# Conservation ofsupermomentum from $$\mathscr I^-$$ to $$\mathscr I^+$$

Kartik Prabhu APS 2019, Denver, CO

# Supertranslations and supermomentum on $$\mathscr I$$

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 $$f_- \otimes f_+$$ # Global conservation (?)

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 (?) # $$i^0$$ is a funny place!

at $$i^0$$:

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

Very low differential structure # Blowup $$i^0$$ — space of directions: $$\mathscr C$$

$$\mathscr C$$: cylinder of directions in $$Ti^0$$; 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: $$\mathcal Q_0[f, S] = \int\limits_S \mathbf u^a \mathbf E_{ab} \mathbf D^b f$$ # Null regularity & matching at $$\mathscr N^\pm$$

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 $$\mathscr N^\pm$$ # Totally fluxless supertranslations on $$\mathscr C$$ and conservation

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 $$\mathcal F[f_-] = \mathcal F[f_+]$$ # Details

1. Maxwell case: JHEP 10 113 (, [arXiv:1808.07863]
2. Supermomentum JHEP 03 148 (, [arXiv:1902.08200]