A Primer on Energy Conditions

Massive Klein-Gordon fields possibly violate the AWEC.

“3. massive Klein-Gordon fields [possibly] (my calculation)”

Cosmological steady-state theories of Bondi and Gold (1948) and Hoyle (1948) necessarily violate the AWEC.

“1. cosmological steady-state theories of Bondi and Gold (1948) and Hoyle (1948) [necessarily] (my calculation)”

Physically traversable wormholes possibly violate the AWEC.

“5. physically traversable wormholes [possibly] (my calculation)”

Classical Dirac fields possibly violate the AWEC.

“3. classical Dirac fields [possibly] (my calculation)”

Conformal scalar fields coupled with dust possibly violate the AWEC.

“5. conformal scalar fields coupled with dust [possibly] (my calculation)”

Typical gauge theories with spontaneously broken symmetries possibly violate the AWEC.

“4. typical gauge theories with spontaneously broken symmetries [possibly] (my calculation)”

Cosmological inflation possibly violates the AWEC.

“2. cosmological inflation [possibly] (my calculation)”

It is a trivial matter to find globally hyperbolic spacetimes that violate any energy condition.

“Now, it is a trivial matter to find globally hyperbolic spacetimes that violate any energy condition.”

Tipler proved that, in a natural sense, the weak energy condition is the weakest local energy condition one can define.

“He first showed that, in a natural sense, the WEC is the weakest local energy condition one can define.”

• cites Tipler (1978)Not checked yet

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The energy that can be negative near a Kerr black hole is the energy defined by the stationary Killing field of the spacetime, not the ordinary energy density as measured by any observer using tools at rest with respect to herself.

“The energy that can be negative near a Kerr black hole is the energy defined by the stationary Killing field of the spacetime, not the ordinary energy density as measured by any observer using tools at rest with respect to herself.”

• cites Penrose 1969Not checked yet

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• cites Penrose and Floyd 1971Not checked yet

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Convergence of null geodesics at a point does not in general imply convergence of all timelike geodesics at that point.

“The interpretations of the geometrical and physical forms of the NEC based on average radial acceleration is undermined by the fact that convergence of null geodesics at a point does not in general imply convergence of all timelike geodesics at that point.”

The strengthened dominant energy condition is logically stronger than the dominant energy condition.

“The second given geometric and physical forms of the SDEC make it manifest that the SDEC is in fact logically stronger than the DEC.”

The weak energy condition implies the null energy condition.

“It is easy to see, by considerations of continuity, that the WEC implies the NEC.”

The negative energy densities in the Casimir effect are inferred from measurement of the Casimir force itself.

“rather, the negative energy densities are inferred from measurement of the Casimir force itself (Brown and Maclay 1969).”

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The physical form of the null energy condition means particles following null geodesics will observe that 'gravity' tends locally to be 'attractive', or at least not repulsive, when acting on nearby particles also following null geodesics.

“the interpretation of the physical form is that particles following null geodesics will observe that 'gravity' tends locally to be 'attractive' (or at least not repulsive) when acting on nearby particles also following null geodesics.”

The effective form of the weak energy condition is essentially equivalent to the null energy condition.

“The effective form of the WEC, therefore, is in fact essentially equivalent to the NEC.”

Energy conditions are not empirical or inductive generalizations for the same reason that they are not laws.

“Neither do they appear to be empirical or inductive generalizations, for the same reason.”

Many profound results in the study of global structure, such as the Hawking Area Theorem, do not depend on the Einstein field equation, but assume only a purely formal condition imposed on the Ricci tensor.

“many of the most profound results in the study of global structuree . g ., the Hawking Area Theorem-do not depend on the Einstein field equation at all, but rather assume only a purely formal condition imposed on the Ricci tensor”

The effective fluid can be neither too stiff nor too lax, but must lie in a middling Goldilocks regime.

“This means that the effective fluid can be neither too 'stiff' nor too 'lax', but must lie in a middling Goldilocks regime.”

The strong energy condition does not imply the weak energy condition.

“The SEC does not imply the WEC, for the SEC can be satisfied even if the ordinary mass-density is negative.”

Except for one special case, the effective formulation of an energy condition is physically equivalent to the geometric and physical formulations, under the assumption that the Einstein field equation holds and matter is not too exotic.

“Thus, except for one special case to be discussed below, the effective formulation should be understood to be in all ways physically equivalent to the geometric and the physical formulations, under the assumption that the Einstein field equation holds, and matter is not too exotic.”

• AssumptionThe authors state this as an assumption: The Einstein field equation holds when relating geometric, physical, and effective formulations of energy conditions.
• AssumptionThe authors state this as an assumption: Matter is not too exotic, for the effective formulation to be physically equivalent to the geometric and physical formulations.

Direct experimental verification of the Casimir effect does not yield direct measurement of negative energy densities.

“Even direct experimental verification of the Casimir effect does not yield direct measurement of negative energy densities, though the Casimir effect relies essentially on the existence of such; rather, the negative energy densities are inferred from measurement of the Casimir force itself (Brown and Maclay 1969).”

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Ordinary energy density is not an observer-independent quantity.

“ordinary energy density is not an observer-independent quantity, and so it can mediate no physical interaction in any way with intrinsic physical significance.”

We have extremely good indirect experimental and observational evidence for violations of several standard energy conditions.

“We do, however, have extremely good indirect experimental and observational evidence for violations of several of them, as I will discuss in § 3.”

• cites Curiel (2014b)Not checked yet

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The strong energy condition implies the null energy condition.

“The SEC does, however, imply the NEC.”

General relativity cannot be comprehended as a theory describing a dynamical 'force' at all.

“Pace particle physicists, general relativity simply cannot be comprehended as a theory describing a dynamical 'force' at all.”

The effective forms of the dominant energy condition and the strengthened dominant energy condition are identical.

“As already noted, the effective forms of the DEC and SDEC are identical.”

Tipler proved that if Tmn ξm ξn is finitely bounded from below for all timelike ξa, then the weak energy condition holds.

