  Recently the photonic golden rule, which predicts that the spontaneous
emission rate of an atom depends on the projected local density of states
(LDOS), was shown to fail in an optical medium with a linear gain amplifier. We
present a classical light-matter theory to fix this widely used spontaneous
emission rate, fully recovering the quantum mechanical rate reported in Franke
et al., Phys. Rev. Lett. 127, 013602 (2021). The corrected classical Purcell
factor, for media containing linear amplifiers, is obtained in two different
forms, both of which can easily be calculated in any standard classical Maxwell
solver. We also derive explicit analytical results in terms of quasinormal
modes, which are useful for studying practical cavity structures in an
efficient way, including the presence of local field effects for finite-size
dipole emitters embedded inside lossy or gain materials (using a real cavity
model). Finally, we derive a full classical correspondence from the viewpoint
of quantized quasinormal modes in the bad cavity limit. Example numerical
calculations are shown for coupled loss-gain microdisk resonators, showing
excellent agreement between few mode expansions and full numerical dipole
simulations.
