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A new intrinsically-relativistic kinetic mechanism for generation of non-isotropic relativistic kinetic equilibria in collisionless N-body systems is pointed out. The theory is developed in the framework of the covariant Vlasov statistical description. The new effect is based on the constraints placed by the conservation laws of neutral single-particle dynamics in prescribed background curved-spacetimes demonstrating existence of Killing tensors. As an illustration, the particular case of the Kerr space-time admitting the so-called Carter constant for the particle geodesic motion is considered. The general functional form of the equilibrium kinetic distribution function (KDF) is determined and an explicit realization in terms of Gaussian-like distributions is provided. It is shown that, due to the Carter constant, these equilibrium KDFs exhibit an anisotropic phase-space functional dependence in terms of the single-particle 4-velocity components, giving rise to corresponding non-isotropic continuum fluid fields. The qualitative properties of the equilibrium stress-energy tensor associated with these systems are discussed, with a particular emphasis on the related occurrence of temperature anisotropy effects. The theory is susceptible of astrophysical applications, including in particular the statistical properties of dark matter halos around stellar-mass or galactic-center black holes., 14 pages

In this paper, we focused on the study of anisotropic strange stars with anisotropic matter distribution in the framework of f ( T , T ) = α T ( r ) 2 + β T ( r ) model, where α , β are arbitrary constants, T and T represent the torsion scalar and trace of energy–momentum tensor, respectively. This model has interesting cosmological properties, in addition to providing an effective dark energy sector explanation (Harko et al., 2014). In order to achieve our goal, we considered the Krori-Barua space–time and utilize the off-diagonal tetrad components for deriving the corresponding set of field equations. We calculate the relation for the evaluation of effective density, effective radial and tangential pressure using the EoS of strange star i.e., MIT bag model p r ( r ) = 1 3 [ ρ ( r ) − 4 B g ] . The unknown constants appearing in the solution have been evaluated by using the conventional matching of interior and exterior space time. To check the feasibility of our proposed model we discuss the anisotropy, energy condition, TOV forces, abreu condition, EoS parameters, adiabatic index, mass function, compactness and redshift for different compact stars models namely, P S R J 1416 − 2230 , 4 U 1608 − 52 , C e n X − 3 , E X O 1785 − 248 and S M C X − 1 .

A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions.

In S. Giombi, I. Klebanov, F. Popov, S. Prakash and G. Tarnopolsky, {\it Phys. Rev.} {\bf D} 98 (2018) 10, 105005, a prismatic tensor model was introduced. We study here the diagrammatics and the double scaling limit of this model, using the intermediate field method. We explicitly exhibit the next-to-leading order Feynman graphs in the $1/N$ expansion. Using appropriate combinatorial tools, we further study the general term of the $1/N$ expansion and we compute the $2-$point function in the double scaling limit., Comment: 40 pages, many figures

We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch., Comment: 24 pages

We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for 4009 nonlinear Schrodinger equations in dimension two, arXiv:1910.08492 ), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrodinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).

We study the semiclassical Einstein field equations with a Klein-Gordon field in ultrastatic and static spacetimes. In both cases, the equations for the spacetime metric become constraint equations. In the ultrastatic case, the Hadamard singular structure can be characterised explicitly, which allows one to in principle give initial data for the Wightman function that has correct distributional singularities, such that the expectation value of the renormalised stress-energy tensor of the solution can be defined, and that hence the semiclassical Einstein equations make sense. Assuming a "positive energy" condition for the Klein-Gordon operator, we characterise the states for which, if the constraints hold for initial data, they hold everywhere in spacetime. These turn out to be time-translation invariant states. The static case is analysed by conformal techniques, effectively reducing the problem to an ultrastatic one., The previous version has been reorganised and expanded and minor errors have been fixed. 25 pages

Traversable wormhole spacetimes are constructed in Palatini version of modified f(R) gravity theory. We impose that the matter supporting these spacetimes satisfies the null energy condition (NEC) and the weak energy condition (WEC). The gravitational field equations in Palatini approach are totally different to those field equations obtained in general relativity and f(R) metric approach. In this framework, by considering $$f(R)= R - \beta R^n$$ which is a given f(R) model that has consistent results with observations, the field equations have been simplified. To explicitly examine the NEC and the WEC, we take three different shape functions $$b(r)=r_0^2/r$$ , $$b(r)=\sqrt{r_0r}$$ , and $$b(r)=r_0^3/r^2$$ . Then, the existence of traversable wormholes in Palatini f(R) gravity is investigated in the absence of exotic matter. It is clearly demonstrated that the matter stress-energy tensor satisfies the NEC and the WEC.

