Classical and semi-classical energy conditions
CClassical and semi-classical energy conditions Prado Martin-Moruno a and Matt Visser ba Departamento de F´ısica Te´orica I, Ciudad Universitaria, Universidad Complutense de Madrid,E-28040 Madrid, Spain b School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand
E-mail: [email protected] ; [email protected] Abstract:
The standard energy conditions of classical general relativity are (mostly)linear in the stress-energy tensor, and have clear physical interpretations in terms ofgeodesic focussing, but suffer the significant drawback that they are often violatedby semi-classical quantum effects. In contrast, it is possible to develop non-standardenergy conditions that are intrinsically non-linear in the stress-energy tensor, and whichexhibit much better well-controlled behaviour when semi-classical quantum effects areintroduced, at the cost of a less direct applicability to geodesic focussing. In thischapter we will first review the standard energy conditions and their various limitations.(Including the connection to the Hawking–Ellis type I, II, III, and IV classification ofstress-energy tensors). We shall then turn to the averaged, nonlinear, and semi-classicalenergy conditions, and see how much can be done once semi-classical quantum effectsare included. Draft chapter, on which the related chapter of the book
Wormholes, Warp Drives and EnergyConditions (to be published by Springer) will be based. a r X i v : . [ g r- q c ] M a r ontents The energy conditions, be they classical, semiclassical, or “fully quantum” are at heartpurely phenomenological approaches to deciding what form of stress-energy is to beconsidered “physically reasonable”. As Einstein put it, the LHS of his field equations G ab = 8 πG N T ab represents the “purity and nobility” of geometry, while the RHSrepresents “dross and uncouth” matter. If one then makes no assumptions regarding thestress-energy tensor, then the Einstein field equations are simply an empty tautology.It is only once one makes some assumptions regarding the stress-energy tensor, (zerofor vacuum, “positive” in the presence of matter), that solving the Einstein equationsbecomes closely correlated with empirical reality. The energy conditions are simplyvarious ways in which one can try to implement the idea that the stress-energy tensorbe “positive” in the presence of matter, and that gravity always attracts. In this chapterwe shall first discuss the standard point-wise energy conditions, and then move on totheir generalizations obtained by averaging along causal curves, and some nonlinearvariants thereof, and finally discuss semi-classical variants.– 1 – Standard energy conditions
General relativity, as well as other metric theories of gravity, proposes a way in whichthe matter content affects the curvature of the spacetime. The theory of gravity itself,however, does not per se tell us anything specific about the kind of matter contentthat we should consider. If we are interested in extracting general features about thespacetime, independently of the form of the stress-energy tensor of matter fields, wehave to consider purely geometric expressions such as the Raychaudhuri equation. Thisequation describes the focussing of a congruence of timelike curves in a given geometry.Particularized to geodesic motion it can be expressed as [1]d θ d s = ω ab ω ab − σ ab σ ab − θ − R ab V a V b , (2.1)where ω ab is the vorticity, σ ab is the shear, θ is the expansion, and V a is the timelikeunit vector tangent to the congruence. Note that for a congruence of time-like geodesicswith zero vorticity, for which V a ∝ ∇ a Φ( x ) with Φ( x ) a scalar field [2], the only term onthe RHS of equation (3.2) that can be positive is the last term. Hence, if R ab V a V b ≥ Time-like convergence condition (TCC): Gravity is always attractive provided R ab V a V b ≥ f or any timelike vector V a . (2.2)Note that although the TCC ensures focussing of timelike geodesics the contrarystatement is not necessarily true, that is R ab V a V b < R ab depends on the stress-energy tensor through theequations of the dynamics, the TCC imposes restrictions on the material content thatdepends on the specific theory of gravity.Analogously, one can take the equation