TheInfoList A gravitational singularity, spacetime singularity or simply singularity is a condition in which
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ... is so intense that
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
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". 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, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
becoming
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American mus ... or, better, by a
geodesic In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
being
incomplete Incomplete may refer to: * Unfinished creative work * Gödel's incompleteness theorems, a specification of logic * Incomplete (Bad Religion song), "Incomplete" (Bad Religion song), 1994 * Incomplete (Sisqó song), "Incomplete" (Sisqó song), 1999 * ...
. Gravitational singularities are mainly considered in the context of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, where
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ... apparently becomes infinite at the center of a
black hole , and within
astrophysics Astrophysics is a science that employs the methods and principles of physics in the study of astronomical objects and phenomena. Among the subjects studied are the Sun, other stars, galaxy, galaxies, extrasolar planets, the interstellar medium and ...
and
cosmology Cosmology (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
as the earliest state of the universe during the
Big Bang The Big Bang theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic ... /
White Hole In general relativity General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. Gener ...
. 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 consisting of a luminous spheroid of plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral) or heliotrope, a mineral aggregate * Quark� ... s this is the
Schwarzschild radius The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius In classical geometry Geometry ...
) would form a black hole, inside which a singularity (covered by an
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. The term was coined by Wolfgang Rindler Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity ...
) would be formed. The
Penrose–Hawking singularity theorems The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the question of when gravitation produces gravitational singularity, singularities. The Penrose ...
define a singularity to have
geodesics In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
that cannot be extended in a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ... manner. The termination of such a geodesic is considered to be the singularity. The initial state of the
universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development ... , 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, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
nor
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
.

# 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 derivative, differentiability or Analyticity ...
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 Classical physics is a group of physics theories that predate modern, more complete, or m ...
, re-normalization, and instability of a hydrogen atom predicted by the
Larmor formula In electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
. 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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, ... 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 every diffeomorphism invariant quantity is finite, in which case spacetime is not smooth at the point of the limit itself. Thus, spacetime looks like a
cone A cone is a three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:π� ... 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... is used. An example of such a conical singularity is a
cosmic string Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a Symmetry breaking, symmetry-breaking phase transition in the early universe when the topology of the Vacuum state, vacuum manifold associated to this ...
and a
Schwarzschild black hole In Albert Einstein, 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 ...
.

## Curvature Solutions to the equations of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
or another theory of
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ... (such as
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory (physics), field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersym ...
) often result in encountering points where the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geomet ...
blows up to infinity. However, many of these points are completely
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ... , 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.
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
s) 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 In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. The term was coined by Wolfgang Rindler Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity ...
. However, spacetime at the event horizon is
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ... . The regularity becomes evident when changing to another coordinate system (such as the Kruskal coordinates), where the metric is perfectly
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ... . 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, 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 ...
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 tensor, metric is an Exact solutions in general relativity, e ...
, the singularity occurs on a ring (a circular line), known as a "
ring singularityA ring singularity or ringularity is the gravitational singularity A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime In physics Physics (from grc, φυσική (ἐπιστήμη), ph ...
". Such a singularity may also theoretically become a
wormhole A wormhole (or Einstein–Rosen bridge or Einstein–Rosen wormhole) is a speculative structure linking disparate points in spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions ... . 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 In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. The term was coined by Wolfgang Rindler Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity ...
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 theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic ... cosmological Cosmology (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
model of the
universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development ... 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, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
hides every singularity behind an
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. The term was coined by Wolfgang Rindler Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity ...
, making naked singularities impossible. This is referred to as the
cosmic censorship hypothesis The weak and the strong cosmic censorship hypotheses are two mathematical conjectures In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures ( ...
. 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 General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General ...
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 In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dime ...
, which is a spinning black hole in a vacuum, if the
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ... ($J$) 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 (mathematics), metric of a Schwarzschild black hole that can be used to express the metric of a Kerr ...
, it can be shown that the coordinate (which is not the radius) of the event horizon is, $r_= \mu \pm \left(\mu^-a^\right)^$, 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 In particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particl ...
, , 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 $r_= \mu \pm \left(\mu^-q^\right)^$, where $\mu = G M / c^$, and $q^ = G Q^/\left(4 \pi \epsilon_ c^\right)$. 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.

# Entropy

Before
Stephen Hawking Stephen William Hawking (8 January 1942 – 14 March 2018) was an English theoretical physicist Theoretical physics is a branch of physics Physics is the that studies , its , its and behavior through , and the related enti ... came up with the concept of
Hawking radiation #REDIRECT Hawking radiation Hawking radiation is black-body radiation that is predicted to be released by black holes, due to quantum effects near the black hole event horizon. It is named after the physicist Stephen Hawking, who provided a theor ...
, 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 establishes the concept of entropy Entropy is a scientific concept, as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term an ...
. 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 Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides ...
of more than one million, rather than the thousand or so since the background radiation formed.

* 0-dimensional singularity:
magnetic monopole In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that departmen ...
* 1-dimensional singularity:
cosmic string Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a Symmetry breaking, symmetry-breaking phase transition in the early universe when the topology of the Vacuum state, vacuum manifold associated to this ...
* 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 polyn ...
* Fuzzball (string theory) * Penrose-Hawking singularity theorems *
White hole In general relativity General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. Gener ...
*
BKL singularity A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe near the initial gravitational singularity A gravitational singularity, spacetime singularity or simply singularity is a condition i ...
*
Initial singularity The initial singularity is a singularity Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", ...

# 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''.