  We consider a pair of dipoles for which direct electrostatic dipole-dipole
interactions may be significantly larger than the coupling to transverse
radiation. We derive a master equation using the Coulomb gauge, which naturally
enables us to include the inter-dipole Coulomb energy within the system
Hamiltonian rather than the interaction. In contrast, the standard master
equation for a two- dipole system, which depends entirely on well-known
gauge-invariant S-matrix elements, is usually derived using the multipolar
gauge, wherein there is no explicit inter-dipole Coulomb interaction. We show
using a generalised arbitrary-gauge light-matter Hamiltonian that this master
equation is obtained in other gauges only if the inter-dipole Coulomb
interaction is kept within the interaction Hamiltonian rather than the
unperturbed part as in our derivation. Thus, our master equation, while still
gauge-invariant, depends on different S-matrix elements, which give
separation-dependent corrections to the standard matrix elements describing
resonant energy transfer and collective decay. The two master equations
coincide in the large separation limit where static couplings are negligible.
We provide an application of our master equation by finding
separation-dependent corrections to the natural emission spectrum of the
two-dipole system.
