One needs a local energy condition sufficient to show the positivity of gravitational mass. Roughly speaking, the usual condition requires the positivity of the local energy density for all observers. On the other hand, there exist quantum effects such that the above condition is not satisfied. As one cannot believe that our Universe becomes unstable by these quantum effects, one desires a new weaker local energy condition. So, this paper is intended as a proposal of a new local energy condition. We restrict our consideration to static asymptotically flat space-times.