Past Singularities Research Articles

Traversable Lorentzian wormhole on the Shtanov-Sahni braneworld with matter obeying the energy conditions

In this paper we have explored the possibility of constructing a traversable wormhole on the Shtanov-Sahni braneworld with a timelike extra dimension. We find that the Weyl curvature singularity at the throat of the wormhole can be removed with physical matter satisfying the NEC ρ + p ≥ 0, even in the absence of any effective Λ-term or any type of charge source on the brane. (The NEC is however violated by the effective matter description on the brane arising due to effects of higher dimensional gravity.) Besides satisfying NEC the matter constituting the wormhole also satisfies the Strong Energy Condition (SEC), ρ + 3p ≥ 0, leading to the interesting possibility that normal matter on the brane may be harnessed into a wormhole. Incidentally, these conditions also need to be satisfied to realize a non-singular bounce and cyclic cosmology on the brane [1] where both past and future singularities can be averted. Thus, such a cyclic universe on the brane, constituted of normal matter can naturally contain wormholes. The wormhole shape function on the brane with a time-like extra dimension represents the tubular structure of the wormhole spreading out at large radial distances much better than in wormholes constructed in a braneworld with a spacelike extra dimension and have considerably lower mass resulting in minimization of the amount of matter required to construct a wormhole. Wormholes in the Shtanov-Sahni (SS) braneworld also have sufficiently low tidal forces, facilitating traversability. Additionally they are found to be stable and exhibit a repulsive geometry. We are left with the intriguing possibility that both types of curvature singularity can be resolved with the SS model, which we discuss at the end of the concluding section.

On a quantum Weyl curvature hypothesis

Roger Penrose's Weyl curvature hypothesis states that the Weyl curvature is small at past singularities, but not at future singularities. We review the motivations for this conjecture and present estimates for the entropy of our Universe. We then extend this hypothesis to the quantum regime by demanding that the initial state of primordial quantum fluctuations be the adiabatic vacuum in a (quasi-)de Sitter space. We finally attempt a justification of this quantum version from a fundamental theory of quantum gravity and speculate on its consequences in the case of a classically recollapsing universe.

A flat FLRW model with dynamical Λ as function of matter and geometry

We revisit the evolution of the scale factor in a flat FLRW spacetime with a new generalized decay rule for the dynamic Λ-term under modified theories of gravity. It analyzes certain cosmological parameters and examines their behaviours in this generalized setting which includes several decay laws in the literature. We have also obtained observational constraints on various model parameters and estimated the present values of cosmological parameters {Ωm0, ΩΛ0, q0,t0, ω0} and have discussed with various observational results. Finite time past and future singularities in this model are also discussed.

Cosmological singularities in conformal Weyl gravity

In this work, we study the issue of the past and future cosmological singularities in the context of the fourth-order conformal Weyl gravity. In particular, we investigate the emergent universe scenario proposed by Ellis et al., and find the stability conditions of the corresponding Einstein static state using the fixed point approach. We show that depending on the values of the parameters of the conformal Weyl gravity theory, there are possibilities for having initially stable emergent states for an FRW universe with both the positive and negative spatial curvatures. This represents that the conformal Weyl gravity can be free of the initial singularity problem. Then, following Barrow et al., we address the possible types of the finite-time future cosmological singularities. We discuss how these singularities also can be avoided in the context of this theory.

