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Black hole

2007 Schools Wikipedia Selection. Related subjects: Space (Astronomy)

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   A black hole is an object predicted by general relativity with a
   gravitational field so strong that nothing can escape it — not even
   light.

   A black hole is defined to be a region of space-time where escape to
   the outside universe is impossible. The boundary of this region is a
   surface called the event horizon. This surface is not a physically
   tangible one, but merely a figurative concept of an imaginary boundary.
   Nothing can move from inside the event horizon to the outside, even
   briefly.

   Theoretically, a black hole can be any size. Astrophysicists expect to
   find black holes with masses ranging between roughly the mass of the
   Sun ("stellar-mass" black holes) to many millions of times the mass of
   the Sun ( supermassive black holes).

   The existence of black holes in the universe is well supported by
   astronomical observation, particularly from studying X-ray emission
   from X-ray binaries and active galactic nuclei. It has also been
   hypothesized that black holes radiate energy due to quantum mechanical
   effects known as Hawking radiation.

History

   The concept of a body so massive that even light could not escape was
   put forward by the English geologist John Michell in a 1784 paper sent
   to Henry Cavendish and published by the Royal Society. At that time,
   the Newtonian theory of gravity and the concept of escape velocity were
   well known. Michell computed that a body with 500 times the radius of
   the Sun and of the same density would have, at its surface, an escape
   velocity equal to the speed of light, and therefore would be invisible.
   In his words:


   Black hole

   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
   (inertial mass), with other bodies, all light emitted from such a body
        would be made to return towards it by its own proper gravity.


   Black hole

   Michell considered the possibility that many such objects that cannot
   be seen might be present in the cosmos.

   In 1796, the French 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). The idea gained
   little attention in the nineteenth century, since light was thought to
   be a massless wave, hence not influenced by gravity.

   In 1915, Albert Einstein developed the theory of gravity called General
   Relativity, having earlier shown that gravity does influence light. 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. Schwarzschild
   himself thought it was not physical. A few months after Schwarzschild,
   a student of Lorentz, Johannes Droste, independently gave the same
   solution for the point mass and wrote more extensively about its
   properties.

   In 1930, Subrahmanyan Chandrasekhar argued that special relativity
   demonstrated that a non-radiating body above 1.44 solar masses, now
   known as the Chandrasekhar limit, would collapse since there was
   nothing known at that time that could stop it from doing so. His
   arguments were opposed by Arthur Eddington, who believed that something
   would inevitably stop the collapse. Both were correct, since a white
   dwarf more massive than the Chandrasekhar limit will collapse into a
   neutron star. However, a neutron star above about three solar masses
   (the Tolman-Oppenheimer-Volkoff limit) will itself become unstable
   against collapse due to similar physics.

   In 1939, Robert Oppenheimer and H. Snyder predicted that massive stars
   could undergo a dramatic gravitational collapse. Black holes could, in
   principle, be formed in nature. Such objects for a while were called
   frozen stars since the collapse would be observed to rapidly slow down
   and become heavily redshifted near the Schwarzschild radius. The
   mathematics showed that an outside observer would see the surface of
   the star frozen in time at the instant where it crosses that radius.
   These hypothetical objects were not the topic of much interest until
   the late 1960s. Most physicists believed that they were a peculiar
   feature of the highly symmetric solution found by Schwarzschild, and
   that objects collapsing in nature would not form black holes.

   Interest in black holes was rekindled in 1967 because of theoretical
   and experimental progress. In 1970, Stephen Hawking and Roger Penrose
   proved that black holes are a generic feature in Einstein's theory of
   gravity, and cannot be avoided in some collapsing objects. Interest was
   renewed in the astronomical community with the discovery of pulsars.
   Shortly thereafter, the expression "black hole" was coined by
   theoretical physicist John Wheeler, being first used in his public
   lecture Our Universe: the Known and Unknown on 29 December 1967. The
   older Newtonian objects of Michell and Laplace are often referred to as
   " dark stars" to distinguish them from the "black holes" of general
   relativity.

Evidence

   A (simulated) Black Hole of ten solar masses as seen from a distance of
   600 km with the Milky Way in the background (horizontal camera opening
   angle: 90°).
   Enlarge
   A (simulated) Black Hole of ten solar masses as seen from a distance of
   600 km with the Milky Way in the background (horizontal camera opening
   angle: 90°).

