Semiclassical Gravity Solutions in Anti-de Sitter Spacetime with a Klein-Gordon Field

Core Concepts

This paper presents a method for obtaining exact semiclassical gravity solutions in Anti-de Sitter spacetime for a Klein-Gordon field with Dirichlet or Neumann boundary conditions, highlighting the simplification of the Hadamard renormalization procedure due to the spacetime's maximal symmetry.

Abstract

Bibliographic Information:

Benito A. Ju´arez-Aubry and Milton C. Mamani-Leqque. (2024). Renormalisation in maximally symmetric spaces and semiclassical gravity in Anti-de Sitter spacetime. arXiv, 2411.06834v1.

Research Objective:

This paper aims to obtain exact semiclassical gravity solutions in the Poincaré fundamental domain of (3+1)-dimensional Anti-de Sitter spacetime (PAdS4) for a Klein-Gordon field with Dirichlet or Neumann boundary conditions.

Methodology:

The authors utilize the Hadamard renormalization procedure to calculate the expectation value of the stress-energy tensor for a Klein-Gordon field in maximally symmetric spacetimes. They exploit the spacetime symmetries to simplify the Hadamard recursion relations and obtain closed-form expressions for the stress-energy tensor in PAdS4.

Key Findings:

• The Hadamard bi-distribution in maximally symmetric spacetimes is invariant under isometries and depends only on the geodesic distance, simplifying renormalization computations.
• Closed-form expressions for the expectation value of the stress-energy tensor are derived for both Dirichlet and Neumann boundary conditions in PAdS4.
• The authors present a method for obtaining semiclassical gravity solutions in PAdS4 by solving an algebraic equation derived from the semiclassical Einstein field equations.

Main Conclusions:

The paper demonstrates the feasibility of obtaining exact semiclassical gravity solutions in PAdS4 with specific boundary conditions. The simplified renormalization procedure due to spacetime symmetries provides a framework for studying semiclassical gravity in other maximally symmetric spacetimes.

Significance:

This research contributes to the understanding of semiclassical gravity in Anti-de Sitter spacetime, which is relevant to the AdS/CFT correspondence and holography. The findings have implications for studying quantum field theory in curved spacetimes and exploring the stability properties of semiclassical AdS.

Limitations and Future Research:

The study focuses on Dirichlet and Neumann boundary conditions. Future research could explore other boundary conditions and their impact on semiclassical gravity solutions. Additionally, investigating the stability of these solutions and extending the analysis to asymptotically AdS spacetimes are promising avenues for further investigation.

Stats

The spacetime dimension considered is n = 4.
The Breitenlohner-Freedman bound is ν = √(9/4 + ℓ²m² - 12ξ) ≥ 0.
For the massless minimally coupled field, Λ attains a minimum value when ℓ = √(29GN/180), for which Λ = -270π/(29GN).

Quotes

Key Insights Distilled From

Renormalisation in maximally symmetric spaces and semiclassical gravity in Anti-de Sitter spacetime

by Beni... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06834.pdf

Deeper Inquiries

How do different boundary conditions, beyond Dirichlet and Neumann, affect the semiclassical gravity solutions in AdS spacetime and their stability?

Answer:
Exploring semiclassical gravity solutions in AdS spacetime with boundary conditions beyond Dirichlet and Neumann is a complex yet fascinating avenue. Here's a breakdown of the potential effects and challenges:
1. Breaking of Symmetry and Computational Complexity:

Robin Boundary Conditions: Robin conditions, a linear combination of Dirichlet and Neumann, are a natural starting point. However, they generally break the maximal symmetry of AdS. This complicates calculations significantly. The Hadamard bi-distribution, for instance, would no longer be a simple function of geodesic distance alone.
Dynamical/Wentzell Boundary Conditions: These conditions involve time derivatives of the field at the boundary, introducing additional degrees of freedom and potentially richer dynamics. They are particularly relevant in holographic renormalization and the study of the dynamical Casimir effect. However, they further increase the complexity of finding exact solutions.
2. Impact on Stress-Energy Tensor and Semiclassical Einstein Equations:

Modified Vacuum Energy: Different boundary conditions alter the vacuum fluctuations of the quantum field. This directly impacts the renormalized stress-energy tensor, which acts as a source in the semiclassical Einstein equations. Consequently, the relationship between the cosmological constant (Λ), the AdS radius of curvature (ℓ), and the field parameters (mass m, coupling ξ) will be modified.
New Solution Space: The space of semiclassical solutions (Λ, ℓ, m, ξ) will be different for each type of boundary condition. It's even possible that certain boundary conditions could lead to solutions that are qualitatively distinct from the Dirichlet and Neumann cases.
3. Stability Considerations:

Perturbative Analysis: Analyzing the stability of these solutions would likely involve perturbing around known solutions (e.g., those with Dirichlet or Neumann conditions) and examining the behavior of the perturbations.
Boundary-Driven Instabilities: New types of instabilities could arise due to the boundary conditions themselves. For example, certain boundary conditions might amplify fluctuations near the boundary, potentially leading to a breakdown of the semiclassical approximation.
4. Holographic Implications:

Dual Field Theories: In the context of the AdS/CFT correspondence, different boundary conditions in the bulk AdS spacetime correspond to different choices of the dual conformal field theory (CFT) living on the boundary. Exploring these less conventional boundary conditions could provide insights into strongly coupled CFTs with unusual properties.
In summary: Going beyond Dirichlet and Neumann boundary conditions in semiclassical AdS gravity is a challenging but potentially rewarding endeavor. It requires tackling increased mathematical complexity but promises to reveal a richer landscape of solutions and deepen our understanding of quantum fields in curved spacetime and holography.

