Comparison of various approaches to the back reaction problem

Abstract

We compare three possible prescriptions for the back-reaction of a quantum mechanical source on the classical gravitational field to which it is coupled. These prescriptions are (i) the transition element 〈out| Tik |in〉 of the energy-momentum operator between asymptotic vacuum states, (ii) the expectation value 〈Ψ| Tik |Ψ〉, and (iii) the Born-Oppenheimer type approximation for back-reaction as used in molecular physics. It is shown that the three approaches match only when the gravitational field is varying adiabatically, and of the three, the use of expectation value provides the most accurate description of the validity of semiclassical Einstein equations. The analysis is carried Out by studying the model of a quantized time-dependent oscillator coupled to a classical particle.