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A Primer on Energy Conditions
Open accessAbstract
An energy condition, in the context of a wide class of spacetime theories (including general relativity), is, crudely speaking, a relation one demands the stress-energy tensor of matter satisfy in order to try to capture the idea that "energy should be positive". The remarkable fact I will discuss in this paper is that such simple, general, almost trivial seeming propositions have profound and far-reaching import for our understanding of the structure of relativistic spacetimes. It is therefore especially surprising when one also learns that we have no clear understanding of the nature of these conditions, what theoretical status they have with respect to fundamental physics, what epistemic status they may have, when we should and should not expect them to be satisfied, and even in many cases how they and their consequences should be interpreted physically. Or so I shall argue, by a detailed analysis of the technical and conceptual character of all the standard conditions used in physics today, including examination of their consequences and the circumstances in which they are believed to be violated.
Overview
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TL;DR
This paper surveys the standard energy conditions in general relativity—their formulations, consequences, violations, and interpretive status—arguing that these simple conditions have profound import for spacetime structure but lack a clear theoretical, epistemic, and physical understanding. It concludes that we face a dilemma: either accept the exotic physics implied by widespread violations or reconstruct fundamental theory to rule them out.
Key findings
- Energy conditions such as NEC, WEC, SEC, DEC, SDEC, and averaged versions (ANEC, AWEC, ASEC) are generic constraints on stress-energy or Ricci curvature that support deep results, yet the paper argues they cannot be derived from other fundamental postulates and are imposed by fiat or checked for particular matter fields.
- The consequences list is dominated by singularity and black-hole theorems, “at 10 each”; the NEC supports results including Penrose’s singularity theorem and Hawking’s Area Theorem, while the SEC mainly supports singularity theorems and the DEC is used in black-hole topology/Killing-structure results and positive-mass theorems.
- Violations are widespread: cosmological evidence indicates the SEC must have been violated in the relatively recent past (redshift \(z < 7\)), and if accelerated expansion/dark energy is correct, the SEC is currently violated; inflation necessarily violates the SEC, while a negative cosmological constant does not violate the NEC.
- Interpretive problems are pervasive: geometric, physical, and effective formulations are logically equivalent under the Einstein equation but have uneven clarity; impressionist conditions lack compelling interpretations or motivations; no standard energy conditions based on the Weyl or Bel-Robinson tensors exist; and the historical Trace Energy Condition was once universally assumed but is now believed violated at densities above 10 times atomic nuclei, where \(p = \rho\).
Methods
- Conceptual, technical, and philosophical review of energy conditions in classical general relativity, with no new empirical data or experiments.
- Surveys formulations (geometric, physical, effective; pointilliste vs. impressionist), catalogues consequences and known violations from the literature, and includes appendices on a failed Raychaudhuri derivation, recent conditions (FEC, DETEC, TOSEC), average radial acceleration, temporal reversibility, and constraints on spacetime theories.
- Main analysis compares interpretations and logical relations among conditions, classifies consequences and violations, and discusses implications for the foundations of spacetime theories.
Limitations
- The paper states it does not provide a decisive answer; it aims only to survey issues, clarify questions, propose conjectures, and open the field, so those hoping for a decisive answer “will leave disappointed.”
- It states there are no compelling geometrical, physical, or effective interpretations or motivations for the impressionist energy conditions, and that many arguments about necessity, measure-zero claims, and foundational status are conjectural or hand-waving.
- Evident from the text: there has been no direct experimental observation of a violation of any standard energy condition; evidence is indirect, many violation entries are marked “possibly” or “necessarily” and some are “my calculation,” and no general proof establishes when energy conditions must hold.
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138 Own interpretation · 108 Not checked yet · 52 No basis given · 10 Own result
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“3. massive Klein-Gordon fields [possibly] (my calculation)”
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“1. cosmological steady-state theories of Bondi and Gold (1948) and Hoyle (1948) [necessarily] (my calculation)”
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“5. physically traversable wormholes [possibly] (my calculation)”
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“3. classical Dirac fields [possibly] (my calculation)”
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“5. conformal scalar fields coupled with dust [possibly] (my calculation)”
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“4. typical gauge theories with spontaneously broken symmetries [possibly] (my calculation)”
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“2. cosmological inflation [possibly] (my calculation)”
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“Now, it is a trivial matter to find globally hyperbolic spacetimes that violate any energy condition.”
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“He first showed that, in a natural sense, the WEC is the weakest local energy condition one can define.”
What the sources say
- Not checked yetcites Tipler (1978)
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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.”
What the sources say
- Not checked yetcites Penrose 1969
- Not checked yetcites Penrose and Floyd 1971
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“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.”
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“The second given geometric and physical forms of the SDEC make it manifest that the SDEC is in fact logically stronger than the DEC.”