“if T mn ξ m ξ n is finitely bounded from below for all timelike ξ a , i . e ., if there exists a b > 0 such that T mn ξ m ξ n ≥ -b for all timelike ξ a , then WEC holds”

• cites Tipler (1978)Not checked yet

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The effective forms of the dominant energy condition and the strengthened dominant energy condition mean pressures are strictly bounded both above and below by the energy density.

“Their interpretation, besides the now-familiar demand that locally measured energy density be non-negative, is that pressures be strictly bounded both above and below by the energy density.”

The proposed interpretations of the SEC's geometric and physical forms, that 'gravity tends to be attractive', are on firmer ground than for the NEC because the convergence of all timelike geodesics at a point does imply the convergence of null geodesics there.

“With regard to the SEC, because the convergence of all timelike geodesics at a point does imply the convergence of null geodesics there, the proposed interpretations of its geometric and physical forms, that 'gravity tends to be attractive', are on firmer ground than for the NEC.”

Energy conditions seem to reach down to a level of structure in our understanding of gravitation and relativistic spacetimes even more fundamental than the Einstein field equation itself.

“In a sense, therefore, energy conditions seem to reach down to and get a hold of a level of structure in our understanding of gravitation and relativistic spacetimes even more fundamental than the Einstein field equation itself.”

The Casimir effect relies essentially on the existence of negative energy densities.

“though the Casimir effect relies essentially on the existence of such”

• cites Brown and Maclay 1969Not checked yet

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There has never been direct experimental observation of a violation of any of the standard energy conditions discussed in the paper.

“It should be noted, however, that, to the best of my knowledge, there has never been direct experimental observation of a violation of any of the standard energy conditions I discuss in § 2.”

There does not exist in general relativity a satisfactory definition for a 'gravitational' stress-energy tensor, one that represents localized stress-energy of purely 'gravitational' systems.

“There does not exist in general relativity a satisfactory definition for a 'gravitational' stress-energy tensor, one that represents localized stress-energy of purely 'gravitational' systems.”

• cites Curiel 2013Not checked yet

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The SEC says nothing at all about the 'effect of gravity' on bodies accelerating under the action of other forces.

“And, again, there is the issue that this condition says nothing at all about the 'effect of gravity' on bodies accelerating under the action of other forces.”

The effective interpretation of the WEC shares the same problem as the NEC: it is unclear what physical significance the relations among energy density and pressure can have when they permit superluminal propagation of physical structure.

“The problems with the effective interpretation of the WEC are much the same as for the NEC: it is not clear what physical significance the given relations among energy density and pressure can have when they permit superluminal propagation of physical structure.”

• AssumptionThe authors state this as an assumption: The effective fluid described by the stress-energy tensor has a well behaved barotropic equation of state, i.e., a fixed invertible relation ρ(p) expressing ρ as a function of the single isotropic pressure p.

There are many results showing when various energy conditions may or must be violated theoretically and observationally, but none showing non-trivially when one must hold.

“There are a plethora of results that show when various energy conditions may or must be violated both theoretically and according to observation, which I discuss in § 3.2, but none that show non-trivially when one must hold.”

The DEC does not preclude superluminal speeds of sound for fields.

“Once again, however, it is clear that the DEC does not preclude superluminal speeds of sound for fields, so it is not clear what work the prohibition on superluminal propagation of stress-energy is doing.”

The one possible exception to the claim that no substantive proposition derives an energy condition is Wall's attempt to derive the averaged null energy condition from the Generalized Second Law of thermodynamics.

“The one possible exception to this claim I know of is the attempt by Wall (2010) to derive the so-called averaged null energy condition (ANEC) from the Generalized Second Law of thermodynamics.”

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Tipler proved that if Tmn ξm ξn is finitely bounded from below for all unit timelike ξa, and T ab is of type i, then the null energy condition holds.

“if T mn ξ m ξ n is finitely bounded from below for all unit timelike ξ a , and T ab is of type i , then the NEC holds.”

• cites Tipler (1978)Not checked yet

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The interpretive flexibility that holds for most fundamental propositions in general relativity does not hold for energy conditions.

“This interpretive flexibility does not seem to hold, however, for energy conditions.”

Inequality (2.3.2) is vacuous, for the righthand side of the inequality can be made as negative as one likes.

“one realizes that the inequality (2.3.2) is vacuous, for the righthand side of the inequality can be made as negative as one likes.”

All known proofs of the most general form of the Zeroth Law rely on the DEC.

“This, of course, is the Zeroth Law of black-hole mechanics, and all known proofs of the most general form of the Zeroth Law rely on the DEC.”

The geometric, physical, and effective forms of the energy conditions are, modulo the assumption of the Einstein field equation, logically equivalent.

“And yet these statements are, modulo the assumption of the Einstein field equation, logically equivalent.”

• AssumptionThe authors state this as an assumption: The Einstein field equation is assumed when comparing the geometric, physical, and effective forms of the energy conditions.

As Borde (1987) points out, Tipler's constraint raises problems for the physical plausibility, or at least possible scope, of the conditions.

“As Borde (1987) points out, this constraint raises problems for the physical plausibility, or at least possible scope, of the conditions.”

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The weakest condition, the NEC, already has remarkably strong consequences.

“The weakest condition, the NEC, already has remarkably strong consequences.”

In every spacetime, one can find at every point a congruence of timelike geodesics that has divergent expansion as one approaches that point.

“When one considers that one can, in every spacetime, find at every point a congruence of timelike geodesics that has divergent expansion as one approaches that point, one realizes that the inequality (2.3.2) is vacuous”

Borde's modifications allow application of the averaged condition to situations in which the total integral may essentially be zero even though there are large and long violations of the relevant pointilliste condition, such as may occur for the SEC during inflationary periods of a spacetime.

“This allows application of the averaged condition to situations in which the total integral may essentially be zero even though there are large and long violations of the relevant pointilliste condition, such as may occur for the SEC during inflationary periods of a spacetime.”

The argument that one can derive the NEC and the SEC from the Raychaudhuri equation fails.