for the focussing of a congruence of nullgeodesics with vanishing vorticity, and formulate the null convergence condition (NCC).This is a particular limit of the TCC and requires that R ab k a k b ≥ k a . It should be noted that hypothetical astronomical objects like wormholes wouldnecessarily violate the NCC [8–10]. Furthermore, this condition might also be violatedduring the early universe if one wishes to avoid the big bang singularity [3, 11, 12]. There have been interesting attempts to derive the NCC from an underlying fundamental frame-work [4–7]. – 2 –ollowing a different spirit, one could consider imposing an assumption on theform of the stress-energy tensor based on our daily experience through some energycondition. This can be done restricting attention to a given theory of gravity. Such isthe case for the strong energy condition which is formulated as:
Strong energy condition (SEC): Gravity is always attractive in GR. (cid:18) T ab − T g ab (cid:19) V a V b ≥ f or any timelike vector V. (2.3)This SEC condition takes into account both the TCC and the Einstein equations inits mathematical formulation. Hence, properly speaking, it only makes sense to referto the SEC in a general relativistic framework.More generally, by looking at the world around us we can also assert certain prop-erties that any “reasonable” material content should satisfy independently of the theoryof gravity describing the curvature of spacetime. These energy conditions can be usedlater to study the geometry once the theory of gravity is fixed. The energy conditionscommonly considered in the literature are [1, 13]: Dominant energy condition (DEC): The energy density measured by any observer isnon-negative and propagates in a causal way. T ab V a V b ≥ , F a F a ≤ . (2.4)Here we have defined the flux 4-vector as F a = − T ab V b , and it is understood that theinequalities contained in expression (2.4) have to be satisfied for any timelike vector. Weak energy condition (WEC): The energy density measured by any observer has tobe non-negative. T ab V a V b ≥ . (2.5)Therefore, if the WEC is not satisfied, then the DEC has to be violated. However, theWEC and the SEC are, in principle, completely independent. Null energy condition (NEC): The SEC and the WEC are satisfied in the limit of nullobservers. T ab k a k b ≥ f or any null vector k a . (2.6)It should be noted that the NCC is completely equivalent to the NEC within theframework of GR. By its very definition if the NEC is violated, then the SEC, on onehand, and the WEC and DEC, on the other hand, cannot be satisfied.Historically, there was also a trace energy condition (TEC), the assertion that thetrace of the stress energy tensor should (in − + ++ signature) be non-positive. Whilefor several decades in the 50s and 60s this was believed to be a physically reasonablecondition, opinion has now shifted, (specifically with the discovery of stiff equations– 3 –f state for neutron star matter), and the TEC has now largely been abandoned [13].(See also reference [14] for uses of the less known subdominant trace energy condition,and reference [15] for the practically unnoticed quasilocal energy conditions.) Thiscautionary tale indicates that one should perhaps take the energy conditions as provi-sional, they are good ways of qualitatively and quantitatively characterizing the levelof weirdness one is dealing with, but they may not actually be fundamental physics. The Hawking–Ellis classification, (more properly called the Segre classification), ofstress-energy tensors is based on the extent to which the ortho-normal components ofthe stress-energy can be diagonalized by local Lorentz transformations: T ab = (canonical type) cd L ca L db . (2.7)Equivalently, one is looking at the Lorentz invariant eigenvalue problemdet (cid:0) T ab − λη ab (cid:1) = 0 , that is , det ( T ab − λδ ab ) = 0 . (2.8)Here as usual η ab = diag( − , , , T ab is not symmetric, sodiagonalization is trickier than one might naively expect. Even the usual Jordan decom-position is not particularly useful, (since for physical reasons one is interested in partialdiagonalization using Lorentz transformations, rather than the more general similaritytransformations which could lead to non-diagonal unphysical T ab ), and so it is moretraditional to classify stress-energy tensors in terms of the spacelike/lightlike/timelikenature of the eigenvectors.Based on (a minor variant of) the discussion of Hawking and Ellis the four types (whenexpressed in an orthonormal basis) are: • Type I: T ab ∼ ρ p p
00 0 0 p . (2.9)This is the generic case, there is one timelike, and 3 spacelike eigenvectors. T ab = ρ u a u b + p n a n b + p n a n b + p n a n b , (2.10)where u a = ( −
1; 0; 0; 0), n a = (0; 1; 0; 0), n a = (0; 0; 1; 0), and n a = (0; 0; 0; 1).The Lorentz invariant eigenvalues of T ab are {− ρ, p , p , p } . Many classicalstress-energy tensors, (for instance, perfect fluids, massive scalar fields, non-nullelectromagnetic fields), are of this type I form. Similarly many semi-classicalstress-energy tensors are of this type I form.– 4 –or a type I stress tensor the classical energy conditions can be summarized (interms of necessary and sufficient constraints) as:NEC ρ + p i ≥ ρ + p i ≥ ρ ≥ ρ + p i ≥ ρ + (cid:80) p i ≥ | p i | ≤ ρ ρ ≥ • Type II: T ab ∼ µ + f f f − µ + f p
00 0 0 p (2.11)This corresponds to one double-null eigenvector k a = (1; 1; 0; 0), so that: T ab = + f k a k b − µ η ab + p n a n b + p n a n b , (2.12)with η ab = diag( − , , , The Lorentz invariant eigenvalues of T ab are now {− µ, − µ, p , p } . Classically, the chief physically significant observed occurrenceof type II stress-energy is when µ = p i = 0, in which case one has T ab = f k a k b . (2.13)This corresponds to classical radiation or null dust.One could also set p i = − µ in which case T ab = − µη ab + f k a k b . (2.14)This corresponds to a superposition of cosmological constant and classical radia-tion or null dust. Finally, if one sets p i = + µ then one has T ab ∼ µ − µ µ
00 0 0 µ + f k a k b . (2.15)This corresponds to a classical electric (or magnetic) field of energy density µ aligned with classical radiation or null dust. For type II Hawking and Ellis choose to set f → ±
1, which we find unhelpful. – 5 –or a type II stress tensor the classical energy conditions can be summarized as:NEC f > µ + p i ≥ f > µ + p i ≥ µ ≥ f > µ + p i ≥ (cid:80) p i ≥ f > | p i | ≤ µ µ ≥ • Type III: T ab ∼ ρ f − ρ f f f − ρ
00 0 0 p (2.16)This corresponds to one triple-null eigenvector k a = (1; 1; 0; 0). T ab = − ρ η ab + f ( n a k b + k a n b ) + p n a n b , (2.17)with η ab = diag( − , , , The Lorentz invariant eigenvalues of T ab are now {− ρ, − ρ, − ρ, p } . Classically, type III does not seem to occur in nature — thereason for which becomes “obvious” once we look at the standard energy condi-tions. Even semi-classically this type III form is forbidden (for instance) by eitherspherical symmetry or planar symmetry — we know of no physical situation wheretype III stress-energy tensors arise.For a type III stress tensor the NEC cannot be satisfied for f (cid:54) = 0, and so theonly way that any of the standard energy conditions can be satisfied for a typeIII stress-energy tensor is if f = 0, in which case it reduces to type I. So if oneever encounters a nontrivial type III stress-energy tensor, one immediately knowsthat all of the standard energy conditions are violated.(This of course implies that any classical type III stress tensor, since it wouldhave to violate all the energy conditions, would likely be grossly unphysical — soit is not too surprising that no classical examples of type III stress-energy havebeen encountered.) • Type IV: T ab ∼ ρ f f − ρ p
00 0 0 p (2.18) For type III Hawking and Ellis choose to set f →
1, which we find unhelpful. – 6 –his corresponds to no timelike or null eigenvectors. The Lorentz invariant eigen-values of T ab are now {− ρ + if, − ρ − if, p , p } . Classically, type IV does not seem to occur in nature. Again, a necessary conditionfor the NEC to be satisfied for a type IV stress tensor is f = 0. The only waythat any of the standard energy conditions can be satisfied for a type IV stress-energy tensor is if f = 0, in which case it reduces to type I. So whenever oneencounters a nontrivial type IV stress-energy tensor, one immediately knows thatall of the standard energy conditions are violated. (Again, this of course impliesthat any classical type IV stress tensor, since it would have to violate all theenergy conditions, would