where
*f*
(*z*) and
*g*
(*z*) are two free functions of one variable ranging from
to
.
*f*
describes ingoing and
*g*
outgoing waves. Regularity at the center
*r*
=0 for all
*t*
requires
*f*
(*z*)=
*g*
(*z*) for
*f*
(*z*) being a smooth function. Physically this means that ingoing
waves move through the center and become outgoing waves. Now we
transform to new coordinates
*x*
and
defined by

and with range
,
. These coordinates are adapted to self-similarity, but unlike
the
*x*
and
introduced in (12) they cover all of Minkowski space with the exception of the
point (*t*
=
*r*
=0). The general solution of the wave equation for
*t*
>
*r*
can formally be written as

through the substitution
and
for
*z*
>0. Similarly, we define
and
for
*z*
<0 to cover the sectors |
*t*
|<
*r*
and
*t*
<-
*r*
. Note that
and
together contain the same information as
*f*
(*z*).

Continuous self-similarity
is equivalent to
and
being constant. Discrete self-similarity requires them to be
periodic in
*z*
with period
. The condition for regularity at
*r*
=0 for
*t*
>0 is
, while regularity at
*r*
=0 for
*t*
<0 requires
. Regularity at
*t*
=
*r*
requires
to vanish, while regularity at
*t*
=-
*r*
requires
to vanish.

We conclude that a self-similar solution (continuous or
discrete), is either zero everywhere, or else it is regular in
only one of three places: At the center
*r*
=0 for
, at the past light cone
*t*
=-
*r*, or at the future light cone
*t*
=
*r*
. We conjecture that other simple matter fields, such as the
perfect fluid, show similar behavior.

Similarly, the scale-invariant or scale-periodic solutions
found in near-critical collapse simulations are regular at both
the past branch of
*r*
=0 and the past light cone (or sound cone, in the case of the
perfect fluid). Once more, in the absence of gravity only the
trivial solution has this property.

I have already argued that the critical solution must be as
smooth on the past light cone as elsewhere, as it arises from the
collapse of generic smooth initial data. No lowering of
differentiability or other unusual behavior should take place
before a curvature singularity arises at the center. As Evans
first realized, this requirement turns the scale-invariant or
scale-periodic ansatz into a boundary value problem between the
past branch of
*r*
=0 and the past sound cone, that is, roughly speaking, between
*x*
=0 and
*x*
=1.

In the CSS ansatz in spherical symmetry suitable for the
perfect fluid, all fields depend only on
*x*, and one obtains an ODE boundary value problem. In a
scale-periodic ansatz in spherical symmetry, such as for the
scalar field, all fields are periodic in
, and one obtains a 1+1 dimensional hyperbolic boundary value
problem on a coordinate square, with regularity conditions at,
say,
*x*
=0 and
*x*
=1, and periodic boundary conditions at
and
. Well-behaved numerical solutions of these problems have been
obtained, with numerical evidence that they are locally unique,
and they agree well with the universal solution that emerges in
collapse simulations (references are given in the column
``Critical solution'' of Table
1). It remains an open mathematical problem to prove existence and
(local) uniqueness of the solution defined by regularity at the
center and the past light cone.

One important technical detail should be mentioned here. In
the curved solutions, the past light cone of the singularity is
not in general
*r*
=-
*t*, or
*x*
=1, but is given by
, or in the case of scale-periodicity, by
, with
periodic in
and initially unknown. The same problem arises for the sound
cone. It is convenient to make the coordinate transformation

so that the sound cone or light cone is by definition at , while the origin is at , and so that the period in is now always . In the DSS case the periodic function and the constant now appear explicitly in the field equations, and they must be solved for as nonlinear eigenvalues. In the CSS case, the constant appears, and must be solved for as a nonlinear eigenvalue.

As an example for a DSS ansatz, we give the equations for the spherically symmetric massless scalar field in the coordinates (12) adapted to self-similarity and in a form ready for posing the boundary value problem. (The equations of [71] have been adapted to the notation of this review.) We introduce the first-order matter variables

which describe ingoing and outgoing waves. It is also useful to replace by

as a dependent variable. In the scalar field wave equation (6) we use the Einstein equations (8) and (9) to eliminate and , and obtain

The three Einstein equations (7, 8, 9) become

As suggested by the format of the equations, they can be
treated as four evolution equations in
and one constraint that is propagated by them. The freedom in
is to be used to make
*D*
=1 at
. Now
and
resemble ``regular singular points'', if we are prepared to
generalize this concept from linear ODEs to nonlinear PDEs. Near
, the four evolution equations are clearly of the form
. That
is also a regular singular point becomes clearest if we replace
*D*
by
. The ``evolution'' equation for
near
then takes the form
, while the other three equations are regular.

This format of the equations also demonstrates how to restrict from a DSS to a CSS ansatz: One simply drops the -derivatives. The constraint then becomes algebraic, and the resulting ODE system can be considered to have three rather than four dependent variables.

Given that the critical solutions are regular at the past
branch of
*r*
=0 and at the past sound cone of the singularity, and that they
are self-similar, one would expect them to be singular at the
future light cone of the singularity (because after solving the
boundary value problem there is no free parameter left in the
solution). The real situation is more subtle as we shall see in
Section
4.5
.

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