The DOI matches its Crossref record; 442 of the 443 references it cites were found in a registry or an index; none of the 222 sources its claims cite is checked for support yet.
Spacetime foam: a review
Open accessAbstract
More than 65 years ago, John Wheeler suggested that quantum uncertainties of the metric would be of order one at the Planck scale, leading to large fluctuations in spacetime geometry and topology, which he termed ‘spacetime foam.’ In this review I discuss various attempts to implement this idea and to test it, both theoretically and, to a lesser extent, observationally.
Overview
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TL;DR
This review surveys theoretical and observational work on Wheeler’s “spacetime foam,” the idea that metric fluctuations become order one at the Planck scale. It finds that foam remains highly model-dependent, with no complete quantum gravity theory, but that path integral, lattice, canonical, and phenomenological approaches suggest possible Planck-scale topology/geometry fluctuations and some observational signatures.
Key findings
- Wheeler’s original estimate gives metric fluctuations \(\delta g \sim \ell_p/L\), becoming order one at the Planck scale; the review stresses that this and all foam claims depend on the approach to quantum gravity.
- In the Euclidean path integral with \(\Lambda<0\), the density of topologies grows superexponentially, \(\rho(\tilde v)\sim e^{a\tilde v\ln\tilde v}\), so the sum over topologies diverges and “entropy” dominates “action”; whether it is Borel summable is unknown.
- Quantum stress-energy fluctuations are mostly negative: in 2D conformal field theory with \(c=1\), 84% of vacuum fluctuations are negative, with a strict lower bound and an infinite positive tail; 4D results are qualitatively similar, with a slower-decaying tail \(\sim e^{-x^{1/3}}\).
- Causal dynamical triangulations excludes topology change but allows large geometry fluctuations; spectral dimension is \(d_S=4\) at large scales, shrinks to \(d_S\approx 2\) at small scales, and is \(d_S\approx 1.5\) for a constant-time slice.
- Proposed observational tests include Lorentz breaking, decoherence/CPT violation, starlight blurring, interferometer noise, spectral-line broadening, and cosmological effects; the review reports no confirmed detection and notes some models are ruled out by existing limits.
Methods
- This is a narrative review, not a primary research paper; it synthesizes prior theoretical and phenomenological literature on spacetime foam.
- It covers Euclidean path integrals and gravitational instantons, lattice methods including causal dynamical triangulations, canonical quantization/Wheeler-DeWitt, lower-dimensional models, and proposed observational tests.
- The main analysis is qualitative and comparative, assessing which approaches permit foamy geometries, topology change, or observable effects, and where sums over topologies or fluctuations diverge.
Limitations
- The paper explicitly calls itself an “incomplete review,” notes that “spacetime foam” is a vague slogan, and states that answers depend on one’s approach to quantum gravity, with no complete theory available.
- The paper states that studies of stress-energy fluctuations neglect correlations between successive fluctuations and do not account for the quantum nature of the gravitational field itself.
- The text makes evident that sums over topologies often diverge without known regularization, that averaging observables in foamy spacetimes is conceptually difficult, and that many observational proposals are model-dependent and not yet confirmed.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
130 Sources not found · 62 Own interpretation · 34 No basis given · 4 Not checked yet · 2 Own method
In the paper
“But a host of other thought experiments [7], many of them more recent, point to the Planck length as the scale at which quantum fluctuations become large.”
§ Spacetime foam: a review > 1 Wheeler's foam
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In the paper
“Thus in this formulation, the only configurations occurring in the path integral are those with H i = H = 0.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“This argument is far from conclusive, of course.For one thing, it requires an extrapolation of the energy density (1.3), derived for small fluctuations of the metric, to order one fluctuations.”
§ Spacetime foam: a review > 1 Wheeler's foam
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In the paper
“In other approaches, though-for instance, the Euclidean path integral [12] or the lattice path sum of causal dynamical triangulations [13]-no such delta functionals occur.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“Classically, the spatial topology of a Lorentzian manifold can change only if the spacetime admits closed timelike curves or has a singular metric [14,15], though the singularity can be quite mild [16]; see [17] for a more thorough discussion.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“In such approaches, the constraints typically have the same status as the rest of the classical field equations; configurations that do not obey the constraints may be dynamically suppressed, but they will still appear in the full path integral.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“In particular, vacuum fluctuations of matter fields will induce quadratic terms in the curvature [41], leading to an effective action”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“The same is true for at least a large class of asymptotically flat three-manifolds.”
§ Spacetime foam: a review > 2.2 Initial data
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In the paper
“Another superexponentially growing class of Einstein manifolds, this time not of constant curvature, can be obtained from hyperbolic manifolds with mild singularities (”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
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In the paper
“sum of two Einstein manifolds need not itself admit an Einstein metric.”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
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In the paper
“Note also that for now, questions about spacetime foam, including the most basic onewhether it exists-will have answers that depend on one's approach to quantum gravity.”
§ Spacetime foam: a review > 1 Wheeler's foam
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In the paper
“For now, we do not know which is the correct choice.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“There is also no consensus over whether fluctuations of topology are possible.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“If one additionally imposes the classical field equations, spatial topology change requires a violation of a positive energy condition [18], and local regions of nontrivial topology evolve to singularities [19,20].”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“We have a wide range of research programs aimed at quantizing gravity [8,9], and although some are more popular than others, it's fair to say that none has yet given us a complete theory.”
