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A gravitational singularity, spacetime singularity or simply singularity is a condition in which
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
is so intense that
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
; therefore, the properties of the singularity cannot be described without an established theory of
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. Trying to find a complete and precise definition of singularities in the theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the current best theory of gravity, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
becoming infinite or, better, by a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
being incomplete. Gravitational singularities are mainly considered in the context of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, where
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
apparently becomes infinite at the center of a
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
, and within
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
and
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
as the earliest state of the universe during the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
/
White Hole In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy-matter, light and information can escape from it. In this sense, it is the reverse of a black ho ...
. Physicists are undecided whether the prediction of singularities means that they actually exist (or existed at the start of the Big Bang), or that current knowledge is insufficient to describe what happens at such extreme densities. General relativity predicts that any object collapsing beyond a certain point (for
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s this is the
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteri ...
) would form a black hole, inside which a singularity (covered by an event horizon) would be formed. The Penrose–Hawking singularity theorems define a singularity to have
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
that cannot be extended in a smooth manner. The termination of such a geodesic is considered to be the singularity. The initial state of the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
, at the beginning of the Big Bang, is also predicted by modern theories to have been a singularity. In this case, the universe did not collapse into a black hole, because currently-known calculations and density limits for gravitational collapse are usually based upon objects of relatively constant size, such as stars, and do not necessarily apply in the same way to rapidly expanding space such as the Big Bang. Neither
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
nor
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
can currently describe the earliest moments of the Big Bang, but in general, quantum mechanics does not permit particles to inhabit a space smaller than their
wavelengths In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
.


Interpretation

Many theories in physics have
mathematical singularities In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For exa ...
of one kind or another. Equations for these physical theories predict that the ball of mass of some quantity becomes infinite or increases without limit. This is generally a sign for a missing piece in the theory, as in the
ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
, re-normalization, and instability of a hydrogen atom predicted by the
Larmor formula In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light. When any charge ...
. In classical field theories including special relativity, but not general relativity, one can say that a solution has a singularity at a particular point in spacetime where certain physical properties become ill defined, with spacetime serving as a background field to locate the singularity. A singularity in general relativity, on the other hand, is more complex because spacetime itself becomes ill defined, and the singularity is no longer part of the regular spacetime manifold. In general relativity, a singularity cannot be defined by "where" or "when". Some theories, such as the theory of
loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
, suggest that singularities may not exist. This is also true for such classical unified field theories as the Einstein–Maxwell–Dirac equations. The idea can be stated in the form that due to
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
effects, there is a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses becomes shorter, or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance.


Types

There are different types of singularities, each with different physical features which have characteristics relevant to the theories from which they originally emerged, such as the different shape of the singularities, ''conical and curved''. They have also been hypothesized to occur without event horizons, structures which delineate one spacetime section from another in which events cannot affect past the horizon; these are called ''naked.''


Conical

A conical singularity occurs when there is a point where the limit of some diffeomorphism invariant quantity does not exist or is infinite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a cone around this point, where the singularity is located at the tip of the cone. The metric can be finite everywhere the
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
is used. An example of such a conical singularity is a cosmic string and a
Schwarzschild black hole In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
.


Curvature

Solutions to the equations of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
or another theory of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
(such as
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
) often result in encountering points where the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
blows up to infinity. However, many of these points are completely regular, and the infinities are merely a result of using an inappropriate coordinate system at this point. In order to test whether there is a singularity at a certain point, one must check whether at this point diffeomorphism invariant quantities (i.e. scalars) become infinite. Such quantities are the same in every coordinate system, so these infinities will not "go away" by a change of coordinates. An example is the Schwarzschild solution that describes a non-rotating, uncharged black hole. In coordinate systems convenient for working in regions far away from the black hole, a part of the metric becomes infinite at the event horizon. However, spacetime at the event horizon is regular. The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly smooth. On the other hand, in the center of the black hole, where the metric becomes infinite as well, the solutions suggest a singularity exists. The existence of the singularity can be verified by noting that the
Kretschmann scalar In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann. Definition The Kretschmann invariant is : K ...
, being the square of the
Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
i.e. R_R^, which is diffeomorphism invariant, is infinite. While in a non-rotating black hole the singularity occurs at a single point in the model coordinates, called a "point singularity", in a rotating black hole, also known as a
Kerr black hole The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...
, the singularity occurs on a ring (a circular line), known as a " ring singularity". Such a singularity may also theoretically become a
wormhole A wormhole ( Einstein-Rosen bridge) is a hypothetical structure connecting disparate points in spacetime, and is based on a special solution of the Einstein field equations. A wormhole can be visualized as a tunnel with two ends at separate ...
. More generally, a spacetime is considered singular if it is geodesically incomplete, meaning that there are freely-falling particles whose motion cannot be determined beyond a finite time, being after the point of reaching the singularity. For example, any observer inside the event horizon of a non-rotating black hole would fall into its center within a finite period of time. The classical version of the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
cosmological model of the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
contains a causal singularity at the start of
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
(''t''=0), where all time-like geodesics have no extensions into the past. Extrapolating backward to this hypothetical time 0 results in a universe with all spatial dimensions of size zero, infinite density, infinite temperature, and infinite spacetime curvature.


