3.2 The black hole threshold

The free data for the system are the two functions Π (r,0) and Φ (r,0). Choptuik investigated several 1-parameter families of such data by evolving the data for many different values of the parameter. Simple examples of such families are Π (r,0) = 0 and a Gaussian for Φ (r,0), with the parameter p taken to be either the amplitude of the Gaussian, with the width and centre fixed, or the width, with position and amplitude fixed, or the position, with width and amplitude fixed. For sufficiently small amplitude (or the peak sufficiently wide), the scalar field will disperse, and for sufficiently large amplitude it will form a black hole.

Generic 1-parameter families behave in this way, but this is difficult to prove in generality. Christodoulou showed for the spherically symmetric scalar field system that data sufficiently weak in a well-defined way evolve to a Minkowski-like spacetime [5861], and that a class of sufficiently strong data forms a black hole [60].

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.

As p → p∗ along the family, the spacetime varies on ever smaller scales. 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, he could determine p ∗ up to a relative precision of 10–15, and make black holes as small as 10–6 times the ADM mass of the spacetime. The power-law scaling (10View Equation) 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 [49Jump To The Next Citation Point]. There were no families of initial data which did not show the universal critical solution and critical exponent. Choptuik therefore conjectured that γ is the same for all 1-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 p − p∗.

It is an empirical fact that typical 1-parameter families cross the threshold only once, so that there is every indication that it is a smooth submanifold, as we assumed in the phase space picture. Taking into account the discussion of mass scaling above, we can formally write the black hole mass as a functional of the initial data z = (ϕ(r,0),Π (r,0)) exactly as

γ M [z] = Q [z]H (P [z ])(P [z]) , (27 )
where P and Q are smooth functions on phase space and H is the Heaviside function. (Q could be absorbed into P.)

In hindsight, polar-radial gauge is well-adapted to self-similarity. In this gauge, DSS corresponds to

( ) Z (r,t) = Z enΔr,enΔt (28 )
for any integer n, where Z stands for any one of the dimensionless quantities a, α or ϕ (and therefore also for rΠ or rΦ). With
( ) --r--- t −-t∗- x = − t − t∗, τ = − ln − L , t < t∗, (29 )
DSS is
Z (x,τ + Δ ) = Z(x,τ ). (30 )
The dimensionful constants t∗ and L depend on the particular 1-parameter family of solutions, but the dimensionless critical fields a ∗, α ∗ and ϕ ∗, and in particular their dimensionless period Δ, are universal. Empirically, Δ ≃ 3.44 for the scalar field in numerical time evolutions, and Δ = 3.445452402 (3) from a numerical construction of the critical solution based on exact self-similarity and analyticity [157Jump To The Next Citation Point].
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