likely be grossly unphysical — so it is not too surprisingthat no classical examples of type IV stress-energy have been encountered.)Semi-classically, however, vacuum polarization effects do often generate type IVstress energy [16]. The renormalized stress-energy tensor surrounding an evapo-rating (Unruh vacuum) black hole will be type IV in the asymptotic region [17],(with spherical symmetry enforcing p = p ). As it will be type I close to thehorizon, there will be a finite radius at which it will be type II. • Euclidean signature:Since we have seen this point cause some confusion, it is worth pointing out thatwhen working in Euclidean signature (in an orthonormal basis) the eigenvalueproblem simplifies to the standard casedet (cid:0) T ab − λδ ab (cid:1) = 0 . (2.20)Since T ab is symmetric it can be diagonalized by ordinary orthogonal 4-matrices.The symmetry group is now O (4), not O (3 , all stress energy tensors are of type I.The other thing that happens in Euclidean signature us that, because one nolonger has the timelike/lightlike/spacelike distinction, both the NEC and DECsimply cannot be formulated. In fact it is best to abandon even the WEC, andconcentrate attention on a Euclidean variant of the SEC: For type IV Hawking and Ellis choose T ab ∼ f f − µ p
00 0 0 p ( µ < f ) . (2.19)We have found the version presented in the text to be more useful. – 7 – Euclidean Ricci convergence condition: R ab V a V b ≥ for all vectors. – Euclidean SEC: ( T ab − T η ab ) V a V b ≥ for all vectors. • Classifying the Ricci and Einstein tensors:Instead of applying the Hawking–Ellis type I — type IV classification to the stress-energy tensor, one could choose to apply these ideas directly to the Ricci tensor(or Einstein tensor). The type I — type IV classification would now become state-ments regarding the number of spacelike/null/eigenvectors of the Ricci/Einsteintensor, and the associated eigenvalues would have a direct interpretation in termsof curvature, (rather than densities, fluxes, and stresses). Formulated in this way,this directly geometrical approach would then lead to constraints on spacetimecurvature that would be independent of the specific gravity theory one workswith.
Sometimes it is useful to side-step the Hawking–Ellis classification and, using onlyrotations, reduce an arbitrary stress-energy tensor to the alternative canonical form T ab ∼ ρ f f f f p f p f p (2.21)The advantage of this alternative canonical form is that it covers all of the Hawking–Ellis types simultaneously. The disadvantage is that it easily provides us with necessaryconditions, but does not easily provide us with sufficient conditions for generic stresstensors. For example, enforcing the NEC requires ρ ± (cid:88) i β i f i + (cid:88) i β i p i ≥ , with (cid:88) i β i = 1 . (2.22)By summing over ± β one obtains ρ + (cid:88) i β i p i ≥ , with (cid:88) i β i = 1 , (2.23)which in turn impliesNEC: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i β i f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:32) ρ + (cid:88) i β i p i (cid:33) ≥ , with (cid:88) i β i = 1 . (2.24)– 8 –hese are necessary and sufficient conditions for the NEC to hold — the quadraticoccurrence of the β i makes it difficult to simplify these further. Similarly for the WECone findsWEC: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i β i f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:32) ρ + (cid:88) i β i p i (cid:33) ≥ , with (cid:88) i β i ≤ . (2.25)For the SEC one obtainsSEC: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i β i f i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:32)(cid:32) ρ + (cid:88) i p i (cid:33) + (cid:88) i β i (cid:32) ρ + p i − (cid:88) j (cid:54) = i p j (cid:33)(cid:33) ≥ , (2.26)again with (cid:80) i β i ≤
1. Finally, the DEC is somewhat messier. The DEC amounts tothe WEC plus the relatively complicated constraint (cid:32) ρ + (cid:88) i β i f i (cid:33) − (cid:88) i ( f i + β i p i ) ≥ , with (cid:88) i β i ≤ . (2.27)This can be slightly massaged to yield (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i β i f i ( ρ − p i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ + (cid:32)(cid:88) i β i f i (cid:33) − (cid:88) i ( f i + β i p i ) ≥ , (2.28)again with (cid:80) i β i ≤
