To study these issues, various 2-dimensional toy models of gravity coupled to scalar field matter have been suggested which are more or less directly linked to a spherically symmetric 4-dimensional situation (see  for a review). In two spacetime dimensions, the quantum expectation value of the matter stress tensor can be determined from the trace anomaly alone, together with the reasonable requirement that the quantum stress tensor is conserved. Furthermore, quantizing the matter scalar field(s) f but leaving the metric classical can be formally justified in the limit of many such matter fields. The two-dimensional gravity used is not the two-dimensional version of Einstein gravity but of a scalar-tensor theory of gravity. , where is called the dilaton, in the 2-dimensional toy model plays essentially the role of r in four spacetime dimensions. There seems to be no preferred 2-dimensional toy model, with arbitrariness both in the quantum stress tensor and in the choice of the classical part of the model. In order to obtain a resemblance of spherical symmetry, a reflecting boundary condition is imposed at a timelike curve in the 2-dimensional spacetime. This plays the role of the curve r =0 in a 2-dimensional reduction of the spherically symmetric 4-dimensional theory.
How does one expect a model of semiclassical gravity to behave when the initial data are fine-tuned to the black hole threshold? First of all, until the fine-tuning is taken so far that curvatures on the Planck scale are reached during the time evolution, universality and scaling should persist, simply because the theory must approximate classical general relativity. Approaching the Planck scale from above, one would expect to be able to write down a critical solution that is the classical critical solution asymptotically at large scales, as an expansion in inverse powers of the Planck length. This ansatz would recursively solve a semiclassical field equation, where powers of (in coordinates x and ) signal the appearances of quantum terms. Note that this is exactly the ansatz (48), but with the opposite sign in the exponent, so that the higher order terms now become negligible as , that is away from the singularity on large scales. On the Planck scale itself, this ansatz would not converge, and self-similarity would break down.
Addressing the question from the side of classical general relativity, Chiba and Siino  write down a 2-dimensional toy model, and add a quantum stress tensor that is determined by the trace anomaly and stress-energy conservation. They note that the quantum stress tensor diverges at r =0. Ayal and Piran  make an ad-hoc modification to these semiclassical equations. They modify the quantum stress tensor by a function which interpolates between 1 at large r, and at small r . They justify this modification by pointing out that the resulting violation of energy conservation takes place only at the Planck scale. It takes place, however, not only where the solution varies dynamically on the Planck scale, but at all times in a Planck-sized world tube around the center r =0, even before the solution itself reaches the Planck scale dynamically. This introduces a nongeometric, background structure, effect at the world-line r =0. With this modification, Ayal and Piran obtain results in agreement with our expectations set out above. For far supercritical initial data, black hole formation and subsequent evaporation are observed. With fine-tuning, as long as the solution stays away from the Planck scale, critical solution phenomena including the Choptuik universal solution and critical exponent are observed . In an intermediate regime, the quantum effects increase the critical value of the parameters p . This is interpreted as the initial data partly evaporating while they are trying to form a black hole.
Researchers coming from the quantum field theory side seem to favor a model (the RST model) in which ad hoc ``counter terms'' have been added to make it soluble. The matter is a conformally rather than minimally coupled scalar field. The field equations are trivial up to an ODE for a timelike curve on which reflecting boundary conditions are imposed. The world line of this ``moving mirror'' is not clearly related to r in a 4-dimensional spherically symmetric model, but seems to correspond to a finite r rather than r =0. This may explain why the problem of a diverging quantum stress tensor is not encountered. Strominger and Thorlacius  find a critical exponent of 1/2, but their 2-dimensional situation differs from the 4-dimensional one in many aspects. Classically (without quantum terms) any ingoing matter pulse, however weak, forms a black hole. With the quantum terms, matter must be thrown in sufficiently rapidly to counteract evaporation in order to form a black hole. The initial data to be fine-tuned are replaced by the infalling energy flux. There is a threshold value of the energy flux for black hole formation, which is known in closed form; recall this is a soluble system. The mass of the black hole is defined as the total energy it absorbs during its lifetime. This black hole mass is given by
where is the difference between the peak value of the flux and the threshold value, and is the quadratic order coefficient in a Taylor expansion in advanced time of the flux around its peak. There is universality with respect to different shapes of the infalling flux in the sense that only the zeroth and second order Taylor coefficients matter.
Peleg, Bose and Parker [107, 17] study the so-called CGHS 2-dimensional model. This (nonsoluble) model does allow for a study of critical phenomena with quantum effects turned off. Again, numerical work is limited to integrating an ODE for the mirror world line. Numerically, the authors find black hole mass scaling with a critical exponent of . They find that the critical solution and the critical exponent are universal with respect to families of initial data. Turning on quantum effects, the scaling persists to a point, but the curve of versus then turns smoothly over to a horizontal line. Surprisingly, the value of the mass gap is not universal but depends on the family of initial data. While this is the most ``satisfactory'' result among those discussed here from the classical point of view, one should keep in mind that all these results are based on mere toy models of quantum gravity.
Rather than using a consistent model of semiclassical gravity, Brady and Ottewill  calculate the quantum stress-energy tensor of a conformally coupled scalar field on the fixed background of the perfect fluid CSS critical solution and treat it as an additional perturbation, on top of the perturbations of the fluid-GR system itself. In doing this, they neglect the coupling between fluid and quantum scalar perturbations through the metric perturbations. From dimensional analysis, the quantum perturbation has a Lyapunov exponent . If this is larger than the positive Lyapunov exponent , it will become the dominant perturbation for sufficiently good fine-tuning, and therefore sufficiently good fine-tuning will reveal a mass gap. For a spherical perfect fluid with equation of state , one finds that for k >0.53, and vice versa. If , the semiclassical approximation breaks down for sufficiently good fine-tuning, and this calculation remains inconclusive.
|Critical Phenomena in Gravitational Collapse
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