# Asymptotic symmetries&chargesat spatial infinity

Kartik Prabhu Brown University, Providence, RI

# Why?

Asymptotic boundaries $$\mathscr I^-$$, $$\mathscr I^+$$ and $$i^0$$

1. $$\mathscr I^\pm$$ have infinite-number of symmetries; BMS group with Lorentz and supertranslations
2. Infinite-number of charges called supermomenta; not conserved but have flux due to radiation

Is there any relation between symmetries and charges on $$\mathscr I^-$$ and $$\mathscr I^+$$?

# Global conservation (?)

1. relate BMS symmetry on $$\mathscr I^-$$ to another one on $$\mathscr I^+$$?
2. incoming flux on $$\mathscr I^-$$ = outgoing flux on $$\mathscr I^+$$?

Infinite-number of global conservation laws; constraints on classical scattering

Quantum S-matrix, soft theorems, information loss (?)

# Global conservation (known results)

(not a complete list)
1. Bondi $$4$$-momentum in full GR. Ashtekar, Magnon-Ashtekar [1979]
2. Angular momentum in stationary spacetimes full GR. Ashtekar, Streubel [1979]
3. Minkowski background: EM Campiglia, Eyheralde [2017], linearised gravity Troessaert [2017], Grgin index (twistor theory literature; very old result)
4. General spacetimes: EM Prabhu [2018], supertranslations Prabhu [2019]

Lorentz charges on general spacetimes?

# How?

1. Solve the scattering problem through the interior of the spacetime
2. Somehow use spatial infinity $$i^0$$

Both methods are do-able in Minkowski spacetime

1. try solve exactly given incoming data (Kirchoff integrals; assuming analyticity)
2. $$i^0$$ is analytic so can “pass through” spatial infinity

# $$i^0$$ is a funny place!

Ashtekar-Hansen structure 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 at $$i^0$$:

• The manifold is $$C^{>1}$$
• Metric $$g_{ab}$$ is $$C^{>0}$$

# Blowup $$i^0$$ — space of directions: $$\mathscr H$$

$$\mathscr H$$: hyperboloid of spatial directions $$\vec\eta$$ in $$Ti^0$$ (a.k.a. deSitter)

• Weyl tensor $$C_{abcd}$$ gives electric and magnetic fields on $$\mathscr H$$:
$$\mathbf E_{ab} = \lim\limits_{\to i^0} \Omega^{1/2}C_{acbd}\eta^c \eta^d$$
$$\mathbf B_{ab} = \lim\limits_{\to i^0} \Omega^{1/2} *C_{acbd}\eta^c \eta^d$$
• Einstein equation:
$$\mathbf D_{[a} \mathbf E_{b]c} = \mathbf D_{[a} \mathbf B_{b]c} = 0$$

# Asymptotic symmetries

diffeos $$\xi^a$$ in the physical spacetime which preserve the asymptotic structure at $$i^0$$

1. preserve $$i^0$$ : $$\xi^a\vert_{i^0} = 0$$
2. preserve $$C^{>1}$$-structure : $$\Omega^{-1/2}\xi^a$$ is direction-dependent
3. preserve the metric and directions: $$(\mathbf f, \mathbf X^a)$$
supertranslation: $$\mathbf f \in C^\infty(\mathscr H)$$
Lorentz: $$\mathbf X^a$$ is a Killing field on $$\mathscr H$$

# Lagrangian and symplectic current

Lagrangian $$L$$ as a $$4$$-form in physical spacetime

• $$\delta L = \mathcal E^{ab} \delta g_{ab} + d \theta(\delta g)$$
• $$\mathcal E^{ab} = 0$$ are equations of motion. $$\theta(\delta g)$$ symplectic potential
• $$\omega(\delta_1 g, \delta_2 g) = \delta_1 \theta(\delta_2 g) - \delta_2 \theta(\delta_1 g)$$ symplectic current
• $$\omega(\delta g, £_\xi g) = d [\delta Q_\xi - \xi \cdot \theta(\delta g)] + EOM$$ Hamiltonian flow of a diffeo is a boundary term
• The boundary term defines a perturbed charge