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“It is easy to see, by considerations of continuity, that the WEC implies the NEC.”
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“rather, the negative energy densities are inferred from measurement of the Casimir force itself (Brown and Maclay 1969).”
What the sources say
- Not checked yetcites Brown and Maclay 1969
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“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.”
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“The effective form of the WEC, therefore, is in fact essentially equivalent to the NEC.”
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“Neither do they appear to be empirical or inductive generalizations, for the same reason.”
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“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”
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“This means that the effective fluid can be neither too 'stiff' nor too 'lax', but must lie in a middling Goldilocks regime.”
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“The SEC does not imply the WEC, for the SEC can be satisfied even if the ordinary mass-density is negative.”
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“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.”
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- The authors state this as an assumption: The Einstein field equation holds when relating geometric, physical, and effective formulations of energy conditions.
- The 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.
In the paper
“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).”
What the sources say
- Not checked yetcites Brown and Maclay 1969
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“ordinary energy density is not an observer-independent quantity, and so it can mediate no physical interaction in any way with intrinsic physical significance.”
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“We do, however, have extremely good indirect experimental and observational evidence for violations of several of them, as I will discuss in § 3.”
What the sources say
- Not checked yetcites Curiel (2014b)
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“The SEC does, however, imply the NEC.”
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“Pace particle physicists, general relativity simply cannot be comprehended as a theory describing a dynamical 'force' at all.”
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“As already noted, the effective forms of the DEC and SDEC are identical.”
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“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”
What the sources say
- Not checked yetcites Tipler (1978)
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“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.”
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“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.”
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“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.”
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“though the Casimir effect relies essentially on the existence of such”
What the sources say
- Not checked yetcites Brown and Maclay 1969
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“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.”
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“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.”
What the sources say
- Not checked yetcites Curiel 2013
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“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.”
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“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.”
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“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.”
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“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.”
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“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.”
What the sources say
- Not checked yetcites Wall (2010)
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“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.”
What the sources say
- Not checked yetcites Tipler (1978)
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“This interpretive flexibility does not seem to hold, however, for energy conditions.”
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“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.”
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“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.”
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“And yet these statements are, modulo the assumption of the Einstein field equation, logically equivalent.”
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In the paper
“As Borde (1987) points out, this constraint raises problems for the physical plausibility, or at least possible scope, of the conditions.”
What the sources say
- Not checked yetcites Borde (1987)
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“The weakest condition, the NEC, already has remarkably strong consequences.”
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“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”
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“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.”
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“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”
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“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.”
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In the paper
“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´”
What the sources say
- Not checked yetcites Kay and Wald 1991
- Not checked yetcites R´ acz and Wald 1992
- Not checked yetcites Wald 1994
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In the paper
“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.”
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In the paper
“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.”
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In the paper
“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.”
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In the paper
“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.”
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In the paper
“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.”
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In the paper
“Another striking feature of the list is that the only important consequences of the SEC (and the ASEC) are singularity theorems”
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“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.”
What the sources say
- Not checked yetcites Wong (2011)
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In the paper
“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.”
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In the paper
“Then, based on Geroch (2010), he shows that physical systems can have well posed initial-value formulations even when the DEC is violated.”
What the sources say
- Not checked yetcites Geroch (2010)
- Not checked yetcites Earman (2012)
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In the paper
“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).”
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In the paper
“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.”
What the sources say
- Not checked yetcites Tipler (1978)
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In the paper
“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.”
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“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.”
What the sources say
- Not checked yetcites Gannon (1975)
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In the paper
“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).”
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In the paper
“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.”
What the sources say
- Not checked yetcites Bardeen, Carter, and Hawking 1973
- Not checked yetcites Wald and Gao 2001
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In the paper
“For a hypersurface-orthogonal congruence, it follows directly from Frobenius's Theorem that ω ab = 0.”
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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, in so far as it comes close to making the SEC both necessary and sufficient for the occurrence of certain types of singularities.”
What the sources say
- Not checked yetcites Ansoldi (2007)
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In the paper
“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”
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In the paper
“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.”
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In the paper
“the presence of a negative cosmological constant does not yield violations of the NEC”
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In the paper
“The most compelling empirical evidence for violations of energy conditions comes from cosmology.”
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In the paper
“Static and globally hyperbolic spacetimes, however, are 'of measure zero' in the space of all spacetimes”
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In the paper
“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 < 7) (Visser 1997a; Visser 1997b; Visser 2005).”
What the sources say
- Not checked yetcites Visser 1997a
- Not checked yetcites Visser 1997b
- Not checked yetcites Visser 2005
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In the paper
“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.”
What the sources say
- Not checked yetcites Tipler 1978
- Not checked yetcites Senovilla 1997
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In the paper
“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 .”