Formation

   General relativity (as well as most other metric theories of gravity)
   not only says that black holes can exist, but in fact predicts that
   they will be formed in nature whenever a sufficient amount of mass gets
   packed in a given region of space, through a process called
   gravitational collapse; as the mass inside the given region of space
   increases, its gravity becomes stronger and (in the language of
   relativity) increasingly deforms the space around it, ultimately until
   nothing (not even light) can escape the gravity; at this point an event
   horizon is formed, and matter and energy must inevitably collapse to a
   density beyond the limits of known physics. For example, if the Sun was
   compressed to a radius of roughly three kilometers (about 1/232,000 its
   present size), the resulting gravitational field would create an event
   horizon around it, and thus a black hole.

   A quantitative analysis of this idea led to the prediction that a
   stellar remnant above about three to five times the mass of the Sun
   (the Tolman-Oppenheimer-Volkoff limit) would be unable to support
   itself as a neutron star via degeneracy pressure, and would inevitably
   collapse into a black hole. Stellar remnants with this mass are
   expected to be produced immediately at the end of the lives of stars
   that are more than 25 to 50 times the mass of the Sun, or by accretion
   of matter onto an existing neutron star.

   Stellar collapse will generate black holes containing at least three
   solar masses. Black holes smaller than this limit can only be created
   if their matter is subjected to sufficient pressure from some source
   other than self-gravitation. The enormous pressures needed for this are
   thought to have existed in the very early stages of the universe,
   possibly creating primordial black holes which could have masses
   smaller than that of the Sun.

   Supermassive black holes are believed to exist in the centre of most
   galaxies, including our own Milky Way. This type of black hole contains
   millions to billions of solar masses, and there are several models of
   how they might have been formed. The first is via gravitational
   collapse of a dense cluster of stars. A second is by large amounts of
   mass accreting onto a "seed" black hole of stellar mass. A third is by
   repeated fusion of smaller black holes. Effects of such supermassive
   black holes on spacetime may be observed in regions as the Virgo
   cluster of galaxies, for example, the location of M87 (see image below)
   and its neighbors.

   Intermediate-mass black holes have a mass between that of stellar and
   supermassive black holes, typically in the range of thousands of solar
   masses. Intermediate-mass black holes have been proposed as a possible
   power source for ultra-luminous X ray sources, and in 2004 detection
   was claimed of an intermediate-mass black hole orbiting the Sagittarius
   A* supermassive black hole candidate at the core of the Milky Way
   galaxy. This detection is disputed.

   Certain models of unification of the four fundamental forces allow the
   formation of micro black holes under laboratory conditions. These
   postulate that the energy at which gravity is unified with the other
   forces is comparable to the energy at which the other three are
   unified, as opposed to being the Planck energy (which is much higher).
   This would allow production of extremely short-lived black holes in
   terrestrial particle accelerators. No conclusive evidence of this type
   of black hole production has been presented, though even a negative
   result improves constraints on compactification of extra dimensions
   from string theory or other models of physics.

Observation

   Formation of extragalactic jets from a black hole's accretion disk
   Enlarge
   Formation of extragalactic jets from a black hole's accretion disk

   In theory, no object within the event horizon of a black hole can ever
   escape, including light. However, black holes can be inductively
   detected from observation of phenomena near them, such as gravitational
   lensing, galactic jets, and stars that appear to be in orbit around
   space where there is no visible matter.

   The most conspicuous effects are believed to come from matter accreting
   onto a black hole, which is predicted to collect into an extremely hot
   and fast-spinning accretion disk. The internal viscosity of the disk
   causes it to become extremely hot, and emit large amounts of X-ray and
   ultraviolet radiation. This process is extremely efficient and can
   convert about 10% of the rest mass energy of an object into radiation,
   as opposed to nuclear fusion which can only convert a few percent of
   the mass to energy. Other observed effects are narrow jets of particles
   at relativistic speeds heading along the disk's axis.

   However, accretion disks, jets, and orbiting objects are found not only
   around black holes, but also around other objects such as neutron stars
   and white dwarfs; and the dynamics of bodies near these non-black hole
   attractors is largely similar to that of bodies around black holes. It
   is currently a very complex and active field of research involving
   magnetic fields and plasma physics to disentangle what is going on.
   Hence, for the most part, observations of accretion disks and orbital
   motions merely indicate that there is a compact object of a certain
   mass, and says very little about the nature of that object. The
   identification of an object as a black hole requires the further
   assumption that no other object (or bound system of objects) could be
   so massive and compact. Most astrophysicists accept that this is the
   case, since according to general relativity, any concentration of
   matter of sufficient density must necessarily collapse into a black
   hole.

