  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.