§ Spacetime foam: a review > 1 Wheeler's foam
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In the paper
“Equating these two expressions, we obtain δg ∼ ℓ p L , (1.4) confirming that metric fluctuations become order one at the Planck scale.”
§ Spacetime foam: a review > 1 Wheeler's foam
What the sources say
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In the paper
“More than 65 years ago, John Wheeler suggested that quantum uncertainties of the metric would be of order one at the Planck scale, leading to large fluctuations in spacetime geometry and topology, which he termed”
- Also stated in: § 1 Wheeler's foam · plain
“He argued that quantum fluctuations of the metric should be of order one at the Planck scale, leading to wild and rapidly varying fluctuations in spacetime geometry and topology, which he called”
§ Spacetime foam: a review > Abstract
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- Also stated in: § 1 Wheeler's foam · plain
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In the paper
“There are, of course, loopholes: if gravity is asymptotically free, for instance, interactions may become weak near the Planck scale.”
§ Spacetime foam: a review > 1 Wheeler's foam
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In the paper
“The condition of vanishing extrinsic curvature is rather strong, but can be understood as part of the requirement for an extremal action [90].”
§ Spacetime foam: a review > • Manifolds with boundary:
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“The sphere S 4 has its usual constant curvature metric, which can be shown to be the Einstein metric with the largest normalized volume ṽ [60].”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
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In the paper
“This is a necessary condition, not a sufficient one; stronger limits have been found by a number of authors [56][57][58][59].”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
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In the paper
“In four dimensions,”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“For Λ < 0, there are an infinite number of compact hyperbolic three-manifolds, whose properties have been extensively, although not completely, characterized [103,105,106].”
§ Spacetime foam: a review > • Manifolds with boundary:
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In the paper
“For the more realistic four-dimensional case, exact results are not available, but there are now good estimates and bounds [117][118][119].The results are qualitatively very similar, although more sensitive to the choice of smearing function.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
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In the paper
“The precise values of correlators will depend on the choice of smearing, but some key qualitative features are largely independent of w.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
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In the paper
“Until recently, though, most of this work focused on small fluctuations that could be treated at low order in perturbation theory.Such fluctuations can have interesting physical effects-see, for instance, [115]-but they are not really spacetime foam.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
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“In fact, for any integer k one can find a four-manifold with k smooth structures that admit (different) Einstein metrics and infinitely many other smooth structures that admit no Einstein metric [39].”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“Simply connected manifolds are under better control [35], providing a practical impetus to focus on such topologies [12].”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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In the paper
“The complex projective plane CP 2 admit an Einstein metric, as do connected sums CP 2 # kCP 2 with 3 ≤ k ≤ 8 [61].”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
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“Einstein manifolds in three dimensions are further restricted: they necessarily have constant curvature (that is, they satisfy (2.9)), and can always be represented as quotients”
§ Spacetime foam: a review > • Manifolds with boundary:
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In the paper
“But we will sometimes find sums for which no obvious regularization exists.”
§ Spacetime foam: a review > 2.4 A caveat on divergent sums
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In the paper
“In particular, there is still a strict lower bound and an infinite positive tail, which now decays slower than exponentially, typically as e -x 1/3 .”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
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In the paper
“For a conformal field theory in two spacetime dimensions, the answer can be calculated exactly [116], and is largely independent of the smearing function [122].”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
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In the paper
“In particular, if one requires stable causality-a slightly stronger version of the condition that there be no closed timelike curves-then local fluctuations in topology are forbidden [93], although a fixed spatial topology may be very complex [94].”
§ Spacetime foam: a review > • Manifolds with boundary:
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In the paper
“Not surprisingly, different approaches to quantum gravity suggest different possibilities for allowed topologies.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Note that three-manifolds are classifiable [36], although this does not necessarily imply that product manifolds R × Σ 3 are.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
What the sources say
- Not matchedcites [36]
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plain- Asserted: passes — The authors put it forward as true.
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In the paper
“For Λ = 0, there are a total of ten flat closed three-manifolds; the three-torus is the simplest, but the sides of a cube can also be identified with various twists [101,103,104].”
§ Spacetime foam: a review > • Manifolds with boundary:
What the sources say
- Not matchedcites [101]
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- Not matchedcites [103]
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- Not matchedcites [104]
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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 not known how much of the space of initial data is covered by this Chrusciel-Isenberg-Pollack construction.”
§ Spacetime foam: a review > 2.2 Initial data
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“But while Birkhoff's theorem implies that a spherically symmetric vacuum spacetime must be locally isometric to some region of Schwarzschild, local patches can be sewn together to form spacetimes that look drastically different [84].”
§ Spacetime foam: a review > 2.2 Initial data
What the sources say
- Not matchedcites [84]
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plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“For the case Λ < 0, some of the known Einstein manifolds look more like Wheeler's foam.Anderson's Dehn surgery construction [53,66], for instance, yields manifolds in which the nontrivial geometry manifests itself in a discrete set of”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
What the sources say
- Not matchedcites [53]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [66]
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plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“A compact, orientable two-manifold-that is, a closed surface-has a topology determined by its genus g, the”
§ Spacetime foam: a review > • Manifolds with boundary:
plaincorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Morrow-Jones, Witt, and Schleich have carried out detailed studies of locally spherically symmetric initial data [82][83][84].”
