Spike statistics

Heinzle, Jakob; Uggla, Claes

doi:10.1007/s10714-013-1523-z

Cited by 5 publications

(7 citation statements)

References 37 publications

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“…Moreover, as also shown in [53], it follows that if (u s ) s∈N is a generic spike induced sequence of eras, then the probability that a randomly chosen era in this sequence possesses length n is given by Probability (length of era…”

Section: Maps and Statisticsmentioning

confidence: 77%

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Recent developments concerning generic spacelike singularities

Uggla

2013

Gen Relativ Gravit

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This is a review of recent progress concerning generic spacelike singularities in general relativity. For brevity the main focus is on singularities in vacuum spacetimes, although the connection with, and the role of, matter for generic singularity formation is also commented on. The paper describes recent developments in two areas and show how these are connected within the context of the conformally Hubble-normalized state space approach. The first area is oscillatory singularities in spatially homogeneous cosmology and the connection between asymptotic behaviour and heteroclinic chains. The second area concerns oscillatory singularities in inhomogeneous models, especially spike chains and recurring spikes. The review also outlines some underlying reasons for why the structures that are the foundation for generic oscillatory behaviour exists at all, which entails discussing how underlying physical principles and applications of solution generating techniques yield hierarchical structures and connections between them. Finally, it is pointed out that recent progress concerning generic singularities motivates some speculations that suggest that a paradigm shift concerning their physical role, and what mathematical issues to address, might be in order.Comment: 32 pages; plenary contribution to ERE201

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Section: Maps and Statisticsmentioning

confidence: 77%

“…Let (u s ) s∈N be a generic spike-induced sequence of eras. Then, as shown in [53], the probability that a randomly chosen element of (u s ) s∈N lies in the interval [m, m + 1) is given bȳ…”

Section: Maps and Statisticsmentioning

confidence: 99%

Recent developments concerning generic spacelike singularities

Uggla

2013

Gen Relativ Gravit

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“…However, it is believed that any spike formation occurs only for a set of time-lines of measure zero [13]. There are many other outstanding questions regarding recurring spike behaviour and generic spacelike singularities [216]. In particular: How and where do spikes form?…”

Section: Spikesmentioning

confidence: 99%

Mathematical general relativity

Coley

2019

Gen Relativ Gravit

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We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein-Yang-Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics.

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“…Recent results indicate that this is not the whole story. There is also a connection between asymptotic silence, generic spacelike singularities, and infinitely recurring oscillating inhomogeneous spikes, described by certain inhomogeneous solutions[2,56,36,35,57].…”

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confidence: 99%

“…Incidentally, we could have introduced the Kasner parameter u, as in said reference. The Kasner map describing how u changes follows from(24),(35) and(36). However, the range and domain of u differ from the critical GR case when v = 1/2.…”

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confidence: 99%

Bifurcations and Chaos in Hořava-Lifshitz Cosmology

Hell,

Lappicy,

Uggla

2020

Preprint

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The nature of generic spacelike singularities in general relativity is connected with first principles, notably Lorentzian causal structure, scale invariance and general covariance. To bring a new perspective on how these principles affect generic spacelike singularities, we consider the initial singularity in spatially homogeneous Bianchi type VIII and IX vacuum models in Hořava-Lifshitz gravity, where relativistic first principles are replaced with anisotropic scalings of Lifshitz type. Within this class of models, General Relativity is shown to be a bifurcation where chaos becomes generic. To describe the chaotic features of generic singularities in Hořava-Lifshitz cosmology, we introduce symbolic dynamics within Cantor sets and iterated function systems.

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Spike statistics

Cited by 5 publications

References 37 publications

Recent developments concerning generic spacelike singularities

Recent developments concerning generic spacelike singularities

Mathematical general relativity

Bifurcations and Chaos in Hořava-Lifshitz Cosmology

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