  Accurately modeling an extreme-mass-ratio inspiral requires knowledge of the
second-order gravitational self-force on the inspiraling small object.
Recently, numerical puncture schemes have been formulated to calculate this
force, and their essential analytical ingredients have been derived from first
principles. However, the \emph{puncture}, a local representation of the small
object's self-field, in each of these schemes has been presented only in a
local coordinate system centered on the small object, while a numerical
implementation will require the puncture in coordinates covering the entire
numerical domain. In this paper we provide an explicit covariant self-field as
a local expansion in terms of Synge's world function. The self-field is written
in the Lorenz gauge, in an arbitrary vacuum background, and in forms suitable
for both self-consistent and Gralla-Wald-type representations of the object's
trajectory. We illustrate the local expansion's utility by sketching the
procedure of constructing from it a numerically practical puncture in any
chosen coordinate system.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena