Scalar Curvature Research Articles

Massive propagating modes of torsion

The dynamics of the torsion field is analyzed in the framework of the Covariant Canonical Gauge Theory of Gravity (CCGG), a De Donder–Weyl Hamiltonian formulation of gauge gravity. The action is quadratic in both, the torsion and the Riemann–Cartan tensor. Since the latter adds the derivative of torsion to the equations of motion, torsion is no longer identical to spin density, as in the Einstein–Cartan theory, but an additional propagating degree of freedom. As torsion turns out to be totally anti-symmetric, it can be parametrised via a single axial vector. It is shown in this paper that, in the weak torsion limit, the axial vector obeys a wave equation with an effective mass term which is partially dependent on the scalar curvature. The source of torsion is thereby given by the fermion axial current which is the net fermionic spin density of the system. Possible measurable effects and approaches to experimental analysis are addressed. For example, neutron star mergers could act as a dipoles or quadrupoles for torsional radiation, and an analysis of radiation of pulsars could lead to a detection of torsion wave background radiation.

Scalar curvature along Ebin geodesics

Abstract Let 𝑀 be a smooth, compact manifold and let N μ \mathcal{N}_{\mu} denote the set of Riemannian metrics on 𝑀 with smooth volume density 𝜇. For g 0 ∈ N μ g_{0}\in{\mathcal{N}_{\mu}} , we show that if dim ( M ) ≥ 5 \dim(M)\geq 5 , then there exists an open and dense subset Y g 0 ⊂ T g 0 ⁢ N μ {\mathcal{Y}_{g_{0}}}\subset T_{g_{0}}{\mathcal{N}_{\mu}} (in the C ∞ C^{\infty} topology) so that, for each h ∈ Y g 0 h\in{\mathcal{Y}_{g_{0}}} , the ( N μ , L 2 ) ({\mathcal{N}_{\mu}},L^{2}) Ebin geodesic γ h ⁢ ( t ) \gamma_{h}(t) with γ h ⁢ ( 0 ) = g 0 \gamma_{h}(0)=g_{0} and γ h ′ ⁢ ( 0 ) = h \gamma_{h}^{\prime}(0)=h satisfies lim t → ∞ R ⁢ ( γ h ⁢ ( t ) ) = − ∞ \lim_{t\to\infty}R(\gamma_{h}(t))=-\infty , uniformly, where R : N μ → C ∞ ⁢ ( M ) R\colon\mathcal{N}_{\mu}\to C^{\infty}(M) is the scalar curvature.

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three

abstract: We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in $\mathbb{R}^{3}$. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a ``Lagrangian'' graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Inflation and reheating in quadratic metric-affine gravity with derivative couplings

Within the framework of metric-affine theories of gravity, where both the metric and connection are treated as independent variables, we consider actions quadratic in the Ricci scalar curvature coupled non-minimally to a scalar field through derivative couplings. Our analysis delves into the inflationary predictions, revealing their consistency with the latest observational constraints across a wide range of parameters. This compatibility permits adjustments such as an increase in the spectral index and a reduction in the tensor-to-scalar ratio. While we do not propose a specific reheating mechanism, our analysis demonstrates that within the quadratic model of inflation, the maximum reheating temperature can reach ∼ 3 × 1015 GeV.

The Einstein-harmonic equations and constant scalar curvature Kähler metrics

Let (M,J) be a compact complex surface. In his paper [9], LeBrun showed that J-invariant solutions of the Einstein-Maxwell equations correspond to conformally Kähler constant scalar curvature metrics whose Ricci tensors are J-invariant. In the present paper, we prove that constant scalar curvature Kähler manifolds of even complex dimension give solutions of Einstein equations with matter fields which we call the Einstein-harmonic equations.

Semiclassical thermodynamic geometry.

In this work the thermodynamic geometry (TG) of semiclassical fluids is analyzed. We present results for two models. The first one is a semiclassical hard-sphere (SCHS) fluid whose Helmholtz free energy is obtained from path-integral Monte Carlo simulations. It is found that, due to quantum contributions in the thermodynamic potential, the anomaly found in TG for the classical hard-sphere fluid related to the sign of the scalar curvature is now avoided in a considerable region of the thermodynamic space. The second model is a semiclassical square-well fluid, described by a SCHS repulsive interaction coupled with a classical attractive square-well contribution. The behavior of the semiclassical curvature scalar as a function of the thermal de Broglie wavelength λ_{B} is analyzed for several attractive-potential ranges. A description of the semiclassical R Widom lines, defined by the maxima of the curvature scalar, is also obtained and results are compared with the corresponding classical systems for different square-well ranges.

Implications of Some Mass-Capacity Inequalities

Applying a family of mass-capacity related inequalities proved in Miao (Peking Math J 2023, https://doi.org/10.1007/s42543-023-00071-7), we obtain sufficient conditions that imply the nonnegativity as well as positive lower bounds of the mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularity.

