Four-dimensional Stationary Algebraically Special Solutions, Weyl Invariants, and Soft Hairs Beyond Large Gauge Transformations

TL;DR

This paper identifies a complete set of four-dimensional stationary, algebraically special, locally asymptotically flat solutions by reducing the Einstein equations to a 2D Laplacian problem on the celestial sphere and parameterizing the solution space with holomorphic/antiholomorphic data $(L(z),\bar{L}(\bar{z}))$ and $(f(z),\bar{f}(\bar{z}))$, with a Virasoro-like structure. Extending to Einstein–Maxwell theory introduces a third pair $(Q,\bar{Q})$ and yields soft electric hair, and a cosmological-constant case provides an exact soft-hairy solution potentially linked to AdS/CFT. The authors define asymptotic charges for the soft modes, show higher modes carry nontrivial supertranslation (and large-$U(1)$) charges while keeping mass/charge fixed, and employ Weyl invariants to demonstrate each mode is a diffeomorphism-inequivalent solution. A robust Weyl-invariant framework is developed to distinguish independent modes, illustrating that soft hairs arise as genuine physical degrees of freedom beyond large gauge transformations. Overall, the work furnishes a concrete, invariant separation of soft-hair modes and paves the way for holographic investigations of soft hair in AdS/CFT contexts.

Abstract

We revisit the Ricci-flat metrics in four dimensions that are stationary and algebraically special, together with the locally asymptotically flat conditions in the generalized Bondi-Sachs framework. We show that the Einstein equation is reduced to Laplacian equation on the celestial sphere. The solutions consist of two pairs of arbitrary holomorphic and antiholomorphic functions analogous to the Virasoro modes. We prove that the higher modes of one pair of the (anti-)holomorphic function contain an infinite tower of soft hairs from the perspectives of the asymptotic supertranslation charges. We verify that different modes of the soft hairs are distinct solutions which cannot be connected by diffeomorphism, using the Weyl invariants associated to the solutions. Extending the construction to Einstein-Maxwell theory introduces a third pair of (anti-)holomorphic functions, arising from the Maxwell tensor which generates soft electric hair. We further present an exact soft-hairy solution of Einstein-Maxwell theory with a cosmological constant, offering a potential understanding of soft hair from the AdS/CFT correspondence.