§ Spacetime foam: a review > 2.2 Initial data
What the sources say
- Not matchedcites [82]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [83]
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- Not matchedcites [84]
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In the paper
“The expectation value ⟨T µν ⟩ of the stress-energy diverges, and must be renormalized.There are a number of ways of doing this; see, for example, [120].”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
What the sources say
- Not matchedcites [120]
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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
“As we will see in §3, though, vacuum fluctuations frequently carry negative energy, so this may not be a problem for the quantum theory.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
hedgedconditional- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“If we choose to allow all topologies, an added complication appears: four-manifolds are not classifiable, in the sense that there is no algorithm that can determine whether two arbitrary four-manifolds are homeomorphic [32,33].”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
What the sources say
- Not matchedcites [32]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [33]
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plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“At the topological level, all three-manifolds have a decomposition as connected sums, unique for orientable manifolds and almost unique for nonorientable ones [87,88].”
§ Spacetime foam: a review > 2.2 Initial data
What the sources say
- Not matchedcites [87]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [88]
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Passive fluctuations have been studied for decades [113], notably through a model known as stochastic gravity [114].”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
What the sources say
- Not matchedcites [113]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [114]
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plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Broadly speaking, quantum fluctuations of the metric come in two forms [112]:”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
What the sources say
- Not matchedcites [112]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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
“On occasion, we will find that certain sums over geometries or topologies diverge, and we will have to make sense of this behavior.”
§ Spacetime foam: a review > 2.4 A caveat on divergent sums
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. It may state two separate claims as one.
In the paper
“Indeed, there are phenomena such as the Casimir effect in which an appropriately regulated sum of vacuum fluctuations gives an experimentally verifiable prediction [111].”
§ Spacetime foam: a review > 2.4 A caveat on divergent sums
What the sources say
- Not matchedcites [111]
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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
“But it could be that we were summing too few terms, and that the addition of more geometries or topologies might give us a physically sensible answer.”
§ Spacetime foam: a review > 2.4 A caveat on divergent sums
hedgedconditionalSingle case- 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’s conclusions may not depend on it. It may state two separate claims as one. The quote only partly states this claim.
In the paper
“The”
§ Spacetime foam: a review > Abstract
What the sources say
- Not matchedcites [91]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: fails — The authors may not put it forward as true.
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In the paper
“As in the compact case, only a few examples of Einstein manifolds with boundary are known for Λ ≥ 0, but large classes are known for Λ < 0.”
§ Spacetime foam: a review > • Manifolds with boundary:
plaincorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“There is some evidence that perturbative quantum gravity is”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
What the sources say
- Not matchedcites [44]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [45]
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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. It may state two separate claims as one.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Notably, Myers' theorem implies that a complete compact Einstein manifold with Λ > 0 necessarily has a finite fundamental group: there are no”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
What the sources say
- Not matchedcites [52]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [62]
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plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Any such manifold is completely determined by its fundamental group π 1 ; it is, in fact, a quotient H 4 /π 1 of hyperbolic space, and the constant negative curvature metric is its unique Einstein metric [63].”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
What the sources say
- Not matchedcites [63]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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 coefficients α and β are expected to be of order ℓ p 2 , and the corrections are likely to be negligible at ordinary scales, but they may become important near the Planck scale.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: The coefficients α and β of the quadratic curvature corrections are of order ℓp^2.
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
“But (2.4) also has other extrema, whose role is not well understood.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Note first that not every four-manifold admits an Einstein metric at all, with any value of Λ.Indeed, even such simple manifolds as T 4 #T 4 , the connected sum of two tori, * and S 1 ×S 3 have no Einstein metrics [52].”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
What the sources say
- Not matchedcites [52]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“2. There is, however, a strict lower bound to the energy of a vacuum fluctuation, determined by quantum energy inequalities [123], with a numerical value determined by the smearing function.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
What the sources say
- Not matchedcites [123]
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plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The topology of three-manifolds is now completely classified, thanks to Perelman's proof of Thurston's geometrization conjecture [36].”
§ Spacetime foam: a review > • Manifolds with boundary:
What the sources say
- Not matchedcites [36]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- 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
“A constant curvature metric necessarily has positive curvature on a sphere (g = 0), vanishing curvature on a torus (g = 1), and negative curvature on a surface of genus g > 1.”
§ Spacetime foam: a review > • Manifolds with boundary:
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. It may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Of course, many routine divergences can be eliminated with, for example, zeta function regularization [109] or Borel summation [110].”
§ Spacetime foam: a review > 2.4 A caveat on divergent sums
What the sources say
- Not matchedcites [109]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [110]
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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 question of what manifolds to allow in a description of spacetime foam is thus complicated, and we seem rather far from a consensus, or even an understanding of how a decision might be reached.”
