Collapse of the wavefunction, the information paradox and backreaction
Eur. Phys. J. C
Collapse of the wavefunction, the information paradox and backreaction
Sujoy K. Modak 1 2
Daniel Sudarsky 0 3
0 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México , Apartado Postal 70-543, 04510 Distrito Federal , Mexico
1 KEK Theory Center, High Energy Accelerator Research Organization (KEK) , Tsukuba, Ibaraki 305-0801 , Japan
2 Facultad de Ciencias, CUICBAS,Universidad de Colima , CP 28045 Colima , Mexico
3 Department of Philosophy, New York University , New York, NY 10003 , USA
We consider the black hole information problem within the context of collapse theories in a scheme that allows the incorporation of the backreaction to the Hawking radiation. We explore the issue in a setting of the two dimensional version of black hole evaporation known as the Russo-Susskind-Thorlacius model. We summarize the general ideas based on the semiclassical version of Einstein's equations and then discuss specific modifications that are required in the context of collapse theories when applied to this model. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Semiclassical CGHS model with backreaction . . . . 3 Review of the RST Model . . . . . . . . . . . . . . . 3.1 Equations of motion . . . . . . . . . . . . . . . . 3.2 Solving semiclassical equations . . . . . . . . . 3.3 Dynamical case of black hole formation and evaporation . . . . . . . . . . . . . . . . . . . . 4 Quantization on RST . . . . . . . . . . . . . . . . . . 5 Incorporating collapse mechanism in the RST model . 5.1 Collapse of the quantum state and Einstein's semiclassical equations . . . . . . . . . . . . . . 5.2 CSL theory . . . . . . . . . . . . . . . . . . . . 5.3 Gravitationally induced collapse rate . . . . . . . 5.4 Spacetime foliation . . . . . . . . . . . . . . . . 5.5 CSL evolution and the modified back reaction . . 6 Recovering the thermal Hawking radiation . . . . . . 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . 8 Appendix A: The renormalized energy-momentum tensor
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Contents
The black hole information question has been with us for
more than four decades, ever since Hawking’s discovery that
black holes emit thermal radiation and therefore evaporate,
leading either to their complete disappearance or to a small
Planck mass scale remnant [
1
]. The basic issue can be best
illustrated by considering an initial setting where an
essentially flat space-time in which a single quantum field is in a
pure quantum state of relative high excitation corresponding
to a spatial concentration of energy, that, when left on its own
will, collapses gravitationally leading to the formation of a
black hole. As the black hole evaporates, the energy that was
initially localized in a small spatial region, ends up in the
form of Hawking radiation that, for much of this evolution
must be almost exactly thermal [
2
]. The point, of course, is
that if this process ends with the complete evaporation of the
black hole (or even if a small remnant is left) the
overwhelming majority of the initial energy content would correspond to
a state of the quantum field possessing almost no information
(except that encoded in the radiation’s temperature) and it is
very difficult to reconcile this with the general expectation
that in any quantum process the initial and final states should
be related by a unitary transformation, and thus all
information encoded in the initial state must be somehow present in
the final one. The issue, of course, is far more subtle and the
above should be taken as only a approximate account of the
problem.
There have been many attempts to deal with this
conundrum, with none of them resulting in a truly satisfactory
resolution of the problem [
3,4
]. In fact there is even a debate
as to the extent to which this is indeed a problem or as some
people like to call it a “paradox” [
5,6
].
In previous works [
7–9
] we helped to clarify the basis of
the dispute, and proposed a scheme where the resolution of
the issue is tied to a proposal to address another lingering
problem of theoretical physics: the so called measurement
problem [
10
] in quantum theory.
The first task was dealt with [
7–9
] by noting that the true
problem arises only when one takes the point of view that a
satisfactory theory of quantum gravity must resolve the
singularity, and that, as a result of such resolution, there will be
no need to introduce a new boundary of space-time in the
region where the classical black hole singularity stood.
Otherwise the problem can be fully understood by noting that the
region in the black hole exterior, at late times corresponding
to those where most of the energy takes the form of thermal
Hawking radiation, contains no Cauchy hypersurfaces and
thus any attempt to provide a full description of the
quantum state in terms of the quantum field modes in the black
hole exterior is simply wrongheaded. In order to provide a
complete description of such late quantum state one nee (...truncated)