The Penrose–Hawking singularity theorems (after
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...
and
Stephen Hawking) are a set of results in
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 ...
that attempt to answer the question of when gravitation produces
singularities. The Penrose singularity theorem is a theorem in
semi-Riemannian geometry
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riem ...
and its general relativistic interpretation predicts a
gravitational singularity
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravitational field, gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer p ...
in black hole formation. The Hawking singularity theorem is based on the Penrose theorem and it is interpreted as a
gravitational singularity
A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravitational field, gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer p ...
in 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 ...
situation. Penrose was awarded the
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity", which he shared with
Reinhard Genzel
Reinhard Genzel http://royalsociety.org/people/reinhard-genzel/ Professor Reinhard Genzel ForMemRS (; born 24 March 1952) is a German astrophysicist, co-director of the Max Planck Institute for Extraterrestrial Physics, a professor at LMU and a ...
and
Andrea Ghez
Andrea is a given name which is common worldwide for both males and females, cognate to Andreas, Andrej and Andrew.
Origin of the name
The name derives from the Greek word ἀνήρ (''anēr''), genitive ἀνδρός (''andrós''), that re ...
.
Singularity
A singularity in
solutions of the Einstein field equations
:''Where appropriate, this article will use the abstract index notation.''
Solutions of the Einstein field equations are metrics of spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field eq ...
is one of two things:
# a situation where matter is forced to be compressed to a point (a space-like singularity)
# a situation where certain light rays come from a region with infinite curvature (a time-like singularity)
Space-like singularities are a feature of non-rotating uncharged
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s as described by the
Schwarzschild metric
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 ...
, while time-like singularities are those that occur in charged or rotating black hole exact solutions. Both of them have the property of
geodesic incompleteness, in which either some light-path or some particle-path cannot be extended beyond a certain proper time or affine parameter (affine parameter being the null analog of proper time).
The Penrose theorem guarantees that some sort of geodesic incompleteness occurs inside ''any'' black hole whenever matter satisfies reasonable
energy conditions
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically-phrased mathem ...
. The energy condition required for the black-hole singularity theorem is weak: it says that light rays are always focused together by gravity, never drawn apart, and this holds whenever the energy of matter is non-negative.
Hawking's singularity theorem is for the whole universe, and works backwards in time: it guarantees that the (classical)
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 ...
has infinite density. This theorem is more restricted and only holds when matter obeys a stronger energy condition, called the ''
dominant energy condition'', in which the energy is larger than the pressure. All ordinary matter, with the exception of a vacuum expectation value of a
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
, obeys this condition. During
inflation
In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
, the universe violates the dominant energy condition, and it was initially argued (e.g. by Starobinsky) that inflationary cosmologies could avoid the initial big-bang singularity. However, it has since been shown that inflationary cosmologies are still past-incomplete, and thus require physics other than inflation to describe the past boundary of the inflating region of spacetime.
It is still an open question whether (classical) general relativity predicts time-like singularities in the interior of realistic charged or rotating black holes, or whether these are artefacts of high-symmetry solutions and turn into spacelike singularities when perturbations are added.
Interpretation and significance
In
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 ...
, a singularity is a place that objects or light rays can reach in a finite time where the curvature becomes infinite, or spacetime stops being a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Singularities can be found in all the black-hole spacetimes, the
Schwarzschild metric
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 ...
, the
Reissner–Nordström metric
In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass ''M''. T ...
, 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 ge ...
and the
Kerr–Newman metric
The Kerr–Newman metric is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mas ...
, and in all cosmological solutions that do not have a scalar field energy or a cosmological constant.
One cannot predict what might come "out" of a big-bang singularity in our past, or what happens to an observer that falls "in" to a black-hole singularity in the future, so they require a modification of physical law. Before Penrose, it was conceivable that singularities only form in contrived situations. For example, in the collapse of a
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 ...
to form a black hole, if the star is spinning and thus possesses some
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 ...
, maybe the
centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
partly counteracts gravity and keeps a singularity from forming. The singularity theorems prove that this cannot happen, and that a singularity will always form once an
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
forms.
In the collapsing star example, since all matter and energy is a source of gravitational attraction in general relativity, the additional angular momentum only pulls the star together more strongly as it contracts: the part outside the event horizon eventually settles down to 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 ge ...
(see
No-hair theorem
The no-hair theorem states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent ''externally'' observable ...