1. Unfortunately this seems to be the best that can generically bedone for the DEC in this alternative canonical form.
For practical computations the energy conditions are most often rephrased in terms ofthe parameters appearing in the type I — type IV classification, as presented in section2.1, with type I stress-energy being the most prominent. (The other types are oftenquietly neglected.) When expressed in this form, the standard energy conditions caneasily be compared with both classical and with published semi-classical estimates ofthe renormalized expectation value of the quantum stress-energy tensor.– 9 –eyond the strong physical intuition in which the energy conditions are based, thereare deep theoretical reasons why one should wish the energy conditions to be satisfied.It is known that violations of the NEC, which is the weaker energy condition we havepresented for the moment, may signal the presence of instabilities and superluminalpropagation [18–21]. Moreover, stress-energy tensors violating the NEC can supporthypothetical configurations as wormholes [9] and warp drives [25] in the framework ofgeneral relativity , which may lead to pathological situations. In addition, the energyconditions are central to proving the positivity of mass [49] and the singularity theorems[1, 3]. In this context, although one might think that violations of the NEC might alwaysallow us to get rid of uncomfortable singularities, this is not the case [50, 51], and theycan instead introduce new kinds of cosmic singularities instead [52–56]. Nevertheless,it is already well-known that the energy conditions are violated in realistic situations,violations of these energy conditions in the presence of semi-classical effects being verycommon. Among many situations where one or more of the energy conditions areviolated we mention: • SEC has to be violated in cosmological scenarios during the early phase of infla-tionary expansion and at the present time [57–61]. • It is easy to find violations of the SEC associated even with quite normal matter[63, 64]. • Non-minimally coupled scalar fields can easily violate all the energy conditionseven classically [62].(However, those fields may be more naturally interpreted as modifications ofgeneral relativity in some cases.) • The Casimir vacuum violates all the mentioned energy conditions [16]. • Numerically estimated renormalized vacuum expectation values of test-field quan-tum stress tensors surrounding Schwarzschild and other black holes [65].(EC violations occur, to one extent or another, for all of the standard quantumvacuum states: Boulware [66], Hartle–Hawking [67], and Unruh [68].) • Explicitly calculable renormalized vacuum expectation values of test-field quan-tum stress tensors in 1+1 dimensional QFTs [69].(Again, energy condition violations occur to one extent or another, for all of thestandard quantum vacuum states: Boulware, Hartle–Hawking, and Unruh.) Note that stable violations of the NEC are now known to be possible when the field has a non-canonical kinetic term [22–24]. See references [26–48] for interesting research along those lines. – 10 –inally, it should be emphasized that the DEC, WEC, and NEC impose restrictionson the form of the stress-energy tensor of the matter fields independently of the theoryof gravity. However, when these conditions are used to extract general features ofthe spacetime, (such as, for example, the occurrence of singularities), one considers aparticular theory of gravitation. Therefore, although the existence of wormholes andbouncing cosmologies required violations of the NEC in general relativity, in modifiedgravity theories the NCC could be violated by material content satisfying the NEC. Inthat case, it is possible to define an effective stress-energy tensor putting together themodifications with respect to general relativity that violates the NEC [62, 70–74]. Asthis effective tensor is not associated with matter, there is no reason to require that ithad some a priori characteristics. Hence, one may obtain wormhole solutions supportedby matter with “positive” energy [75–77], as in some of the examples reviewed in thefirst part of this book.
