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We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact Cauchy surfaces satisfying the strong energy condition. If the spacetime contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant $\Lambda > 0$. Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors., Comment: 16 pages; v2 minor changes and reference added

In this article, we extend a construction of [6] to obtain a large class of vacuum cosmological spacetimes that do not contain any CMC Cauchy surfaces. The allowed spatial topologies for these examples are of the form $M \# M$, where $M$ is any closed, oriented, irreducible $3$-manifold which is not spherical. This complements the recent results of [19], where, instead of initial data methods, global spacetime gluing arguments were used. The study of such examples is sure to yield insight into Bartnik's cosmological splitting conjecture [1]., Comment: 8 pages; v2: minor corrections. To appear in the Proceedings of the XI International Meeting on Lorentzian Geometry

Utilizing some of Sbierski's recent $C^0$-inextendibility techniques [13], we prove the $C^0$-inextendibility of a class of spatially flat FLRW spacetimes without particle horizons., Comment: 7 pages

The inflationary scenario, which states that the early universe underwent a brief but dramatic period of accelerated spatial expansion, has become the current paradigm of early universe cosmology. Although inflationary cosmology has its many successes, it does not (as of yet) have the status of an established physical theory. In this paper, we provide mathematical support for the inflationary scenario in a class of anisotropic spacetimes by generalizing the work in [21]. These anisotropic spacetimes satisfy certain initial conditions so that they are perfectly isotropic at the big bang but become less isotropic as time progresses. The resulting inflationary eras are a consequence of the initial conditions which force the energy-momentum tensor to be dominated by a cosmological constant at the big bang., Comment: 20 pages, 3 figures, proceeding article for Minkowski meeting 2023 volume

In this paper, we study rigidity aspects of Penrose's singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose's singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship., Comment: 18 pages. v3 agrees with published version

The Reissner-Weyl-Nordstr\"om (RWN) spacetime of a point nucleus features a naked singularity for the empirically known nuclear charges $Ze$ and masses $M = A(Z,N)m_{\mathrm{p}}$, where $m_{\mathrm{p}}$ is the proton mass and $A(Z,N)\approx Z+N$ the atomic mass number, with $Z$ the number of protons and $N$ the number of neutrons in the nucleus. The Dirac Hamiltonian for a test electron with mass $m_{\mathrm{e}}$, charge $-e$, and anomalous magnetic moment $\mu_a (\approx - \frac{1}{4\pi}\frac{e^3}{m_{\mathrm{e}} c^2})$ in the electrostatic RWN spacetime of such a 'naked point nucleus' is known to be essentially self-adjoint, with a spectrum that consists of the union of the essential spectrum $(-\infty, m_{\mathrm{e}} c^2]\cup[m_{\mathrm{e}} c^2, \infty)$ and a discrete spectrum of infinitely many eigenvalues in the gap $(-m_{\mathrm{e}} c^2,m_{\mathrm{e}} c^2)$, having $m_{\mathrm{e}} c^2$ as accumulation point. In this paper the discrete spectrum is characterized in detail for the first time, for all $Z\leq 45$ and $A$ that cover all known isotopes. The eigenvalues are mapped one-to-one to those of the traditional Dirac Hydrogen spectrum. Numerical evaluations that go beyond $Z=45$ into the realm of not-yet-produced hydrogenic ions are presented, too. A list of challenging open problems concludes this publication., Comment: 40 pages, 1 figure, submitted for publication

CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, in [2] Bartnik constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, in [9], Chru\'sciel, Isenberg, and Pollack constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology was also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik's construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact oriented Euclidean or hyperbolic three-manifold with any another compact oriented Euclidean or hyperbolic three-manifold. We work with the Tolman-Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models., Comment: 25 pages, 2 figures; final version agrees with published version

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 physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions., Comment: 62 pages, 6 figures; v2: minor corrections and references added, matches published version

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 some $\varepsilon > 0$. This asymptotic assumption implies $a(\tau) = \tau + o(\tau)$. In this paper, we show that $a(\tau) = \tau + o(\tau)$ is not sufficient to achieve an extension through $\tau = 0$, but it is necessary provided its derivative converges as $\tau \to 0$. We also show that the $\varepsilon$ in $a(\tau) = \tau + o(\tau^{1+\varepsilon})$ is not necessary to achieve an extension through $\tau = 0$., Comment: 14 pages, 1 figure

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 asymptotically (locally) AdS setting., Comment: 24 pages, 1 figure. v2 accepted to JMP

Milne-like spacetimes are a class of $k = -1$ FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper [30], 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 properties with Milne-like spacetimes but without the strong spatially isotropic assumption associated with them. We show how our results yield a "quasi de Sitter" expansion for the early universe which could have applications to inflationary scenarios., Comment: 16 pages, 3 figures, article for the conference "Singularity theorems, causality, and all that; a tribute to Roger Penrose." Final version to appear in GRG

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 integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed., Comment: 62 pages, 13 figures

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., Comment: 13 pages. To appear in Int. J. Mod. Phys. D

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 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., Comment: Comments are welcome

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., Comment: 13 pages; v2: minor corrections/clarifications. To appear in the Proceedings of the X International Meeting on Lorentzian Geometry

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., Comment: 24 pages

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., Comment: 40 pages, 6 figures. v2: references fixed and proof of prop 3.7 fixed. v3: added acknowledgement to Mittag-Leffler program. v4: to appear in GRG

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., Comment: 46 pages, 20 figures, v3 to appear in Foundations of Physics

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., Comment: 7 pages; (v2) proof of Proposition 3 (and associated claims) corrected, references and several comments added; (v3) to appear in GRG

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., Comment: 9 pages, 4 figures; v2: figure corrected, references added. Version to appear in CQG

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 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 Poincar\'e 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 $\Omega$. 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., Comment: 33 pages. v2: corrections made and references added. This paper supersedes arXiv:1706.01408

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 $C^{0,\alpha}$ spacetimes ($\alpha <1$) 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., Comment: 7 pages, 1 figure. Version 2 accepted to CQ&G

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., Comment: Withdrawn. This paper is superceded by arXiv:1803.00174v2

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., Comment: 8 pages; v2: minor changes, version to appear in CMP

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 S\"amann [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., Comment: v2: pertinent citations added and minor changes, v3: small refinements and improvements; theorem added, v4: added figure, minor changes, version accepted for publication in CMP

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., Comment: 22 pages, v2: minor changes and clarifications; reference added. To appear in Annales Henri Poincare