diffeos \(\xi^a\) in the physical spacetime which preserve the asymptotic structure at \(i^0\)
preserve \(i^0\) : \(\xi^a\vert_{i^0} = 0\)
preserve \(C^{>1}\)-structure : \(\Omega^{-1/2}\xi^a\) is direction-dependent
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
\(\omega(\delta g, £_\xi g) = d [\delta Q_\xi - \xi \cdot \theta(\delta g)] + EOM\) 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)
\(\mathbf Q_a = \delta \mathbf W_{ab} {}^\star \mathbf X^b - \tfrac{1}{8} \delta \mathbf E \mathbf K \mathbf X_{a}\)