Narrow Results

Recently, there has been an interest in exploring black holes that are regular in the sense that the central curvature singularity is avoided. Here, we depict a method to obtain a regular black hole (RBH) spacetime from the unhindered gravitational collapse beginning with regular initial data of a spherically symmetric perfect fluid. In other words, we obtain the equilibrium (static) spacetime $(\mathcal{ℳ},\tilde{g})$ as a limiting case of the time-evolving (nonstationary) spacetime $(\mathcal{ℳ},g)$. In the spirit of P. S. Joshi, D. Malafarina and R. Narayan, Class. Quantum Grav. 31 (2014) 015002, our description of gravitational collapse is implicit in nature in the sense that we do not describe the data at each time-slice. Rather, we impose a condition in terms of geometric and matter variables for the collapse to have an end-state that is devoid of incomplete geodesics but admits a marginally trapped surface (MTS). The admission of MTS causally disconnects two mutually exclusive regions ${\widehat{\mathcal{ℳ}}}_1$ and ${\widehat{\mathcal{ℳ}}}_2\subset \mathcal{ℳ}$ in the sense that $\forall p\in {\widehat{\mathcal{ℳ}}}_2$, the causal past of p does not intersect ${\widehat{\mathcal{ℳ}}}_1$. While the classic Oppenheimer–Snyder collapse model necessarily produces a black hole with a Schwarzschild singularity at the center, we show here that there are classes of regular initial conditions for which the collapse gives rise to a RBH.

In this research paper, we discuss the energy–momentum-squared gravity model ${ℱ}({ℛ},{{𝒯}}^2)$ coupled with perfect fluid. We obtain the equation of state for the perfect fluid in ${ℱ}({ℛ},{{𝒯}}^2)$-gravity model. Furthermore, we deal with the energy–momentum-squared gravity model ${ℱ}({ℛ},{{𝒯}}^2)$ coupled with perfect fluid, which admits the Ricci solitons with a conformal vector field. We provide a clue in this series to determine the density and pressure in the radiation and phantom barrier periods, respectively. In addition, we investigate the different energy conditions, black holes and singularity conditions for perfect fluid attached to ${ℱ}({ℛ},{{𝒯}}^2)$-gravity in terms of Ricci soliton. Finally, we derive the Schrödinger equation for energy–momentum-squared gravity model ${ℱ}({ℛ},{{𝒯}}^2)$ coupled with perfect fluid.

In practice, hypersurfaces are defined via functions of coordinates. As a result, causal classification of hypersurfaces bounding charts can be frame-dependent. We prove that if there exists any frame such that, at some (limiting) point, the metric is diagonal and a basis vector lies on a null hypersurface (defined by that frame) then the metric is degenerate and $C^0$-inextendible there. We apply this to Schwarzschild spacetimes. We show that volume-forms of all 1-submanifolds vanish in the limit of approaching the event horizon, then prove that any point of a 1-submanifold at which volume-forms would vanish (by continuity) cannot be contained in the 1-submanifold, or the manifold itself. This yields a geometric obstruction to the $C^0$-extendibility of curves and the manifold from the black hole exterior to the event horizon. The Ingoing Eddington–Finkelstein coordinate map is shown to fail to be a diffeomorphism at the event horizon, and thus cannot be used to extend the exterior. We discuss the implications for gravitational collapse spacetimes, and show that the geometry of curves inherited from the manifold precludes worldlines from ‘leaving the manifold’, providing new physical intuition for their incompleteness. Finally, the framework is shown not to further constrain extendibility of FLRW spacetimes.

This paper will prove the uniqueness theorem for 3-layered complex-valued neural networks where the threshold parameters of the hidden neurons can take non-zeros. That is, if a 3-layered complex-valued neural network is irreducible, the 3-layered complex-valued neural network that approximates a given complex-valued function is uniquely determined up to a finite group on the transformations of the learnable parameters of the complex-valued neural network.

Let X be a reduced pure n-dimensional complex analytic set in ℂ^N with an isolated singularity at 0. We obtain some L²-existence and regularity results for the -operator on a deleted neighborhood of the singular point 0 in X.

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

This paper investigates the existence and uniqueness of solutions for singular second-order boundary value problem on time scales by using mixed monotone method. The theorems obtained are very general and complement the previous known results. When the time scale 𝕋 is chosen as ℝ or ℤ, the problem will be the corresponding continuous or discrete boundary value problem.

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ². As a sample of our results, we prove that N is biholomorphic to ℂ² if H²(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form where λ₁/λ₂ ∈ ℚ₊. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R³ are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss–Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.

Let ℳ_ℙ³(c₁,c₂) be the moduli space of stable rank-2 vector bundles on ℙ³ with Chern classes c₁, c₂. We prove the following results: (1) Let k, β, γ be three integers such that k > 0, 0 ≤ β < γ, γ ≥ 2, kγ - (k + 1)β > 0; then the moduli space ℳ_ℙ³(0, kγ² - (k + 1)β²) is singular (the case k = 2, β = 0 was previously proved by M. Maggesi).

(2) Let k, β, γ be three integers, with β and γ odd, such that k > 0, 0 < β < γ, γ ≥ 5, kγ - (k + 1)β + 1 > 0; then the moduli space ℳ_ℙ³(-1,k(γ/2)² - (k + 1)(β/2)²) + 1/4) is singular.

In particular ℳ_ℙ³(0,5), ℳ_ℙ³(-1,6) are singular.

A de-Sitter gauge theory of the gravitational field is developed using a spherical symmetric Minkowski space–time as base manifold. The gravitational field is described by gauge potentials and the mathematical structure of the underlying space–time is not affected by physical events. The field equations are written and their solutions without singularities are obtained by imposing some constraints on the invariants of the model. An example of such a solution is given and its dependence on the cosmological constant is studied. A comparison with results obtained in General Relativity theory is also presented.

