User:Mpatel/sandbox/Black hole



A black hole is a region of space where mass is so concentrated that nearby matter or electromagnetic radiation (such as visible light) cannot escape the overwhelming gravitational field. The event horizon of a black hole marks the 'point of no return', a boundary that, if crossed, inevitably leads falling matter and radiation towards the central singularity.

Several types of black hole are thought to exist, their existence being theoretically predicted to arise as solutions of Einstein's field equations of general relativity, the currently accepted theory of gravitation. Black holes are generally believed to form from the collapse of massive stars. Others are thought to arise from quantum mechanical processes in the early universe.

Quantum mechanics research also indicates that, rather than holding captured matter forever, black holes may slowly leak a form of thermal energy called Hawking radiation and may well have a finite life. Together with white holes, black holes are speculated to form wormholes, shortcuts through space and time. The as yet unknown theory of quantum gravity is believed to give the fully correct description of black holes.

Black holes are believed to reveal their presence through interaction with orbiting material. Recent observations of orbiting stars and gas indicate evidence for black holes. Such observations have resulted in the general scientific consensus that, barring a breakdown in our understanding of nature, black holes do exist in our universe.

History
The idea of a body so massive that even light could not escape was put forward by geologist John Michell in a letter written to Henry Cavendish in 1783 to the Royal Society: If the semi-diameter of a sphere of the same density as the Sun were to exceed that of the Sun in the proportion of 500 to 1, a body falling from an infinite height towards it would have acquired at its surface greater velocity than that of light, and consequently supposing light to be attracted by the same force in proportion to its vis inertiae, with other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity.

This assumes that light is influenced by gravity in the same way as massive objects.

In 1796, mathematician Pierre-Simon Laplace promoted the same idea in the first and second editions of his book Exposition du système du Monde (it was removed from later editions).

Such "dark stars" were largely ignored in the nineteenth century, since light was then thought to be a massless wave and therefore not influenced by gravity. Unlike the modern black hole concept, the object behind the horizon is assumed to be stable against collapse.

In 1915, Albert Einstein developed the theory of gravity called general relativity, having earlier shown that gravity does influence light (although light has zero rest mass, it is not the rest mass that is the source of gravity but the energy). A few months later, Karl Schwarzschild gave the solution for the gravitational field of a point mass and a spherical mass, showing that a black hole could theoretically exist. The Schwarzschild radius is now known to be the radius of the event horizon of a non-rotating black hole, but this was not well understood at that time, for example Schwarzschild himself thought it was not physical. Johannes Droste, a student of Lorentz, independently gave the same solution for the point mass a few months after Schwarzschild and wrote more extensively about its properties.

In 1930, astrophysicist Subrahmanyan Chandrasekhar calculated using general relativity that a non-rotating body above 1.44 solar masses (the Chandrasekhar limit) would collapse, since no mechanism was known to prevent this. His arguments were opposed by Arthur Eddington, who believed that something would inevitably stop the collapse. Eddington was partly correct: a white dwarf slightly more massive than the Chandrasekhar limit will collapse into a neutron star. But in 1939, Robert Oppenheimer and others predicted that stars above approximately three solar masses (the Tolman-Oppenheimer-Volkoff limit) would collapse into black holes for the reasons presented by Chandrasekhar.

Oppenheimer and his co-authors used Schwarzschild's system of coordinates (the only coordinates available in 1939), which produced mathematical singularities at the Schwarzschild radius, in other words the equations broke down at the Schwarzschild radius because some of the terms were infinite. This was interpreted as indicating that the Schwarzschild radius was the boundary of a "bubble" in which time "stopped". The collapsed stars were briefly known as "frozen stars", as the calculations indicated that an outside observer would see the surface of the star frozen in time at the instant where its collapse takes it inside the Schwarzschild radius. But many physicists could not accept the idea of time standing still inside the Schwarzschild radius, and there was little interest in the subject for over 20 years.

In 1958, David Finkelstein broke the deadlock over "stopped time" and introduced the concept of the event horizon by presenting Eddington-Finkelstein coordinates, which enabled him to show that "The Schwarzschild surface r = 2 m is not a singularity but acts as a perfect unidirectional membrane: causal influences can cross it in only one direction". All theories up to this point, including Finkelstein's, covered only non-rotating, uncharged black holes.

In 1967, astronomers discovered pulsars, and within a few years could show that the known pulsars were rapidly rotating neutron stars. Until that time, neutron stars were also regarded as just theoretical curiosities. So the discovery of pulsars awakened interest in all types of ultra-dense objects that might be formed by gravitational collapse.

Physicist John Wheeler is widely credited with coining the term black hole in his 1967 public lecture Our Universe: the Known and Unknown, as an alternative to the more cumbersome "gravitationally completely collapsed star". However, Wheeler insisted that someone else at the conference had coined the term and he had merely adopted it as useful shorthand. The term was also cited in a 1964 letter by Anne Ewing to the AAAS:

"According to Einstein’s general theory of relativity, as mass is added to a degenerate star a sudden collapse will take place and the intense gravitational field of the star will close in on itself. Such a star then forms a 'black hole' in the universe."

The phrase had already entered the language years earlier as the Black Hole of Calcutta incident of 1756 in which 146 Europeans were locked up overnight in punishment cell of barracks at Fort William by Siraj ud-Daulah, and all but 23 perished.

What makes it impossible to escape from black holes?
Popular accounts commonly try to explain the black hole phenomenon by using the concept of escape velocity, the speed needed for a vessel starting at the surface of a massive object to completely clear the object's gravitational field. It follows from Newton's law of gravity that a sufficiently dense object's escape velocity will equal or even exceed the speed of light. Citing that nothing can exceed the speed of light they then infer that nothing would be able to escape such a dense object. However, the argument has a flaw in that it does not explain why light would be affected by a gravitating body or why it would not be able to escape. Nor does it give a satisfactory explanation for why a powered spaceship would not be able to break free.