Could the simplified renormalization procedure presented be extended to other quantum field theories in maximally symmetric spacetimes beyond the Klein-Gordon field?

Answer:
Yes, the simplified renormalization procedure based on the symmetry properties of maximally symmetric spacetimes has the potential to be extended to other quantum field theories beyond the Klein-Gordon field. Here's a breakdown of the key points and potential challenges:
Reasons for Extensibility:

Isometry Invariance: The core principle underlying the simplification is the invariance of the Hadamard bi-distribution under the isometries of the spacetime. This property is not unique to the Klein-Gordon field. In maximally symmetric spacetimes, the Hadamard bi-distribution for any quantum field theory obeying the appropriate wave equation should inherit this symmetry.

Dependence on Geodesic Distance: As a consequence of isometry invariance, the Hadamard bi-distribution can be expressed as a function of the geodesic distance (or Synge's world function) alone. This significantly reduces the complexity of the Hadamard recursion relations and simplifies the renormalization procedure.

Potential Applications:

Free Scalar Fields with Different Spins: The extension to free scalar fields with higher spins (e.g., spin-1 Proca fields, spin-2 linearized gravity) should be relatively straightforward. The Hadamard bi-distributions for these fields would have a similar structure, and the renormalization of their stress-energy tensors would follow a similar pattern.

Free Fermionic Fields: Extending the procedure to free fermionic fields (e.g., Dirac spinors) is more involved but feasible. The key difference is that the Hadamard bi-distribution for fermions is a bi-spinor object. However, the underlying symmetry arguments still apply, and one could construct the bi-spinor Hadamard bi-distribution using the appropriate Green's functions and taking advantage of the spacetime's symmetries.

Interacting Theories (Challenges): Extending the method to interacting quantum field theories poses significant challenges. The presence of interactions typically breaks the explicit dependence of the Hadamard bi-distribution on geodesic distance alone. However, in certain cases with conformal symmetry or supersymmetry, simplifications might still be possible.

In conclusion: The simplified renormalization procedure based on the symmetries of maximally symmetric spacetimes holds promise for application to various quantum field theories beyond the Klein-Gordon field. While extensions to interacting theories present significant hurdles, the method provides a valuable tool for studying free fields in these highly symmetric backgrounds.

How does the presence of quantum fields and their backreaction on the spacetime geometry influence the causal structure and holographic properties of AdS spacetime?

Answer:
The backreaction of quantum fields on the geometry of AdS spacetime leads to intriguing consequences for both its causal structure and its holographic properties:
1. Modifications to Causal Structure:

Shifting of the Boundary: The presence of a quantum field's stress-energy tensor, even in a seemingly stable AdS background, can subtly alter the effective cosmological constant and AdS radius. This shift can be viewed as a "quantum-corrected" AdS boundary.
Potential for Singularities: In more extreme cases, if the backreaction becomes sufficiently strong (e.g., due to strong field fluctuations or instabilities), it could potentially lead to the formation of singularities, altering the global causal structure of the spacetime. This is an area of active research, particularly in the context of understanding the stability of AdS spacetime.
2. Impact on Holographic Properties:

Dual CFT Interpretation: In the AdS/CFT correspondence, the backreaction of the quantum field in the bulk AdS spacetime corresponds to quantum corrections to the dual conformal field theory (CFT) living on the boundary.
Changes in Correlation Functions: These quantum corrections modify the correlation functions of operators in the CFT. Studying these changes provides insights into the behavior of strongly coupled CFTs.
Entanglement Entropy and Boundary Effects: The entanglement entropy of a region in the CFT is related to the geometry of extremal surfaces in the bulk AdS spacetime. Backreaction from quantum fields can modify these surfaces, leading to corrections to entanglement entropy calculations and revealing subtle correlations between the bulk and boundary theories.
Holographic Renormalization and Boundary Conditions: The process of holographic renormalization, which relates divergences in the bulk gravity theory to boundary counterterms in the CFT, is intimately connected to the choice of boundary conditions for the quantum fields. Backreaction effects further highlight the interplay between bulk dynamics and boundary conditions in the holographic dictionary.
3. Open Questions and Future Directions:

Stability of AdS: A major open question is whether the backreaction of quantum fields could render AdS spacetime unstable, potentially leading to a dynamical evolution towards a different spacetime.
Quantum Gravity Effects: Understanding the full backreaction of quantum fields, especially in regimes where quantum gravity effects become important, remains a significant challenge. This is crucial for addressing questions about the stability and ultimate fate of AdS spacetime in a complete theory of quantum gravity.
In summary: The backreaction of quantum fields on AdS spacetime is a rich and multifaceted phenomenon with profound implications for both the spacetime's causal structure and its holographic interpretation. It provides a window into the interplay between gravity, quantum fields, and the holographic principle, offering valuable insights into the nature of quantum gravity and strongly coupled quantum field theories.