CoChat
Sign Up
Cited by 223Relativity and Gravitational TheoryCosmology and Gravitation TheoriesBlack Holes and Theoretical Physics

Energy conditions in general relativity and quantum field theory

Open access
Kontou, Eleni-Alexandra, Sanders, Ko·2020·Classical and Quantum Gravity
DOI 10.1088/1361-6382/ab8fcf
Open this paper in CoChat — free accountOpen PDF

Abstract

This review summarizes the current status of the energy conditions in general relativity and quantum field theory. We provide a historical review and a summary of technical results and applications, complemented with a few new derivations and discussions. We pay special attention to the role of the equations of motion and to the relation between classical and quantum theories. Pointwise energy conditions were first introduced as physically reasonable restrictions on matter in the context of general relativity. They aim to express e.g. the positivity of mass or the attractiveness of gravity. Perhaps more importantly, they have been used as assumptions in mathematical relativity to prove singularity theorems and the non-existence of wormholes and similar exotic phenomena. However, the delicate balance between conceptual simplicity, general validity and strong results has faced serious challenges, because all pointwise energy conditions are systematically violated by quantum fields and also by some rather simple classical fields. In response to these challenges, weaker statements were introduced, such as quantum energy inequalities and averaged energy conditions. These have a larger range of validity and may still suffice to prove at least some of the earlier results. One of these conditions, the achronal averaged null energy condition, has recently received increased attention. It is expected to be a universal property of the dynamics of all gravitating physical matter, even in the context of semiclassical or quantum gravity.

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1088/1361-6382/ab8fcf

Cites

176 of 177 references verified0 citation errors · 1 not checkable

Statistics

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Within standard general relativity, the NCC is equivalent to the NEC, and the TCC is equivalent to the SEC.

    “Within standard general relativity the NCC is equivalent to the NEC, and the TCC is equivalent to the SEC.”

    From Generic warp drives violate the null energy condition (2022) · cites this paper as [45]

  • NEC violation is known in the case of the Casimir effect and has been shown for tunnelling between degenerate vacua.

    “An important feature related to the above finite-volume effects is the violation of the Null Energy Condition (NEC - see [18, 19] for reviews), which is known in the case of the Casimir effect, and which has been shown in [12, 13] for tunnelling between degenerate vacua.”

    From Double-well instantons in finite volume (2024) · cites this paper as [19]

  • The physicality of warp metrics is established through energy conditions imposed on the stress-energy tensor components.

    “The physicality of warp metrics is established through a set of conditions, known as energy conditions, which are imposed on the stress-energy tensor components [10].”

    From Analyzing warp drive spacetimes with warp factory (2024) · cites this paper as [10]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Cites

Flagged first, then by citations

Research works printed but not linked: 24

  1. Casimir H B G 1948 Proc. Koninklijke Nederl. Akademie Wetenschappen 51 793
  2. Casimir H B G 1997 Proc. Koninklijke Nederl. Akademie Wetenschappen 100N3-4 61 (Engl. transl.)
  3. Ceyhan F and Faulkner T 2018 Recovering the QNEC from the ANEC (arXiv:1812.04683 [hep-th]) · arXiv
  4. Witten E 2019 Light rays, singularities, and all that (arXiv:1901.03928 [hep-th]) · arXiv
  5. Everett A and Roman T 2012 Time Travel and Warp Drives: A Scientific Guide to Shortcuts through Time and Space (Chicago, IL: University of Chicago Press)
  6. Weinberg S 1972 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity vol 1 (New York: Wiley)
  7. Maeda H and Martinez C 2018 Energy conditions in arbitrary dimensions (arXiv:1810.02487 [gr-qc]) · arXiv
  8. Ehlers J 1961 Akad. Wiss. Lit. Abh. Math.-Natwiss. Kl. (Mainz) 11 792-837 (Engl. transl.) · DOI
  9. Feynman R P, Morinigo F B and Wagner W G 1995 Feynman Lectures on Gravitation (Reading, MA: Addison-Wesley)
  10. Pirani F A E 1956 Acta Phys. Pol. 15 389-405 (Engl. transl.) · DOI
  11. Martín-Moruno P and Visser M 2017 Classical and Semi-Classical Energy Conditions (Cham: Springer) pp 193-213
  12. Choquet-Bruhat Y 2009 General Relativity and the Einstein Equations (Oxford: Oxford University Press)
  13. Haag R 2012 Local Quantum Physics: Fields, Particles, Algebras (Berlin: Springer)
  14. Fewster C J 2017 Quantum Energy Inequalities (Cham: Springer) pp 215-54
  15. Ford L H and Roman T A 1995 Averaged energy conditions and quantum inequalities Phys. Rev. D 51 4277-86 · DOI · Paper page
  16. Graham N and Olum K D 2004 Phys. Rev. D 69 109901 (erratum) · DOI
  17. Fewster C J and Roman T A 2009 Phys. Rev. D 80 069903 (erratum) · DOI · Paper page
  18. Mandelstam L and Tamm I 1991 The Uncertainty Relation between Energy and Time in Non- Relativistic Quantum Mechanics (Berlin: Springer) pp 115-23
  19. Kontou E-A 2015 Averaged null energy condition and quantum inequalities in curved spacetime PhD Thesis Tufts University
  20. Kelly W R and Wall A C 2015 Phys. Rev. D 91 069902 (erratum) · DOI
  21. Wald R M 1994 Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, IL: University of Chicago Press)
  22. Lesourd M 2017 Hawking's area theorem with a weaker energy condition (arXiv:1711.06480 [gr-qc]) · arXiv
  23. Friedman J L, Schleich K and Witt D M 1995 Phys. Rev. Lett. 75 1872 (erratum) · DOI
  24. Hawking S W 1976 Commun. Math. Phys. 46 206 (erratum) · DOI · Paper page

Cited by

Sorted by citations

Related

Authors
Kontou, E.-A., & Sanders, K.
Affiliations
  • College of the Holy Cross (US)
  • Dublin City University (IE)
ORCID
Venue
Classical and Quantum Gravity
Publisher
IOP Publishing
Published
2020-09-18
Volume
37
Issue
19
Pages
193001
ISSN
  • 0264-9381 (ISSN-L)
  • 1361-6382
DOI
10.1088/1361-6382/ab8fcf
OpenAlex
W3009127927
Cited by
223
References
172
Citations by year
2026: 33 · 2025: 38 · 2024: 56 · 2023: 42 · 2022: 29 · 2021: 22 · 2020: 3
FWCI
10.70
Type
journal article
Language
en
Open access
Licence
Locations
  • Classical and Quantum Gravity (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • UvA-DARE (University of Amsterdam) (submittedVersion)
  • Data Archiving and Networked Services (DANS) (submittedVersion)
  • arXiv (Cornell University)
Funders
H2020 Marie Skłodowska-Curie Actions (744037)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
energy conditions, quantum inequalities, averaged null energy condition, singularity theorems, wormholes, exotic matter, semiclassical gravity, quantum field theory in curved spacetime, equations of motion

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review