This paper aims to discuss the viable and analytic solution of the spherically symmetric gravastar model under the influence of modification of Gauss–Bonnet gravity, i.e., f(G) gravity, where G is the Gauss–Bonnet curvature term. For this purpose, we evaluate the field equations in corresponding theory and conservation equation with the help of an effective energy–momentum tensor. A mathematical formalism of the gravastar’s three regions, i.e., interior de-Sitter region, thin shell, and the exterior Schwarzschild vacuum region have been discussed. We, then analyze different realistic features, in particular energy, entropy, and length of the shell. The viability of these physical features is then examined through the graphical representations separately. Within the framework of an alternative theory, we have obtained the exact and singularity free model of gravastar.

In this paper, we study the thick brane system in the so-called f(Q) gravity, where the gravitational interaction was encoded by the nonmetricity Q like scalar curvature R in general relativity. With a special choice of $$f(Q)=Q-b Q^n$$ f ( Q ) = Q - b Q n , we find that the thick brane system can be solved analytically with the first-order formalism, where the complicated second-order differential equation is transformed to several first-order differential equations. Moreover, the stability of the thick brane system under tensor perturbation is also investigated. It is shown that the tachyonic states are absent and the graviton zero mode can be localized on the brane. Thus, the four-dimensional Newtonian potential can be recovered at low energy. Besides, the corrections of the massive graviton Kaluza–Klein modes to the Newtonian potential are also analyzed briefly.

In this study, we are interested to investigate the dissipative gravitational collapse of anisotropic spherically symmetric radiating star, which satisfies the initially static Karmarkar condition in f ( R , T ) gravity. Moreover, we have chosen a linear model f ( R , T ) = f ( R ) + f ( T ) , where R is Ricci scalar, T is trace of energy-momentum tensor and λ is dimensionless coupling parameter. The terms f ( R ) = R and f ( T ) = 2 λ T describe the geometry and matter of the radiating star, respectively. The complete set of the field equations for the radiating star has been given by the Lagrangian matter L m = − 1 3 ( P R + 2 P ⊥ ) . Firstly, we have established the static core because the collapse proceeds from initially static inner spacetime and further developed by the energy dissipation (non-adiabatically) in terms of heat flux in terms of Vaidya exterior metric. For the anisotropic radiating model, we have verified that the energy conditions and other physical properties are well-behaved in the background of f ( R , T ) theory. Eventually, by setting parameter λ = 0 , the results of our model are restored in General Relativity.

We have achieved the isovector and isoscalar tensor form factors of the N ( 1535 ) state in the framework of the light-cone QCD sum rule approach. In numerical computations, we have considered the most general forms of the interpolating current for the nucleon and the tensor current together with the distribution amplitudes of the N ( 1535 ) resonance. We have shown that the squared momentum transfer, Q 2 , dependence of N ( 1535 ) tensor form factors is characterized by a multipole fit function. Our results provide a perspective of the momentum dependence of the tensor form factors and can help to improve our understanding of the internal structure of nucleon states.

In this paper, we study the modified gravity theory, where and represent the scalar-torsion and trace of energy-momentum tensor, respectively. According to our cosmological study, we show that the gravity theory has a good agreement with observations. Function type for can be determined by several factors. The most important motivation for determining is that the Lagrangian of model follows the Noether continues symmetry; therefore, according to the Noether approach, we try to determine the function of . Moreover, the proposed model should behave in agreement with observations in terms of phenomenology. As a result, we compared the predictions of the model with observation constrains, for example, the cosmic microwave background and Hubble diagram can be mentioned. In this paper, we study the gravity in the FRW space-time. So, we achieved the effective Lagrangian with respect to independent variables, scale factor , scalar-torsion and trace of energy-momentum tensor . After imposing the Noether symmetry to Lagrangian, we can introduce the suitable form for the function. . Then, we calculated the Noether conservative quantity for the model.

