Aspects of black hole physics in different quantum field theories

Barman, Subhajit (2020) Aspects of black hole physics in different quantum field theories. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

The Hawking effect is one of the most remarkable results of quantum field theory in curved spacetime and happens to be a highly studied topic. Usually, the Hawking effect is understood using Bogoliubov transformation between ingoing and outgoing field modes in a black hole spacetime. These field modes are described in terms of the advanced and retarded null coordinates. However, null coordinates are not suitable to construct true matter field Hamiltonian. Therefore, to provide a canonical derivation of the Hawking effect, a set of near-null coordinates were introduced. We have used these near-null coordinates to provide a Hamiltonian based derivation of the Hawking effect for non-extremal Kerr black holes, which have particular significance as they are among astrophysically observed objects. Furthermore, it is believed that extremal black holes do not emit Hawking radiation as understood by taking extremal limits of non-extremal black holes. However, it is debated whether one can make such conclusion reliably starting from an extremal black hole, as the associated Bogoliubov coefficients which relate ingoing and outgoing field modes do not satisfy the required consistency condition. We have addressed this issue in the canonical approach. Starting from extremal Kerr black holes we have shown that the required consistency condition is satisfied in the canonical derivation and it produces zero number density for Hawking particles. We have also pointed out the reason behind the reported failure of Bogoliubov coefficients to satisfy the required condition. The Hawking effect is considered to be a cornerstone in understanding the black hole thermodynamics, where one mesmerizing postulate is the area law of entropy. It has been speculated that this area law should have some quantum mechanical origin and there is a prospect of understanding this area law in terms of the entanglement entropy of quantum fields. The area dependence of entanglement entropy of a free scalar field is often understood in terms of coupled harmonic oscillators. In Schrödinger quantization, the Gaussian nature of ground state wave-function for these oscillators is sufficient to provide the exact form of the reduced density matrix and its eigenvalues, thus giving the entanglement entropy. However, in polymer quantization, a quantization technique used in loop quantum gravity, the ground state is not Gaussian, and the formalism, which can provide the exact analytical form of the reduced density matrix is not yet known. In order to address this issue, we have treated the interaction between two coupled harmonic oscillators in perturbative approach and evaluated the entanglement entropy in Schrödinger and polymer quantization. In contrary to Schrödinger quantization, we have shown that in high-frequency regime the entanglement entropy decreases for polymer quantization keeping the ratio of coupling strength to the square of individual oscillator frequency fixed. Furthermore, for a free scalar field, we have validated the area dependence of entanglement entropy in Fock quantization and demonstrated that polymer quantization produces the similar area law. On the other hand, it is believed that the inclusion of extra spatial dimensions has the potential to resolve the hierarchy problem, which is related to the enormous difference between the gravitational scale and the Electro-Weak scale. It is predicted that there is a possibility of detecting higher-dimensional black holes in the high energy particle colliders. Although these micro black holes are not yet observed in the Large Hadron Collider of CERN, they remain interesting arenas to venture in for their enthralling properties coming from the inclusion of extra dimensions. The simplest of these higher dimensional black holes are the Schwarzschild Tangherlini black holes, which are higher dimensional generalizations of the four dimensional static Schwarzschild black holes. Therefore, we have considered a n dimensional Schwarzschild Tangherlini black hole spacetime with massless minimally coupled free scalar fields in its bulk and 3−brane. The bulk scalar field equation is separable using the higher dimensional spherical harmonics on (n − 2)−sphere. First, using the Hamiltonian formulation with the help of the near-null coordinates we have obtained the expected temperature of the Hawking effect, identical for both bulk and brane localized scalar fields. Second, it is known that the spectrum of the Hawking effect as seen at asymptotic future does not correspond to a perfect black body and it is properly represented by a greybody distribution. We have calculated the bounds on this greybody factor for the scalar field in both bulk and 3−brane. Furthermore, we have reaffirmed that these bounds predict a decrease in the greybody factor as the spacetime dimensionality n increases and also reaffirmed that for a large number of extra dimensions, the Hawking quanta is mostly emitted in the brane. Furthermore, in the four-dimensional Schwarzschild case, the polymer quantization predicts the Hawking effect to become shortlived. In this case, the Hawking effect becomes few milliseconds for a solar mass black hole whereas it is few years for an ultra-massive black hole. Consequently, it provides a new way to resolve the so-called information loss paradox.

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