We consider the vacuum polarisation of a quantum scalar �field on a black hole background. We introduce additional space-like dimensions and study the vacuum polarisation
outside the event horizon. We consider a massless, quantum scalar field on the static, hyperspherically symmetric Schwarzschild-Tangherlini metric with zero cosmological constant. We calculate all results for a Hartle-Hawking vacuum state. The brane and bulk cases are considered separately; the metric on the brane is the four dimensional projection resulting from a bisection of the general black hole metric. On the brane we extend previous work on the Schwarzschild metric, for a conformally coupled �field, such that we choose to include up to seven extra dimensions in the bulk. To conclude the
work on the brane we present numerical results for the vacuum polarisation and outside the event horizon. For the bulk with �five dimensions we present a complete methodology from initial set up to test results that demonstrates a way in which such a calculation
can be completed for the �first time. This is achieved by the introduction of non-physical Minkowski terms into the renormalisation scheme. Finally we discuss the prospects for extending our methodology to calculating the vacuum polarisation in the bulk for more than five dimensions
