Lindblad resonances have been suggested as an important mechanism for angular momentum transport and heating in discs in binary black hole systems. We present the basic equations for the torque and heating rate for relativistic thin discs subjected to a perturbation. The Lindblad resonance torque is written explicitly in terms of metric perturbations for an equatorial disc in a general axisymmetric, time-stationary space–time with a plane of symmetry. We show that the resulting torque formula is gauge-invariant. Computations for the Schwarzschild and Kerr space–times are presented in the companion paper (Paper II).