§ Spacetime foam: a review > 2 Constructing foamy spacetimes
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
“At the geometrical level, though, the structure of the”
§ Spacetime foam: a review > 2.2 Initial data
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
“For Λ ≥ 0, the known examples of Einstein metrics fit one aspect of Wheeler's definition of spacetime foam: they have complicated topologies that may contribute to the path integral in an amount comparable to the simplest S 4 topology.But viewing these as Planck scale fluctuations is not so obvious.The topological structure seems more global; in particular, by Myers' theorem, there are no wormhole”
§ Spacetime foam: a review > 2.1 Gravitational Instantons
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“This may tame the divergence-while the number of compact Einstein manifolds grows superexponentially with ṽ, this need not imply that the number of Einstein manifolds with fixed (K, h) does.”
§ Spacetime foam: a review > 4.1 Path integrals
What the sources say
- Not matchedcites [29]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [90]
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hedgedconditional- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Quantum gravity has no fixed background structure that can be used to measure the passage of time, and finding a general enough “relational” time is not easy.”
§ Spacetime foam: a review > 4.3 Canonical quantization
What the sources say
- Not matchedcites [157]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [158]
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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 instance, scattering off quantum foam is likely to be energy dependent, plausibly modifying dispersion relations for light.”
§ Spacetime foam: a review > 5.1 Propagation
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“Moreover, the Einstein-Hilbert action can be rewritten as a Chern-Simons action [168][169][170], a form in which the path integral (4.2) can often be computed exactly.”
§ Spacetime foam: a review > 4.4 Lower dimensions
What the sources say
- Not matchedcites [168]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [169]
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- Not matchedcites [170]
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plaindirect- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“They find that recoil off such solitons induces an energy dependent vacuum refractive index, which might be astrophysically observable.”
§ Spacetime foam: a review > 5.1 Propagation
What the sources say
- Not matchedcites [188]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [189]
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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
“It remains possible, though, that the cumulative effect of many Planck scale fluctuations might be observable.”
§ Spacetime foam: a review > 6 Observational tests
hedgedconditional- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“It was noted long ago [236][237][238] that quantum gravitational fluctuations at the Planck scale might “smear” the light cone, potentially eliminating the source of divergences.”
§ Spacetime foam: a review > 5.3 Divergences
What the sources say
- Not matchedcites [236]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [237]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [238]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- Asserted: fails — The authors may not put it forward as true.
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Probably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it.
- Not matched
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
“This decoherence can pick out a preferred direction of time, leading to CPT violation.”
§ Spacetime foam: a review > 6.2 Some proposed tests
plaindirect- 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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It may state two separate claims as one.
In the paper
“Friedman et al. analyzed fermion propagation on manifolds with nonorientable handles [205], and concluded that fermions would acquire Planck-scale masses unless either the particles had a finite size (larger than that of typical quantum fluctuations) or the path integral included a sum over spin structures and an arbitrary restriction to only CP-reversing handles.”
§ Spacetime foam: a review > 5.1 Propagation
What the sources say
- Not matchedcites [205]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In [174], the partition function for an arbitrary number of wormholes was computed, in the saddle point approximation for Λ > 0 and with one loop corrections included for Λ < 0. The sum over wormholes was shown to diverge, and to not even be Borel summable.”
§ Spacetime foam: a review > • Lorentzian path integrals in 2+1 dimensions
What the sources say
- Not matchedcites [174]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Not matched
In the paper
“We can treat these as a stochastic source term for the Einstein equations, a procedure that has been shown to be a particular next order extension of semiclassical gravity [114].”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
What the sources say
- Not matchedcites [114]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“These results should probably not be taken too literally: they neglect the correlations between successive fluctuations, and do not account for the quantum nature of the gravitational field itself.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
hedged- 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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It may state two separate claims as one.
In the paper
“In causal dynamical triangulations [13], a time slicing is chosen in a way that fixes the spatial topology, prohibiting topology change and at the same time providing an unambiguous direction of time that makes the spacetime more “Lorentzian.””
§ Spacetime foam: a review > 4.2 Lattice methods
What the sources say
- Not matchedcites [13]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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
“Of course, the distribution of fluctuations obtained from (3.2) depends on a smearing length; this might most naturally taken to be the Planck length, but one can explore the effects of changing the scale.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
hedgedcorrelational- 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. It may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Moreover, virtual handles and similar nontrivial fluctuations of topology can lead to effective nonlocal interactions, which can perhaps act as a thermal bath [201,222].”
§ Spacetime foam: a review > 5.2 Decoherence and unitarity
What the sources say
- Not matchedcites [201]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [222]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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. It may state two separate claims as one.
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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
“While vacuum fluctuations of matter certainly affect the small scale structure of spacetime, the real motivation for quantum foam comes from quantum gravity.”
§ Spacetime foam: a review > 4 Quantization
plaindirect- 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
“Such a loss of coherence would lead to the disappearance of diffraction patterns-Airy rings-from extragalactic sources, as well as angular broadening and blurring of distant point sources.”
§ Spacetime foam: a review > 6.2 Some proposed tests
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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It may state two separate claims as one.
In the paper
“A similar “dimensional reduction” occurs in a number of other approaches to quantum gravity [150]; it is not known whether the phenomenon is generically connected to spacetime foam”
§ Spacetime foam: a review > 5.3 Divergences
What the sources say
- Not matchedcites [150]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedcorrelational- 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. It may state two separate claims as one.
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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
“-Components of tensors depend on a choice of basis, which can vary from point to point.”
