# On the evolution of the spacetime Bartnik mass

## TL;DR

This paper investigates the evolution of the spacetime Bartnik mass for surfaces in initial data sets, deriving a formula for its rate of change and showing it is non-decreasing under outward flow, assuming a key conjecture.

## Contribution

It provides a formula for the derivative of the spacetime Bartnik mass along evolving surfaces under the dominant energy condition, supporting its monotonicity in outward flows.

## Key findings

- Derived an expression for the derivative of the Bartnik mass.
- Proved monotonicity of the Bartnik mass under outward flow.
- Supported the conjecture relating Bartnik mass to ADM mass of stationary manifolds.

## Abstract

It is conjectured that the full (spacetime) Bartnik mass of a surface $\Sigma$ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by $\Sigma$. Assuming this holds true for a 1-parameter family of surfaces $\Sigma_t$ evolving in an initial data set {with the dominant energy condition}, we compute an expression for the derivative of the Bartnik mass along these surfaces. An immediate consequence of this formula is that the Bartnik mass of $\Sigma_t$ is monotone non-decreasing whenever $\Sigma_t$ flows outward. It is our pleasure to dedicate this paper to Robert Bartnik on the occasion of his $60$th birthday.