Abstract:
This thesis is devoted to study the dynamical instability of some relativistic collapsing
self-gravitating structures with both Newtonian and post-Newtonian approximations
in metric f(R) gravity. In this setting, we consider evolution of spherical, cylin-
drical and restricted axial stellar systems ¯lled with expansion and expansion-free
adiabatic/non-adiabatic matter con¯gurations. We analyze the role of adiabatic index
(sti®ness parameter) in the instability constraints of these self-gravitating structures.
We construct dynamical equation using contracted Bianchi identities of the e®ective
dark sources as well as usual matter distribution. The perturbation approach is ap-
plied on physical variables and then formulate modi¯ed versions of collapse equation
which leads to instability constraints at both N and pN regimes.
We ¯rst consider spherically and cylindrically symmetric spacetimes ¯lled with
charged and uncharged expansion-free anisotropic matter distributions. It is found
that the adiabatic index does not have any role in the expansion-free evolution within
N and pN approximations. Rather, this range is governed by pressure anisotropy, ra-
dial pro¯le of system energy density, f(R) model and electric charge (for charged
distribution). We also explore the role of heat radiations in non-viscous charged
spherical and cylindrical systems as well as shearing viscous uncharged axial relativis-
tic interior. It is found that heat radiations try to decrease stability of the evolving
systems, while viscosity tends to increase system stability. The electromagnetic ¯eld
decreases instability regions for charged spherical systems while its opposite e®ects
have been observed for cylindrical collapsing self-gravitating systems. We conclude
that f(R) dark energy sources coming up from the well-known f(R) models a®ect
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the whole dynamical behavior of collapsing systems due to its repulsive nature.
Finally, we study the factors involved in the energy density irregularities of rela-
tivistic planar °uid distribution in the presence of Palatini f(R) corrections. For this
purpose, we develop a link between the Weyl scalar and structural properties of the
system by a couple of di®erential equations. We also investigate the e®ects of Palatini
f(R) terms in the formulation of structure scalars obtained by orthogonal splitting
of the Riemann tensor in general relativity. We then identify the parameters which
produce energy density irregularities in expansion and expansion-free dissipative as
well as non-dissipative matter distributions. It is found that particular combinations
of the matter variables lead to irregularities in an initially homogeneous °uid dis-
tribution. We conclude that Palatini f(R) extra corrections tend to decrease the
inhomogeneity, thereby imparting stability to the self-gravitating system.