

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 [109
]. 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
where the scale factor satisfies the modified Friedmann equation
(see below),
The dust matter and the dark radiation evolve as
where
is the epoch when the cloud started to collapse. The
proper radius from the centre of the cloud is
. The collapsing boundary surface
is given in the interior comoving coordinates as a
free-fall surface, i.e.
, so
that
.
We can rewrite the modified Friedmann equation on
the interior side of
as
where the “physical mass”
(total energy per proper
star volume), the total “tidal charge”
, and the “energy”
per unit mass
are given by
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 (137). 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
be continuous (there
are no intrinsic stresses on
). The extrinsic curvature is
continuous if the metric is continuous and if
is continuous. We therefore need to match the
metrics and
across
.
The most general static spherical metric that
could match the interior metric on
is
We need two conditions to determine the functions
and
. Now
is a freely falling surface
in both metrics, and the radial geodesic equation for the exterior
metric gives
where
is a constant and the dot denotes a
proper time derivative, as above. Comparing this with
Equation (162) gives one condition.
The second condition is easier to derive if we change to null
coordinates. The exterior static metric, with
The interior Robertson-Walker metric takes the form [109]
where
Comparing Equations (167) and (168) on
gives the second condition. The two conditions
together imply that
is a constant, which we can take as
without loss of generality (choosing
), and that
In the limit
, we recover the 4D
Schwarzschild solution. In the general brane-world case,
Equations (166) and (169) imply that the brane
Ricci scalar is
However, this contradicts the field equations (137), which require
It follows that a static exterior is only possible if
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
- 5D bulk graviton stresses, which transmit
effects nonlocally from the interior to the exterior, and
- the non-vanishing of the effective pressure at
the boundary, which means that dynamical information from the
interior can be conveyed outside via the 4D matching
conditions.
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.

