Ground state for a massive scalar field in the BTZ spacetime with Robin boundary conditions

We consider a real, massive scalar field in Bañados-Teitelboim-Zanelli spacetime, a $2+1$-dimensional black hole solution of Einstein’s field equations with a negative cosmological constant. First, we analyze the space of classical solutions in a mode decomposition, and we characterize the collection of all admissible boundary conditions of Robin type which can be imposed at infinity. Second, we investigate whether, for a given boundary condition, there exists a ground state by constructing explicitly its two-point function. We demonstrate that for a subclass of the boundary conditions it is possible to construct a ground state that locally satisfies the Hadamard property. In all other cases, we show that bound state mode solutions exist and, therefore, such construction is not possible.

Figure 1

Penrose diagrams of the BTZ black hole for the rotating $0<r_{-}<r_{+}$ (left) and the static $0=r_{-}<r_{+}$ (right) cases.

Figure 2

Real and imaginary part of the bound state frequency ${\tilde{\omega }}_{\zeta }$ as a function of the parameter $\zeta$ defining the Robin boundary condition for a BTZ black hole with $\ell =1$, $r_{+}=5$, and $r_{-}=3$ and a scalar field with ${\mu }^2=-0.65$ and $k=1$. The bound state mode solutions exist for values of $\zeta$ between ${\zeta }_{*}\approx 0.5625\pi$ and $\pi$.

Figure 3

Contour for the integral representation of the identity operator in (a2).