Effects of nonlinearity of f(R) gravity and perfect fluid in Kaluza\u2013Klein models with spherical compactification

Ezgi Canay,Maxim Eingorn,Alexander Zhuk

doi:10.1140/epjc/s10052-020-7979-6

Abstract

We study the effects associated with nonlinearity of f(R) gravity and of the background perfect fluid manifested in the Kaluza–Klein model with spherical compactification. The background space-time is perturbed by a massive gravitating source which is pressureless in the external space but has an arbitrary equation of state (EoS) parameter in the internal space. As characteristics of a nonlinear perfect fluid, the squared speeds of sound are not equal to the background EoS parameters in the external and internal spaces. In this setting, we find exact solutions to the linearized Einstein equations for the perturbed metric coefficients. For nonlinear models with f^{prime prime }(R_0)ne 0, we show that these coefficients acquire correction terms in the form of two summed Yukawa potentials and that in the degenerated case, the solutions are reduced to a single Yukawa potential with some “corrupted” prefactor (in front of the exponential function), which, in addition to the standard 1/r term, contains a contribution independent of the three-dimensional distance r. In the linear f''(R)=0 model, we generalize the previous studies to the case of an arbitrary nonlinear perfect fluid. We also investigate the particular case of the nonlinear background perfect fluid with zero speed of sound in the external space and demonstrate that a non-trivial solution exists only in the case of f''(R_0)=0.

Highlights

Energy and the dark matter problem and neither to unify all fundamental interactions into a single theory
The static background metric defined on the product manifold M = M4 × Md is perturbed by a compact gravitating mass and the perturbed metric coefficients are investigated in the weak field limit
In linear models with spherical compactification, corrections to the metric coefficients caused by the gravitating mass acquire the form of the Yukawa potential with the Yukawa mass defined by the radius a of the sphere: mrad ∼ 1/a

Summary

Introduction

It is appealing to study models which combine both approaches, in other words, to consider nonlinear f (R) models in multidimensional space-time. In linear models with spherical compactification, corrections to the metric coefficients caused by the gravitating mass acquire the form of the Yukawa potential with the Yukawa mass defined by the radius a of the sphere: mrad ∼ 1/a. This scalar degree of freedom (so-called gravexcitons/radions [36,37]) is a result of the variations in the internal space volume. We generalize the linear model f (R) = R + 2κΛ6 previously considered in [31,32,33] to the case of an arbitrary nonlinear background perfect fluid. In Appendix we collect the formulas for the perturbations of the Ricci tensor which we use to construct the linearized Einstein equations

Basic equations

Conclusion