A Primer on Energy Conditions

In the paper

“6. 'hyper-fast' travel 83 [possibly] (my calculation)”

In the paper

“2. a positive cosmological constant ( e . g ., anti-de Sitter Space) [necessarily] (my calculation)”

In the paper

“1. a negative cosmological constant, as in de Sitter spacetime, and the 'dark energy' postulated to drive the observed accelerated expansion of the universe [necessarily] (my calculation)”

In the paper

“Proof: pick your favorite globally hyperbolic spacetime and some open set in it; from the formulæ in Wald (1984, Appendix D), it follows that one can always find a conformal transformation of the metric that is the identity outside the open set and non-trivial inside such that at some point in the set the transformed stress-energy tensor will yield whatever one wants on contraction with a timelike”

What the sources say

• Not checked yetcites Wald (1984, Appendix D) source

In the paper

“The one possible exception to this claim is a null fluid, which has a stress-energy tensor of the form T ab = ρk a k b + p 1 x a x b + p 2 y a y b , where k a is null and x a and y a are unit spacelike vectors orthogonal to k a and to each other.”

In the paper

“Now, most propositions of a fundamental character in general relativity admit of interpretation as either a postulate of the theory or as a derived consequence from some other propositions taken as postulates.”

In the paper

“This interpretive flexibility does not seem to hold, however, for energy conditions. I know of no substantive proposition that, starting from some set of other important 'fundamental postulates', has as its consequence an energy condition. One either imposes an energy condition by fiat, or one shows that it holds for stress-energy tensors associated with particular forms of matter fields.”

In the paper

“Even if we put that point aside, though, there are other problems, as Earman (2012) argues, claiming the DEC ought not be interpreted as prohibiting superluminal propagation of stress-energy.”

What the sources say

• Not checked yetcites Earman (2012)

In the paper

“the Einstein tensor itself, the fundamental constituent of the Einstein field equation, has no simple, natural geometrical interpretation”

In the paper

“The SDEC, as the name suggests, is slightly stronger in that it requires energy-momentum flux as measured by any timelike observer to be strictly timelike for non-trivial stress-energy distributions.”

In the paper

“The existence of a negative cosmological constant is a case in which NEC is satisfied, but, by the failure of the SEC, there is still divergence of timelike geodesics”

In the paper

“In every case, the physical formulation is logically equivalent to the geometric formulation if the Einstein field equation is assumed to hold.”

In the paper

“The DEC (and a fortiori the SDEC) are, therefore, standardly taken to rule out 'superluminal propagation of stress-energy'.”

What the sources say

• Not checked yetcites Wald 1984, p. 219 source

In the paper

“the geometric form of the NEC requires that null geodesic congruences tend to be convergent in sufficiently small neighborhoods of every spacetime point (or at least not divergent).”

In the paper

“Ordinary mass-energy density must be non-negative as experienced by any observer traversing a timelike curve, and the pressure in any given spacelike direction can never be so negative as to dominate that value.”

In the paper

“They seem in no sense to be laws, under any standard account in the literature, for none of them holds for all known 'physically reasonable' types of matter, and each of them is in fact violated in what seem to be physically important circumstances.”

In the paper

“Thus, the WEC is not the weakest condition in a logical sense one can impose, but it is the weakest in a loose, physical sense: one cannot do better by imposing further natural restrictions.”

In the paper

“It also says, however, that ordinary mass-energy density cannot be negatively dominated by the sum of the individual pressures (momentum fluxes) at any point, as determined by an observer traversing a timelike curve.”

In the paper

“Averaged radial acceleration is, after all, only an average, factitious quantity. That it be negative does not say that individual freely falling ordinary bodies cannot in fact accelerate away from each other for no apparent reason, only that, on average, they do not do so.”

In the paper

“Classically, some fluids such as water are known to exhibit negative pressures in some regimes as measured by observers traversing timelike curves ( viz ., us), but these negative pressures are never large enough to dominate the fluid's mass-energy.”

In the paper

“the geometric form of the SEC requires that timelike geodesic congruences tend to be convergent in sufficiently small neighborhoods of every spacetime point.”

In the paper

“The interpretations of their physical forms are apparent: every timelike observer will measure ordinary mass-energy density to be non-negative, and will also measure total flux of energymomentum to be causal, with the flow oriented in the same direction as the observer's proper time.”

In the paper

“The interpretation of the effective form of the NEC is that the natural measure either of massenergy or of pressure in any given spacelike direction can be negative as determined by an observer traversing a null curve, but not both, and, if either is negative, it must be less so than the other is positive.”

