Vortices at the Edge of the Sky: Measuring the Charge of Superrotations ✛ ✎

At the edge of spacetime, where all light rays end, live symmetries that physicists have debated for half a century. They are called superrotations, transformations of the celestial sphere that diverge at isolated points, like vortices in superfluid helium. For a long time, it seemed impossible to measure the charge of these vortices: the integrals that should yield conserved quantities blow up to infinity.

This new work returns to an old idea of Roger Penrose: add a boundary to spacetime, null infinity, where light rays end up. On this boundary, charges can be defined as integrals over cross-sections, spheres at a given time. The Komar formula, adapted by Geroch and Winicour, provides a way to compute the charge for ordinary BMS symmetries. But for superrotations, which are singular at the poles, the integral diverges. The authors apply the Flanagan-Nichols regularization: they exclude neighborhoods of the poles and then take the limit. It turns out that the integrand remains finite despite inverse powers of the conformal factor.

The key result is that the superrotation charge is expressed as a finite integral over a cross-section of null infinity. A test with the example $Y^A$ = $z^2$ ∂_z yields a finite value. The charge is covariant and satisfies the continuity condition across cross-sections proposed by Chen, Wald, and Yau. The superrotation flux is also defined by a local formula that generalizes the result of Geroch and Winicour.

This result removes an obstacle to using superrotations as physical symmetries. Unlike approaches that rely on coordinate expansions or algebraic topology, this one uses Penrose's purely geometric method, which gives explicit covariance. The work connects superrotations to the Wald-Zoupas program for defining conserved quantities in general relativity and opens a path to understanding the infrared structure of gravity and possible holography at null infinity. The method can be extended to other extensions of the BMS group, for example, to diffeomorphisms of the sphere or superrotations in higher dimensions. It remains to explore the connection with soft graviton theorems and the Virasoro algebra, as well as to apply the technique to black holes and the analysis of gravitational wave memory.