Gravitational energy cannot become negative

An analytic proof that the Bondi four-momentum, which defines the space-time curvature at large to small distances from, in this case, a gravitational source cannot become negative is presented. A theorem is presented of a space-time which is asymptotically flat at future null infinity, the dominant energy condition holds, and a nonsingular spacelike surface exists which asymptotically approaches the null cone of constant retarded time near null infinity. A proof is provided that shows that the remaining Bondi energy at any time must be future-directed and vanishes when the stress-energy tensor of the nonsingular space-time curvature surface becomes zero. The results show that an isolated system can not radiate more energy than it originally has as defined by its total Arnowitt-Deser-Misner energy.