In the paper

“ordinary energy density, not being an observer-independent quantity, is not a particularly natural concept in general relativity.”

In the paper

“It should be clear that the NEC does not require that c s ≤ 1; in other words, stuff can satisfy the NEC while still permitting superluminal propagation of physically significant structure.”

In the paper

“the interpretation of the physical form is that observers following timelike geodesics will see that 'gravity' tends locally to be 'attractive' in its action on stuff following both timelike and null geodesics.”

In the paper

“Since all known 'physically reasonable' classical fields (and indeed many unreasonable ones) have associated stress-energy tensors of type i , this is no serious restriction.”

In the paper

“No physical system will 'experience' ordinary energy density at all.”

In the paper

“He first argues for the positive conclusion that the proper way to conceive of a prohibition on superluminal propagation is the existence of a well posed (in the sense of Hadamard) initial-value formulation for all fields on spacetime.”

What the sources say

• Not checked yetcites Earman (2012)

In the paper

“The interpretation of the physical form of the WEC is straightforward: the (ordinary) total energy density of all matter fields, as measured in a natural way by any observer traversing a timelike curve, is never negative.”

In the paper

“the effective form requires that ordinary mass-energy density at any point cannot be negatively dominated by momentum fluxes in any given spacelike direction as determined by an observer traversing a null curve”

In the paper

“Clearly, the SDEC implies the DEC, which implies the WEC.”

In the paper

“The singularity theorems following from the SEC are the most physically salient both because they tend to have the weakest ancillary assumptions, and because they apply to physically important situations, both for collapsing bodies and for cosmology.”

In the paper

“For the AWEC, if Γ contains enough timelike geodesics and the spacetime is well behaved, then there may be null geodesics that are limit curves of sub-families of Γ; in this case, the relevant characteristic integral will be non-negative for those null geodesics, and the AWEC with the fixed Γ can be said to imply the ANEC for the family of limiting null geodesics.”

In the paper

“The ASEC does not imply either the AWEC or the ANEC.”

In the paper

“Otherwise, I know of no compelling interpretation of it, as it allows energy density to be unboundedly negative, so long as the absolute value of pressure is not too great.”

In the paper

“The first term on the righthand side is manifestly negative, as is the second, since σ ab is spacelike in both indices, and so σ mn σ mn ≥ 0.”

In the paper

“I am sorry to say the discussion of the possible interpretations of, or even just motivations for, the standard impressionist energy conditions is a simple one to have: there are no compelling geometrical, physical or effective interpretations of these conditions, not even hand-waving, rough or approximate ones, and no compelling physical or philosophical motivations for them.”

In the paper

“Even in well behaved spacetimes, however, there may be null geodesics that are not the limit of any family of timelike geodesics, so in general the AWEC does not imply the ANEC.”

In the paper

“none of the quantities constrained by the impressionist conditions enter the equations of motion or the field equations of any known kinds of physical system, and, correlatively, no couplings between any known kinds of physical system are mediated by those quantities”

In the paper

“Even if one could find natural, compelling ways to restrict attention to a privileged class of congruences in such a way as to resolve the technical problems I raised for this kind of argument, there would still be interpretative problems with this kind of argument.”

In the paper

“Indeed, it was the epoch-making result of Penrose (1965) showing that a singularity would inevitably result from gravitational collapse in an open universe that first demonstrated the power that the qualitative abstraction of energy conditions gives in proving farreaching results of great physical importance.”

What the sources say

• Not checked yetcites Penrose (1965)

In the paper

“If the total sum on the righthand side is negative, then the expansion of the congruence is decreasing with proper time, i . e ., the geodesics in the congruence are everywhere converging on each other.”

In the paper

“This is a weaker form of the Zeroth Law, in so far as it is not known whether the event horizons of all 'physically reasonable' black holes are sufficiently regular in the sense required”

In the paper

“The result of Ansoldi (2007), however, that black holes with singularity-free interiors necessarily violate the SEC, may push one towards the opposite view”

What the sources say

• Not checked yetcites Ansoldi (2007)

In the paper

“I conclude this section with an observation of what is not here: there are no standard energy conditions based on the Weyl conformal tensor C a bcd or on the Bel-Robinson tensor T abcd .”

In the paper

“Clearly, the NEC, WEC and SEC respectively imply the ANEC, AWEC and ASEC.”