What the sources say
- Not checked yetcites Weatherall (2012)
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In the paper
“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.”
What the sources say
- Not checked yetcites Malament (2012a)
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In the paper
“Weatherall (2012) strengthened the result by showing that the SDEC is necessary for the geodesic to be timelike, not spacelike.”
What the sources say
- Not checked yetcites Weatherall (2012)
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In the paper
“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.”
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In the paper
“That some of the impressionist energy conditions can be used to prove essentially identical theorems”
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In the paper
“all the assumptions used in proving essentially all of them have strong observational or theoretical support except the energy conditions”
What the sources say
- Not checked yetcites Sciama (1976)
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In the paper
“and so being free of singularities is, under the ancillary conditions, unstable under arbitrarily small perturbations.”
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In the paper
“already shows that satisfaction of the pointilliste conditions is not necessary for validity of at least some of the theorems.”
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In the paper
“The most precise, rigorous and strongest geodesic theorems (Geroch and Jang 1975; Ehlers and Geroch 2004) both assume the SDEC.”
What the sources say
- Not checked yetcites Geroch and Jang 1975
- Not checked yetcites Ehlers and Geroch 2004
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In the paper
“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.”
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In the paper
“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).”
What the sources say
- Not checked yetcites Joshi 2003
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In the paper
“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.”
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In the paper
“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.”
What the sources say
- Not checked yetcites Geroch (1966)
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In the paper
“The Lorentzian splitting theorems may be thought of as rigidity theorems for singularity theorems invoking the SEC”
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In the paper
“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.”
What the sources say
- Not checked yetcites Joshi 2003
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In the paper
“it is particularly striking that Penrose (1990) and Ashtekar and Penrose (1990) were able to prove positivity of ADM mass using only it”
What the sources say
- Not checked yetcites Penrose (1990)
- Not checked yetcites Ashtekar and Penrose (1990)
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In the paper
“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).”
What the sources say
- Not checked yetcites Roman 1988
- Not checked yetcites Senovilla 1997
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In the paper
“Almost all of the first category invoke the NEC for their proof.”
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In the paper
“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.”
What the sources say
- Not checked yetcites Senovilla 2007
- Not checked yetcites Senovilla 2008
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In the paper
“the splitting theorems show that, under certain other assumptions, there will be no singularities only when the spacetime is static and globally hyperbolic.”
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In the paper
“All known proofs of the positivity of the Bondi mass do require the DEC”
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In the paper
“8. 'hyper-fast' travel 74 [necessarily] (Alcubierre 1994; Krasnikov 1998; Olum 1998)”
What the sources say
- Not checked yetcites Alcubierre 1994
- Not checked yetcites Krasnikov 1998
- Not checked yetcites Olum 1998
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In the paper
“7. closed timelike curves [possibly] (Visser 1996)”
What the sources say
- Not checked yetcites Visser 1996
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In the paper
“1. conformally coupled massless and massive scalar fields [possibly] (Visser and Barcel´ o 2000; Barcel´ o and Visser 2002)”
What the sources say
- Not checked yetcites Visser and Barceló 2000
- Not checked yetcites Barceló and Visser 2002
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In the paper
“7. future-eternal inflationary cosmologies [possibly] (Borde and Vilenkin 1997)”
What the sources say
- Not checked yetcites Borde and Vilenkin 1997
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“This may give reason to doubt the stability of at least some of the singularity theorems in regimes where the energy conditions fail.”
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In the paper
“4. cosmological steady-state theories of Bondi and Gold (1948) and Hoyle (1948) 72 [necessarily]”
What the sources say
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“6. naked singularities [possibly] (Penrose 1979; Barcel´ o and Visser 1999; Joshi 2003)”
What the sources say
- Not checked yetcites Penrose 1979
- Not checked yetcites Barceló and Visser 1999
- Not checked yetcites Joshi 2003
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In the paper
“4. 'big rip' singularities 68 [necessarily] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Cattoën and Visser 2005
- Not checked yetcites Cattoën and Visser 2008
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In the paper
“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)”
What the sources say
- Not checked yetcites Flanagan and Wald 1996
- Not checked yetcites Visser and Barceló 2000
- Not checked yetcites Barceló and Visser 2002
hedgedconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedged- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, it is only the strict symmetry of the Bekenstein models that precludes singularities.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“2. closed timelike curves [possibly] (Visser 1996)”
What the sources say
- Not checked yetcites Visser 1996
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“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”
What the sources say
- Not checked yetcites Bekenstein (1975)
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
What the sources say
- Not checked yetcites Tipler (1978)
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“5. sudden future singularities 69 [possibly] (Barrow 2004a; Barrow 2004b; Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Barrow 2004a
- Not checked yetcites Barrow 2004b
- Not checked yetcites Cattoën and Visser 2005
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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None of the sources cited for this claim have been checked yet.