Two concepts introduced by Albert Einstein are needed to explain the phenomenon. The first is that time and space are not two independent concepts, but are interrelated forming a single continuum, spacetime. This continuum has some special properties. An object is not free to move around spacetime at will; it must always move forward in time and cannot change its position in space faster than the speed of light. This is the main result of the theory of special relativity.

The second concept is the base of general relativity; mass deforms the structure of this spacetime. The effect of a mass on spacetime can informally be described as tilting the direction of time towards the mass. As a result, objects tend to move towards masses. This is experienced as gravity. This tilting effect becomes more pronounced as the distance to the mass becomes smaller. At some point close to the mass, the tilting becomes so strong that all the possible paths an object can take lead towards the mass. This implies that any object that crosses this point can no longer get further away from the mass, not even using powered flight. This point is called the event horizon.

Formation
From the exotic nature of black holes, it is natural to question if such bizarre objects could actually exist in nature or whether they are merely pathological solutions to Einstein's equations. In 1970, Hawking and Penrose showed using their singularity theorems that under generic conditions, black holes are expected to form. Apart from the gravitational collapse of heavy objects such as stars, black holes could in principle also arise by more exotic means.

Gravitation collapse
Gravitational collapse occurs when an object's internal pressure is insufficient to resist its own gravity. For stars this usually occurs either because there is not enough "fuel" remaining to maintain its temperature, or because a star which would have been stable receives a lot of extra matter in a way which does not raise its core temperature. In either case, the star's temperature is no longer high enough to prevent it from collapsing under its own weight (the ideal gas law explains the connection between pressure, temperature, and volume).

The collapse may be stopped by the degeneracy pressure of the star's constituents, condensing the matter into an exotic denser state. The result is a type of compact star. the type depending on the mass of the remnant - the matter left over after changes triggered by the collapse (such as supernova or pulsations leading to a planetary nebula) have blown away the outer layers. This can be substantially less than the original star - remnants exceeding 5 solar masses are produced by stars which were over 20 solar masses before the collapse.

If the mass of the remnant exceeds ~2-4 solar masses (the Tolman-Oppenheimer-Volkoff limit), even the degeneracy pressure of neutrons is insufficient to stop the collapse. After this, no known mechanism (except possibly quark degeneracy pressure, see quark star) is powerful enough to stop the collapse and the object will inevitably become a black hole.

This gravitational collapse of heavy stars is assumed to be responsible for the formation of most (if not all) stellar mass black holes.

Primordial black holes
Gravitational collapse requires great densities. Nowadays, such high densities are only found in stars, but in the early universe shortly after the big bang densities were much greater, possibly allowing for the creation of black holes. The high density alone is not enough to allow the formation of black holes since a uniform mass distribution will not allow the mass to bunch up. In order for primordial black holes to form in such a dense medium, there must be initial density perturbations which can then grow under their own gravity. Different models for the early universe vary widely in their predictions of the size of these perturbations. Various models predict the creation of black holes, ranging from a Planck mass to hundreds of thousands of solar masses. Primordial black holes could thus account for the creation of any type of black hole.

Production in high energy collisions
Besides gravitational collapse, black holes could also be created in high energy collisions that produce sufficient density. Since classically black holes can be of any mass, micro black holes would be expected to form in any such process (no matter how low the energy). However, to date, no such events have been detected as a deficiency of the mass balance in particle accelerator experiments. This suggests that there must be a lower limit for the mass of black holes.

Theoretically this boundary is expected to lie around the Planck mass (~1019 GeV/c2), where quantum effects are expected to make the theory of general relativity break down completely. This would put the creation of black holes firmly out of reach of any high energy process occurring on or near the Earth. Certain developments in quantum gravity however suggest that this bound could be much lower. Some braneworld scenarios for example put the Planck mass much lower, may be even as low as 1 TeV. This would make it possible for micro black holes to be created in the high energy collisions occurring when cosmic rays hit the earth's atmosphere, or even maybe in the new Large Hadron Collider at CERN. These theories are however very speculative, and the creation of black holes in these processes is deemed unlikely by many specialists.

Growth
Once formed, a black hole can continue growing by absorbing additional matter. Any black hole will continually absorb interstellar dust from its direct surroundings and omnipresent cosmic background radiation, but neither of these processes should significantly affect the mass of a stellar black hole. More significant contributions can occur when the black hole is formed in a binary star system. After formation, the black hole can then absorb significant amounts of matter from its companion.

Much larger contributions can be obtained when a black hole merges with other stars or compact objects. The supermassive black holes suspected to exist in the center of most galaxies are expected to have formed from the coagulation of many smaller objects. The process has also been proposed as the origin of some intermediate-mass black holes.

The No-hair theorem
According to the "No Hair" theorem a black hole has only three independent physical properties: mass, charge and angular momentum. Any two black holes that share the same values for these properties are indistinguishable. This contrasts with other astrophysical objects such as stars, which have very many—possibly infinitely many—parameters. Consequently, a great deal of information is lost when a star collapses to form a black hole. As information is preserved in most physical theories, the loss of information in black holes is puzzling. Physicists refer to this as the black hole information paradox.

The No Hair theorem makes assumptions about the nature of our universe and the matter within. Other assumptions lead to different conclusions. For example, if nature allows magnetic monopoles to exist—which appears to be theoretically possible, but has never been observed—then it should also be possible for black holes to have magnetic charge. If the universe has more than four dimensions (as string theories, a controversial but apparently possible class of theories, would require), or has a global anti-de Sitter structure, the theorem could fail completely, allowing many sorts of "hair". However, in our apparently four-dimensional, very nearly flat universe, the theorem should hold.