In the paper

“Thus, if we assume that 'gravity is everywhere attractive', and we interpret this to mean that congruences of timelike geodesics which have vanishing twist should always converge, then, in order to ensure that the total righthand side of equation (2.3.1) is always negative, we require that R mn ξ m ξ n ≥ 0, which is just the geometrical form of the SEC.”

In the paper

“Finally, the restriction to geodesics has no compelling physical or philosophical basis that I can see, but appears to be dictated by pragmatic considerations about the technical tractability of required calculations.”

In the paper

“There is, as is to be expected, no simple interpretation of its geometric form.”

In the paper

“Raychaudhuri's equation expresses the rate of change of the scalar expansion of a congruence of geodesics, as one sweeps along the congruence, as a function of the expansion itself, of the congruence's shear and twist tensors, and of the Ricci tensor.”

What the sources say

• Not checked yetcites Wald (1984) source

In the paper

“It is, moreover, manifestly weaker than all the other standard energy conditions, as its characteristic non-linearity (most easily seen in the second given articulations of its geometric and physical forms, and in its effective form) ensures that essentially no limit is placed on the possible negativity of the ordinary mass-energy of matter.”

In the paper

“Without that interpretation, however, one has little motivation for invoking Raychaudhuri's equation in the first place without ancillary physical justification.”

In the paper

“The DEC occupies yet more treacherous ground, in so far as the geometric form has no clear interpretation, the physical interpretation (as Earman's and Wong's arguments show) is muddled at best, and the effective is only partially unproblematic.”

In the paper

“In fact, all one can conclude from the demand that the righthand side of equation (2.3.1) be non-positive (when ω ab = 0) is that”

In the paper

“Because the pointilliste energy conditions, moreover, constrain the behavior of curvature terms only at individual points, and that by reference to all timelike or null (or both) vectors at those points”

In the paper

“The Bel-Robinson tensor, moreover, may usefully be thought of as a measure of a kind of 'super-energy' associated with purely gravitational phenomena, and directly measures in a precise sense the intensity of gravitational radiation in infinitesimal regions.”

What the sources say

• Not checked yetcites Penrose and Rindler (1984)
• Not checked yetcites Senovilla (2000, 2002)

In the paper

“Of course, this is not the SEC, but only a weaker form of the geometric formulation, one that sets a non-constant lower bound on 'how negative' mass-energy and momentum-energy flux can get (invoking the physical form of the condition).”

In the paper

“In no case can one allow arbitrary parametrizations for null curves in the defining integral, as that would simply reduce the ANEC to the NEC.”

In the paper

“Consider the NEC: on the face of it, the geometric form has a relatively unproblematic interpretation, whereas the interpretations of the physical and effective forms are beset with more serious problems.”

In the paper

“No reasonable impressionist analogue of either of the pointilliste dominant conditions are known.”

In the paper

“I know of no rigorous proofs of the stability of an event horizon under generic small perturbations.”

In the paper

“These conditions were all constructed by reverse-engineering-an investigator looked for the weakest condition she could impose on the averaged behavior of some quantity depending on curvature or stress-energy in order to derive the result of interest to her.”

In the paper

“Nonetheless, geodesic congruences seem to be about the only structure one has naturally available to work with, if one wants to try to constrain the behavior of curvature as measured by the contractions of curvature tensors with causal vectors.”

In the paper

“So long as one wants to work with geodesic congruences, therefore, it seems one must find some way to restrict the class one allows as relevant to those that are 'physically significant' in some important and clear way.”

In the paper

“The simplest interpretation of its physical form is that the total flux of energy-momentum as measured by any timelike observer is always causal, albeit the temporal direction of the flux is not restricted.”

In the paper

“Very recently, Abreu, Barcel´ o, and Visser (2011) introduced a new classical energy condition:”

What the sources say

• Not checked yetcites Abreu, Barceló, and Visser (2011)

In the paper

“This may not sound so bad at first, until one realizes it means that, for a spacetime to satisfy the constrained condition, every curve in the fixed class must eventually traverse only regions of non-negative energy density, both to the past and the future: violations of the WEC are to be allowed only in bounded regions in the interior of spacetime, so to speak.”

In the paper

“Because isotropic tachyonic gases always satisfy ρ < 1 3 p , with weaker bounds for anisotropic tachyonic material, the effective form may be interpreted as ruling out the possibility of tachyonic matter.”

In the paper

“The case is just the opposite for the WEC: the geometric form has no clear interpretation, whereas the physical form and at least part of the effective form (the positivity of energy density) are relatively unproblematic.”

