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) TWELFTH MARCEL GROSSMANN MEETING Paris, July 17, 2009

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. According to Maeda, the gravitational field equations in the bulk read

Using the Gauss’ relations on the brane and making we get that the modified Einstein field equations on the brane read 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 is the standard two- dimensional Euclidian geometry. Assuming that the energy-momentum tensor of our model is given by where and, we get that the Codazzi’s conditions read

Assuming that, it is straightforward to check that the unique solution for the modified Einstein field equations on the brane is still 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 implying that, the dynamical equation for the scale factor reads

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 and the dynamical equation for the scale factor reads

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. Hawking Temperature: where

5-Black Hole Termodynamics (Quasi-Extremal Configuration): 5-A: Geometrical Approach Expanding the polynomial, assuming that we get that Defining 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.