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Eric Ling

Publications and source records attributed to Eric Ling.

32 records · Page 2Linked to original sources
arXiv2021

Remarks on the existence of CMC Cauchy surfaces

As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in solving the Einstein equations, both in solving the contraints and the evolution equations. In this paper we review the CMC existence result obtained by the authors in [10] and consider some new existence results motivated by a conjecture of Dilts and Holst [8]. We also address some issues concerning the conformal structure of cosmological spacetimes.

gr-qc↗
arXiv2021

Topological censorship in spacetimes compatible with $Λ> 0$

Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant $Λ=0$ or $Λ<0$. Here, we prove a topological censorship theorem that is compatible with $Λ>0$ and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there possibly being nontrivial topology in the universe.

gr-qc↗
arXiv2020

Aspects of $C^0$ causal theory

This paper serves as an introduction to $C^0$ causal theory. We focus on those parts of the theory which have proven useful for establishing spacetime inextendibility results in low regularity -- a question which is motivated by the strong cosmic censorship conjecture in general relativity. This paper is self-contained; prior knowledge of causal theory is not assumed.

gr-qc↗
arXiv2020

A conformal infinity approach to asymptotically $\text{AdS}_2\times S^{n-1}$ spacetimes

It is well known that the spacetime $\text{AdS}_2\times S^2$ arises as the `near horizon' geometry of the extremal Reisser-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence. Motivated by a conjectural viewpoint of Juan Maldacena, the authors in [4] studied the rigidity of asymptotically $\text{AdS}_2\times S^2$ spacetimes satisfying the null energy condition. In this paper, we take an entirely different and more general approach to the asymptotics based on the notion of conformal infinity. This involves a natural modification of the usual notion of timelike conformal infinity for asymptotically anti-de Sitter spacetimes. As a consequence we are able to obtain a variety of new results, including similar results to those in [4] (but now allowing both higher dimensions and more than two ends) and a version of topological censorship.

gr-qc↗
arXiv2020

The Big Bang is a Coordinate Singularity for $k = -1$ Inflationary FLRW Spacetimes

We show that the big bang is a coordinate singularity for a large class of $k = -1$ inflationary FLRW spacetimes which we have dubbed 'Milne-like.' By introducing a new set of coordinates, the big bang appears as a past boundary of the universe where the metric is no longer degenerate -- a result which has already been investigated in the context of vacuum decay [12]. We generalize their results and approach the problem from a more mathematical perspective. Similar to how investigating the geometrical properties of the $r = 2m$ event horizon in Schwarzschild led to a better understanding of black holes, we believe that investigating the geometrical properties of the big bang coordinate singularity for Milne-like spacetimes could lead to a better understanding of cosmology. We show how the mathematics of these spacetimes may help illuminate certain issues associated with dark energy, dark matter, and the universe's missing antimatter.

gr-qc↗
arXiv2018

Weakly trapped surfaces in asymptotically de Sitter spacetimes

It is a standard fact that trapped or marginally trapped surfaces are not visible from conformal infinity, under the usual set of conditions on matter fields and the conformal completion, provided that the cosmological constant is non-positive. In this note we show that the situation is more delicate in the presence of a positive cosmological constant: we present examples of visible marginally trapped surfaces, and we provide a set of natural conditions which guarantee non-visibility.

gr-qc↗
arXiv2018

Milne-like spacetimes and their role in Cosmology

Milne-like spacetimes are a class of FLRW models which admit $C^0$ spacetime extensions through the big bang. The boundary of a Milne-like spacetime can be identified with a null cone in the extension. We find that the comoving observers all emanate from a single point in the extension. This suggests that something physical may have happened there. Next we ask what role these spacetimes have in modern cosmology. We find that Milne-like spacetimes are inconsistent with classical cosmology theory in which a radiation-dominated era follows immediately after the big bang. However, Milne-like spacetimes are consistent with inflationary theory. In fact the very nature of Milne-like spacetimes solves the horizon problem. We note that this has already been observed for the Milne universe in [2] and [25]. Next we show, assuming a finite initial condition on the energy density, that Milne-like spacetimes naturally exhibit a slow-roll inflationary era. This makes Milne-like spacetimes good models for inflationary theory. We also show that Milne-like spacetimes still admit $C^0$ extensions under well-behaved perturbations.

gr-qc↗
arXiv2018

Milne-like Spacetimes and their Symmetries

When developing a quantum theory for a physical system, one determines the system's symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincaré group, $\mathbb{R}^4 \rtimes \text{O}(1,3)$, and the irreducible unitary representations are interpreted as elementary particles which determine the particle's mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be $\mathbb{R} \times \text{O}(1,3)$ and classify their irreducible unitary representations. Again they represent particles with mass and spin. Unlike the classification for the Poincaré group, we do not obtain any faster-than-light particles. The factor $\mathbb{R}$ corresponds to cosmic time translations. These generate a mass Casimir operator which yields a Lorentz invariant Dirac equation on Milne-like spacetimes. In fact it's just the original Dirac equation multiplied by a conformal factor $Ω$. Therefore many of the invariants and symmetries still hold. We offer a new interpretation of the negative energy states and propose a possible solution to the matter-antimatter asymmetry problem in our universe.

gr-qc↗
arXiv2018

Topology and singularities in cosmological spacetimes obeying the null energy condition

We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3+1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise connection between the topology of the Cauchy surfaces and the occurrence of past singularities. In addition to (a refinement of) the Penrose singularity theorem, the proof makes use of some recent advances in the topology of 3-manifolds and of certain fundamental existence results for minimal surfaces.

gr-qc↗
arXiv2018

Maximizers in Lipschitz spacetimes are either timelike or null

We prove that causal maximizers in $C^{0,1}$ spacetimes are either timelike or null. This question was posed in [17] since bubbling regions in $C^{0,α}$ spacetimes ($α<1$) can produce causal maximizers that contain a segment which is timelike and a segment which is null, cf. [3]. While $C^{0,1}$ spacetimes do not produce bubbling regions, the causal character of maximizers for spacetimes with regularity at least $C^{0,1}$ but less than $C^{1,1}$ was unknown until now. As an application we show that timelike geodesically complete spacetimes are $C^{0,1}$-inextendible.

gr-qc↗
arXiv2017

Timelike completeness as an obstruction to $C^0$-extensions

The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak solution. In this paper we show that a timelike complete and globally hyperbolic Lorentzian manifold is $C^0$-inextendible. For the proof we make use of the result, recently established by Sämann [17], that even for \emph{continuous} Lorentzian manifolds that are globally hyperbolic, there exists a length-maximizing causal curve between any two causally related points.

gr-qc↗
arXiv2017

Some Remarks on the $C^0$-(in)extendibility of Spacetimes

The existence, established over the past number of years and supporting earlier work of Ori [14], of physically relevant black hole spacetimes that admit $C^0$ metric extensions beyond the future Cauchy horizon, while being $C^2$-inextendible, has focused attention on fundamental issues concerning the strong cosmic censorship conjecture. These issues were recently discussed in the work of Jan Sbierski [17], in which he established the (nonobvious) fact that the Schwarschild solution in global Kruskal-Szekeres coordinates is $C^0$-inextendible. In this paper we review aspects of Sbierski's methodology in a general context, and use similar techniques, along with some new observations, to consider the $C^0$-inextendibility of open FLRW cosmological models. We find that a certain special class of open FLRW spacetimes, which we have dubbed `Milne-like,' actually admit $C^0$ extensions through the big bang. For spacetimes that are not Milne-like, we prove some inextendibility results within the class of spherically symmetric spacetimes.

math.DG↗