The Formation of Black Holes in General Relativity

Shortly after Einstein’s discovery of the gravitation laws, Schwarzschild discovered their particular solution. The Schwarzschild solution, a Lorentzian metric on a four-manifold, depends on one parameter (the so-called mass) and is characterized as the unique spherically symmetric vacuum solution. After a suitable “changes of coordinates”, one can remove the so-called non-true singularity of this metric. It was realized that this non-true singularity is actually an event horizon, i.e. a boundary of such a space-time region, the points of which can be linked to “the infinity” via a causal geodesic. In the 60s, R. Penrose formalized the concept of the infinity, introduced the notion of a trapped surface and was able to define future event horizons in a proper way. Moreover, he proved his Incompleteness Theorem shortly after. The presence of a trapped surface implies the existence of a so-called black hole in the studied universe. A possible forming of trapped surfaces is investigated in the book using an analysis of the dynamics of gravitational collapse. Formation of trapped surfaces is established using focusing of gravitational waves. The main tool used in this case is based on the short pulse method for general hyperbolic Euler-Lagrange systems. The monograph is written in a fairly technical way, theorems are stated carefully and computations are presented almost in their full length. The book is recommended for physicists or mathematicians working in general relativity, as well as for mathematicians interested in long time range asymptotics for partial differential equations on manifolds.

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