Made available in DSpace on 2021-06-25T11:00:24Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-08-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) We present a five dimensional cosmological metric that reveals a four-dimensional energy-momentum tensor. We analyse three cases for the resulting field equations: null, positive and negative cosmological constant Λ. For the case with null cosmological constant, we obtain a solution able to describe the radiation-dominated era of the universe. The positive five-dimensional cosmological constant case yields a bounce cosmological model. In the negative Λ case, the scale factor for the line element is obtained as a(t)=c5sinh([Formula presented]|Λ|t), where c5 is a constant. This solution can remarkably describe not only the late-time cosmic acceleration but also the non-accelerated stages of the cosmic expansion, namely the radiation and matter dominated epochs in a continuous form. We obtain an analytical equation of state capable of describing the different epochs of the universe and it reads as p=ω(z)ρ with ω(z)=[Formula presented][1−[Formula presented]], with p being the pressure of the universe, ρ its density and z is the redshift. In our model the constant c5 is related to the Hubble constant as H0=[Formula presented]coth[arcsinh([Formula presented])2] and this satisfactorily fits the observational data for the low redshift sample of the experimental measurements of the Hubble parameter, which results in H0=72.2−5.5+5.3 km s−1 Mpc−1 and c5=0.600−0.058+0.061. ITA - Instituto Tecnológico de Aeronáutica - Departamento de Física Universidade de São Paulo Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Rua do Matão 1226, Cidade Universitária Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva - R. Geraldo Alckmin 519 Departamento de Física Instituto de Ciências Ambientais Químicas e Farmacêuticas (ICAQF), Universidade Federal de São Paulo - UNIFESP, Rua São Nicolau no.210, Centro Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva - R. Geraldo Alckmin 519 FAPESP: 2013/26258-2 FAPESP: 2015/08476-0 FAPESP: 2017/05859-0 FAPESP: 2018/20689-7 CNPq: 313236/2018-6 CNPq: 438562/2018-6 CAPES: 88881.309870/2018-01 CAPES: 88882.446980/2019-01

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on $$\mathbb {R}^2\times S^2$$ and $$\mathbb {R}^2 \times H^2$$ . We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor q.

In recent few years, the Gauss-Bonnet f ( G , T ) theory of gravity has fascinated considerable researchers owing to its coupling of trace of the stress-energy tensor T with the Gauss-Bonnet term G . In this context, we focuss ourselves to study bouncing universe with in f ( G , T ) gravity background. Some important preliminaries are presented along with the discussion of cosmological parameters to develop a minimal background about f ( G , T ) theory of gravity. The exact bouncing solutions with physical analysis are provided with the choice of two equation of state parameters. It is shown that the results do agree with the present values of deceleration, jerk and snap parameters. Moreover, it is concluded that the model parameters are quite important for the validity of conservation equation (as the matter coupled theories do not obey the usual conservation law).

A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

We present a six-dimensional $\mathcal{N}=(1,0)$ supersymmetric higher gauge theory in which self-duality is consistently implemented by physically trivial additional fields. The action contains both $\mathcal{N}=(1,0)$ tensor and vector multiplets and is non-trivially interacting. The tensor multiplet part is loosely related to a recently proposed action by Sen that leads to on-shell self-duality in an elegant way. As we also show, Sen's action finds a very natural and direct interpretation from a homotopy algebraic perspective., 29 pages, typos fixed, clarifications added, published version

The gauge theory of the de Sitter group, SO$(1,4)$, in the ambient space formalism has been considered in this article. This method is essential to constructing the de Sitter super-conformal gravity and Quantum gravity. $10$ gauge vector fields are needed, corresponding to $10$ generators of the de Sitter group. Using the gauge-invariant Lagrangian, the field equation of these vector fields has been obtained. The gauge vector field solutions are recalled. Then, the spin-$2$ gauge potentials are constructed from the gauge vector field. There are two possibilities for presenting this tensor field: rank-$2$ symmetric and mixed symmetry rank-$3$ tensor fields. To preserve the conformal transformation, a spin-$2$ field must be represented by a mixed symmetry rank-$3$ tensor field, $\mathcal {K}_{\alpha\beta\gamma}$. This tensor field has been rewritten using a generalized polarization tensor field and a de Sitter plane wave. This generalized polarization tensor field has been calculated as a combination of vector polarization, $\mathcal {E}_{\alpha}$, and tensor polarization of rank-$2$, $\mathcal {E}_{\alpha\beta}$, which can be used in the gravitational wave consideration. For the construction of this polarization tensor, the arbitrary constant vector fields appear. We fix it so that, in the limit, $H=0$, one obtains the polarization tensor in Minkowski space-time. It has been shown that under some simple conditions, the spin-$2$ mixed symmetry rank-$3$ tensor field can be simultaneously transformed by the unitary irreducible representation of de Sitter and conformal groups, SO$(2,4)$., Comment: 11 pages, typos corrected

