# Angular momentum at$$\mathscr I$$inEinstein-Maxwell theory

Kartik Prabhu
with Béatrice Bonga & Alex Grant
APS 2020, Virtual!!

# Maxwell stress-energy

$T_{ab} = F_{ac} F_b{}^c - \tfrac{1}{4} g_{ab} F^2, \quad \nabla^b T_{ab} = 0$ (gauge invariant, conserved) current $$T^a{}_b\xi^b$$ for Killing field $$\xi^a$$
1. energy/momentum when $$\xi^a$$ is timelike/spacelike
2. angular momentum when $$\xi^a$$ is rotational

# Stress-energy flux at $$\mathscr{I}$$

At $$\mathscr{I}$$, stress-energy flux $$\mathcal{F}_T = \int n_a T^a{}_b\xi^b$$ for BMS symmetries $$\xi^a \equiv (f, X^a)$$

1. supertranslations $$(f, X^a = 0)$$: $$\int f \mathcal{E}^a \mathcal{E}_a \geq 0$$
2. Lorentz $$(f=0, X^a)$$: $$\int \mathcal{E}^a (F_{ab} X^b + 2$$$$\Re[\varphi_1]$$$$X_a)$$

$$\mathcal{E}_a$$ radiative d.o.f but $$\Re[\varphi_1]$$ Coulombic d.o.f

Angular momentum flux depends on the Coulombic d.o.f (!!) refs?

# Einstein-Maxwell theory — Wald-Zoupas charges & fluxes

General prescription for charges and symmetries at $$\mathscr{I}$$ in any local-covariant theory.

• Symplectic current from Einstein-Maxwell Lagrangian: $$\int \delta_1 N^{ab} \delta_2 \sigma_{ab} + \delta_1 \mathcal{E}^a \delta_2 A_a - (1 \leftrightarrow 2)$$ (radiative)
• Symplectic potential on $$\mathscr{I}$$: $$\Theta = N^{ab} \delta \sigma_{ab} + \mathcal{E}^a \delta A_a$$
stationary solutions: $$N_{ab} = \mathcal{E}_a = 0$$ implies $$\Theta = 0$$ for all perturbations

# Wald-Zoupas flux

(radiative) $\mathcal{F} = \int \Theta(\delta_\xi\Phi) = \int N^{ab} \delta_\xi \sigma_{ab} + \mathcal{E}^a £_\xi A_a$ $\,\,\, = \mathcal{F}_{GR} + \mathcal{F}_T + \Delta \left( \int_S \Re[\varphi_1] X^a A_a \right)$
• $$\mathcal{F}_{GR}$$ is usual flux formula in vacuum GR.
• Note additional “boundary” Maxwell term for Lorentz.

# Wald-Zoupas charge

$$\mathcal{F} = \Delta \mathcal{Q}$$; with $$\beta = f + \tfrac{1}{2}u \mathscr D_a X^a$$

$$\mathcal{F}_{GR} + \mathcal{F}_T = \Delta \int_S \big[ - \xi^a ($$$$\Omega^{-1}C_{abcd}) l^b l^c n^d$$$$+ \tfrac{1}{2} \beta \sigma^{ab} N_{ab} + X^a \sigma_{ab} \mathscr D_c \sigma^{bc} - \tfrac{1}{4} \sigma_{ab} \sigma^{ab} \mathscr D_c X^c \big]$$

$$\mathcal{F} = \Delta \big( \mathcal Q_{GR} + \int_S$$ $$\Re[\varphi_1]$$ $$X^a A_a \big)$$

1. WZ-charge in Einstein-Maxwell is the usual GR-charge + additional Maxwell term for Lorentz.
2. charge has Coulombic stuff but flux is purely radiative

# Gauge invariance

$$A_a \mapsto A_a + \nabla_a \lambda$$ with $$£_n \lambda = 0$$
• $$\mathcal Q \mapsto \mathcal Q + \int_S \Re[\varphi_1] £_X \lambda$$
• Lorentz charge shifts by a Maxwell charge.
• $$(Lorentz \ltimes s.translations) \ltimes Maxwell .transformation$$

# Summary

 Stress-energy flux (Maxwell) WZ flux (Einstein-Maxwell) depends on Coulombic d.o.f. purely radiative no charge formula on a cross-section WZ charge formula gauge invariant gauge covariant according to symm. group not Hamiltonian (when Coulombic $$\neq 0$$) Hamiltonian

# Other things

• Kerr spacetime (in axisymmetry gauge choice): extra term vanishes; usual answer for angular momentum.
• Thin, spherical charged shell: extra term contains time-dependent dipole moment.
• WZ flux can be quantized since it is Hamiltonian.
• WZ charge can be matched to ADM angular momentum at spatial infinity (?)