Higher spin fluctuations on spinless 4D BTZ black hole

We construct linearized solutions to Vasiliev’s four-dimensional higher spin gravity on warped AdS3 ×ξS1 which is an Sp(2) × U(1) invariant non-rotating BTZ-like black hole with ℝ2 × T2 topology. The background can be obtained from AdS4 by means of identifications along a Killing boost K in the region where ξ2 ≡ K2 ≥ 0, or, equivalently, by gluing together two Bañados-Gomberoff-Martinez eternal black holes along their past and future space-like singularities (where ξ vanishes) as to create a periodic (non-Killing) time. The fluctuations are constructed from gauge functions and initial data obtained by quantizing inverted harmonic oscillators providing an oscillator realization of K and of a commuting Killing boost tilde{K} . The resulting solution space has two main branches in which K star commutes and anti-commutes, respectively, to Vasiliev’s twisted-central closed two-form J. Each branch decomposes further into two subsectors generated from ground states with zero momentum on S1. We examine the subsector in which K anti-commutes to J and the ground state is mathrm{U}{(1)}_Ktimes mathrm{U}{(1)}_{tilde{K}} -invariant of which U(1)K is broken by momenta on S1 and mathrm{U}{(1)}_{tilde{K}} by quasi-normal modes. We show that a set of mathrm{U}{(1)}_{tilde{K}} -invariant modes (with n units of S1 momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at {tilde{K}}^2=1 . We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.

Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝn + 1 remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1, and convex. Our results generalize the well‐known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.

Exact solutions, finite time singularities and non-singular universe models from a variety of Λ(t) cosmologies

Cosmological models with time-dependent [Formula: see text] (read as [Formula: see text]) have been investigated widely in the literature. Models that solve background dynamics analytically are of special interest. Additionally, the allowance of past or future singularities at finite cosmic time in a specific model signals for a generic test on its viabilities with the current observations. Following these, in this work we consider a variety of [Formula: see text] models focusing on their evolutions and singular behavior. We found that a series of models in this class can be exactly solved when the background universe is described by a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) line element. The solutions in terms of the scale factor of the FLRW universe offer different universe models, such as power-law expansion, oscillating, and the singularity free universe. However, we also noticed that a large number of the models in this series permit past or future cosmological singularities at finite cosmic time. At last we close the work with a note that the avoidance of future singularities is possible for certain models under some specific restrictions.

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.

A three-dimensional coupled Nitsche and level set method for electrohydrodynamic potential flows in moving domains

In this paper we present a new algorithm for computing three-dimensional electrohydrodynamic flow in moving domains which can undergo topological changes. We consider a non-viscous, irrotational, perfect conducting fluid and introduce a way to model the electrically charged flow with an embedded potential approach. To numerically solve the resulting system, we combine a level set method to track both the free boundary and the surface velocity potential with a Nitsche finite element method for solving the Laplace equations. This results in an algorithmic framework that does not require body-conforming meshes, works in three dimensions, and seamlessly tracks topological change. Assembling this coupled system requires care: while convergence and stability properties of Nitsche's methods have been well studied for static problems, they have rarely been considered for moving domains or for obtaining the gradients of the solution on the embedded boundary. We therefore investigate the performance of the symmetric and non-symmetric Nitsche formulations, as well as two different stabilization techniques. The global algorithm and in particular the coupling between the Nitsche solver and the level set method are also analyzed in detail. Finally we present numerical results for several time-dependent problems, each one designed to achieve a specific objective: (a) The oscillation of a perturbed sphere, which is used for convergence studies and the examination of the Nitsche methods; (b) The break-up of a two lobe droplet with axial symmetry, which tests the capability of the algorithm to go past flow singularities such as topological changes and preservation of an axi-symmetric flow, and compares results to previous axi-symmetric calculations; (c) The electrohydrodynamical deformation of a thin film and subsequent jet ejection, which will account for the presence of electrical forces in a non-axi-symmetric geometry.

Further holographic investigations of big bang singularities

We further explore the quantum dynamics near past cosmological singularities in anisotropic Kasner-AdS solutions using gauge/gravity duality. The dual description of the bulk evolution involves $$ \mathcal{N}=4 $$ super Yang-Mills on the contracting branch of an anisotropic de Sitter space and is well defined. We compute two-point correlators of Yang-Mills operators of large dimensions using spacelike geodesics anchored on the boundary. The correlator between two points separated in a direction with negative Kasner exponent p always exhibits a pole at horizon scales, in any dimension, which we interpret as a dual signature of the classical bulk singularity. This indicates that the geodesic approximation selects a non-normalizable Yang-Mills state.

