Angular momentum at \(\mathscr I\)
in Einstein-Maxwell
theory
APS 2020, Virtual!!
Maxwell stress-energy
- energy/momentum when \(\xi^a\) is timelike/spacelike
- 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 \(\xi^a \equiv (f, X^a)\)
- supertranslations \((f, X^a = 0) \): \( \int f \mathcal{E}^a \mathcal{E}_a \geq 0 \)
- 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
- \(\mathcal{F}_{GR}\) is usual flux formula in vacuum GR.
- Note additional “boundary” Maxwell term for Lorentz.
Wald-Zoupas charge
\(\mathcal{F} = \Delta \mathcal{Q}\); with \(\beta = f + \tfrac{1}{2}u \mathscr D_a X^a\)
\( \mathcal{F}_{GR} + \mathcal{F}_T = \Delta \int_S \big[ - \xi^a (\)\(\Omega^{-1}C_{abcd}) l^b l^c n^d\)\(+ \tfrac{1}{2} \beta \sigma^{ab} N_{ab} + X^a \sigma_{ab} \mathscr D_c \sigma^{bc} - \tfrac{1}{4} \sigma_{ab} \sigma^{ab} \mathscr D_c X^c \big] \)
\( \mathcal{F} = \Delta \big( \mathcal Q_{GR} + \int_S \) \(\Re[\varphi_1]\) \(X^a A_a \big) \)
- WZ-charge in Einstein-Maxwell is the usual GR-charge + additional Maxwell term for Lorentz.
- charge has Coulombic stuff but flux is purely radiative
Gauge invariance
- \(\mathcal Q \mapsto \mathcal Q + \int_S \Re[\varphi_1] £_X \lambda\)
- 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 (?)