The adjoint method for the study of runaway electron dynamics in momentum space Liu et al (2016 Phys. Plasmas23 010702) is rederived using the Green's function method, for both the runaway probability function (RPF) and the expected loss time (ELT). The RPF and ELT obtained using the adjoint method are presented, both with and without the synchrotron radiation reaction force. The adjoint method is then applied to study the runaway electron avalanche. Both the critical electric field and the growth rate for the avalanche are calculated using this fast and novel approach.