Black holes (BH) are likely the most puzzling prediction of general relativity (GR). Their prediction marks the limit of the theory itself, since in GR, at the center of the BH lies a singularity. Singular points in physics are the indication that the limit of validity of the theory is reached. Indeed, a singular point leads to infinities, and thus no quantitative predictions can be made.
The black hole singularity is hidden behind an event horizon, roughly defined as the boundary of the region of space-time that is causally disconnected from the rest of the spacetime. See Wald’s book ‘Gravitation’ for a formal and precise definition. Rotating black holes further have a ergoregion, which has a lot of amazing properties, such as allowing to ‘give’ energy to particles. Between the ergoregion and the event horizon, spacetime is dragged in the direction of rotation of the black hole, such that particles that would be stationary with respect to distant observer would have local velocity higher than the speed of light (Lense-Thirring effect). This is related to the existence of negative energy in the ergosphere, allowing energy extraction such as the Penrose process. See http://inspirehep.net/record/1341250.
Black Holes in Astrophysics
Black holes are vacuum GR solutions, everywhere except at the center of the black hole, where the source is a massive point. In this sense, they are GR’s elementary particles. However, they usually form from a very massive dying star, that undergoes collapse. The star once all its nuclear fuel is burnt will undergo a collapse that can be stopped by (i) the electron degeneracy pressure, (ii) the neutron degeneracy pressure, (iii) nothing, depending on the initial mass of the star. When it comes to the final collapse, a shock waves bounces on the forming objects, and ejects a large part of matter far away. This is in close relation with the observed abundance of heavy elements in the universe, which would otherwise be mainly constituted of hydrogen.
In the first case, the equilibrium configuration is a white dwarf, mainly made of iron. The second case, which we will talk about extensively elsewhere is a neutron star, while the last one are black holes.
Once the collapsing matter crosses the horizon, it is causally disconnected of the rest of the space-time. All the information contained in the star is lost and the external space-time is described by a 2 (3) parameter family. This is related to the so-called information paradox.
At the center of galaxies, it is believed that supermassive black holes are present. They have masses of around 10^6 – 10^10 solar masses and are supposed to be formed from merger of smaller black hole and absorption of other material. They are of crucial importance because they drive the galactic evolution. They are surrounded by an accretion disk, consisting mainly of a hot iron plasma with a velocity close to that of light. This hot plasma emits a X-ray spectrum, mainly from the Kalpha line of iron. The accretion disk allows to assess the potential presence of the black hole, and its shape and dynamics is intimely related to that of the black hole.
Solar mass black holes instead have masses of the order of solar masses and result from a heavy star collapse. They are much more difficult to detect and their presence is usually deduced when they belong to a binary.
Thermodynamics of Black Holes, AdS/CFT
A very interesting property of black holes is that they obey the laws of thermodynamics. The horizon of a black hole can only grow under classical process, similar to the second law of thermodynamics. It is possible to assign a temperature and an entropy to the black hole. By the way, black hole emits though Hawking radiation a black body spectrum with the exact same temperature than that of the black hole. In the presence of a cosmological constant, the thermodynamics properties extend even further with a pressure term.
In AdS (negative cosmological constant) and higher dimensions, black holes provide a link between gravity and quantum field theory, from the celebrated AdS/CFT duality.
In this context, black holes play a crucial role because they allow to deal with a finite temperature CFT, given precisely by the temperature of the black hole. Phase diagrams and thermodynamical properties also transports from the bulk (AdS) to the boundary (CFT). This becomes particularly intersting when studying systems consisting of matter fiels (e.g. scalars, fermions…) and/or black holes admitting phase transitions between a state with black hole and a regular solution. This is in essence the case for the so-called holographic superconductors.