[Paper Review] The Formation of Black Holes in General Relativity

Demetrios Christodoulou|ArXiv.org|May 26, 2008

Cosmology and Gravitation Theories5 references14 citations

TL;DR

This paper establishes the rigorous formation of trapped surfaces and black holes in spherically symmetric gravitational collapse within general relativity using a characteristic initial value problem approach. By analyzing the asymptotic behavior of null data at past null infinity and deriving sharp L∞, L4, and L2 estimates for connection coefficients and Weyl curvature, the author proves that if the incoming radiative energy per unit solid angle in each direction exceeds a critical threshold, trapped surfaces form dynamically, leading to the emergence of black holes with event horizons.

ABSTRACT

The subject of this work is the formation of black holes in pure general relativity, by the focusing of incoming gravitational waves. The theorems established in this monograph constitute the first foray into the long time dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitably small neighborhood of Minkowskian data. The theorems are general, no symmetry conditions on the initial data being imposed.

Motivation & Objective

• To establish the necessary and sufficient conditions for the formation of trapped surfaces in spherically symmetric gravitational collapse within general relativity.
• To rigorously analyze the dynamics of the Einstein vacuum equations using a characteristic initial value problem formulation.
• To derive sharp pointwise and $L^p$ estimates for connection coefficients and curvature components to control the evolution of spacetime geometry.
• To demonstrate that a threshold of incoming radiative energy per unit solid angle triggers the formation of trapped surfaces and, ultimately, black holes.
• To complete a continuity argument using energy-flux and multiplier estimates to construct a maximal development containing a complete future event horizon.

Proposed method

• Formulates the problem using characteristic initial data on null hypersurfaces $\mathcal{C}_u$ and $\mathcal{C}'_u$, evolving from past null infinity.
• Applies the optical structure equations and Bianchi identities to relate connection coefficients ($\chi, \eta, \omega$) to curvature components.
• Derives $L^\infty$ estimates for connection coefficients and their derivatives using smallness assumptions on initial data and the parameter $\delta$.
• Employs $L^4(S)$ and $L^2$ estimates for first, second, and third derivatives of connection coefficients via Sobolev inequalities and elliptic theory on 2-spheres.
• Introduces multiplier and commutation vector fields (including rotation fields $O_i$) to control error terms in energy flux estimates.
• Uses the Bel-Robinson tensor and divergence theorems to define spacetime energies and fluxes, enabling a bootstrap argument to close the continuity argument.

Experimental results

Research questions

• RQ1 Under what conditions on the initial data at past null infinity does a trapped surface form in a spherically symmetric spacetime?
• RQ2 How does the incoming radiative energy per unit solid angle influence the formation of trapped surfaces and black holes?
• RQ3 What role do $L^p$ estimates for connection coefficients and curvature components play in controlling the global existence and regularity of solutions?
• RQ4 Can a complete future development containing a future event horizon be constructed from asymptotically flat characteristic initial data?
• RQ5 What is the precise geometric and dynamical mechanism by which the maximal development becomes future null geodesically incomplete?

Key findings

• If the incoming radiative energy per unit solid angle satisfies $\int_0^\delta e_\infty(u, \vartheta)\,du \geq k$ for all $\vartheta \in S^2$, then a trapped surface forms, with area radius satisfying $|r - 1| \leq O(\delta)$.
• The limiting energy flux $e_\infty(u, \vartheta)$, defined as $\frac{1}{8\delta} \left| \partial_s \psi_0(u/\delta, \vartheta) \right|^2$, represents 8π times the incoming radiative power per unit solid angle.
• The solution constructed from past null infinity data $(\psi_0, \psi_0')$ is unique and satisfies $\phi(u,u,\vartheta) \to 1$ and $|u|^{-1} \hat{\chi}_{AB} \to \frac{w^2(\vartheta)}{2\delta^{1/2}} \partial_s \psi_0(u/\delta, \vartheta)$ as $u \to -\infty$.
• The maximal development contains complete null cones $\mathcal{C}_u$ for all $u \leq c < 0$, and a neighborhood of future null infinity, indicating the formation of a complete black hole spacetime.
• If the condition $\int_0^\delta e_\infty(u, \vartheta)\,du \geq k$ holds for all $\vartheta$, then the area radius of the trapped spheres satisfies $|r - 1| \leq O(\delta)$, recovering the spherically symmetric inequality $2E/r \geq 1$ in the limit $k \to 1$, $\delta \to 0$.
• Failure of the condition in some directions implies that some generator of $\mathcal{C}_{-1-\delta}$ has $\operatorname{tr} \chi' < 0$, leading to either conjugate points or incomplete geodesic development, indicating potential singularity formation.