Soft Hair on Schwarzschild: A Wrinkle in Birkhoff's Theorem

Abstract

The double null form of the Schwarzschild metric is usually arrived at by demanding Eddington-Finkelstein (EF) conditions at the horizon. This leads to certain logarithmic fall-offs that are too slow along null directions at $\mathscr{I}$, resulting in divergences in the covariant surface charges. These coordinates are therefore $not$ asymptotically flat. In this paper, we find a natural alternative double null form for Schwarzschild that is adapted to $\mathscr{I}^{+}$ or $\mathscr{I}^{-}$ instead of the horizon. In its final form, the metric has only power law fall-offs and fits into the recently introduced Special Double Null (SDN) gauge, with finite surface charges. One remarkable feature of SDN gauge is that spherical symmetry and vacuum Einstein equations allow an infinite number of asymptotic integration constants in the metric, on top of the mass. This is an apparent violation of Birkhoff's theorem. We note however that all except two of these new parameters are absent in the charges, and therefore correspond to trivial hair. The remaining two parameters do show up in the charges, depending on the choice of allowed fall-offs. We provide an understanding of this observation -- Birkhoff's theorem fixes Schwarzschild only $up$ $to$ $diffeomorphisms$, but diffeomorphisms need not vanish at infinity and can in principle become global symmetries. If such asymptotic diffeomorphisms are spherically symmetric, their associated soft modes can become Birkhoff hair. The relevant global symmetries here are certain hypertranslation shifts in the $v$-coordinate at $\mathscr{I}^{+}$ (and $u$ at $\mathscr{I}^{-}$), which are inaccessible in other gauges.