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 $c=d=0$ at some retarded time $u_0$. This condition ensures that asymptotically null hypersurfaces near $u=u_0$ are asymptotically null initial data sets of order 2 and the positive energy theorem for null infinity can be applied.