Quantum scalar field theory on a rotating black hole in five dimensions

In this thesis, we study a massive scalar field on a five-dimensional, rotating, asymptotically anti-de Sitter black hole spacetime. We introduce an enhanced symmetry in the spacetime by taking the two angular momentum parameters equal. We picked this setup because the geometry possesses additional symmetries that simplify analysis of mode solutions of the scalar field equation and the stress–energy tensor.

In fact, when the angular momentum of the black hole is sufficiently small (so we are slowly rotating), we find that there is no speed-of-light surface. This allows us to identify a Killing vector field that is timelike everywhere outside the event horizon. Numerous advantages come from this setup, one of these is the absence of classical superradiance, which extremely simplifies the calculation.

We then use quantum field theory on curved spacetime to perform the canonical quantisation of the massive scalar field on this background. We study separately the angular and radial parts of the Klein–Gordon equation. We identify the radial differential equation as a Heun equation, while the angular part takes the form of a spin-weighted spherical harmonics differential equation.

We also present addition theorems for spin-weighted spherical harmonics, generalising previous results for scalar (spin-zero) spherical harmonics. These new results simplify the numerical analysis of the stress–energy tensor of a scalar field on a Kerr–AdS5 black hole, but could be generalised to other scenarios.

After this, we construct the quantum states to evaluate observables. In particular, we construct analogues of the Boulware and Hartle–Hawking quantum states for the quantum scalar field. Finally, we compute the differences in expectation values of the square of the quantum scalar field operator and the stress–energy tensor operator between these two quantum states, and present their numerical evaluation.