Energy Conditions  

General Idea > s.a. quantum field theory effects; singularities; energy-momentum.
* Idea: We try to define in some way the notion of positivity of the local energy density, even without having a definition of energy density.
* Remark: If we try to think of the Einstein equation as giving Tab once we specify some gab, the problem is that the solution will in general not satisfy the energy conditions.
* Negative energy densities: They are predicted for quantum fields in black-hole radiation; In addition, all of the local conditions below have been experimentally tested in the lab, and shown not to hold for the Casimir effect (s.a. refs on Lorentzian wormholes); It is not clear whether the averaged WEC holds in those cases, but it seems that it could be violated as well.

Weak Energy Condition > s.a. cosmological expansion [constraint on history].
* Idea: Energy density and pressure satisfy + p 0.
$ Def: A stress-energy tensor Tab satisfies the weak energy condition if

Tab t a t b 0 ,   for any causal vector t a.

@ References: Roman PRD(86) [in quantum field theory]; Bellucci & Faraoni NPB(02)ht/01 [non-minimal scalar field, and definition of Tab].

Averaged Null / Weak Energy Condition > s.a. anomalies.
$ Def: A stress-energy tensor Tab satisfies the averaged energy condition if

gamma Tab l a l b d 0 ,

for any inextendible null geodesic with tangent vector l a.
@ General references: in Visser PRD(90); Yurtsever CQG(90); Fewster & Osterbrink PRD(06)gq [non-minimally coupled scalar].
@ In quantum field theory: Yurtsever PRD(95)gq/94, PRD(95)gq; Verch JMP(00)mp/99 [2D]; Fewster & Roman PRD(03)gq/02; Fewster et al PRD(07)gq/06 [spacetimes with boundaries].
@ Variations: Hayward PRD(95)gq/94, CQG(94)gq [quasilocal]; Graham & Olum PRD(05)ht, Graham JPA(06)in [in Casimir effect situations]; Graham & Olum PRD(07)-a0705 [achronal averaged null energy condition].

Dominant Energy Condition
* Idea: Energy density and pressure satisfy 0 and |p| .
$ Def: A stress-energy tensor Tab satisfies the dominant energy condition if

Tab t a t'b 0 ,   for any two future directed causal vectors t a, t'a.

* Relationships: This condition implies the WEC, and is stronger that the positivity of the local energy seen by any observer; It is equivalent to requiring that the local four-momentum Tab t a seen by any observer be a future-directed timelike or null vector (the speed of energy flow does not exceed the speed of light).

Strong Energy Condition
$ Def: A stress-energy tensor Tab is said to satisfy the strong energy condition if (T:= T aa)

Tab t a t b – T,   for any unit timelike vector t a.

* Relationships: The strong energy condition does not imply the WEC, unless in the definition of the latter we replace "... any timelike vector t" by "... any null vector t", but the former does appear to be a stronger physical requirement.
* Applications: Observations suggest that it was violated sometime between galaxy formation and the present.

Other References > s.a. causality violations; tests of general relativity with light.
@ General: Visser & Barceló gq/00-in; Carter gq/02-in [and vacuum stability]; Barceló & Visser IJMPD(02)gq-GRF; Santos et al PRD(07)-a0708 [in f(R) gravity].
@ Operationally, with detectors: Helfer gq/96, CQG(98)gq/97.
@ And cosmology: Tippett & Lake gq/04 [at bounces]; Gong & Wang PLB(07)-a0705 [acceleration]; Lima et al a0812-in.
@ Worldline quantum inequalities: Fewster CQG(00)gq/99, PRD(04)gq, & Verch CMP(02)mp/01 [Dirac fields in curved spacetime].

Violations > s.a. cosmic strings; QED; quantum field theory effects [negative en density]; quantum field theory effects in curved spacetime.
* In quantum field theory in curved spacetime: One issue is that the gravitational field will produce vacuum polarization, and the corresponding stress-energy tensor may not satisfy the energy conditions.
@ In cosmology: Borde & Vilenkin PRD(97) [inflation]; Visser PRD(97)gq; Barceló & Visser gq/00-in [implications]; Aref'eva & Volovich TMP(08)ht/06 [consistency of models]; Santos et al PRD(07)ap/07, PRD(07)-a0706 [and conditions on expansion], Lima et al PLB(08)-a0808 [and acceleration, supernova data]; Cattoën & Visser CQG(08) [parameters].
@ Nec violation and instabilities: Buniy & Hsu PLB(06)ht/05; Dubovsky et al JHEP(06)ht/05; Creminelli et al JHEP(06)ht [violation without instabilities and cosmology]; Buniy et al PRD(06)ht; Urban Olum a0910 [anec violation in conformally flat spacetime].
@ In quantum field theory in curved spacetime: Visser PRD(96)gq [Hartle-Hawking vacuum], PRD(96)gq [Boulware vacuum], PRD(96)gq [1+1 Schwarzschild], PRD(97)gq [Unruh vacuum]; Xiong & Zhu IJMPA(07)gq/06 [strong energy condition in lqg].
@ In semiclassical general relativity: Flanagan & Wald PRD(96) [back-reaction and ANEC]; Visser gq/97-in.
@ Wormholes: Barceló & Visser CQG(00)gq, NPB(00)ht [brane world]; Kar et al Pra(04)gq [quantification]; Roman gq/04-in.
@ And second law: Ford & Roman PRD(01)gq/00; Davies & Ottewill PRD(02)gq.

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 1 nov 2009