Investigation of the quantum mechanical behavior of the spinning relativistic particles in the massive gravity theories

Black holes have an important position in the modern physics since they are cosmological laboratories to test an idea about the quantization of the gravity. Moreover, the most important evidence in order to form a consistent quantum gravity theory is a black hole radiation, known as Hawking radiation. In this connection, in the first part of the thesis, Hawking radiation from new-type and warped-AdS$_{3}$ black holes obtained in the framework of of 2+1 dimensional new massive gravity, are scrutinized by using the quantum tunnelling method for spin-0, spin-1/2 and spin-1 particles. Additionally, Hawking radiation of the extended particles of spin-0 and spin-1/2 depending on black hole's properties as well as total angular momentum, energy and mass of tunnelling particles is calculated. At the same time, while Hawking radiation from extended particles of spin-0 and spin-1/2 is different, it observed to be same for point-like particles of spin-0 and spin-1/2. On the other hand, according to the recent observational data, our universe has an accelerating expansion. The source of accelerating expansion is still accepted to be a major problem in the modern cosmology. Therefore, in the second part of the study, by coupling the gravitational field in the context of the 2+1 dimensional new massive gravity theory with the fermionic and bosonic fields, expansion problem of the universe is investigated.