Naked singularity

Until the early 1990s, it was widely believed that
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship hypothesis. However, in 1991, physicists Stuart Shapiro and Saul Teukolsky performed computer simulations of a rotating plane of dust that indicated that general relativity might allow for "naked" singularities. What these objects would actually look like in such a model is unknown. Nor is it known whether singularities would still arise if the simplifying assumptions used to make the simulation were removed. However, it is hypothesized that light entering a singularity would similarly have its geodesics terminated, thus making the
naked singularity In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. In a black hole, the singularity is completely enclosed by a boundary known as the event horizon, inside which the curvature of s ...
look like a black hole. Disappearing event horizons exist in the 
Kerr metric The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...
, which is a spinning black hole in a vacuum, if the 
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
 (J) is high enough. Transforming the Kerr metric to 
Boyer–Lindquist coordinates In the mathematical description of general relativity, the Boyer–Lindquist coordinates are a generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole. The H ...
, it can be shown that the coordinate (which is not the radius) of the event horizon is, r_= \mu \pm (\mu^-a^)^, where \mu = G M / c^, and a=J/M c. In this case, "event horizons disappear" means when the solutions are complex for r_, or \mu^ < a^. However, this corresponds to a case where J exceeds GM^/c (or in Planck units, , i.e. the spin exceeds what is normally viewed as the upper limit of its physically possible values. Similarly, disappearing event horizons can also be seen with the  Reissner–Nordström geometry of a charged black hole if the charge (Q) is high enough. In this metric, it can be shownHobson, et al., ''General Relativity an Introduction for Physicists'', Cambridge University Press 2007, p. 320-325 that the singularities occur at r_= \mu \pm (\mu^-q^)^, where \mu = G M / c^, and q^ = G Q^/(4 \pi \epsilon_ c^). Of the three possible cases for the relative values of \mu and q, the case where \mu^ < q^ causes both r_ to be complex. This means the metric is regular for all positive values of r, or in other words, the singularity has no event horizon. However, this corresponds to a case where Q/\sqrt exceeds M\sqrt (or in Planck units, , i.e. the charge exceeds what is normally viewed as the upper limit of its physically possible values. Also, actual astrophysical black holes are not expected to possess any appreciable charge. A black hole possessing the lowest M value consistent with its J and Q values and the limits noted above, i.e., one just at the point of losing its event horizon, is termed
extremal In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
.


Entropy

Before Stephen Hawking came up with the concept of Hawking radiation, the question of black holes having entropy had been avoided. However, this concept demonstrates that black holes radiate energy, which conserves entropy and solves the incompatibility problems with the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
. Entropy, however, implies heat and therefore temperature. The loss of energy also implies that black holes do not last forever, but rather evaporate or decay slowly. Black hole temperature is inversely related to mass. All known black hole candidates are so large that their temperature is far below that of the cosmic background radiation, which means they will gain energy on net by absorbing this radiation. They cannot begin to lose energy on net until the background temperature falls below their own temperature. This will occur at a cosmological redshift of more than one million, rather than the thousand or so since the background radiation formed.


See also

* 0-dimensional singularity: magnetic monopole * 1-dimensional singularity: cosmic string * 2-dimensional singularity:
domain wall A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with pol ...
*
Fuzzball (string theory) Fuzzballs are theorized by some superstring theory scientists to be the true quantum description of black holes. The theory attempts to resolve two intractable problems that classic black holes pose for modern physics: # The information parad ...
* Penrose-Hawking singularity theorems *
White hole In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy-matter, light and information can escape from it. In this sense, it is the reverse of a black ho ...
*
BKL singularity A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe near the initial gravitational singularity, described by an anisotropic, chaotic solution of the Einstein field equation of gravitatio ...
*
Initial singularity The initial singularity is a singularity predicted by some models of the Big Bang theory to have existed before the Big Bang and thought to have contained all the energy and spacetime of the Universe. The instant immediately following the initi ...


Notes


References

* * * §31.2 The nonsingularity of the gravitational radius, and following sections; §34 Global Techniques, Horizons, and Singularity Theorems * * (Free access.) * * Penrose, Roger (1996)
"Chandrasekhar, Black Holes, and Singularities"
''ias.ac.in''. * Penrose, Roger (1999)
"The Question of Cosmic Censorship"
''ias.ac.in''. * Singh, T.P. (1999
"Gravitational Collapse, Black Holes and Naked Singularities"
''ias.ac.in''.


Further reading

* ''
The Elegant Universe ''The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory'' is a book by Brian Greene published in 1999, which introduces string and superstring theory, and provides a comprehensive though non-technical assess ...
'' by
Brian Greene Brian Randolph Greene (born February 9, 1963) is a American theoretical physicist, mathematician, and string theorist. Greene was a physics professor at Cornell University from 19901995, and has been a professor at Columbia University sinc ...
. This book provides a
layman In religious organizations, the laity () consists of all members who are not part of the clergy, usually including any non-ordained members of religious orders, e.g. a nun or a lay brother. In both religious and wider secular usage, a layper ...
's introduction to string theory, although some of the views expressed are already becoming outdated. His use of common terms and his providing of examples throughout the text help the layperson understand the basics of string theory. {{Portal bar, Physics, Astronomy, Stars, Spaceflight, Outer space, Solar System General relativity Lorentzian manifolds Physical paradoxes Physical phenomena Concepts in astronomy