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 . 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  could verify the mass scaling law with a relatively simple code, due to the fact that it holds even quite far from criticality. Garfinkle  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  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
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