Studies in thermal aspects of Schwarzschild spacetime

Singha, Chiranjeeb (2020) Studies in thermal aspects of Schwarzschild spacetime. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Chiranjeeb Singha (12IP026)) 12IP026.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

The Hawking effect is one of the most extensively studied topics in the literature. Yet it remains relatively under-explored within the framework of the canonical quantization. The key difficulty lies in the fact that the hyper-surfaces for fixed Schwarzschild time are not always spacelike and thus Hamiltonian dynamics are not well-posed in such coordinates. Also, in the standard derivation of the Hawking effect, one needs to find the relation between the ingoing and outgoing massless scalar field modes as seen by two asymptotic observers in the past and the future null infinity. Naturally, these field modes are described using null coordinates. However, for using null coordinates, one can show that the corresponding Hamiltonian is identically zero. Therefore, null coordinates do not lead to a true Hamiltonian that describes the evolution of these modes. To overcome these difficulties in a canonical formulation, in this thesis, we introduce a set of near-null coordinates which allows one to perform an exact Hamiltonian-based derivation of the Hawking effect. These near-null coordinates lead to a non-trivial matter Hamiltonian that describes the dynamics of the field modes. These coordinates being structurally closer to the null coordinates, allow one to follow the basic tenets of the Hawking effect very closely. However, while using the near-null coordinates, one faces the difficulty of having to deal with non-vanishing matter diffeomorphism generator as the spatial decomposition involves a non-zero shift vector. Next, we introduce another set of coordinates which allows an exact canonical derivation of the Hawking effect and in terms of these coordinates, the spacetime decomposition into spatial hyper-surfaces does not involve any shift vector. Therefore, using these coordinates, one can have a much simpler Hamiltonian-based derivation of the Hawking effect. These derivations open up an avenue to explore the Hawking effect in the framework of different canonical quantization methods such as the polymer quantization. The standard derivation of the Hawking effect in Schwarzschild spacetime implies a thermal nature in the outgoing particles. So one may ask whether such thermal radiation can be detected in principle? Now to investigate the manifestation of the thermal nature of black hole spacetimes, in this thesis, we study resonance Casimir- Polder interaction in the Schwarzschild spacetime. As it has been recently proposed, the resonance Casimir-Polder interaction could lead to the detection of Gibbons-Hawking radiation or the thermal nature of de Sitter spacetime. In our case, the same study leads to an insightful distinction between the Schwarzschild spacetime and thermal Minkowski spacetime. Subsequently, we show that, for Schwarzschild spacetime, beyond a characteristic length scale which is proportional to the inverse of the surface gravity κ, the resonance Casimir-Polder interaction is temperature-dependent and is characterized by a 1/L2 power-law provided the atoms are located close to the horizon. However, for thermal Minkowski spacetime, the resonance Casimir-Polder interaction does not depend on temperature and is characterized by a 1/L power-law decay always. So it seems that the spacetimes can be distinguished from each other using resonance Casimir-Polder interaction behavior. We note that the length scale limit beyond the characteristic value is not compatible with the local flatness of the spacetime.

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