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 [66] 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 [34] 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 [4] 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 [115] 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 [23] 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
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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