   One important observable difference between black holes and other
   compact massive objects is that any infalling matter will eventually
   collide with the latter at relativistic speeds, leading to emission as
   the kinetic energy of the matter is thermalized. In addition
   thermonuclear "burning" may occur on the surface as material builds up.
   These processes produce irregular intense flares of X-rays and other
   hard radiation. Thus the lack of such flare-ups around a compact
   concentration of mass is taken as evidence that the object is a black
   hole, with no surface onto which matter can collect.

Suspected black holes

   An artist depiction of two black holes merging.
   Enlarge
   An artist depiction of two black holes merging.

   There is now a great deal of indirect astronomical observational
   evidence for black holes in two mass ranges:
     * stellar mass black holes with masses of a typical star (4–15 times
       the mass of our Sun), and
     * supermassive black holes with masses ranging from on the order of
       10^5 to 10^10 solar masses.

   Additionally, there is some evidence for intermediate-mass black holes
   (IMBHs), those with masses of a few hundred to a few thousand times
   that of the Sun. These black holes may be responsible for the emission
   from ultraluminous X-ray sources (ULXs).

   Candidates for stellar-mass black holes were identified mainly by the
   presence of accretion disks of the right size and speed, without the
   irregular flare-ups that are expected from disks around other compact
   objects. Stellar-mass black holes may be involved in gamma ray bursts
   (GRBs); short duration GRBs are believed to be caused by colliding
   neutron stars, which form a black hole on merging. Observations of long
   GRBs in association with supernovae suggest that long GRBs are caused
   by collapsars; a massive star whose core collapses to form a black
   hole, drawing in the surrounding material. Therefore, a GRB could
   possibly signal the birth of a new black hole, aiding efforts to search
   for them.

   Candidates for more massive black holes were first provided by the
   active galactic nuclei and quasars, discovered by radioastronomers in
   the 1960s. The efficient conversion of mass into energy by friction in
   the accretion disk of a black hole seems to be the only explanation for
   the copious amounts of energy generated by such objects. Indeed the
   introduction of this theory in the 1970s removed a major objection to
   the belief that quasars were distant galaxies — namely, that no
   physical mechanism could generate that much energy.

   From observations in the 1980s of motions of stars around the galactic
   centre, it is now believed that such supermassive black holes exist in
   the centre of most galaxies, including our own Milky Way. Sagittarius
   A* is now generally agreed to be the location of a supermassive black
   hole at the centre of the Milky Way galaxy. The orbits of stars within
   a few AU of Sagittarius A* rule out any object other than a black hole
   at the centre of the Milky Way assuming the current standard laws of
   physics are correct.
   The jet emitted by the galaxy M87 in this image is thought to be caused
   by a supermassive black hole at the galaxy's centre
   Enlarge
   The jet emitted by the galaxy M87 in this image is thought to be caused
   by a supermassive black hole at the galaxy's centre

   The current picture is that all galaxies may have a supermassive black
   hole in their centre, and that this black hole accretes gas and dust in
   the middle of the galaxies generating huge amounts of radiation — until
   all the nearby mass has been swallowed and the process shuts off. This
   picture may also explain why there are no nearby quasars.

   Although the details are still not clear, it seems that the growth of
   the black hole is intimately related to the growth of the spheroidal
   component — an elliptical galaxy, or the bulge of a spiral galaxy — in
   which it lives.

   In 2002, the Hubble Telescope identified evidence indicating that
   intermediate size black holes exist in globular clusters named M15 and
   G1. The evidence for the black holes stemmed from the orbital velocity
   of the stars in the globular clusters; however, a group of neutron
   stars could cause similar observations.

Recent discoveries

   In 2004, astronomers found 31 candidate supermassive black holes from
   searching obscured quasars. The lead scientist said that there are from
   two to five times as many supermassive black holes as previously
   predicted.

   In June 2004 astronomers found a super-massive black hole, Q0906+6930,
   at the centre of a distant galaxy about 12.7 billion light years away.
   This observation indicated rapid creation of super-massive black holes
   in the early universe.

   In November 2004 a team of astronomers reported the discovery of the
   first intermediate-mass black hole in our Galaxy, orbiting three
   light-years from Sagittarius A*. This medium 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 February 2005, a blue giant star SDSS J090745.0+24507 was found to
   be leaving the Milky Way at twice the escape velocity (0.0022 of the
   speed of light), having been catapulted out of the galactic core which
   its path can be traced back to. The high velocity of this star supports
   the hypothesis of a super-massive black hole in the centre of the
   galaxy.

