Abstract
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.
Highlights
When the gravitational-wave detector LISA launches in the early 2030s, one of its key sources will be extreme-mass-ratio inspirals (EMRIs) [1, 2], in which stellar-mass compact objects spiral into massive black holes in galactic nuclei [3]
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
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
Summary
When the gravitational-wave detector LISA launches in the early 2030s, one of its key sources will be extreme-mass-ratio inspirals (EMRIs) [1, 2], in which stellar-mass compact objects spiral into massive black holes in galactic nuclei [3]. To be sufficiently accurate for LISA science, self-force calculations must be carried beyond linear perturbation theory, to second order in m/M [4, 5] They need to be carried out in the spacetime of an astrophysically realistic, spinning, Kerr black hole. Using the simple model problem of a static point mass in flat spacetime, we carry out the GHZ procedure and illuminate its relationship to previous methods, highlighting (i) the half-string singularity structure in the GHZ solution, (ii) how it can be transformed to the no-string gauge and how it provides additional information beyond current no-string calculations, and (iii) why the gauge singularities in both the half-string solution and no-string solution render the second-order field equations ill defined.
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