In the paper
“3. physically traversable wormholes [necessarily] (Morris and Thorne 1988; Visser 1989b; Visser 1989a)”
What the sources say
- Not checked yetcites Morris and Thorne 1988
- Not checked yetcites Visser 1989b
- Not checked yetcites Visser 1989a
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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In the paper
“5. classical Dirac fields [possibly] (Wald 1984)”
What the sources say
- Not checked yetcites Wald 1984 source
hedgedconditional- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“9. any fluid with a barotropic index w < -1 70 (such as those postulated in so-called phantom cosmologies) [necessarily] (Visser 1996; Dabrowski and Denkiewicz 2009)”
What the sources say
- Not checked yetcites Visser 1996
- Not checked yetcites Dabrowski and Denkiewicz 2009
boosteddirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“Of course, a marginally trapped surface would form only if there were deviations from homogeneity.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“6. a positive cosmological constant ( e . g ., anti-de Sitter Space) 73 [necessarily] (Hawking and Ellis 1973; Visser 1996)”
What the sources say
- Not checked yetcites Hawking and Ellis 1973 source
- Not checked yetcites Visser 1996
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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None of the sources cited for this claim have been checked yet.
In the paper
“3. 'big bang' and 'big crunch' singularities 67 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Cattoën and Visser 2005
- Not checked yetcites Cattoën and Visser 2008
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“First and perhaps foremost, they imply violations of all other pointilliste energy conditions.”
plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“1. 'big bang' and 'big crunch' singularities 80 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Cattoën and Visser 2005
- Not checked yetcites Cattoën and Visser 2008
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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None of the sources cited for this claim have been checked yet.
In the paper
“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)”
What the sources say
- Not checked yetcites Hawking and Ellis 1973 source
- Not checked yetcites Caldwell 2002
- Not checked yetcites Caldwell, Kamionkowski, and Weinberg 2003
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“5. spatially closed, expanding, singularity-free spacetimes [necessarily] (Senovilla 1997)”
What the sources say
- Not checked yetcites Senovilla 1997
boosteddirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“2. massless and massive non-minimally coupled scalar fields [possibly] (Flanagan and Wald 1996; Dubovsky, Gr´ egoire, Nicolis, and Rattazzi 2006)”
What the sources say
- Not checked yetcites Flanagan and Wald 1996
- Not checked yetcites Dubovsky, Grégoire, Nicolis, and Rattazzi 2006
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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In the paper
“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.”
What the sources say
- Not checked yetcites Bondi (1957)
hedged- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“11. minimally coupled massless and massive scalar fields [possibly] (Visser and Barcel´ o 2000; Barcel´ o and Visser 2002)”
What the sources say
- Not checked yetcites Visser and Barceló 2000
- Not checked yetcites Barceló and Visser 2002
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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None of the sources cited for this claim have been checked yet.
In the paper
“4. just before or just after a cosmological 'inflexion' 78 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Cattoën and Visser 2005
- Not checked yetcites Cattoën and Visser 2008
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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None of the sources cited for this claim have been checked yet.
In the paper
“1. 'big bang' and 'big crunch' singularities 75 [possibly] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Cattoën and Visser 2005
- Not checked yetcites Cattoën and Visser 2008
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“2. sudden future singularities 81 [possibly] (Barrow 2004a; Barrow 2004b; Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Barrow 2004a
- Not checked yetcites Barrow 2004b
- Not checked yetcites Cattoën and Visser 2005
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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None of the sources cited for this claim have been checked yet.
In the paper
“10. physically traversable wormholes [necessarily] (Molina-Paris and Visser 1999; Hochberg, Molina-Paris, and Visser 1999)”
What the sources say
- Not checked yetcites Molina-Paris and Visser 1999
- Not checked yetcites Hochberg, Molina-Paris, and Visser 1999
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“8. asymptotically flat black holes with regular (non-singular) interiors [necessarily] (Ansoldi 2007)”
What the sources say
- Not checked yetcites Ansoldi 2007
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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In the paper
“12. massive Klein-Gordon fields [possibly] (Visser 1996)”
What the sources say
- Not checked yetcites Visser 1996
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Generally speaking, for the latter, the regions where the energy conditions are violated can be 'localized' to a neighborhood of the 'event'.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“13. typical gauge theories with spontaneously broken symmetries [possibly] (Tipler 1978)”
What the sources say
- Not checked yetcites Tipler 1978
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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In the paper
“4. traversable wormholes (Morris, Thorne, and Yurtsever 1988) [possibly]”
What the sources say
- Not checked yetcites Morris, Thorne, and Yurtsever 1988
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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In the paper
“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.”
