Showing papers on "Scalar curvature published in 2021"

On the constant scalar curvature Kähler metrics (I)—A priori estimates

Abstract: In this paper, we derive apriori estimates for constant scalar curvature K\\\"ahler metrics on a compact K\\\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\\\"ahler potential. We also discuss some local versions of these estimates which can be of independent interest.

On the constant scalar curvature Kähler metrics (II)—Existence results

Abstract: In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\\\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of K\\\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\\mathcal{E}^1, d_1)$ are smooth.

An anisotropic version of Tolman VII solution in f(R, T) gravity via gravitational decoupling MGD approach

Abstract: In this work, we have adopted gravitational decoupling by minimal geometric deformation (MGD) approach and have developed an anisotropic version of well-known Tolman VII isotropic solution in the framework of f(R, T) gravity, where R is Ricci scalar and T is trace of energy momentum tensor. The set of field equations has been developed with respect to total energy momentum tensor, which combines effective energy momentum tensor in f(R, T) gravity and additional source $$\phi _{ij}$$ . Following MGD approach, the set of field equations has been separated into two sections. One section represents f(R, T) field equations, while the other is related to the source $$\phi _{ij}$$ . The matching conditions for inner and outer geometry have also been discussed, and an anisotropic solution has been developed using mimic constraint for radial pressure. In order to check viability of the solution, we have considered observation data of three different compact star models, named PSR J1614-2230, PSR 1937+21 and SAX J1808.4-3658, and have discussed thermodynamical properties analytically and graphically. The energy conditions are found to be satisfied for the three compact stars. The stability analysis has been presented through causality condition and Herrera’s cracking concept, which ensures physical acceptability of the solution.

Rescaled Einstein-Hilbert gravity from f ( R ) gravity: Inflation, dark energy, and the swampland criteria

Abstract: We consider the scenario that the effective gravitational Lagrangian of a minimally coupled scalar field at large curvatures is described by a rescaled Einstein-Hilbert gravity, with the Ricci scalar being multiplied by a dimensionless parameter $\ensuremath{\alpha}$. Such Lagrangian densities might originate, for example, by a class of exponential models of $f(R)$ gravity, in the presence of a canonical scalar field, for which at early times the effective Lagrangian of the theory becomes that of a rescaled canonical scalar field with the Einstein-Hilbert term becoming $\ensuremath{\sim}\ensuremath{\alpha}R$, with $\ensuremath{\alpha}$ a dimensionless constant, and the resulting theory is an effective theory of a Jordan frame $f(R,\ensuremath{\phi})$ theory. This rescaled Einstein-Hilbert canonical scalar field theory at early times has some interesting features, since it alters the inflationary phenomenology of well-known scalar field models of inflation, but more importantly, in the context of this rescaled theory, the swampland criteria are easily satisfied, assuming that the scalar field is slowly rolling. We consider two models of inflation to exemplify our study, a fiber inflation model and a model that belongs to the general class of supergravity $\ensuremath{\alpha}$-attractor models. The inflationary phenomenology of the models is demonstrated to be viable, and for the same set of values of the free parameters of each model which ensure their inflationary viability, all the known swampland criteria are satisfied too, and we need to note that we assumed that the first swampland criterion is marginally satisfied by the scalar field, so $\ensuremath{\phi}\ensuremath{\sim}{M}_{p}$ during the inflationary era. Finally, we examine the late-time phenomenology of the fiber inflation potential in the presence of the full $f(R)$ gravity, and we demonstrate that the resulting model produces a viable dark energy era, which resembles the $\mathrm{\ensuremath{\Lambda}}$-cold-dark-mater model. Thus, in the modified gravity model we present, the Universe is described by a rescaled Einstein-Hilbert gravity at early times; hence, in some sense, the modified gravity effect is minimal primordially, and the scalar field controls mainly the dynamics with a rescaled Ricci scalar gravity. However, the effect of $f(R)$ gravity becomes stronger at late times, where it controls the dynamics, synergistically with the scalar field.

Dynamical Cobordism and Swampland Distance Conjectures

Abstract: We consider spacetime-dependent solutions to string theory models with tadpoles for dynamical fields, arising from non-trivial scalar potentials. The solutions have necessarily finite extent in spacetime, and are capped off by boundaries at a finite distance, in a dynamical realization of the Cobordism Conjecture. We show that as the configuration approaches these cobordism walls of nothing, the scalar fields run off to infinite distance in moduli space, allowing to explore the implications of the Swampland Distance Conjecture. We uncover new interesting scaling relations linking the moduli space distance and the SDC tower scale to spacetime geometric quantities, such as the distance to the wall and the scalar curvature. We show that walls at which scalars remain at finite distance in moduli space correspond to domain walls separating different (but cobordant) theories/vacua; this still applies even if the scalars reach finite distance singularities in moduli space, such as conifold points. We illustrate our ideas with explicit examples in massive IIA theory, M-theory on CY threefolds, and 10d non-supersymmetric strings. In 4d $$ \mathcal{N} $$ = 1 theories, our framework reproduces a recent proposal to explore the SDC using 4d string-like solutions.