§ Spacetime foam: a review > 6.1 The problem of averaging
plaindirect- 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.
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
“But they are clearly relevant to the understanding of macroscopic effects of spacetime foam-both for averaging over geometries and, for a fixed inhomogeneous geometry, for comparing different locations and times.”
§ Spacetime foam: a review > 6.1 The problem of averaging
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Classically, the initial data of §2.2 leads to singularities: the “necks” of the connected sums typically contain marginally outer trapped surfaces, essentially black hole horizons, and the standard singularity theorems apply [161].”
§ Spacetime foam: a review > 4.3 Canonical quantization
What the sources say
- Not matchedcites [161]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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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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 discussed in §5.1, spacetime foam could provide a preferred reference frame, allowing for a violation of Lorentz invariance.”
§ Spacetime foam: a review > 6.2 Some proposed tests
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. It may state two separate claims as one.
In the paper
“something like the Wheeler-DeWitt equation [129] is likely to hold”
§ Spacetime foam: a review > 4 Quantization
What the sources say
- Not matchedcites [129]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedged- 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’s conclusions may not 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
“Thus most fluctuations are negative, but there is a”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
plainSingle case- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“In a somewhat different approach, a matrix model inspired by causal dynamical triangulations ( §4.2) can be used to obtain transition amplitudes between spatial slices [176].While the slices have a fixed S 2 topology at weak coupling, in the strong coupling regime they “disintegrate” into a foam of multiple components connected by wormholes.”
§ Spacetime foam: a review > • Lorentzian path integrals in 2+1 dimensions
What the sources say
- Not matchedcites [176]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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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It may state two separate claims as one.
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None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“I think it is fair to say that most physicists currently working on quantum gravity expect that the process is ultimately unitary, but the question is still hotly debated [218,219].”
§ Spacetime foam: a review > 5.2 Decoherence and unitarity
What the sources say
- Not matchedcites [218]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [219]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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. It may state two separate claims as one.
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In the paper
“Fields in a foamy spacetime may behave as open systems, with the Planck scale fluctuations of the spacetime acting as an environment; if the fields are observed and the spacetime is not, the result can be a loss of coherence [221].”
§ Spacetime foam: a review > 5.2 Decoherence and unitarity
What the sources say
- Not matchedcites [221]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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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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
“But while fluctuations of topology are excluded, large fluctuations of geometry can certainly occur.”
§ Spacetime foam: a review > 4.2 Lattice methods
plain- 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.
It may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“A number of relatively ad hoc smearings have investigated [239][240][241][242], and do indeed cut off divergences; Ohanian has argued that this behavior is independent of the details of the smearing [243].”
§ Spacetime foam: a review > 5.3 Divergences
What the sources say
- Not matchedcites [239]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [240]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [241]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [242]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [243]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Not matched
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
“This eliminates one version of spacetime foam, the conjectured quantum fluctuations in topology, although the initial spatial slice may have an arbitrarily complicated topology.”
§ Spacetime foam: a review > 4.2 Lattice methods
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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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
“But it seems likely that whatever the eventual final theory of quantum gravity, some echo of this equation will remain.”
§ Spacetime foam: a review > 4.3 Canonical quantization
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
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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. 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
“If the standard picture of spacetime foam is right, quantum fluctuations of spacetime only become large near the Planck scale.”
§ Spacetime foam: a review > 6 Observational tests
hedgedconditional- 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.
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Probably not a substantial claim. The paper may give no reason why it holds.
In the paper
“In his first proposal of spacetime foam, Wheeler suggested that . . . it is essential to allow for fluctuations in the metric and gravitational interactions in any proper treatment of the compensation problem-the problem of compensation of “infinite” energies that is so central to the physics of fields and particles [3].”
§ Spacetime foam: a review > 5.3 Divergences
What the sources say
- Not matchedcites [3]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- 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.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Similarly, for light with wavelength λ one might expect random phase fluctuations on the order [210,211] δϕ ∼ 2πδℓ/λ ∼ 2π λ ℓ 1-α ℓ α p .(5.3)”
§ Spacetime foam: a review > 5.1 Propagation
What the sources say
- Not matchedcites [210]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [211]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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 now, though, different approaches to quantum gravity may give very different answers to questions about spacetime foam.”
§ Spacetime foam: a review > 4 Quantization
hedgedconditional- 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.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The Wheeler-DeWitt equation can be solved in a WKB approximation, and one can look for formally time-independent stationary states.Such states exist, and they frequently give high probabilities for “foamy” geometries.”
§ Spacetime foam: a review > 4.3 Canonical quantization
What the sources say
- Not matchedcites [79]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- 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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It may state two separate claims as one.
- Not matched
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
“But both the definition of the slice (say as a slice of constant mean curvature) and the geometry of the geodesic balls will depend on the metric, making it difficult to compare results among different manifolds, or even different locations in the same manifold.”
§ Spacetime foam: a review > 6.1 The problem of averaging
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.
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It may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“The results so far indicate that foamy structures with rapidly varying geometries occur with fairly high probability, though there remain some unsettled questions about the proper probability measure.”
