Analytic studies and toy models

4.5 Analytic studies and toy models

4.5.1 Exact solutions of Einstein–Klein–Gordon

A number of authors have explored the possibility of finding critical phenomena with CSS (rather than DSS) massless scalar critical solutions. The Roberts 1-parameter family of 3+1 solutions [184 ] has been analyzed along this line in [173 , 32 , 206 , 137 ]. This family contains black holes whose masses (with a suitable matching to an asymptotically flat solution) scale as $(p − 1)1∕2$ for $p ≳ 1$ , but such a special family of solutions has no direct relevance for collapse from generic data. Its generalization to other dimensions has been considered in [75 ]. A fully analytic construction of all (spherical and nonspherical) linear perturbations of the Roberts solution by Frolov [73 , 74 ] has shown that there is a continuum of unstable spherical modes filling a sector of the complex plane with $Re λ ≥ 1$ , so that it cannot be a critical solution. Interestingly, all nonspherical perturbations decay.

Frolov [76 ] has also suggested that the critical (p = 1) Roberts solution, which has an outgoing null singularity, plus its most rapidly growing (spherical) perturbation mode would evolve into the Choptuik solution, which would inherit the oscillation in $τ$ with a period 4.44 of that mode.

A similar transition within a single 1-parameter family of solutions has been pointed out in [170 ] for the Wyman solution [207 ].

Hayward [121 , 66 ] and Clement and Fabbri [64 , 65 ] have also proposed critical solutions with a null singularity, and have attempted to construct black hole solutions from their linear perturbations. This is probably irrelevant to critical collapse, as the critical spacetime does not have an outgoing null singularity. Rather, the singularity is naked but first appears in a point. The future light cone of that point is not a null singularity but a CH with finite curvature.

Other authors have attempted analytic approximations to the Choptuik solution. Pullin [181 ] has suggested describing critical collapse approximately as a perturbation of the Schwarzschild spacetime. Price and Pullin [180 ] have approximated the Choptuik solution by two flat space solutions of the scalar wave equation that are matched at a “transition edge” at constant self-similarity coordinate x. The nonlinearity of the gravitational field comes in through the matching procedure, and its details are claimed to provide an estimate of the echoing period $Δ$ .

4.5.2 Toy models

Birmingham and Sen [12 ] considered the formation of a black hole from the collision of two point particles of equal mass in 2+1 gravity. Peleg and Steif [175 ] have investigated the collapse of a dust ring. In both cases the mass of the black holes is a known function of the parameters of the initial condition, giving a “critical exponent” 1/2, but no underlying self-similar solution is involved.

Mahajan et al. [152 ] expand the initial data for Einstein clusters in powers of the radius and, assuming that there are no shell crossings, find a mass scaling exponent of 3/2 for two such expansion coefficients. Universality is not demonstrated, and so the connection with a CSS solution discussed by Harada and Mahajan [118 ] is unclear. Update

Frolov [78 ], and Frolov, Larsen and Christensen [79 ] consider a stationary 2+1-dimensional Nambu–Goto membrane held fixed at infinity in a stationary 3+1-dimensional black hole background spacetime. The induced 2+1 metric on the membrane can have wormhole, black hole, or Minkowski topology. The critical solution between Minkowski and black hole topology has 2+1 CSS. The mass of the apparent horizon of induced black hole metrics scales with $γ$ = 2/3, superimposed with a wiggle of period $√ -- 3π∕ 7$ in $lnp$ . The mass scaling is universal with respect to different background black hole metrics, as they can be approximated by Rindler space in the mass scaling limit.

Horowitz and Hubeny [130 ], and Birmingham [11 ] have attempted to calculate the critical exponent in toy models from the adS-CFT correspondence. Álvarez-Gaumé et al. have attempted to use the adS-CFT correspondence for calculating $γ$ from the QCD side for the spherical massless scalar field in 5 dimensions [5 ], and the spherical perfect fluid with $P = kρ$ for k = 1/(d–1) in d = 5, 6, 7 dimensions [7 ].

Álvarez-Gaumé et al. [6 ] have calculated critical exponents for the formation of an apparent horizon in the collision of two gravitational shock waves. Update

Burko [37 ] considers the transition between existence and non-existence of a null branch of the singularity inside a spherically symmetric charged black hole with massless scalar field matter thrown in.

Wang [205 ] has constructed homothetic cylindrically symmetric solutions of 3+1 Einstein–Klein–Gordon and studied their cylindrically symmetric perturbations. It is not clear how these are related to a critical surface in phase space.