International Journal of Geometric Methods in Modern Physics

The aim of this work is to provide a general description of the corpuscular theory of gravity. After reviewing some of the major conceptual issues emerging from the semiclassical and field theoretic approaches to Einstein’s gravity, we present a synthetic overview of two novel (and extremely intertwined) perspectives on quantum mechanical effects in gravity: the horizon quantum mechanics (HQM) formalism and the classicalization scheme. After this preliminary discussion, we then proceed with implementing the latter to several different scenarios, namely self-gravitating systems, the early Universe, and galactic dynamics. Concerning the first scenario, we start by describing the generation of the Newtonian potential as the result of a coherent state of toy (scalar) gravitons. After that we employ this result to study some features of the gravitational collapse and to argue that black holes can be thought of a self-sustained quantum states, at the critical point, made of a large number of soft virtual gravitons. We then refine this simplified analysis by constructing an effective theory for the gravitational potential of a static spherical symmetric system up to the first post-Newtonian correction. Additionally, we employ the HQM formalism to study the causal structure emerging from the corpuscular scenario. Finally, we present a short discussion of corpuscular black holes in lower dimensional spaces. After laying down the basics of corpuscular black holes, we present a generalization of the aforementioned arguments to cosmology. Specifically, we first introduce a corpuscular interpretation of the de Sitter spacetime. Then we use it as the starting point for a corpuscular formulation of the inflationary scenario and to provide an alternative viewpoint on the dark components of the $\mathrm{\Lambda }$CDM model. The key message of this work is that the corpuscular theory of gravity offers a way to unify most of the experimental observations (from astrophysical to galactic and cosmological scales) in a single framework, solely based on gravity and baryonic matter.

The investigation of approximate symmetries of reparametrization invariant Lagrangians of $n+1$ degrees of freedom and quadratic velocities is presented. We show that extra conditions emerge which give rise to approximate and conditional Noether symmetries of such constrained actions. The Noether symmetries are the simultaneous conformal Killing vectors of both the kinetic metric and the potential. In order to recover these conditional symmetry generators which would otherwise be lost in gauge fixing the lapse function entering the perturbative Lagrangian, one must consider the lapse among the degrees of freedom. We establish a geometric framework in full generality to determine the admitted Noether symmetries. Additionally, we obtain the corresponding first integrals (modulo a constraint equation). For completeness, we present a pedagogical application of our method.

We present a general approach for the formulation of equations of motion for astrophysical compact objects in general relativistic theories. The objects are modeled as relativistic particles, which are assumed to be moving in a geometric background which in turn is asymptotically flat. The background can be affected by the compact object; so that our approach can be applied to binary systems; by concentrating on the main contributions coming from back reaction effects due to gravitational radiation.

Our premises are different from the traditional post-Newtonian and self-force approaches since we make strong use of the asymptotic properties of the self geometry of the particle; as is described below. In particular, we use a center of mass setting, and relativistic individual dynamical times for each object. We expect our model to complement the other approaches in different regimes.

We compare the gravitational collapse of homogeneous perfect fluid with various equations of state in the framework of General Relativity and in $R^2$ gravity. We make our calculations using dimensionless time with characteristic timescale $t_g\sim {(G\rho )}^{-1/2}$, where $\rho$ is a density of collapsing matter. The cases of matter, radiation and stiff matter are considered. We also account the possible existence of vacuum energy and its influence on gravitational collapse. In a case of $R^2$ gravity, we have additional degree of freedom for initial conditions of collapse. For barotropic equation of state (EoS) $p=w\rho$, the result depends from the value of parameter $w$: for $w>1/3$ the collapse occurs slowly in comparison with General Relativity while for $w<1/3$, we have the opposite situation. Vacuum energy as expected slows down the rate of collapse and for some critical density gravitational contraction may change to expansion. It is interesting to note that for General Relativity such expansion is impossible. We also consider the collapse in the presence of so-called phantom energy. For description of phantom energy, we use Lagrangian in the form $-X-V$ (where $X$ and $V$ are the kinetic and potential energy of the field, respectively) and consider the corresponding Klein–Gordon equation for phantom scalar field.

