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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 {xa} and some metric tensor gab. The paths of light rays are described by null (i.e. lightlike) geodesics, which are computed using the geodesic equation

where Da is the covariant derivative for the metric gab 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 wab and shear sab, 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 q0<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 gravity k 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 gravity k 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.

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A black hole unobscured by dust

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