Search Results (24)

A new conformally invariant gravitational generalization of the Born–Infeld (BI) model is proposed and analyzed from the point of view of symmetries. Taking a geometric identity involving the determinant functions det$f\left(B_{\mu \nu },F_{\mu \nu }\right)$ with the Bach $B_{\mu \nu }$ and the electromagnetic field $F_{\mu \nu }$ tensors (with the 4-dimensional Greek letter indexes), two characteristic geometrical Lagrangian densities (Lagrangians) are derived: the first Lagrangian being the square root of the determinant function $\sqrt{\left|det\left(B_{\mu \nu }+F_{\mu \nu }\right)\right|}$ (reminiscent of the standard BI model) and the second Lagrangian being the fourth root $\sqrt[4]{gdet\left(B_{\alpha \gamma }{B}_{\beta }^{\gamma }+F_{\alpha \gamma }{F}_{\beta }^{\gamma }\right)}$. It is shown, after explicit computation of the gravitational equations, that the square-root model is incompatible with the inclusion of the electromagnetic tensor, consequently forcing the nullity of $F_{\mu \nu }$. In sharp contrast, the traceless fourth-root model is fully compatible and a natural ansatz of the type $B_{\mu \rho }{B}_{\nu }^{\rho }\propto \mathrm{\Omega }\left(x\right)g_{\mu \nu }$ (conformal-Killing), with $\mathrm{\Omega }$ the conformal factor and x the 4-coordinate, can be considered. Among other essential properties, the geometrical conformal Lagrangian of the fourth-root type is self-similar with respect to the determinant g of the metric tensor $g_{\mu \nu }$ and can be extended to non-Abelian fields in a way similar to the model developed by the author earlier. This self-similarity is related to the conformal properties of the model, such as the Bach currents or flows presumably of a topological origin. Possible applications and comparisons with other models are briefly discussed. Full article

The subject of this study is almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds. The considerations are restricted to a special class of these manifolds, namely those of the Sasaki-like type, because of their geometric construction and the explicit expression of their classification tensor by the pair of B-metrics. Here, each of the two B-metrics is considered as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form. The soliton potential is chosen to be a conformal Killing vector field (in particular, concircular or concurrent) and then a generalization of the notion of conformality using contact conformal transformations of B-metrics. The resulting manifolds, equipped with the introduced almost solitons, are geometrically characterized. In the five-dimensional case, an explicit example on a Lie group depending on two real parameters is constructed, and the properties obtained in the theoretical part are confirmed. Full article

This paper explores the geometry of 3-dimensional quasi Sasakian manifolds under CL-transformations. We construct both infinitesimal and $CL$-transformation and demonstrate that the former does not necessarily yield projective killing vector fields. A novel invariant tensor, termed the $CL$-curvature tensor, is introduced and shown to remain invariant under $CL$-transformations. Utilizing this tensor, we characterize $CL$-flat, $CL$-symmetric, $CL$-$\phi$ symmetric and $CL$-$\phi$ recurrent structures on such manifolds by mean of differential equations. Furthermore, we investigate conditions under which a Ricci soliton exists on a CL-transformed quasi Sasakian manifold, revealing that under flat curvature, the structure becomes Einstein. These findings contribute to the understanding of curvature dynamics and soliton theory within the context of contact metric geometry. Full article

In this paper, we introduce and conduct a comprehensive study of pseudo-Schouten symmetric manifolds ${\left(PSS\right)}_n$. We establish necessary and sufficient conditions for such a manifold to be Einstein and quasi-Einstein, respectively. Next, we examine pseudo-Schouten symmetric spacetimes within the framework of general relativity. Furthermore, we investigate their role in relativistic spacetime models by considering Einstein’s field equations with and without a cosmological constant. We also show that pseudo-Schouten symmetric spacetimes satisfying Einstein’s equations with a quadratic Killing energy–momentum tensor or a Codazzi-type energy–momentum tensor cannot have non-zero constant scalar curvature. Finally, the existence of pseudo-Schouten symmetric spacetime is shown by constructing an explicit non-trivial example. Full article

