Gravity Degrees of Freedom on a Null Surface

TL;DR

Addresses the canonical structure of gravity on a null boundary without gauge fixing by deriving the full set of bulk and boundary canonical pairs from the covariant phase-space formalism. Using a double-null foliation and a boost-covariant framework, it expresses the symplectic potential and boundary action entirely in terms of intrinsic null geometry, including a redshift factor and a boost-invariant normal frame. A central result is the identification of spin-2, spin-1, and spin-0 degrees of freedom as physical, with the conformal shear as the bulk momentum and additional boundary momenta associated with the twist and expansion-surface-gravity combination. The work also provides a Lagrangian boundary term with corner contributions, connecting to recent null-boundary formalisms and enabling future exploration of soft gravitons and information flow across finite null regions.

Abstract

A canonical analysis for general relativity is performed on a null surface without fixing the diffeomorphism gauge, and the canonical pairs of configuration and momentum variables are derived. Next to the well-known spin-2 pair, also spin-1 and spin-0 pairs are identified. The boundary action for a null boundary segment of spacetime is obtained, including terms on codimension two corners.

Figures (3)

• Figure 1: A typical situation where the symplectic structure on the null surface $B$ is of interest is when $B$ is part of the boundary of the spacetime region $R$ under consideration. The other parts of the boundary are spacelike surfaces $\Sigma_i$.
• Figure 2: The geometry of our setup is depicted. The null hypersurface $B$ is a member of the foliation $\phi^1 = \text{const.}$ that need not be null everywhere. It is ruled into codimension two surfaces $S$ by a second foliation $\phi^0=\text{const.}$ The vectors ${{\ell}}$ and ${\bar{\ell}}$ are null and normal to $S$. ${{\ell}}$ is normal also to $B$, and since $B$ is null it is at the same time tangential to $B$. ${\bar{\ell}}$ is transverse to $B$, and the vectors are normalized as ${{\ell}}^a {\bar{\ell}}^b g_{ab} = 1$.
• Figure 3: The observer $v = - g^{-1}({\mathrm{d}} \phi^0)$ crosses the null surface $B$, and measures the frequency $\nu$ of light rays propagating along $B$. The redshift, i.e., the relative change of frequency, per unit time $\phi^0$ is given in the geodesic lightcone frame by: $z = (\nu(\phi^0) - \nu(\phi^0 + {\mathrm{d}} \phi^0)) / \nu(\phi^0) = D_0 h \cdot {\mathrm{d}} \phi^0$