The Bondi–Sachs Formalism for the Einstein Scalar Field Equations with the Zero Cosmological Constant

Inspired by interaction of gravitational waves and dark matters, we study the Bondi–Sachs formalism for Einstein massless scalar field with zero cosmological constant. We provide asymptotic expansions for the Bondi–Sachs metrics as well as the scalar fields and prove the peeling property. We also prove the positivity of the Bondi energy–momentum under condition \(c=d=0\) $c=d=0$ at some retarded time \(u_0\) $u_0$ . This condition ensures that asymptotically null hypersurfaces near \(u=u_0\) $u=u_0$ are asymptotically null initial data sets of order 2 and the positive energy theorem for null infinity can be applied.