scholarly journals New Cosmological Solutions of a Nonlocal Gravity Model

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 3
Author(s):  
Ivan Dimitrijevic ◽  
Branko Dragovich ◽  
Zoran Rakic ◽  
Jelena Stankovic
Keyword(s):  
Dark Energy ◽  
Eigenvalue Problem ◽  
Gravity Model ◽  
Radiation Effects ◽  
Equations Of Motion ◽  
Flat Space ◽  
Nonlocal Operator ◽  
Cosmological Solutions

A nonlocal gravity model (2) was introduced and considered recently, and two exact cosmological solutions in flat space were presented. The first solution is related to some radiation effects generated by nonlocal dynamics on dark energy background, while the second one is a nonsingular time symmetric bounce. In the present paper, we investigate other possible exact cosmological solutions and find some the new ones in nonflat space. Used nonlocal gravity dynamics can change the background topology. To solve the corresponding equations of motion, we first look for a solution of the eigenvalue problem □(R−4Λ)=q(R−4Λ). We also discuss possible extension of this model with a nonlocal operator, symmetric under □⟷□−1, and its connection with another interesting nonlocal gravity model.

scholarly journals Some Cosmological Solutions of a New Nonlocal Gravity Model

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 917
Author(s):  
Ivan Dimitrijevic ◽  
Branko Dragovich ◽  
Alexey S. Koshelev ◽  
Zoran Rakic ◽  
Jelena Stankovic
Keyword(s):  
General Relativity ◽  
Dark Energy ◽  
Analytic Function ◽  
Cosmological Constant ◽  
Explicit Form ◽  
Gravity Model ◽  
Necessary Conditions ◽  
Equations Of Motion ◽  
Nonlocal Operator ◽  
Cosmological Solutions

In this paper, we investigate a nonlocal modification of general relativity (GR) with action S = 1 16 π G ∫ [ R − 2 Λ + ( R − 4 Λ ) F ( □ ) ( R − 4 Λ ) ] − g d 4 x , where F ( □ ) = ∑ n = 1 + ∞ f n □ n is an analytic function of the d’Alembertian □. We found a few exact cosmological solutions of the corresponding equations of motion. There are two solutions which are valid only if Λ ≠ 0 , k = 0 , and they have no analogs in Einstein’s gravity with cosmological constant Λ . One of these two solutions is a ( t ) = A t e Λ 4 t 2 , that mimics properties similar to an interference between the radiation and the dark energy. Another solution is a nonsingular bounce one a ( t ) = A e Λ t 2 . For these two solutions, some cosmological aspects are discussed. We also found explicit form of the nonlocal operator F ( □ ) , which satisfies obtained necessary conditions.

Positivity results for a nonlocal elliptic equation

1998 ◽  
Vol 128 (4) ◽  
pp. 697-715 ◽  
Author(s):  
Pedro Freitas ◽  
Guido Sweers
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Parabolic Equations ◽  
Elliptic Systems ◽  
Stationary Solutions ◽  
Second Order ◽  
Real Eigenvalue ◽  
Nonlocal Operator ◽  
Nonlocal Parabolic Equations ◽  
Positive Eigenfunction

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

THE VILKOVISKY EFFECTIVE ACTION IN THE EVEN-DIMENSIONAL QUANTUM GRAVITY

1989 ◽  
Vol 04 (07) ◽  
pp. 633-644 ◽  
Author(s):  
I. L. BUCHBINDER ◽  
E. N. KIRILLOVA ◽  
S. D. ODINTSOV
Keyword(s):  
Numerical Calculation ◽  
Effective Potential ◽  
Effective Action ◽  
Equations Of Motion ◽  
Flat Space ◽  
Einstein Gravity ◽  
Quantum Field ◽  
Dimensional Torus ◽  
Spontaneous Compactification ◽  
The One

The one-loop Vilkovisky effective potential which is not dependent on a gauge and a parametrization of quantum field, is investigated. We have considered Einstein gravity on a background manifold of (flat space) × (d−4- sphere) or × (d−4- dimensional torus ), d is even, and of R3 × (1- sphere ), where R3 is flat space. The numerical calculation for the cases R4 × Td−4 (d = 6,8,10) and R3 × S1 is done. The solution to the one-loop corrected equations of motion is found, although the spontaneous compactification is not stable in these cases.

