Phys. Rev. D, 113, 024037 (2026)

Four-dimensional stationary algebraically special solutions, Weyl invariants, and soft hairs beyond large gauge transformations

H. Lü^1,2,3 and Pujian Mao¹

1 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China.

2 The International Joint Institute of Tianjin University, Fuzhou, Tianjin University, Tianjin 300350, China.

3 Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China.

* mrhonglu@gmail.com, pjmao@tju.edu.cn

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.