connects the non-time-symmetric data $(\gamma, H, P)$ to a time-symmetric data set $(\tilde{\gamma}, \tilde{H}, 0)$. This allows the authors to leverage existing time-symmetric estimates (e.g., from Mantoulidis-Schoen) to bound the non-time-symmetric mass.

3. Key Contributions and Results

A. Existence of Extensions (Theorem A / Corollary 4.17)

The authors prove that for Bartnik data $(\Sigma, \gamma, H_o, P_o, 0)$ with constant $H_o, P_o$ and strictly positive scalar curvature, an admissible extension exists if the data satisfies specific inequalities involving the area radius $r_o$ and geometric constants $\alpha, \beta$ (which measure how close $\gamma$ is to a round metric).

• Condition: $H_o^2 < \frac{4}{r_o^2} \frac{\beta}{\alpha}$.
• Result: There exists an extension satisfying the DEC with no trapped surfaces.

B. Upper Bound Estimates (Theorem B)

Using the direct construction with a specific choice of the function $x(r) = B/\sqrt{r}$, they derive an explicit upper bound for the Bartnik mass:
$$m_B \leq m_H + \frac{H_o r_o^2 \sqrt{\alpha}(4 - H_o^2 r_o^2)}{8\sqrt{4\beta - (\alpha+1)H_o^2 r_o^2}}$$
where $m_H$ is the Hawking mass of the boundary. This bound reduces to known time-symmetric results when $P_o = 0$.

C. Reduction Method (Theorem C / Corollary 6.2)

By deforming the data to a time-symmetric state, they provide an alternative estimate:
$$m_B \leq \left( 1 + \sqrt{\frac{\alpha r_o^2 H_o^2}{4\beta - (1+\alpha)r_o^2 H_o^2}} \right) m_H$$
This result relies on the condition $R_\gamma > \frac{3}{2}H_o^2$ (where $H_o = \sqrt{H_o^2 - P_o^2}$ is the spacetime mean curvature).

D. Handling of $\omega_\perp$

The authors explicitly restrict their analysis to the case $\omega_\perp \equiv 0$. They note that while the ansatz can be modified to include non-zero $\omega_\perp$, the resulting expressions become cumbersome, and they leave this for future work. However, they cite Lin's thesis [8] as a precedent for handling Marginally Outer Trapped Surfaces (MOTS) where $\omega_\perp$ can be managed.

4. Significance

• Breaking Time-Symmetry: This is one of the first works to provide concrete existence results and mass estimates for Bartnik data where the extrinsic curvature $K$ is non-zero (specifically $P \neq 0$).
• Physical Relevance: By relaxing the time-symmetry assumption, the results apply to more physically realistic scenarios involving momentum and non-static gravitational fields.
• Technical Robustness: The paper successfully extends the "collar" gluing technique to the full Einstein constraint equations (including the momentum constraint and DEC), not just the scalar curvature constraint.
• Connection to Penrose Inequality: The work aligns with recent efforts to understand the Penrose inequality in the non-time-symmetric setting, offering a pathway to verify if the Bartnik mass of a MOTS equals its Hawking mass (as conjectured).

In summary, McCormick and Wolff provide a rigorous framework for extending non-time-symmetric Bartnik data, establishing the existence of valid spacetime extensions, and deriving concrete upper bounds for the quasi-local mass, thereby advancing the understanding of mass in general relativity beyond the static limit.