Bondi mass limits and mass loss for boosted Kerr stability

Let $(\gamma,k)$ be the initial data from Theorem~, with ADM mass $\mathfrak m_0$ and vanishing linear momentum, let $g$ be the resulting spacetime metric, and set $\chi=\frac12\|\mathsf S\|_{L^\infty}$. Let $M_B$ denote the Bondi mass defined using the level sets of $t_*$ on $\mathscr I^+$, and let $\slashed\pi_0$ and $\slashed g$ denote the asymptotic radiation field and metric. Bondi mass conjecture. The Bondi mass of $\phi_\mathsf S^*g$ satisfies

$$\lim_{t_*\to-\infty}M_B(\phi_\mathsf S^*g;t_*)=\mathfrak m_0,\qquad \lim_{t_*\to+\infty}M_B(\phi_\mathsf S^*g;t_*)=\mathfrak m\cosh\chi,$$

and obeys

$$\frac{\mathrm d}{\mathrm d u}M_B(\phi_\mathsf S^*g;u)=-\frac{1}{32\pi}\int_{\mathscr I^+\cap\{t_*=u\}}\left|r\partial_{t_*}\slashed\pi_0(\phi_\mathsf S^*h)\right|_{\slashed g}^2\,\mathrm d\slashed g.$$

The conjecture would imply $\mathfrak m\cosh\chi\leq\mathfrak m_0$, expressing that the final Bondi energy cannot exceed the initial energy after accounting for energy carried away by gravitational radiation. Establishing the formula requires higher-order asymptotic computations than those used for nonlinear stability, and the source leaves it open.