In the paper

“This suggests that the NEC is necessary for these theorems”

In the paper

“Again, the lack of such results depending on impressionist conditions suggests that the pointilliste conditions are necessary”

In the paper

“Finally, if any model of inflationary cosmology is correct, then we know that the SEC was necessarily violated at least during that period and, depending on the particulars of the model, possibly the ASEC as well.”

In the paper

“we already know that essentially all energy conditions are promiscuously violated when quantum effects are taken into account.”

In the paper

“That there is no impressionist analogue to the DEC may suggest that the DEC is necessary for these results.”

In the paper

“there is no weaker energy condition that one could attempt to replace it with (except perhaps the FEC, if it turns out to be viable-see § 2.4)”

In the paper

“In fact, if the current consensus that the expansion of the universe is accelerating is correct, and so some form of 'dark energy' exists, then we know that the SEC is currently being violated on cosmological scales, entirely independently of any assumptions about the nature of the fields entering into the stress-energy tensor or cosmological constant”

What the sources say

• Not checked yetcites Visser and Barceló 2000
• Not checked yetcites Barceló and Visser 2002
• Not checked yetcites Visser 2005

In the paper

“It is tempting to relate Gannon's singularity theorem to Topology Censorship, especially in so far as the latter requires only the ANEC, which the NEC implies.”

In the paper

“Every consequence of the DEC pertaining to black holes is of the kind that constrains topological or Killing structure of the horizons.”

In the paper

“Because the presence of naked singularities would seem to herald a spectacular breakdown in predictability and even determinism associated with dynamical evolution in general relativity”

In the paper

“many attempts to make the Hypothesis precise focus on the initial-value formulation of general relativity.”

In the paper

“the only real addition the DEC makes to the NEC and the WEC, that energy-momentum flux be causal”

In the paper

“With regard to entropy bounds such as that of Bousso (1999a, 1999b), if in fact the NEC or DEC were necessary for their validity, this could spell serious trouble for many programs in quantum gravity, or at least for the ways that research in such programs are currently being carried out”

What the sources say

• Not checked yetcites Bousso (1999a, 1999b)

In the paper

“Now, the relation of the energy conditions to the status of the Cosmic Censorship Hypothesis is complicated, first and foremost, by the fact that there are a multitude of different formulations of the Hypothesis”

In the paper

“If one assumes that the singularity predicted by Gannon's theorem will be hidden behind an event horizon, then the theorem gives some insight into why non-trivial spatial topological structure will always (quickly come to be) hidden inside a black hole.”

In the paper

“The proofs of many of those results, moreover, tend to have the same structure: very broadly speaking, one assumes the result is not true and then derives a contradiction with the fact that null geodesic congruences, by dint of the NEC, must be convergent (or at least not divergent).”

In the paper

“no such results are known to follow from any of the impressionist energy conditions.”

In the paper

“and that Penrose, Sorkin, and Woolgar (1993) were able to prove it using the even weaker ANEC, and not the significantly stronger DEC, as all other known proofs require.”

What the sources say

• Not checked yetcites Penrose, Sorkin, and Woolgar (1993)

In the paper

“With the possible exception of the First Law of black-hole mechanics (for asymptotically flat black holes), every fundamental result about black holes requires an energy condition for its proof, with the majority relying either on the NEC or the DEC.”

In the paper

“If the DEC were to fail, then it seems plausible that the Bondi energy could become negative, if negative mass-energy radiated to null infinity.”

In the paper

“The NEC is also used to prove many results about the kinds of properties required to characterize black holes (the constituents of the 'No Hair' Theorems), viz ., that stationary black holes can be entirely characterized by three parameters, mass, angular momentum and electric charge.”

In the paper

“because the NEC does not require the convergence of timelike geodesics (as I discussed in § 2.1), and so does not entail that 'gravity be attractive' for bodies traversing such curves”

In the paper

“The work of Malament (2012a) and Weatherall (2012), however, show that, at best, such approaches to the foundations of general relativity must be more subtle where the Geodesic Principle is concerned, and, at worst, that the Principle may in fact not be suitable at all for playing a fundamental role in giving an interpretation of the theory.”

What the sources say

• Not checked yetcites Malament (2012a)
• Not checked yetcites Weatherall (2012)

In the paper

“Roughly speaking, the results pertaining to black holes fall into three categories: those constraining the topological and Killing structure of horizons; those constraining the kinds of property black holes can possess; and those contraining the relations among the horizon and the properties.”

