Gravitational vacuum polarization. III. Energy conditions in the (1+1)-dimensional Schwarzschild spacetime
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Building on techniques developed in the two preceding papers, I investigate the various pointwise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field in the (1+1)-dimensional Schwarzschild spacetime. Because the stress-energy tensors are analytically known, I can get exact results for the Hartle-Hawking, Boulware, and Unruh vacua. This exactly solvable model serves as a useful check on my (3+1)-dimensional investigations wherein I had to resort to a mixture of analytic approximations and numerical techniques. The key results in 1+1 dimensions are that (1) NEC is satisfied outside the event horizon for the Hartle-Hawking vacuum and violated for the Boulware and Unruh vacua, and (2) DEC is violated everywhere in the spacetime (for any quantum state, not just the standard vacuum states).
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There are signs that the weak and null energy conditions may be violated at the quantum-microscopic scale, although they seem to be satisfied at macroscopic scales.
“Itis worth mentioning that there are signs that WEC and NECmaybeviolated at the quantum-microscopic scale [4049], although they seem to be satisfied at the macroscopic scales.”
From Alcubierre warp drive in spherical coordinates with some matter configurations (2023) · cites this paper as [43]
The paper’s summary (p. 1) reports NEC violations in the Boulware and Unruh vacua, and its Hartle–Hawking summary (p. 4) reports WEC violations in part of the spacetime. It does not establish a microscopic-versus-macroscopic distinction: the Discussion (pp. 6–7) instead describes violations throughout the exterior region in the Boulware vacuum. The claim that WEC and NEC seem satisfied at macroscopic scales is therefore not supported as a general statement. (LLM, conf=0.85)
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References 17 · Cited by 50
- Authors
- Visser, M.
- Affiliations
- Washington University in St. Louis (US)
- Corresponding author
- Visser, Matt
- ORCID
- Visser, Matt
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 1996-10-15
- Volume
- 54
- Issue
- 8
- Pages
- 5123-5128
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2064478437
- Cited by
- 50
- References
- 17
- Citations by year
- 2025: 3 · 2024: 2 · 2023: 2 · 2022: 1 · 2021: 2 · 2020: 1 · 2019: 5 · 2018: 2 · 2017: 4 · 2016: 2 · 2015: 2 · 2014: 4 · 2013: 2 · 2012: 3
- FWCI
- 1.62
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Electrodynamics and Casimir Effect
- Keywords
- energy conditions, quantum stress-energy tensor, conformally coupled scalar field, Schwarzschild metric, 1+1 dimensional gravity, averaged null energy condition, dominant energy condition, Hartle-Hawking vacuum, Boulware vacuum, Unruh vacuum, event horizon, quantum field theory in curved spacetime
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