Black Hole Hair in Higher Dimensions

We study the property of matter in equilibrium with a static, spherically symmetric black hole in D-dimensional space time. It requires that this kind of matter has an equation of state omega equivalent to pr/rho = -n/(n + 2k), k, n is an element of N (where n > 1 corresponds to a mixture of vacuum matter and "hair" matter), which seems to be independent of D. However, when we associate this result with specific models, we find that these hairy black holes can live only in some special dimensional space time: (i) D = 2 + 2k/n while the black hole is surrounded by cosmic strings, which requires D is even or D is an element of N, depending on the value of n, this is consistent with some important results in superstring theory, it might reveal the relation between cosmic string and superstring in another aspect; (ii) the black hole can be surrounded by linear dilaton field only in 4-dimensional space time. In both cases, D = 4 is special. We also present some examples of such hairy black holes in higher dimensions, including a toy model with negative energy density.