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4.2 Taylor expansion into the bulk

One can use a Taylor expansion, as in Equation (82View Equation), 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,
2 ~gmn(x, y) = ~gmn(x,0)- Emn(x,0+)y2 - -Emn(x, 0+)|y|3 [ l ] 1-- 32- ab a 4 + 12 []Emn - l2 Emn + 2RmanbE + 6Em Ean y + ... (142) y=0+
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,
2 ~gmndxmdxn = - F (r)dt2 + -dr-- + r2d_O_2, (143) H(r)
the projected Weyl term on the brane is given by
F [ 1 - H ] E00 = -- H' - ------ , (144) r [ r ] -1-- F-' 1---H- Err = - rH F - r , (145) ( ' ') E = - 1 + H + rH F--+ H-- . (146) hh 2 F H
These components allow one to evaluate the metric coefficients in Equation (142View Equation). For example, the area of the 5D horizon is determined by ~ghh; defining y(r) as the deviation from a Schwarzschild form for H, i.e.,
2m-- H(r) = 1 - r + y(r), (147)
where m is constant, we find
( 2 ) 1 [ 1 ] ~ghh(r,y) = r2 - y' 1 + --|y| y2 + --2- y'+ --(1 + y')(ry' - y)' y4 + ... (148) l 6r 2
This shows how y and its r-derivatives determine the change in area of the horizon along the extra dimension. For the black string y = 0, and we have 2 ~ghh(r,y) = r. For a large black hole, with horizon scale » l, we have from Equation (41View Equation) that
2 y ~~ - 4ml- . (149) 3r3
This implies that ~ghh 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].

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