Petrov Classification and holographic reconstruction of spacetime

TL;DR

The paper develops a constructive boundary-to-bulk procedure that yields exact four-dimensional Einstein spacetimes with algebraic specialness (Petrov types II–D–N–III–O) from carefully chosen boundary data. By coupling a 3D boundary metric with a symmetric, traceless, conserved energy–momentum tensor and enforcing a shearless boundary congruence, the authors define a resummation ansatz that produces a bulk metric solving Einstein’s equations with negative cosmological constant. The Petrov type of the bulk is dictated by a pair of complex reference tensors T^± built from the boundary energy–momentum and Cotton tensors, and exact bulk solutions emerge when boundary constraints C-con and T-con are satisfied. Concrete realizations recover Robinson–Trautman spacetimes and, with boundary vorticity, the Plebański–Demiański family, thereby linking boundary data to a broad class of non-perturbative, holographically dual geometries and offering a boundary-driven solution-generating framework for integrable sectors of gravity.

Abstract

Using the asymptotic form of the bulk Weyl tensor, we present an explicit approach that allows us to reconstruct exact four-dimensional Einstein spacetimes which are algebraically special with respect to Petrov's classification. If the boundary metric supports a traceless, symmetric and conserved complex rank-two tensor, which is related to the boundary Cotton and energy-momentum tensors, and if the hydrodynamic congruence is shearless, then the bulk metric is exactly resummed and captures modes that stand beyond the hydrodynamic derivative expansion. We illustrate the method when the congruence has zero vorticity, leading to the Robinson-Trautman spacetimes of arbitrary Petrov class, and quote the case of non-vanishing vorticity, which captures the Plebanski-Demianski Petrov D family.