“Even though I think the argument fails, it is of interest to try to pinpoint exactly why it fails, as it sheds light on why it appears to be difficult to derive the energy conditions from other fundamental principles”

The opposite is true for the pointilliste conditions, whose constrained quantities promiscuously appear in equations of motion, field equations and couplings for many if not most known kinds of physical system.

“the opposite is true for the pointilliste conditions, whose constrained quantities promiscuously appear in equations of motion, field equations and couplings for many if not most known kinds of physical system.”

There are weaker forms of the Zeroth Law that require no energy condition for their proof.

“The qualification 'most general' is required because there are weaker forms of the Zeroth Law that require no energy condition for their proof: any sufficiently regular Killing horizon must be bifurcate, and the appropriate generalization of surface gravity for a bifurcate Killing horizon must be constant on the entire horizon, without the need to impose any energy condition (Kay and Wald 1991; R´”

• cites Kay and Wald 1991Not checked yet

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• cites R´ acz and Wald 1992Not checked yet

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• cites Wald 1994Not checked yet

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One can always find geodesic congruences that are as badly behaved as one wants, in just about any way one wants to make that idea precise, with respect to how various measures of curvature evolve along the congruences.

“one can always find geodesic congruences that are as badly behaved as wants, in just about any way one wants to make that idea precise, with respect to how various measures of curvature evolve along the congruences.”

The roles the assumption of the Zeroth Law plays in the proofs of the two versions of the First Law differ significantly.

“The roles the assumption of the Zeroth Law plays in the proofs of the two versions of the First Law differ significantly, moreover, so it is not clear one could give a single principled answer to the question of whether or not the First Law is a consequence of the DEC that covered both versions at once.”

All the most compelling arguments in favor of a reasonably broad kind of stability of an event horizon assume constraints on the form of the matter causing the perturbation, constraints that usually look a lot like energy conditions.

“All the most compelling arguments in favor of a reasonably broad kind of stability I know, however, do assume constraints on the form of the matter causing the perturbation, constraints that usually look a lot like energy conditions.”

Tipler's constraint rules out cases where the integral equals zero because the relevant curve passes endlessly in and out of regions of positive and negative energy density.

“Then Tipler's constraint rules out cases where the integral equals zero because the relevant curve passes endlessly in and out of regions of positive and negative energy density.”

The Weyl tensor is not directly constrained by the stress-energy tensor of matter, in the sense that it may be non-zero even when T_ab is zero.

“Because the Weyl tensor is not directly constrained by the stress-energy tensor of matter, in the sense that it may be non-zero even when T ab is zero, it is often thought to represent 'purely gravitational' degress of freedom.”

The only important consequences of the SEC (and the ASEC) are singularity theorems.

“Another striking feature of the list is that the only important consequences of the SEC (and the ASEC) are singularity theorems”

Wong (2011) points out that the theorem often cited in support of the idea that the DEC prohibits superluminal propagation of stress-energy shows only that the DEC prohibits 'the edge of a vacuum' from propagating superluminally, not arbitrary stress-energy distributions.

“Wong, I think rightly, points out that this theorem in fact shows only that DEC prohibits 'the edge of a vacuum' (or vacuum fluctuations, in a quantum context) from propagating superluminally, not arbitrary stress-energy distributions.”

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The standard pointilliste energy conditions do not directly constrain the behavior of anything one may want to think of as gravitational stress-energy.

“The standard pointilliste energy conditions do not directly constrain the behavior of anything one may want to think of as gravitational stress-energy, and yet one may still want to try to do so.”

Based on Geroch (2010), Earman shows that physical systems can have well posed initial-value formulations even when the DEC is violated.

“Then, based on Geroch (2010), he shows that physical systems can have well posed initial-value formulations even when the DEC is violated.”

• cites Geroch (2010)Not checked yet

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• cites Earman (2012)Not checked yet

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The First Law of black-hole mechanics (for asymptotically flat black holes) does not require for its validity the assumption of any energy condition, unlike the other three Laws.

“There is a striking absentee from the list of consequences: strictly speaking, the First Law of black-hole mechanics (for asymptotically flat black holes)-conservation of mass-energy-does not require for its validity the assumption of any energy condition (unlike the other three Laws).”

Tipler (1978) required the additional constraint that the characteristic integral of the averaged condition at issue can equal zero for any curve only if its integrand equals zero along the entire curve.

“Tipler (1978), which if my history is not mistaken was the first use of an averaged condition to prove results of any depth, required the additional constraint that the characteristic integral of the averaged condition at issue can equal zero for any curve only if its integrand ( e . g ., T mn ξ m ξ n for the physical AWEC) equals zero along the entire curve.”

• cites Tipler (1978)Not checked yet

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The interpretive problems of the different forms of the energy conditions are not all of the same type.

“I don't think so, for, if we could, then surely the forms that had interpretive problems would all be of the same type, but that is not the case here.”

Gannon's theorem: in any asymptotically flat spacetime with a non-simply connected Cauchy surface, a singularity is bound to form (assuming the NEC).

“Among the singularity theorems it supports, to my mind the most astonishing is the one due to Gannon (1975): in any asymptotically flat spacetime with a non-simply connected Cauchy surface, a singularity is bound to form.”

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Topological complexity by itself, with the only constraint on metrical structure being the mild one of the NEC, suffices for the formation of singularities (in the guise of the incompleteness of a causal geodesic).

“Topological complexity by itself, with the only constraint on metrical structure being the mild one of the NEC, suffices for the formation of singularities (in the guise of the incompleteness of a causal geodesic).”

All the most rigorous and the most physically compelling derivations of the First Law assume that the surface gravity of the black hole is constant on the event horizon.

“The delicacy arises from the fact that all the most rigorous and the most physically compelling derivations of the Law I know (Bardeen, Carter, and Hawking 1973; Wald and Gao 2001) assume that the surface gravity of the black hole is constant on the event horizon.”

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• cites Wald and Gao 2001Not checked yet

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For a hypersurface-orthogonal congruence, it follows directly from Frobenius's Theorem that ω_ab = 0.

