Divergence Of Curvature Research Articles

Geometric thermodynamics of collapse of gels.

Stimulus-induced volumetric phase transition in gels may be potentially exploited for various bioengineering and mechanical engineering applications. Since the discovery of the phenomenon in the 1970s, extensive experimental research has helped understand the phase transition and related critical phenomena. However, little insight is available on the evolving microstructure. In this article, we aim at unravelling certain geometric aspects of the micromechanics underlying discontinuous phase transition in polyacrylamide gels. Towards this, we use geometric thermodynamics and a Landau-Ginzburg type free energy functional involving a squared gradient, in conjunction with Flory-Huggins theory. We specifically exploit Ruppeiner's approach of Riemannian geometry-enriched thermodynamic fluctuation theory, which was previously employed to investigate phase transitions in van der Waals fluids and black holes. The framework equips us with a scalar curvature that is typically indicative of certain aspects of the microstructure during phase transition. Since previous studies have indicated that curvature divergence relates to correlation length divergence, we infer that the gel possesses a heterogeneous microstructure during phase transition, i.e., at critical points. Curvature also provides an insight into the universality class of phase transition and the nature of polymer-polymer interactions.

Thermodynamic geometry of charged rotating black holes in Einsteinian cubic gravity

We study the thermodynamic geometry of two types of asymptotically anti-de Sitter black holes in four-dimensional Einsteinian cubic gravity, including the uncharged rotating and charged non-rotating black holes, applying the Ruppeiner thermodynamic geometry method. The divergence of the scalar curvature of the Ruppeiner metric ([Formula: see text]) is found to be related to the vanishing of the heat capacity. The stability of the solutions is shown to be affected by the Einsteinian cubic gravity coupling constant [Formula: see text]. In the unstable phase of the rotating black hole, the [Formula: see text] appeared to possess additional diverging points, where the number of these points and the behavior of [Formula: see text] are controlled by the angular momentum parameter. Also, for the charged non-rotating black hole case, we show that depending on the values of [Formula: see text], the solution may enjoy two types of phase transition and [Formula: see text] behaviors. The characteristic behavior of [Formula: see text] of these two types of black holes enabled us to recognize the types of interactions between microscopic degrees of freedom.

On the moduli space curvature at infinity

We analyse the scalar curvature of the vector multiplet moduli space MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document} of type IIA string theory compactified on a Calabi-Yau manifold X. While the volume of MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document} is known to be finite, cases have been found where the scalar curvature diverges positively along trajectories of infinite distance. We classify the asymptotic behaviour of the scalar curvature for all large volume limits within MXVM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_X^{\ extrm{VM}} $$\\end{document}, for any choice of X, and provide the source of the divergence both in geometric and physical terms. Geometrically, there are effective divisors whose volumes do not vary along the limit. Physically, the EFT subsector associated to such divisors is decoupled from gravity along the limit, and defines a rigid N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 field theory with a non-vanishing moduli space curvature Rrigid. We propose that the relation between scalar curvature divergences and field theories decoupled from gravity is a common trait of moduli spaces compatible with quantum gravity.

Hypersonic shock wave and turbulent boundary layer interaction in a sharp cone/flare model

A direct numerical simulation of hypersonic Shock wave and Turbulent Boundary Layer Interaction(STBLI) at Mach 6.0 on a sharp 7° half-angle circular cone/flare configuration at zero angle of attack is performed. The flare angle is 34° and the momentum thickness Reynolds number based on the incoming turbulent boundary layer on the sharp circular cone is Reθ = 2506. It is found that the mean flow is separated and the separation bubble occurring near the corner exhibits unsteadiness. The Reynolds analogy factor changes dramatically across the interaction, and varies between 1.06 and 1.27 in the downstream region, while the QP85 scaling factor has a nearly constant value of 0.5 across the interaction. The evolution of the reattached boundary layer is characterized in terms of the mean profiles, the Reynolds stress components, the anisotropy tensor and the turbulence kinetic energy. It is argued that the recovery is incomplete and the near-wall asymptotic behavior does not occur for the hypersonic interaction. In addition, mean skin friction decomposition in an axisymmetric turbulent boundary layer is carried out for the first time. Downstream of the interaction, the contributions of transverse curvature and body divergence are negligible, whereas the positive contribution associated with the turbulence kinetic energy production and the negative spatial-growth contribution are dominant. Based on scale decomposition, the positive contribution is further divided into terms with different spanwise length scales. The negative contribution is analyzed by comparing the convective term, the streamwise-heterogeneity term and the pressure gradient term.

