Quantum Field Theory of Black Hole Perturbations with Backreaction VI. Apparent Horizons, Quasi-Local Mass and Effective Classical Metrics

Abstract: In a recent series of papers we developed a first-principle and gauge invariant approach to black hole perturbation theory valid to any order. We included back reaction effects to tackle the situation of evaporating black holes and obtained an explicit expression for the dynamics of the reduced phase space to second order. The physics of evaporating black holes is in particular encoded by apparent horizons, an observer dependent generalisation of the event horizon. We determine the shape of the apparent horizon to second order in the perturbations. The area of the apparent horizon is an interesting observable which is expected to decrease in the quantum theory due to Hawking evaporation. We show how the full four dimensional metric can be reconstructed in terms of the reduced phase space variables. In the quantum theory, taking expectation values of this metric, we obtain an effective classical metric, whose causal structure can then be visualised in a quantum corrected Penrose diagram. We conclude with an outlook into the quantisation procedure in the reduced phase space formalism and the implications on the area of the apparent horizon.

• The paper presents a gauge-invariant perturbative framework that explicitly reconstructs black hole observables, including apparent horizons up to second order in perturbations.
• It derives analytic expressions for the quasi-local mass, linking local energy contributions with backreaction while maintaining diffeomorphism invariance.
• The formalism extends to incorporate electromagnetic fields and paves the way for quantum-corrected metrics to understand horizon evolution in evaporating black holes.

Gauge-Invariant Black Hole Perturbation Theory with Backreaction: Apparent Horizons, Quasi-Local Mass, and Effective Classical Metrics

Overview

The paper "Quantum Field Theory of Black Hole Perturbations with Backreaction VI. Apparent Horizons, Quasi-Local Mass and Effective Classical Metrics" (2605.13714) continues a rigorous program developing a manifestly gauge-invariant, first-principles formalism for black hole perturbation theory including dynamical backreaction. The framework enables the explicit reconstruction of physically relevant observables, particularly in the context of black hole evaporation—where backreaction plays a critical role inaccessible to traditional semi-classical analyses.

In this volume, the authors focus on the explicit determination of apparent horizons up to second order in the perturbation series and derive analytic expressions for the quasi-local mass associated with the evolving black hole spacetime. Furthermore, the paper reconstructs the four-dimensional spacetime metric in terms of reduced phase space degrees of freedom and discusses the emergence of effective (quantum-corrected) classical metrics, enabling the visualization of modified causal structures via Penrose diagrams.

Formalism: Gauge-Invariant Reduced Phase Space

The approach leverages a separation of phase space variables according to the underlying spherical symmetry, splitting degrees of freedom into symmetric (background) and non-symmetric (gravitational wave) components. To obtain physically meaningful, gauge-independent predictions, the full diffeomorphism invariance of general relativity is resolved through an explicit construction of Dirac observables. The entire dynamical content resides in the reduced phase space, coordinatized by observable degrees of freedom, and evolution is generated by a physical Hamiltonian, constructed up to third order in prior work (Neuser et al., 2024, Neuser et al., 8 Feb 2026).

A central technical advancement is the systematic treatment of lapse and shift in the chosen foliation (Gullstrand-Painlevé (GP) gauge), ensuring that all metric components are reconstructed explicitly in terms of true physical degrees of freedom. Boundary terms, fall-off conditions, and regularity at the horizon are handled meticulously to guarantee mathematical consistency at every order.

Apparent Horizon and Quasi-Local Mass: Second Order Construction

The apparent horizon—defined quasi-locally as the outermost marginally trapped surface on a given spacelike hypersurface—is perturbatively located by solving the vanishing expansion condition for outgoing null geodesics. The authors derive the angular profile of the apparent horizon up to second order in perturbations, treating odd and even parity sectors via canonical transformations to master variables.

