Energy conditions in general relativity and quantum field theory
Eleni-Alexandra Kontou · Ko Sanders
Abstract and summary · read the original at the source
In one page
Einstein’s equation on its own puts no limit on what shape spacetime can take. Choose any geometry you like — a wormhole, a warp bubble — and you can always solve backwards for the matter that would produce it. Energy conditions are the rulebook physicists added to stop that from being free: short statements such as ‘the energy an observer measures is never negative’, strong enough to prove the singularity theorems and the black-hole results. Eleni-Alexandra Kontou and Ko Sanders review the whole subject, and their central point is that every one of the original point-by-point conditions is broken systematically by quantum fields, and by some quite ordinary classical fields too. The field answered with weaker and more durable rules: quantum energy inequalities, which say exactly how far the energy may dip below the ambient vacuum level and for how long, and averaged conditions taken along a whole light ray. One survivor stands out, the achronal averaged null energy condition, which the authors expect to hold universally.
Why it matters hereThis is the rulebook chapter 4 is written against. Every warp and wormhole geometry on this site is finally scored on how much below-ambient-vacuum energy it needs and over what region, and this review is the single place where all the versions of that constraint, and all the loopholes still open in them, are set out together with the theorems attached. Chapter 2 gets the mechanism underneath — the reason the point-by-point rules fail at all is that a structured quantum vacuum is entitled to negative expectation values — and chapter 6 gets the accounting language, because the same inequalities govern what a vacuum device may draw and give back. Read it beside the smeared null energy condition at /library/stm-75affcb30e, the four-dimensional null result at /library/stm-d1e51c2294, and the Ford and Roman inequalities at /library/stm-5e822858ce.
What it claims
01Every pointwise energy condition is systematically violated by quantum fields. The argument goes back to Epstein, Glaser and Jaffe: a self-adjoint local operator whose vacuum expectation value is zero either vanishes identically or must admit negative measurement values, which for the energy density means the density is somewhere negative in some state.Sect. 3, Theorems 3.1 and 3.2 and the paragraph that follows them, p. 17
Settled physics02The first quantum energy inequality was Ford’s in 1978: if the magnitude of a negative energy flux and the time it lasts are tied together, with the flux bounded by the inverse square of the duration, then no macroscopic violation of the second law of thermodynamics can be arranged from it. Free quantum scalar fields in two-dimensional flat spacetime were shown to obey a bound of that form even though they violate the classical conditions.Sect. 3, Eq. 47 and the discussion of Ford’s 1978 paper preceding it
Settled physics03Semiclassical exotic geometries were first made concrete with a Casimir plate system as the below-ambient-vacuum source, in Morris, Thorne and Yurtsever’s wormhole. Ford and Roman then used quantum energy inequalities to bound them: a static wormhole is possible semiclassically only if its throat is close to Planck size, or the negative energy is concentrated in bands many orders of magnitude smaller than the throat; for an Alcubierre bubble the wall thickness is constrained to a few hundred Planck lengths, and for a Krasnikov tube a metre long and a metre across the total negative energy comes out at the order of ten to the sixteenth galactic masses.Sect. 5.3, the paragraphs on Ford and Roman, Everett and Roman, and Ford and Pfenning
Published and peer-reviewed04The review names what those bounds leave open rather than closing the subject: atomic-size warp bubbles, acquiring the required negative energy from several quantum fields at once, and the minimum violations of the averaged conditions quantified by Visser, Kar and Dadhich. Counter-examples to the averaged null energy condition in semiclassical gravity are also known, so certain possibilities remain.Sect. 5.3, the paragraph beginning ‘Even though these results significantly constrain exotic spacetimes’
What to watch05The sharpest single theorem in the applications is Graham and Olum’s, quoted here as Theorem 5.9: an asymptotically flat, globally hyperbolic spacetime that obeys the on-shell self-consistent achronal averaged null energy condition together with the generic condition cannot have a compactly generated Cauchy horizon. The authors note one class that lies outside its reach — long wormholes, such as the Maldacena, Milekhin and Popov solution, where the passage takes longer than the ambient route and the geodesics through it stay chronal.Sect. 5.3, Theorem 5.9 and the closing paragraph on long wormholes
Published and peer-reviewed06The authors’ own forward question is the achronal averaged null energy condition itself: it is the only condition that may hold in full generality on-shell, for classical scalar fields it holds for any coupling once trans-Planckian values are excluded, and in the semiclassical treatment with transversal smearing over Planck-scale distances no counter-example is known. They expect a deeper reason for it in quantum gravity, and say the search for that reason may be an important path towards understanding quantum gravity itself.Sect. 6, Outlook
What to watch
Read it · abstract
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.
Eleni-Alexandra Kontou, Department of Mathematics, University of York, and Department of Physics, College of the Holy Cross; Ko Sanders, School of Mathematical Sciences and Centre for Astrophysics and Relativity, Dublin City University. Classical and Quantum Gravity 37, 193001 (2020).
(Abstract only. The complete review is free to read at https://arxiv.org/abs/2003.01815 and via https://doi.org/10.1088/1361-6382/ab8fcf — see the rights note for why the full text is not reproduced here.)
The way in
https://doi.org/10.1088/1361-6382/ab8fcfPublished as Classical and Quantum Gravity 37, 193001 (2020), a fifty-page invited review. LICENCE. Checked directly rather than taken from an aggregator label, and no Creative Commons statement was found on either side: the Crossref record for this DOI carries only IOP’s standard publishing licence and its text-and-data-mining page, the Unpaywall record returns oa_status bronze with a null licence, and the arXiv posting of the same manuscript, 2003.01815v2 of 5 June 2020, is filed under arXiv’s non-exclusive distribution licence, which grants arXiv distribution rights only and no re-use. So this page carries the summary, the claims and the authors’ own abstract, and sends the reader to the full text at the source. TEXT. The summary and the claims below were written from the complete arXiv manuscript, whose abstract is word-for-word the published one; section, theorem and equation numbers in the locators are the review’s own.
How to cite it
Eleni-Alexandra Kontou, Ko Sanders (2020) Energy conditions in general relativity and quantum field theory. doi:10.1088/1361-6382/ab8fcf
Where it sits in the curriculum
The metric, warp drives and wormholesWhat the vacuum isEnergy from the vacuum