What the sources say
- Not checked yetcites Gannon (1975, 1976)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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In the paper
“6. cosmological inflation [necessarily] (Visser 1996)”
What the sources say
- Not checked yetcites Visser 1996
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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None of the sources cited for this claim have been checked yet.
In the paper
“3. cosmological 'bounces' 77 [necessarily] (Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Cattoën and Visser 2005
- Not checked yetcites Cattoën and Visser 2008
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“9. closed timelike curves [possibly] (Visser 1996)”
What the sources say
- Not checked yetcites Visser 1996
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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None of the sources cited for this claim have been checked yet.
In the paper
“15. 'hyper-fast' travel 79 [necessarily] (Krasnikov 1998; Olum 1998; Alcubierre 1994)”
What the sources say
- Not checked yetcites Krasnikov 1998
- Not checked yetcites Olum 1998
- Not checked yetcites Alcubierre 1994
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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None of the sources cited for this claim have been checked yet.
In the paper
“1. massless conformally coupled scalar fields 82 [possibly] (Visser and Barcel´ o 2000; Barcel´ o and Visser 2002)”
What the sources say
- Not checked yetcites Visser and Barceló 2000
- Not checked yetcites Barceló and Visser 2002
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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In the paper
“3. closed timelike curves [possibly] (Visser 1996)”
- Also stated in: § AWEC · p. 31 · hedged
“4. closed timelike curves [possibly] (Visser 1996)”
What the sources say
- Not checked yetcites Visser 1996
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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- Also stated in: § AWEC · p. 31 · hedged
In the paper
“14. conformal scalar fields coupled with dust [possibly] (Bekenstein 1975)”
What the sources say
- Not checked yetcites Bekenstein 1975
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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In the paper
“3. classical Dirac fields [necessarily] (Penrose and Rindler 1984)”
What the sources say
- Not checked yetcites Penrose and Rindler 1984 source
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.'”
What the sources say
- Not checked yetcites Geroch and Horowitz (1979)
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedged- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
boosted- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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).”
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: unclear — It is unclear whether it could be checked against evidence.
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None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
- Not checked yetcites Oppenheimer and Volkoff (1939)
- Not checked yetcites Harrison, Wakano, and Wheeler (1957)
hedged- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Clearly, then, T ′ ab = T ab .”
boosted- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The metric stays the same.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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”
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None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
- Not checked yetcites Oppenheimer and Volkoff (1939)
- Not checked yetcites Harrison, Wakano, and Wheeler (1957)
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In the paper
“(The same holds as well for all the more recently proposed energy conditions discussed in § 2.4.)”
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In the paper
“It is a concept at the level of differential topology and conformal structure.”
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In the paper
“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.”
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In the paper
“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.'”
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In the paper
“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.”
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In the paper
“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.”
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In the paper
“a white hole will violate an energy condition if and only if its time-reversed black hole does so.”
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In the paper
“Temporal orientation is not a metrical concept.”
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In the paper
“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.”
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In the paper
“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.”
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In the paper
“They also seem to be largely independent of topological structure (except in so far as it is required to define differential structure).”
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In the paper
“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”
What the sources say
- Not checked yetcites Ringström 2009 source
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Probably not a substantial claim. 94
- It may state two separate claims as one. 55
- The quote only partly states this claim. 2
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“6. 'hyper-fast' travel 83 [possibly] (my calculation)”
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In the paper
“2. a positive cosmological constant ( e . g ., anti-de Sitter Space) [necessarily] (my calculation)”
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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)”
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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
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- Rests on an assumption: Conformal transformations preserve causal structure.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
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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.”
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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.”
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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)
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Probably not a substantial claim.
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In the paper
“the Einstein tensor itself, the fundamental constituent of the Einstein field equation, has no simple, natural geometrical interpretation”
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Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
- The authors state this as an assumption: The Einstein field equation holds when relating geometric, physical, and effective formulations of energy conditions.
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
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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).”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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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.”
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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.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
boosted- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“ordinary energy density, not being an observer-independent quantity, is not a particularly natural concept in general relativity.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
boosted- Asserted: passes — The authors put it forward as true.
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- The 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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“No physical system will 'experience' ordinary energy density at all.”
plain- Asserted: passes — The authors put it forward as true.
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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)
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Clearly, the SDEC implies the DEC, which implies the WEC.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The ASEC does not imply either the AWEC or the ANEC.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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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.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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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”
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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.”
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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)
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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.”
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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”
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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)
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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 .”
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In the paper
“Clearly, the NEC, WEC and SEC respectively imply the ANEC, AWEC and ASEC.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: Gravity is everywhere attractive, interpreted as meaning that congruences of timelike geodesics with vanishing twist should always converge.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“There is, as is to be expected, no simple interpretation of its geometric form.”
