For any function \(f(\mathbb S^2)\) on \(\mathscr I^-\) or \(\mathscr I^+\)

1. supermomentum: \(\mathcal Q [f, S] = \int\limits_S f (Re \psi_2 - \tfrac{1}{2} \sigma^{ab} N_{ab}) \)
2. conservation law: flux of charges
   \(\mathcal F[f] = \int f N_{ab} N^{ab} + N_{ab} \mathscr D^a \mathscr D^b f\)

full supertranslation group \(f_- \otimes f_+\)

1. relate supertranslations \(\lim\limits_{\to i^0} f_-(\mathbb S^2) = \lim\limits_{\to i^0} f_+( - \mathbb S^2) \) ?
2. total \(\mathcal F[f_-] = \) total \(\mathcal F[f_+]\)?

Constrains allowed classical scattering

Quantum S-matrix, information loss (?)

at \(i^0\):

• \(\nabla_a\Omega = 0\); \(\nabla_a \nabla_b\Omega = 2 g_{ab}\) (cannot use Bondi frame)
• if \(mass \neq 0\): \(\partial_a g_{bc}\) is direction-dependent

Very low differential structure

\(\mathscr C\): cylinder of directions in \(Ti^0\); conformal to Ashtekar-Hansen hyperboloid

\(\mathscr N^\pm\): spheres of null directions

• direction-dependent “electric” Weyl tensor: \(\mathbf E_{ab}\)
• Einstein equation: equations for \(\mathbf E_{ab}\) on \(\mathscr C\)
• Spi-supertranslations: \(f(\mathscr C)\)
• Spi-supermomenta: \(\mathcal Q_0[f, S] = \int\limits_S \mathbf u^a \mathbf E_{ab} \mathbf D^b f \)

Assume regularity of fields and symmetries at \(\mathscr N^\pm\)

• \(N_{ab} \sim O(1/u^{1+\epsilon})\) (Bondi frame)\(\implies\)
• BMS-supermomentum from \(\mathscr I^\pm\) \(=\) Spi-supermomentum on \(\mathscr C\) at \(\mathscr N^\pm\)

1. Einstein equation \(\implies\) (rescaled) \(\mathbf E_{ab}\) evolves antipodally from \(\mathscr N^-\) to \(\mathscr N^+\)
2. demand total flux \(\mathcal F[f; \mathscr C] = 0 \iff f\vert_{\mathscr N^-} \equiv - f\vert_{\mathscr N^+}\) antipodally

Totally fluxless symmetries on \(\mathscr C\) match antipodally i.e. \(\lim\limits_{\to i^0} f_-(\mathbb S^2) = - \lim\limits_{\to i^0} f_+(- \mathbb S^2) \) and \(\mathcal F[f_-] = \mathcal F[f_+]\)