By physical properties
The simplest black hole is one that has mass but neither charge nor angular momentum. These black holes are often referred to as Schwarzschild black holes after the physicist Karl Schwarzschild who discovered this solution in 1915. It was the first non-trivial exact solution to the Einstein equations to be discovered, and according to Birkhoff's theorem, the only vacuum solution that is spherically symmetric. For real world physics this means that there is no observable difference between the gravitational field of such a black hole and that of any other spherical object of the same mass—for example a spherical star or planet—once one is in the empty space outside the object. The popular notion of a black hole "sucking in everything" in its surroundings is therefore incorrect; the external gravitational field, far from the event horizon, is essentially like that of ordinary massive bodies.

Electrically charged non-rotating black holes are described by the Reissner-Nordström solution, developed by Hans Reissner and Gunnar Nordström.

Mathematician Roy Kerr presented the Kerr solution in 1963, showing how this could be used to describe a rotating black hole. In addition to its theoretical interest, Kerr's work made black holes more believable for astronomers, since black holes are formed from stars and all known stars rotate.

The most general known stationary black hole solution is the Kerr-Newman metric, discovered by physicist Ezra Newman et al having both charge and angular momentum. All these general solutions share the property that they converge to the Schwarzschild solution at distances that are large compared to the ratio of charge and angular momentum to mass (in natural units).

While the mass of a black hole can take any positive value, the charge and angular momentum are constrained by the mass. In natural units, the total charge Q and the total angular momentum J are expected to satisfy Q2+(J/M)2 ≤ M2 for a black hole of mass M. Black holes saturating this inequality are called extremal. Solutions of Einstein's equations violating the inequality do exist, but do not have a horizon. These solutions have naked singularities and are deemed unphysical, as the cosmic censorship hypothesis rules out such singularities due to the generic gravitational collapse of realistic matter. This is supported by numerical simulations.

Due to the relatively large strength of the electromagnetic force, black holes forming from the collapse of stars are expected to retain the nearly neutral charge of the star. Rotation, however, is expected to be a common feature of compact objects, and the black-hole candidate binary X-ray source GRS 1915+105 appears to have an angular momentum near the maximum allowed value.

By mass
Black holes are commonly classified according to their mass, independent of angular momentum J. The size of a black hole, as determined by the radius of the event horizon, or Schwarzschild radius, is proportional to the mass $$M\,$$ through $$r_{sh} \approx 2.95\, M/M_\bigodot \;\mathrm{km,}$$ where $$r_{sh}\,$$ is the Schwarzschild radius and $$M_\bigodot$$ is the mass of the Sun. A black hole's size and mass are thus simply related independent of rotation. According to this criterion, black holes are classed as:


 * Supermassive - contain hundreds of thousands to billions of solar masses. and are thought to exist in the center of most galaxies, including the Milky Way. They are believed to be responsible for active galactic nuclei, and presumably form either from the coalescence of smaller black holes, or by the accretion of stars and gas onto them. The largest known supermassive black hole is located in OJ 287 weighing in at 18 billion solar masses.


 * Intermediate - contain thousands of solar masses. They have been proposed as a possible power source for ultraluminous X-ray sources. There is no known mechanism for them to form directly, so they likely form via collisions of lower mass black holes, either in the dense stellar cores of globular clusters or galaxies.  Such creation events should produce intense bursts of gravitational waves, which may be observed soon.  The boundary between super- and intermediate-mass black holes is a matter of convention.  Their lower mass limit, the maximum mass for direct formation of a single black hole from collapse of a massive star, is poorly known at present.
 * Stellar-mass black holes - have masses ranging from a lower limit of about 1.5–3.0 solar masses (the Tolman-Oppenheimer-Volkoff limit for the maximum mass of neutron stars) up to perhaps 15–20 solar masses. They are created by the collapse of individual stars, or by the coalescence (inevitable, due to gravitational radiation) of binary neutron stars. Stars may form with initial masses up to ≈100 solar masses, or possibly even higher, but these shed most of their outer massive layers during earlier phases of their evolution, either blown away in stellar winds during the red giant, AGB, and Wolf-Rayet stages, or expelled in supernova explosions for stars that turn into neutron stars or black holes. Being known mostly by theoretical models for late-stage stellar evolution, the upper limit for the mass of stellar-mass black holes is somewhat uncertain at present.  The cores of still lighter stars form white dwarfs.
 * Micro (also mini black holes) - have masses much less than that of a star. At these sizes, the effects of quantum mechanics are expected to come into play. There is no known mechanism for them to form via normal processes of stellar evolution, but certain inflationary scenarios predict their production during the early stages of the evolution of the universe. According to some theories of quantum gravity they may also be produced in the highly energetic reaction produced by cosmic rays hitting the atmosphere or even in particle accelerators such as the Large Hadron Collider. The theory of Hawking radiation predicts that such black holes will evaporate in bright flashes of gamma radiation. NASA's Fermi Gamma-ray Space Telescope satellite (formerly GLAST) launched in 2008 is searching for such flashes.

Event horizon
The defining feature of a black hole is the event horizon, a surface in spacetime that marks a point of no return. Once an object crosses this surface, it cannot return to the other side. Consequently, anything inside this surface is completely hidden from outside observers. Other than this, the event horizon is a completely normal part of space with no special features that would allow someone falling into the black hole to know when they would cross the horizon. The event horizon is not a solid surface, and does not obstruct or slow down matter or radiation that is traveling towards the region within the event horizon.

Outside the event horizon, the gravitational field is identical to the field produced by any other spherically symmetric object of the same mass. The popular conception of black holes as "sucking" things in is misleading: objects can orbit black holes indefinitely, provided they stay outside the photon sphere (described below), and also ignoring the effects of gravitational radiation which causes orbiting objects to lose energy (similar to the effect of electromagnetic radiation).