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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
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Without that interpretation, however, one has little motivation for invoking Raychaudhuri's equation in the first place without ancillary physical justification.”
plainconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
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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”
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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”
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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)
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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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).”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“No reasonable impressionist analogue of either of the pointilliste dominant conditions are known.”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“I know of no rigorous proofs of the stability of an event horizon under generic small perturbations.”
hedged- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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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.”
hedged- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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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.”
plain- Asserted: passes — The authors put it forward as true.
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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)
plain- Asserted: passes — The authors put it forward as true.
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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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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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.”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“This suggests that the NEC is necessary for these theorems”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Again, the lack of such results depending on impressionist conditions suggests that the pointilliste conditions are necessary”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“we already know that essentially all energy conditions are promiscuously violated when quantum effects are taken into account.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“That there is no impressionist analogue to the DEC may suggest that the DEC is necessary for these results.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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)”
hedged- Asserted: passes — The authors put it forward as true.
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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
boosteddirect- Asserted: passes — The authors put it forward as true.
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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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“many attempts to make the Hypothesis precise focus on the initial-value formulation of general relativity.”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“the only real addition the DEC makes to the NEC and the WEC, that energy-momentum flux be causal”
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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)
hedgedconditional- Asserted: passes — The authors put it forward as true.
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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”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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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).”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“no such results are known to follow from any of the impressionist energy conditions.”
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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)
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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”
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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)
hedgedconditional- Asserted: passes — The authors put it forward as true.
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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.”
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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”
plaindirect- Asserted: passes — The authors put it forward as true.
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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)
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“I have no explanation for why the DEC should be used in almost no singularity theorems”
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since stress-energy tensors not of type i are generally considered 'unphysical'”
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
hedged- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
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
plaindirect- Asserted: passes — The authors put it forward as true.
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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
hedgedconditional- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may be too vague to check against evidence. The paper may give no reason why it holds.
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
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“cosmological inflation necessarily violates SEC and, depending on particulars of the model, possibly ASEC.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
hedgedconditional- Asserted: passes — The authors put it forward as true.
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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
boosteddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“1. naked singularities [possibly] (Ford and Roman 1992)”
What the sources say
- Not checked yetcites Ford and Roman 1992
hedgedconditional- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
boosted- Asserted: passes — The authors put it forward as true.
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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
hedged- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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”
hedged- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We do not yet have indubitable evidence for the existence of a fundamental scalar field in nature.”
plain- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
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.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.”
hedged- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
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)
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As for the singularity theorems, they also fall roughly into two classes, which for lack of better terms I will refer to as pinpointing and not.”
hedged- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“2. sudden future singularities 76 [possibly] (Barrow 2004a; Barrow 2004b; Catto¨ en and Visser 2005; Catto¨ en and Visser 2008)”
What the sources say
- Not checked yetcites Barrow 2004a
- Not checked yetcites Barrow 2004b
- Not checked yetcites Cattoën and Visser 2005
hedgedconditional- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“Such theorems demonstrate the existence of singularities associated with closed, trapped surfaces (for singularities contained in asymptotically flat black holes: Penrose 1965; Hawking and Ellis 1973), or with trapping surfaces (for singularities contained in generalized black holes: Hayward 1994a; Hayward 1994b), or with the 'boundaries' of spacetime (such as big bang and big crunch singularities”
What the sources say
- Not checked yetcites Penrose 1965
- Not checked yetcites Hawking and Ellis 1973 source
- Not checked yetcites Hayward 1994a
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A temporal orientation then is logically equivalent to the existence of a continuous timelike vector field ξ a ; by convention, the future lobe of the null cone at each point is that into which ξ a points, and a causal vector is itself future-directed if it points into or lies tangent to the future lobe.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This equation, conjoined with the definition of a spacetime model ( M , g ab ), constitutes the entirety of general relativity as a formal theory.”
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As its name suggests, the stress-energy tensor encodes for matter only information about what we normally think of as its energy, momentum and stress content.”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
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)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“General relativity, then, assumes that what we normally think of as stress-energy content completely determines the relation of spacetime structure to matter-no other property of matter 'couples' with spacetime structure at all, except in so far as it may have a part in determining the stress-energy of the matter.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In fact, it is widely believed today that matter at densities above 10 times that of atomic nuclei, as we expect to find in the interior of neutron stars, behaves in exactly that manner (Shapiro and Teukolsky 1983, ch. 8).”
What the sources say
- Not checked yetcites Shapiro and Teukolsky 1983
boosteddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“General relativity assumes the existence of a single object, the stress-energy tensor T ab , that encodes, for all fields of matter, all properties relevant to determining the relationship of the matter to the geometrical structure of spacetime.”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“It was not seriously questioned until the work of Zel'dovich in the early 1960s, in which he showed that a natural solution for a quantum field theory relevant to modeling the matter in neutron stars leads to macroscopic equations of state of the form p = ρ .”
