Angular momentum at \(\mathscr I\)
in Einstein-Maxwell

Kartik Prabhu
with Béatrice Bonga & Alex Grant
Phys. Rev. D101 044013 (,
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.
  • \((Lorentz \ltimes s.translations) \ltimes Maxwell .transformation\)


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 (?)