On Modified Dispersion Relations

It is getting more and more seriously that general relativity do not have a quantum description as the development of the big bang cosmology. It leads to a so called quantum gravity theory has not yet finished. Therefore a phenomenology research at this area is important and necessary. The paradox between quantum gravity and special relativity give birth to the so called non-linear or doubly special relativity, and the main part of this thesis which called modified dispersion relations (MDRs) is an important idea of it. MDRs has been greatly researched in recent years. In this thesis, we investigated two different parts, black hole thermodynamics and the threshold anomaly of ultra-high energy cosmic rays (UHE cosmic rays).Chapter 1 is a brief summary of how it comes the doubly special relativity and the modified dispersion relations. In detail, quantum gravity believes that there exists a minimal scale - Planck scale, and it is disagree with special relativity. Doubly special relativity can partly solve the problem which requires that Planck scale is a constant at any frames as well as the speed of light. This kind of theory straightly leads to a modified dispersion relation.In chapter 2, which is our main part, we investigate the Hawking temperature of black holes, and show that black holes can have a peaceful destiny at late times. In general believe, the temperature of the black hole goes to infinity as they evaporate then may end up with an explosion. Some papers demonstrate that there will not be an explosion in Schwarzschild black holes in which the modified dispersion relations plays an important role. Different from these papers, we extent the situation into (A)dS black holes. The temperature and the entropy of the black hole is calculated, more important, we find that the heat capacity goes to zero when the temperature goes up. It means at some temperature the black hole will stop evaporate. It prevents black holes from total evaporation. In this thesis, we used an extended uncertainty principle (EUP) and a generalized EUP (GEUP) instead of the usual Heisenberg uncertainty principle because we have to take the cosmological constant into account in large scale structure.Chapter 3, which is our another main part, we discuss the threshold anomaly of UHE cosmic rays. Greisen, Zatsepin and Kuz'min came up a theory that says UHE cosmic rays with energy above some level (it shows 5×10¹⁹ eV) will scatter with the microwave background photons such that they can never be detected in the earth. It is usually called the GZK-cutoff. But some of the UHECRs of several orders higher than 5×10¹⁹eV has been detected. In this chapter we demonstrate that even a modified term with tiny parameter in modified dispersion relations will change the threshold greatly such that the threshold can be some other value much higher than before. We also showed that in ordinary laboratory we do not notice the higher threshold because this kind of effect only becomes observable in super high energy scale. Special relativity is good enough to describe nuclear reaction in ordinary laboratory. We have to point out that even though recently HiRes first observed the GZK-cutoff, it is still worth to find out that what MDRs leads to.Chapter 4 is our summary of this thesis.