§ Spacetime foam: a review > 4.3 Canonical quantization
What the sources say
- Not matchedcites [79]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [80]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedged- 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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It may state two separate claims as one.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“but induced Planck-size λφ 4 interactions, and possibly Planck-scale tachyonic masses, for scalar fields.”
§ Spacetime foam: a review > 5.1 Propagation
What the sources say
- Not matchedcites [200]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [204]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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It may state two separate claims as one.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“The idea that spacetime foam could lead to a failure of unitarity in quantum field theory dates back to work by Hawking [216,217], who argued that virtual black holes could form coherently and then evaporate thermally.”
§ Spacetime foam: a review > 5.2 Decoherence and unitarity
What the sources say
- Not matchedcites [216]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [217]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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. It may state two separate claims as one.
- Not matched
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
“With our choice of renormalization, the vacuum expectation value of the stress-energy tensor is zero (up to the possible higher order curvature corrections discussed briefly in §2), but the fluctuations can be very large.”
§ Spacetime foam: a review > 3 Quantum fluctuations of the stress-energy tensor
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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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
“But spacetime is also the arena in which other fields propagate, and it is possible that a foamy structure could influence that propagation.”
§ Spacetime foam: a review > 5 Fields in foamy spacetimes
hedgedconditional- 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
“Finally, though, even if canonical quantization allows complicated spatial geometry and topology, it is not at all clear that it allows topology change.”
§ Spacetime foam: a review > 4.3 Canonical quantization
hedged- 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.
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
“Even our most energetic probes have wavelengths far larger than ℓ p , and thus cannot resolve individual Planck scale events.”
§ Spacetime foam: a review > 6 Observational tests
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: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds. 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
“This is not necessarily a problem, since in an inhomogeneous spacetime the “foam” provides a background structure.”
§ Spacetime foam: a review > 5.1 Propagation
plaincorrelational- 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.
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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
“For Λ > 0, the set of Einstein metrics is even more poorly understood, and much less can be said.”
§ Spacetime foam: a review > 4.1 Path integrals
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.
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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
“the construction of an inner product on the space of solutions is a deep and largely unsolved problem [154].”
§ Spacetime foam: a review > 4.3 Canonical quantization
What the sources say
- Not matchedcites [154]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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.
- Not matched
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
“In practice, any observations we can make will probe a region much larger than Planck size.”
§ Spacetime foam: a review > 6.1 The problem of averaging
plain- 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: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
In the paper
“The three-dimensional version due to Boulatov can be made Borel summable [179], but it is not currently known whether this extends to higher dimensions.”
§ Spacetime foam: a review > 4.5 Other approaches
What the sources say
- Not matchedcites [179]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- 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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It may state two separate claims as one.
- Not matched
In the paper
“it is plausible that some form of the “holographic” bulk-boundary duality [130] will survive.”
§ Spacetime foam: a review > 4 Quantization
What the sources say
- Not matchedcites [130]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
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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.
- Not matched
In the paper
“Building on [171], the exact path integral was computed in [173] for one infinite class of such manifolds, the lens spaces.The sum fails to converge, even with zeta function regularization, although again this might change if more topologies are incorporated.”
§ Spacetime foam: a review > 4.4 Lower dimensions
What the sources say
- Not matchedcites [173]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedged- 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.
It may state two separate claims as one.
- Not matched
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 do not have an answer to any of these questions, but the answer to the first is plausibly “yes.””
§ Spacetime foam: a review > 4.3 Canonical quantization
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check 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. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The quote only partly states this claim.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“In a more explicitly “foamy” construction, Crane and Smolin have considered the effect of a gas of virtual black holes on field propagation [198,247], arguing that their presence alters the spectrum of perturbations in a way that imposes a high energy cutoff on the density of states.”
§ Spacetime foam: a review > 5.3 Divergences
What the sources say
- Not matchedcites [198]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [247]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Coleman has argued that spacetime foam can change this [67,225].”
§ Spacetime foam: a review > (7.1)
What the sources say
- Not matchedcites [67]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [225]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“The observed absence of such effects has been used to rule out certain phenomenological models of spacetime foam [210,[265][266][267][268], although other models suggest that the effects may still be unobservably small [269].”
§ Spacetime foam: a review > 6.2 Some proposed tests
What the sources say
- Not matchedcites [210,[265][266][267][268]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [269]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- 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.
It may state two separate claims as one.
- Not matched
In the paper
“If axions exist, their shift symmetry can be broken by spacetime fluctuations, in particular by wormholes [280].”
§ Spacetime foam: a review > 6.2 Some proposed tests
What the sources say
- Not matchedcites [280]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainconditional- 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.
- Not matched
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 photons can lose energy through interactions with spacetime foam, one effect should be a broadening of spectral lines.”
§ Spacetime foam: a review > 6.2 Some proposed tests
hedgedconditional- 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: fails — The paper may give no reason why it holds.
Probably not a substantial claim. 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
“The problem is that “all forms of energy” should include the energy of vacuum fluctuations, which should manifest as a cosmological constant Λ.”
§ Spacetime foam: a review > 7 The cosmological constant
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. 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
“As we have seen, spacetime foam could take many different forms.”
§ Spacetime foam: a review > 8 Conclusion
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check against evidence.
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- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.