To a distant observer clocks near a black hole appear to tick more slowly than those further away from the black hole. Due to this effect (known as gravitational time dilation) the distant observer will see an object falling into a black hole slow down as it approaches the event horizon, taking an infinite time to reach it. At the same time all process on this object slow down causing emitted light to appear redder an dimmer, an effect known as gravitational red shift. Eventually, the falling object becomes so dim that it can no longer be seen, at a point just before it reaches the event horizon.

From the falling object's viewpoint, distant objects generally appear blue-shifted due to the gravitational field of the black hole. This effect may be partly (or even entirely) negated by the Doppler red shift caused by the velocity with the distant object. For the object, nothing particularly special happens at the event horizon. In fact, there is no (local) way to determine whether the horizon has been passed or not. An infalling object takes a finite proper time (i.e. measured by its own clock) to fall past the event horizon.

<!--==Black hole parameters==

Astrophysical black holes are characterized by two parameters: their mass and their angular momentum (or spin). The mass parameter M is equivalent to a characteristic length GM/c2=1.48 km(M/M0), or a characteristic timescale GM/c³=4.93 x 10-6(M/M0) , where M0 denotes the mass of the Sun. These scales, for example, give the order of magnitude of the radii and periods of near-hole orbits. The timescale also applies to the process in which a developing horizon settles into its asymptotically stationary form. For a stellar mass hole this is of order 10-5 sec, while for a supermassive hole of 108 M0 , it is thousands of seconds.

For Schwarzschild holes, and approximately for Kerr holes, the horizon is at radius RH=2GM/c². At the horizon the "acceleration of gravity" has no meaning, since a falling observer cannot stop at the horizon to be weighed. What is relevant at the horizon is the tidal stresses that stretch and distort the falling observer. This tidal stretching is given by the same expression, the gradient of the gravitational acceleration, as in Newtonian theory: 2GM/RH3=c6/(4G2M2).

In the case of a solar mass black hole the tidal stress (acceleration per unit length) is enormous at the horizon, on the order of :  3 x 109(M/M0)2 sec-2 : that is, a person would experience a differential gravitational field of about 109 Earth gravities, enough to rip apart ordinary materials. For a supermassive hole, by contrast, the tidal force at the horizon is smaller by a typical factor 1010-16 and would be easily survivable. However, at the central singularity, deep inside the event horizon, the tidal stress is infinite. In addition to its mass M, the Kerr spacetime is described with a spin parameter 'a' defined by the dimensionless expression a/M= cJ/GM2 where J is the angular momentum of the hole. For the Sun (based on surface rotation) this number is about 0.2, and is much larger for many stars. Since angular momentum is ubiquitous in astrophysics, and since it is expected to be approximately conserved during collapse and black hole formation, astrophysical holes are expected to have significant values of a/M, from several tenths up to and approaching unity.

The value of a/M can be unity (an "extreme" Kerr hole), but it cannot be greater than unity. In the mathematics of general relativity, exceeding this limit replaces the event horizon with an inner boundary on the spacetime where tidal forces become infinite. Because this singularity is "visible" to observers, rather than hidden behind a horizon, as in a black hole, it is called a naked singularity. Toy models and heuristic arguments suggest that as a/M approaches unity it becomes more and more difficult to add angular momentum. The conjecture that such mechanisms will always keep a/M below unity is called cosmic censorship.

The inclusion of angular momentum changes details of the description of the horizon, so that, for example, the horizon area becomes Horizon area= 4πG2/c4[{M+(M²-a²)1/2}²+a²]

This modification of the Schwarzschild (a=0) result is not significant until a/M becomes very close to unity. For this reason, good estimates can be made in many astrophysical scenarios with a ignored.-->

Singularity
According to general relativity, there is a space-time singularity at the center of a spherical black hole, which means an infinite space-time curvature. It means that, from the viewpoint of an observer falling into a black hole, in a finite time (at the end of his fall) a black hole's mass becomes entirely compressed into a region with zero volume, so its density becomes infinite. This zero-volume, infinitely dense region at the center of a black hole is called a gravitational singularity.

The singularity in a non-rotating black hole is a point, in other words it has zero length, width, and height. The singularity of a rotating black hole is smeared out to form a ring shape lying in the plane of rotation. The ring still has no thickness and hence no volume.

The appearance of singularities in general relativity is commonly perceived as signaling the breakdown of the theory. This breakdown is not unexpected, as it occurs in a situation where quantum mechanical effects should become important, since densities are high and particle interactions should thus play a role. Unfortunately, to date it has not been possible to combine quantum and gravitation effects in a single theory. It is however quite generally expected that a theory of quantum gravity will feature black holes without singularities.

Formation of the singularity takes a finite (and very short) time only from the viewpoint of an infalling observer. From the standpoint of a distant observer, it takes infinite time to do so due to gravitational time dilation.

The difference in gravitational field at two points results in a tidal force, which, in a very strongly varying field causes an object to experience a large differential force in the direction of the gravitational source. Near black holes, the enormous tidal forces are expected to literally stretch and rip the object down to it's constituent atoms and elementary particles, a process called spaghettification. The strength of the tidal force of a black hole depends on how gravitational attraction changes with distance, rather than on the absolute force being felt. This means that small black holes cause spaghettification while infalling objects are still outside their event horizons, whereas objects falling into large, supermassive black holes may not be deformed or otherwise feel excessively large forces before passing the event horizon.

Photon sphere
The photon sphere is a spherical boundary of zero thickness such that photons moving along tangents to the sphere will be trapped in a circular orbit. For non-rotating black holes, the photon sphere has a radius 1.5 times the Schwarzschild radius. The orbits are dynamically unstable, hence any small perturbation (maybe caused by some infalling matter) will grow over time, either setting it on an outward trajectory escaping the black hole or on an inward spiral eventually crossing the event horizon.

While light can still escape from inside the photon sphere, any light that crosses the photon sphere on an inbound trajectory will be captured by the black hole. Hence any light reaching an outside observer from inside the photon sphere must have been emitted by objects inside the photon sphere but still outside of the event horizon.