The dynamics of chiral p-forms can be captured by a lower-dimensional parity-violating action motivated by a Kaluza-Klein reduction on a circle. The massless modes are (p-1)-forms with standard kinetic terms and Chern-Simons couplings to the Kaluza-Klein vector of the background metric. The massive modes are p-forms charged under the Kaluza-Klein vector and admit parity-odd first-order kinetic terms. Gauge invariance is implemented by a Stueckelberg-like mechanism using (p-1)-forms. A Chern-Simons term for the Kaluza-Klein vector is generated at one loop by massive p-form modes. These findings are shown to be consistent with anomalies and supersymmetry for six-dimensional supergravity theories with chiral tensor multiplets., Comment: 4 pages, 1 figure

In this paper, we have investigated the exact solutions of a perfect fluid Bianchi type $$-I$$ cosmological model in the context of $$f(R, T) = R + \lambda T$$ gravity, where R is Ricci scalar, and T is a trace of the energy-momentum tensor. We show that the generalized Friedman equation’s exact solution for the average scale factor a involves the hypergeometric function. This approach is novel, and it has not been previously carried out by other authors. Consequently, a set of three cosmological models, namely dust, radiation and dark energy, have been investigated. The physical and kinematical parameters in each model are discussed. It is shown that the modeled universe asymptotically becomes isotropic at late times.

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible K\"ahler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a K\"ahler structure to generalise some classical result on co-adjoint orbits., Comment: Revised and accepted version, 24 pages

We investigate the coupling of the minimal surface equation with a spinor of harmonic type. This arises as the Euler–Lagrange equations of the sum of the volume functional and the Dirac action, defined on an appropriated Dirac bundle. The solutions show a relation to Dirac-harmonic maps under some constraints on the energy-momentum tensor, extending the relation between Riemannian minimal surface and harmonic maps.

We discuss a one-parameter family of states in two-dimensional holographic conformal field theories which are constructed via the Euclidean path integral of an effective theory on a family of hyperbolic slices in the dual bulk geometry. The effective theory in question is the CFT flowed under a $T\overline{T}$ deformation, which "folds" the boundary CFT towards the bulk time-reflection symmetric slice. We propose that these novel Euclidean path integral states in the CFT can be interpreted as continuous tensor network (CTN) states. We argue that these CTN states satisfy a Ryu-Takayanagi-like minimal area upper bound on the entanglement entropies of boundary intervals, with the coefficient being equal to $\frac{1}{4G_N}$; the CTN corresponding to the bulk time-reflection symmetric slice saturates this bound. We also argue that the original state in the CFT can be written as a superposition of such CTN states, with the corresponding wavefunction being the bulk Hartle-Hawking wavefunction., 38 pages, 6 figures

In this paper, we give a new lower bound for the eigenvalues of the Dirac operator defined on the Spin Riemannian hypersurface manifold endowed with 2-tensor, in terms of the Energy-Momentum tensor, scalar curvature and extrinsic curvature. Then this estimate is improved in two different ways by considering the conformal invariance of the Dirac operator. The first is given in term of the first eigenvalue of the Yamabe operator. The latter, is given in terms of the the area of a topological 2-sphere., TUBITAK The Scientific and Technological Research Council of Turkey [120F109]; TUBITAK (The Scientific and Technological Research Council of Turkey), This study was supported by TUBITAK The Scientific and Technological Research Council of Turkey (Project Number: 120F109). Also, The authors would like to appreciate TUBITAK (The Scientific and Technological Research Council of Turkey) for supporting this research.

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.

In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras., 32 pages, comments are welcome

Journal of high energy physics 04(4), 130 (1-34) (2021). doi:10.1007/JHEP04(2021)130, In this work we construct the crossing symmetry equations for mixed correlators of two long and two BPS operators in 4D $ \mathcal{N} $ = 1 SCFTs. The analysis presented here illustrates how our general group theoretic approach to long superblocks and tensor structures of superconformal algebras can be applied to give explicit ready-to-use expressions. In the case at hand, we obtain a system of four crossing symmetry equations for the relevant OPE coefficients. One of these four equations coincides with the equation found and analysed by Li, Meltzer and Stergiou by restricting to the superprimary component of the long multiplets. The other three equations are new and they provide powerful additional constraints on the same OPE data., Published by SISSA, [Trieste]