   The formation of micro black holes on Earth in particle accelerators
   has been tentatively reported, but not yet confirmed. So far there are
   no observed candidates for primordial black holes.

Features and theories

   Black holes require the general relativistic concept of a curved
   spacetime: their most striking properties rely on a distortion of the
   geometry of the space surrounding them.

Gravitational field

   The gravitational field outside a black hole 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 false:
   objects can orbit around black holes indefinitely without getting any
   closer. The strange properties of spacetime only become noticeable
   closer to the black hole.

Event horizon

   The "surface" of a black hole is the so-called event horizon, an
   imaginary surface surrounding the mass of the black hole. Stephen
   Hawking proved that the topology of the event horizon of a non-spinning
   black hole is a sphere. At the event horizon, the escape velocity is
   more than the speed of light. This is why anything inside the event
   horizon, including a photon, is prevented from escaping across the
   event horizon by the extremely strong gravitational field. Particles
   from outside this region can fall in, cross the event horizon, and will
   never be able to leave.

   Since external observers cannot probe the interior of a black hole,
   according to classical general relativity, black holes can be entirely
   characterised according to three parameters: mass, angular momentum,
   and electric charge. This principle is summarised by the saying, coined
   by John Archibald Wheeler, " black holes have no hair" meaning that
   there are no features that distinguish one black hole from another,
   other than mass, charge, and angular momentum.

Inside the event horizon

   Spacetime inside the event horizon of an uncharged non-rotating black
   hole is peculiar in that the singularity is in every observer's future,
   so all particles within the event horizon move inexorably towards it (
   Penrose and Hawking). This means that there is a conceptual inaccuracy
   in the non-relativistic concept of a black hole as originally proposed
   by John Michell in 1783. In Michell's theory, the escape velocity
   equals the speed of light, but it would still, for example, be
   theoretically possible to hoist an object out of a black hole using a
   rope. General relativity eliminates such loopholes, because once an
   object is inside the event horizon, its time-line contains an end-point
   to time itself, and no possible world-lines come back out through the
   event horizon. A consequence of this is that a pilot in a powerful
   rocket ship that had just crossed the event horizon who tried to
   accelerate away from the singularity would reach it sooner in his
   frame, since geodesics (unaccelerated paths) are paths that maximise
   proper time.

   As the object continues to approach the singularity, it will be
   stretched radially with respect to the black hole and compressed in
   directions perpendicular to this axis. This phenomenon, called
   spaghettification, occurs as a result of tidal forces: the parts of the
   object closer to the singularity feel a stronger pull towards it
   (causing stretching along the axis), and all parts are pulled in the
   direction of the singularity, which is only aligned with the object's
   average motion along the axis of the object (causing compression
   towards the axis).

Singularity

   At the centre of the black hole, well inside the event horizon, general
   relativity predicts a singularity, a place where the curvature of
   spacetime becomes infinite and gravitational forces become infinitely
   strong.

   It is expected that future refinements or generalisations of general
   relativity (in particular quantum gravity) will change what is thought
   about the nature of black hole interiors. Most theorists interpret the
   mathematical singularity of the equations as indicating that the
   current theory is not complete, and that new phenomena must come into
   play as one approaches the singularity.

   The cosmic censorship hypothesis asserts that there are no naked
   singularities in general relativity. This hypothesis is that every
   singularity is hidden behind an event horizon and cannot be probed.
   Whether this hypothesis is true remains an active area of theoretical
   research.

Rotating black holes

   An artist's impression of a black hole with a closely orbiting
   companion star that exceeds its Roche limit. In-falling matter forms an
   accretion disk, with some of the matter being ejected in highly
   energetic polar jets.
   Enlarge
   An artist's impression of a black hole with a closely orbiting
   companion star that exceeds its Roche limit. In-falling matter forms an
   accretion disk, with some of the matter being ejected in highly
   energetic polar jets.

   According to theory, the event horizon of a black hole that is not
   spinning is spherical, and its singularity is expected to be a single
   point where the curvature becomes infinite. If the black hole carries
   angular momentum (inherited from a star that is spinning at the time of
   its collapse), it begins to drag space-time surrounding the event
   horizon in an effect known as frame-dragging. This spinning area
   surrounding the event horizon is called the ergosphere and has an
   ellipsoidal shape. Since the ergosphere is located outside the event
   horizon, objects can exist within the ergosphere without falling into
   the hole. However, because space-time itself is moving in the
   ergosphere, it is impossible for objects to remain in a fixed position.
   Objects grazing the ergosphere could in some circumstances be
   catapulted outwards at great speed, extracting energy (and angular
   momentum) from the hole, hence the Greek name ergosphere ("sphere of
   work") because it is capable of doing work.