Other compact objects, such as neutron stars, can also have photon spheres. This follows because the gravitational field of an object does not depend on its actual size, hence any object that is smaller than 1.5 times the Schwarzschild radius corresponding to its mass will have a photon sphere.

Ergosphere


Rotating black holes are surrounded by a region of spacetime in which it is impossible to stand still, called the ergosphere. This is the result of a process known as frame-dragging; general relativity predicts that any rotating mass will tend to slightly "drag" along the spacetime immediately surrounding it. Any object near the rotating mass will tend to start moving in the direction of rotation. For a rotating black hole this effect becomes so strong near the event horizon that an object would have to move faster than the speed of light in the opposite direction to just stand still.

The ergosphere of black hole is bounded by
 * on the outside, an oblate spheroid, which coincides with the event horizon at the poles and is noticeably wider around the "equator". This boundary is sometimes called the "ergosurface", but it is just a boundary and has no more solidity than the event horizon. At points exactly on the ergosurface, spacetime is "dragged around at the speed of light."
 * on the inside, the (outer) event horizon.

Within the ergosphere, space-time is dragged around faster than light—general relativity forbids material objects to travel faster than light (so does special relativity), but allows regions of space-time to move faster than light relative to other regions of space-time.

Objects and radiation (including light) can stay in orbit within the ergosphere without falling to the center. But they cannot hover (remain stationary, as seen by an external observer), because that would require them to move backwards faster than light relative to their own regions of space-time, which are moving faster than light relative to an external observer.

Objects and radiation can also escape from the ergosphere. In fact the Penrose process predicts that objects will sometimes fly out of the ergosphere, obtaining the energy for this by "stealing" some of the black hole's rotational energy. If a large total mass of objects escapes in this way, the black hole will spin more slowly and may even stop spinning eventually.

Hawking radiation
In 1971, Stephen Hawking showed that the total area of the event horizons of any collection of classical black holes can never decrease, even if they collide and swallow each other; that is merge. This is remarkably similar to the Second Law of Thermodynamics, with area playing the role of entropy. As classical objects with zero temperature it was assumed that black holes had zero entropy; the second law of thermodynamics would then be violated by an entropy-laden material entering the black hole, resulting in a decrease of the total entropy of the universe. Therefore, Jacob Bekenstein proposed that a black hole should have an entropy, and that it should be proportional to its horizon area. Since black holes do not classically emit radiation, the thermodynamic viewpoint seemed simply an analogy, since zero temperature implies infinite changes in entropy with any addition of heat, which implies infinite entropy. However, in 1974, Hawking applied quantum field theory to the curved spacetime around the event horizon and discovered that black holes emit Hawking radiation, a form of thermal radiation, allied to the Unruh effect, which implied they had a positive temperature. This strengthened the analogy between black hole dynamics and thermodynamics: using the first law of black hole mechanics, it follows that the entropy of a non-rotating black hole is one quarter the horizon area. This is a universal result and is extended to cosmological horizons such as in de Sitter space. It was later suggested that black holes are maximum-entropy objects, meaning that the maximum possible entropy of a region of space is the entropy of the largest black hole that can fit into it. This led to the holographic principle.

Hawking radiation reflects a characteristic temperature of the black hole, which can be calculated from its entropy. The more its temperature falls, the more massive a black hole becomes: the more energy a black hole absorbs, the colder it gets. A black hole with roughly the mass of the planet Mercury would have a temperature in equilibrium with the cosmic microwave background radiation (about 2.73 K). More massive than this, a black hole will be colder than the background radiation, and it will gain energy from the background faster than it gives energy up through Hawking radiation, becoming even colder still. However, for a less massive black hole the effect implies that the mass of the black hole will slowly evaporate with time, with the black hole becoming hotter and hotter as it does so. Although these effects are negligible for black holes massive enough to have been formed astronomically, they would rapidly become significant for hypothetical smaller black holes, where quantum-mechanical effects dominate. Indeed, small black holes are predicted to undergo runaway evaporation and eventually vanish in a burst of radiation.

Although general relativity can be used to perform a semi-classical calculation of black hole entropy, this situation is theoretically unsatisfying. In statistical mechanics, entropy is understood as counting the number of microscopic configurations of a system which have the same macroscopic qualities (such as mass, charge, pressure, etc.). But without a satisfactory theory of quantum gravity, one cannot perform such a computation for black holes. Some promise has been shown by string theory, however. There one posits that the microscopic degrees of freedom of the black hole are D-branes. By counting the states of D-branes with given charges and energy, the entropy for certain supersymmetric black holes has been reproduced. Extending the region of validity of these calculations is an ongoing area of research.

In 1974, Stephen Hawking showed that black holes are not entirely black but emit small amounts of thermal radiation. He got this result by applying quantum field theory in a static black hole background. The result of his calculations is that a black hole should emit particles in a perfect black body spectrum. This effect has become known as Hawking radiation. Research from various directions suggests the theoretical validity of the effect.

The temperature of the emitted black body spectrum is proportional to the surface gravity of the black hole. For a Schwarzschild black hole this is inversely proportional to the mass. Consequently, large black holes are very cold and emit very little radiation. A stellar black hole of 10 solar masses, for example, would have a Hawking temperature of several nanokelvin, much less than the 2.7K produced by the Cosmic Microwave Background. Micro black holes on the other hand could be quite bright producing high energy gamma rays.

Due to low Hawking temperature of stellar black holes, Hawking radiation has never been observed at any of the black hole candidates.

If Hawking's theory of black hole radiation is correct, then black holes are expected to emit a thermal spectrum of radiation, and thereby lose mass, by the equivalence of mass and energy. Black holes will thus shrink and evaporate over time. The temperature of this spectrum (Hawking temperature) is proportional to the surface gravity of the black hole, which in turn is inversely proportional to the mass. Large black holes thus emit less radiation than small black holes.

