  Heat supplied to a metal is absorbed by the electrons and then transferred to
the lattice. In conventional metals energy is released to the lattice by
phonons emitted from the Lindhard continuum. However in a `bad' metal, with
short mean free path, the low energy Lindhard continuum is destroyed.
Furthermore in a `slow' metal, with Fermi velocity less than the sound
velocity, particle-hole pairs are kinematically unable to emit phonons. To
describe energy transfer to the lattice in these cases we obtain a general Kubo
formula for the energy relaxation rate in terms of the electronic density
spectral weight $\text{Im} \, G^R_{nn}(\omega_k,k)$ evaluated on the phonon
dispersion $\omega_k$. We apply our Kubo formula to the high temperature
Hubbard model, using recent data from quantum Monte Carlo and experiments in
ultracold atoms to characterize $\text{Im} \, G^R_{nn}(\omega_k,k)$. We
furthermore use recent data from electron energy-loss spectroscopy to estimate
the energy relaxation rate of the cuprate strange metal to a high energy
optical phonon. As a second, distinct, application of our formalism we consider
`slow' metals. These are defined to have Fermi velocity less than the sound
velocity, so that particle-hole pairs are kinematically unable to emit phonons.
We obtain an expression for the energy relaxation rate of a slow metal in terms
of the optical conductivity.