“For a hypersurface-orthogonal congruence, it follows directly from Frobenius's Theorem that ω ab = 0.”

The result of Ansoldi (2007) comes close to making the SEC both necessary and sufficient for the occurrence of certain types of singularities.

“The result of Ansoldi (2007), however, that black holes with singularity-free interiors necessarily violate the SEC, may push one towards the opposite view, in so far as it comes close to making the SEC both necessary and sufficient for the occurrence of certain types of singularities.”

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The NEC is relevant for the results about the topological and Killing structure of horizons associated with asymptotically flat black holes.

“One can perhaps see why the NEC is relevant for the results about the topological and Killing stucture of horizons associated with asymptotically flat black holes”

In such cases, one can show that the focusing effects the energy condition induces in geodesic congruences actually contributes directly to the lack of an event horizon.

“In such cases, one can show that the focusing effects the energy condition induces in geodesic congruences actually contributes directly to the lack of an event horizon.”

The presence of a negative cosmological constant does not yield violations of the NEC.

“the presence of a negative cosmological constant does not yield violations of the NEC”

The most compelling empirical evidence for violations of energy conditions comes from cosmology.

“The most compelling empirical evidence for violations of energy conditions comes from cosmology.”

Static and globally hyperbolic spacetimes are 'of measure zero' in the space of all spacetimes.

“Static and globally hyperbolic spacetimes, however, are 'of measure zero' in the space of all spacetimes”

Strongly substantiated cosmographic arguments comparing best estimates for the age of the oldest stars to the epoch of galaxy formation show that the SEC must have been violated in the relatively recent cosmological past (redshift z < 7).

“For instance, strongly substantiated cosmographic arguments comparing best estimates for the age of the oldest stars to the epoch of galaxy formation show that the SEC must have been violated in the relatively recent cosmological past (redshift z &lt; 7) (Visser 1997a; Visser 1997b; Visser 2005).”

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The existence of cosmological (i.e., non-collapse) singularities in both spatially open and closed universes can be shown under the assumption of the ASEC, without the full SEC.

“Likewise, the existence of cosmological ( i . e ., non-collapse) singularities in both spatially open and closed universes can be shown under the assumption of the ASEC (Tipler 1978; Senovilla 1997), without the full SEC.”

• cites Tipler 1978Not checked yet

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• cites Senovilla 1997Not checked yet

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The example Weatherall (2012) produces to show that a null curve can satisfy all of the theorem's conditions relies on a stress-energy tensor not of Hawking-Ellis type i.

“It is perhaps important that the example Weatherall (2012) produces to show that a null curve can satisfy all of the theorem's conditions relies on a stress-energy tensor not of Hawking-Ellis type i .”

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Under the assumptions used to prove the theorem of Geroch and Jang (1975), Malament (2012a) showed that the SDEC is necessary for the body to follow a geodesic, and not just any timelike curve.

“Under the assumptions used to prove the theorem of Geroch and Jang (1975), Malament (2012a) showed that the SDEC is necessary for the body to follow a geodesic, and not just any timelike curve.”

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Weatherall (2012) strengthened the result by showing that the SDEC is necessary for the geodesic to be timelike, not spacelike.

“Weatherall (2012) strengthened the result by showing that the SDEC is necessary for the geodesic to be timelike, not spacelike.”

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There is no common thread to the role the DEC plays in the proofs of the various results about black holes it is assumed for, analogous to the way that the NEC plays essentially the same role in the proofs of many of its consequences.

“There is, however, no common thread to the role the DEC plays in the proofs of the various results about black holes it is assumed for, analogous to the way that the NEC plays essentially the same role in the proofs of many of its consequences.”

Some of the impressionist energy conditions can be used to prove essentially identical theorems.

“That some of the impressionist energy conditions can be used to prove essentially identical theorems”

All the assumptions used in proving essentially all singularity theorems have strong observational or theoretical support except the energy conditions.

“all the assumptions used in proving essentially all of them have strong observational or theoretical support except the energy conditions”

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Being free of singularities is, under the ancillary conditions, unstable under arbitrarily small perturbations.

“and so being free of singularities is, under the ancillary conditions, unstable under arbitrarily small perturbations.”

Satisfaction of the pointilliste conditions is not necessary for validity of at least some of the theorems.

“already shows that satisfaction of the pointilliste conditions is not necessary for validity of at least some of the theorems.”

The most precise, rigorous and strongest geodesic theorems (Geroch and Jang 1975; Ehlers and Geroch 2004) both assume the SDEC.

“The most precise, rigorous and strongest geodesic theorems (Geroch and Jang 1975; Ehlers and Geroch 2004) both assume the SDEC.”

• cites Geroch and Jang 1975Not checked yet

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• cites Ehlers and Geroch 2004Not checked yet

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The proof that a given globally hyperbolic extension is maximal works by showing that the extension cannot be extended because to do so would result 'immediately' in singularities, contradicting the assumption of extendibility.

“roughly speaking, the proof works by showing that the given globally hyperbolic extension cannot be extended (and so is maximal) because to do so would result 'immediately' in singularities, contradicting the assumption of extendibility.”

There are in fact cases where satisfaction of an energy condition actually seems to aid the development of a naked singularity after gravitational collapse, e.g., the WEC in the case of the self-similar collapse of a body of perfect fluid.

“there are in fact cases where satisfaction of an energy condition actually seems to aid the development of a naked singularity after gravitational collapse, e . g ., the WEC in the case of the self-similar collapse of a body of perfect fluid (Joshi 2003).”

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Visser predicted on purely theoretical grounds that the most likely culprit for violation of SEC in the recent cosmological past must be a negative cosmological constant.

“Visser predicted on purely theoretical grounds that the most likely culprit for violation of SEC in the recent cosmological past must be a negative cosmological constant.”

The Lorentzian splitting theorems go some way towards proving the conjecture of Geroch (1966) that essentially all spatially closed spacetimes either have singularities or do not satisfy the SEC.

“Thus, they go some way towards proving the conjecture of Geroch (1966) that essentially all spatially closed spacetimes either have singularities or do not satisfy the SEC.”

