Brown-York charges at null boundaries

TL;DR

This work extends the Brown–York boundary stress tensor to subregions bounded by null hypersurfaces, deriving the simple mixed-index form $T^i{}_j = -rac{1}{8\, ext{ } \, ext{ } G} (W^i{}_j - W \, ext{ } \, \delta^i{}_j)$ where $W^i{}_j$ is the null surface shape operator. The authors show that, when paired with a Dirichlet-type null boundary data yielding a Carroll structure, the resulting charges are covariantly conserved with respect to a rigged connection and coincide with canonical Wald–Zoupas charges for transformations that act covariantly on the boundary data; for anomalous transformations, the charges differ by an intrinsic boundary functional tied to the anomaly. They connect the conservation law to the null constraint equations (Raychaudhuri and Damour–Navier–Stokes) and discuss implications for asymptotically flat spacetimes and celestial holography, including how holographic counterterms preserve charge equivalence under covariant renormalization. The paper also compares the null BY tensor to other proposals, clarifying how different choices of intrinsic boundary structures influence the resulting stress tensor and charges. Potential future directions include extending the framework to future null infinity in 4D, exploring covariant holographic counterterms, and linking to generalized BMS symmetries and celestial CFT data.

Abstract

The Brown-York stress tensor provides a means for defining quasilocal gravitational charges in subregions bounded by a timelike hypersurface. We consider the generalization of this stress tensor to null hypersurfaces. Such a stress tensor can be derived from the on-shell subregion action of general relativity associated with a Dirichlet variational principle, which fixes an induced Carroll structure on the null boundary. The formula for the mixed-index tensor $T{}^i{}_j$ takes a remarkably simple form that is manifestly independent of the choice of auxiliary null vector at the null surface, and we compare this expression to previous proposals for null Brown-York stress tensors. The stress tensor we obtain satisfies a covariant conservation equation with respect to any connection induced from a rigging vector at the hypersurface, as a result of the null constraint equations. For transformations that act covariantly on the boundary structures, the Brown-York charges coincide with canonical charges constructed from a version of the Wald-Zoupas procedure. For anomalous transformations, the charges differ by an intrinsic functional of the boundary geometry, which we explicity verify for a set of symmetries associated with finite null hypersurfaces. Applications of the null Brown-York stress tensor to symmetries of asymptotically flat spacetimes and celestial holography are discussed.