Collapse of the wavefunction, the information paradox and backreaction

The European Physical Journal C, Jul 2018

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.

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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 - 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)


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Sujoy K. Modak, Daniel Sudarsky. Collapse of the wavefunction, the information paradox and backreaction, The European Physical Journal C, 2018, pp. 556, Volume 78, Issue 7, DOI: 10.1140/epjc/s10052-018-6032-5