Spherically symmetric curved space times from quantum fields backreaction corrections in two dimensional analogue

Aim of the paper is to obtain 2d analogue of the backreaction equation which will be useful to study final state of quantum perturbed spherically symmetric curved space times. Thus we take Einstein-massless-scalar $\psi$ tensor gravity model described on class of spherically symmetric curved space times. We rewrite the action functional in 2d analogue in terms of dimensionless dilaton-matter field $(\chi=\Phi\psi)$ where dilaton field $\Phi$ is conformal factor of 2-sphere. Then we seek renormalized expectation value of quantum dilaton-matter field stress tensor operator by applying Hadamard rennormalization prescription. Singularity of the Green function is assumed to be has logarithmic form. Covariantly conservation condition on the renormalized quantum dilaton-matter stress tensor demands to input a variable cosmological parameter $\lambda(x)$. Energy conditions (weak, strong and null) is studied on the obtained renormalized stress tensor leading to dynamical equations for $\lambda(x), \Phi$ and quantum vacuum state $W_0(x)=<0|\hat{\chi}^2|0>_{ren}.$ In weak quantum field limits our obtained trace anomaly corresponds to one which obtained from zeta function regularization method. Setting null-like apparent horizon equation $\nabla_c\Phi\nabla^c\Phi=0,$ our procedure predicts that physically correct value of the parameter in the anomaly trace $\frac{1}{24\pi}\{R-\alpha\frac{\nabla_{c}\nabla^{c}\Phi}{\Phi}+(\alpha-6)\frac{\nabla_c\Phi\nabla^c\Phi}{\Phi^2} \}$ should be $\alpha=6.$ At last we solved the backreaction equation and obtained explicitly metric field solution in the slow varying limits of the quantum and dilaton fields which has black holes topology and its singularity is covered by apparent horizon hypersurface.