A disaffinity vector on a Riemannian manifold $(M,g)$ is a vector field whose affinity tensor vanishes. In this paper, we observe that nontrivial disaffinity functions offer obstruction to the topology of M and show that the existence of a nontrivial disaffinity function on M does not allow M to be compact. A characterization of the Euclidean space is also obtained by using nontrivial disaffinity functions. Further, we study properties of disaffinity vectors on M and prove that every Killing vector field is a disaffinity vector. Then, we prove that if the potential field $\zeta$ of a Ricci soliton $\left(M,g,\zeta ,\lambda \right)$ is a disaffinity vector, then the scalar curvature is constant. As an application, we obtain conditions under which a Ricci soliton $\left(M,g,\zeta ,\lambda \right)$ is trivial. Finally, we prove that a Yamabe soliton $\left(M,g,\xi ,\lambda \right)$ with a disaffinity potential field $\xi$ is trivial. Full article

In this paper, the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space $V_4$. In the case where the group of motions $G_3$ acts in a simply transitive way in the homogeneous space $V_4$, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space $V_3$ of the group $G_3$. In this case, the metric tensor of the space $V_3$ can be given by a nonholonomic reper consisting of three independent vectors ${\ell }_{\left(a\right)}^{\alpha }$, which define the generators of the group $G_3$ of finite transformations in the space $V_3$. The representation of the metric tensor of $V_4$ spaces by means of vector fields ${\ell }_{\left(a\right)}^{\alpha }$ has a great physical meaning and makes it possible to substantially simplify the equations of mathematical physics in such spaces. Therefore, the Petrov classification should be complemented by the classification of vector fields ${\ell }_{\left(a\right)}^{\alpha }$ connected to Killing vector fields. For homogeneous spaces, this problem has been largely solved. A complete solution of this problem is presented in the present paper, where I refine the Petrov classification for homogeneous spaces in which the group $G_3$, which belongs to type $VIII$ according to the Petrov classification, acts simply transitively. In addition, this paper provides the complete classification of vector fields ${\ell }_{\left(a\right)}^{\alpha }$ for space $V_4$ in which the group $G_3$ acts simply transitivity on isotropic hypersurfaces. Full article

We highlight some properties of a class of distinguished vector fields associated to a $(1,1)$-tensor field and to an affine connection on a Riemannian manifold, with a special view towards the Ricci vector fields, and we characterize them with respect to statistical, almost Kähler, and locally product structures. In particular, we provide conditions for these vector fields to be closed, Killing, parallel, or semi-torse forming. In the gradient case, we give a characterization of the Euclidean sphere. Among these vector fields, the Ricci and torse-forming-like vector fields are particular cases. Full article

A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT. Full article

Thanks to the recent advent of the event horizon telescope (EHT), we now have the opportunity to test the physical ramifications of the strong-field near-horizon regime for astrophysical black holes. Herein, emphasizing the trade-off between tractability and generality, the authors discuss a particularly powerful three-function distortion of the Kerr spacetime, depending on three arbitrary functions of the radial coordinate r, which on the one hand can be fit to future observational data, and on the other hand is sufficiently general so as to encompass an extremely wide class of theoretical models. In all of these spacetimes, both the timelike Hamilton–Jacobi (geodesic) and massive Klein–Gordon (wave) equations separate, and the spacetime geometry is asymptotically Kerr; hence, these spacetimes are well-suited to modeling real astrophysical black holes. The authors then prove the existence of Killing horizons for this entire class of spacetimes, and give tractable expressions for the angular velocities, areas, and surface gravities of these horizons. We emphasize the validity of rigidity results and zeroth laws for these horizons. Full article

We consider autonomous holonomic dynamical systems defined by equations of the form ${\ddot{q}}^a=-{\mathsf{\Gamma }}_{bc}^a\left(q\right){\dot{q}}^b{\dot{q}}^c$$-Q^a\left(q\right)$, where ${\mathsf{\Gamma }}_{bc}^a\left(q\right)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^a\left(q\right)$ are the generalized forces. We prove a theorem which for these systems determines autonomous and time-dependent first integrals (FIs) of any order in a systematic way, using the ’symmetries’ of the geometry defined by the dynamical equations. We demonstrate the application of the theorem to compute linear, quadratic, and cubic FIs of various Riemannian and non-Riemannian dynamical systems. Full article

