Null boundary phase space: slicings, news and memory

TL;DR

<3-5 sentence high-level summary> We construct and analyze the boundary phase space for D-dimensional Einstein gravity with a fixed null boundary, showing the boundary symmetry algebra is a semi-direct sum $Diff({\cal N}) \inplus Weyl({\cal N})$ generated by $D$ towers of surface charges that live on the boundary. The charges can be rendered integrable by suitable slicings of the phase space, with the Heisenberg slicing yielding a direct-sum algebra $Heisenberg \oplus Diff({\cal N}_v)$ and other slicings giving $Diff({\cal N}) \inplus Weyl({\cal N})$, highlighting the slicing-dependence of the charge algebra. The paper also introduces null-surface expansion and spin memory effects, showing how gravitational waves crossing the boundary imprint persistent changes in the boundary charges and deriving corresponding balance equations. These results generalize prior 3D analyses to arbitrary dimensions and clarify how boundary degrees of freedom encode bulk gravitational dynamics through flux, integrability, and memory phenomena.

Abstract

We construct the boundary phase space in $D$-dimensional Einstein gravity with a generic given co-dimension one null surface ${\cal N}$ as the boundary. The associated boundary symmetry algebra is a semi-direct sum of diffeomorphisms of $\cal N$ and Weyl rescalings. It is generated by $D$ towers of surface charges that are generic functions over $\cal N$. These surface charges can be rendered integrable for appropriate slicings of the phase space, provided there is no graviton flux through $\cal N$. In one particular slicing of this type, the charge algebra is the direct sum of the Heisenberg algebra and diffeomorphisms of the transverse space, ${\cal N}_v$ for any fixed value of the advanced time $v$. Finally, we introduce null surface expansion- and spin-memories, and discuss associated memory effects that encode the passage of gravitational waves through $\cal N$, imprinted in a change of the surface charges.

Figures (5)

• Figure 1: Section of null hypersurface $\cal N$ at $r=0$ in $rv$-plane. Infalling null rays traverse $\cal N$ at different values of advanced time $v$. Each point on the red line corresponds to a transverse surface ${\cal N}_v$.
• Figure 2: Depiction of co-dimension one null boundary $\mathcal{N}$. ${\cal N}$ has the topology of $\mathbb{R}_v\ltimes {\cal N}_v$ where the transverse surface ${\cal N}_v$ is typically a $D-2$ dimensional spacelike compact surface.
• Figure 3: Penrose diagram for shockwave entering black hole. Shaded oval denotes absorption (not in solution space). Dashed orange (green) line is initial (final) horizon $\color{orange}{\boldsymbol{\mathcal{H}}^+}$ ($\color{darkgreen}{\tilde{\boldsymbol{\mathcal{H}}}^+}$).
• Figure 4: Solution space, schematically. Each point represents a solution, labeled by surface charges. On the right, a non-trivial diffeomorphism moves along some (coadjoint) orbit of the symmetry algebra from solution sol$_3$ to solution sol$_4$. On the left, additionally genuine or fake fluxes are switched on, moving from one orbit (sol$_1$) to another (sol$_2$).
• Figure 5: Null boundary ${\cal N}$ and segments $\Sigma_1,\Sigma_2$ on it.