Thermodynamic scalar curvature and topological classification in accelerating charged AdS black holes under rainbow gravity

In this article, we studied δ-almost Yamabe solitons within the framework of paracontact metric manifolds. First, we proved that for a paracontact metric manifold M , if a paracontact metric g represents a δ-almost Yamabe soliton associated with the potential vector field Z being an infinitesimal contact transformation, then Z is Killing, and if the potential vector field Z is collinear with ξ, then the manifold M is K -paracontact. Next, if we take a K -paracontact metric manifold admitting δ-almost Yamabe soliton with the potential vector field Z parallel to the characteristic vector field and with constant scalar curvature then either the scalar curvature will vanish or g becomes a δ-Yamabe soliton under a certain condition. We established some results on K -paracontact manifolds admitting δ-almost gradient Yamabe solitons. Moreover, we consider a (k, µ)-paracontact metric manifold admitting a non-trivial δ-almost gradient Yamabe soliton. We have shown that the potential vector field Z is parallel to ξ. We have also discussed δ-almost gradient Yamabe solitons on the para-Sasakian manifold. Finally, we consider a para-cosymplectic manifold with a δ-almost Yamabe soliton. In the end, we construct two examples of K -paracontact metric manifolds with δ-almost Yamabe soliton.

It is known that the Lemaitre–Tolman–Bondi (LTB) spacetimes describe spherically symmetric radially inhomogeneous models of the universe. Such models have widely been used to study gravitational collapse and late-time cosmic acceleration. In particular, its non-static and inhomogeneous nature can yield ideal models that can be applied in modified gravitational theories. In this paper, we explore non-static LTB spacetimes within the framework of [Formula: see text] gravity, where [Formula: see text] is a function of the scalar curvature [Formula: see text]. To tackle non-linearity of the field equations and to simplify the problem, we restrict the problem to linear model of the [Formula: see text] gravity. In addition to the linear model, we use an algebraic classification of the spacetime components which further help to explore a variety of solutions. In this way, we come across some particular classes of exact LTB solutions to the gravitational field equations in [Formula: see text] gravity. With the help of these solutions, we formulate a geodesic Lagrangian and consequently explore Noether Vector Fields (NVFs). The NVFs link symmetries of the associated Lagrangian to conserved quantities. These conserved quantities may be applied to simplify the dynamical system. For each solution, we compute the corresponding NVFs and note that the dimension of the Noether symmetry algebra for the LTB spacetimes under linear [Formula: see text] gravity is either 7, 9, or 17.

The canonical quantization of gravity in general relativity is greatly simplified by the artificial decomposition of space time into a 3 + 1 formalism. Such a simplification appears to come at the cost of general covariance. This quantization procedure requires tangential and perpendicular infinitesimal diffeomorphisms generated by the symmetry group under the Legendre transformation of the given action. This gauge generator, along with the fact that Weyl curvature scalars may act as “intrinsic coordinates” (or a dynamical reference frame) that depend only on the spatial metric (gab) and the conjugate momenta (pcd), allows for an alternative approach to canonical quantization of gravity. In this paper, we present the tensorial solution of the set of Weyl scalars in terms of canonical phase-space variables.

Abstract We review notions of mass of asymptotically locally Anti-de Sitter three-dimensional spacetimes, and apply them to some known solutions. For two-dimensional general relativistic initial data sets the mass is not invariant under asymptotic symmetries, but a unique mass parameter can be obtained either by minimisation, or by a monodromy construction, or both. We give an elementary proof of positivity, and of a Penrose-type inequality, in a natural gauge. We carry-out a gluing construction at infinity to time-symmetric asymptotically locally hyperbolic vacuum initial data sets and derive mass/entropy formulae for the resulting manifolds.
Finally, we show that all mass aspect functions can be realised by constant scalar curvature metrics on complete manifolds which are smooth except for at most one conical singularity.

