Quantum field theory on the Bertotti-Robinson space-time

We consider the problem of quantum field theory on the Bertotti-Robinson space-time, which arises naturally as the near-horizon geometry of an extremal Reissner-Nordström black hole, but can also arise in certain near-horizon limits of nonextremal Reissner-Nordström space-time. The various vacuum states have been considered in the context of ${\mathrm{AdS}}_2$ black holes [M. Spradlin and A. Strominger, J. High Energy Phys. 11 (1999) 021] where it was shown that the Poincaré vacuum, the global vacuum and the Hartle-Hawking vacuum are all equivalent, while the Boulware vacuum and the Schwarzschild vacuum are equivalent. We verify this by explicitly computing the Green’s functions in closed form for a massless scalar field corresponding to each of these vacua. Obtaining a closed form for the Green’s function corresponding to the Boulware vacuum is nontrivial, and the novel computational technique employed may well be useful in deriving closed form Green’s functions in other space-times. Having obtained the propagator for the Boulware vacuum, which is a zero-temperature Green’s function, we can then consider the case of a scalar field at an arbitrary temperature by an infinite image imaginary-time sum, which yields the Hartle-Hawking propagator upon setting the temperature to the Hawking temperature. Finally, we compute the renormalized stress-energy tensor for a massless scalar field in the various quantum vacua.

Figure 1

Penrose diagram showing the various coordinate patches on the Bertotti-Robinson space-time. It is clear from the diagram that the time, $t$, only covers the part of the timelike boundary from $-\pi /2<\tau <\pi /2$ leading to a black hole interpretation in $(t,\rho )$ coordinates.

Figure 2

Plots of the renormalized stress-energy tensor components for the conformally coupled field for the various quantum states.

Figure 3

Plots of the renormalized stress-energy tensor components for the minimally coupled field for the various quantum states.