  Inspirals of stellar-mass objects into massive black holes will be important
sources for the space-based gravitational-wave detector LISA. Modelling these
systems requires calculating the metric perturbation due to a point particle
orbiting a Kerr black hole. Currently, the linear perturbation is obtained with
a metric reconstruction procedure that puts it in a "no-string" radiation gauge
which is singular on a surface surrounding the central black hole. Calculating
dynamical quantities in this gauge involves a subtle procedure of "gauge
completion" as well as cancellations of very large numbers. The singularities
in the gauge also lead to pathological field equations at second perturbative
order. In this paper we re-analyze the point-particle problem in Kerr using the
corrector-field reconstruction formalism of Green, Hollands, and Zimmerman
(GHZ). We clarify the relationship between the GHZ formalism and previous
reconstruction methods, showing that it provides a simple formula for the
"gauge completion". We then use it to develop a new method of computing the
metric in a more regular gauge: a Teukolsky puncture scheme. This scheme should
ameliorate the problem of large cancellations, and by constructing the linear
metric perturbation in a sufficiently regular gauge, it should provide a first
step toward second-order self-force calculations in Kerr. Our methods are
developed in generality in Kerr, but we illustrate some key ideas and
demonstrate our puncture scheme in the simple setting of a static particle in
Minkowski spacetime.