   The singularity inside a rotating black hole is expected to be a ring,
   rather than a point, though the interior geometry of a rotating black
   hole is currently not well understood. While the fate of an observer
   falling into a non-rotating black hole is spaghettification, the fate
   of an observer falling into a rotating black hole is much less clear.
   For instance, in the Kerr geometry, an infalling observer can
   potentially escape spaghettification by passing through an inner
   horizon. However, it is unlikely that the actual interior geometry of a
   rotating black hole is the Kerr geometry due to stability issues, and
   the ultimate fate of an observer falling into a rotating black hole is
   currently not known.

Entropy and 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.
   This sounded remarkably similar to the Second Law of Thermodynamics,
   with area playing the role of entropy. Classically, one could violate
   the second law of thermodynamics by material entering a black hole
   disappearing from our universe and 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 was simply an analogy. However, in 1974,
   Hawking applied quantum field theory to the curved spacetime around the
   event horizon and discovered that black holes can emit Hawking
   radiation, a form of thermal radiation. Using the first law of black
   hole mechanics, it follows that the entropy of a black hole is one
   quarter of the area of the horizon. This is a universal result and can
   be extended to apply to cosmological horizons such as in de Sitter
   space. It was later suggested that black holes are maximum-entropy
   objects, meaning that the maximum 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.

   The Hawking radiation reflects a characteristic temperature of the
   black hole, which can be calculated from its entropy. This temperature
   in fact 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.
   If ultra-high-energy collisions of particles in a particle accelerator
   can create microscopic black holes, it is expected that all types of
   particles will be emitted by black hole evaporation, providing key
   evidence for any grand unified theory. Above are the high energy
   particles produced in a gold ion collision on the RHIC.
   Enlarge
   If ultra-high-energy collisions of particles in a particle accelerator
   can create microscopic black holes, it is expected that all types of
   particles will be emitted by black hole evaporation, providing key
   evidence for any grand unified theory. Above are the high energy
   particles produced in a gold ion collision on the RHIC.

   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.

Black hole unitarity

   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 appears from
   the outside to become more and more red-shifted as it approaches (but
   never ultimately appears to reach) the event horizon.

   Ideas of quantum gravity, on the other hand, suggest that there can
   only be a limited finite entropy (ie 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, representing new nondeterministically
   created information. However, on 21 July 2004 he presented a new
   argument, reversing his previous position. On this new calculation, the
   entropy associated with the black hole itself would still be
   inaccessible to external observers; and in the absence of this
   information, 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. However, if the black hole
   evaporates completely, then such an identification can be made, and
   unitarity is preserved. It is not clear how far even the specialist
   scientific community is yet persuaded by the mathematical machinery
   Hawking has used (indeed many regard all work on quantum gravity so far
   as highly speculative); 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

   Black holes are predictions of Albert Einstein's theory of general
   relativity. There are many known solutions to the Einstein field
   equations which describe black holes, and they are also thought to be
   an inevitable part of the evolution of any star of a certain size. In
   particular, they occur in the Schwarzschild metric, one of the earliest
   and simplest solutions to Einstein's equations, found by Karl
   Schwarzschild in 1915. This solution describes the curvature of
   spacetime in the vicinity of a static and spherically symmetric object,
   where the metric is,

          ds^2 = - c^2 \left( 1 - {2Gm \over c^2 r} \right) dt^2 + \left(
          1 - {2Gm \over c^2 r} \right)^{-1} dr^2 + r^2 d\Omega^2 ,

   where d\Omega^2 = d\theta^2 + \sin^2\theta\; 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 — 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 × 10^30 kg/m^3, a supermassive black hole of 10^9
   solar masses has a density of around 20 kg/m^3, 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 × 10^26 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 visible 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.

   More general black holes are also predicted by other solutions to
   Einstein's equations, such as the Kerr metric for a rotating black
   hole, which possesses a ring singularity. Then we have the
   Reissner-Nordström metric for charged black holes. Last the Kerr-Newman
   metric is for the case of a charged and rotating 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[G]. The
   location of the marginally stable orbit, thought to be close to the
   inner edge of an accretion disk, is at 6r[G] for a Schwarzschild black
   hole.

Alternative models

   Several alternative models, which behave like a black hole but avoid
   the singularity, have been proposed. But 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.

   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 Msolar) black holes.

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