Eddington–Born–Infeld cosmology: a cosmographic approach, a tale of doomsdays and the fate of bound structures

The Eddington-inspired-Born-Infeld scenario (EiBI) can prevent the Big Bang singularity for a matter content whose equation of state is constant and positive. In a recent paper we showed that, on the contrary, it is impossible to smooth a big rip in the EiBI setup. In fact the situations are still different for other singularities. In this paper we show that a big freeze singularity in general relativity (GR) can in some cases be smoothed to a sudden or a type IV singularity under the EiBI scenario. Similarly, a sudden or a type IV singularity in GR can be replaced in some regions of the parameter space by a type IV singularity or a loitering behaviour, respectively, in the EiBI framework. Furthermore, we find that the auxiliary metric related to the physical connection usually has a smoother behaviour than that based on the physical metric. In addition, we show that bound structures close to a big rip or a little rip will be destroyed before the advent of the singularity and will remain bound close to a sudden, big freeze or type IV singularity. We then constrain the model following a cosmographic approach, which is well-known to be model-independent, for a given Friedmann-Lema\^itre-Robertson-Walker geometry. It turns out that among the various past or present singularities, the cosmographic analysis can pick up the physical region that determines the occurrence of a type IV singularity or a loitering effect in the past. Moreover, to determine which of the future singularities or doomsdays is more probable, observational constraints on higher order cosmographic parameters is required.

Dynamical behavior in f (T, TG) cosmology

The class of gravitational modification, based on the quadratic torsion scalar T as well as on the new quartic torsion scalar TG, which is the teleparallel equivalent of the Gauss–Bonnet term, is a novel theory, different from both f (T) and ones. We perform a detailed dynamical analysis of a spatially flat universe governed by the simplest non-trivial model of gravity which does not introduce a new mass scale. We find that the universe can result in dark-energy dominated, quintessence-like, cosmological-constant-like, or phantom-like solutions, according to the parameter choices. Additionally, it may result in a dark energy–dark matter scaling solution; thus it can alleviate the coincidence problem. Finally, the analysis ‘at infinity’ reveals that the universe may exhibit future, past, or intermediate singularities, depending on the parameters.

GIP integrators for Matrix Riccati Differential Equations

Matrix Riccati Differential Equations (MRDEs) are initial value problems of the form:X′=A21-XA11+A22X-XA12X,X(0)=X0.These equations arise frequently throughout applied mathematics, science, and engineering. It can happen that even when the Aij are smooth functions of t or constant, the solution X may have a singularity or even infinitely many singularities.This paper shows several classes of numerical algorithms, which we call GIP integrators, that can solve for X past its singularities. Furthermore, none of the algorithms require knowledge of the placement or even existence of singularities in X. Also, it is shown how embedded Runge–Kutta methods can be used to construct GIP integrators to not only approximate X past singularities but also provide for error estimation to allow efficient time stepping. Finally, several examples are shown to validate the theory.

Anisotropic Bianchi Type-II Viscous Fluid Models with Time-Dependent Gravitational and Cosmological Constants

This paper deals with totally anisotropic Bianchi type-II cosmological models filled with a bulk viscous fluid in the presence of time-varying gravitational and cosmological constants. Exact solutions of the field equations are obtained by applying a special law of variation for Hubble’s parameter which yields a constant value of the deceleration parameter. Two different physically viable models of the universe are presented in two types of cosmologies, one with power-law expansion and other one with exponential expansion. Cosmological model with power-law expansion has an initial big-bang type singularity at t = 0, whereas the model with exponential expansion has a singularity in infinite past. The physical and dynamical properties of the models are discussed.

How many singularities are near and how will they disrupt human history?

