Ultrarelativistic limit of the Kerr theorem

The original Kerr theorem provides the foundation for Kerr-Schild transformations by classifying all shear-free and geodesic null congruences in flat spacetime; the key ingredient of the Kerr-Schild ansatz. However, due to the high level of degeneracy of the outcome, it is often less practical than its symmetric refinements, which may single out congruences leading to physically significant spacetimes by imposing relevant symmetries. An illustrative example is the stationary axisymmetric version of Kerr theorem, which has been shown to lead directly and uniquely to the Kerr black hole in vacuum. In this work, we propose a new symmetric refinement of the Kerr theorem by boosting the stationary symmetry into its ultrarelativistic limit to achieve invariance under null translations, while keeping axisymmetry. Under these assumptions, the classification yields only two distinct congruences. The first congruence is covariantly constant and, through the Kerr-Schild ansatz, evidently yields an axisymmetric $\mathit{pp}$-wave. The vacuum axisymmetric profile of this $\mathit{pp}$-wave displays a logarithmic dependence on the polar radius, characteristic of the exterior gravitational field of the Bonnor light beam, and includes as a special case the Aichelburg-Sexl ultrarelativistic limit of the Schwarzschild black hole. The Kerr-Schild transformation of the second congruence gives rise to a nontrivial vacuum solution recently reported in [Phys. Rev. D 112, 024020 (2025)]. Using circularity and appropriately fixing the reparametrization invariance of the orthogonal manifold to the Killing fields, we show that the latter solution corresponds to the well-known Taub-NUT spacetime with planar topology. These results emphasize how symmetry-based refinements of the Kerr theorem constitute a powerful tool to constructing physically essential spacetimes.