The explicit expression for the area $A$ of the apparent horizon to second order reveals a correction term involving an integral over the local energy density $E(r)$, derived from (gauge-invariant) master variables and canonical momenta, evaluated within the unperturbed Schwarzschild radius. This leads to a transparent formula for the quasi-local mass:

$M_0 = M + \frac{1}{16\pi} \int^{r_s} E(r) dr$

where $M$ is the initial black hole mass and $E(r)$ encodes perturbative energy contributions. Notably, the area and quasi-local mass are shown to be invariant under changes in the foliation up to the considered order, consistent with the underlying diffeomorphism invariance.

A strong claim is the precise identification of the second-order correction as a local (rather than global) quantity, fully determined by the observable sector of phase space. This provides, for the first time, an operationally meaningful notion of horizon area and mass in dynamical, evaporating contexts where the global event horizon is not accessible.

Extensions: Einstein-Maxwell Sector and Higher-Order Effects

The formalism is extended to incorporate electromagnetic fields, yielding a straightforward generalization to Reissner-Nordström black holes. Both inner and outer horizons arise within the formal solution, and the apparent horizon area and quasi-local mass formulae are derived analogously, including precise electromagnetic corrections.

Third-order corrections and even more general matter couplings (e.g., scalar, neutrino fields) are outlined, with explicit recursion relations provided to facilitate further perturbative treatments, ensuring the scalability and extensibility of the method.

Dynamical Evolution, Energy Flux, and Area Laws

The dynamical evolution of the quasi-local mass is obtained by computing its Poisson bracket with the physical Hamiltonian. For the classical theory, it is shown that $\dot{M}_0 \geq 0$, reflecting the standard area theorem (classical irreversibility of horizon area). This result is robust in the classical regime; violations are expected in the quantum theory through known mechanisms (e.g., quantum energy inequality violations), suggesting a concrete path to analyze black hole evaporation beyond the semi-classical level.

Explicit expressions for local energy currents ($j^0$, $j^3$) are constructed using the master variable formalism, providing a link between local fluxes and global mass change. The formalism naturally supports conservation laws and continuity equations for energy, even in the presence of dynamical horizons and evolving spacetime geometry.

Effective Classical Metrics and Quantum-Corrected Penrose Diagrams

Having constructed the full 4-metric in the physical gauge, the paper details the procedure for extracting quantum-corrected classical metrics via expectation values in the quantum theory. This enables, in principle, the construction of modified Penrose diagrams visualizing changes in causal structure induced by backreaction and quantum effects.

While the explicit quantum computation is left for future work, the authors establish the methodology for angular averaging and spherically symmetric reduction, facilitating practical computation of effective radial null geodesics and the visualization of the evolving horizon in compactified spacetime diagrams.

Implications and Future Directions

The presented formalism provides a concrete route for analyzing quantum gravitational backreaction in black hole dynamics, addressing limitations in traditional perturbation theory that neglects dynamical modification of the background. The identification of a local, operationally defined quasi-local mass and its evolution opens avenues for rigorous treatments of evaporation, horizon area reduction, and information loss—the main issues in the quantum gravity of black holes.

Practically, this framework could be extended to analyze Hawking evaporation in regimes where semi-classical assumptions break down, incorporate higher-spin matter, or serve as the foundation for non-perturbative quantization (e.g., via Fock or other representations suggested by the underlying physical Hamiltonian). Future developments may also resolve the global structure of evaporating black hole spacetimes, rendering detailed, physically meaningful predictions accessible for observable signatures and the study of horizon dynamics.

Conclusion

This paper provides a comprehensive perturbative framework for analyzing black hole evaporation including dynamical backreaction, yielding precise, gauge-invariant expressions for local observables such as the apparent horizon and quasi-local mass up to second order. The reconstruction of all metric components from Dirac observables and the explicit approach to quantum-corrected geometry position the formalism as a powerful tool in addressing foundational open questions regarding the end stages of black hole evaporation, with potential future impact on both theoretical and phenomenological studies in quantum gravity (2605.13714).