Anisotropic Spherically Symmetric Collapsing Star From Higher Order Derivative Gravity Theory

Adding linear combinations $R^2,R_{\mu\nu}R^{\mu\nu}$ and $R_{\mu\nu\eta\delta}R^{\mu\nu\eta\delta}$ with Einstein-Hilbert action we obtain interior metric of an an-isotropic spherically symmetric collapsing (ASSC) stellar cloud. We assume stress tensor of the higher order geometrical terms to be treat as an-isotropic imperfect fluid with time dependent density function $\rho(t)$ and radial and tangential pressures $p_r(t)$ and $p_t(t)$ respectively. We solved linearized metric equation via perturbation method and obtained 12 different kinds of metric solutions. Calculated Ricci and Kretschmann scalars of our metric solutions are non-singular at beginning of the collapse for 2 kinds of them only. Event and apparent horizons are formed at finite times for two kinds of singular metric solutions while 3 metric solutions exhibit with event horizon only with no formed apparent horizon. There are obtained 3 other kinds of the metric solutions which exhibit with apparent horizon with no formed event horizon. Furthermore 3 kinds of our metric solutions do not exhibit with horizons. Barotropic index of all 12 kinds of metric solutions are calculated also. They satisfy different regimes such as domain walls (6 kinds), cosmic string (2 kinds), dark matter (2 kinds), anti-matter (namely negative energy density) (1 kind) and stiff matter (1 kind). Time dependent radial null geodesics expansion parameter $\Theta(t)$ is also calculated for all 12 kinds of our metric solutions. In summary 4 kinds of our solutions take absolutely positive value $\Theta>0$ which means the collapse ended to a naked singularity but 8 kinds of our metric solutions ended to a covered singularity at end of the collapse where $\Theta\leq0$ and so trapped surfaces are appeared.