### 4.2 Taylor expansion into the bulk

One can use a Taylor expansion, as in Equation (82), in order to probe properties of a static black hole
on the brane [73]. (An alternative expansion scheme is discussed in [50].) For a vacuum brane metric,
This shows in particular that the propagating effect of 5D gravity arises only at the fourth order of the
expansion. For a static spherical metric on the brane,
the projected Weyl term on the brane is given by
These components allow one to evaluate the metric coefficients in Equation (142). For example, the area of
the 5D horizon is determined by ; defining as the deviation from a Schwarzschild form for ,
i.e.,
where is constant, we find
This shows how and its -derivatives determine the change in area of the horizon along the extra
dimension. For the black string , and we have . For a large black hole, with horizon
scale , we have from Equation (41) that
This implies that is decreasing as we move off the brane, consistent with a pancake-like shape of the
horizon. However, note that the horizon shape is tubular in Gaussian normal coordinates [113].