Mathematical modelling and analysis of gravitational collapse in curved geometry

•Modified gravity.•Gravitational collapse.•Singularities formation.•Spatial curvature. Background and objectives: In the visible universe, it is believed that mass and energy are interchangeable. However, the physical and chemical processes in the hidden world put the scientist into the thought of matter and energy contents that are responsible for these phenomena. These are regarded as dark matter and dark energy. In this article, we study the effects of spacetime curvature on the gravitational collapse of dark energy in modified gravity, considering the collapse of the spherically symmetric star, which is composed of perfect and homogeneous fluid. We studied the collapse for closed, flat and hyperbolic geometry. Method: As a result of mathematical modeling, we achieved highly non-linear differential equations. For the solution, we needed the assumption of physical significance. Specifically, we have taken the dark energy collapse. Then we achieved a simple system and solved for the analytic solutions of the field equations. Results: It is shown that the possible collapse is visibly influenced by spatial curvature. The collapse time is advanced for closed spacetime, delayed for the hypersurface, and the flat space behaves intermediately. We have taken here the equation of state in linear form to discuss the exhibition of fluid profile and a specific necessary criterion for the occurrence of spacetime singularity. Conclusion: In this paper, we study the mathematical model of gravitational collapse in modified gravity, which derives the field equations using the principle of least action. The significant outcomes are the influences of the spatial curvature on the collapsing process and the time of formation of spacetime singularity. The matching of boundary and the fundamental continuity of the 1-form and 2-form are discussed.