for the black hole mass
*M*
in the limit
(but
). Choptuik found
. The second is the appearance of a highly complicated,
scale-periodic solution for
. The logarithmic scale period of this solution,
, 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.

Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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