

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 [72]. (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].

