%1281 slov
\section{Gravitational collapse}
A star is a self-gravitating collection of atoms supported by thermal
pressure. Eventually, fusion at the core must stop, after which
the star cools and contracts. For $M < 1.4M_\odot = M_\text C$
(Chandrasekhar limit) (adopted from \cite[chapter 1]{townsend},
but \cite[chapter 24]{mtw} gives the value 1.2 solar masses), the star
is supported by electron degeneracy pressure, and ends as a white
dwarf.

For $M > M_\text C$ the star must continue to contract until
nucleon-nucleon repulsion halts the implosion. The result is
a~neutron star ($M_\text{max} \sim 3M_\odot$), unless the core's
mass is so great, that gravity overcomes neutron-degeneracy
pressure and pulls the star on in to form a black hole. The
original analysis of the collapse of a star with uniform density
and zero pressure was given by Oppenheimer and Snyder in 1939.


\section{Black holes}
\subsection{Theoretical history}
The idea of a dark star has its origins with John Mitchell \cite {history} who presented this concept to the Royal Society in 1783. By using  Newton's theory of 
gravitation with his corpuscular theory of light, he found
that for a star of the same  density as the Sun, the escape velocity would be the speed of light if the star had
 a radius $R = 486R_\odot$ and mass $M =
1.2 \times 10^8M_\odot$. The French mathematician and philosopher Pierre Simon
Laplace also independently
published the idea of such a star.
These objects were at the time not believed to exist.

In 1915, Einstein presented his theory of General Relativity.
His theory was widely accepted once it was established that it explained the precession
of Mercury's orbit, and predicted the bending of starlight by the Sun.
One of the first solutions to Einstein's equations in a vacuum was determined by Karl Schwarzschild. This was a~spherically symmetric exact solution. It has a special property of having a~so called Schwarzschild Radius $r = 2M$ which is the radius below which the gravitational attraction between the particles of a body must cause it to undergo irreversible gravitational collapse. The Schwarzschild solution represents the empty space-time external to a spherically symmetric object. It has a~singularity at $r=0$ where $r$ is the natural radial coordinate.

In the 1920's to the 1960's there was considerable development of the theory of General Relativity as it related to the idea of black holes (see \cite{chandrasekhar} or summary in \cite{sharp}).  Subsequent to the measurement of the bending of light by the expedition led by Sir Arthur Eddington there was much research into the theory of Einstein.
In his book ``The Internal Constitution of Stars" Eddington
maintained that all stars ended their life cycle by collapsing
into white dwarfs. After applying quantum physics and Relativity
to Eddington's calculations, astrophysicist S.~Chandrasekhar
showed that only stars of mass below a certain limit could become white
dwarfs. This mass is known as the Chandrasekhar limit. This notion was not accepted by Eddington and was the cause of dispute for some time thereafter. In fact if the mass of a star is more than $1.2M_\odot$ then the star cannot stabilize and further collapse is inevitable.
Indeed, Baade and Zwicky predicted that further collapse strips atoms of their electrons resulting in the idea of a neutron star. These stars have enormous densities.

The first idea of a black hole occurs in the work of Oppenheimer and Snyder who calculated that if a star has sufficient mass it must collapse and cut itself off from communication with the rest of the universe. Not even neutron degeneracy pressure, which gives rise to neutron stars, can stop the collapse. It was realised that for the Schwarzschild radius $r = 2M$ the Schwarzschild solution was not singular but that if  the mass was contained inside this ``horizon" boundary no light could escape. If the Schwarzschild radius was contained inside the star then this was not relevant.

On physical grounds it is expected that black holes should rotate and have a oblate ``shape" which requires different solutions to the equations of General Relativity other than that of Schwarzschild. The unique solution which has all the correct properties was obtained by Roy Kerr \cite{kerr}.  It is a so called Petrov Type D solution. The metric is readily interpretable when written in terms of the so called Boyer-Lindquist coordinates. The geodesics of this metric can be explicitly calculated because of the existence of the fourth constant of the motion in addition to the Lie symmetries that are known to be present which correspond to time translation and axial symmetry.This space-time has been well studied with one principal result being that a collapsing star loses all its individual characteristics and its final state is determined by mass and angular momentum only. Therefore, the final state is determined by a Kerr metric with these parameters. 


\subsection{Observational history}
It was not until in 1963 when Maarten Schmidt \cite{schmidt} discovered  extremely
luminous distant objects called quasars that the existence of
black holes was generally acknowledged. Their enormous energy
output from small volumes of space suggested that they were
powered by accretion of matter onto very massive black holes
residing in galactic nuclei. From observations in the 1980's of
motions of stars around the galactic center, it is now believed
that such supermassive black holes (SMBHs) with masses in the range $10^6$ to $10^{10}$ solar masses exist in the center of
most galaxies (\cite{0401527, 0212164, marconi}). \cite{pbh} argue
that SMBHs may be of pregalactic origin. If that is true
primordial black holes would be the centers for growing
protogalaxies.


The first observational evidence for the existence of a stellar
massive black hole was given in the early 70's by the observations of
the binary X-ray source Cygnus X-1 (\cite{gursky, sc, connors1}).
Today, there are about 20 stellar mass black hole candidates that
range in mass from 3 to 15 solar masses \\(\cite{done, kcpz}).
Additionally, there is some evidence for intermediate-mass black
holes, (those with masses of a few thousand times of the Sun.)  %\\
(\cite{gmp})

The big boost for  black hole research came when the Hubble Space
Telescope (HST) was launched (1994). Over the last decade the
combination of HST and ground-based data has led to the
measurement of the black-hole masses of some 40 inactive galaxies
in the nearby Universe and the last few years has also seen a new
industry emerge in estimating the black-hole masses of active
galaxies (\cite{mclure}).





