Four-dimensional Stationary Algebraically Special Solutions and Soft Hairs

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 Bondi-Sachs framework. 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 show that the higher modes of one pair of the (anti-)holomorphic function contain an infinite tower of soft hairs from the perspective of the asymptotic supertranslation charges. Within our general ansatz, we obtain the complete set of algebraic Petrov type-D solutions of five parameters. We show that the type-D solution is diffeomorphism to the Kerr-Taub-NUT solution.