Establish a fundamental Lagrangian for the hairy Kerr theory

Establish a Lagrangian or action principle from which the rotating hairy Kerr black-hole solution, including its effective matter sector, can be derived.

Background

The rotating hairy Kerr geometry is constructed within the gravitational decoupling framework by introducing an effective conserved energy–momentum tensor rather than specifying a fundamental matter theory. The additional matter sector may represent dark-sector or other unknown fields, but its microscopic origin is not fixed by the model.

A fundamental action would provide a dynamical explanation for the effective energy–momentum tensor and clarify which matter fields or modified-gravity theory support the hairy black-hole solution. The paper explicitly identifies the absence of such an action as an unresolved theoretical issue.

References

It is important to note that, at present, a Lagrangian formulation of the hairy gravitational theory considered here is not known. Within the gravitational decoupling framework, one instead works directly at the level of the Einstein field equations by introducing an additional effective energy-momentum tensor, $T_{\mu\nu}$, which deforms a known seed solution in a controlled manner.

— Probing hairy Kerr black holes through quasi-periodic oscillations II: a study based on the resonance models  (2610.11378 - Dasgupta et al., 8 Oct 2026) in Section 2, paragraph following the definition of the fundamental frequencies