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Register a new userQuantum thermodynamics in the interior of a Reissner-Nordstrom black-hole
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We study the interior of a Reissner-Nordstrom Black-Hole (RNBH) using Relativistic Quantum Geometry, which was introduced in some previous works. We found discrete energy levels for a scalar field from a polynomial condition for the Heun Confluent functions expanded around the effective causal radius $r_*$. From the solutions it is obtained that the uncertainty principle is valid for each energy level of space-time, in the form: $E_n, r_{*,n}=hbar/2$, and the charged mass is discretized and distributed in a finite number of states. The classical RNBH entropy is recovered as the limit case where the number of states is very large, and the RNBH quantum temperature depends on the number of states in the interior of the RNBH. This temperature, depending of the number of states of the RNBH, is related with the Bekeinstein-Hawking (BH) temperature: $T_{BH} leq T_{N} < 2,T_{BH}$.
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We derive explicit expressions for the two-point function of a massless scalar field in the interior region of a Reissner-Nordstrom black hole, in both the Unruh and Hartle-Hawking quantum states. The two-point function is expressed in terms of the standard $lmomega$ modes of the scalar field (those associated with a spherical harmonic $Y_{lm}$ and a temporal mode $e^{-iomega t}$), which can be conveniently obtained by solving an ordinary differential equation, the radial equation. These explicit expressions are the internal analogs of the well known results in the external region (originally derived by Christensen and Fulling), in which the two-point function outside the black hole is written in terms of the external $lmomega$ modes of the field. They allow the computation of $<Phi^{2}>_{ren}$ and the renormalized stress-energy tensor inside the black hole, after the radial equation has been solved (usually numerically). In the second part of the paper, we provide an explicit expression for the trace of the renormalized stress-energy tensor of a minimally-coupled massless scalar field (which is non-conformal), relating it to the dAlembertian of $<Phi^{2}>_{ren}$. This expression proves itself useful in various calculations of the renormalized stress-energy tensor.
We investigate spherically symmetric, steady state, adiabatic accretion onto a Tangherlini-Reissner-Nordstrom black hole in arbitrary dimensions by using $D$-dimensional general relativity. We obtain basic equations for accretion and determine analytically the critical points, the critical fluid velocity, and the critical sound speed. We lay emphasis on the condition under which the accretion is possible. This condition constrains the ratio of mass to charge in a narrow limit, which is independent of dimension for large dimension. This condition may challenge the validity of the cosmic censorship conjecture since a naked singularity is eventually produced as the magnitude of charge increases compared to the mass of black hole.
For the Schwarzschild black hole the Bekenstein-Hawking entropy is proportional to the area of the event horizon. For the black holes with two horizons the thermodynamics is not very clear, since the role of the inner horizons is not well established. Here we calculate the entropy of the Reissner-Nordstrom black hole and of the Kerr black hole, which have two horizons. For the spherically symmetric Reissner-Nordstrom black hole we used several different approaches. All of them give the same result for the entropy and for the corresponding temperature of the thermal Hawking radiation. The entropy is not determined by the area of the outer horizon, and it is not equal to the sum of the entropies of two horizons. It is determined by the correlations between the two horizons, due to which the total entropy of the black hole and the temperature of Hawking radiation depend only on mass $M$ of the black hole and do not depend on the black hole charge $Q$. For the Kerr and Kerr-Newman black holes it is shown that their entropy has the similar property: it depends only on mass $M$ of the black hole and does not depend on the angular momentum $J$ and charge $Q$.
In classical General Relativity, the values of fields on spacetime are uniquely determined by their values at an initial time within the domain of dependence of this initial data surface. However, it may occur that the spacetime under consideration extends beyond this domain of dependence, and fields, therefore, are not entirely determined by their initial data. This occurs, for example, in the well-known (maximally) extended Reissner-Nordstrom or Reissner-Nordstrom-deSitter (RNdS) spacetimes. The boundary of the region determined by the initial data is called the Cauchy horizon. It is located inside the black hole in these spacetimes. The strong cosmic censorship conjecture asserts that the Cauchy horizon does not, in fact, exist in practice because the slightest perturbation (of the metric itself or the matter fields) will become singular there in a sufficiently catastrophic way that solutions cannot be extended beyond the Cauchy horizon. Thus, if strong cosmic censorship holds, the Cauchy horizon will be converted into a final singularity, and determinism will hold. Recently, however, it has been found that, classically this is not the case in RNdS spacetimes in a certain range of mass, charge, and cosmological constant. In this paper, we consider a quantum scalar field in RNdS spacetime and show that quantum theory comes to the rescue of strong cosmic censorship. We find that for any state that is nonsingular (i.e., Hadamard) within the domain of dependence, the expected stress-tensor blows up with affine parameter, $V$, along a radial null geodesic transverse to the Cauchy horizon as $T_{VV} sim C/V^2$ with $C$ independent of the state and $C eq 0$ generically in RNdS spacetimes. This divergence is stronger than in the classical theory and should be sufficient to convert the Cauchy horizon into a strong curvature singularity.
The Reissner-Nordstrom-de Sitter (RN-dS) spacetime can be considered as a thermodynamic system. Its thermodynamic properties are discussed that the RN-dS spacetime has phase transitions and critical phenomena similar to that of the Van de Waals system or the charged AdS black hole. The continuous phase transition point of RN-dS spacetime depends on the position ratio of the black hole horizon and the cosmological horizon. We discuss the critical phenomenon of the continuous phase transition of RN-dS spacetime with Landau theory of continuous phase transition, that the critical exponent of spacetime is same as that of the Van de Waals system or the charged AdS black hole, which have universal physical meaning. We find that the order parameters are similar to those introduced in ferromagnetic systems. Our universe is an asymptotically dS spacetime, thermodynamic characteristics of RN-dS spacetime will help us understand the evolution of spacetime and provide a theoretical basis to explore the physical mechanism of accelerated expansion of the universe.
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