# Supertranslation charge & flux on null surfaces

Venkatesa Chandrasekaran, Éanna É. Flanagan, EGM 20, PSU, PA# Null boundary data

Spacetime \((M, g_{ab})\) solving Einstein’s equations.

Let \(N = \partial M\) be a null boundary with normal \(l_a\)

\(l^a \hat{=} g^{ab}l_b\) is null, tangent to \(N\) and geodesic.

Null boundary structure: \((N, [l^a])\) with \(l^a \sim e^{\alpha} l^a\) for \(\alpha \in \mathbb R\)

# Null boundary data: perturbations

One-parameter family of metrics \(g_{ab}(\lambda)\) with \(g_{ab}(0) = g_{ab}\) such that for each \(\lambda\)

- \(N\) is null (common null boundary)
- \(l^a(\lambda) \in [l^a]\) (common null direction)

perturbations \(\delta g_{ab} = \left.\frac{d}{d\lambda}~g_{ab}(\lambda)\right\vert_{\lambda=0}\)

# Symplectic potential and current

- Lagrangian form: \(L\)
- symplectic potential: \(\delta L = E^{ab} \delta g_{ab} + d \theta(\delta g)\)
- symplectic current: \( \omega(g; \delta_1 g, \delta_2 g) = \delta_1 \theta(\delta_2 g) - \delta_2 \theta(\delta_1 g) \)

# Boundary symmetries

for diffeos \(X^a\) and a spacelike surface \(\Sigma\) where \(S = \Sigma \cap N\)

- \( \int_\Sigma \omega(g;\delta g, £_X g) = \int_S \delta Q_X - X \cdot \theta(\delta g) = I_X(S,\delta g) \)
- \(X^a \sim X'^a \) iff \(X^a \mathop{\hat{=}} X'^a\) and \(I_X(S, \delta g) = I_{X'}(S,\delta g)\) for all \(S\) and \(\delta g\).
- boundary symmetry: \([X^a] = X^a/\sim \)

# Null boundary supertranslation

- \(X^a \hat{=} f l^a\) with \(£_l f \hat{=} 0 \) (abelian Lie subalgebra)
- there are other symmetries with non-trivial Lie algebra structure

Then, the supertranslation charge \(\mathscr Q_X\) just defined by \[ \delta\mathscr Q_X = \int_S \delta Q_X - X \cdot \theta(\delta g) = I_X(S,\delta g) \]

# Integrability in phase space

*Well, no!*

Consider \[ \delta_1 I_X(S, \delta_2 g) - \delta_2 I_X(S, \delta_1 g) \\ = - \int_S X \cdot \omega(\delta_1 g, \delta_2 g) \neq 0 \]

\(I_X(S, \delta g)\) cannot be written as \(\delta (something)\) i.e. naïve definition of charge is not integrable in phase space!

# Wald-Zoupas “conserved quantity”

- Find a boundary symplectic potential \(\Theta\) for the pullback of \(\omega\) to \(N\) \[ \underset{\leftarrow}{\omega}(g; \delta_1 g, \delta_2 g) = \delta_1 \Theta(g; \delta_2 g) - \delta_2 \Theta(g; \delta_1 g) \]
- WZ charge \(\mathscr Q_X\) defined by (always integrable in phase space) \[ \delta \mathscr Q_X = \int_S \delta Q_X - X \cdot \theta(\delta g) + X \cdot \Theta(\delta g) \]
- Flux is \(\mathscr F_X = \int \Theta(£_X g)\).

# WZ charge of supertranslations

\[ \mathscr Q_X = \tfrac{1}{8\pi}\int_S \varepsilon^{(2)} f \left( \theta - \tfrac{1}{2}~ \kappa \right) \] \[ \mathscr F_X = \tfrac{1}{8\pi} \int \varepsilon^{(3)} f \left( \sigma_{ab}\sigma^{ab} - \tfrac{1}{2}~ \theta^2 - \tfrac{1}{2}~ \kappa \theta + \tfrac{1}{2}~ £_l \kappa \right) \]\(\kappa\) is surface gravity, \(\sigma_{ab}\) is shear and \(\theta\) is expansion of \(N\) relative to \(l^a\)

# Memory effect on horizon

\(N\) is stationary spherically symmetric black hole horizon. Let a finite burst of radiation enter \(N\).

- Let \(\Psi\) be the electric part of the shear \(\sigma_{ab}\) integrated through the radiation period, then \[ \Delta q_{AB} = 2 (D_A D_B - \tfrac{1}{2}~ q_{AB} D^2)\Psi \]
- The \(\Psi\) is related to a supertranslation \(f\) given by \[ (D^2 + 2) \Psi = -\kappa f \] (using the transformation of the transverse shear under a supertranslation)