Notes on Black Hole Fluctuations and Back-Reaction

The idea of viewing a black hole (particle detector) interacting with a quantum field as a dissipative system, and the Hawking [1]- Unruh [2] radiation as a manifestation of a fluctuction-dissipation relation was first proposed by Candelas and Sciama [3, 4]. Even though, as we will soon see, the fluctuations in the thought of these earlier authors are not the correct ones and the relations proposed were not really addressing the back-reaction of quantum fields in a classical black hole spacetime, the idea remains attractive. Indeed one of us (BLH) found it so attractive that he launched a systematic investigation into the statistical mechanical properties of particle/spacetime and quantum field interactions. This involved the introduction of statistical mechanical ideas such as quantum open sytems [5] and field-theoretical methods such as the influence functional [6] and Schwinger-Keldysh formalisms [7] for the establishment of a quantum statistical field theory in curved spacetime (for a review, see [8, 9]). It was found that the back-reaction of quantum fields (through processes like particle creation) on a classical background spacetime can be described by an Einstein-Langevin equation [10], which is a generalization of the semiclassical Einstein equation to include stochastic sources due to created particles. Indeed it was also found from first principles that the back-reaction can be encapsulated in the form of a Fluctuation-Dissipation Relation (FDR) [12, 13], which takes into account the mutual influence of the quantum field and the background spacetime (or detector, in the case of Unruh radiation). We expect it to hold also for black hole systems, both in the familiar static condition where black hole thermodynamics based on the Bekenstein-Hawking relation was constructed, and for dynamical collapse problems. This is the major theme in our current program of research.