Some aspects of virtual black holes
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Some Aspects of Virtual Black Holes1 M. Faizal Department of Mathematics, University of Durham, DH1 3LE, Durham, United Kingdom email: [email protected] Received September 25, 2011
Abstract—We first consider consistently thirdquantize modified gravity. We then analyze certain aspects of virtual black holes in this thirdquantized modified gravity. We see how a statistical mechanical origin for the Bekenstein–Hawking entropy naturally arises in this model. Furthermore, the area and hence the entropy of a real macroscopic black hole is quantized in this model. Virtual black holes cause a loss of quantum coher ence, which gives an intrinsic entropy to all physical systems that can be used to define a direction of time and hence provide a solution to the problem of time. DOI: 10.1134/S1063776112020045 1
1. INTRODUCTION It is expected that quantum fluctuations of space time can cause the topology of spacetime to change at the Planck scale, giving it a foamlike structure called the spacetime foam [1, 2]. Spacetime form has largely been discussed via the formation of baby universes that render the spacetime multiply connected [3–5]. In this model, the spacetime manifold therefore has a large value of the first Betti number B1 and the second Betti number vanishes, B2 = 0. The problem with this model is that it predicts a wrong value of the QCD θparameter [6] and the cosmological constant [7, 8]. However, there is an alternative model of spacetime foam that seems to predict a correct value of the QCD θparameter [9]. In this model, the topology of space time changes by the formation of virtual black holes and thus the spacetime remains single connected [10, 11]. In this model, the spacetime manifold has a large value of the second Betti number B2 and the first and third Betti numbers vanish, B1 = B3 = 0. In this pic ture, there is also an elegant way to describe black hole evaporation without the appearance of a naked singu larity. Macroscopic real black holes evaporate to the Planck size by emitting Hawking radiation. At this stage, they are left with no energy or charge. They then disappear in a sea of virtual black holes. Because this picture of spacetime foam seems to be more realistic, we analyze certain aspects of it in this paper. To study the physical effects of virtual black holes, we should analyze the collision of particles with energy less than the Planck energy in a small region contain ing a virtual black hole. For this, we would need to find a Euclidean solution for this process. But it is very dif ficult to find such a solution, and we therefore analyze virtual black holes via third quantization. The third quantization has been discussed implicitly in [12, 13] 1
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and explicitly in [14, 15]. The modification of the Wheeler–De Witt equation by the addition of nonlin ear terms and the third quantization of the resultant theory was formally analyzed in [18]. Third quantiza tion of Brans–Dicke th
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