Taylor expansion into the bulk

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,

2 &tidle;gμν(x, y) = &tidle;gμν(x,0) − ℰμν(x,0+ )y2 − -ℰμν(x, 0+)|y|3 [ ℓ ] 1-- 32- αβ α 4 + 12 □ ℰμν − ℓ2 ℰμν + 2R μανβℰ + 6ℰ μ ℰαν 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,

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 ) θθ 2 F H

These components allow one to evaluate the metric coefficients in Equation (142

). For example, the area of the 5D horizon is determined by $g&tidle;θθ$ ; defining $ψ (r)$ as the deviation from a Schwarzschild form for $H$ , i.e.,

where $m$ is constant, we find

This shows how $ψ$ and its $r$ -derivatives determine the change in area of the horizon along the extra dimension. For the black string $ψ = 0$ , and we have $2 &tidle;gθθ(r,y) = r$ . For a large black hole, with horizon scale $≫ ℓ$ , we have from Equation (41

) that

This implies that $&tidle;gθθ$ 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 ].