1.3 Plan of this review1 Introduction1.1 Definition of the topic

1.2 Historical introduction

In 1987 Christodoulou, who was studying the spherically symmetric Einstein-scalar model analytically [45, 46Jump To The Next Citation Point In The Article, 47Jump To The Next Citation Point In The Article, 48, 49Jump To The Next Citation Point In The Article], suggested to Matt Choptuik, who was investigating the same system numerically, the following question [38Jump To The Next Citation Point In The Article]: Consider a generic smooth one-parameter family of asymptotically flat smooth initial data, such that for large values of the parameter p a black hole is formed, and no black hole is formed for small p . If one makes a bisection search for the critical value tex2html_wrap_inline2221 where a black hole is just formed, does the black hole have finite or infinitesimal mass? After developing advanced numerical methods for this purpose, Choptuik managed to give highly convincing numerical evidence that the mass is infinitesimal. Moreover he found two totally unexpected phenomena [37Jump To The Next Citation Point In The Article]: The first is the now famous scaling relation

  equation16

for the black hole mass M in the limit tex2html_wrap_inline2225 (but tex2html_wrap_inline2227). Choptuik found tex2html_wrap_inline2229 . The second is the appearance of a highly complicated, scale-periodic solution for tex2html_wrap_inline2225 . The logarithmic scale period of this solution, tex2html_wrap_inline2233, is a second dimensionless number coming out of the blue. As a third remarkable phenomenon, both the ``critical exponent'' and ``critical solution'' are ``universal'', that is the same for all one-parameter families ever investigated. Similar phenomena to Choptuik's results were quickly found in other systems too, suggesting that they were limited neither to scalar field matter nor to spherical symmetry. Most of what is now understood in critical phenomena is based on a mixture of analytical and numerical work.

Critical phenomena are arguably the most important contribution from numerical relativity to new knowledge in general relativity to date. At first researchers were intrigued by the appearance of a complicated ``echoing'' structure and two mysterious dimensionless numbers in the evolution of generic smooth initial data. Later it was realized that critical collapse also provides a natural route to naked singularities, and that it constitutes a new generic strong field regime of classical general relativity, similar in universal importance to the black hole end states of collapse.



1.3 Plan of this review1 Introduction1.1 Definition of the topic

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
© Max-Planck-Gesellschaft. ISSN 1433-8351
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