Curvature Blow-up and Mass Inflation in Spherically Symmetric Collapse to a Schwarzschild Black Hole

We study the black hole interiors of spacetimes arising from gravitational collapse within the spherically symmetric Einstein-scalar field system. By investigating the precise blow-up rates of curvature and mass at the spacelike singularity, near timelike infinity, we give an answer to whether the interior metric converges to a Schwarzschild metric. We show, in particular, that the Kretschmann scalar blows up faster than in the Schwarzschild setting, due to mass inflation. Moreover, the blow-up rate is not constant and converges to the Schwarzschild rate towards timelike infinity and it depends on the precise late-time polynomial behaviour of the scalar field along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism.