Energy Conditions  

General Idea > s.a. quantum field theory effects; singularities; energy-momentum tensor.
* 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; There are indications that the energy conditions should be thought of as conditions on the geometry, rather than the matter.
* 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.
@ General references: Visser & Barceló gq/00-conf; Carter gq/02-conf [and vacuum stability]; Barceló & Visser IJMPD(02)gq-GRF; Curiel a1405 [primer, conceptual]; Wall PRL(17)-a1701 [novel method for deriving energy conditions, classical and quantum]; Martín-Moruno & Visser a1702-ch [and semiclassical]; Kontou & Sanders a2003-CQG [rev].
@ Extensions: Martín-Moruno & Visser PRD(13)-a1305, JHEP(13)-a1306 [flux energy conditions and other semiclassical replacements]; Martín-Moruno & Visser a1510-MG14 [non-linear energy conditions and their quantum extensions]; Maeda & Martínez a1810 [in arbitrary dimensions].
@ In modified gravity: Santos et al PRD(07)-a0708 [f(R) theories]; Capozziello et al PLB(14)-a1312; Capozziello et al PRD(15)-a1407 [in extended theories]; Zubair & Waheed ASS(15)-a1502 [f(T) gravity]; Parikh & van der Schaar PRD(15)-a1406 [derivation of null energy condition from worldsheet string theory]; Bamba et al GRG(17) [f(G) gravity]; Capozziello et a PLB(18)-a1803 [f(R) cosmology]; Ilyas IJGMP(19)-a1907 [in non-local gravity].
@ Operationally, with detectors: Helfer gq/96, CQG(98)gq/97.
@ At bounces: Tippett & Lake gq/04; Giovannini PRD(17)-a1708 [averaged energy conditions].
@ Other cosmology: Gong & Wang PLB(07)-a0705 [acceleration]; Lima et al proc(10)-a0812.
@ Worldline quantum inequalities: Fewster CQG(00)gq/99, PRD(04)gq, & Verch CMP(02)mp/01 [Dirac fields in curved spacetime].
> Related topics: see causality violations; tests of general relativity with light; types of higher-order gravity theories.
> Online resources: see Wikipedia page.

Null Energy Condition
Def: A stress-energy tensor \(T_{ab}\) satisfies the null energy condition if

\[ T_{ab}\,l^al^b \ge 0\;,\quad{\rm for\ any\ null\ vector}\ l^a\;. \]

@ Proofs: Parikh & Svesko PRD(17)-a1511 [from the second law of thermodynamics]; Parikh IJMPD(15)-a1512 [from string theory and thermodynamics]; Koeller & Leichenauer PRD(16)-a1512 [holographic].
@ Quantum null energy condition: Bousso et al PRD(16)-a1509; Fu et al CQG(17)-a1706 [in curved space]; Balakrishnan et al a1706 [general proof]; Fu & Marolf PRL(18)-a1711; Malik & Lopez-Mobilia a1910 [free fermionic field theories].

Weak Energy Condition > s.a. cosmological expansion [constraint on history].
* Idea: Energy density and pressure satisfy ρ + 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

γ Tab l a l b dλ ≥ 0 ,

for any inextendible null geodesic γ with tangent vector l a.
@ General references: in Visser PRD(90); Yurtsever CQG(90); Fewster & Osterbrink PRD(06)gq [non-minimally coupled scalar]; Kontou PhD-a1507 [and quantum inequalities].
@ 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]; Kelly & Wall PRD(14)-a1408 [holographic proof]; Hartman et al JHEP(17)-a1610 [from microcausality].
@ 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]; Urban & Olum PRD(10)-a1002 [and violations].

Dominant Energy Condition
* Idea: Energy density and pressure satisfy ρ ≥ 0 and |p| ≤ ρ.
$ 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 ≥ − \(1\over2\)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.
@ References: Zaslavskii PLB(10)-a1004 [and regular spherical black holes].

Violations > s.a. cosmic strings; QED; quantum field theory effects [negative en density]; quantum field theory effects in curved spacetime.
* Of nec: The nec can be violated in a consistent way in models with unconventional kinetic terms, such as Galileon theories and their generalizations.
* 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; Visser & Barceló gq/00-conf [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]; Jamil et al GRG(12)-a1211 [FLRW models in generalized teleparallel gravities].
@ 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; Rubakov PRD(13)-a1305, PU(14)-a1401 [and universe creation in the laboratory]; Elder et al PRD(14) [solution evolving between satisfying and violating the null energy condition]; Krommydas a1806-MS [and implications].
@ Averaged null energy condition: Urban & Olum PRD(10)-a0910 [violation in conformally flat spacetime]; Kontou & Olum PRD(15)-a1507 [proof].
@ 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-MG8.
@ Wormholes: Barceló & Visser CQG(00)gq, NPB(00)ht [brane world]; Kar et al Pra(04)gq [quantification]; Roman gq/04-MGX.
@ And second law: Ford & Roman PRD(01)gq/00; Davies & Ottewill PRD(02)gq.

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