Scalar-Field collapse and the emergence of a de Sitter bounce core

We investigate the fully covariant dynamics of a massless scalar field undergoing spherical collapse in Painlevé–Gullstrand coordinates. The Einstein–scalar system reduces to a closed 1+1D evolution scheme for (τ,v,M), with the Misner–Sharp mass satisfying v2=2GM/r. We identify a genuinely dynamical, time-dominated regime in which ∣τ˙∣≫∣vτ′′∣+∣(2v/r+v′)τ′∣, the scalar gradient becomes timelike, and the stress tensor approaches a local de Sitter equation of state p=−ρ=12τ˙2. In this regime the interior geometry tends toward a de Sitter core with Λeff=4πG τ˙2. Tracking the areal radius R(t) of collapsing shells yields the exact radial equation R¨=−GM/R2−4πGRp. A bounce requires a turning point R˙=0 with R¨>0, leading to the dynamical condition τ˙2>3M/(4πR3). This criterion is necessary but not sufficient: it guarantees outward acceleration at a turning point but does not ensure that such a point forms dynamically.