[Paper Review] Curvature blow-up rates in spherically symmetric gravitational collapse to a Schwarzschild black hole

Xinliang An, Dejan Gajic|arXiv (Cornell University)|Apr 24, 2020

Cosmology and Gravitation Theories41 references5 citations

TL;DR

This paper investigates curvature blow-up rates in spherically symmetric gravitational collapse leading to a Schwarzschild black hole, showing that the Kretschmann scalar blows up faster than in the static Schwarzschild case due to mass inflation. Using precise late-time polynomial behavior of the scalar field along the event horizon, the authors derive that blow-up rates are not constant and converge to the Schwarzschild rate toward timelike infinity, revealing a PDE-driven blow-up mechanism distinct from ODE-based models.

ABSTRACT

We study the black hole interiors of spacetimes arising from gravitational collapse in the spherically symmetric Einstein-scalar field setting, and we investigate the precise blow-up rates of curvature and mass at the spacelike singularity near timelike infinity. 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.

Motivation & Objective

• To quantify the blow-up rates of curvature invariants, particularly the Kretschmann scalar, in dynamical black hole interiors formed by spherically symmetric gravitational collapse.
• To investigate how mass inflation and late-time behavior of the scalar field along the event horizon affect the strength and rate of curvature singularities.
• To establish that the blow-up rate of curvature is not constant but converges to the Schwarzschild rate as timelike infinity is approached.
• To demonstrate that the blow-up mechanism is driven by a PDE system rather than an ODE mechanism, distinguishing it from classical singularities.

Proposed method

• The study employs double null coordinates to parametrize the spacetime metric and define the Hawking mass and curvature invariants.
• It uses a system of PDEs derived from the Einstein-scalar field equations under spherical symmetry, with initial data satisfying inverse polynomial bounds on the scalar field's late-time behavior at the event horizon.
• The authors apply refined estimates for $ r ilde{\partial}_v r $ and $ r ilde{\partial}_u r $, showing convergence of renormalized dynamical quantities to their Schwarzschild counterparts at timelike infinity.
• A reverse Grönwall inequality is used to control lower bounds on the Hawking mass, ensuring monotonicity and blow-up in the trapped region.
• The Kretschmann scalar is estimated via lower and upper bounds derived from the Hawking mass and radial derivatives, revealing non-constant blow-up rates.
• The analysis combines asymptotic expansions and weighted energy estimates to control the behavior near the spacelike singularity.

Experimental results

Research questions

• RQ1 How does the Kretschmann scalar blow-up rate in dynamical black hole interiors compare to that in the static Schwarzschild solution?
• RQ2 What role does the late-time polynomial decay of the scalar field along the event horizon play in determining the curvature blow-up rate?
• RQ3 Does the blow-up rate of curvature remain constant or evolve toward the Schwarzschild rate as the spacetime approaches timelike infinity?
• RQ4 Is the curvature blow-up mechanism in dynamical collapse driven by PDE dynamics rather than ODE-like behavior?
• RQ5 To what extent do renormalized dynamical quantities converge to their Schwarzschild values in the black hole interior?

Key findings

• The Kretschmann scalar blows up faster than $ r^{-6} $ in dynamical black hole interiors, exceeding the Schwarzschild rate due to mass inflation.
• The blow-up rate is not constant; it asymptotically approaches the Schwarzschild $ r^{-6} $ rate as $ v \to \infty $, i.e., toward timelike infinity.
• The precise late-time polynomial behavior of the scalar field along the event horizon directly determines the curvature blow-up rate.
• The Hawking mass blows up at the spacelike singularity, and this blow-up is responsible for the enhanced curvature singularity.
• Renormalized dynamical quantities, such as $ r\partial_v r $, converge to their Schwarzschild values as $ v \to \infty $, indicating asymptotic approach to Schwarzschild behavior.
• The blow-up mechanism is fundamentally PDE-driven, as the blow-up rate depends on the solution’s evolution rather than a fixed ODE structure.