Perfect fluid matter

4.2 Perfect fluid matter

4.2.1 Spherical symmetry

Evans and Coleman [72 ] performed the first simulations of critical collapse with a perfect fluid with EOS $p = k ρ$ (where $ρ$ is the energy density and p the pressure) for k = 1/3 (radiation), and found a CSS critical solution with a mass-scaling critical exponent $γ ≈ 0.36$ . Koike, Hara and Adachi [138 ] constructed that critical solution and its linear perturbations from a CSS ansatz as an eigenvalue problem, computing the critical exponent to high precision. Independently, Maison [153 ] constructed the regular CSS solutions and their linear perturbations for a large number of values of k, showing for the first time that the critical exponents were model-dependent. As Ori and Piran before [171 , 172 ], he claimed that there are no regular CSS solutions for k > 0.89, but Neilsen and Choptuik [161 , 162 ] have found CSS critical solutions for all values of $k$ right up to 1, both in collapse simulations and by making a CSS ansatz. The difficulty comes from a change in character of the sonic point, which becomes a nodal point for k > 0.89, rather than a focal point, making the ODE problem associated with the CSS ansatz much more difficult to solve. Harada [114 ] has also found that the critical solution becomes unstable to a “kink” (discontinuous at the sonic point of the background solution) mode for k > 0.89, but because it is not smooth it does not seem to have any influence on the numerical simulations of collapse. On the other hand, the limit k → 1 leading to the stiff EOS $p = ρ$ is singular in that during evolution the fluid 4-velocity can become spacelike and the density $ρ$ negative. The stiff fluid equations of motion are in fact equivalent to the massless-scalar field, but the critical solutions can differ, dependending on how one deals with the issue of negative density [35 ]. Summarizing, it is possible to construct the Evans–Coleman CSS critical (codimension-1) solution for all values 0 < k < 1. This solution can be identified in the general classification of CSS perfect-fluid solutions as the unique spacetime that is analytic at the center and at the sound cone, is ingoing near the center, and outgoing everywhere else [40 , 41 , 42 ]. There is even a Newtonian counterpart of the critical solution: the Hunter (a) solution [115 ]. Álvarez-Gaumé et al. [7 ] have calculated $γ$ using perturbation theory for the spherically symmetric perfect fluid with $P = k ρ$ for $k ≲ 1∕4$ in d = 5, 6, 7 dimensions. Update

$p = kρ$ is the only EOS compatible with exact CSS (homothetic) solutions for perfect fluid collapse [38 ] and therefore we might think that other equations of state would not display critical phenomena, at least of type II. Neilsen and Choptuik [161 ] have given evidence that for the ideal gas EOS $p = k ρ0�$ (where $ρ0$ is the rest mass density and $�$ is the internal energy per rest mass unit) the black hole threshold also contains a CSS attractor, and that it coincides with the CSS exact critical solution of the ultrarelativistic case with the same $k$ . This is interpreted a posteriori as a sign that the critical CSS solution is highly ultrarelativistic, $ρ = (1 + �)ρ0 ≃ �ρ0 ≫ ρ0$ , and hence rest mass is irrelevant. Novak [167 ] has also shown in the case k = 1, or even with a more general tabulated EOS, that type II critical phenomena can be found by velocity-induced perturbations of static TOV solutions. A thorough and much more precise analysis by Noble and Choptuik [165 , 166 ] of the possible collapse scenarios of the stiff k = 1 ideal gas has confirmed this surprising result, and again the critical solution (and hence the critical exponent) is that of the ultrarelativistic limit problem. Parametrizing, as usual, the TOV solutions by the central density $ρc$ , they find that for low-density initial stars it is not possible to form a black hole by velocity-induced collapse; for intermediate initial values of $ρ c$ , it is possible to induce type II criticality for large enough velocity perturbations; for large initial central densities they always get type I criticality, as we might have anticipated.

Noble and Choptuik [165 ] have also investigated the evolution of a perfect fluid interacting with a massless scalar field indirectly through gravity. By tuning of the amplitude of the pulse it is possible to drive a fluid star to collapse. For massive stars type I criticality is found, in which the critical solution oscillates around a member of the unstable TOV branch. For less massive stars a large scalar amplitude is required to induce collapse, and the black hole threshold is always dominated by the scalar field DSS critical solution, with the fluid evolving passively.

4.2.2 Nonspherical perturbations

Non-spherically symmetric perturbations around the spherical critical solution for the perfect fluid can be used to study angular momentum perturbatively. All nonspherical perturbations of the perfect fluid critical solution decay for equations of state $p = k ρ$ with k in the range 1/9 < k < 0.49 [101 ], and so the spherically symmetric critical solution is stable under small deviations from spherical symmetry. Infinitesimal angular momentum is carried by the axial parity perturbations with angular dependence l = 1. From these two facts one can derive the angular momentum scaling law at the black hole threshold [99 , 103 ],

which should be valid in the range 1/9 < k < 0.49. The angular momentum exponent $μ(k)$ is related to the mass exponent $γ (k )$ by

where

is the growth or decay rate of the dominant l = 1 axial perturbation mode. In particular for the value k = 1/3, where $γ ≃ 0.3558$ , $μ = (5∕2 )γ ≃ 0.8895$ .

Ori and Piran [172 ] have pointed out that there exists a CSS perfect fluid solution for 0 < k < 0.036 generalizing the Larson–Penston solution of Newtonian fluid collapse, and which has a naked singularity for 0 < k < 0.0105. Harada and Maeda [113 , 115 ] have shown that this solution has no growing perturbative modes in spherical symmetry and hence a naked singularity becomes a global attractor of the evolution for the latter range of k. This is also true in the limit k = 0, which can be considered as the Newtonian limit [116 , 117 ]. Their result has been confirmed with very high precision numerics by Snajdr [189 ]. This seems to violate cosmic censorship, as generic spherical initial data would create a naked singularity. However, the exact result (41 ) holds for any regular CSS spherical perfect fluid solution, and so all such solutions with k < 1/9 have at least one unstable nonspherical perturbation. Therefore the naked singularity is unstable to infinitesimal perturbations with angular momentum when one lifts the restriction to spherical symmetry.

4.2.3 Cosmological applications

In the early universe, quantum fluctuations of the metric and matter can be important, for example providing the seeds of galaxy formation. Large enough fluctuations will collapse to form primordial black holes. As large quantum fluctuations are exponentially more unlikely than small ones, $P(δ) ∼ e− δ2$ , where $δ$ is the density contrast of the fluctuation, one would expect the spectrum of primordial black holes to be sharply peaked at the minimal $δ$ that leads to black hole formation, giving rise to critical phenomena [163 ]. See also [94 , 209 ].

An approximation to primordial black hole formation is a spherically symmetric distribution of a radiation gas ( $p = ρ∕3$ ) with cosmological rather than asymptotically flat boundary conditions. In [163 , 164 ] type II critical phenomena were found, which would imply that the mass of primordial black holes formed are much smaller than the naively expected value of the mass contained within the Hubble horizon at the time of collapse. The boundary conditions and initial data were refined in [119 , 160 ], and a minimum black hole mass of $∼$ 10^–4 of the horizon mass was found, due to matter accreting onto the black hole after strong shock formation. However, when the initial data are constructed more realistically from only the growing cosmological perturbation mode, no minimum mass is found [177 , 159 ]. Update