In the paper
“In a later paper [298], they suggest that a large negative bare cosmological constant could combine with a positive contribution from spacetime foam to produce a “microcyclic” structure to the Universe, with Planck sized regions independently undergoing oscillations but with a net overall expansion.”
§ Spacetime foam: a review > (7.1)
What the sources say
- Not matchedcites [298]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
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
“Vacuum fluctuations of bosonic and fermionic fields contribute to the cosmological constant with opposite signs, and quantum field theoretical considerations do not tell us whether Λ should be positive or negative.”
§ Spacetime foam: a review > (7.1)
plaindirect- 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. It may state two separate claims as one.
In the paper
“It has been suggested that this mechanism could explain the absence of a large negative cosmological constant [51].”
§ Spacetime foam: a review > (7.1)
What the sources say
- Not matchedcites [51]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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: passes — The paper gives a 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.
- Not matched
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
“While this is consistent with our present knowledge of quantum field theory, it requires extrapolations to scales far beyond what can be probed by current observations.”
§ Spacetime foam: a review > • Particle propagation
plain- 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.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“As noted in §5.1, a particular approach to spacetime foam, a partial sum over Euclidean instantons, seems to predict Planck scale masses and interactions for the Higgs boson and fermions [200,204,205], and is thus ruled out by observation.”
§ Spacetime foam: a review > • Particle propagation
What the sources say
- Not matchedcites [200]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [204]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [205]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- 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.
It may state two separate claims as one.
- Not matched
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, as described in §3, one still expects enormous fluctuations in the vacuum energy, leading again to a very highly inhomogeneous spacetime.”
§ Spacetime foam: a review > (7.1)
plaindirect- 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.
In the paper
“*In the alternative reduced phase space quantization, one first solves the constraints for a set of “true” degrees of freedom and then quantizes these.While this program can sometimes be carried out in lower dimensions[153], in 3+1 dimensions the resulting Hamiltonian is highly nonlocal, and fairly little progress has been made.”
§ Spacetime foam: a review > 8 Conclusion
What the sources say
- Not matchedcites [153]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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.
It may state two separate claims as one.
- Not matched
In the paper
“There are also arguments that the Coleman mechanism is inconsistent with the AdS/CFT correspondence of string theory [292], suggesting that certain topologies might be excluded in a complete quantum theory of gravity.”
§ Spacetime foam: a review > (7.1)
What the sources say
- Not matchedcites [292]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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. It may state two separate claims as one.
- Not matched
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
“Quantum fluctuations might affect topology, or just geometry.They might affect four-geometry, or spatial threegeometry, or the more abstract Euclidean geometry of path integral instantons.”
§ Spacetime foam: a review > 8 Conclusion
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check 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. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“In the early 1980s, Baum [287] and Hawking [288] argued that the Euclidean path integral in the presence of a positive cosmological constant is dominated by the contribution of the four-sphere, giving Z ∼ e 3π/ℓ 2 p Λ .”
§ Spacetime foam: a review > 7 The cosmological constant
What the sources say
- Not matchedcites [287]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [288]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“There are debates about the sign in the exponent (for instance, [135,289]), but if (7.1) is correct and Λ is positive, it suggests a very high probability for a very small cosmological constant.”
§ Spacetime foam: a review > (7.1)
What the sources say
- Not matchedcites [135]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [289]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgedconditional- 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.
- Not matched
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 discussed in §4.1, the sum over topologies (4.6) fails to converge: any process that would normally push Λ more negative instead drives the production of new topologies.”
§ Spacetime foam: a review > (7.1)
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.
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 ideas of spacetime foam and “minimum length” are logically distinct, although both could arise from Planck scale quantum fluctuations.”
§ Spacetime foam: a review > • Particle propagation
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. 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
“In Wheeler's original picture of spacetime foam, large fluctuations at the Planck scale average out to be nearly invisible at macroscopic scales.”
§ Spacetime foam: a review > (7.1)
plaindirect- 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.
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 is not quite right, of course-in standard general relativity, Λ is a fixed parameter, not a variable in a probability amplitude.”
§ Spacetime foam: a review > (7.1)
plain- 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: fails — The paper may give no reason why it holds.
Probably not a substantial claim. 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
“Moreover, these “foamy” regions are long-lasting-the probability current for moving out of such regions is tiny-lending support to the idea that small macroscopic expansion may persist even with a large Λ.”
§ Spacetime foam: a review > (7.1)
plainconditional- 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
“Unruh and Wang argue that these fluctuations average to almost zero, but that a parametric resonance produces an overall expansion rate that is nonzero but exponentially smaller than the naive expectation.”
§ Spacetime foam: a review > (7.1)
What the sources say
- Not matchedcites [297]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- 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. It may state two separate claims as one.
- Not matched
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 a danger that it would lead to “giant wormholes”; the role of configurations that are only “nearly” saddle points is unclear; and the results depend crucially on the controversial sign in the exponent (7.1).”
§ Spacetime foam: a review > (7.1)
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.
Probably not a substantial claim. 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
“Even then, though, the initial data described here can be evolved with the standard second order Einstein equations, yielding spacetimes that remain solutions to quadratic gravity.”
§ Spacetime foam: a review > 8 Conclusion
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: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds. 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
“They might lead to a wide range of possible observational effects, but alternatively might be too small to see at the scales we can observe.”
