The recent discovery of a black hole at the core of an active galactic nucleus by the Hubble Space Telescope confirms that black holes are in fact real astrophysical objects. It is now commonly believed by astronomers that black holes are located in the nuclei of most quasars and galaxies, including our own. If this is true, then black holes are ubiquitous objects in the universe. It is important that our physical understanding of black holes be as complete as possible.
A black hole is the result of total gravitational collapse of a star. The interesting feature of a black hole is the event horizon which completely surrounds the black hole. The event horizon acts as a one way membrane, allowing matter and radiation to enter the black hole but allowing nothing to leave it. The interior is thus a mysterious region since any information about the inside is trapped inside the black hole. Despite this, it is still possible to make predictions about the interior. The exterior region up to the event horizon is well understood. This allows the specification of initial conditions at the event horizon. The Einstein equations can then be used to evolve the initial conditions through the interior, giving a mathematical description of the black hole interior.
A typical black hole found in nature will be rotating. The well known Kerr solution for a rotating black hole has an interesting feature, called a Cauchy horizon contained in its interior. The Cauchy horizon is a light-like surface which is the boundary of the domain of validity of the Cauchy problem. What this means is that it is impossible to use the laws of physics to predict the structure of the region after the Cauchy horizon. This breakdown of predictability has led physicists to hypothesize that a singularity should form at the Cauchy horizon forcing the evolution of the interior to stop at the Cauchy horizon, rendering the idea of a region after it as meaningless.
Recently this hypothesis was tested in a simple black hole model. A spherically symmetric black hole with a point electric charge has the same essential features as a rotating black hole. It was shown that in the spherical model that the Cauchy horizon does develop a scalar curvature singularity. It was a also found that the mass of the black hole measured near the Cauchy horizon diverges exponentially as the Cauchy horizon is approached. This led to this phenomena being dubbed "mass inflation".
Unfortunately the result for spherical symmetry required rather special boundary conditions. Some of the research that i have done has shown that these special assumptions can be abandoned for more physically realistic boundary conditions. The result for spherical symmetry was found to be unchanged by this introduction.
The main focus of my thesis is the development of a general solution for a black hole interior which does not assume that the black hole is spherical. A real astrophysical black hole will not be exactly spherical, will rotate, and a complex system of matter and radiation will be flowing into it. The solution that we are working on will hold for any physical flows into the black hole. The main result that is coming out of this research is that there is a singularity forming along the Cauchy horizon, as before. The structure of this singularity is of a more complex nature than that for the spherical black hole. This analysis is, of course, quite difficult, but it is simplified a great deal by a 2+2 splitting formalism which we have recently developed.

2+2 Splitting of the Einstein Equations
In 1962 Arnowitt, Deser and Misner (ADM) introduced a formalism which allows us to view gravity as "geometrodynamics". In the ADM formalism spacetime is treated as three-dimensional space plus time by singling out a specific choice of time. Spacetime is then foliated by a series of spacelike three-dimensional hypersurfaces, one at each value of the time coordinate "t". The three-dimensional geometry on an initial hypersurface is then seen to evolve to later hypersurfaces through the Einstein equations, hence the term geometrodynamics. This approach to gravity has proven to be quite fruitful in the study of quantum cosmology.
There are instances in general relativity when null hypersurfaces occur, such as at the horizons of a black hole, which can not be treated easily by a 3+1 splitting. In these cases it would be conceptually simpler to use a 2+2 splitting. In our formalism we consider two different foliations of spacetime by intersecting null hypersurfaces. The intersections of the null surfaces are on spacelike two-dimensional surfaces (with the two null generators of the null hypersurfaces as normals). We then have as geometric variables the metric on the spacelike surfaces, a "lapse" function, two shift vectors, and the extrinsic curvatures of the surface, which describe the embedding of the surface in both of the null directions. It is then possible to write the Einstein equations in a simple form in terms of the geometric variables, Lie derivatives with respect to the null generators, and two-dimensionally covariant derivatives. The final form of the Einstein equations is reminiscent of the ADM form which is nice conceptually.

 

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