Shear-free collapsing compact star in pseudo-flat 5-D spacetime emitting radiation

Abstract A shear-free, collapsing compact object, described by an embedding class I solution and emitting radiation, is presented here. Because of the properties of embedding class I, this four-dimensional interior spacetime can be embedded in pseudo-flat five-dimensional space. Starting with an initially static and shear-free gravitating body through the Karmarkar condition, we have explored the time evolution of the system using the time-dependent Einstein field equations. The interior solution is smoothly matched with the Vaidya outgoing solution across a time-like hypersurface which yields the temporal behaviour of the model. Further, a thorough physical analysis of the thermodynamic variables has been presented spatially and temporally. As the time progresses into the future, the surface compactness factor approaches the Buchdahl limit until it crosses this to proceed with the collapse. Before $$2m_\Sigma /r_\Sigma \rightarrow 8/9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mn>8</mml:mn> <mml:mo>/</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> </mml:math> , the equation of state adjusts itself like a self-immune system to hold the system by increasing $$\Gamma _{rc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mrow> <mml:mi>rc</mml:mi> </mml:mrow> </mml:msub> </mml:math> (i.e. increasing stiffness). All the physical quantities, e.g., density, pressure anisotropy, radiation, luminosity, temperature, are non-singular throughout the interior of the star at a finite time slice. As the time progresses, these quantities point towards a future singularity. Throughout the evolution, the surface density and surface pressure always remain lower than their corresponding central values. Similarly, the causal temperature also always remains higher than the non-causal counterpart throughout the interior as well as during the evolution.