Cosmological perturbations in Palatini formalism

Abstract: We investigate cosmological perturbations of scalar-tensor theories in Palatini formalism. First we introduce an action where the Ricci scalar is conformally coupled to a function of a scalar field and its kinetic term and there is also a k-essence term consisting of the scalar and its kinetic term. This action has three frames that are equivalent to one another: the original Jordan frame, the Einstein frame where the metric is redefined, and the Riemann frame where the connection is redefined. For the first time in the literature, we calculate the quadratic action and the sound speed of scalar and tensor perturbations in three different frames and show explicitly that they coincide. Furthermore, we show that for such action the sound speed of gravitational waves is unity. Thus, this model serves as dark energy as well as an inflaton even though the presence of the dependence of the kinetic term of a scalar field in the non-minimal coupling, different from the case in metric formalism. We then proceed to construct the L3 action called Galileon terms in Palatini formalism and compute its perturbations. We found that there are essentially 10 different(inequivalent) definitions in Palatini formalism for a given Galileon term in metric formalism. We also see that,in general, the L3 terms have a ghost due to Ostrogradsky instability and the sound speed of gravitational waves could potentially deviate from unity, in sharp contrast with the case of metric formalism. Interestingly, once we eliminate such a ghost, the sound speed of gravitational waves also becomes unity. Thus, the ghost-free L3 terms in Palatini formalism can still serve as dark energy as well as an inflaton, like the case in metric formalism.

The moduli space of two-convex embedded spheres

Abstract: We prove that the moduli space of $2$-convex embedded $n$-spheres in $\mathbb{R}^{n+1}$ is path-connected for every $n$. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques’ influential proof of the path-connectedness of the moduli space of positive scalar curvature metrics on three-manifolds.

Constrained deformations of positive scalar curvature metrics, II

Abstract: We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean-convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.

Biharmonic submanifolds of Kaehler product manifolds

Abstract: In this paper, the authors have established the necessary and sufficient conditions for the submanifolds of Kaehler product manifolds to be biharmonic. Moreover, the magnitude of scalar curvature for the hypersurfaces in a product of two unit spheres has been derived. Also, for the same product, the magnitude of the mean curvature vector for Lagrangian submanifolds has been estimated. Finally, the non-existence condition for totally complex Lagrangian submanifolds in a product of unit sphere and a hyperbolic space has been proved.

Spacetime as a quantum circuit

Abstract: We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the “complexity equals volume” conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $$ T\overline{T} $$ , we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action.

Scalar and mean curvature comparison via the Dirac operator

Abstract: We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing $\widehat{\mathrm{A}}$-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

Black hole scalarization with Gauss-Bonnet and Ricci scalar couplings

Abstract: Spontaneous scalarization is a gravitational phenomenon in which deviations from general relativity arise once a certain threshold in curvature is exceeded, while being entirely absent below that threshold. For black holes, scalarization is known to be triggered by a coupling between a scalar and the Gauss-Bonnet invariant. A coupling with the Ricci scalar, which can trigger scalarization in neutron stars, is instead known to not contribute to the onset of black hole scalarization, and has so far been largely ignored in the literature when studying scalarized black holes. In this paper, we study the combined effect of both these couplings on black hole scalarization. We show that the Ricci coupling plays a significant role in the properties of scalarized solutions and their domain of existence. This work is an important step in the construction of scalarization models that evade binary pulsar constraints and have general relativity as a cosmological late-time attractor, while still predicting deviations from general relativity in black hole observations.

Ground and thermal states for the Klein-Gordon field on a massless hyperbolic black hole with applications to the anti-Hawking effect

Abstract: On an $n$-dimensional, massless, topological black hole with hyperbolic sections, we construct the two-point function both of a ground state and of a thermal state for a real, massive, free scalar field arbitrarily coupled to scalar curvature and endowed with Robin boundary conditions at conformal infinity. These states are used to compute the response of an Unruh-DeWitt detector coupled to them for an infinite proper time interval along static trajectories. As an application, we focus on the massless conformally coupled case, and we show, numerically, that the anti-Hawking effect, which is manifest on the three-dimensional case, does not occur if we consider a four-dimensional massless hyperbolic black hole. On the one hand, we argue that this result is compatible with what happens in the three- and four-dimensional Minkowski spacetime, while, on the other hand, we stress that it generalizes existing results concerning the anti-Hawking effect on black hole spacetimes.