What the sources say
- Not checked yetcites Zel'dovich and Novikov (1971)
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, once again, we seem pushed toward the view that energy conditions play some fundamental role or other in any reasonably broad conception of spacetime theories, or at least any such conception that would include a notion of dynamical evolution.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: unclear — It is unclear whether it could be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“all stress-energy is fungible, is inter-changeable, in the strong sense that the form it takes (electromagnetic, visco¨ elastic, thermal, etc .), and so a fortiori any property or quality it may have idiosyncratic to that form, is irrelevant to its gravitational effects, both locally and globally.”
plain- Asserted: fails — The authors may not put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this case, negative mass plays the role of an anti-inertial charge.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“(Note that no problem arises with parametrization of spacelike curves: there is no natural or preferred sense for their parametrization in the first place.)”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Now, if one accepts my ordering, or anything close to it, energy conditions do not seem to fit anywhere neatly in it.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is exactly this feature of general relativity that affords the energy conditions their power.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The inability to derive the energy conditions from other propositions of a fundamental character constitutes an essential part of what pushes one to conceive of them as structure 'at a deeper level' than many other elements on the list, perhaps even deeper than causality conditions (many of which can be derived from other fundamental assumptions), and so applicable across a very wide range of”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“No matter what else is the case, so long as definitional dependence (what one needs in place already to define or characterize structure of a particular sort) is one criterion used in ordering such a list, it seems that energy conditions must be not so fundamental as differential structure: one needs differential structure in order to write down any tensor, and so a fortiori to write down a stress”
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The authors state this as an assumption.
The Einstein field equation holds when relating geometric, physical, and effective formulations of energy conditions.
“if the Einstein field equation is assumed to hold”
“under the assumption that the Einstein field equation holds”
The 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.
“and matter is not too exotic”
The paper relies on this without support.
The conditions analyzed are all the standard energy conditions used in physics today.
“all the standard conditions used in physics today”
The authors state this as an assumption.
The discussion is restricted to the context of general relativity until § 5, unless explicitly stated otherwise.
“From hereon until § 5, unless explicitly stated otherwise, the discussion should be understood to be restricted to the context of general relativity.”
The authors state this as an assumption.
The Einstein field equation is assumed when comparing the geometric, physical, and effective forms of the energy conditions.
“modulo the assumption of the Einstein field equation, logically equivalent.”
The 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.
“Assuming a well behaved barotropic equation of state for the effective fluid described by the stress-energy tensor, i . e ., a fixed relation ρ ( p ) expressing ρ as an invertible function of the single isotropic pressure p”
The authors state this as an assumption.
For the effective form, the stress-energy tensor is of Hawking-Ellis type i and is diagonal in the fixed 1+3 orthonormal frame.
“the stress-energy tensor (assumed, recall, for the effective form, to be of Hawking-Ellis type i ) is diagonal.”
The authors state this as an assumption.
Gravity is everywhere attractive, interpreted as meaning that congruences of timelike geodesics with vanishing twist should always converge.
“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”
The paper relies on this without support.
The stationary Killing field is spacelike in the ergosphere, so no observer can have its orbits as worldline.
“Because the stationary Killing field is spacelike in the ergosphere, no observer can have its orbits as worldine.”
The paper relies on this without support.
The energy defined by a stationary Killing field is observer-independent and obeys both a local and an integral conservation law.
“The energy defined by a stationary Killing field, however, is observer-independent and so has prima facie physical significance, even more so given that it obeys both a local and an integral conservation law.”
The paper relies on this without support.
Stress-energy tensors not of Hawking-Ellis type i are generally considered unphysical.
“Since stress-energy tensors not of type i are generally considered 'unphysical'”
The authors state this as an assumption.
Any physically reasonable spacetime theory should 'look enough like' general relativity so as to make all the elements of the list make sense in its context.
“In constructing the list, I have been guided by the tenet that any physically reasonable spacetime theory should 'look enough like' general relativity so as to make all the elements of the list make sense in its context.”
The authors state this as an assumption.
The list of structures and its ordering by fundamentality is of interest for the sake of argument.
“Now, granting the interest of the list for the sake of argument, where, if at all, should one place energy conditions on it?”
The authors state this as an assumption.
No global feature of the geometry, as constrained by a theory's field equations, should depend on anything but purely qualitative properties of stress-energy; a fortiori, no global feature of the geometry should depend on the species of matter present, so long as that species manifests a relevant qualitative property.
“In particular, no global feature of the geometry, as constrained by a theory's field equations, should depend on anything but purely qualitative properties of stress-energy; a fortiori , no global feature of the geometry should depend on the species of matter present, so long as that species manifests a relevant qualitative property.”