Holographic insulator/superconductor phase transition via matching method and thermodynamic geometry approach

This paper employs two analytical techniques, namely, the matching method and the thermodynamic geometry approach to investigate the insulator/superconductor phase transition holographically in the presence of a five-dimensional AdS soliton background. This is a sequel of our earlier work [D. Parai, S. Gangopadhyay and D. Ghorai, Ann. Phys. 403, 59 (2019)] where we carried out a similar analysis using the Sturm–Liouville eigenvalue approach. We have first used the matching method to obtain the critical chemical potential, condensation operator and the charge density. Next, we investigate the free energy and thermodynamic geometry of this model. This investigation of the thermodynamic geometry leads to the critical chemical potential of the system from the condition of the divergence of the scalar curvature. We have then compared the value of the critical chemical potential [Formula: see text] obtained from these two different methods. The findings are then compared with the numerical and analytical results obtained from the Sturm–Liouville technique.

What actually happens when you approach a gravitational singularity?

Roger Penrose’s 2020 Nobel Prize in Physics recognizes that his identification of the concepts of “gravitational singularity” and an “incomplete, inextendible, null geodesic” is physically very important. The existence of an incomplete, inextendible, null geodesic does not say much, however, if anything, about curvature divergence, nor is it a helpful definition for performing actual calculations. Physicists have long sought for a coordinate independent method of defining where a singularity is located, given an incomplete, inextendible, null geodesic, that also allows for standard analytic techniques to be implemented. In this essay, we present a solution to this issue. It is now possible to give a concrete relationship between an incomplete, inextendible, null geodesic and a gravitational singularity, and to study any possible curvature divergence using standard techniques.

Generating rotating spacetime in Ricci-based gravity: naked singularity as a black hole mimicker

Motivated by the lack of rotating solutions sourced by matter in General Relativity as well as in modified gravity theories, we extend a recently discovered exact rotating solution of the minimal Einstein-scalar theory to its counterpart in Eddington-inspired Born-Infeld gravity coupled to a Born-Infeld scalar field. This is accomplished with the implementation of a well-developed mapping between solutions of Ricci-Based Palatini theories of gravity and General Relativity. The new solution is parametrized by the scalar charge and the Born-Infeld coupling constant apart from the mass and spin of the compact object. Compared to the spacetime prior to the mapping, we find that the high-energy modifications at the Born-Infeld scale are able to suppress but not remove the curvature divergence of the original naked null singularity. Depending on the sign of the Born-Infeld coupling constant, these modifications may even give rise to an additional timelike singularity exterior to the null one. In spite of that, both of the naked singularities before and after the mapping are capable of casting shadows, and as a consequence of the mapping relation, their shadows turn out to be identical as seen by a distant observer on the equatorial plane.Even though the scalar field induces a peculiar oblateness to the appearance of the shadow with its left and right endpoints held fixed, the closedness condition for the shadow contour sets a small upper bound on the absolute value of the scalar charge, which leads to observational features of the shadow closely resembling those of a Kerr black hole.

Axially Symmetric Type N Space-Time with Causality Violating Curves and the von Zeipel Cylinder

An axially symmetric nonvacuum solution of the Einstein field equations, regular everywhere and free from curvature divergence is presented. The matter-energy content is a the pure radiation field satisfying the energy conditions, and the metric is of type N in the Petrov classification scheme. The space-time develops circular closed timelike curves everywhere outside a finite region of space i.e., beyond a null curve. Furthermore, the physical interpretation of the solution based on the study of the equations of geodesic deviation is presented. Finally, the von Zeipel cylinders with respect to the Zero Angular Momentum Observers (ZAMOs) is discussed. In addition, circular null and timelike geodesic of pace-time are also presented.