Renormalization group equations of Higgs-R 2 inflation

Abstract: We derive one- and two-loop renormalization group equations (RGEs) of Higgs-R2 inflation. This model has a non-minimal coupling between the Higgs and the Ricci scalar and a Ricci scalar squared term on top of the standard model. The RGEs derived in this paper are valid as long as the energy scale of interest (in the Einstein frame) is below the Planck scale. We also discuss implications to the inflationary predictions and the electroweak vacuum metastability.

Wormhole solutions in modified f(R,φ,X) gravity

Abstract: This work aims to investigate the wormhole solutions in the background of f(R,φ,X) theory of gravity, where R is Ricci scalar, φ is scalar potential, and X is the kinetic term. We consider spherica...

Static spacetimes haunted by a phantom scalar field. I. Classification and global structure in the massless case

Abstract: We discuss various novel features of $n(\ensuremath{\ge}4)$-dimensional spacetimes sourced by a massless (non)phantom scalar field in general relativity. Assuming that the metric is a warped product of static two-dimensional Lorentzian spacetime and an ($n\ensuremath{-}2$)-dimensional Einstein space ${K}^{n\ensuremath{-}2}$ with curvature $k=0,\ifmmode\pm\else\textpm\fi{}1$, and that the scalar field depends only on the radial variable, we present a complete classification of static solutions for both signs of kinetic term. Contrary to the case with a nonphantom scalar field, the Fisher solution is not unique, and there exist two additional metrics corresponding to the generalizations of the Ellis-Gibbons solution and the Ellis-Bronnikov solution. We explore the maximal extension of these solutions in detail by the analysis of null/spacelike geodesics and singularity. For the phantom Fisher and Ellis-Gibbons solutions, we find that there inevitably appear parallelly propagated (p.p) curvature singularities in the parameter region where there are no scalar curvature singularities. Interestingly, the areal radius blows up at these p.p curvature singularities, which are nevertheless accessible within a finite affine time along the radial null geodesics. It follows that only the Ellis-Bronnikov solution describes a regular wormhole in the two-sided asymptotically flat spacetime. Using the general transformation relating the Einstein and Jordan frames, we also present a complete classification of solutions with the same symmetry coupled to a conformal scalar field. Additionally, by solving the field equations in the Jordan frame, we prove that this classification is genuinely complete.

Construction of inflationary scenarios with the Gauss–Bonnet term and nonminimal coupling

Abstract: Inflationary models with a scalar field nonminimally coupled both with the Ricci scalar and with the Gauss–Bonnet term are studied. We propose the way of generalization of inflationary scenarios with the Gauss–Bonnet term and a scalar field minimally coupled with the Ricci scalar to the corresponding scenarios with a scalar field nonminimally coupled with the Ricci scalar. Using the effective potential, we construct a set of models with the same values of the scalar spectral index $$n_s$$ and the amplitude of the scalar perturbations $$A_s$$ and different values of the tensor-to-scalar ratio r.

On the fill-in of nonnegative scalar curvature metrics

Abstract: In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

General approach to the Lagrangian ambiguity in f(R, T) gravity

Abstract: The f(R, T) gravity is a theory whose gravitational action depends arbitrarily on the Ricci scalar, R, and the trace of the stress–energy tensor, T; its field equations also depend on matter Lagrangian, $$\mathscr {L}_{m}$$ . In the modified theories of gravity where field equations depend on Lagrangian, there is no uniqueness on the Lagrangian definition and the dynamics of the gravitational and matter fields can be different depending on the choice performed. In this work, we have eliminated the $$\mathscr {L}_{m}$$ dependence from f(R, T) gravity field equations by generalizing the approach of Moraes in Ref. [1]. We also propose a general approach where we argue that the trace of the energy–momentum tensor must be considered an “unknown” variable of the field equations. The trace can only depend on fundamental constants and few inputs from the standard model. Our proposal resolves two limitations: first the energy–momentum tensor of the f(R, T) gravity is not the perfect fluid one; second, the Lagrangian is not well-defined. As a test of our approach we applied it to the study of the matter era in cosmology, and the theory can successfully describe a transition between a decelerated Universe to an accelerated one without the need for dark energy.

Gravitationally induced uncertainty relations in curved backgrounds

Abstract: This paper aims at investigating the influence of space-time curvature on the uncertainty relation. In particular, relying on previous findings, we assume the quantum wave function to be confined to a geodesic ball on a given spacelike hypersurface whose radius is a measure of the position uncertainty. On the other hand, we concurrently work out a viable physical definition of the momentum operator and its standard deviation in the nonrelativistic limit of the $3+1$ formalism. Finally, we evaluate the uncertainty relation, which to second order depends on the Ricci scalar of the effective 3-metric and the corresponding covariant derivative of the shift vector. For the sake of illustration, we apply our general result to a number of examples arising in the context of both general relativity and extended theories of gravity.

Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

Abstract: In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.