The notation of Moore-Penrose inverse of matrices has been extended from matrix space to even-order tensor space with Einstein product. In this paper, we give the numerical study on the Moore-Penrose inverse of tensors via the Einstein product. More precisely, we transform the calculation of Moore-Penrose inverse of tensors via the Einstein product into solving a class of tensor equations via the Einstein product. Then, by means of the conjugate gradient method, we obtain the approximate Moore-Penrose inverse of tensors via the Einstein product. Finally, we report some numerical examples to show the efficiency of the proposed methods and testify the conclusion suggested in this paper.

We study an interesting alternative of modified gravity theory, namely, the unimodular f(R, T) gravity in which R is the Ricci scalar and T is the trace of the stress–energy tensor. We study the viability of the model by using the energy conditions. We discuss the strong, weak, null and dominant energy conditions in terms of deceleration, jerk and snap parameters. We investigate energy conditions for reconstructed unimodular f(R, T) models and give some constraints on the parametric space of the model. We observe that by setting appropriately free parameters, energy conditions can be satisfied. Furthermore, we study the stability of the solutions in perturbations framework. In this case, we investigate stability conditions for de Sitter and power law solutions and we examine viability of cosmological evolution of these perturbations. The results show that for some values of the input parameters, for which energy conditions are satisfied, de Sitter and power-law solutions may be stable.

General $\mathcal{N}=(1,0)$ supergravity-matter systems in six dimensions may be described using one of the two fully fledged superspace formulations for conformal supergravity: (i) $\mathsf{SU}(2)$ superspace; and (ii) conformal superspace. With motivation to develop rigid supersymmetric field theories in curved space, this paper is devoted to the study of the geometric symmetries of supergravity backgrounds. In particular, we introduce the notion of a conformal Killing spinor superfield $\epsilon^\alpha$, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector $\xi^a$ and tensor $\zeta^{a(n)}$ superfields. The former parametrise conformal isometries of supergravity backgrounds, which in turn yield symmetries of every superconformal field theory. Meanwhile, the conformal Killing tensors of a given background are associated with higher symmetries of the hypermultiplet. By studying the higher symmetries of a non-conformal vector multiplet we introduce the concept of a Killing tensor superfield. We also analyse the problem of computing higher symmetries for the conformal d'Alembertian in curved space and demonstrate that, beyond the first-order case, these operators are defined only on conformally flat backgrounds., Comment: 62 pages

In this paper, we define a (1, 3)-tensor field T(X, Y)Z on Kenmotsu manifolds and give a necessary and sufficient condition for T to be a curvature-like tensor. Next, we present some properties related to the curvature-like tensor T and prove that $$M^{2m+1}$$ is an $$\eta $$ -Einstein–Kenmotsu manifold if and only if $$\sum ^{m}_{j=1}T( \varphi (e_j), e_j) X = 0$$ . Besides, we define a (1, 4)-tensor field t on the Kenmotsu manifold M which determines when M is a Chaki T-pseudo-symmetric manifold. Then, we obtain a formula for the covariant derivative of the curvature tensor of Kenmotsu manifold M. We also find some conditions under which an $$\eta $$ -Einstein–Kenmotsu manifold is a Chaki T-pseudo-symmetric. Finally, we give an example to verify our results and prove that every three-dimensional Kenmotsu manifold is a generalized pseudo-symmetric manifold.

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat gradient steady Ricci solitons. More precisely, we prove that any $n$-dimensional complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well., 10 pages; final version to appear in Proc. Amer. Math. Soc. arXiv admin note: text overlap with arXiv:1105.3163

