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 *T*_{ab}
once we specify some g_{ab}, 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].

@ __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-a1803 [*f*(*R*) cosmology].

@ __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}

*T*_{ab} *l*^{ a} *l*^{ b} ≥ 0
, for any null vector *l*^{ a}.

@ __Proofs__: Parikh & Svesko a1511 [from the second law of thermodynamics];
Parikh IJMPD(15)-a1512 [from string theory and thermodynamics];
Koeller & Leichenauer 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 a1711.

**Weak Energy Condition** > s.a. cosmological expansion [constraint on history].

* __Idea__: Energy density
and pressure satisfy *ρ* + *p* ≥ 0.

$ __Def__: A stress-energy
tensor *T*_{ab} satisfies
the weak energy condition if

*T*_{ab} *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 *T*_{ab}].

**Averaged Null / Weak Energy Condition** > s.a. anomalies.

$ __Def__: A stress-energy
tensor *T*_{ab} satisfies
the averaged energy condition if

∫_{γ}* T*_{ab} *l*^{ a} *l*^{ b} d*λ* ≥ 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 *T*_{ab} satisfies the dominant energy condition if

*T*_{ab} *t*^{ a }*t' ^{b}* ≥ 0
, for any two future directed causal vectors

* __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 *T*_{ab} *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
*T*_{ab} is said to satisfy the strong energy
condition if (*T*:= *T*^{ a}_{a})

*T*_{ab} *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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