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

By means of the scattering angles, we obtain an effective metric of spinless binaries with radiation-reaction effects up to fourth post-Minkowskian order, which is the foundation of the effective-one-body theory. We note that there are freedoms for the parameters of the effective metric because one equation corresponds to two parameters for each post-Minkowskian order. Accordingly, in order to construct a self-consistent effective-one-body theory in which the Hamiltonian, radiation-reaction forces and waveforms for the plus" and cross" modes of the gravitational wave should be based on the same physical model, we can fix these freedoms by requiring the null tetrad component of the gravitationally perturbed Weyl tensor ${\mathrm{\Psi }}_4^B$ to be decoupled in the effective spacetime.