The position-dependent non-conservative forces are called curl forces introduced recently by Berry and Shukla (J Phys A 45:305201, 2012). The aim of this paper is to study mainly the curl force dynamics of non-conservative central force $$\ddot{x} = -xg(x,y)$$ and $$\ddot{y} = -yg(x,y)$$ connected to higher-order saddle potentials. In particular, we study the dynamics of the type $$\ddot{x}_i = -x_ig \big (\frac{1}{2}(x_{1}^{2} - x_{2}^{2}) \big )$$ , $$i=1,2$$ and its application towards the trapping of ions. We also study the higher-order saddle surfaces, using the pair of higher-order saddle surfaces and rotated saddle surfaces by constructing a generalized rotating shaft equation. The complex curl force can also be constructed by using this pair. By the direct computation, we show that all these motions of higher-order saddles are completely integrable due to the existence of two conserved quantities, viz. energy function and the Fradkin tensor. The Newtonian system $$\ddot{x} = {{\mathcal {X}}}(x,y)$$ , $$\ddot{y} = {{\mathcal {Y}}}(x,y)$$ has also been studied with an energy like first integral $$I(\mathbf{x}, \dot{\mathbf{x}}) = \frac{1}{2}\dot{\mathbf{x}}^TM(\mathbf{x})\dot{\mathbf{x}} + U(\mathbf{x})$$ , where $$M(\mathbf{x})$$ is a $$(2 \times 2)$$ matrix of which the components are polynomials of degree less than or equal to two and the condition on $${{\mathcal {X}}}$$ and $${{\mathcal {Y}}}$$ for which the curl is non-vanishing is also obtained.

This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor $$\mathcal {D}(u)$$ D ( u ) to the 3D Navier–Stokes equations in Lorentz spaces. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that the norm $$\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} $$ ‖ λ 2 + ‖ L p , ∞ ( 0 , T ; L q , ∞ ( R 3 ) ) with $$ {2}/{p}+{3}/{q}=2$$ 2 / p + 3 / q = 2 $$( {3}/{2} ( 3 / 2 < q ≤ ∞ ) is small. This generalizes the corresponding works of Neustupa–Penel and Miller.

The classical theory of radiation of charged particles is currently a complete theory. It has wide applications in the creation of various sources of radiation: synchrotron, undulator, etc. At the same time, the issues of radiation of the intrinsic magnetic moment of particles have not been fully investigated. In this paper, we describe a way to construct radiation fields of a point source within the framework of the Hertz tensor formalism. The derivation of the potentials and field strengths of a point radiation source is given.

The energy-momentum tensor (EMT) for a spin-3/2 baryon is related to \emph{seven} mechanical quantities. In this work, we provide the general form of the gravitational form factors (GFFs) for a spin-3/2 baryon by using the multipole expansion and interesting relations between the EMT densities and the GFFs. To verify those general relations, we study the nucleon and the $\Delta$ GFFs within the SU(2) Skyrme model based on the large $N_{c}$ limit., Comment: 20 pages, 7 figures

The dilemma with the deficiencies of the nonlocal kernel functions as the building blocks of the Eringen’s nonlocal theory has been of concern. The aim of the current work is to provide a remedy for the calculation of the components of the nonlocal moduli tensor pertinent to face center cubic (fcc) crystals accounting for their true symmetry group. To this end, three new distinct nonlocal kernel functions which are the discrete atomistic Green’s functions in the stress space are obtained through the nonlocal dispersion relations associated with the longitudinal and shear waves in fcc crystals combined with the corresponding ones calculated via ab initio based on density functional perturbation theory (DFPT). In other words, based on nonlocal continuum and quantum mechanical considerations different kernel functions pertinent to each component of the elastic moduli tensor are calculated for the first time. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameter associated with Al, Cu, Ni, Pd, and Ag fcc crystals have been computed using ab initio density functional theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a screw dislocation inside an infinitely extended fcc medium has been calculated; the solutions for two cases where the dislocation line is along the $\left [ 0\ 0\ 1 \right ]$ and $\left [ 1\ 0\ 1 \right ]$ directions are compared. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated.

We seek the cosmological outcome of a ghost dark energy (GDE) model with sign-changeable interaction in $$f(R,T)$$ modified theory of gravity. We assume that the universe is filled with two fluid, a barotropic fluid, and ghost dark energy. With the particular choice of the functional $$f(R,T)=R+2f(T)$$ , where $$R$$ is the Ricci scalar and $$T$$ is the trace of the stress-energy tensor, we obtain exact solutions of the model by using the energy density of generalized GDE and a law of variation for the Hubble parameter $$H$$ . Interestingly, at late times, the equation of state (EoS) parameter in our model corresponds to a phantom dark energy model showing consistency with the recent observational data. A study of the statefinder parameters demonstrates that our model corresponds to the $$\Lambda$$ CDM model at certain time. A correspondence between GDE and Phantom DE is established. The phantom dynamics of the potential and the scalar field is reconstructed, which illustrates the accelerated phase of the Universe. The physical and geometric properties of the model are discussed.

