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  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
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