Eric Ling's research while affiliated with University of Toronto and other places
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Publications (35)
A bstract
Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have...
The time separation function (or Lorentzian distance function) is a fundamental object used in Lorentzian geometry. For smooth spacetimes it is known to be lower semicontinuous, and in fact, continuous for globally hyperbolic spacetimes. Moreover, an axiom for Lorentzian length spaces - a synthetic approach to Lorentzian geometry - is the existence...
Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important...
![Left: the Milne universe, a(τ)=τ\documentclass[12pt]{minimal}...](publication/369552786/figure/fig1/AS:11431281156703372@1683601389350/Left-the-Milne-universe-attdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
Milne-like spacetimes are a class of hyperboloidal FLRW spacetimes which admit continuous spacetime extensions through the big bang, \(\tau = 0\). In a previous paper [27], it was advocated that the existence of this big bang extension could have applications to fundamental problems in cosmology, which illustrates the physical importance of such ex...
In this paper, we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymp...
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zero- G limit (z GKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the z GKN spacetime. The spectrum is characterized by a triplet of integers,...
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...
Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, $\tau = 0$. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies $a(\tau) = \tau + o(\tau^{1+\varepsilon})$ as $\tau \to 0$ for...
Milne-like spacetimes are a class of $$k = -1$$ k = - 1 FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper Ling (Found. of Phys. 50:385–428, 2020), it was shown that the cosmological constant appears as an initial condition for Milne-like spacetimes. In this paper, we generalize this statement to s...
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 i...
In this paper we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asympt...
Milne-like spacetimes are a class of $k = -1$ FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper [29], it was shown that the cosmological constant appears as an initial condition for Milne-like spacetimes. In this paper, we generalize this statement to spacetimes which share similar geometrical pro...
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr--Newman spacetime is determined in the zero-$G$ limit (z$G$KN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the z$G$KN spacetime. The spectrum is characterized by a triplet of intege...
In this short paper, we review the Dirac equation on the zero-gravity Kerr-Newman spacetime. Our main objective is to provide a correspondence between the classification of the bound states for the zGKN spectrum and the usual hydrogenic states $1s_{1/2}$, $2s_{1/2}$, etc. of the Hydrogen atom.
Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant $\Lambda=0$ or $\Lambda<0$. Here, we prove a topological censorship theorem that is compatible with $\Lambda>0$ and which can be applied to whole universes containing possibly multiple collections of black hole...
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...
It is well known that the spacetime AdS2×S2 arises as the ‘near-horizon’ geometry of the extremal Reissner–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, Galloway and Graf (Adv Theor Math Phys 23(2):403–435, 2019) studied the rigidit...
![The coordinate system ϕ:Uε→Rn+1\documentclass[12pt]{minimal}...](publication/342018316/figure/fig3/AS:961489997398033@1606248487436/The-coordinate-system-phUeRn-1documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Left: The C0\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/342018316/figure/fig4/AS:961489997426714@1606248487640/Left-The-C0documentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![A future set F⊂J+(Σ)\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/342018316/figure/fig5/AS:961489997398055@1606248487724/A-future-set-FJ-Sdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
This paper serves as an introduction to C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0$$\end{document} causal theory. We focus on those parts of the theory which...
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...
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...
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 the...
We present a physics model for a time-symmetric Milne-like universe based on the $q$-theory approach to the cosmological constant problem, supplemented by an assumed vacuum-matter energy exchange possibly due to quantum-dissipative effects. Without fine-tuning of the initial vacuum energy density, we obtain a harmless big bang singularity (with fin...
We show that the big bang is just a coordinate singularity for $k = -1$ inflationary FLRW spacetimes. That is, it can be removed by introducing a set of coordinates in which the big bang appears as a past Cauchy horizon where the metric is no longer degenerate. In fact this past Cauchy horizon is just the future lightcone at the origin of a spaceti...
In this note we present a result establishing the existence of a compact CMC Cauchy surface from a curvature condition related to the strong energy condition.
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 const...
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...
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 establish that e...