# Symplectic current: limit to $$i^0$$

asymptotic conditions on metric perturbations: $$\Omega^{3/2}\omega$$ has a direction-dependent limit

pullback to $$\mathscr H$$: $$\underleftarrow{\mathbf \omega} = \mathbf \varepsilon_3 ( \delta_1 \mathbf E \delta_2 \mathbf K - \delta_2 \mathbf E \delta_1 \mathbf K)$$

$$\mathbf E$$ and $$\mathbf K_{ab}$$ are potentials for the Weyl tensor: $\mathbf E_{ab} = -\tfrac{1}{4}~ (\mathbf D_a \mathbf D_b \mathbf E + \mathbf h_{ab} \mathbf E) \quad \mathbf B_{ab} = - \tfrac{1}{4}~ \mathbf \varepsilon_{cda} \mathbf D^c \mathbf K^d{}_b$

# Symplectic current: Symmetries

• Can be written as a total derivative (computation to show this) $$\underleftarrow{\mathbf \omega} (\delta g, \delta_{(\mathbf f, \mathbf X)}g) = \mathbf \varepsilon_3 \mathbf D^a \mathbf Q_a(\delta g, (\mathbf f, \mathbf X)) + EOM$$
• Use $$\mathbf Q_a$$ to define a charge on cross-sections of $$\mathscr H$$

# Charge: Supertranslations

• $$\mathbf Q_a = \delta (\mathbf E \mathbf D_a \mathbf f - \mathbf f \mathbf D_a \mathbf E)$$
• Supermomentum: $$\mathcal Q[\mathbf f; S] = \int\limits_S \mathbf \varepsilon_2 \mathbf u^a (\mathbf E \mathbf D_a \mathbf f - \mathbf f \mathbf D_a \mathbf E)$$
• Can be matched up to the supermomentum on $$\mathscr I^\pm$$

# Charge: Lorentz (perturbed)

• $$\mathbf Q_a = \delta \mathbf W_{ab} {}^\star \mathbf X^b - \tfrac{1}{8} \delta \mathbf E \mathbf K \mathbf X_{a}$$
• $$\mathbf{W}_{ab} = \mathbf{\beta}_{ab} + \tfrac{1}{8}~ \mathbf{\varepsilon}_{cd(a} \mathbf D^{c} \mathbf{E} \mathbf K^d{}_{b)} - \tfrac{1}{16}~ \mathbf{\varepsilon}_{abc}\mathbf{K}\mathbf D^{c}\mathbf{E}$$
• $$\mathbf \beta_{ab}$$ is the subleading magnetic Weyl tensor
• Note that $$\mathbf Q_a$$ is not $$\delta (something)$$. This can be fixed using the Wald-Zoupas prescription (whole another talk!!)

# Charge: Lorentz

• Do the Wald-Zoupas stuffs
• $$\mathcal Q[\mathbf{X}^{a};S] = \int_S\mathbf{\varepsilon}_{2}~ \mathbf u^{a} [ \mathbf{W}_{ab} {}^\star{\mathbf{X}}^{b} - \tfrac{1}{8} \mathbf{K} \mathbf{E} \mathbf{X}_{a}]$$
• Same as the charge found by Ashtekar-Hansen when $$\mathbf K_{ab} = 0$$
• Same as the charge found by Compère-Dehouck when $$\mathbf K = 0$$

# In the oven

• Does the Lorentz charge at $$i^0$$ match the ones from $$\mathscr I^\pm$$?
• implications for scattering and conservation laws in quantum theory?
• Since $$\mathscr H$$ is the same as deSitter, is there any relation to deSitter holography?

# Details

1. KP and I. Shehzad, Asymptotic symmetries and charges at spatial infinity in general relativity, [arXiv:1912.04305].
2. KP, Conservation of asymptotic charges from past to future null infinity: Maxwell fields, JHEP 10 113 (, [arXiv:1808.07863].
3. KP, Conservation of asymptotic charges from past to future null infinity: Supermomentum in general relativity, JHEP 03 148 (, [arXiv:1902.08200].