# Conservation of

asymptotic charges

from \(\mathscr I^-\) to \(\mathscr I^+\)

APS 2018, Columbus, OH
# Charges and fluxes on \(\mathscr I\)

\((M, g)\) 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 \(\mathscr I^+\)

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

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

# Global conservation (?)

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

Constrains allowed classical scattering

Quantum S-matrix, information loss (?)

# Global conservation (known results)

(not a complete list)- Bondi \(4\)-momentum in full GR. Ashtekar, Magnon-Ashtekar [1979]
- Angular momentum in stationary spacetimes full GR. Ashtekar, Streubel [1979]
*all charges*on Minkowski background: EM Campiglia, Eyheralde [2017] and linearised gravity Troessaert [2017]

all charges on general spacetimes?

# \(i^0\) is a funny place!

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

# Blowup \(i^0\) — space of directions: \(\mathscr C\)

\(\mathscr C\): 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: \(\mathcal Q_0[\alpha, S] = \int\limits_S \alpha \mathbf E_a \mathbf u^a \)

# Null regularity & matching at \(\mathscr N^\pm\)

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

# Totally fluxless symmetries on \(\mathscr C\) and conservation

- Maxwell equations \(\implies\) (rescaled) \(\mathbf E_a \mathbf u^a\) evolves
*antipodally*from \(\mathscr N^-\) to \(\mathscr N^+\) *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 \(\mathcal F[\alpha_-] = \mathcal F[\alpha_+]\)

# Summary

EM on general asymp. flat spacetimes

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

uses

- Blowup of \(i^0\) to cylinder \(\mathscr C\)
- regularity of fields and symmetries in null directions
- Maxwell equations; totally fluxless symmetries