Geodesic deviation analysis of time conformal Schwarzschild like black hole

The astrophysical phenomena, for instance, the growth or decay of black holes (BHs), gravitational waves may continuously change the curvature of spacetimes. In addition, such phenomena also affect the thermodynamic structure of the sources over time. In this research, we examined the insertion of the time conformal factor e & varepsilon;f(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e<^>{\epsilon f(t)}$$\end{document} in the Renormalization Group Improved (RGI) Schwarzschild BH, without violating symmetry structure. We demonstrated that the curvature invariants, which are responsible for the spacetime structure around the time conformal Renormalization Group Improved Schwarzschild (TCRGIS) BH, depend explicitly on this time conformal factor. The parameter gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} that appeared in these invariants gives a complete radial profile of the square of the Ricci tensor, the Ricci scalar and the Kretschmann scalar. For positive values of gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} the behavior of the curvature of TCRGIS-BH is similar to both the RGI-BH and the Schwarzschild BH as it becomes infinite at center for gamma=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document}. Moreover, we have analyzed that the curvature invariants for TCRGIS BH decrease more rapidly as compared to both the RGI Schwarzschild BH and regular Schwarzschild BH, as a function of time.