\((M, g)\) conformal completion of \((\hat M, \hat g)\); \(g_{ab} = \Omega^2 \hat g_{ab}\)

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

1. charges: \(\mathcal Q [\alpha, S] = \int\limits_S \alpha F_{ab} l^a n^b \)
2. conservation law: \(\mathcal Q [\alpha, S_2] - \mathcal Q [\alpha, S_1] = \mathcal F[\alpha]\)

full symmetry group \(\alpha_- \otimes \alpha_+\)

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

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 \(charge \neq 0\): \(\Omega F_{ab}\) is direction-dependent
• if \(mass \neq 0\): \(\partial_a g_{bc}\) is direction-dependent

Very low differential structure

\(\mathscr C\): cylinder of directions in \(Ti^0\)

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

• direction-dependent “electric field”: \(\mathbf E_a\)
• Maxwell equations: equations for \(\mathbf E_a\) on \(\mathscr C\)
• symmetries: \(\alpha(\mathscr C)\)
• charges: \(\mathcal Q_0[\alpha, S] = \int\limits_S \alpha \mathbf E_a \mathbf u^a \)

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

• Limits from \(\mathscr I^{\pm}\) \(=\) limits to \(\mathscr N^\pm\) on \(\mathscr C\)
• \(\implies\) charges from \(\mathscr I^\pm\) \(=\) charges on \(\mathscr C\) at \(\mathscr N^\pm\)

regular symmetry group \(\alpha(\mathscr C)\): \(\alpha_\pm(\mathbb S^2) = \alpha(\mathscr C)\vert_{\mathscr N^\pm}\)

1. Maxwell equations \(\implies\) (rescaled) \(\mathbf E_a \mathbf u^a\) evolves antipodally from \(\mathscr N^-\) to \(\mathscr N^+\)
2. demand total flux \(\mathcal F[\alpha; \mathscr C] = 0 \iff \alpha\vert_{\mathscr N^-} \equiv \alpha\vert_{\mathscr N^+}\) antipodally

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