4.5 Critical phenomena and naked 4 Extensions of the basic 4.3 Approximate self-similarity and universality

4.4 Gravity regularizes self-similar matter 

One important aspect of self-similar critical solutions is that they have no equivalent in the limit of vanishing gravity. The critical solution arises from a time evolution of smooth, even analytic initial data. It should therefore itself be analytic outside the future of its singularity. Self-similar spherical matter fields in spacetime are singular either at the center of spherical symmetry (to the past of the singularity), or at the past characteristic cone of the singularity. Only adding gravity makes solutions possible that are regular at both places. As an example we consider the spherical massless scalar field.

4.4.1 The massless scalar field on flat spacetime

It is instructive to consider the self-similar solutions of a simple matter field, the massless scalar field, in spherical symmetry without gravity. The general solution of the spherically symmetric wave equation is of course


where f (z) and g (z) are two free functions of one variable ranging from tex2html_wrap_inline2927 to tex2html_wrap_inline2929 . 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 tex2html_wrap_inline2337 defined by


and with range tex2html_wrap_inline2947, tex2html_wrap_inline2949 . These coordinates are adapted to self-similarity, but unlike the x and tex2html_wrap_inline2337 introduced in (12Popup Equation) 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 tex2html_wrap_inline2959 and tex2html_wrap_inline2961 for z >0. Similarly, we define tex2html_wrap_inline2965 and tex2html_wrap_inline2967 for z <0 to cover the sectors | t |< r and t <- r . Note that tex2html_wrap_inline2975 and tex2html_wrap_inline2977 together contain the same information as f (z).

Continuous self-similarity tex2html_wrap_inline2981 is equivalent to tex2html_wrap_inline2983 and tex2html_wrap_inline2985 being constant. Discrete self-similarity requires them to be periodic in z with period tex2html_wrap_inline2377 . The condition for regularity at r =0 for t >0 is tex2html_wrap_inline2995, while regularity at r =0 for t <0 requires tex2html_wrap_inline3001 . Regularity at t = r requires tex2html_wrap_inline3005 to vanish, while regularity at t =- r requires tex2html_wrap_inline3009 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 tex2html_wrap_inline3013, 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.

4.4.2 The self-gravitating massless scalar field

The presence of gravity changes this singularity structure qualitatively. Dimensional analysis applied to the metric (23Popup Equation) or (28Popup Equation) shows that tex2html_wrap_inline2603 [the point (t = r =0)] is now a curvature singularity (unless the self-similar spacetime is Minkowski). But elsewhere, the solution can be more regular. There is a one-parameter family of exact spherically symmetric scalar field solutions found by Roberts [112Jump To The Next Citation Point In The ArticlePopup Equation] that is regular at both the future and past light cone of the singularity, not only at one of them. (It is singular at the past and future branch of r =0.) The only solution without gravity with this property is tex2html_wrap_inline3025 . The Roberts solution will be discussed in more detail in Section  4.5 below.

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 tex2html_wrap_inline2337, 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 tex2html_wrap_inline2537 and tex2html_wrap_inline3045 . 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 tex2html_wrap_inline3051, or in the case of scale-periodicity, by tex2html_wrap_inline3053, with tex2html_wrap_inline3055 periodic in tex2html_wrap_inline2337 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 tex2html_wrap_inline3059, while the origin is at tex2html_wrap_inline3061, and so that the period in tex2html_wrap_inline3063 is now always tex2html_wrap_inline3065 . In the DSS case the periodic function tex2html_wrap_inline3067 and the constant tex2html_wrap_inline2377 now appear explicitly in the field equations, and they must be solved for as nonlinear eigenvalues. In the CSS case, the constant tex2html_wrap_inline3055 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 (12Popup Equation) adapted to self-similarity and in a form ready for posing the boundary value problem. (The equations of [71Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline2323 by


as a dependent variable. In the scalar field wave equation (6Popup Equation) we use the Einstein equations (8Popup Equation) and (9Popup Equation) to eliminate tex2html_wrap_inline2251 and tex2html_wrap_inline2255, and obtain


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


As suggested by the format of the equations, they can be treated as four evolution equations in tex2html_wrap_inline3079 and one constraint that is propagated by them. The freedom in tex2html_wrap_inline3067 is to be used to make D =1 at tex2html_wrap_inline3059 . Now tex2html_wrap_inline3061 and tex2html_wrap_inline3059 resemble ``regular singular points'', if we are prepared to generalize this concept from linear ODEs to nonlinear PDEs. Near tex2html_wrap_inline3061, the four evolution equations are clearly of the form tex2html_wrap_inline3093 . That tex2html_wrap_inline3059 is also a regular singular point becomes clearest if we replace D by tex2html_wrap_inline3099 . The ``evolution'' equation for tex2html_wrap_inline3101 near tex2html_wrap_inline3059 then takes the form tex2html_wrap_inline3105, 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 tex2html_wrap_inline3063 -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 .

4.5 Critical phenomena and naked 4 Extensions of the basic 4.3 Approximate self-similarity and universality

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
© Max-Planck-Gesellschaft. ISSN 1433-8351
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