Entanglement and Black Holes

Abstract

Black Holes have an entropy that can be shown to depend on the area of its horizon. In this context, we look at the possibility of entanglement of quantum field modes across the horizon being the source of this entropy and look at the entanglement entropies of various quantum fields. In this thesis, we start by looking at two methods to calculate entanglement entropy for the scalar quantum field theory and study their various properties. The first method relies majorly on numerical simulations while the second method (covariance matrix approach) gives analytical results. The numerical method can help show the area law dependence of the entropy of a scalar field. Through the covariance matrix approach we see that the entanglement entropy of a scalar field at leading order behaves differently in different regimes of a parameter $\zeta$. It is shown that for Neumann boundary conditions, the entanglement entropy diverges as $\log \log$ in the regime $\zeta \ll 1$ and diverges as $\log$ in the regime $\zeta \gg 1$. Then, we try to generalize the above results for a fermionic field, with the goals of seeing area law dependence, the above crossover in behaviour of entanglement entropy and obtaining Hawking temperature in this theory. Lastly, we also have a look at Black Hole thermodynamics and study the concept of cosmological particle creation, which leads to concepts like the Unruh effect and Hawking Radiation.