In the paper

“Although the proposition that a given globally hyperbolic extension of a spacetime is the maximal such extension depends for its only known proof on the assumption of the SEC, this is not really a counter-example to my claim”

In the paper

“He showed as well that the SDEC is not strong enough to ensure that the curve not be null: there is a spacetime with a null geodesic satisfying all the conditions of the Geroch-Jang Theorem.”

What the sources say

• Not checked yetcites Weatherall (2012)

In the paper

“I have no explanation for why the DEC should be used in almost no singularity theorems”

In the paper

“Since stress-energy tensors not of type i are generally considered 'unphysical'”

In the paper

“So far as I know, there is no proof that gravitational collapse will lead to singularities in the case of spatially closed spacetimes under the weaker assumption of an impressionist energy condition.”

In the paper

“both a cosmological 'bounce' and a Tolman wormhole (perhaps the two most natural possible replacements for an initial big bang singularity) require violation only of the SEC (Hochberg, Molina-Paris, and Visser 1999; MolinaParis and Visser 1999), not any of the other energy conditions”

What the sources say

• Not checked yetcites Hochberg, Molina-Paris, and Visser 1999
• Not checked yetcites Molina-Paris and Visser 1999

In the paper

“Second, they already would seem to allow not only violations of the ordinary Second Law of thermodynamics (Ford 1978; Davies and Ottewill 2002), but of the Generalized Second Law as well”

What the sources say

• Not checked yetcites Ford 1978
• Not checked yetcites Davies and Ottewill 2002

In the paper

“the pions that mediate the strong nuclear force can to a very high degree of approximation be represented by just such scalar fields.”

In the paper

“The theorems predicting big bang and big crunch singularities face one more problem peculiar to them alone: all such theorems invoke energy conditions of various kinds, mostly the SEC, and yet one can show that, depending on the characteristics of a given big bang or big crunch singularity, the presence of the singularity itself implies a violation of the relevant energy condition.”

In the paper

“We would expect, however, on physical grounds, that even slight deviations from homogeneity could lead to the development of marginally trapped surfaces.”

In the paper

“10. 'hyper-fast' travel 71 [possibly] (Visser, Bassett, and Liberati 2000)”

What the sources say

• Not checked yetcites Visser, Bassett, and Liberati 2000

In the paper

“cosmological inflation necessarily violates SEC and, depending on particulars of the model, possibly ASEC.”

In the paper

“Thus, Bekenstein argues, nuclear matter in the very early, dense stages of the actual universe may not have satisfied the SEC, which may suggest that the initial singularity in standard BigBang models may be avoidable.”

In the paper

“8. Tolman wormholes and Einstein-Rosen bridges [necessarily] (Barcel´ o and Visser 1999)”

What the sources say

• Not checked yetcites Barceló and Visser 1999

In the paper

“1. naked singularities [possibly] (Ford and Roman 1992)”

What the sources say

• Not checked yetcites Ford and Roman 1992

In the paper

“Many meson fields in the Standard Model (pions, kaons and many other mesons, including their 'charmed', 'truth' and 'beauty' correlates), moreover, are modeled to an extraordinarily high degree of accuracy as scalar fields, even though we believe they in fact consist of bound states of quark-antiquark pairs.”

In the paper

“It is also widely believed that the so-called 'strong CP problem', the fact that no CP-violation in strong nuclear interactions has ever been observed, is best solved by the postulation of a scalar field called the axion (Peccei and Quinn 1977)”

What the sources say

• Not checked yetcites Peccei and Quinn 1977

In the paper

“though to the best of my knowledge it is not known whether any classical models of the axion violate any of the energy conditions”

In the paper

“We do not yet have indubitable evidence for the existence of a fundamental scalar field in nature.”

In the paper

“It follows from a theorem Tipler proves that, if a black hole (marginally trapped surface) develops in one of Bekenstein's spacetimes, then, because they do satisfy the WEC, a singularity would necessarily develop.”

In the paper

“The recently discovered Higgs field is without question phenomenologically a scalar field, but the jury is still out on whether or not it is a composite, bound state of underlying non-scalar entities.”

In the paper

“In particular, negative masses uniformly repel all other mass, irrespective of the sign of the other masses, and likewise that positive masses uniformly attract all other masses, and so, most strikingly, a system consisting of one positive and one negative mass will uniformly accelerate.”

What the sources say

• Not checked yetcites Bondi (1957)