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This paper develops a step‑by‑step, perturbative construction of the Kerr black hole metric. The authors treat the metric as a function of two small parameters: Newton’s constant G (which controls the usual weak‑field expansion in GM/r) and the Kerr spin parameter a (a length scale equal to the source’s angular momentum divided by its mass). Working in harmonic coordinates, they set up recursion rules that generate the metric order by order in these two parameters and study how, and when, the resulting series can be summed back into the known closed form of the Kerr solution.
To make the calculation practical they do most of the work in momentum space. They use the Landau–Lifshitz “gothic” variables (a way to package the metric that simplifies the equations) and introduce auxiliary field variables so they do not have to repeatedly invert series expansions. The Einstein equations then produce algebraic recursion relations: the next order in G is built by “gluing” together lower‑order pieces. The authors say these recursion relations can be solved to arbitrarily high order in both G and a, and they construct explicit metric corrections up to fourth post‑Minkowskian order (fourth order in G) while keeping all orders in the spin parameter a.
A striking practical feature of the approach is how it handles causality and integrations. Working in momentum space lets them use retarded Green functions systematically and reuse modern loop‑integral techniques familiar from particle physics. However, Fourier transforms in this procedure can produce divergences. To manage them the authors adopt dimensional regularization (continuing the number of spatial dimensions away from three). They explain in detail that some algebraic identities true in exactly three dimensions do not survive this continuation, and that special care is needed when taking limits back to three dimensions.
Why this matters: the Kerr metric is the rotating black hole solution of Einstein’s equations and is central to both theory and observations. Showing how it emerges order by order makes the structure of general relativity more explicit. The method also connects to recent amplitude and worldline approaches, and it provides a concrete laboratory to test whether post‑Minkowskian (weak‑field) expansions might converge in classical gravity. That question matters because a convergent expansion for single sources would support the use of perturbative methods for physically relevant observables.
Important caveats are stated in the paper. The harmonic coordinates used do not cover the whole spacetime: the coordinate map discussed applies only outside r̄>GM. Re‑summing the perturbative series into the known closed form requires adding redundant terms that reflect gauge freedom — in plain terms, one must choose harmonic functions to shift coordinates to reach the compact exact form. Dimensional regularization introduces ambiguities because identities valid in three dimensions do not automatically continue to d=3−2ϵ, so the authors propose and document a prescription to handle these issues consistently. Finally, the authors do not claim that post‑Minkowskian series converge for all problems; they explore the extent of summability in this specific, single‑source setting and provide explicit high‑order checks rather than a general proof of convergence.