This paper reviews a large number of approaches that have been used for considering technologically driven profound societal change. We agree with Vinge's suggestion for naming events that are “capable of rupturing the fabric of human history” (or leading to profound societal changes) as a “singularity”. This is a useful terminology especially since a mathematically rigorous singularity seems impossible for technological and related societal change. The overview of previous work is done within the context of a broader look at the role of technological change within human history. The review shows that a wide variety of methods have been used and almost all point to singularities in the present century particularly in the middle of the century. The diversity of the methods is reassuring about the potential robustness of these predictions. However, the subjectivity of labeling events as singularities (even well studied past events) is a concern about all of the methods and thus one must carefully pause when relying in any way on these predictions. The general lack of empirical research in this area is also a concern. Quantitative considerations (by proponents and opponents) about past singularities or future singularities often confound two types of metrics. The first type is essentially related to diffusion of technologies (or bundles of technologies) where the logistic curve is empirically well established as the proper time dependence. The second type of metric is for technological capability where hyper-exponentials are empirically well established for their time dependence. In this paper, we consider two past singularities (arguably with important enough social change to qualify) in which the basic metric is alternatively of one type or another. The globalization occurring under Portuguese leadership of maritime empire building and naval technological progress is characterized by a metric describing diffusion. The revolution in time keeping, on the other hand, is characterized by a technological capability metric. For these two cases (and thus robust to the choice of metric type), we find that: • People undergoing profound technologically-driven societal change do not sense a singularity. • The societal impacts depend in complicated ways on human needs, institutional variables and other more uncertain factors and thus are particularly hard to project; • The societal impact is apparently not determined by the rate of progress on either type of metric or by projections to mathematical points with either kind of metric. This finding supports the existing concept that social change due to technology is a more holistic phenomenon than can be characterized by any technical metric. In the final section, we use these empirical findings as the basis for exploring the possibilities for and nature of future singularities. In this we speculate that the potential for a future strong singularity based upon computational capability does not appear particularly probable but that one may already be occurring and is not fully noticed by those (us) going through it. Other possible 21st century singularities (life extension and fossil fuel elimination are two examples considered) may also be already underway rather than waiting for the predicted mid-century changes.

COSMOLOGICAL HISTORY OF THE VACUUM

In this contribution, the author wishes to express his profound friendship and intellectual admiration towards Mario Castagnino. The playful tone of this contribution serves to underline some key elements of the longstanding scientific collaboration between the author and MC. These are used to construct a bootstrap-like cosmological history. The Universe appears to be its own cause, resulting from a prior stage which is nothing but itself. This leads to a self-consistent multiverse devoid of past or future singularities.

Quasi-optimal and robust a posteriori error estimates in $L^{\infty}(L^{2})$ for the approximation of Allen-Cahn equations past singularities

Quasi-optimal a posteriori error estimates in L ∞ (0,T; L 2 (Ω)) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.

No mass drop for mean curvature flow of mean convex hypersurfaces

A possible evolution of a compact hypersurface in Rn+1 by mean curvature past singularities is defined via the level set flow. In the case where the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. As a consequence, we obtain the fact that no mass drop can occur along such a flow. A further application of the techniques used above is to give a new variational formulation for mean curvature flow of mean convex hypersurfaces

Past singularities in phantom theories

FLRW models filled with just dark energy are shown to have a finite past, since causal geodesics cannot be extended beyond a certain proper time. It is shown that curvature measured along causal geodesics becomes infinity on travelling to the past, though curvature scalars tend to zero. Furthermore the time measured by free-falling observers from coincidence time to Big Rip is shown to be as short as wished by increasing their linear momentum.

Does the Big Rip survive quantization?

It is known that certain quantum cosmological models present quantum behavior for large scale factors. Since quantization can suppress past singularities, it is natural to inquire whether quantum effects can prevent future singularities. To this end, a Friedmann–Robertson–Walker quantum cosmological model dominated by a phantom energy fluid is investigated. The classical model displays accelerated expansion ending in a Big Rip. The quantization is performed in three different ways, which turn out to lead to the same result, namely there is a possibility that quantum gravitational effects could not remove the Big Rip.