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4.5 Oppenheimer–Snyder collapse gives a non-static black hole

The simplest scenario in which to analyze gravitational collapse is the Oppenheimer–Snyder model, i.e., collapsing homogeneous and isotropic dust [110Jump To The Next Citation Point]. The collapse region on the brane has an FRW metric, while the exterior vacuum has an unknown metric. In 4D general relativity, the exterior is a Schwarzschild spacetime; the dynamics of collapse leaves no imprint on the exterior.

The collapse region has the metric

a (τ )2[dr2 + r2dΩ2 ] ds2 = − dτ2 + -----------2---2---, (159 ) (1 + kr ∕4 )
where the scale factor satisfies the modified Friedmann equation (see below),
2 ( ) ˙a--= 8πG-ρ 1 + -ρ- + ρℰ- . (160 ) a2 3 2 λ ρ
The dust matter and the dark radiation evolve as
( ) ( ) a0 3 a0 4 ρ = ρ0 -μ- , ρℰ = ρℰ 0 μ-- , (161 )
where a0 is the epoch when the cloud started to collapse. The proper radius from the centre of the cloud is R (τ) = ra(τ )∕ (1 + 1kr2) 4. The collapsing boundary surface Σ is given in the interior comoving coordinates as a free-fall surface, i.e. r = r = const. 0, so that R (τ) = r a(τ)∕(1 + 1kr2) Σ 0 4 0.

We can rewrite the modified Friedmann equation on the interior side of Σ as

2 R˙2 = 2GM---+ 3GM----+ Q--+ E, (162 ) R 4π λR4 R2
where the “physical mass” M (total energy per proper star volume), the total “tidal charge” Q, and the “energy” per unit mass E are given by
4 πa30r30ρ0 M = ------1--2-3, (163 ) 3(1 + 4kr0) --ρℰ0a40r40--- Q = (1 + 1kr2)4, (164 ) 4 0 ----kr20---- E = − (1 + 1kr2 )2 > − 1. (165 ) 4 0
Now we assume that the exterior is static, and satisfies the standard 4D junction conditions. Then we check whether this exterior is physical by imposing the modified Einstein equations (137View Equation). We will find a contradiction.

The standard 4D Darmois–Israel matching conditions, which we assume hold on the brane, require that the metric and the extrinsic curvature of Σ be continuous (there are no intrinsic stresses on Σ). The extrinsic curvature is continuous if the metric is continuous and if R˙ is continuous. We therefore need to match the metrics and R˙ across Σ.

The most general static spherical metric that could match the interior metric on Σ is

[ ] 2 2 2Gm (R ) 2 dR2 2 2 ds = − F(R ) 1 − ----R---- dt + 1-−-2Gm--(R-)∕R-+ R dΩ . (166 )
We need two conditions to determine the functions F (R ) and m (R). Now Σ is a freely falling surface in both metrics, and the radial geodesic equation for the exterior metric gives R˙2 = − 1 + 2Gm (R )∕R + E&tidle;∕F (R )2, where &tidle;E is a constant and the dot denotes a proper time derivative, as above. Comparing this with Equation (162View Equation) gives one condition. The second condition is easier to derive if we change to null coordinates. The exterior static metric, with
dR dv = dt + ---------------, (1 − 2Gm ∕R )F

( ) ds2 = − F2 1 − 2Gm-- dv2 + 2F dvdR + R2 dΩ2. (167 ) R
The interior Robertson–Walker metric takes the form [110]
τ2 [1 − (k + a˙2)R2 ∕a2]dv2 ds2 = − -,v-----------------------+ ∘--2τ,vdvdR----+ R2d Ω2, (168 ) (1 − kR2 ∕a2) 1 − kR2 ∕a2
( 1 ) dR dτ = τ,vdv + 1 + --kr2 ---------1---2. 4 ra˙− 1 + 4kr

Comparing Equations (167View Equation) and (168View Equation) on Σ gives the second condition. The two conditions together imply that F is a constant, which we can take as F (R ) = 1 without loss of generality (choosing &tidle; E = E + 1), and that

3M 2 Q m (R ) = M + ------3 + -----. (169 ) 8 πλR 2GR
In the limit λ −1 = 0 = Q, we recover the 4D Schwarzschild solution. In the general brane-world case, Equations (166View Equation) and (169View Equation) imply that the brane Ricci scalar is
μ 9GM--2- R μ = 2πλR6 . (170 )
However, this contradicts the field equations (137View Equation), which require
Rμ μ = 0. (171 )
It follows that a static exterior is only possible if M ∕λ = 0, which is the 4D general relativity limit. In the brane-world, collapsing homogeneous and isotropic dust leads to a non-static exterior. Note that this no-go result does not require any assumptions on the nature of the bulk spacetime, which remains to be determined.

Although the exterior metric is not determined (see [123] for a toy model), we know that its non-static nature arises from

The result suggests that gravitational collapse on the brane may leave a signature in the exterior, dependent upon the dynamics of collapse, so that astrophysical black holes on the brane may in principle have KK “hair”. It is possible that the non-static exterior will be transient, and will tend to a static geometry at late times, close to Schwarzschild at large distances.

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