A stellar black hole of 5 solar masses has a Hawking temperature of about 12 nanoKelvin. This is far less than the 2.7 K produced by the cosmic microwave background (CMB) radiation. Stellar mass (and larger) black holes thus receive more mass from the CMB than they emit through Hawking radiation and will thus grow instead of shrink. In order to have a Hawking temperature larger than 2.7 K (and thus be able to evaporate) a black hole needs to be lighter than the Moon (and thus have diameter of less than a tenth of a millimeter).

Rdaiation effects are expected to be very strong for very small black holes. Even a black hole that is heavy compared to a human would evaporate in an instant. A black hole the weight of a car (~ 10−24 m) would only take a nanosecond to evaporate, during which time it would briefly have a luminosity more than 200 times that of the sun. Lighter black holes are expected to evaporate even faster, for example a black hole of mass 1 TeV/c2 would take less than 10−88 seconds to evaporate completely. Of course, for such a small black hole quantum gravitation effects are expected to play an important role and could —although current developments in quantum gravity do not indicate so— hypothetically make such a small black hole stable.

Information loss paradox
An open question in fundamental physics is the so-called information loss paradox, or black hole unitarity paradox. Classically, the laws of physics are the same run forward or in reverse. That is, if the position and velocity of every particle in the universe were measured, we could (disregarding chaos) work backwards to discover the history of the universe arbitrarily far in the past. In quantum mechanics, this corresponds to a vital property called unitarity, which has to do with the conservation of probability.

Black holes, however, might violate this rule. The position under classical general relativity is subtle but straightforward: because of the classical no hair theorem, we can never determine what went into the black hole. However, as seen from the outside, information is never actually destroyed, as matter falling into the black hole takes an infinite time to reach the event horizon.

Ideas about quantum gravity, on the other hand, suggest that there can only be a limited finite entropy (i.e. a maximum finite amount of information) associated with the space near the horizon; but the change in the entropy of the horizon plus the entropy of the Hawking radiation is always sufficient to take up all of the entropy of matter and energy falling into the black hole.

Many physicists are concerned however that this is still not sufficiently well understood. In particular, at a quantum level, is the quantum state of the Hawking radiation uniquely determined by the history of what has fallen into the black hole; and is the history of what has fallen into the black hole uniquely determined by the quantum state of the black hole and the radiation? This is what determinism, and unitarity, would require.

For a long time Stephen Hawking had opposed such ideas, holding to his original 1975 position that the Hawking radiation is entirely thermal and therefore entirely random, containing none of the information held in material the hole has swallowed in the past; this information he reasoned had been lost. However, on 21 July 2004 he presented a new argument, reversing his previous position. On this new calculation, the entropy (and hence information) associated with the black hole escapes in the Hawking radiation itself. However, making sense of it, even in principle, is difficult until the black hole completes its evaporation. Until then it is impossible to relate in a 1:1 way the information in the Hawking radiation (embodied in its detailed internal correlations) to the initial state of the system. Once the black hole evaporates completely, such identification can be made, and unitarity is preserved.

By the time Hawking completed his calculation, it was already very clear from the AdS/CFT correspondence that black holes decay in a unitary way. This is because the fireballs in gauge theories, which are analogous to Hawking radiation, are unquestionably unitary. Hawking's new calculation has not been evaluated by the specialist scientific community, because the methods he uses are unfamiliar and of dubious consistency; but Hawking himself found it sufficiently convincing to pay out on a bet he had made in 1997 with Caltech physicist John Preskill, to considerable media interest. <!--==Mathematical theory of non-rotating, uncharged black holes==

Black holes in 3 dimensions
A black hole solution is also known for 3 dimensional spacetimes. This so called BTZ black hole exists however only for a negative cosmological constant.

In general relativity, there are many known solutions of the Einstein field equations which describes different types of black holes. The Schwarzschild metric is one of the earliest and simplest solutions. This solution describes the curvature of spacetime in the vicinity of a static and spherically symmetric uncharged object, where the metric is,


 * $$ \mathrm{d}s^2 = - c^2 \left( 1 - {2Gm \over c^2 r} \right) \mathrm{d}t^2 + \left( 1 - {2Gm \over c^2 r} \right)^{-1} \mathrm{d}r^2 + r^2 \mathrm{d}\Omega^2 $$,

where $$\mathrm{d}\Omega^2 = \mathrm{d}\theta^2 + \sin^2\theta\; \mathrm{d}\phi^2$$ is a standard element of solid angle.

According to general relativity, a gravitating object will collapse into a black hole if its radius is smaller than a characteristic distance, known as the Schwarzschild radius. (Indeed, Buchdahl's theorem in general relativity shows that in the case of a perfect fluid model of a compact object, the true lower limit is somewhat larger than the Schwarzschild radius.) Below this radius, spacetime is so strongly curved that any light ray emitted in this region, regardless of the direction in which it is emitted, will travel towards the centre of the system. Because relativity forbids anything from traveling faster than light, anything below the Schwarzschild radius – including the constituent particles of the gravitating object – will collapse into the centre. A gravitational singularity, a region of theoretically infinite density, forms at this point. Because not even light can escape from within the Schwarzschild radius, a classical black hole would truly appear black.

The Schwarzschild radius is given by


 * $$r_{\rm S} = {2\,Gm \over c^2} $$

where G is the gravitational constant, m is the mass of the object, and c is the speed of light. For an object with the mass of the Earth, the Schwarzschild radius is a mere 9 millimeters&mdash;about the size of a marble.

