The purpose of this thesis is to calculate the relativistic correction to the gravitational waves produced by compact binaries in the inspiral phase. The correction is up to the next to leading order, the so-called first post-Newtonian order (1PN), which are correctional terms proportional to $(v/c)^2$ compared to leading order, Newtonian, terms. These corrections are well known in the literature, even going beyond the first order corrections, so why is it computed again here? In later years, an alternative approach for computing these terms using effective field theory has emerged. This thesis investigates this approach by replicating it, and attempts to make this approach more accessible to those not familiar with effective field theories. It has been claimed that this approach greatly simplifies the complicated calculations of gravitational waveforms, and even provides the required intuition for 'physical understanding'. By this master student that was found not to be entirely correct. The calculations were made easier for those with a rich background in quantum field theory, but for those who are not well acquainted with quantum field theory this was not the case. It was, however, found to be a worthwhile method as a means for deepening one's understanding of gravity, and might provide a shorter route for some alternative theories of gravity to testable predictions.
First post-Newtonian correction to gravitational waves produced by compact binaries: How to compute relativistic corrections to gravitational waves using Feynman diagrams