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
Taz Home Page 2
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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