Historical research guide · 0400 · 2020
Energy conditions in general relativity and quantum field theory
Eleni-Alexandra Kontou · Ko Sanders
In one minute
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.
Connection
This 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.
From the source
Every 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
The 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
Semiclassical 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
The 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’
The 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
The 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
The way in
https://doi.org/10.1088/1361-6382/ab8fcfHow to cite it
Eleni-Alexandra Kontou, Ko Sanders (2020) Energy conditions in general relativity and quantum field theory doi:10.1088/1361-6382/ab8fcf