  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 $\Psi_4^B$ to be decoupled in the
effective spacetime.
