Energy density bounds for black strings

The conserved charge called Y-ADM mass density associated with asymptotically translational Killing-Yano tensor gives us an appropriate physical meaning about the energy density of $p$ brane spacetimes or black strings. We investigated the positivity of energy density in black string spacetimes, using the spinorial technique introduced by Witten. Recently, the positivity of Y-ADM mass density in p brane spacetimes was discussed. In this paper, we will extend this discussion to the transversely asymptotically flat black string spacetimes containing an apparent horizon. We will give the sufficient conditions for the Y-ADM mass density to become positive in such spacetimes.