The mean density inside the Schwarzschild radius decreases as the mass of the black hole increases, so while an earth-mass black hole would have a density of 2 × 1030 kg/m³, a supermassive black hole of 109 solar masses has a density of around 20 kg/m³, less than water! The mean density is given by


 * $$\rho=\frac{3\,c^6}{32\pi m^2G^3}.$$

Since the Earth has a mean radius of 6371 km, its volume would have to be reduced 4 × 1026 times to collapse into a black hole. For an object with the mass of the Sun, the Schwarzschild radius is approximately 3 km, much smaller than the Sun's current radius of about 696,000 km. It is also significantly smaller than the radius to which the Sun will ultimately shrink after exhausting its nuclear fuel, which is several thousand kilometers. More massive stars can collapse into black holes at the end of their lifetimes.

The formula also implies that any object with a given mean density is a black hole if its radius is large enough. The same formula applies for white holes as well. For example, if the observable universe has a mean density equal to the critical density, then it is a white hole, since its singularity is in the past and not in the future as should be for a black hole.

There is also the Black Hole Entropy formula:


 * $$S = \frac{Akc^3}{4\hbar G}.$$

Where A is the area of the event horizon of the black hole, $$\hbar$$ is Dirac's constant (the "reduced Planck constant"), k is the Boltzmann constant, G is the gravitational constant, c is the speed of light and S is the entropy.

A convenient length scale to measure black hole processes is the "gravitational radius", which is equal to
 * $$r_{\rm G} = {Gm \over c^2} .$$

When expressed in terms of this length scale, many phenomena appear at integer radii. For example, the radius of a Schwarzschild black hole is two gravitational radii and the radius of a maximally rotating Kerr black hole is one gravitational radius. The location of the light circularization radius around a Schwarzschild black hole (where light may orbit the hole in an unstable circular orbit) is $$3r_{\rm G}$$. The location of the marginally stable orbit, thought to be close to the inner edge of an accretion disk, is at $$6r_{\rm G}$$ for a Schwarzschild black hole. -->

Accretion disks and gas jets
Most accretion disks and gas jets are not clear proof that a stellar-mass black hole is present, because other massive, ultra-dense objects such as neutron stars and white dwarfs cause accretion disks and gas jets to form and to behave in the same ways as those around black holes. But they can often help by telling astronomers where it might be worth looking for a black hole.

On the other hand, extremely large accretion disks and gas jets may be good evidence for the presence of supermassive black holes, because as far as we know any mass large enough to power these phenomena must be a black hole.

Strong radiation emissions


Steady X-ray and gamma ray emissions also do not prove that a black hole is present, but can tell astronomers where it might be worth looking for one - and they have the advantage that they pass fairly easily through nebulae and gas clouds.

But strong, irregular emissions of X-rays, gamma rays and other electromagnetic radiation can help to prove that a massive, ultra-dense object is not a black hole, so that "black hole hunters" can move on to some other object. Neutron stars and other very dense stars have surfaces, and matter colliding with the surface at a high percentage of the speed of light will produce intense flares of radiation at irregular intervals. Black holes have no material surface, so the absence of irregular flares around a massive, ultra-dense object suggests that there is a good chance of finding a black hole there.

Intense but one-time gamma ray bursts (GRBs) may signal the birth of "new" black holes, because astrophysicists think that GRBs are caused either by the gravitational collapse of giant stars or by collisions between neutron stars, and both types of event involve sufficient mass and pressure to produce black holes. But it appears that a collision between a neutron star and a black hole can also cause a GRB, so a GRB is not proof that a "new" black hole has been formed. All known GRBs come from outside our own galaxy, and most come from billions of light years away so the black holes associated with them are actually billions of years old.

Astrophysicists such as Winter believe that some ultraluminous X-ray sources may be the accretion disks of intermediate-mass black holes.

Quasars are thought to be the accretion disks of supermassive black holes, since no other known object is powerful enough to produce such strong emissions. Quasars produce strong emission across the electromagnetic spectrum, including UV, X-rays and gamma-rays and are visible at tremendous distances due to their high luminosity. Between 5 and 25% of quasars are "radio loud," so called because of their powerful radio emissions.

Gravitational lensing


A gravitational lens is formed when the light from a very distant, bright source (such as a quasar) is "bent" around a massive object (such as a black hole) between the source object and the observer. The process is known as gravitational lensing, and is one of the predictions of general relativity.

A source image behind the lens may appear as multiple images to the observer. In cases where the source, massive lensing object, and the observer lie in a straight line, the source will appear as a ring behind the massive object.

Gravitational lensing can be caused by objects other than black holes, because any very strong gravitational field will bend light rays. Some of these multiple-image effects are probably produced by distant galaxies.

Objects orbiting possible black holes


Objects orbiting black holes probe the gravitational field around the central object. An early example, discovered in the 1970s, is the accretion disk orbiting the putative black hole responsible for Cygnus X-1, a famous X-ray source. While the material itself cannot be seen directly, the X rays flicker on a millisecond time scale, as expected for hot clumpy material orbiting a ~10 solar-mass black hole just prior to accretion. The X-ray spectrum exhibits the characteristic shape expected for a disk of orbiting relativistic material, with an iron line, emitted at ~6.4 keV, broadened to the red (on the receding side of the disk) and to the blue (on the approaching side).

Another example is the star S2, seen orbiting the Galactic Center. Here the star is several light hours from the ~3.5×106 solar mass black hole, so its orbital motion can be plotted. Nothing is observed at the center of the observed orbit, the position of the black hole itself——as expected for a black object.

Determining the mass of black holes
Quasi-periodic oscillations can be used to determine the mass of black holes. The technique uses a relationship between black holes and the inner part of their surrounding disks, where gas spirals inward before reaching the event horizon. As the gas collapses inwards, it radiates X-rays with an intensity that varies in a pattern that repeats itself over a nearly regular interval. This signal is the Quasi-Periodic Oscillation, or QPO. A QPO’s frequency depends on the black hole’s mass; the event horizon lies close in for small black holes, so the QPO has a higher frequency. For black holes with a larger mass, the event horizon is farther out, so the QPO frequency is lower.