We look at some dynamic geometries produced by scalar fields with both the "right" and the "wrong" signs of the kinetic energy. We start with anisotropic homogeneous universes with closed, open and flat spatial sections. A non-singular solution to the Einstein field equations representing an open anisotropic universe with the ghost field is found. This universe starts collapsing from t→-∞ and then expands to t→∞ without encountering singularities on its way. We further generalize these solutions to those describing inhomogeneous evolution of the ghost fields. Some interesting solutions with the plane symmetry are discussed. These have a property that the same line element solves the Einstein field equations in two mirror regions |t|≥z and |t|≤z, but in one region the solution has the right and in the other, the wrong sign of the kinetic energy. We argue, however, that a physical observer cannot reach the mirror region in a finite proper time. Self-similar collapse/expansion of these fields are also briefly discussed.

We study conformal field theory on two-dimensional orbifolds and show this to be an effective way to analyze physical effects of geometric singularities with angular deficits. They are closely related to boundaries and cross caps. Representative classes of singularities can be described exactly using generalizations of boundary states. From this we compute correlation functions and derive the spectra of excitations localized at the singularities.

We explore the gravitational collapse of a spherically symmetric dust cloud in the Einstein–Gauss–Bonnet gravity without a cosmological constant, and obtain three families of LTB-like solutions. It is shown that the Gauss–Bonnet term has a profound influence on the nature of singularities, and the global structure of spacetime changes drastically from the analogous general relativistic case. Interestingly, the formation of a naked, massive and uncentral singularity, allowed in five-dimensional spacetime, is forbidden if D≥6. Moreover, such singularity is gravitational strong and a serious counterexample to CCH.

Gravitational collapse of a spherically symmetric homogeneous perfect barotropic fluid with linear as well as polytropic-type Equation of State (EoS) has been investigated in the framework of a linear model of $f(R,T)$ gravity. This modified gravity has the potential to explain the observed cosmic acceleration. The calculations have been done taking the transformed time coordinate $t\to \sqrt{\frac{{\rho }_0}{3}}t$, where ${\rho }_0$ is the initial density of the fluid. For linear EoS $p=\omega \rho$, the condition for being a true singularity, along with sufficient condition for the formation of apparent horizon covering the singularity has been derived. For a polytrope having the EoS $p=K{\rho }^{1+\frac{1}{n}}$, the scale factor $(A)$ as a function of fluid density $(\rho )$ has been obtained which is then used to study the dynamics of the fluid. Role of the polytropic index $(n)$ and the constant of proportionality $(K)$ in the dynamics of the fluid is also studied. A new type of exotic matter field having varied dependence of scale factor on the density, and having the potential to give rise to bouncing cosmology, provided it is the dominating fluid in the universe, is obtained in this domain and is investigated. Energy conditions are discussed.

The Hawking–Penrose theorem is not covariant under field redefinitions. Should the invariance under such transformations be a true principle in nature, spacetime singularities become dubious objects. We here review the concept of covariant singularities, that is, singularities that are invariant under both spacetime diffeomorphisms and field redefinitions.

Power Maxwell nonlinear electrodynamics has become one of the well-known nonlinear models. Initially, it was proposed to make Maxwell’s theory conformally invariant in higher dimensions. Therefore, it was not expected from the model to remove the electric field’s singularity at the point charges’ location. Such singularity removal was the core of the very early model known as the Born–Infeld model. In the literature, this point has been highlighted very recently in [B. E. Panah, EPL  134, 20005 (2021)] where a variable power Maxwell model has been introduced to remove the field’s singularity. Here, in this short study, we shed more light on the issue by giving a physically acceptable mechanism for removing the singularity of the point charge at its location by applying the power Maxwell’s nonlinear electrodynamics.

We give a method for constructing generalized null Cartan helices, which may have singularities, by using regular space-like plane curves and smooth functions in Minkowski 3-space. We also study the principle normal worldsheet, which is deeply related to generalized null Cartan helices from the viewpoint of singularity theory.

Modeling the worldsheets along the trajectory (a string) of a point particle (a 0-brane) as two timelike surfaces in 3-subspace of Minkowskian spacetime, respectively, the work aims at analyzing the topological structures of these two worldsheets along the trajectory of the point particle from the view point of singularity theory. Different from the regular curves, the traveling trajectory (modeled as framed timelike curve) of the particle is allowed to be singular, two worldsheets are generated by the traveling trajectories of the particle. As applications of singularity theory, we classify the singularities of these two worldsheets along the traveling trajectory of the particle. Using the approach of the unfolding theory in singularity theory, we find two new geometric invariants which are useful for characterizing the local topological structures of singularities of these two worldsheets along the particle. It is revealed that there exist cuspidal edge type and swallowtail type of singularities for these two worldsheets under the appropriate conditions of geometric invariants. Meanwhile, it is also pointed out that the types of singularities of these two worldsheets have a close relationships to the order of contacts between these two worldsheets and two timelike planes, respectively. Finally, some examples are presented to interpret our theoretical results.

AdS dynamics for massive scalar field is studied both by solving exactly the equation of motion and by constructing bulk-boundary propagator. A Robertson–Walker-like metric is deduced from the familiar SO(2, n) invariant metric. The metric allows us to present a timelike Killing vector, which is not only invariant under spacelike transformations but also invariant under the isometric transformations of SO(2, n) in certain sense. A horizon appears in this coordinate system. Singularities of field variables at boundary are demonstrated explicitly. It is shown that there is a one-to-one correspondence among the exact solutions and the bulk fields obtained by using the bulk-boundary propagator.