4.1 Matter obeying wave equations

4.1.1 2-dimensional nonlinear σ model

We have already discussed in Section 3.6 the effects of adding a potential to the evolution of the scalar field. An alternative generalization is a modification of the kinetic term in the Lagrangian, with the general form

1- μν K I J 2 g GIJ (ϕ )∇ μϕ ∇ νϕ (35 )
for an N-dimensional vector field I ϕ (x) with I = 1,...,N, and where K GIJ (ϕ ) is a fixed nonlinear function acting as a metric on the so-called target space of the fields ϕI. Such a system is called a nonlinear σ model, harmonic map or wave map. The fields ϕI and metric GIJ are dimensionless, and this allows the introduction of dimensionless parameters in the system, which cannot be asymptotically neglected using the arguments of Section 2.6. (Compare with the potential V (ϕI ), which has dimensions (length)–2, and hence requires dimensionful parameters.)

The case N = 1 gives nothing new, and so Hirschmann and Eardly (HE from now on) studied the N = 2 case with a target manifold with constant curvature [124Jump To The Next Citation Point], proportional to a real dimensionless constant κ. Using a single complex coordinate ϕ the action of the system can be written as

∫ [ 2 ] d4x√g-- R- − ---|∇-ϕ|---- . (36 ) 2 (1 − κ|ϕ|2)2
For κ ≥ 0 this system is equivalent to the problem of a real massless scalar field coupled to Brans–Dicke (BD) gravity (with the BD coupling constant given by 8ωBD = − 12 + κ−1). Liebling and Choptuik [150Jump To The Next Citation Point] have shown that there is a smooth transition in the BD system from DSS criticality for low κ (the flat target space case κ = 0 is equivalent to a self-gravitating complex massless scalar field, whose critical solution is the original DSS spacetime found by Choptuik) to CSS criticality for larger κ (the case κ = 1 is equivalent to the axion-dilaton system, and has been shown to display CSS criticality in [110Jump To The Next Citation Point]).

Generalizing their previous results for κ =0 [123122Jump To The Next Citation Point], HE constructed for each κ a CSS solution based on the ansatz ϕ (τ,x) = eiω τϕ(x) for the critical scalar field. Studying its perturbations HE concluded that this solution is critical for κ > 0.0754, but has three unstable modes for κ < 0.0754 and even more for κ < –0.28. Below 0.0754 a DSS solution takes over, as shown in the simulations of Liebling and Choptuik [150Jump To The Next Citation Point], and HE conjectured that the transition is a Hopf bifurcation, such that the DSS cycle smoothly shrinks with growing κ, collapsing onto the CSS solution at the transition and then disappearing with a finite value of the echoing period Δ.

The close relation between the CSS and DSS critical solutions is also manifest in the construction of their global structures. In particular, the results of [122] and [69Jump To The Next Citation Point] for the CSS κ =0 and κ =1 solutions respectively show that the Cauchy horizon of the singularity is almost but not quite flat, exactly as was the case with the Choptuik DSS spacetime (see Section 3.3).

4.1.2 Spherical Einstein–SU(2) sigma model

This is an N = 3 sigma model, and it also displays a transition between CSS and DSS criticality, but this is a totally different type of transition, in particular showing a divergence in the echoing period Δ [131Jump To The Next Citation Point].

In a reduction to spherical symmetry, the effective action is

∫ [ ( 2 ) ] d4x √g-- R-− η |∇ ϕ|2 + 2sin--ϕ , (37 ) 2 r2
where r is the area radius, and the coupling constant η is dimensionless. It has been shown (numerically [27] and then analytically [28]) that for 0 ≤ η < 1∕2, there is an infinite sequence of CSS solutions ϕn labelled by a nodal number n, and having n growing modes. (The case η = 0, in which the sigma field decouples from gravity, will be revisited below.) The n-th solution is always regular in the past light cone of the singularity, but is regular up to the future light cone only for η < η n where η0 ≃ 0.0688, η1 ≃ 0.1518, and ηn < ηn+1 < 1 ∕2. For larger couplings an apparent horizon develops and the solution cannot be smoothly continued. These results suggest that ϕ0 is a stable naked singularity for η < η0, and ϕ1 acts as a critical solution between naked singularity formation and dispersal for η < η0 and between black hole formation and dispersal for η < η < η 0 1. The numerical experiments agree with this scenario in the range 0 ≤ η < 0.14. Other CSS solutions of this system are investigated in [25], and the possibility of chaos in [195].

Aichelburg and collaborators [131144] have shown that for η ≳ 0.2 there is clear DSS type II criticality at the black hole threshold. The period Δ depends on η, monotonically decreasing towards an asymptotic value for η → ∞. Interesting new behavior occurs in the intermediate range 0.14 < η < 0.2 that lies between clear CSS and clear DSS. With decreasing η the overall DSS includes episodes of approximate CSS [197], of increasing length (measured in the log-scale time τ). As η → ηc ≃ 0.170 from above the duration of the CSS epochs, and hence the overall DSS period Δ diverges. For 0.14 < η < 0.17 time evolutions of initial data near the black hole threshold no longer show overal DSS, but they still show CSS episodes. Black hole mass scaling is unclear in this regime.

It has been conjectured that this transition from CSS to DSS can be interpreted, in the language of the theory of dynamical systems, as the infinite-dimensional analogue of a 3-dimensional Shil’nikov bifurcation [145]. High-precision numerics in [3] further supports this picture: For η > ηc a codimension-1 CSS solution coexists in phase space with a codimension-1 DSS attractor such that the (1-dimensional) unstable manifold of the DSS solution lies on the stable manifold of the CSS solution. For η close to ηc the two solutions are close and the orbits around the DSS solution become slower because they spend more time in the neighbourhood of the CSS attractor. A linear stability analysis predicts a law Δ ≃ − 2λ log(η − ηc) + b for some constant b, where λ is the Lyapunov exponent of the CSS solution. For η = ηc both solutions touch and the DSS cycle dissapears.

4.1.3 Einstein–Yang–Mills

Choptuik, Chmaj and Bizoń [52Jump To The Next Citation Point] have found both type I and type II critical collapse in the spherical Einstein–Yang–Mills system with SU(2) gauge potential, restricting to the purely magnetic case, in which the matter is described by a single real scalar field. The situation is very similar to that of the massive scalar field, and now the critical solutions are the well-known static n = 1 Bartnik–McKinnon solution [10Jump To The Next Citation Point] for type I and a DSS solution (later constructed in [97Jump To The Next Citation Point]) for type II. In both cases the black holes produced in the supercritical regime are Schwarzschild black holes with zero Yang–Mills field strength, but the final states (and the dynamics leading to them) can be distinguished by the value of the Yang–Mills final gauge potential at infinity, which can take two values, corresponding to two distinct vacuum states.

Choptuik, Hirschmann and Marsa [56Jump To The Next Citation Point] have investigated the boundary in phase space between formation of those two types of black holes, using a code that can follow the time evolutions for long after the black hole has formed. This is a new “type III” phase transition whose critical solution is an unstable static black hole with Yang–Mills hair [14Jump To The Next Citation Point202Jump To The Next Citation Point], which collapses to a hairless Schwarzschild black hole with either vacuum state of the Yang–Mills field, depending on the sign of its one growing perturbation mode. This “coloured” black hole is actually a member of a 1-parameter family parameterized by its apparent horizon radius and outside the horizon it approaches the corresponding BM solution. When the horizon radius approaches zero the three critical solutions meet at a “triple point”. What happens there deserves further investigation.

Millward and Hirschmann [158Jump To The Next Citation Point] have further coupled a Higgs field to the Einstein–Yang–Mills system. New possible end states appear: regular static solutions, and stable hairy black holes (different from the coloured black holes referred to above). Again there are type I or type II critical phenomena depending on the initial conditions.

It is known that the spherical critical solutions within the magnetic ansatz become more unstable when other components of the gauge field are taken into account, and so they will not be critical in the general case.

4.1.4 Vacuum 4+1

Bizoń, Chmaj and Schmidt [20] have found a way of constructing asymptotically flat vacuum spacetimes in 4+1 dimensions which are spherically symmetric while containing gravitational waves (Birkhoff’s theorem does not hold in more than 3+1 dimensions). Recall that in (3+1 dimensional) Bianchi IX cosmology the manifold is M1 × S3 where the S3 is equipped with an SU(2) invariant (homogeneous but anisotropic) metric

ds2 = L21σ21 + L22σ23 + L23σ23, (38 )
where the σi are the SU(2) left-invariant 1-forms, and the Li are functions of time only. Similarly, the spacetimes of Bizoń, Chmaj and Schmidt are of the product form M2 × S3 where the Li now depend only on r and t. This gives rise to nontrivial dynamics, including a threshold between dispersion and black hole formation. With the additional U(1) symmetry L1 = L2 (biaxial solutions) there is only one dynamical degree of freedom. At the black hole threshold, type II critical phenomena are seen with Δ ≃ 0.49 and γ ≃ 0.3289.