The key to the averaged energy conditions is simply to integrate the Raychaudhuriequation along a timelike geodesic θ f − θ i = (cid:90) fi (cid:26) ω ab ω ab − σ ab σ ab − θ − R ab V a V b (cid:27) ds. (3.1)Assuming zero vorticity, and noting that both σ ab σ ab ≥ θ ≥ θ f ≤ θ i − (cid:90) fi R ab V a V b ds. (3.2)That is, a constraint on the integrated timelike convergence condition, (cid:82) R ab V a V b ds , issufficient to control the convergence of timelike geodesics. In applications, the integralgenerally runs from some base point along a timelike geodesic into the infinite future.We formulate the ATCC (averaged timelike convergence condition): Averaged time-like convergence condition (ATCC): Gravity attracts provided (cid:90) ∞ R ab V a V b ds ≥ along any timelike geodesic . (3.3)In standard general relativity the Einstein equations allow one to reformulate this as Averaged strong condition (ASEC): Gravity in GR attracts on average. (cid:90) ∞ (cid:18) T ab − T g ab (cid:19) V a V b ds ≥ along any timelike geodesic . (3.4)– 11 –imilar logic can be applied to null geodesic congruences and the Raychaudhuri equa-tion along null geodesics to deduce Averaged null convergence condition (ANCC): Gravity attracts provided (cid:90) ∞ R ab k a k b dλ ≥ along any null geodesic . (3.5)In standard general relativity the Einstein equations allow one to reformulate this as Averaged null energy condition (ANEC): A necessary requirement for gravity to attractin GR on average is (cid:90) ∞ T ab k a k b dλ ≥ along any null geodesic . (3.6)Note that the integral must be performed using an affine parameter — with arbi-trary parameterization these conditions would be vacuous. This ANEC has attractedconsiderable attention over the years to prove singularity theorems [85, 86] and is forinstance the basis of the topological censorship theorem [87, 88] forbidding a large classof traversable wormholes. However, note that even the ANEC can be violated by con-formal anomalies [89], so to have any hope of a truly general derivation of the ANECfrom more fundamental principles, one would need to enforce some form of conformalanomaly cancellation. The idea behind nonlinear energy conditions is inspired to some extent by the qual-itative difference between the NEC/WEC/SEC and the DEC. The NEC/WEC/SECare strictly linear — any sum of stress-energy tensors satisfying these conditions willalso satisfy the same condition. This linearity fails however for the DEC, which can berephrased as the two conditions T ab V a V b ≥
0; ( − T ac η cd T db ) V a V b ≥ ∀ timelike V. (4.1)The second quadratic condition is imply the statement that the flux F be non spacelike.We can further combine these into one quadratic condition( − [ T ac − (cid:15)η ac ] η cd [ T db − (cid:15)η db ]) V a V b ≥ (cid:15) ; ∀ (cid:15) > ∀ timelike V. (4.2) Interesting studies include (but are not limited to) references [78–84]. – 12 –hat is, the DEC (suitably rephrased) is a nonlinear quadratic constraint on the stress-energy, and it is this version of the DEC that can naturally be integrated along timelikegeodesics to develop an ADEC. Furthermore this observation opens up the possibilitythat other nonlinear constraints on the stress-energy might be physically interesting.Among such possibilities, we mention the flux energy condition [90, 91], determinantenergy condition, and trace-of-square energy condition [17].
