In this paper, we construct the canonical harmonic coordinates of the Kerr metric within the multipolar post-Minkowskian (MPM) formalism to the fourth post-Minkowskian (4PM) order. Based on the well-known Geroch-Hansen moments of the Kerr metric and Gürsel’s theorem, we derive the exact canonical MPM moments ${\mathrm{M}}_L$, ${\mathrm{S}}_L$, which are free of any gauge moments. With these moments, we iteratively compute the gothic metric perturbation $h_{\mathrm{can}}^{\mu \nu }$ up to 4PM order and compute the 4PM canonical metric $g_{\mu \nu }^{\mathrm{can}}$. The resulting spatial and time components of these metrics are even functions of the spin parameter $a$ while the mixed components are odd. This parity property distinguishes the canonical coordinates from other harmonic coordinates. To contrast this minimal-gauge construction, we also extract the 1PM source moments of the Kerr metric in the Jiang-Lin coordinates. We find that the Jiang-Lin representation possesses nonvanishing gauge moments starting from the 1PM order, whereas in the canonical representation gauge moments vanish to all orders. This comparison highlights the canonical coordinates as the most gauge-pure representation of the Kerr metric in the MPM framework. The complete canonical metric for the Schwarzschild case is also computed to all PM orders. A recent independent construction by Damgaard et al. using momentum-space recursion yields 4PM equivalent results expressed as a power series in $a$, providing a cross-validation of our closed-form 4PM canonical metric. The coordinate transformation linking the canonical Kerr coordinates to previously known harmonic Kerr coordinates remains an open problem.

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• Classical black holes
• General relativity
• General relativity equations & solutions