An R<sup>4</sup>spacetime with a Cauchy surface which is not R<sup>3</sup>

Newman, Richard P. A. C.; Clark, C J S

doi:10.1088/0264-9381/4/1/008

Cited by 11 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There is a very rich literature on these topics on specialized books [1,3,10,12,14,[16][17][18] and on the original papers (see also [61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]), but we thought it was important to analyze them in a single paper. This choice of arguments helps also nonexpert readers in gaining familiarity with concepts and techniques frequently used in classical gravity and which also find application in quantum gravity, as it happens for the theory of spinors and for Ashtekar's variables.…”

Section: Discussionmentioning

confidence: 99%

Mathematical Structures of Space-Time

Esposito

1992

Fortschr. Phys.

View full text Add to dashboard Cite

Abstract. At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables.Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves 1 Mathematical Structures of Space-Time whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christoffel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Namely, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity.Our work should be compared with important previous papers. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy-momentum tensor, and we emphasize the role played by the full extrinsic curvature tensor and by the variation formulae. * Fortschritte der Physik, 40, 1-30 (1992). 2Mathematical Structures of Space-Time

show abstract

Section: Discussionmentioning

confidence: 99%

Mathematical Structures of Space-Time

Esposito

1992

Fortschr. Phys.

View full text Add to dashboard Cite

show abstract

“…However to gain this covariance the technical assumption that the Cauchy surface M be compact as in Definiton 4.1 is essential. For example, there exists a space-time diffeomorphic to ℝ 4 admitting Cauchy surfaces which are even not homeomorphic [18]. One of them is the standard ℝ 3 while the other is a so-called Whithead space [30]: an open contractible 3-manifold W which is not homeomorphic (hence not diffeomorphic) to ℝ 3 ; there are uncountable many pairwise non-homeomorphic Whitehead spaces but it is known [16] that every Whithead space W satisfies that the product W × ℝ is always diffeomorphic to ℝ 4 .…”

Section: Appendix: a Temporal Approach To General Relativitymentioning

confidence: 99%

A Set-Theoretic Analysis of the Black Hole Entropy Puzzle

Etesi

2023

Found Phys

View full text Add to dashboard Cite

Motivated by the known mathematical and physical problems arising from the current mathematical formalization of the physical spatio-temporal continuum, as a substantial technical clarification of our earlier attempt (Etesi in Found Sci 25:327–340, 2020), the aim in this paper is twofold. Firstly, by interpreting Chaitin’s variant of Gödel’s first incompleteness theorem as an inherent uncertainty or fuzziness present in the set of real numbers, a set-theoretic entropy is assigned to it using the Kullback–Leibler relative entropy of a pair of Riemannian manifolds. Then exploiting the non-negativity of this relative entropy an abstract Hawking-like area theorem is derived. Secondly, by analyzing Noether’s theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a single, stationary, asymptotically flat black hole is modeled by the set of real numbers in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole (proportional to the area of its “instantaneous” event horizon), indicating that this apparently physical quantity might have a pure set-theoretic origin, too.

show abstract

“…The existence of many non-homeomorphic Whitehead continua has interesting consequences in the initial value formulation, too-see e.g. [20]. 2.…”

Section: Riemannian Considerationsmentioning

confidence: 99%

Exotica and the status of the strong cosmic censor conjecture in four dimensions

Etesi

2017

Class. Quantum Grav.

View full text Add to dashboard Cite

An immense class of physical counterexamples to the four dimensional strong cosmic censor conjecture-in its usual broad formulation-is exhibited. More precisely, out of any closed and simply connected 4-manifold an open Ricci-flat Lorentzian 4-manifold is constructed which is not globally hyperbolic and no perturbation of it, in any sense, can be globally hyperbolic. This very stable non-global-hyperbolicity is the consequence of our open spaces having a "creased end" i.e., an end diffeomorphic to an exotic R 4 . Open manifolds having an end like this is a typical phenomenon in four dimensions.The construction is based on a collection of results of Gompf and Taubes on exotic and selfdual spaces, respectively, as well as applying Penrose' non-linear graviton construction (i.e., twistor theory) to solve the Riemannian Einstein's equation. These solutions then are converted into stably non-globally-hyperbolic Lorentzian vacuum solutions. It follows that the plethora of vacuum solutions we found cannot be obtained via the initial value formulation of the Einstein's equation because they are "too long" in a certain sense (explained in the text). This different (i.e., not based on the initial value formulation but twistorial) technical background might partially explain why the existence of vacuum solutions of this kind has not been realized so far in spite of the fact that, apparently, their superabundance compared to the well-known globally hyperbolic vacuum solutions is overwhelming.

show abstract

An R⁴spacetime with a Cauchy surface which is not R³

Cited by 11 publications

References 14 publications

Mathematical Structures of Space-Time

Mathematical Structures of Space-Time

A Set-Theoretic Analysis of the Black Hole Entropy Puzzle

Exotica and the status of the strong cosmic censor conjecture in four dimensions

Contact Info

Product

Resources

About

An R4spacetime with a Cauchy surface which is not R3

Cited by 11 publications

References 14 publications

Mathematical Structures of Space-Time

Mathematical Structures of Space-Time

A Set-Theoretic Analysis of the Black Hole Entropy Puzzle

Exotica and the status of the strong cosmic censor conjecture in four dimensions

Contact Info

Product

Resources

About

An R⁴spacetime with a Cauchy surface which is not R³