We calculate a sheaf line in ${CP}^3$ which is the real line supporting sheaf points on ${CP}^3$ of $SL(2,C)$ Yang–Mills (YM) instanton (or $SU(2)$ complex YM instanton) sheaves for some given ADHM data we obtained previously. We found that this sheaf line is indeed a special jumping line over $S^4$ spacetime. In addition, we calculate the singularity structure of the connection $A$ and the field strength $F$ at the corresponding singular point on $S^4$ of this sheaf line. We found that the order of singularity at the singular point on $S^4$ associated with the sheaf line in ${CP}^3$ is higher than those of other singular points associated with normal jumping lines. We conjecture that this is a general feature for sheaf lines among jumping lines.

The purpose of this paper is to study normal $GCR$-lightlike submanifolds of indefinite nearly Kaehler manifolds. We find some necessary and sufficient conditions for an isometrically immersed $GCR$-lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal $GCR$-lightlike submanifold. Further, we derive a characterization theorem for holomorphic bisectional curvature of a normal $GCR$-lightlike submanifold of an indefinite nearly Kaehler manifold.

In this paper, we investigate a tunneling mechanism of massless particles from the Schwarzschild black hole (S-BH) in the framework of polymer quantum mechanics. According to the corresponding invariant Liouville volume, we determine the tunneling rate from S-BH by the polymeric quantization procedure. In this regard, we show that the temperature and tunneling radiation of the black hole receive new corrections in such a way that the exact radiant spectrum is no longer precisely thermal.

If $M$ is a 3-dimensional contact metric manifold such that $Q\phi =\phi Q$ which admits a Yamabe soliton $(g,V)$ with the flow vector field $V$ pointwise collinear with the Reeb vector field $\xi$, then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if $M$ is endowed with a Yamabe soliton $(g,V)$, then either $M$ is flat or it has constant scalar curvature and the flow vector field $V$ is Killing. Furthermore, we show that if $M$ is non-flat, then either $M$ is a Sasakian manifold of constant curvature $1$ or $V$ is an infinitesimal automorphism of the contact metric structure on $M$.

The paper is concerned with the Kaluza–Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.

This paper deals with the exploration of collapse rate and Tolman mass function for spherical system under the influence of $f(R)$ modified gravity theory after using Herrera technique. We have taken the matter configuration to be anisotropic as well as dissipative due to diffusion and streaming out radiations. The modified field equations and dynamical equations are explored for a systematic investigation of Tolman mass. An explicit link among the matter profile, Weyl tensor and mass function is exhibited in this scenario. We found that the anisotropic pressure and density inhomogeneity tends to increase the Tolman mass within the sphere radius, while $f(R)$ terms stabilize the system.

Following our recent work on braneworld mimetic gravity, in this paper, we study an extension of braneworld mimetic gravity to the case that the gravitational sector on the brane is modified in the spirit of $f(R)$ theories. We assume the physical 5D bulk metric in the Randall–Sundrum II braneworld scenario consists of a 5D scalar field (which mimics the dark sectors on the brane) and an auxiliary 5D metric. We find the 5D Einstein’s field equations and the 5D equation of motion of the bulk scalar field in this setup. By using the Gauss–Codazzi equations, we obtain the induced Einstein’s field equations on the brane. Finally, by adopting the FRW background, we find the Friedmann equation on the brane in this $f(R)$ mimetic braneworld setup.

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

In this paper, we study the dynamics of a scalar–tensor model of dark energy in which a scalar field that plays the role of dark energy, non-minimally coupled to the Gauss–Bonnet invariant in four dimensions. We utilize the dynamical system method to extract the critical points of the model and to conclude about their stability, we investigate the sign of the corresponding eigenvalues of the perturbation matrix at each point numerically. For exponential form of the scalar field potential and coupling function, we find five stable points among the critical points of the autonomous system. We also find four scaling attractor solutions with the property that the ratio of dark energy to dark matter density parameters are of order one. These solutions give the hope to alleviate the well-known coincidence problem in cosmology.

