Quantum Field Theory, Black Holes and Holography

Abstract

These notes are an expanded version of lectures given at the Croatian School on Black Holes at Trpanj, June 21-25, 2010. The aim is to provide a practical introduction to quantum field theory in curved spacetime and related black hole physics, with AdS/CFT as the loose motivation.

Figures (17)

• Figure 1: Rindler geometry is (the right wedge of) Minkowski space, as foliated by constant acceleration trajectories. The hyperboloid denotes a constant acceleration foliation and captures the Rindler spatial coordinate $\xi$. Rindler time along it is measured by $\eta$.
• Figure 2: Unruh's analytic continuation in the $-u$ plane. Solid line is the branch cut of $\ln(-u)$.
• Figure 3: Analyticity structure of zero-temperature Green functions. $G_-(t,{\bf x}, {\bf y})$ is analytic in the upper half $t$-plane and $G_+(t,{\bf x}, {\bf y})$ is analytic in the lower half plane. The difference between the two on the real $t$-axis is the commutator (see main text). Therefore when the commutator is non-zero, there is a branch cut on the real $t$ axis. But when it is zero, one can analytically continue $G_+$ to $G_-$. This happens when $|t| < |{\bf x}-{\bf y}|$, corresponding to correlators computed in the region outside the lightcone.
• Figure 4: The finite temperature analog of the previous figure. At finite temperature, i.e., $\beta < \infty$, there are new branch cuts that have moved in from infinity, and they are periodic according to the KMS condition. Close to the real axis, the analyticity structure and the continuations are as in the zero temperature case: the strips of analyticity of $G_+^\beta (t)$ and $G_-^\beta (t)$ alternate from there. All of these can be determined as the boundary values close to the cuts of the a unique function that is defined on the imaginary $t$ axis.
• Figure 5: Penrose diagram of collapsing dust.