A bstract We study the scalar curvature R of the vector moduli space of 5d $$\mathcal{N}=1$$ supergravities, obtained by compactifying M-theory on a Calabi-Yau three-fold. We find that R can only diverge at points where some gauge interactions go to infinite coupling in Planck units and become SCFTs or LSTs decoupled from gravity and other vector multiplets. For 5d SCFTs of rank r ≤ 2 divergences occur if, additionally, the SCFT still couples to the vevs of such vector multiplets, so that along its Coulomb branch its gauge kinetic matrix and/or string tensions depend on some non-dynamical parameters. If the strong coupling singularity is better understood as a 6d (1 , 0) SCFT, as in some decompactification limits, then divergences in R arise when the SCFT is endowed with a non-Abelian gauge group.

Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature

Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity

Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability

Abstract The existence of black holes in the Universe is nowadays established on the grounds of a blench of astrophysical observations, most notably those of gravitational waves from binary mergers and the imaging of supermassive objects at the heart of M87 and Milky Way galaxies. However, this success of Einstein's General Relativity (GR) to connect theory of black holes with observations is also the source of its doom, since Penrose's theorem proves that, under physically sensible conditions, the development of a space-time singularity (as defined by the existence of a focal point for some geodesic paths in finite affine time) within black holes as described by GR is unavoidable. In this work, we thoroughly study how to resolve space-time singularities in spherically symmetric black holes. To do it so we find the conditions on the metric functions required for the restoration of geodesic completeness without any regards to the specific theory of the gravitational and matter fields supporting the amended metric. Our discussion considers both the usual trivial radial coordinate case and the bouncing radial function case and arrives to two mechanisms for this restoration: either the focal point is displaced to infinite affine distance or a bounce prevents the focusing of geodesics. Several explicit examples of well known (in)complete space-times are given. Furthermore, we consider the connection of geodesic (in)completeness with another criterion frequently used in the literature to monitor singular space-times: the blow up of (some sets of) curvature scalars and the infinite tidal forces they could bring with them, and discuss the conditions required for the harmlessness upon physical observers according to each criterion.

Abstract We give obstructions for a noncompact manifold to admit a complete Riemannian metric with (nonuniformly) positive scalar curvature. We treat both the finite volume and infinite volume cases.

In Kähler geometry, Calabi extremal metrics serves as a class of more available special metrics than Kähler metrics with constant scalar curvatures, as a generalization of Kähler Einstein metrics. In recent years, Maxwell–Einstein metrics (or conformally Kähler Einstein–Maxwell metrics) appeared as another alternative choice for Calabi extremal metrics. It turns out that some similar metrics defined by Futaki and Ono have similar roles in the Kähler geometry. In this paper, we prove that for some completions of certain line bundles, there is at least one k-generalized Maxwell–Einstein metric defined by Futaki and Ono conformally related to a metric in any given Kähler class for any integer 3≤k≤13.

Abstract We construct a noncommutative (NC) $$\hbox {AdS}_4$$ AdS 4 -charged black hole with a planar horizon topology. The NC effects of this geometry are captured by a Gaussian distribution of black hole mass codified in a fluid-like energy–momentum tensor. A natural bound in radial coordinate is established, below which the scalar curvature changes its sign and defines a NC cutoff that embeds the point singularity. We study in detail the thermodynamic structure of this scenario, finding a well-defined black hole mass and an analytic criterion for its stability. Focusing on the $$\hbox {AdS}_{2}$$ AdS 2 structure near the horizon, we find a novel effective curvature radius with dependency on the NC cutoff. These results motivate us to explore the holographic superconducting system in terms of the nearness from the cutoff. The behavior of the magnetic field in the deep IR geometry is studied and we found semi-analytical novel expressions for the upper critical magnetic fields of a dual type-II superconductor in the canonical and grand canonical ensembles. The condensation in the form of hair is studied in terms of the bound states of the associated Schrödinger potential of the scalar field, interpreted as the dual to the density of Cooper pairs. The NC effects increase the hair formation due to a steeper $$\hbox {AdS}_2$$ AdS 2 throat comparable to the commutative case. Finally, we obtain the effective IR scalar field equation on the near horizon and near extremal NC Schwarzschild $$\hbox {AdS}_2$$ AdS 2 geometry and confirm that NC effects promote bound states that the commutative version forbids.