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The Lorentzian splitting theorems may be thought of as rigidity theorems for singularity theorems invoking the SEC.

“The Lorentzian splitting theorems may be thought of as rigidity theorems for singularity theorems invoking the SEC”

The most common formulations invoke either the DEC or the WEC as a constraint on the matter fields permissible for the initial-value formulation of general relativity.

“The most common formulations invoke either the DEC or the WEC (Joshi 2003) as a constraint on the matter fields permissible for the initial-value formulation of general relativity.”

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Penrose (1990) and Ashtekar and Penrose (1990) were able to prove positivity of ADM mass using only the NEC.

“it is particularly striking that Penrose (1990) and Ashtekar and Penrose (1990) were able to prove positivity of ADM mass using only it”

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The original singularity theorem, the demonstration by Penrose (1965) that singularities should form after gravitational collapse in spatially open universes, holds under the weaker assumption of the ANEC.

“The original singularity theorem, the demonstration by Penrose (1965) that singularities should form after gravitational collapse in spatially open universes, holds under the weaker assumption of the ANEC (Roman 1988; Senovilla 1997).”

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Almost all of the first category invoke the NEC for their proof.

“Almost all of the first category invoke the NEC for their proof.”

The only type of singularity theorem in which the DEC is used is the only one to make substantive, explicit assumptions about the distribution of stress-energy, in this case in the demand for non-zero averaged spatial energy density.

“The one type of singularity theorem (Senovilla 2007, 2008) it is used in, moreover, is the only one to make substantive, explicit assumptions (over and above the energy conditions themselves) about the distribution of stress-energy, in this case in the demand for non-zero averaged spatial energy density.”

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The splitting theorems show that, under certain other assumptions, there will be no singularities only when the spacetime is static and globally hyperbolic.

“the splitting theorems show that, under certain other assumptions, there will be no singularities only when the spacetime is static and globally hyperbolic.”

All known proofs of the positivity of the Bondi mass do require the DEC.

“All known proofs of the positivity of the Bondi mass do require the DEC”

'Hyper-fast' travel necessarily violates the SEC.

“8. 'hyper-fast' travel 74 [necessarily] (Alcubierre 1994; Krasnikov 1998; Olum 1998)”

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Closed timelike curves possibly violate the NEC.

“7. closed timelike curves [possibly] (Visser 1996)”

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Conformally coupled massless and massive scalar fields possibly violate the NEC.

“1. conformally coupled massless and massive scalar fields [possibly] (Visser and Barcel´ o 2000; Barcel´ o and Visser 2002)”

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Future-eternal inflationary cosmologies possibly violate the SEC.

“7. future-eternal inflationary cosmologies [possibly] (Borde and Vilenkin 1997)”

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This may give reason to doubt the stability of at least some of the singularity theorems in regimes where the energy conditions fail.

“This may give reason to doubt the stability of at least some of the singularity theorems in regimes where the energy conditions fail.”

Cosmological steady-state theories of Bondi and Gold (1948) and Hoyle (1948) necessarily violate the SEC.

“4. cosmological steady-state theories of Bondi and Gold (1948) and Hoyle (1948) 72 [necessarily]”

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• cites Hoyle 1948 sourceNot checked yet

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Naked singularities possibly violate the NEC.

“6. naked singularities [possibly] (Penrose 1979; Barcel´ o and Visser 1999; Joshi 2003)”

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• cites Barceló and Visser 1999Not checked yet

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• cites Joshi 2003Not checked yet

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'Big rip' singularities necessarily violate the NEC.

“4. 'big rip' singularities 68 [necessarily] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Cattoën and Visser 2005Not checked yet

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• cites Cattoën and Visser 2008Not checked yet

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Generically non-minimally coupled massless and massive scalar fields possibly violate the NEC.

“2. generically non-minimally coupled massless and massive scalar fields [possibly] (Flanagan and Wald 1996; Visser and Barcel´ o 2000; Barcel´ o and Visser 2002; Dubovsky, Gr´ egoire, Nicolis, and Rattazzi 2006)”

• cites Flanagan and Wald 1996Not checked yet

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• cites Visser and Barceló 2000Not checked yet

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• cites Barceló and Visser 2002Not checked yet

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The only safe conclusion seems to be that, at least in the case of these kinds of singularity, violations of salient energy conditions need not preclude their existence.

“The only safe conclusion seems to be that, at least in the case of these kinds of singularity, violations of salient energy conditions need not preclude their existence.”

Thus, it is only the strict symmetry of the Bekenstein models that precludes singularities.

“Thus, it is only the strict symmetry of the Bekenstein models that precludes singularities.”

Closed timelike curves possibly violate the SEC.

“2. closed timelike curves [possibly] (Visser 1996)”

• cites Visser 1996Not checked yet

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Bekenstein (1975) gave an example of a massive conformal scalar field coupled with dust in an example of how to construct a nonsingular FLRW model, exploiting the fact that the system can be made to violate the SEC.

“The particular example of a massive conformal scalar field coupled with dust given by Bekenstein (1975) in an example of how to construct a nonsingular FLRW model, exploiting the fact that the system can be made to violate the SEC”

• cites Bekenstein (1975)Not checked yet

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Tipler (1978) has pointed out an amusing poignancy in the role that homogeneity (high symmetry) plays in Bekenstein's construction of nonsingular FLRW spacetimes that violate the SEC.

“Tipler (1978), in a line of argument intended to mitigate such doubts, has pointed out an amusing poignancy in the role that homogeneity (high symmetry) plays in Bekenstein's construction of nonsingular FLRW spacetimes that violate the SEC.”

• cites Tipler (1978)Not checked yet

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Using classical scalar fields alone, without even having to resort to quantum weirdness, it is relatively easy to engineer violations of even the weakest conditions, the NEC and the ANEC.

“Indeed, using classical scalar fields alone, without even having to resort to quantum weirdness, it is relatively easy to engineer violations of even the weakest conditions, the NEC and the ANEC, as the list of violations shows.”

Sudden future singularities possibly violate the NEC.

