arXiv:2404.18956v1 Announce Type: new
Abstract: In a seminal work, Hawking showed that natural states for free quantum matter fields on classical spacetimes that solve the spherically symmetric vacuum Einstein equations are KMS states of non-vanishing temperature. Although Hawking’s calculation does not include backreaction of matter on geometry, it is more than plausible that the corresponding Hawking radiation leads to black hole evaporation which is in principle observable.
Obviously, an improvement of Hawking’s calculation including backreaction is a problem of quantum gravity. Since no commonly accepted quantum field theory of general relativity is available yet, it has been difficult to reliably derive the backreaction effect. An obvious approach is to use black hole perturbation theory of a Schwarzschild black hole of fixed mass and to quantise those perturbations. But it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations.
In a recent work we proposed a new approach to this problem that applies when the physical situation has an approximate symmetry, such as homogeneity (cosmology), spherical symmetry (Schwarzschild) or axial symmetry (Kerr). The idea, which is surprisingly feasible, is to first construct the non-perturbative physical (reduced) Hamiltonian of the reduced phase space of fully gauge invariant observables and only then to apply perturbation theory directly in terms of observables. The task to construct observables is then disentangled from perturbation theory, thus allowing to unambiguosly develop perturbation theory to arbitrary orders.
In this first paper of the the series we outline and showcase this approach for spherical symmetry and second order in the perturbations for Einstein-Klein-Gordon-Maxwell theory. Details and generalisation to other matter and symmetry and higher orders will appear in subsequent companion papers.
Introduction
In this article, we explore the conclusions of a recent work that presents a new approach to the problem of incorporating backreaction in Hawking’s calculation of black hole evaporation. The authors propose a method that applies perturbation theory directly in terms of observables, disentangling the task of constructing observables from perturbation theory. This approach allows for the unambiguous development of perturbation theory to arbitrary orders.
Challenges and Opportunities
The proposed method offers potential challenges and opportunities for future research in the field of quantum gravity.
- Challenge 1: Lack of Accepted Quantum Field Theory of General Relativity – One major challenge in improving Hawking’s calculation is the absence of a commonly accepted quantum field theory of general relativity. This poses a barrier to reliably deriving the backreaction effect. Future research should focus on developing a quantum field theory that incorporates the principles of general relativity.
- Challenge 2: Reconciling Perturbation Theory with Gauge Invariance – The authors mention that it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations. This challenge must be addressed in order to fully understand and apply the proposed approach. Researchers should explore innovative solutions or alternative frameworks that can accommodate higher order perturbations while maintaining gauge invariance.
- Opportunity 1: Constructing Non-Perturbative Physical Hamiltonian – The authors emphasize the importance of constructing the non-perturbative physical Hamiltonian of the reduced phase space of fully gauge invariant observables. This presents an opportunity for future research to develop robust methods and techniques for constructing observables in various physical situations with approximate symmetry.
- Opportunity 2: Generalization to Other Matter and Symmetry – The current work focuses on spherical symmetry and second order perturbations in Einstein-Klein-Gordon-Maxwell theory. This presents an opportunity for future research to generalize the proposed approach to other matter models, different types of symmetry (e.g., homogeneity, axial symmetry), and higher orders of perturbation. Such generalizations would enhance the applicability of the method and broaden our understanding of black hole evaporation.
Roadmap for the Future
Based on the conclusions and opportunities identified in the article, the following roadmap is suggested for readers and researchers interested in this field:
- Continue researching and developing quantum field theories that unite general relativity and quantum mechanics. This will provide a solid foundation for further investigations into the backreaction effect.
- Explore innovative approaches or alternative frameworks to reconcile perturbation theory with gauge invariance beyond linear perturbations. This will overcome the limitations mentioned by the authors and allow for more comprehensive analyses.
- Investigate methods for constructing non-perturbative physical Hamiltonians of reduced phase spaces with fully gauge invariant observables. This will be crucial in implementing the proposed approach and advancing our understanding of black hole evaporation.
- Extend the proposed approach to other matter models, symmetry types, and higher orders of perturbation. This will provide a more comprehensive understanding of black hole evaporation in various physical situations.
- Read the subsequent companion papers by the authors, which will provide further details and generalizations of the proposed approach.
Conclusion
The recent work outlined in this article presents a new approach to incorporating backreaction in black hole evaporation calculations. While challenges such as the absence of a commonly accepted quantum field theory of general relativity and reconciling perturbation theory with gauge invariance remain, there are opportunities to advance our understanding through innovations in theory construction and generalization to different scenarios. By following the suggested roadmap for future research, readers can contribute to the progress of this field and potentially uncover new insights.