| Popis: |
We have presented a gauge theoretic approach for gravity in $2+1$-D, starting from the Chern-Simons action with the $SO(2,2)$ and $ISO(2,1)$ Lie group structures. With these groups, we are able to construct some known results from general relativity namely the Banados, Teteilboim and Zanelli (BTZ) solution and the spinning point particle solution respectively. We obtain the canonical structure in each case. To get a more physical picture we probe the canonical structure by embedding the Arnowitt-Deser-Misner (ADM) metric into the corresponding spacetime. After several canonical transformations we found in each case that the spacetime is described by a two dimensional phase space with two degrees of freedom, the mass and the angular momentum. These are stationary solutions. In the second part of this thesis, we have extended this approach to non-rotating dynamical collapse. During a spherical collapse the end state does not always lead to a black hole. Instead, due to quantum effects, collapsing shells in the exterior of the apparent horizon are accompanied by outgoing Unruh radiation in its interior. Both collapsing shells and the outgoing Unruh radiation appear to stop at the apparent horizon. This solution is obtained by solving the Wheeler-DeWitt equation (WDW) which is the Hamiltonian constraint, elevated to operator form by applying Dirac's quantization procedure. As a consequence, we can expect that the collapse process would end in a quasi-stable, static compact object without forming a black hole. We have shown that, using Einstein equations, such a stable configuration is possible where the BTZ horizon radius is $87\%$ of the boundary radius, so the BTZ horizon lies inside the boundary. In the last part, we consider a scalar field model of dark matter(DM) which forms a Bose Einstein condensate. We coupled a non-relativistic scalar field with $\phi^4$ interactions to linearized gravity in the Gravitoelectromagnetic formulation. This model predicts that halo sizes vary significantly depending on the mass of the scalar field particles and the nature of the self-interaction. Dark matter halos ranging from asteroid size to galaxy size are possible within this model. This was one of the major challenges in the Cold Dark Matter (CDM) paradigm. We examined the stability of the halos by studying small oscillations about the equilibrium using a collective coordinates approach with both types of self-interaction. We found oscillations about the minima of the energy with a time period of $10$ billion years for attractive interactions and $1.3$ billion years for repulsive interactions. |