“5. sudden future singularities 69 [possibly] (Barrow 2004a; Barrow 2004b; Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Barrow 2004aNot checked yet

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• cites Barrow 2004bNot checked yet

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• cites Cattoën and Visser 2005Not checked yet

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Physically traversable wormholes necessarily violate the SEC.

“3. physically traversable wormholes [necessarily] (Morris and Thorne 1988; Visser 1989b; Visser 1989a)”

• cites Morris and Thorne 1988Not checked yet

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• cites Visser 1989bNot checked yet

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• cites Visser 1989aNot checked yet

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Classical Dirac fields possibly violate the SEC.

“5. classical Dirac fields [possibly] (Wald 1984)”

• cites Wald 1984 sourceNot checked yet

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Any fluid with a barotropic index w < -1 (such as those postulated in so-called phantom cosmologies) necessarily violates the NEC.

“9. any fluid with a barotropic index w &lt; -1 70 (such as those postulated in so-called phantom cosmologies) [necessarily] (Visser 1996; Dabrowski and Denkiewicz 2009)”

• cites Visser 1996Not checked yet

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• cites Dabrowski and Denkiewicz 2009Not checked yet

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A marginally trapped surface would form only if there were deviations from homogeneity.

“Of course, a marginally trapped surface would form only if there were deviations from homogeneity.”

A positive cosmological constant (e.g., anti-de Sitter Space) necessarily violates the SEC.

“6. a positive cosmological constant ( e . g ., anti-de Sitter Space) 73 [necessarily] (Hawking and Ellis 1973; Visser 1996)”

• cites Hawking and Ellis 1973 sourceNot checked yet

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• cites Visser 1996Not checked yet

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'Big bang' and 'big crunch' singularities possibly violate the NEC.

“3. 'big bang' and 'big crunch' singularities 67 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Cattoën and Visser 2005Not checked yet

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• cites Cattoën and Visser 2008Not checked yet

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Violations of the NEC imply violations of all other pointilliste energy conditions.

“First and perhaps foremost, they imply violations of all other pointilliste energy conditions.”

'Big bang' and 'big crunch' singularities possibly violate the DEC.

“1. 'big bang' and 'big crunch' singularities 80 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Cattoën and Visser 2005Not checked yet

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• cites Cattoën and Visser 2008Not checked yet

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A negative cosmological constant, as in de Sitter spacetime, and the 'dark energy' postulated to drive the observed accelerated expansion of the universe necessarily violate the ANEC.

“7. a negative cosmological constant, as in de Sitter spacetime, and the 'dark energy' postulated to drive the observed accelerated expansion of the universe [necessarily] (Hawking and Ellis 1973; Caldwell 2002; Caldwell, Kamionkowski, and Weinberg 2003; Dabrowski, Stachowiak, and Szydglyph[suppress] lowski 2003)”

• cites Hawking and Ellis 1973 sourceNot checked yet

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• cites Caldwell 2002Not checked yet

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• cites Caldwell, Kamionkowski, and Weinberg 2003Not checked yet

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Spatially closed, expanding, singularity-free spacetimes necessarily violate the ANEC.

“5. spatially closed, expanding, singularity-free spacetimes [necessarily] (Senovilla 1997)”

• cites Senovilla 1997Not checked yet

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Massless and massive non-minimally coupled scalar fields possibly violate the AWEC.

“2. massless and massive non-minimally coupled scalar fields [possibly] (Flanagan and Wald 1996; Dubovsky, Gr´ egoire, Nicolis, and Rattazzi 2006)”

• cites Flanagan and Wald 1996Not checked yet

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• cites Dubovsky, Grégoire, Nicolis, and Rattazzi 2006Not checked yet

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Active and passive gravitational mass is not necessarily equal in general relativity, at least when negative mass is allowed.

“One possible mechanism for producing anti-inertial charge is strongly suggested by the arguments of Bondi (1957) showing that active and passive gravitational mass is not necessarily equal in general relativity, at least when negative mass is allowed.”

• cites Bondi (1957)Not checked yet

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Minimally coupled massless and massive scalar fields possibly violate the ASEC.

“11. minimally coupled massless and massive scalar fields [possibly] (Visser and Barcel´ o 2000; Barcel´ o and Visser 2002)”

• cites Visser and Barceló 2000Not checked yet

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• cites Barceló and Visser 2002Not checked yet

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Just before or just after a cosmological 'inflexion' possibly violates the ANEC.

“4. just before or just after a cosmological 'inflexion' 78 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Cattoën and Visser 2005Not checked yet

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• cites Cattoën and Visser 2008Not checked yet

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'Big bang' and 'big crunch' singularities possibly violate the ANEC.

“1. 'big bang' and 'big crunch' singularities 75 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Cattoën and Visser 2005Not checked yet

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• cites Cattoën and Visser 2008Not checked yet

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Sudden future singularities possibly violate the DEC.

“2. sudden future singularities 81 [possibly] (Barrow 2004a; Barrow 2004b; Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Barrow 2004aNot checked yet

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• cites Barrow 2004bNot checked yet

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• cites Cattoën and Visser 2005Not checked yet

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Physically traversable wormholes necessarily violate the ASEC.

“10. physically traversable wormholes [necessarily] (Molina-Paris and Visser 1999; Hochberg, Molina-Paris, and Visser 1999)”

• cites Molina-Paris and Visser 1999Not checked yet

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• cites Hochberg, Molina-Paris, and Visser 1999Not checked yet

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Energy condition violations fall into two classes: those associated with a type of physical system and those associated with a type of 'event'.

“First, roughly speaking, the violations fall into one of two classes, being associated with a type of physical system ( e . g ., conformally coupled scalar field, classical Dirac field) or with a type of 'event' (very loosely construed, e . g ., traversable wormhole, closed timelike curve, or big rip singularity).”

Asymptotically flat black holes with regular (non-singular) interiors necessarily violate the ANEC.

“8. asymptotically flat black holes with regular (non-singular) interiors [necessarily] (Ansoldi 2007)”

• cites Ansoldi 2007Not checked yet

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Massive Klein-Gordon fields possibly violate the ASEC.