The paper relies on this without support.
Conformal transformations preserve causal structure.
“since conformal transformations preserve causal structure, the transformed spacetime is still globally hyperbolic.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ physical · p. 7
“i . e ., the supremum of all such b is 0”
§ physical · p. 7
“there exists a b > 0 such that T mn ξ m ξ n ≥ -b for all timelike ξ a”
Citations
Cites
The Large Scale Structure of Space-Time
Hawking, S. W., & Ellis, G. F. R. (1973) · doi:10.1017/cbo9780511524646 · Cited by 10,624
Bekenstein, J. D. (1973) · Physical Review D · doi:10.1103/physrevd.7.2333 · Cited by 7,555
Wald, R. M. (1984) · doi:10.7208/chicago/9780226870373.001.0001 · Cited by 7,188
CP Conservation in the Presence of Pseudoparticles
Peccei, R. D., & Quinn, H. R. (1977) · Physical Review Letters · doi:10.1103/physrevlett.38.1440 · Cited by 7,029
Wald, R. M. (1984) · Cited by 6,274
Black Holes, White Dwarfs, and Neutron Stars
Shapiro, S. L., & Teukolsky, S. A. (1983) · Wiley eBooks · doi:10.1002/9783527617661 · Cited by 4,833
Susskind, L. (1995) · Journal of Mathematical Physics · doi:10.1063/1.531249 · Cited by 4,085
Oppenheimer, J. R., & Volkoff, G. M. (1939) · Physical Review · doi:10.1103/physrev.55.374 · Cited by 4,035
The four laws of black hole mechanics
Bardeen, J. M., Carter, B., & Hawking, S. W. (1973) · Communications in Mathematical Physics · doi:10.1007/bf01645742 · Cited by 3,932
Die Grundlage der allgemeinen Relativitätstheorie
Einstein, A. (1916) · Annalen der Physik · doi:10.1002/andp.19163540702 · Cited by 3,468
Research works printed but not linked: 3
- Harrison, B., M. Wakano, and J. Wheeler (1957). Report. In La Structure et l’Evolution de l’Univers, Onzieme Conseil de Physique Solvay. Brussels: Editions Stoops.
- Penrose, R. (1979). Singularities and time-asymmetry. See [108], pp. 581–638.
- ’t Hooft, G. (1988). On the quantization of space and time. In M. Markov, V. Berezin, and V. Frolov (Eds.), Quantum Gravity, pp. 551–567. Singapore: World Scientific Press.
Cited by
Sorted by citations
First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric
Akiyama, K., Alberdi, A., Alef, W., Algaba, J. C., Anantua, R., Asada, K., Azulay, R., Bach, U., Baczko, A.-K., Ball, D. R., et al. (2022) · The Astrophysical Journal Letters · doi:10.3847/2041-8213/ac6756 · Cited by 862
The 1965 Penrose singularity theorem
Senovilla, J. M. M., & Garfinkle, D. (2015) · Classical and Quantum Gravity · doi:10.1088/0264-9381/32/12/124008 · Cited by 248
Revisiting the mechanical properties of the nucleon
Lorcé, C., Moutarde, H., & Trawiński, A. P. (2019) · The European Physical Journal C · doi:10.1140/epjc/s10052-019-6572-3 · Cited by 274
Energy conditions in general relativity and quantum field theory
Kontou, E.-A., & Sanders, K. (2020) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ab8fcf · Cited by 223
Phenomenological aspects of black holes beyond general relativity
Carballo-Rubio, R., Filippo, F. D., Liberati, S., & Visser, M. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.98.124009 · Cited by 193
Metadata
- Authors
- Curiel, E.
- Affiliations
- Harvard University (US)
- Corresponding author
- Curiel, Erik
- ORCID
- Curiel, Erik
- Venue
- Einstein Studies
- Publisher
- Springer New York
- Published
- 2017
- Pages
- 43-104
- ISSN
- 2381-5833 (ISSN-L)
- 2381-5841
- OpenAlex
- W2133796620
- Semantic Scholar
- 8a5f31a78962cd7f1c14ee29d694bf83f150c4ff
- Cited by
- 226
- References
- 246
- Citations by year
- 2026: 25 · 2025: 26 · 2024: 25 · 2023: 32 · 2022: 26 · 2021: 23 · 2020: 14 · 2019: 13 · 2018: 20 · 2017: 8
- FWCI
- 62.39
- Type
- book chapter
- Language
- de
- Open access
- green
- arxiv.org
- Locations
- Einstein studies (publishedVersion)
- PhilSci-Archive (University of Pittsburgh) (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- Open access LMU (Ludwid Maxmilian's Universitat Munchen) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Keywords
- energy conditions, energy-momentum tensor, positive energy, general relativity, spacetime structure, epistemology of physics
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