The determination of the first integrals (FIs) of a dynamical system and the subsequent assessment of their integrability or superintegrability in a systematic way is still an open subject. One method which has been developed along these lines for holonomic autonomous dynamical systems with dynamical equations ${\ddot{q}}^a=-{\mathsf{\Gamma }}_{bc}^a\left(q\right){\dot{q}}^b{\dot{q}}^c-Q^a\left(q\right)$, where ${\mathsf{\Gamma }}_{bc}^a\left(q\right)$ are the coefficients of the Riemannian connection defined by the kinetic metric of the system and $-Q^a\left(q\right)$ are the generalized forces, is the so-called direct method. According to this method, one assumes a general functional form for the FI I and requires the condition $\frac{dI}{dt}=0$ along the dynamical equations. This results in a system of partial differential equations (PDEs) to which one adds the necessary integrability conditions of the involved scalar quantities. It is found that the final system of PDEs breaks into two sets: a. One set containing geometric elements only and b. A second set with geometric and dynamical quantities. Then, provided the geometric quantities are known or can be found, one uses the second set to compute the FIs and, accordingly, assess the integrability of the dynamical system. The ‘solution’ of the system of PDEs for quadratic FIs (QFIs) has been given in a recent paper (M. Tsamparlis and A. Mitsopoulos, J. Math. Phys. 61, 122701 (2020)). In the present work, we consider the application of this ‘solution’ to Newtonian autonomous conservative dynamical systems with three degrees of freedom, and compute integrable and superintegrable potentials $V(x,y,z)$ whose integrability is determined via autonomous and/or time-dependent QFIs. The geometric elements of these systems are the ones of the Euclidean space $E^3$, which are known. Setting various values for the parameters determining the geometric elements, we determine in a systematic way all known integrable and superintegrable potentials in $E^3$ together with new ones obtained in this work. For easy reference, the results are collected in tables so that the present work may act as an updated review of the QFIs of Newtonian autonomous conservative dynamical systems with three degrees of freedom. It is emphasized that, by assuming different values for the parameters, other authors may find more integrable potentials of this type of system. Full article

We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For a zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics. Full article

A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on Hadamard manifolds. Full article

Recently, the current authors have formulated and extensively explored a rather novel Painlevé–Gullstrand variant of the slow-rotation Lense–Thirring spacetime, a variant which has particularly elegant features—including unit lapse, intrinsically flat spatial 3-slices, and a separable Klein–Gordon equation (wave operator). This spacetime also possesses a non-trivial Killing tensor, implying separability of the Hamilton–Jacobi equation, the existence of a Carter constant, and complete formal integrability of the geodesic equations. Herein, we investigate the geodesics in some detail, in the general situation demonstrating the occurrence of “ultra-elliptic” integrals. Only in certain special cases can the complete geodesic integrability be explicitly cast in terms of elementary functions. The model is potentially of astrophysical interest both in the asymptotic large-distance limit and as an example of a “black hole mimic”, a controlled deformation of the Kerr spacetime that can be contrasted with ongoing astronomical observations. Full article

$f(\mathcal{R},T)$-gravity is a generalization of Einstein’s field equations $\left({EFE}_s\right)$ and $f\left(\mathcal{R}\right)$-gravity. In this research article, we demonstrate the virtues of the $f(\mathcal{R},T)$-gravity model with Einstein solitons $\left(ES\right)$ and gradient Einstein solitons $\left(GES\right)$. We acquire the equation of state of $f(\mathcal{R},T)$-gravity, provided the matter of $f(\mathcal{R},T)$-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a $f(\mathcal{R},T)$-gravity filled with perfect fluid admits an Einstein soliton $(g,\rho ,\lambda )$ and the Einstein soliton vector field $\rho$ of $(g,\rho ,\lambda )$ is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the $f(\mathcal{R},T)$-gravity model. Next, we prove that if a $f(\mathcal{R},T)$-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the $f(\mathcal{R},T)$-gravity model together with gradient Einstein soliton. Full article