Feldtheoretische Konstruktion der Jordan—Brans—Dicke-Theorie / Fieldtheoretic Construction of the Jordan-Brans-Dicke-Theory

1971 ◽  
Vol 26 (4) ◽  
pp. 599-622
Author(s):  
H. von Grünberg
Keyword(s):  
Gauge Group ◽  
Tensor Field ◽  
Mathematical Proof ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Flat Space ◽  
Momentum Tensor ◽  
Field Equations ◽  
Tensor Theory ◽  
Generally Covariant

Abstract In the framework of Lorentz invariant theories of gravitation the fieldtheoretic approach of the generally covariant Jordan-Brans-Dicke-theory is investigated.It is shown that a slight restriction of the gauge group of Einstein's linear tensor theory leads to the linearized Jordan-Brans-Dicke-theory. The problem of the inconsistency of the field equations and the equations of motion is solved by introducing the Landau-Lifschitz energy momentum tensor of the gravitational field as an additional source term into the field equations. The second order of the theory together with the corresponding gauge group are calculated explicitly. By means of the structure of the gauge group of the tensor field it is possible to identify the successive orders of the scalar-tensor theory as an expansion of the Jordan-Brans-Dicke-theory in flat space-time. The question of the uniqueness of the procedure is answered by showing that the structure of the gauge group of the tensor field is predetermined by the linear equations of motion. The mathematical proof of this fact confirms formally the meaning of the equations of motion for the geometry of space.

scholarly journals LOOP VARIABLES AND THE INTERACTING OPEN STRING IN A CURVED BACKGROUND

2005 ◽  
Vol 20 (14) ◽  
pp. 1037-1045
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Gauge Invariance ◽  
Equations Of Motion ◽  
Interaction Strength ◽  
Flat Space ◽  
Open String ◽  
Target Space ◽  
Gauge Invariant ◽  
Free Case ◽  
New Feature ◽  
Generally Covariant

Applying the loop variable proposal to a sigma model (with boundary) in a curved target space, we give a systematic method for writing the gauge and generally covariant interacting equations of motion for the modes of the open string in a curved background. As in the free case described in an earlier paper, the equations are obtained by covariantizing the flat space (gauge invariant) interacting equations and then demanding gauge invariance in the curved background. The resulting equation has the form of a sum of terms that would individually be gauge invariant in flat space or at zero interaction strength, but mix amongst themselves in curved space when interactions are turned on. The new feature is that the loop variables are deformed so that there is a mixing of modes. Unlike the free case, the equations are coupled, and all the modes of the open string are required for gauge invariance.

scholarly journals Cosmological solutions of F (R, T) gravity model with k -essence

2019 ◽  
Vol 1391 ◽  
pp. 012163 ◽  
Author(s):  
Koblandy Yerzhanov ◽  
Bekdaulet Meirbekov ◽  
Gulnur Bauyrzhan ◽  
Ratbay Myrzakulov
Keyword(s):  
Gravity Model ◽  
Cosmological Solutions

scholarly journals Remarks on a nonlinear nonlocal operator in Orlicz spaces

2019 ◽  
Vol 9 (1) ◽  
pp. 305-326 ◽  
Author(s):  
Ernesto Correa ◽  
Arturo de Pablo
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Sobolev Inequality ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Laplacian Operator ◽  
Nonlocal Operator ◽  
Generalized Eigenvalue

Abstract We study integral operators $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$in a bounded domain $\Omega ,$and boundary condition u ≡ 0 on ${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem $f\left( u \right)=\lambda \psi \left( u \right).$

scholarly journals The correct and unusual coordinate transformation rules for electromagnetic quadrupoles

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170652 ◽  
Author(s):  
J. Gratus ◽  
T. Banaszek
Keyword(s):  
Coordinate Transformation ◽  
Equations Of Motion ◽  
Flat Space ◽  
Polar Coordinates ◽  
Cartesian Coordinates ◽  
Transformation Rules ◽  
Integral Transformations ◽  
Second Derivatives ◽  
Electric Dipoles ◽  
Derivatives Of

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.

RECONSTRUCTION OF 5D COSMOLOGICAL MODELS FROM THE EQUATION OF STATE OF DARK ENERGY

2006 ◽  
Vol 15 (02) ◽  
pp. 215-224 ◽  
Author(s):  
LI XIN XU ◽  
HONG YA LIU ◽  
CHENG WU ZHANG
Keyword(s):  
Dark Energy ◽  
Equation Of State ◽  
Arbitrary Function ◽  
Equations Of State ◽  
Cosmological Models ◽  
Cosmological Solutions ◽  
The Universe

We consider a class of five-dimensional cosmological solutions which contain two arbitrary function μ(t) and ν(t). We find that the arbitrary function μ(t) contained in the solutions can be rewritten in terms of the redshift z as a new arbitrary function f(z). We further show that this new arbitrary function f(z) can be solved for four known parameterized equations of state of dark energy. Then 5D models can be reconstructed and the evolution of the density and deceleration parameters of the universe can be determined.

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