Flux energy condition (FEC): The energy density measured by any observer propagatesin a causal way. F a F a ≤ . (4.3)It has to be emphasized that the FEC does not assume anything about the sign ofthe energy density. Moreover, by its very definition the DEC is just the combinationof the WEC and the FEC. For the different types of stress energy tensor we haveType I ρ ≥ p i —Type II µ ≥ p i µf ≥ f (cid:54) = 0) types III and IV. The FEC isa weakening of the DEC, and is equivalent to the single quadratic constraint( T ac η cd T db ) V a V b ≤ ∀ timelike V. (4.4)Once phrased in this way, it becomes clear how to formulate an averaged version of theFEC. Averaged flux energy condition (AFEC): (cid:90) ∞ ( T ac η cd T db ) V a V b ds ≤ along all timelike geodesics . (4.5)On the other hand, among other non-linear combinations of the stress-energy ten-sor, we might consider the following energy conditions: Determinant energy condition (DETEC): The determinant of the stress-energy tensoris nonnegative. det (cid:0) T ab (cid:1) ≥ . (4.6)In terms of the type decomposition we have– 13 –ype I ρp p p ≥ − µ p p ≥ ρ p ≥ − ( ρ + f ) p p ≥ ρp p p − f p p − f p p − f p p ≥ Trace-of-square energy condition (TOSEC): The trace of the squared stress-energytensor is nonnegative. T ab T ab ≥ . (4.7)In terms of the type decomposition we haveType I ρ + (cid:80) p i ≥ µ + (cid:80) p i ≥ ρ + p ≥ ρ − f ) + (cid:80) p i ≥ (cid:80) f i ≤ ( ρ + (cid:80) i p i )Since the TOSEC is basically a sum of squares, it is only in Lorentzian signaturethat it can possibly be nontrivial — and even in Lorentzian signature only a type IVstress-energy tensor can violate the TOSEC.Some comments about the fulfilment of these conditions are in order [17]: • DETEC is violated in cosmological scenarios during the early inflationary phaseand at the present time in the framework of GR. • FEC is violated by the Casimir vacuum, but it can be satisfied by the renormalizedstress-energy tensor of the Boulware vacuum of a Schwarzschild spacetime. • TOSEC can be violated by the Unruh vacuum of a Schwarzschild spacetime.
We now note that when considering semi-classical effects the ECs may be violated,although the relevant inequalities are typically not satisfied only by a small amount .This fact led us to formulate the semi-classical energy conditions in a preliminary andsomewhat vague way, to allow them to quantify the violation of the classical ECs,before considering a particular more specific formulation [17, 91].– 14 – uantum WEC (QWEC): The energy density measured by any observer should not beexcessively negative.
The QWEC, therefore, allows the energy density to be negative in semi-classical sit-uations but its value has to be bounded from below. Now, assuming that this bounddepends on characteristics of the system, such as the number of fields N , the system4-velocity U a , and a characteristic distance L , we can formulate the QWEC as T ab V a V b ≥ − ζ (cid:126) NL ( U a V a ) . (5.1)Here ζ is a parameter of order unity. For the QWEC we have:Type I ρ ≥ − ζ (cid:126) N/L ρ + p i ≥ − ζ (cid:126) N/L —Type II µ ≥ − ζ (cid:126) N/L p i ≥ − ζ (cid:126) N/L f ≥ − ζ (cid:126) N/L Type III ρ ≥ − ζ (cid:126) N/L p ≥ − ζ (cid:126) N/L | f | ≤ ζ (cid:126) N/L Type IV ρ ≥ − ζ (cid:126) N/L p i ≥ − ζ (cid:126) N/L | f | ≤ ζ (cid:126) N/L In a similar way, other quantum (semi-classical) energy conditions have been formu-lated.
Quantum FEC (QFEC): The energy density measured by any observer either propa-gates in a causal way, or does not propagate too badly in an acausal way.