Supermassive black holes at the centers of galaxies


According to the American Astronomical Society, every large galaxy has a supermassive black hole at its center. The black hole’s mass is proportional to the mass of the host galaxy, suggesting that the two are linked very closely. The Hubble Space Telescope and ground-based telescopes in Hawaii were used in a large survey of galaxies.

For decades, astronomers have used the term "active galaxy" to describe galaxies with unusual characteristics, such as unusual spectral line emission and very strong radio emission. However, theoretical and observational studies have shown that the active galactic nuclei (AGN) in these galaxies may contain supermassive black holes. The models of these AGN consist of a central black hole that may be millions or billions of times more massive than the Sun; a disk of gas and dust called an accretion disk; and two jets that are perpendicular to the accretion disk.

Although supermassive black holes are expected to be found in most AGN, only some galaxies' nuclei have been more carefully studied in attempts to both identify and measure the actual masses of the central supermassive black hole candidates. Some of the most notable galaxies with supermassive black hole candidates include the Andromeda Galaxy, M32, M87, NGC 3115, NGC 3377, NGC 4258, and the Sombrero Galaxy.

Astronomers are confident that our own Milky Way galaxy has a supermassive black hole at its center, in a region called Sagittarius A*:
 * A star called S2 (star) follows an elliptical orbit with a period of 15.2 years and a pericenter (closest) distance of 17 light hours from the central object.
 * The first estimates indicated that the central object contains 2.6M (2.6 million) solar masses and has a radius of less than 17 light hours. Only a black hole can contain such a vast mass in such a small volume.
 * Further observations strengthened the case for a black hole, by showing that the central object's mass is about 3.7M solar masses and its radius no more than 6.25 light-hours.

Intermediate-mass black holes in globular clusters
In 2002, the Hubble Space Telescope produced observations indicating that globular clusters named M15 and G1 may contain intermediate-mass black holes. This interpretation is based on the sizes and periods of the orbits of the stars in the globular clusters. But the Hubble evidence is not conclusive, since a group of neutron stars could cause similar observations. Until recent discoveries, many astronomers thought that the complex gravitational interactions in globular clusters would eject newly-formed black holes.

In November 2004 a team of astronomers reported the discovery of the first well-confirmed intermediate-mass black hole in our Galaxy, orbiting three light-years from Sagittarius A*. This black hole of 1,300 solar masses is within a cluster of seven stars, possibly the remnant of a massive star cluster that has been stripped down by the Galactic Centre. This observation may add support to the idea that supermassive black holes grow by absorbing nearby smaller black holes and stars.

In January 2007, researchers at the University of Southampton in the United Kingdom reported finding a black hole, possibly of about 10 solar masses, in a globular cluster associated with a galaxy named NGC 4472, some 55 million light-years away.

Stellar-mass black holes in the Milky Way


The Milky Way galaxy contains several probable stellar-mass black holes which are closer to Earth than the suspected supermassive black hole in the Sagittarius A* region. These candidates are all members of X-ray binary systems in which the denser object draws matter from its partner via an accretion disk. The probable black holes in these pairs range from three to more than a dozen solar masses. The most distant stellar-mass black hole detected is a member of a binary system located in the Messier 33 galaxy.

In 1971, Louise Webster and Paul Murdin, at the Royal Greenwich Observatory, and Charles Thomas Bolton, working independently at the University of Toronto's David Dunlap Observatory, observed HDE 226868 wobble, as if orbiting around an invisible but massive companion. Further analysis led to the declaration that the companion, Cygnus X-1, was in fact a black hole.

Micro black holes
There is theoretically no smallest size for a black hole. Stephen Hawking theorized that primordial black holes could evaporate and shrink to become micro black holes. Searches for evaporating primordial black holes are proposed for the GLAST satellite, launched in 2008. However, if micro black holes can be created by other means, such as by cosmic ray impacts or in colliders, that does not imply that they must evaporate.

The formation of black hole analogs on Earth in particle accelerators has been reported. These analogs are not the same as gravitational black holes, but are vital testing grounds for quantum theories of gravity.

They act like black holes because of the correspondence between the theory of the strong nuclear force, which has nothing to do with gravity, and the quantum theory of gravity. They are similar because both are described by string theory. So the formation and disintegration of a fireball in quark gluon plasma can be interpreted in black hole language. The fireball at the Relativistic Heavy Ion Collider (RHIC) is a phenomenon which is closely analogous to a black hole, and many of its physical properties can be correctly predicted using this analogy. The fireball, however, is not a gravitational object. It is presently unknown whether the much more energetic Large Hadron Collider (LHC) would be capable of producing the speculative large extra dimension micro black hole, as many theorists have suggested.

Alternative models
Several alternative models, which behave like a black hole but avoid the singularity, have been proposed. However, most researchers judge these concepts artificial, as they are more complicated but do not give near term observable differences from black holes (see Occam's razor). The most prominent alternative theory is the Gravastar.

In March 2005, physicist George Chapline at the Lawrence Livermore National Laboratory in California proposed that black holes do not exist, and that objects currently thought to be black holes are actually dark-energy stars. He draws this conclusion from some quantum mechanical analyses. Although his proposal currently has little support in the physics community, it was widely reported by the media. A similar theory about the non-existence of black holes was later developed by a group of physicists at Case Western Reserve University in June 2007.

Among the alternate models are magnetospheric eternally collapsing objects, clusters of elementary particles (e.g., boson stars ), fermion balls, self-gravitating, degenerate heavy neutrinos and even clusters of very low mass (~0.04 solar mass) black holes.

University textbooks and monographs

 * , the lecture notes on which the book was based are available for free from Sean Carroll's website.

Research papers

 * . Stephen Hawking's purported solution to the black hole unitarity paradox, first reported at a conference in July 2004.
 * . More accurate mass and position for the black hole at the centre of the Milky Way.
 * . Lecture notes from 2005 SLAC Summer Institute.