We prove that causal maximizers in $C^{0,1}$ spacetimes are either strictly timelike or strictly null. This question was posed in [16] since bubbling regions in $C^{0,\alpha}$ spacetimes ($\alpha <1$) can produce causal maximizers which contain a segment which is timelike and a segment which is null, cf. [3]. While $C^{0,1}$ spacetimes do not produ...
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 Poincar\'e group, $\mathbb{R}^4 \rtimes \text{O}(1,3)$, and the irreducible unitary representations are elementary particles which determine the particle's mas...
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 C0- inextendible. For the proof we make use of the re...
Milne-like spacetimes are FLRW spacetimes 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 w...
The discovery over the past number of years 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 work of Jan Sbierski [16...
Citations
... It was shown in [28] that this holds for nonhomogeneous versions of Milne-like spacetimes and hence it's a consequence of the geometry of the big bang extension. Moreover, there are analogous results for flat FLRW spacetimes which are asymptotically de Sitter [14]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/277938059661312-1443277005080_Q64/Ghazal-Geshnizjani.jpg)
... Specifically, we will mostly explore spacetimes that are flat past-asymptotically dS and derive conditions on their extendibility. There will be many parallels with the works of [38,[46][47][48][49] on the topic of metric extendibility of Milne-like spacetimes, which are homogeneous and isotropic universes with negative (as opposed to zero here) constant spatial curvature. ...
... 8,9,13 In reference 9 it was shown that the discrete spectrum of the Dirac Hamiltonian on zGKN is nonempty. In an upcoming paper 10 we classify the discrete spectrum and show that the spectrum is indexed by three integers. See Theorem 2.1 below. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272572868722710-1441997844570_Q64/Michael-Kiessling-2.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/756055705399297-1557269135239_Q64/A-Shadi-Tahvildar-Zadeh.jpg)
... There have been many different attempts to either prove or find a counterexample to Bartnik's conjecture, and partial results have been obtained under additional requirements [1,8,11,[14][15][16][17][18]. Now, a standard argument using the compactness of the Cauchy hypersurfaces and the Limit Curve Lemma [3,Cap. ...
... ; it's used in this paper and in [14,26]. A proof showing openness of I + (p) is given in [24,Thm. ...
... Therefore X is a continuous extension of 1 Ω u to M ∪ {O}. Since g(u, u) = −1 (by definition of a perfect fluid), continuity implies g(X, X) = −1 at O. Using [72,Lem. 2.9] and applying the Gram-Schmidt orthogonalization process appropriately, for any 0 < ε < 1, we can assume that the coordinates (x 0 , . . . ...
... Specifically, we will mostly explore spacetimes that are flat past-asymptotically dS and derive conditions on their extendibility. There will be many parallels with the works of [38,[46][47][48][49] on the topic of metric extendibility of Milne-like spacetimes, which are homogeneous and isotropic universes with negative (as opposed to zero here) constant spatial curvature. ...
... Globally hyperbolic spacetimes do not always admit CMC Cauchy surfaces and the problem of determining them in a generic sense is still open. Several existence results for compact Cauchy surfaces can be found in the literature, like the fundamental one put forward by Bartnik [9], which in turn has motivated others in the same line, like the one put forward by Galloway and Ling in [31], where the existence of the CMC Cauchy surface is consequence of an ambient curvature hypothesis related to the strong energy condition. Naturally, the study of CMC and maximal hypersurfaces in twisted product ambients is also an issue of remarkable interest. ...
... There are many others recent works regarding trapped surfaces in a broad way. Such works consider several backgrounds and deal with different aspects of the surfaces: geometrical properties, rigidity, representation, non-existence results, classification in some spacetimes, causality properties, etc (see for instance [2,4,8,9,11,12,25] and references therein). ...
... Recently, Galloway, Ling and Sbierski [6] have proved the following result Theorem 1.1. A smooth (at least C 2 ) Lorentzian spacetime that is timelike geodesically complete and globally hyperbolic is C 0 -inextendible. ...