Scalar quantum particle in (1+2)-dimensions Gurses space–time and the energy–momentum distributions

In this article, we investigate the relativistic quantum dynamics of a scalar particle without any potentials in (1+2)-dimensions Gurses space–time (Gurses, 1994), a perfect fluid stationary solution of the Einstein’s field equations. This Gurses space–time in 3D is free-from curvature divergence and admits closed time-like curves everywhere outside a finite region space. Finally, the energy–momentum distrbutions using the known complexes of Landau–Lifshitz’s, Einstein and Papapetrou, are evaluated and the result is quite interesting.

Axisymmetric Pure Radiation Space–Time with Causality-Violating Geodesics

We present a stationary axisymmetric space–time admitting circular closed timelike geodesics everywhere within a finite region of space. The space–time is free from curvature divergence and is locally isometric to a nonvacuum pp-wave space–time. The matter–energy content is a pure radiation field and satisfies the null energy condition (NEC), and the metric is of type N in the Petrov classification scheme. Finally, we demonstrate the existence of timelike and null circular geodesic paths for this metric.

Local divergence and curvature divergence in first order optics

The far-field divergence of a light beam propagating through a first order optical system is presented as a square root of the sum of the squares of the local divergence and the curvature divergence. The local divergence is defined as the ratio of the beam parameter product to the beam width whilst the curvature divergence is a ratio of the space-angular moment also to the beam width. It is established that the beam’s focusing parameter can be defined as a ratio of the local divergence to the curvature divergence. The relationships between the two divergences and other second moment-based beam parameters are presented. Their various mathematical properties are presented such as their evolution through first order systems. The efficacy of the model in the analysis of high power continuous wave laser-based welding systems is briefly discussed.

A type N radiation field solution with Lambda

An anti-de Sitter background four-dimensional type N solution of the Einstein’s field equations, is presented. The matter-energy content pure radiation field satisfies the null energy condition (NEC), and the metric is free-from curvature divergence. In addition, the metric admits a non-expanding, non-twisting and shear-free geodesic null congruence which is not covariantly constant. The space-time admits closed time-like curves which appear after a certain instant of time in a causally well-behaved manner. Finally, the physical interpretation of the solution, based on the study of the equation of the geodesics deviation, is analyzed.

A stationary cylindrically symmetric spacetime which admits CTCs and its physical interpretation

Here we present a stationary cylindrically symmetric spacetime which is free from curvature divergence, and satisfy pure radiation field as matter–energy content with positive constant energy density. The metric admits circular closed timelike curves (CTCs) everywhere outside a finite region and the stability of these curves under small linear perturbations is studied, and found to be linearly stable. The metric is of type N in the Petrov classification scheme, and it has a shearfree geodesic null congruence. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It is demonstrated that, this solution can be understood as exact transverse gravitational waves with amplitudes Ψ4.

Type N Einstein space Time Machine spacetime

We present an Einstein space axially symmetry spacetime admitting closed timelike curves (CTCs) which appear after a certain instant of time, i.e., a time machine spacetime. The spacetime is a four-dimensional generalization of flat Misner space in curved spacetime, free-from curvature divergences and belongs to type N in the Petrov classification. The spacetime admits a non-expanding, non-twisting, and shear-free null geodesic congruence.

Non-linear effects on the holographic free energy and thermodynamic geometry

We have analytically investigated the effects of non-linearity on the free energy and thermodynamic geometry of holographic superconductors in -dimensions. The non-linear effect is introduced by considering the coupling of the massive charged scalar field with Born-Infeld electrodynamics. We then calculate the relation between critical temperature and charge density by two different methods, namely, the matching method and the divergence of the scalar curvature which is obtained by investigating the thermodynamic geometry of the model. Both the results indicate a decrease in the critical temperature with increase in the Born-Infeld parameter b.

