Unstable propagation of black strings

We investigate the five-and six-dimensional black strings within Einstein-Maxwell theory. The extremal black string solutions are endowed with the null Killing symmetry. Using this symmetry, we show that one graviton is a propagating mode, while both the Maxwell (F) and three-form (H) fields are non-propagating modes. Further we discuss the stability and classical hair of the extremal black strings. It turns out that for any mode with l greater than or equal to 0, this propagation is unstable and there is no time-dependent classical hair.