and the matter equation is

Note that the matter equation of motion is contained within the contracted Bianchi identities. Choptuik chose Schwarzschild-like coordinates

where
is the metric on the unit 2-sphere. This choice of coordinates
is defined by the radius
*r*
giving the surface area of 2-spheres as
, and by
*t*
being orthogonal to
*r*
(polar-radial coordinates). One more condition is required to
fix the coordinate completely. Choptuik chose
at
*r*
=0, so that
*t*
is the proper time of the central observer.

In the auxiliary variables

the wave equation becomes a first-order system,

In spherical symmetry there are four algebraically independent components of the Einstein equations. Of these, one is a linear combination of derivatives of the other and can be disregarded. The other three contain only first derivatives of the metric, namely , and . Choptuik chose to use the equations giving and for his numerical scheme, so that only the scalar field is evolved, but the two metric coefficients are calculated from the matter at each new time step. (The main advantage of such a numerical scheme is its stability.) These two equations are

and they are, respectively, the Hamiltonian constraint and the slicing condition. These four first-order equations totally describe the system. For completeness, we also give the remaining Einstein equation,

But what happens in between? Choptuik found that in all
1-parameter families of initial data he investigated he could
make arbitrarily small black holes by fine-tuning the parameter
*p*
close to the black hole threshold. An important fact is that
there is nothing visibly special to the black hole threshold. One
cannot tell that one given data set will form a black hole and
another one infinitesimally close will not, short of evolving
both for a sufficiently long time. ``Fine-tuning'' of
*p*
to the black hole threshold proceeds by bisection: Starting with
two data sets one of which forms a black hole, try a third one in
between along some one-parameter family linking the two, drop one
of the old sets and repeat.

With
*p*
closer to
, the spacetime varies on ever smaller scales. The only limit
was numerical resolution, and in order to push that limitation
further away, Choptuik developed numerical techniques that
recursively refine the numerical grid in spacetime regions where
details arise on scales too small to be resolved properly. In the
end, Choptuik could determine
up to a relative precision of
, and make black holes as small as
times the ADM mass of the spacetime. The power-law scaling (1) was obeyed from those smallest masses up to black hole masses
of, for some families, 0.9 of the ADM mass, that is, over six
orders of magnitude [38]. There were no families of initial data which did not show the
universal critical solution and critical exponent. Choptuik
therefore conjectured that
is the same for all one-parameter families of smooth,
asymptotically flat initial data that depend smoothly on the
parameter, and that the approximate scaling law holds ever better
for arbitrarily small
.

Choptuik's results for individual 1-parameter families of data
suggest that there is a smooth hypersurface in the
(infinite-dimensional) phase space of smooth data which divides
black hole from non-black hole data. Let
*P*
be any smooth scalar function on the space so that
*P*
=0 is the black hole threshold. Then, for any choice of
*P*, there is a second smooth function
*C*
on the space so that the black hole mass as a function of the
initial data is

The entire unsmoothness at the black hole threshold is now captured by the non-integer power. We should stress that this formulation of Choptuik's mass scaling result is not even a conjecture, as we have not stated on what function space it is supposed to hold. Nevertheless, considering 1-parameter families of initial data is only a tool for numerical investigations of the the infinite-dimensional space of initial data, and a convenient way of expressing analytic approximations.

Clearly a collapse spacetime which has ADM mass 1, but settles
down to a black hole of mass (for example)
has to show structure on very different scales. The same is true
for a spacetime which is as close to the black hole threshold,
but on the other side: The scalar wave contracts until curvature
values of order
are reached in a spacetime region of size
before it starts to disperse. Choptuik found that all
near-critical spacetimes, for all families of initial data, look
the same in an intermediate region, that is they approximate one
universal spacetime, which is also called the critical solution.
This spacetime is scale-periodic in the sense that there is a
value
of
*t*
such that when we shift the origin of
*t*
to
, we have

for all integer
*n*
and for
, and where
*Z*
stands for any one of
*a*,
or
(and therefore also for
or
). The accumulation point
depends on the family, but the scale-periodic part of the
near-critical solutions does not.

This result is sufficiently surprising to formulate it once
more in a slightly different manner. Let us replace
*r*
and
*t*
by a pair of auxiliary variables such that one of them is the
logarithm of an overall spacetime scale. A simple example is

(
has been defined so that it increases as
*t*
increases and approaches
from below. It is useful to think of
*r*,
*t*
and
*L*
as having dimension length in units
*c*
=
*G*
=1, and of
*x*
and
as dimensionless.) Choptuik's observation, expressed in these
coordinates, is that in any near-critical solution there is a
spacetime region where the fields
*a*,
and
are well approximated by their values in a universal solution,
as

where the fields , and of the critical solution have the property

The dimensionful constants
and
*L*
depend on the particular one-parameter family of solutions, but
the dimensionless critical fields
,
and
, and in particular their dimensionless period
, are universal.

The evolution of near-critical initial data starts resembling
the universal critical solution beginning at some length scale
that is related (with some factor of order one) to the initial
data scale. A slightly supercritical and a slightly subcritical
solution from the same family (so that
*L*
and
are the same) are practically indistinguishable until they have
reached a very small scale where the one forms an apparent
horizon, while the other starts dispersing. If a black hole is
formed, its mass is related (with a factor of order one) to this
scale, and so we have for the range
of
on which a near-critical solution approximates the universal
one

where the unknown factors of order one give rise to the
unknown constant. As the critical solution is periodic in
with period
for the number
*N*
of scaling ``echos'' that are seen, we then have the
expression

Note that this holds for both supercritical and subcritical solutions.

Choptuik's results have been repeated by a number of other
authors. Gundlach, Price and Pullin [79] could verify the mass scaling law with a relatively simple
code, due to the fact that it holds even quite far from
criticality. Garfinkle [58] used the fact that recursive grid refinement in near-critical
solutions is not required in arbitrary places, but that all
refined grids are centered on
, in order to use a simple fixed mesh refinement on a single
grid in double null coordinates:
*u*
grid lines accumulate at
*u*
=0, and
*v*
lines at
*v*
=0, with (*v*
=0,
*u*
=0) chosen to coincide with
. Hamadé and Stewart [81] have written an adaptive mesh refinement algorithm based on a
double null grid (but using coordinates
*u*
and
*r*), and report even higher resolution than Choptuik. Their
coordinate choice also allowed them to follow the evolution
beyond the formation of an apparent horizon.

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