Gravitational self-force in a radiation gauge

In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin weight $\pm 2$ perturbed Weyl scalar (${\psi }_0$ or ${\psi }_4$) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from ${\psi }_0$ in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate $r_0$ of the particle in any convenient gauge. The resulting metric perturbation is singular only on the trajectory of the particle. A mode-sum method is then used to renormalize the self-force. Gralla shows that the renormalized self-force can be used to find the correction to a geodesic orbit in a gauge for which the leading, $O({\rho }^{-1})$, term in the metric perturbation has spatial components even under a parity transformation orthogonal to the particle trajectory, and we verify that the metric perturbation in a radiation gauge satisfies that condition. We show that the singular behavior of the metric perturbation and the expression for the bare self-force have the same power-law behavior in $L=\ell +1/2$ as in a Lorenz gauge (with different coefficients). We explicitly compute the singular Weyl scalar and its mode-sum decomposition to subleading order in $L$ for a particle in circular orbit in a Schwarzschild geometry and obtain the renormalized field. Because the singular field can be defined as this mode sum, the coefficients of each angular harmonic in the sum must agree with the large $L$ limit of the corresponding coefficients of the retarded field. One may therefore compute the singular field by numerically matching the retarded field to a power series in $L$. To check the accuracy of the numerical method, we analytically compute leading and subleading terms in the singular expansion of ${\psi }_0$ and compare the numerical and analytic values of the renormalization constants, finding agreement to high precision. Details of the numerical computation of the perturbed metric, the self-force, and the quantity $h_{\alpha \beta }{u}^{\alpha }{u}^{\beta }$ (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.