In evolutions with the general ansatz where all Li are different (triaxial solutions) [21], the U(1) symmetry is recovered dynamically in the approach to the critical surface. However, each biaxial solution, and in particular the critical solution, exists in three copies obtained by permutation of the σ i. Therefore, in the triaxial case, the critical surface contains three critical solutions. The boundaries, within the critical surface, between their basins of attraction contain in turn codimension-two DSS attractors. It is conjectured that there is in fact a countable family of DSS solutions with n unstable modes.

Szybka and Chmaj [196] give numerical evidence that these boundaries within the critical surface are fractal (in contrast to the critical surface itself, which is smooth as in all other known systems.) UpdateJump To The Next Update Information

A similar ansatz can be made in other odd spacetime dimensions, and in 8+1 dimensions type II critical behaviour is again observed [19].

4.1.5 Scalar field collapse in 2+1

Spacetime in 2+1 dimensions is flat everywhere where there is no matter, so that gravity is not acting at a distance in the usual way. There are no gravitational waves, and black holes can only be formed in the presence of a negative cosmological constant (see [39] for a review).

Scalar field collapse in circular symmetry was investigated numerically by Pretorius and Choptuik [178Jump To The Next Citation Point], and Husain and Olivier [134Jump To The Next Citation Point]. In a regime where the cosmological constant is small compared to spacetime curvature they find type II critical phenomena with a universal CSS critical solution, and γ = 1.20 ± 0.05 [178]. The value γ ≃ 0.81 [134] appears to be less accurate.

Looking for the critical solution in closed form, Garfinkle [82] found a countable family of exact spherically symmetric CSS solutions for a massless scalar field with Λ = 0, but his results remain inconclusive. The q = 4 solution appears to match the numerical evolutions inside the past light cone, but its past light cone is also an apparent horizon. The q = 4 solution has three growing modes although the top one would give γ = 8∕7 ≃ 1.14 if only the other two could be ruled out [86]. An attempt at this [125] seems unmotivated. At the same time, it is possible to embed the Λ = 0 solutions into a family of Λ < 0 ones [64Jump To The Next Citation Point65Jump To The Next Citation Point43], which can be constructed along the lines of Section 2.6, so that Garfinkle’s solution could be the leading term in an expansion in e− τ(− Λ )−1∕2.

4.1.6 Scalar field collapse in higher dimensions

Critical collapse of a massless scalar field in spherical symmetry in 5+1 spacetime dimensions was investigated in [83Jump To The Next Citation Point]. Results are similar to 3+1 dimensions, with a DSS critical solution and mass scaling with γ ≃ 0.424. Birukou et al [13Jump To The Next Citation Point133] have developed a code for arbitrary spacetime dimension. They confirm known results in 3+1 (γ ≃ 0.36) and 5+1 (γ ≃ 0.44) dimensions, and investigate 4+1 dimensions. Without a cosmological constant they find mass scaling with γ ≃ 0.41 for one family of initial data and γ ≃ 0.52 for another. They see wiggles in the ln M versus ln(p − p∗) plot that indicate a DSS critical solution, but have not investigated the critical solution directly. With a negative cosmological constant and the second family, they find γ = 0.49. Bland and Kunstatter [29] have made a more precise determination: γ = 0.4131 ± 0.0001. This was motivated by an attempt to explain this exponent using an holographic duality between the strong coupling regime of 4+1 gravity and the weak coupling regime of 3+1 QCD [5Jump To The Next Citation Point], which had predicted γ = 0.409552.

Kol [140] relates a solution that is related to the Choptuik solution to a variant of the critical solution in the black-string black hole transition, and claims to obtain analytic estimates for γ and Δ. This has motivated a numerical determination of γ and Δ for the spherical massless scalar field in noninteger dimension up to 14 [19030].

