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By means of the scattering angles, we obtain an effective metric of spinlessbinaries with radiation-reaction effects up to fourth post-Minkowskian order,which is the foundation of the effective-one-body theory. We note that thereare freedoms for the parameters of the effective metric because one equationcorresponds to two parameters for each post-Minkowskian order. Accordingly, inorder to construct a self-consistent effective-one-body theory in which theHamiltonian, radiation-reaction forces and waveforms for the ``plus" and``cross" modes of the gravitational wave should be based on the same physicalmodel, we can fix these freedoms by requiring the null tetrad component of thegravitationally perturbed Weyl tensor $\Psi_4^B$ to be decoupled in theeffective spacetime.

Effective metric of spinless binaries with radiation-reaction effect up to fourth Post-Minkowskian order in effective-one-body theory