On stability and scalar curvature rigidity of quaternion-Kähler manifolds

Abstract Q-map spaces form an important class of quaternionic Kähler manifolds of negative scalar curvature. Their one-loop deformations are always inhomogeneous and have been used to construct cohomogeneity one quaternionic Kähler manifolds as deformations of homogeneous spaces. Here we study the group of isometries in the deformed case. Our main result is the statement that it always contains a semidirect product of a group of affine transformations of $$\mathbb {R}^{n-1}$$ R n - 1 with a Heisenberg group of dimension $$2n+1$$ 2 n + 1 for a q-map space of dimension 4n. The affine group and its action on the normal Heisenberg factor in the semidirect product depend on the cubic affine hypersurface which encodes the q-map space.

Abstract The Euclidean Einstein-Hilbert action is well-known to be unbounded below and thus to raise many questions regarding the definition of the gravitational path integral. A variety of works since the late 1980’s have suggested that this problem disappears when one fixes a foliation of the spacetime and imposes the corresponding gravitational constraints. However, we show here that this approach fails with various classes of boundary conditions imposed on the foliation: compact slices without boundary, asymptotically flat, or asymptotically locally anti-de Sitter slices. We also discuss the idea of fixing the scalar curvature and Wick-rotating the conformal factor, and show that it also fails to produce an action bounded from below.

Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

This paper consists of two parts. In the first part, we attempt to find the intermolecular force of charged AdS black hole (BH). Starting from the fact that the equation of states of BH and the van der Waals (vdW) equation have the same compressibility factors , we determine the intermolecular force of BH. We find that this force can always be written as the sum of the topological force created by the topological charge, and the electrostatic force created by the conducting microsphere charged with the electric charge. This is the intermolecular force for all systems whose phase transition possesses the same compressibility factor . In part 2 we begin with the equation of state of white hole (WH) whose temperature is negative and find that its compressibility factor is equal to, and, at the same time, we establish the anti- vdW equation with compressibility factor . This is the main factor for us to determine the intermolecular force of WH. This force is composed of two terms. The first term is the repulsive force, created by the topological charge, and the second term exhibits the attractive electrostatic force, created by two quasi- Cooper pairs (similar to Cooper pairs in the superconductors) consisting of two charged spheric molecules with electric charge. The formation of quasi-Cooper pairs is by BH a quantum effect which was realized in the process of quantum tunneling from BH to WH. At high temperature, the quasi-Cooper pairs are broken, leading to the cancellation of the attractive force, and the repulsive force will push all molecules of WH further and further away. The behaviors of BH force and WH force are totally suppoted by the corresponding scalar curvatures of the thermodynamic geometry. Keywords: Intermolecular force, charged AdS black hole, white hole, quantum tunneling, topological force, phase transition.

In this paper, two concircular invariants of a nearly Kähler manifold are considered. It is proved that a nearly Kähler manifold is concircularly flat if and only if the first concircular invariant is zero. A formula for calculating the second concircular invariant is obtained, and a subclass of nearly Kähler manifolds is distinguished, called the class of concircular-paraKähler manifolds. A concircular-paraKähler manifold of zero scalar curvature is isometric to the complex Euclidean space $\mathbb{C}^n$ equipped with the standard Hermitian metric. The class of concircular-paraKähler manifolds of nonzero constant type coincides with the class of six-dimensional proper nearly Kähler manifolds. It is proved that a concircular-paraKähler nearly Kähler manifold is a Riemannian manifold of constant nonnegative scalar curvature. In this case, its scalar curvature is zero if and only if it is a Kähler manifold. A complete local characterization of concircular-paraKähler nearly Kähler manifolds and concircular-recurrent nearly Kähler manifolds is obtained.

UDC 514.7 We study pointwise hemislant submersions as a generalization of pointwise slant submersions and hemislant submersions from cosymplectic manifolds onto Riemannian manifolds. We investigate the integrability of distributions and the geometry of totally geodesic foliations arising from the definition of these submersions. Moreover, we study the $\phi$-pluriharmonicity of these maps and obtain some inequalities connecting the Ricci curvature with the scalar curvature, depending on whether $\xi$ is vertical or horizontal, for pointwise hemislant submersions from cosymplectic space forms onto Riemannian manifolds.