“12. massive Klein-Gordon fields [possibly] (Visser 1996)”

• cites Visser 1996Not checked yet

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For violations associated with a type of 'event', the regions where the energy conditions are violated can generally be localized to a neighborhood of the event.

“Generally speaking, for the latter, the regions where the energy conditions are violated can be 'localized' to a neighborhood of the 'event'.”

Pinpointing singularity theorems allow one to say where in spacetime the singularities occur, in the sense that one can localize the singularities.

“Roughly speaking, pinpointing theorems, as the name suggests, in certain ways allow one to say where in spacetime the singularities occur, and so in a sense one can 'localize' the singularities.”

Typical gauge theories with spontaneously broken symmetries possibly violate the ASEC.

“13. typical gauge theories with spontaneously broken symmetries [possibly] (Tipler 1978)”

• cites Tipler 1978Not checked yet

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Traversable wormholes possibly violate the AWEC.

“4. traversable wormholes (Morris, Thorne, and Yurtsever 1988) [possibly]”

• cites Morris, Thorne, and Yurtsever 1988Not checked yet

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Non-pinpointing singularity theorems, such as those of Gannon (1975, 1976), merely demonstrate the existence of incomplete, inextendible geodesics without giving information about where the geodesic is in spacetime.

“Singularity theorems that are not pinpointing, such as those of Gannon (1975, 1976), merely demonstrate the existence of incomplete, inextendible geodesics without giving one any information about 'where' the geodesic is in spacetime.”

• cites Gannon (1975, 1976)Not checked yet

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Cosmological inflation necessarily violates the ANEC.

“6. cosmological inflation [necessarily] (Visser 1996)”

• cites Visser 1996Not checked yet

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Cosmological 'bounces' necessarily violate the ANEC.

“3. cosmological 'bounces' 77 [necessarily] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”

• cites Cattoën and Visser 2005Not checked yet

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• cites Cattoën and Visser 2008Not checked yet

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Closed timelike curves possibly violate the ASEC.

“9. closed timelike curves [possibly] (Visser 1996)”

• cites Visser 1996Not checked yet

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'Hyper-fast' travel necessarily violates the ASEC.

“15. 'hyper-fast' travel 79 [necessarily] (Krasnikov 1998; Olum 1998; Alcubierre 1994)”

• cites Krasnikov 1998Not checked yet

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• cites Olum 1998Not checked yet

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• cites Alcubierre 1994Not checked yet

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Massless conformally coupled scalar fields possibly violate the AWEC.

“1. massless conformally coupled scalar fields 82 [possibly] (Visser and Barcel´ o 2000; Barcel´ o and Visser 2002)”

• cites Visser and Barceló 2000Not checked yet

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• cites Barceló and Visser 2002Not checked yet

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Closed timelike curves possibly violate the AWEC.

“3. closed timelike curves [possibly] (Visser 1996)”

• Also stated in: § AWEC · p. 31 · hedged

  “4. closed timelike curves [possibly] (Visser 1996)”

• cites Visser 1996Not checked yet

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Conformal scalar fields coupled with dust possibly violate the ASEC.

“14. conformal scalar fields coupled with dust [possibly] (Bekenstein 1975)”

• cites Bekenstein 1975Not checked yet

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Classical Dirac fields necessarily violate the DEC.

“3. classical Dirac fields [necessarily] (Penrose and Rindler 1984)”

• cites Penrose and Rindler 1984 sourceNot checked yet

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One reason to think energy conditions should form part of any broad conception of what constitutes a spacetime theory rests on the remark of Geroch and Horowitz that without energy conditions the Einstein field equation 'has no content.'

“One reason to think they should form part of any broad conception of what constitutes a spacetime theory rests on the remark of Geroch and Horowitz I quoted on page 3, that without energy conditions the Einstein field equation 'has no content.'”

• cites Geroch and Horowitz (1979)Not checked yet

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The temporal orientation is determined by how one parametrizes temporal curves.

“The temporal orientation is determined by how one parametrizes temporal curves (which in turn, of course, depends on whether one can do so in a way that consistently singles out a choice of 'future lobe of null cone' at every point of the manifold in the first place).”

Except for the impressionist conditions, energy conditions do not require a projective structure.

“Energy conditions, however, do not seem to require a notion of temporal orientability, as the discussion of § 4 strongly suggests, and, except for the impressionist conditions, neither do they require a projective structure.”

If one fixes a 1+3 tetrad such that the metric at a point can be expressed as a sum of coefficients times tetrad vectors, then reversing the sign of the timelike tetrad vector does not change the metric.

“If one fixes a 1 + 3 tetrad { µ ξ a } µ ∈{ 0 , 1 , 2 , 3 } (not necessarily orthonormal) such that the metric at a point can be expressed as ∑ µ α µ µ ξ a µ ξ b , for some real coefficients α µ , then reversing the sign of 0 ξ a clearly does not change the metric.”

This fact poses a serious problem for any attempt to formulate a notion of dynamical evolution that would support any minimal notion of predictability or determinism.

“Now, this fact poses a serious problem for any attempt to formulate a notion of dynamical evolution that would support any minimal notion of predictability or determinism.”

Standard proofs showing existence and uniqueness of solutions to the initial-value problem of general relativity require matter-fields that yield quasi-linear, hyperbolic equations of motion satisfying something very much like the DEC.

“So does the fact that the standard proofs showing existence and uniqueness of solutions to the initial-value problem of general relativity require matter-fields that yield quasi-linear, hyperbolic equations of motion satisying something very much like the DEC (Hawking and Ellis 1973; Wald 1984).”

• cites Hawking and Ellis 1973 sourceNot checked yet

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• cites Wald 1984 sourceNot checked yet

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Holding on to everything in the list except for the Einstein field equation, so long as whatever field equations do hold depend only on something like the stress-energy tensor that does not depend on idiosyncratic features of particular kinds of matter, one will likely face the same problem.