So, the QFEC allows the flux 4-vector to be (somewhat) space-like in semi-classicalsituations but its norm has to be bounded from above. It can be written as F a F a ≤ ζ (cid:18) (cid:126) NL (cid:19) ( U a V a ) . (5.2)Here ζ is again a parameter of order unity. For the QFEC we have:Type I ρ − p i ≥ − ζ ( (cid:126) N/L ) — —Type II µ − p i ≥ − ζ ( (cid:126) N/L ) µf ≥ − ζ ( (cid:126) N/L ) —Type III ρ − p ≥ − ζ ( (cid:126) N/L ) | ρf | ≤ ζ ( (cid:126) N/L ) | f | ≤ ζ (cid:126) N/L Type IV ρ − p i ≥ − ζ ( (cid:126) N/L ) | ρf | ≤ ζ ( (cid:126) N/L ) | f | ≤ ζ (cid:126) N/L Analogously with the classical energy conditions, the quantum DEC (QDEC) wouldbe satisfied in situations where both the QWEC and QFEC are fulfilled. That is, theQDEC states the following
Quantum DEC (QDEC): The energy density measured by any observer should not beexcessively negative, and it either propagates in a causal way or does not propagate toobadly in an acausal way. – 15 – pecifically T ab V a V b ≥ − ζ W (cid:126) NL ( U a V a ) , (5.3) F a F a ≤ ζ F (cid:18) (cid:126) NL (cid:19) ( U a V a ) . (5.4)In order to get the relevant inequalities for the QDEC to be satisfied for types I —IV stress-energy tensors, one needs just to combine the inequalities of the QWEC andQFEC tables, taking into account that the two ζ ’s should be kept distinct.(For a recent rediscovery of a quantum energy condition along the lines investigatedin references [17, 91, 92] and presented here see reference [93]. For non-local quantumenergy inequalities see references [94, 95].)Regarding the semi-classical quantum energy conditions it should be noted that: • QECs are, of course, satisfied in classical situations where their classical counter-parts are satisfied. • The Casimir vacuum satisfies the QECs presented here [17]. • QFEC can be satisfied by the Boulware and Unruh vacuum even in situationswhere the QWEC is violated [91]. • The QECs can be used as a way of quantifying/minimizing the violation of theECs.Concerning the last point it should be clarified we are referring to quantifying thedegree of violation of an energy condition for a given amount of matter [92, 96–98], andnot to minimizing the quantity of matter violating the energy condition [99–101].
As we have seen, the energy conditions (both classical and semi-classical) are numerousand varied, and depending on the context can give one rather distinct flavours of bothqualitative and quantitative information — either concerning the matter content, orconcerning the (attractive) nature of gravity. Variations on the original classical point-wise energy conditions are still under development and investigation, and the status ofthe energy conditions as fundamental physics should still be considered provisional.There is no strict ordering on the set of energy conditions, at best a partial ordering,so there is no strictly “weakest” energy condition. Nonetheless, perhaps the weakest ofthe usual energy conditions (in terms of the constraint imposed on the stress energy)– 16 –s the ANEC, which makes it the strongest energy condition in terms of proving the-orems. Attempts at developing a general proof of the ANEC are ongoing — part ofthe question is exactly what one might mean by “proving an energy condition” frommore fundamental principles. It should be noted, however, that even the ANEC canbe violated by conformal anomalies, so to have any hope of a truly general derivationof the ANEC from more fundamental principles, one would need to enforce some formof conformal anomaly cancellation.On the other hand, it should be emphasized that FEC is completely independent ofthe NEC. Hence, there could be in principle realistic situations where the stress-energytensor violates the ANEC but satisfies the FEC. Although, as we have reviewed, theFEC can be violated by semi-classical effects, its quantum counterpart is satisfied bythe renormalized stress-energy tensor of quantum vacuum states.An alternative approach is to formulate curvature conditions directly on the Ricci orEinstein tensors, and then use global analysis, (based, for instance, on the Raychaud-huri equation or its generalized variants) to extract information regarding curvaturesingularities and/or the Weyl tensor.
Acknowledgements
PMM acknowledges financial support from the Spanish Ministry of Economy and Com-petitiveness through the postdoctoral training contract FPDI-2013-16161, and throughthe project FIS2014-52837-P. MV acknowledges financial support via the Marsden Fundadministered by the Royal Society of New Zealand.
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