Quantum Field Theory of Black Hole Perturbations with Backreaction V. Beyond Second Order Perturbations

TL;DR

The paper develops a gauge-invariant, higher-order perturbation theory for black holes by employing a reduced phase space approach in Gullstrand–Painlevé gauge, where constraints are reformulated as finite polynomials. It expands the Hamiltonian and diffeomorphism constraints to all orders in the non-symmetric true degrees of freedom, then solves the symmetric and non-symmetric constraints up to third order to obtain a physical Hamiltonian that encodes backreaction between gravitational perturbations. A detailed odd-parity (axial) analysis at third order illustrates the structure of the interaction Hamiltonian and its dependence on angular momentum recouplings, while a comparison with Lagrangian perturbation theory clarifies how different relational observables yield related but gauge-dependent reduced Hamiltonians. The framework sets the stage for gauge-invariant, non-linear black hole perturbations and provides a foundation for extending to matter and quantization, with important implications for quasi-normal mode analysis and gravitational-wave physics.

Abstract

Black hole perturbation theory beyond second order is not well understood because typically one defines the meaning of gauge invariance order by order which is ambiguous. In this series of works we therefore developed a new approach which disentangles the meaning of gauge invariance from the perturbative order. It is based on the reduced phase space approach to the Hamiltonian formulation of General Relativity and constructs a non-perturbative, albeit implicit, formulation of the dynamics of only observables that are gauge invariant to all orders. To obtain explicit expressions, perturbation theory is then employed, but now only perturbations are considered that are gauge invariant to all orders. There are both spherically symmetric and non-symmetric observables and the formulation takes the (perturbative) backreaction between those fully into account. The formulation has access to both the exterior and interior of the dynamical horizon. In previous papers of this series we have introduced the general formalism and performed consistency checks with second order results obtained in other approaches. The real virtue of our approach starts emerging at higher than second order where we expect differences from previous works both due to backreaction effects and because we work with observables that are gauge invariant to all orders, not only up to a given order. In this paper, we consider the third order. Also new to our approach is that we start from a non-perturbative, namely polynomial, version of the constraints which therefore are finite polynomials in all degrees of freedom before reducing, rather than an infinite series. This allows for an exact and non-perturbative, while implicit, solution of the constraints which does not need to truncate the series and thus is of tremendous technical advantage.