“Holding on to everything in my list except for the Einstein field equation, so long as whatever field equations do hold depend only on something like the stress-energy tensor that does not depend on idiosyncratic features of particular kinds of matter, I strongly suspect that one will likely face the same problem.”

Before 1961, it was more or less universally believed in the general relativity community that the Trace Energy Condition would always be satisfied, even under the most extreme physical conditions.

“Before 1961, it seemed to have been more or less universally believed in the general relativity community that this condition would always be satisfied, even under the most extreme physical conditions.”

• cites Oppenheimer and Volkoff (1939)Not checked yet

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• cites Harrison, Wakano, and Wheeler (1957)Not checked yet

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The time-reversed spacetime has stress-energy tensor T'_ab = T_ab.

“Clearly, then, T ′ ab = T ab .”

The conditions one needs to impose to make the initial-value problem of general relativity merely consistent-the Gauss-Codazzi constraints-look very much like conditions on the allowed forms of types of matter.

“The conditions one needs to impose to make the initial-value problem of general relativity merely consistent-the so-called Gauss-Codazzi constraints-look very much like conditions on the allowed forms of types of matter.”

The metric stays the same under time-reversal.

“The metric stays the same.”

The rule for constructing the time-reverse of a (temporally orientable) relativistic spacetime is to leave everything the same except for the sense of parametrization of timelike (and null) curves, which should be reversed.

“So, in sum, I claim the rule for constructing the time-reverse of a (temporally orientable) relativistic spacetime is to leave everything the same except for the sense of parametrization of timelike (and null) curves, which should be reversed.”

Standard proofs of the well-posedness of the initial-value formulation for general relativity rely on the DEC.

“The fact that the only known proof of the theorem that a given globally hyperbolic extension of a spacetime is the maximal such extension requires the WEC (Ringstr¨ om 2009), in conjunction with the fact that standard proofs of the well-posedness of the initial-value formulation for general relativity rely on the DEC, suggest that it may be the energy conditions that intervene to ensure a cogent”

The Trace Energy Condition is assumed without argument, or even remark, in the seminal papers of Oppenheimer and Volkoff (1939) and Harrison, Wakano, and Wheeler (1957) on possible equations of state for neutron stars.

“It is, for instance, assumed without argument, or even remark, in the seminal papers of Oppenheimer and Volkoff (1939) and Harrison, Wakano, and Wheeler (1957) on possible equations of state for neutron stars.”

• cites Oppenheimer and Volkoff (1939)Not checked yet

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• cites Harrison, Wakano, and Wheeler (1957)Not checked yet

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The same time-reversal invariance holds for all the more recently proposed energy conditions discussed in §2.4.

“(The same holds as well for all the more recently proposed energy conditions discussed in § 2.4.)”

Temporal orientation is a concept at the level of differential topology and conformal structure.

“It is a concept at the level of differential topology and conformal structure.”

The inequality of passive and active gravitational mass already constitutes a violation of the principle of equivalence, at least in one of its guises.

“Arguably, the inequality of passive and active gravitational mass already constitutes a violation of the principle of equivalence, at least in one of its guises.”

If one cannot give principled reasons for why exactly one spacetime and no other is the natural result of dynamical evolution off a Cauchy surface according to the Einstein field equation, then one has captured one sense in which the Einstein field equation may 'have no content.'

“If one cannot give principled reasons for why exactly one of those spacetimes and no other is the natural result of dynamical evolution off the Cauchy surface according to the Einstein field equation, then one has captured one sense in which the Einstein field equation may 'have no content.'”

Globally hyperbolic spacetimes represent in a natural way possible solutions to the initial-value problem of general relativity as it is normally posed.

“First, note that globally hyperbolic spacetimes represent in a natural way possible solutions to the initial-value problem of general relativity as it is normally posed.”

Because all the standard energy conditions (and pretty much all the nonstandard ones) rely on the distinction between causal and non-causal vectors in general, and often on the distinction between null and timelike, it seems likely that energy conditions will be less fundamental than conformal structure.

“Because all the standard energy conditions (and pretty much all the nonstandard ones), rely on the distinction between causal and non-causal vectors in general, and often on the distinction between null and timelike, it seems likely that energy conditions will be less fundamental than conformal structure as well.”

A white hole will violate an energy condition if and only if its time-reversed black hole does so.

“a white hole will violate an energy condition if and only if its time-reversed black hole does so.”

Temporal orientation is not a metrical concept.

“Temporal orientation is not a metrical concept.”

A spacetime satisfies any one of the standard energy conditions listed in §2 if and only if the time-reverse of the spacetime does as well.

“It is a simple matter to verify that a spacetime satisfies any one of the standard energy conditions listed in § 2 if and only if the time-reverse of the spacetime does as well.”

Energy conditions do not seem to require a notion of temporal orientability, as the discussion of §4 strongly suggests.

“Energy conditions, however, do not seem to require a notion of temporal orientability, as the discussion of § 4 strongly suggests, and, except for the impressionist conditions, neither do they require a projective structure.”

Energy conditions seem to be largely independent of topological structure (except in so far as it is required to define differential structure).

“They also seem to be largely independent of topological structure (except in so far as it is required to define differential structure).”

The only known proof of the theorem that a given globally hyperbolic extension of a spacetime is the maximal such extension requires the WEC.

“The fact that the only known proof of the theorem that a given globally hyperbolic extension of a spacetime is the maximal such extension requires the WEC (Ringstr¨ om 2009), in conjunction with the fact that standard proofs of the well-posedness of the initial-value formulation for general relativity rely on the DEC, suggest that it may be the energy conditions that intervene to ensure a cogent”

• cites Ringström 2009 sourceNot checked yet

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It may be the energy conditions that intervene to ensure a cogent notion of dynamical evolution that supports some minimal notion of predictability or determinism.

“The fact that the only known proof of the theorem that a given globally hyperbolic extension of a spacetime is the maximal such extension requires the WEC (Ringstr¨ om 2009), in conjunction with the fact that standard proofs of the well-posedness of the initial-value formulation for general relativity rely on the DEC, suggest that it may be the energy conditions that intervene to ensure a cogent”