4.1.7 Other systems obeying wave equations

Choptuik, Hirschmann and Liebling [53Jump To The Next Citation Point] have presented perturbative indications that the static solutions found by van Putten [199Jump To The Next Citation Point] in the vacuum Brans–Dicke system are critical solutions. They have also performed full numerical simulations, but only starting from small deviations with respect to those solutions.

Ventrella and Choptuik [200Jump To The Next Citation Point] have performed numerical simulations of collapse of a massless Dirac field in a special state: an incoherent sum of two independent left-handed zero-spin fields having opposite orbital angular momentum. This is prepared so that the total distribution of energy-momentum is spherically symmetric. The freedom in the system is then contained in a single complex scalar field obeying a modified linear wave equation in spherical symmetry. There are clear signs of CSS criticality in the metric variables, and the critical complex field exhibits a phase of the form eiωτ for a definite ω (the Hirschmann and Eardly ansatz for the complex scalar field critical solutions), which can be considered as a trivial form of DSS.

Garfinkle, Mann and Vuille [90Jump To The Next Citation Point] have found coexistence of types I and II criticality in the spherical collapse of a massive vector field (the Proca system), the scenario being almost identical to that of a massive scalar field. In the self-similar phase the collapse amplifies the longitudinal mode of the Proca field with respect to its transverse modes, which become negligible, and the critical solution is simply the gradient of the Choptuik DSS spacetime.

Sarbach and Lehner [185] find type I critical behaviour in q+3-dimensional spacetimes with U(1) × SO(q+1) symmetry in Einstein–Maxwell theory at the threshold between dispersion and formation of a black string.

Table 1: Critical collapse in spherical symmetry.
Matter Type Collapse Critical solution Perturbations of
simulations critical solution
Perfect fluid, p = kρ II [72Jump To The Next Citation Point, 161Jump To The Next Citation Point] CSS [72Jump To The Next Citation Point, 153Jump To The Next Citation Point, 161Jump To The Next Citation Point] [153Jump To The Next Citation Point, 139, 101Jump To The Next Citation Point, 103Jump To The Next Citation Point]
– in 4+1, 5+1, 6+1 II   CSS [7Jump To The Next Citation Point] [7Jump To The Next Citation Point]
Vlasov I? [183Jump To The Next Citation Point, 168Jump To The Next Citation Point] [155Jump To The Next Citation Point]  
Real scalar field:
– massless, minimally coupled II [47, 48Jump To The Next Citation Point, 49Jump To The Next Citation Point] DSS [95] [98Jump To The Next Citation Point, 154Jump To The Next Citation Point]
– massive I [34] oscillating [186Jump To The Next Citation Point]  
II [49] DSS [112Jump To The Next Citation Point, 106Jump To The Next Citation Point] [112, 106Jump To The Next Citation Point]
– conformally coupled II [48] DSS
– 4+1 II [13]    
– 5+1 II [83]    
Massive complex scalar field I, II [120Jump To The Next Citation Point] [186] [120]
Massless scalar electrodynamics II [126] DSS [106Jump To The Next Citation Point] [106]
Massive vector field II [90Jump To The Next Citation Point] DSS [90Jump To The Next Citation Point] [90]
Massless Dirac II [200Jump To The Next Citation Point] CSS [200]
Vacuum Brans–Dicke I [53Jump To The Next Citation Point] static [199] [53]
2-d sigma model:
– complex scalar (κ = 0) II [46] DSS [98Jump To The Next Citation Point] [98]
– axion-dilaton (κ = 1) II [110Jump To The Next Citation Point] CSS [69, 110Jump To The Next Citation Point] [110]
– scalar-Brans–Dicke (κ > 0) II [150, 147] CSS, DSS
– general κ including κ < 0 II CSS, DSS [124Jump To The Next Citation Point] [124]
SU(2) Yang–Mills I [52Jump To The Next Citation Point] static [10] [143]
II [52] DSS [97Jump To The Next Citation Point] [97]
“III” [56] coloured BH [14, 202] [193, 201, 18]
SU(2) Yang–Mills–Higgs (idem) [158] (idem)  
SU(2) Skyrme model I [17Jump To The Next Citation Point] static [17Jump To The Next Citation Point] [17]
II [22Jump To The Next Citation Point] DSS [22]
SO(3) Mexican hat II [148] DSS  

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