Transcript Apresentação do PowerPoint

TWELFTH MARCEL GROSSMANN MEETING
Paris, July 17, 2009
Forming Nonsingular Black Holes from
Dust Collapse
by
R. Maier
(Centro Brasileiro de Pesquisas Físicas-Rio de Janeiro)
I. Damião Soares
(Centro Brasileiro de Pesquisas Físicas-Rio de Janeiro)
1-Introduction:
1.A-Motivation:
Fundamental Theorems (Israel and Carter):
The final stage of a general collapse of uncharged matter is typically a Kerr
black hole, which has an involved singularity structure.
Best Theoretical Evidence (Wald):
The interior of the black hole thus formed is analogous to the interior of a
Schwarzschild black hole with a global spacelike singularity.
Alternative Theory: Braneworld Theory
Our Aim:
Proposal of a bulk-brane model (where Einstein equations on the brane have corrections
terms due to the bulk-brane interaction) that avoids such a singularity.
1.B-General Topics:
• General Theory
• The Interior Solution and Its Matching with the Exterior Geometry
• Experimental Tests and Hawking Temperature
• Black Hole Thermodynamics
• Conclusions
2- The General Theory:
In the frame of a bulk-brane theory, we assume a 5-dimensional bulk in a
1+3 brane embedded in the bulk.
1
According to Maeda, the gravitational field equations in the bulk read
1: Physical Re view D, Volume 62, 024012
Using the Gauss’ relations on the brane and making
modified Einstein field equations on the brane read
we get that the
where
,
,
and
.
3- The Interior Solution and Its Matching with the Exterior
Geometry :
Let us consider a spherically symmetric geometry in comoving coordinates
given by
.
where
and
are arbitrary functions and
dimensional Euclidian geometry.
is the standard two-
Assuming that the energy-momentum tensor of our model is given by
where
we get that the Codazzi’s conditions read
and
,
Assuming that
, it is straightforward to check that the unique solution for the
modified Einstein field equations on the brane is still
•
2
According to Shtanov-Sahni it is always possible to embed such a geometry in a five
dimensional pure de Sitter bulk spacetime (
).
Therefore, from Codazzi’s equations we get
dynamical equation for the scale factor reads
2: arXiv : gr  qc / 0208047
v3 9 Feb 2003
implying that, the
Finally, assuming the following initial conditions
we get that
, as long as we take
.
and
We determine the spherically symmetric metric outside the collapsing star from its
matching to the FRW metric, at the surface defined by
in comoving coordinates.
To this end let us transform the comoving coordinates of the original geometry to
Schwarzschild coordinates
through the equations
If F[S(r,t)] is an arbitrary function of S, where S(r,t) satisfies
,
we automatically guarantee that
.
Assuming k > 0, it’s easy to check that the solution for S(r,t) is given by
For the physical domain of parameters to be considered here, S turns out a
monotonous function of which can be properly inverted to express
in terms of S, for an
explicit range of . The remaining task is to choose F in terms of S. A choice can be
suitably made for the case k > 0 so that at the surface of matching
we obtain
where
is the total mass of the collapsing dust.
Now, one may define
in such a way that
the dynamical equation for the scale factor reads
and
On the other hand, we define
in
such a way that the condition for horizon formation reads
Fig1: Plot of the
polynomial P(R) for
dust masses
(no black hole),
(extremal black hole)
and
(black
hole with outer horizon
and inner
horizon
). The
figure corresponds to
=0.05, in units
G =c=1.
The maximal analytical extension of the geometry:
The condition
is sufficient to guarantee that:
4- Experimental Tests and Hawking Temperature:
•
Planetary Perihelia Precession (per revolution):
where
•
Bending of Light:
where R is the radius of the body.
•
where
Hawking Temperature:
5-Black Hole Termodynamics (Quasi-Extremal Configuration):
5-A: Geometrical Approach
Expanding the polynomial
assuming that
Defining
,
we get that
it’s easy to check that
However, in this case
We can therefore associate the horizon area of the quasi-extremal black hole
with the entropy, in accordance to Bekenstein's definition
The above equation corresponds to an extended Second Law of Thermodynamics with
an extra work term connected to the variation of the brane tension.
5-Conclusions:
•
The dynamics of the gravitational collapse is examined in the realm of string
based formalism of D-branes that encompasses General Relativity as a low energy
limit. A complete analytical solution is given to the spherically symmetric collapse of a
pure dust star, including its matching with a corrected Schwarzschild exterior
spacetime.
•
The collapse forms a black hole (an exterior event horizon) enclosing not a
singularity but perpetually bouncing matter in the infinite chain of spacetime maximal
analytical extensions inside the outer event horizon. This chain of analytical extensions
has a structure analogous to that of the Reissner-Nordstrom solution, except that the
timelike singularities are avoided by bouncing barriers.
•
For the exterior geometry of the nonsingular black hole, we examine the
corrections on the Hawking temperature and on the experimental tests of General
Relativity.
•
In the case of a quasi-extremal black hole, we reproduce Bekenstein's results of
black hole thermodynamics.
•
It's worth to remark that the interior trapped bouncing matter has the possibility of
being expelled by disruptive nonlinear resonance mechanisms.