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In this study, beginning with the 4D asymptotically flat Kerr black holesolution in the Boyer-Lindquist (BL) coordinate system, then by using thetechnique of frame-dragging and some coordinate transformation to incorporatethe linear-order supertranslation correction, we finally obtain a 4Dasymptotically flat black hole solution with an arbitrary $a$ in thelinear-order supertranslated BL coordinate system. We can confirm that thissatisfies the Einstein equation. At the end, we calculate the angular momentumof the spacetime we obtained in this study including the linear-ordercorrection. Since the supertranslation is a type of the general coordinatetransformation, which is usually defined by linear-order, the corrections weincorporate in this study is sufficient, considering the fact that $a$ isarbitrary. This work is interesting as a generalization of the supertranslatedspacetime to the rotating system.

A 4D asymptotically flat rotating black hole solution including supertraslation correction