On Curvature Pinching for Submanifolds with Parallel Normalized Mean Curvature Vector

In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature R≥c, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and p=2.

Conformally flat almost cosymplectic 3-manifolds

In this paper, we consider an almost cosymplectic 3-manifold M such that its scalar curvature is invariant along the Reeb vector field. We prove that if the Reeb vector field of M is harmonic, then M is conformally flat if and only if it is locally isometric to the product ℝ × N 2(c), where N 2(c) is a Kähler surface of constant sectional curvature c. Almost cosymplectic 3-manifolds on which the Reeb vector field satisfies the h-a condition together with conformal flatness are also classified.

The instability of the inner horizon of the quantum-corrected black hole

We analyse the stability of the inner horizon of the quantum-corrected black hole which is proposed in loop quantum gravity as the exterior of the quantum Oppenheimer–Snyder and Swiss Cheese models. It is shown that the flux and energy density of a test scalar field measured by free-falling observers are both divergent near the Cauchy horizon. By considering the generalized Dray–’t Hooft–Redmond relation which is independent of the field equation, we find that the mass inflation always happens and the scalar curvature and Kretschmann scalar are also divergent on the inner horizon. These suggest that the inner horizon is unstable and will probably turn into a null singularity.The results support the strong cosmic censorship hypothesis.However, this also implies that the quantum corrected model may not be the definitive endpoint as a regular black hole.Besides, it further proposes that it may be challenging to observe the astronomical phenomenon which depends on the existence of the inner horizon of the black hole

Reconstruction schemes of scalar field models for the Power Law Entropy Corrected Holographic Dark Energy model with Ricci scalar cut-off

In this work, we examine the cosmological characteristics of the Power Law Entropy Corrected Holographic Dark Energy (PLECHDE) model with infrared (IR) cut-off, which is determined by the curvature parameter k, the time derivative of H, and the average radius of the Ricci scalar curvature R, which varies with the Hubble parameter H squared. We obtain the deceleration parameter q and the Equation of State (EoS) parameter of Dark Energy (DE) ωD. Additionally, we derive the Hubble parameter H and the scale factor a expressions as functions of the cosmic time t. Additionally, we examine the limiting scenario that pertains to a flat Dark Dominated Universe. Furthermore, we establish a correspondence between the DE model considered and some scalar fields, in particular the Generalized Chaplygin Gas, the Modified Chaplygin Gas, the Modified Variable Chaplygin Gas, the New Modified Chaplygin Gas, the Viscous Generalized Chaplygin Gas, the Dirac–Born–Infeld, the Yang–Mills, and the Non Linear Electrodynamics scalar field models.

Anisotropic generalized polytropic spheres: Regular 3D black holes

We model gravitating relativistic 3D spheres composed of an anisotropic fluid in which the radial and transverse components of the pressure correspond to the vacuum energy and a generalized polytropic equation-of-state, respectively. By using the generalized Tolman–Oppenheimer–Volkoff (TOV) equation, and solving the complete system of equations for these anisotropic generalized polytropic spheres, for a given range of model parameters, we find three novel classes of asymptotically AdS black hole solutions with regular core. We show the regularity of the solutions using curvature scalars and the formalism of geodesic completeness. Then, using the eigenvalues of the Riemann curvature tensor, we consider the effects of repulsive gravity in the three static 3D regular black holes, concluding that their regular behavior can be explained as due to the presence of repulsive gravity near the center of the objects. Finally, we study the stability of the regular black holes under the flow of the energy through the Cauchy horizon.

Scalar and mean curvature comparison via the Dirac operator

Scalar and mean curvature comparison via the Dirac operator

Clairaut Conformal Submersions

The aim of this paper is to introduce Clairaut conformal submersions between Riemannian manifolds. First, we find necessary and sufficient conditions for conformal submersions to be Clairaut conformal submersions. In particular, we obtain Clairaut relation for geodesics on the total manifolds of conformal submersions, and prove that Clairaut conformal submersions have constant dilation along their fibers, which are totally umbilical, with mean curvature being gradient of a function. Further, we calculate the scalar and Ricci curvatures of the vertical distributions of the total manifolds. Moreover, we find a necessary and sufficient condition for Clairaut conformal submersions to be harmonic. For a Clairaut conformal submersion we find conformal changes of the metric on its domain or image, that give a Clairaut Riemannian submersion, a Clairaut conformal submersion with totally geodesic fibers, or a harmonic Clairaut submersion. Finally, we give two non-trivial examples of Clairaut conformal submersions to illustrate the theory and present a local model of every Clairaut conformal submersion with integrable horizontal distribution.