The P1 - curvature tensor defined from W3 - curvature tensor has been studied in the space time of general relativity. The Bianchi like differential identity is satisfied by P1 - tensor if and only if the Ricci tensor is of Codazzi type. It is shown that Einstein like field equations can be expressed with the help of the contracted part of P1 - tensor, which is conserved if the energy momentum tensor is Codazzi type. Considering P1 -flat space time satisfying Einstein’s field equations with cosmological term, the existence of Killing vector field ξ is shown if and only if the Lie derivative of the energy-momentum tensor vanishes with respect to ξ, as well as admitting a conformal Killing vector field is established if and only if the energy-momentum tensor has the symmetry inheritance property. Finally for a P1 - flat perfect fluid space time satisfying Einstein’s equations with cosmological term, some results are obtained.

Herein, a spin 1 particle with anomalous magnetic moment in an external Coulomb field is studied. We start with the relativistic tensor system of the Proca type in Cartesian coordinates. In these equations the Γ parameter is present related to an additional characteristic of the particle. In the case of an external magnetic field, it is interpreted as an anomalous magnetic moment. In the presence of an external electric field, additional interaction terms are presented as well; moreover, the terms of the first and second orders in parameter Γ appear. The case of an external Coulomb field is considered in detail. In the nonrelativistic approximation a Pauli type equation is obtained. In the nonrelativistic equation the separation of the variables with the use of spherical vectors is realized. One separate 2-nd order differential equation is found, in which additional interaction terms are missing. Besides, we derive systems of two coupled 2-nd order equations wherein linear and quadratic in parameter Γ interaction terms are presented. Previously, another approach was developed for analyzing the vector particle with anomalous magnetic moment. It was based on the use of tetrad formalism and separation of the variables in the Duffin – Kemmer equation with the help of the Wigner function. The nonrelativistic approximation was performed directly in the system of radial equations. Besides, previously formal Frobenius type solutions for an arising 4-th order differential equation were constructed; however, physically interpretable energy spectra were not found. We have proved that the radial equations derived by different methods are the same up to a simple liner transformation over two radial functions. In this paper, we have obtained a simpler 4-th order equation, the construction of Frobenius solutions becomes technically easier, but physical energy spectra are not found either.

In this paper, we demonstrate that at leading order in post Minkowskian (PM) expansion, the stress-energy tensor of Kerr-Newman can be recovered to all orders in spin from three sets of minimal coupling: the electric and gravitational minimal coupling for higher-spin particles, and the "minimal coupling" for massive spin-2 decay. These couplings are uniquely defined from kinematic consideration alone. This is shown by extracting the classical piece of the one-loop stress-energy tensor form factor, which we provide a basis that is valid to all orders in spin. The 1 PM stress tensor, and the metric in the harmonic gauge, is then recovered from the classical spin limit of the form factor., Comment: 19 pages, 3 figures

We consider the renormalized Bochner Laplacian acting on tensor powers of a positive line bundle on a compact symplectic manifold. We derive an explicit local formula for the spectral density function in terms of coefficients of the Riemannian metric and symplectic form., v2: several typos corrected, v3: final version

We prove the linear stability of subextremal Reissner–Nordstrom spacetimes as solutions to the Einstein–Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations derived in our previous works (Giorgi in Ann Henri Poincar, 21: 24852580, 2020; Giorgi in Class Quantum Grav 36:205001, 2019), where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range $$|Q

A classification of metrics and electromagnetic potentials is carried out for the case when the Hamilton–Jacobi equation admits a complete separation of variables in a space with a complete set consisting of one vector and two tensor Killing fields. The classification is supplemented by solution of the vacuum Maxwell equations.

The Lohe hierarchy is a hierarchy of finite-dimensional aggregation models consisting of the Kuramoto model, the complex Lohe sphere model, the Lohe matrix model and the Lohe tensor model. In contrast, the Schr\"odinger-Lohe model is the only known infinite-dimensional Lohe aggregation model in literature. In this paper, we provide an explicit connection between the Schr\"odinger-Lohe model and the complex Lohe sphere model, and then by exploiting this explicit relation, we construct infinite-dimensional liftings of the Lohe matrix and the Lohe tensor models. In this way, we establish the Schr\"odinger-Lohe hierarchy which corresponds to the infinite-dimensional extensions of the Lohe hierarchy. For the proposed hierarchy, we provide sufficient frameworks leading to the complete aggregation in terms of coupling strengths and initial configurations.