Criticality and instability of black hole spacetimes

Ziel

My proposed project lies at the interface of analysis, geometry and partial differential equations. Specifically, the project focuses on rigorously investigating foundational questions and phenomena in gravitational physics within the mathematical theory of general relativity formulated by Einstein in 1915. My emphasis is on understanding the critical formation and turbulent instabilities of black hole spacetimes.
I Criticality. The first major objective of the project is to advance our understanding of criticality and the role
of extremal black holes. In their seminal 1973 work on black hole thermodynamics, Bardeen, Carter and Hawking
conjectured that extremal black holes cannot form from collapsing matter and represent merely an idealized,
unattainable limit. This conjecture was recently disproved through the construction of matter configurations that
collapse under their own gravitational attraction to form extremal black holes. Moreover, such configurations can
exhibit the phenomenon of extremal critical collapse: they lie exactly at the threshold separating collapse to a
black hole from non-collapse. The first objective is to explore the stability and universality of this novel critical
phenomenon.
II Instability. The second major objective of the project concerns black holes in the presence of a negative
cosmological constant Λ < 0. Such black holes have taken a very prominent role in high energy physics but a
rigorous understanding of their nonlinear dynamical behavior upon perturbation has remained elusive due to
turbulence phenomena suggested by heuristics. Starting from a model case, the ultimate goal of the project is to
go beyond the heuristics and mathematically prove that these types of black holes exhibit turbulent instabilities,
provide insights into their long time dynamics as well as give precise criteria for the presence of nonlinear stability.

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