§ Spacetime foam: a review > 8 Conclusion
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: fails — It may be too vague to check 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. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. 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
“If the top quark is the heaviest particle, this gives a factor of about 10 58 ; if new particles continue up to the Planck mass, one again obtains 10 122 .”
§ Spacetime foam: a review > 8 Conclusion
plaindirect- 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.
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
“†As noted above, quantum corrections may lead to higher order field equations, for which it is not known whether the initial value problem is well posed.”
§ Spacetime foam: a review > 8 Conclusion
hedgedconditional- 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: fails — The paper may give no reason why it holds.
Probably not a substantial claim. 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
“But the cosmological constant we are interested in comes from Planck scale fluctuations, and on that scale the universe is certainly not homogeneous.”
§ Spacetime foam: a review > (7.1)
boosteddirect- 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.
The authors state this as an assumption.
Matter effects at Planck-scale distances are negligible except perhaps very near a singularity, so vacuum solutions can be considered.
“Omitting matter is a physically reasonable assumption.We are interested in geometry at the Planck scale, and except perhaps very near a singularity, the effects of matter at such short distances are negligible; even a neutron star has a density of only 10 -79 in Planck units.”
The paper relies on this without support.
The coefficients α and β of the quadratic curvature corrections are of order ℓp^2.
“The coefficients α and β are expected to be of order ℓ p 2 , and the corrections are likely to be negligible at ordinary scales, but they may become important near the Planck scale.”
The paper relies on this without support.
We do not yet have a complete quantum theory of gravity.
“Unfortunately, we do not yet have a complete quantum theory of gravity.”
The paper relies on this without support.
Spacetime foam is not a terribly well-defined notion.
“Of course, as noted earlier, “spacetime foam” is not a terribly well-defined notion.”
The paper relies on this without support.
The Planck scale is very small, and structures at that scale would plausibly wash out over larger distances.
“This is not entirely clear-the Planck scale is very small, and it is plausible that structures at that scale would wash out over larger distances”
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
§ Spacetime foam: a review > 5.1 Propagation
“Note that the dispersion relations (5.1) are not Lorentz invariant.”
§ Spacetime foam: a review > 5.1 Propagation
“the corresponding fluctuations of energy and momentum could lead to modified dispersion relations of the form [208,209] E 2 ∼ p 2 ± ϵ E E p α (5.2)”
Citations
Cites
The cosmological constant problem
Weinberg, S. (1989) · Reviews of Modern Physics · doi:10.1103/revmodphys.61.1 · Cited by 7,341
Quantum Theory of Gravity. I. The Canonical Theory
DeWitt, B. S. (1967) · Physical Review · doi:10.1103/physrev.160.1113 · Cited by 3,385
Hartle, J. B., & Hawking, S. W. (1983) · Physical Review D · doi:10.1103/physrevd.28.2960 · Cited by 3,027
2 + 1 dimensional gravity as an exactly soluble system
Witten, E. (1988) · Nuclear Physics B · doi:10.1016/0550-3213(88)90143-5 · Cited by 2,562
On gravity's role in quantum state reduction
Penrose, R. (1996) · General Relativity and Gravitation · doi:10.1007/bf02105068 · Cited by 1,872
Gravitation, gauge theories and differential geometry
Eguchi, T., Gilkey, P., & Hanson, A. J. (1980) · Physics Reports · doi:10.1016/0370-1573(80)90130-1 · Cited by 1,809
Bousso, R. (2002) · Reviews of Modern Physics · doi:10.1103/revmodphys.74.825 · Cited by 1,741
General relativity without coordinates
Regge, T. (1961) · Il Nuovo Cimento · doi:10.1007/bf02733251 · Cited by 1,477
A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories
Achúcarro, A., & Townsend, P. K. (1986) · Physics Letters B · doi:10.1016/0370-2693(86)90140-1 · Cited by 1,267
Jackiw, R. (1985) · Nuclear Physics B · doi:10.1016/0550-3213(85)90448-1 · Cited by 1,111
Research works printed but not linked: 6
- Wheeler, 1981, The lesson of the black hole, Proc. Am. Phil. Soc., 125, 25
- Thorpe, 1969, Some remarks on the Gauss-Bonnet formula, J. Math. Mech., 18, 779
- Gromov, 2010, Visions in Mathematics: Geom. Funct. Anal. 2000 Special Volume, Part I, p 118
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Metadata
- Authors
- Carlip, S.
- Affiliations
- University of California, Davis (US)
- Corresponding author
- Carlip, S
- ORCID
- Carlip, S
- Venue
- Reports on Progress in Physics
- Publisher
- IOP Publishing
- Published
- 2023-05-05
- Volume
- 86
- Issue
- 6
- Pages
- 066001
- ISSN
- 0034-4885 (ISSN-L)
- 1361-6633
- OpenAlex
- W4366468183
- Semantic Scholar
- 4a713e81123a3b6bb97d0c27385cbcc8f369a700
- Cited by
- 41
- References
- 286
- Citations by year
- 2026: 9 · 2025: 15 · 2024: 7 · 2023: 9
- FWCI
- 1.48
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Reports on Progress in Physics (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy (DE-FG02-91ER40674)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- spacetime foam, Planck scale, quantum gravity, spacetime topology, John Wheeler
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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