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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 [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,
2 &tidle;gμνdx μdxν = − F (r)dt2 + -dr-- + r2dΩ2, (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 ) θθ 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 g&tidle;θθ; defining ψ (r) as the deviation from a Schwarzschild form for H, i.e.,
2m-- H (r) = 1 − r + ψ (r), (147 )
where m is constant, we find
( 2 ) 1 [ 1 ] &tidle;gθθ(r,y) = r2 − ψ ′ 1 + --|y| y2 + --2- ψ ′ +--(1 + ψ ′)(rψ ′ − ψ )′ y4 + ... (148 ) ℓ 6r 2
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 (41View Equation) that
2 ψ ≈ − 4m--ℓ . (149 ) 3r3
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].
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