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 } ) \neq 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 ^ { \prime \prime } ( 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 ^ { \prime \prime } ( R _ { 0 } ) = 0 .