# Supertranslation charge & flux on null surfaces

Venkatesa Chandrasekaran, Éanna É. Flanagan, Kartik Prabhu 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 $$\lambda$$

1. $$N$$ is null (common null boundary)
2. $$l^a(\lambda) \in [l^a]$$ (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)