Since the Hubble
Space Telescope was launched in 1990, there have been many
observations of what are believed to be black holes, including the photograph
below of a suspected black hole in the heart of the galaxy NGC 6251.
But the study of black holes began in theoretical
physics long before there were any observations of these objects by
astronomers. Not just an interesting physical phenomenon, black holes
are extreme geometrical objects with fascinating mathematical properties
that have posed serious challenges to the foundations of classical and
quantum physics.

Black hole geometry

What makes a black hole so special is the extreme
effect it has on the propagation of light. Suppose we have a black hole
spacetime described in general relativity by some set of coordinates
{x^{a}} and some metric tensor g_{ab}. The paths of
light rays are described by null (i.e. lightlike) geodesics, which are
computed using the geodesic equation

where D_{a} is the covariant derivative for the metric g_{ab}
and

is the tangent vector to the null geodesic in question, and t
is the distance parameter along the geodesic, the analog of time along
a ray of light.
The possible transverse (orthogonal to the propagation
direction) deformations of a bundle of null geodesics can be reduced
to three types: the expansion q,
rotation w_{ab} and shear
s_{ab}, computed as the
trace, antisymmetric part and symmetric part, respectively, of the covariant
derivative of the geodesic tangent vector

Taking the derivative of the expansion q
along a null geodesic leads to what is called the focusing
equation

If we're in a spacetime with no rotation, and
the matter and energy density is positive, then we arrive at a very
important inequality for q
that is the key to all the mysterious and interesting properties of
black holes:

The quantity q measures how
light rays expand or converge, in other words q
measures the focusing of light by gravity. According to our sign convention,
if q is negative, it means the
light rays are being focused together instead of spread apart by the
spacetime geometry. The above inequality tells us that once light rays
start being converged by gravity with some value q_{0}<0,
then in a finite distance along the light ray, nearby light rays will
be focused to a point, such that they cross each other with zero transverse
area A

This is bad news if these light rays all emanated
from a single source, because it means the light is being infinitely
focused into a singularity, and the concept of a geodesic has broken
down. When q turns negative for
both "incoming" and "outgoing" light rays, it means
that the light has been trapped, that
the escape velocity from that gravitational field has become greater
than the speed of light.
When q
is zero or negative for both incoming and outgoing null geodesics orthogonal
to a smooth spacelike surface, that surface is called a
trapped surface, and any closed trapped surface must lie inside
a black hole. This an abstract general
definition of a black hole that is independent of any coordinate system
used to describe it. Gravity bends light like a lens, and a black hole
can be thought of as a very peculiar type of lens, one that bends light
so that it can never be seen.
Black holes have four very important properties
which have become known as the Four Laws of
Black Hole Physics of classical general relativity.

The Four Laws of Black Hole Physics

0

The surface
gravityk at the
event horizon is constant: it has the same value
everywhere on the event horizon.

1

The change in
mass of a black hole is proportional to the surface gravity times
the change in area. dM = (k/8p)
dA

2

The surface
area of the event horizon of a black hole can only
increase, never decrease. (This means that two black holes
can join to make a bigger black hole, but one black hole can never
split in two.)

3

It is impossible
to lower the surface gravityk
at the event horizon to zero through
any physical process.

Note that according to the second law property,
it is impossible for black holes to decay and go away, because a black
hole cannot get smaller or split into smaller black holes. This is going
to be changed when we add quantum mechanics to the theory in the next
section. If these laws look
familiar somehow, there's a good reason. This is a tremendously
important similarity that will also be discussed in the next
section.

The Singularity Within

The problem with the type of focusing of light
that defines the presence of a black hole is that once it starts, the
focusing equation says that it ends in utter disaster. Once a bundle
of null geodesics becomes trapped by crossing to q<0,
within a finite distance along each geodesic, q> -Infinity,
the geodesics will cross at a point, and the transverse area of the
bundle will go to zero. When this happens, the necessary conditions
for the existence and uniqueness of these geodesics are violated, and
it's no longer possible to use the geodesic equations to predict what
happens to the geodesics after they cross.
The spacetime will then exhibit one of the two
possible behaviors:
1. The spacetime curvature in this region remains finite for all observers,
but notion of predictability for the spacetime breaks down, and evolution
of the spacetime can no longer be uniquely predicted from a set of initial
data.
2. The spacetime curvature in this region becomes infinite for all or
some observers, so that there simply is no possibility of extending
geodesics past the point where they cross, they simply end there. The
spacetime as a whole retains its predictability but the region contains
a spacetime singularity where the paths of observers simply end their
existence, and spacetime itself can no longer be defined.

Is there a Cosmic Censor?

So gravity can focus light so powerfully that
it can spontaneously end the existence of observers, destroy the definition
of the spacetime itself, or spoil the unique time evolution in a spacetime
based on a sensible set of initial data? What is to protect us then
from the pathological possibilities of strong gravitation fields?
The Cosmic Censorship
Conjecture proposes that in the context of the theory of general
relativity, in a spacetime where the total energy density is positive,
pathologies such as spacetime singularities and breakdowns in causality
and predictability are always hidden behind the event horizons of black
holes.