  We develop a method for computing the linearized gravitational backreaction
for Nambu-Goto strings using a fully covariant formalism. We work with
equations of motion expressed in terms of a higher dimensional analog of the
geodesic equation subject to self-generated forcing terms. The approach allows
arbitrary spacetime and worldsheet gauge choices for the background and
perturbation.
  The perturbed spacetime metric may be expressed as an integral over a
distributional stress-energy tensor supported on the string worldsheet. By
formally integrating out the distribution, this quantity may be re-expressed in
terms of an integral over the retarded image of the string. In doing so, one
must pay particular attention to contributions that arise from the field point
and from non-smooth regions of the string. Then, the gradient of the perturbed
metric decomposes into a sum of boundary and bulk terms. The decomposition
depends upon the worldsheet coordinates used to describe the string, but the
total is independent of those considerations.
  We illustrate the method with numerical calculations of the self-force at
every point on the worldsheet for loops with kinks, cusps and
self-intersections using a variety of different coordinate choices. For field
points on smooth parts of the worldsheet the self-force is finite. As the field
point approaches a kink or cusp the self-force diverges, but is integrable in
the sense that the displacement of the worldsheet remains finite. As a
consistency check, we verify that the period-averaged flux of energy-momentum
at infinity matches the direct work the self-force performs on the string.
  The methodology can be applied to address many fundamental questions for
string loop evolution.