The relativistic diffusion process and friction have been studied, especially in the framework of $f(R)$-gravity theory. The study of relativistic diffusion and friction processes based on $f(R)$-gravity is an alternative solution to solve the incompatibility problem emerging in the attempt to couple between the Fokker–Planck equation [FPE] to the Einstein field equation [EFE] encountered by Calogero. The energy–momentum tensor of the cosmological scalar field as proposed by Calogero is replaced by the presence of additional terms in the field equation of $f(R)$-gravity. The additional energy–momentum tensor in the field equation of $f(R)$-gravity in this context is regarded to compensate for the presence of the diffusion and another process like friction. The additional energy–momentum tensor is also regarded as due to the so-called curvature fluid or background fluid. Here we assume the presence of interaction between matter and the background fluid in the form of physical processes like diffusion, friction, etc. We also assume that there is ‘interplay’ between diffusion process and friction. In other words, the diffusion process and friction are not independent. As examples, we consider some viable models of $f(R)$ that satisfy both cosmological and local gravity constraints, i.e. $f(R)=R+\mathrm{\Lambda }{R}^2,f(R)=R-\frac{{{ℳ}}^4}{R}$, and $f(R)=R-{ℰ}{r}_{c}ln(1+\frac{R}{r_c})$. Furthermore, we apply it to explain the diffusion and friction processes in the expanding universe by considering the Friedmann–Lemaitre–Robertson–Walker (FLRW) model.

This paper is aimed to evaluate the existence of wormholes in viable $f(R,T)$ gravity models (where $R$ is the scalar curvature and $T$ is the trace of stress–energy tensor of matter). The exact solutions for energy–momentum tensor components depending on different shapes and redshift functions are calculated without some additional constraints. To investigate this, we consider static spherically symmetric geometry with matter contents as anisotropic fluid and formulate the Einstein field equations for three different $f(R,T)$ models. For each model, we derive expression for weak and null energy conditions and graphically analyzed its violation near the throat. It is really interesting that wormhole solutions do not require the presence of exotic matter — like that in general relativity. Finally, the stability of the solutions for each model is presented using equilibrium condition.

We derive the weak field limit for a gravitational Lagrangian density $L_g=(R+a_0{R}^2+{\sum }_{k=1}^p{a}_{k}R{\square }^{k}R)\sqrt{-g}$, where higher-order derivative terms in the Ricci scalar $R$ are taken into account. The interest for this kind of effective theories comes out from the consideration of the infrared and ultraviolet behaviors of gravitational field and, in general, from the formulation of quantum field theory in curved spacetimes. Here, we obtain solutions in weak field regime both in vacuum and in the presence of matter and derive gravitational waves considering the contribution of $R{\square }^{k}R$ terms. By using a suitable set of coefficients $a_k$, it is possible to find up to $(p+2)$ normal modes of oscillation with six polarization states with helicity 0 or 2. Here $p$ is the higher-order term in the $\square$ operator appearing in the gravitational Lagrangian. More specifically: the mode ${\omega }_1$, with $k^2=0$, has transverse polarizations ${𝜖}_{\mu \nu }^{(+)}$ and ${𝜖}_{\mu \nu }^{(\times )}$ with helicity 2; the $(p+1)$ modes ${\omega }_m$, with $k^2\ne 0$, have transverse polarizations ${𝜖}_{\mu \nu }^{(1)}$ and non-transverse ones ${𝜖}_{\mu \nu }^{(\text{TT})}$, ${𝜖}_{\mu \nu }^{(\text{TS})}$, ${𝜖}_{\mu \nu }^{(L)}$ with helicity 0.

Motivated by condensed matter physical systems, in which a finite-time singularity indicates that the topology of the system changes, we critically examine the possibility of the Universe’s topology change at a finite-time cosmological singularity. We emphasize on Big Rip and Type II and IV cosmological singularities, which we classify to future spacelike and timelike singularities. For the Type IV and Type II singularities, since no geodesics incompleteness occurs, no topological change is allowed, by using Geroch’s theorem arguments. However, for the Big Rip case, Tipler’s arguments allow a topology change, if the spacetime in which the topology change occurs is non-compact and the boundary of this region are two topologically distinct three-dimensional spacelike partial Cauchy hypersurfaces. Also, some additional requirements must hold true, among which the weak energy condition, which can be satisfied in a geometric way in the context of a modified gravity. We critically examine Tipler’s arguments for the Big Rip case, and we discuss the mathematical implications of such a topological change, with regard to the final hypersurface on which geodesics incompleteness occurs.

We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang–Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example, we consider the ‘gluon Lagrangian’, i.e. a Yang–Mills Lagrangian on the $(1,1)$-order gauge-natural bundle of $SU(3)$-principal connections, and canonically define a ‘gluon’ classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.