PWE-Based Radar Equation to Predict Backscattering of Millimeter-Wave in a Sand and Dust Storm

A parabolic wave equation-based radar equation is proposed to compute the backscattered power of millimeter-wave in a sand and dust storm, in which the height profiles of particle-size distribution and total number density are accounted for. The conventional radar equation is also improved by incorporating the effects of earth curvature and beam divergence. The improvement of accuracy by using these two radar equations, especially when the specific attenuation is a function of height, is verified by simulations.

Toward robust prediction of crossflow-wave instability in hypersonic boundary layers

Abstract Traveling and stationary crossflow-wave instabilities in a laminar Mach 6 boundary layer are investigated on a 38.1% scale model of the Fifth Hypersonic International Flight Research Experiment (HIFiRE-5) elliptic cone at zero angle of attack and yaw. The Langley Stability and Transition Analysis Code (LASTRAC) was used to analyze the crossflow dominated boundary layer in the mid-span region near the downstream end of the model. Disturbance growth rates, wave angles, and phase speeds are computed with LASTRAC using quasi-parallel Linear Stability Theory (LST), Linear Parabolized StabilityEquations (LPSE), and two-plane or surface marching LPSE (2pLPSE). The predicted wave angles and phase speeds are validated using experimental data, and are found to be in better agreement than previous computations. Further numerical analysis is conducted using the Spatial BiGlobal technique (SBG), which simultaneously accounts for wall-normal and spanwise gradients in the mean boundary layer at a particular axial station. For the first time in the literature, a comparison is made between crossflow wave growth rates computed using LST, LPSE, and 2pLPSE and those computed using SBG, accounting for curvature and geometric divergence of the elliptic cone. The agreement between LST, LPSE, and SBG is fair at best, but excellent agreement is realized between 2pLPSE and SBG. This result constitutes a co-verification of the LASTRAC and SBG stability codes, and provides evidence that 2pLPSE accurately models the physics of the traveling-crossflow instability.

Geodesic completeness and the lack of strong singularities in effective loop quantum Kantowski–Sachs spacetime

Resolution of singularities in the Kantowski–Sachs model due to non-perturbative quantum gravity effects is investigated. Using the effective spacetime description for the improved dynamics version of loop quantum Kantowski–Sachs spacetimes, we show that even though expansion and shear scalars are universally bounded, there can exist events where curvature invariants can diverge. However, such events can occur only for very exotic equations of state when pressure or derivatives of energy density with respect to triads become infinite at a finite energy density. In all other cases curvature invariants are proved to remain finite for any evolution in finite proper time. We find the novel result that all strong singularities are resolved for arbitrary matter. Weak singularities pertaining to above potential curvature divergence events can exist. The effective spacetime is found to be geodesically complete for particle and null geodesics in finite time evolution. Our results add to a growing evidence for generic resolution of strong singularities using effective dynamics in loop quantum cosmology by generalizing earlier results on isotropic and Bianchi-I spacetimes.

Black hole solutions in functional extensions of Born-Infeld gravity

We consider electrovacuum black hole spacetimes in classical extensions of Eddington-inspired Born-Infeld gravity. By rewriting Born-Infeld action as the square root of the determinant of a matrix $\hat{\Omega}$, we consider the family of models $f (|\hat{\Omega}|)$, and study black hole solutions for a power-law family of models labelled by a simple parameter. We show how the innermost structure of the corresponding black holes is modified as compared to their General Relativity counterparts, discussing in which cases a wormhole structure replaces the point-like singularity. We go forward to argue that in such cases a geodesically complete and thus non-singular spacetime is present, despite the existence of curvature divergences at the wormhole throat.

Impact of curvature divergences on physical observers in a wormhole space–time with horizons

The impact of curvature divergences on physical observers in a black hole space–time, which, nonetheless, is geodesically complete is investigated. This space–time is an exact solution of certain extensions of general relativity coupled to Maxwell’s electrodynamics and, roughly speaking, consists of two Reissner–Nordström (or Schwarzschild or Minkowski) geometries connected by a spherical wormhole near the center. We find that, despite the existence of infinite tidal forces, causal contact is never lost among the elements making up the observer. This suggests that curvature divergences may not be as pathological as traditionally thought.