Product Inequalities for $\mathbb T^\rtimes$-Stabilized Scalar Curvature

We study metric invariants of Riemannian manifolds $X$ defined via the $\mathbb T^\rtimes$-stabilized scalar curvatures of manifolds $Y$ mapped to $X$ and prove in some cases additivity of these invariants under Riemannian products $X_1\times X_2$.

PeV-Scale SUSY and Cosmic Strings from F-Term Hybrid Inflation

We consider F-term hybrid inflation (FHI) and SUSY breaking in the context of a B−L extension of the MSSM that largely respects a global U(1)R symmetry. The hidden sector Kaehler manifold enjoys an enhanced SU(1,1)/U(1) symmetry, with the scalar curvature determined by the achievement of a SUSY-breaking de Sitter vacuum without undesirable tuning. FHI turns out to be consistent with the data, provided that the magnitude of the emergent soft tadpole term is confined to the range (1.2–100) TeV, and it is accompanied by the production of B−L cosmic strings. If these are metastable, they are consistent with the present observations from PTA experiments on the stochastic background of gravitational waves with dimensionless tension Gμcs≃(1−9.2)·10−8. The μ parameter of the MSSM arises by appropriately adapting the Giudice–Masiero mechanism and facilitates the out-of-equilibrium decay of the R saxion at a reheat temperature lower than about 71 GeV. Due to the prolonged matter-dominated era, the gravitational wave signal is suppressed at high frequencies. The SUSY mass scale turns out to lie in the PeV region.

Algebraic classification of 2+1 geometries: a new approach

We present a convenient method of algebraic classification of 2+1 spacetimes into the types I, II, D, III, N and O, without using any field equations. It is based on the 2+1 analogue of the Newman–Penrose curvature scalars ΨA of distinct boost weights, which are specific projections of the Cotton tensor onto a suitable null triad. The algebraic types are then simply determined by the gradual vanishing of such Cotton scalars, starting with those of the highest boost weight. This classification is directly related to the specific multiplicity of the Cotton-aligned null directions and to the corresponding Bel–Debever criteria. Using a bivector (that is 2-form) decomposition, we demonstrate that our method is fully equivalent to the usual Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem and determining the respective canonical Jordan form of the Cotton–York tensor. We also derive a simple synoptic algorithm of algebraic classification based on the key polynomial curvature invariants. To show the practical usefulness of our approach, we perform the classification of several explicit examples, namely the general class of Robinson–Trautman spacetimes with an aligned electromagnetic field and a cosmological constant, and other metrics of various algebraic types.

Fill-ins with scalar curvature lower bounds and applications to positive mass theorems

Given a constant C and a smooth closed (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional Riemannian manifold (Σ,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma , g)$$\\end{document} equipped with a positive function H, a natural question to ask is whether this manifold can be realised as the boundary of a smooth n-dimensional Riemannian manifold with scalar curvature bounded below by C and boundary mean curvature H. That is, does there exist a fill-in of (Σ,g,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma ,g,H)$$\\end{document} with scalar curvature bounded below by C? We use variations of an argument due to Miao and the author (Int Math Res Not 7:2019, 2019) to explicitly construct fill-ins with different scalar curvature lower bounds, where we permit the fill-in to contain another boundary component provided it is a minimal surface. Our main focus is to illustrate the applications of such fill-ins to geometric inequalities in the context of general relativity. By filling in a manifold beyond a boundary, one is able to obtain lower bounds on the mass in terms of the boundary geometry through positive mass theorems and Penrose inequalities. We consider fill-ins with both positive and negative scalar curvature lower bounds, which from the perspective of general relativity corresponds to the sign of the cosmological constant, as well as a fill-in suitable for the inclusion of electric charge.

Cosmological FLRW phase transitions and micro-structure under Kaniadakis statistics

This article studies the thermodynamics phase transitions and critical phenomena of an FLRW cosmological model under truncated Kaniadakis's statistics. The EoS is derived from the corrected Friedmann field equations and the thermodynamics unified first law. To approach thermodynamics, we use an approximation for KSAH≪1, on the cosmological equation, and also check the validity of our approximation. Then for different values from Kaniadakis parameter K, order O(10−37) for the relic-abundance, O(10−84) for 7Li-abundance and O(10−125) in the recombination era, the EoS reveals non-trivial critical points where a first-order phase transition occurs a sort of a van der Waals fluid. Interestingly, the numerical values of the critical exponents are the same as those of the van der Waals system. Besides, to obtain more insights into the thermodynamics description, the so-called Ruppeiner's geometry is studied through the normalized scalar curvature, disclosing this invariant zone where the system undergoes repulsive/attractive interactions. Near the critical point, this curvature provides again the same critical exponent and universal constant value as for van der Waals fluid. Despite the similarity, both systems are quite different because the present one considers a relativistic entropy (Kaniadakis entropy) while the Van der Waals gas responds to a classical entropy (Maxwell-Boltzmann).