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In this, the first of two companion papers, we present a method for findingthe gravitational self-force in a modified radiation gauge for a particlemoving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of anearlier result by Wald is used to show the spin-weight $\pm 2$ perturbed Weylscalar ($\psi_0$ or $\psi_4$) determines the metric perturbation outside theparticle up to a gauge transformation and an infinitesimal change in mass andangular momentum. A Hertz potential is used to construct the part of theretarded metric perturbation that involves no change in mass or angularmomentum from $\psi_0$ in a radiation gauge. The metric perturbation iscompleted by adding changes in the mass and angular momentum of the backgroundspacetime outside the radial coordinate $r_0$ of the particle in any convenientgauge. The resulting metric perturbation is singular on the trajectory of theparticle and discontinuous across the sphere $r=r_0$. A mode-sum method can beused to renormalize the self-force, but the justification given in thepublished version of this paper \cite{sf2} referred to work by Sam Gralla\cite{gralla10} to justify the use of the renormalized self-force, and theradiation gauge we use does not satisfy the regularity conditions required byGralla. Instead we show that the renormalized self-force, computed either fromthe retarded field for $r>r_0$ or for $r<r_0$ gives the correct equations ofmotion in a gauge smoothly related to a Lorenz gauge; and Pound et al.\cite{pmb13} argue that the average of the self-force obtained in the waydescribed in our paper for $r>r_0$ and for $r<r_0$ gives the correct equationof motion for our gauge (what Pound et al. call the no-string gauge).

Gravitational Self-force in a Radiation Gauge