). The part inside the event horizon necessarily has a singularity somewhere. The proof is somewhat constructiveit shows that the singularity can be found by following light-rays from a surface just inside the horizon. But the proof does not say what type of singularity occurs, spacelike, timelike,
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
, jump discontinuity in the metric. It only guarantees that if one follows the time-like geodesics into the future, it is impossible for the boundary of the region they form to be generated by the null geodesics from the surface. This means that the boundary must either come from nowhere or the whole future ends at some finite extension.
An interesting "philosophical" feature of general relativity is revealed by the singularity theorems. Because general relativity predicts the inevitable occurrence of singularities, the theory is not complete without a specification for what happens to matter that hits the singularity. One can extend general relativity
to a unified field theory, such as the Einstein–Maxwell–Dirac system, where no such singularities occur.
Elements of the theorems
In history, there is a deep connection between the curvature of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
and its
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. The
Bonnet–Myers theorem
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:
In the special case of ...
states that a complete Riemannian manifold that has
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
everywhere greater than a certain positive constant must be
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
. The condition of positive Ricci curvature is most conveniently stated in the following way: for every geodesic there is a nearby initially parallel geodesic that will bend toward it when extended, and the two will intersect at some finite length.
When two nearby parallel
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 connection. ...
s intersect, the extension of either one is no longer the shortest path between the endpoints. The reason is that two parallel geodesic paths necessarily collide after an extension of equal length, and if one path is followed to the intersection then the other, you are connecting the endpoints by a non-geodesic path of equal length. This means that for a geodesic to be a shortest length path, it must never intersect neighboring parallel geodesics.
Starting with a small sphere and sending out parallel geodesics from the boundary, assuming that the manifold has a
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
bounded below by a positive constant, none of the geodesics are shortest paths after a while, since they all collide with a neighbor. This means that after a certain amount of extension, all potentially new points have been reached. If all points in a connected manifold are at a finite geodesic distance from a small sphere, the manifold must be compact.
Roger Penrose argued analogously in relativity. If
null geodesic
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a fre ...
s, the paths of
light rays, are followed into the future, points in the future of the region are generated. If a point is on the boundary of the future of the region, it can only be reached by going at the speed of light, no slower, so null geodesics include the entire boundary of the proper future of a region. When the null geodesics intersect, they are no longer on the boundary of the future, they are in the interior of the future. So, if all the null geodesics collide, there is no boundary to the future.
In relativity, the Ricci curvature, which determines the collision properties of geodesics, is determined by the
energy tensor, and its projection on light rays is equal to the null-projection of the energy–momentum tensor and is always non-negative. This implies that the volume of a
congruence of parallel null geodesics once it starts decreasing, will reach zero in a finite time. Once the volume is zero, there is a collapse in some direction, so every geodesic intersects some neighbor.
Penrose concluded that whenever there is a sphere where all the outgoing (and ingoing) light rays are initially converging, the boundary of the future of that region will end after a finite extension, because all the null geodesics will converge. This is significant, because the outgoing light rays for any sphere inside the horizon of a
black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
solution are all converging, so the boundary of the future of this region is either compact or comes from nowhere. The future of the interior either ends after a finite extension, or has a boundary that is eventually generated by new light rays that cannot be traced back to the original sphere.
Nature of a singularity
The singularity theorems use the notion of geodesic incompleteness as a stand-in for the presence of infinite curvatures. Geodesic incompleteness is the notion that there are
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 connection. ...
s, paths of observers through spacetime, that can only be extended for a finite time as measured by an observer traveling along one. Presumably, at the end of the geodesic the observer has fallen into a singularity or encountered some other pathology at which the laws of general relativity break down.
Assumptions of the theorems
Typically a singularity theorem has three ingredients:
# An
energy condition on the matter,
# A condition on the
global structure of spacetime,
# Gravity is strong enough (somewhere) to trap a region.
There are various possibilities for each ingredient, and each leads to different singularity theorems.
Tools employed
A key tool used in the formulation and proof of the singularity theorems is the
Raychaudhuri equation
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.
The equation is important as a fundamental lemma for the Penrose–Hawking singularity the ...
, which describes the divergence
of a
congruence (family) of geodesics. The divergence of a congruence is defined
as the derivative of the log of the determinant of the congruence volume. The Raychaudhuri
equation is
:
where
is the shear tensor of the congruence and
is also known as the Raychaudhuri scalar (see the
congruence page for details). The key point is that
will be non-negative provided that the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
hold and
* the
null energy condition holds and the geodesic congruence is null, or
* the
strong energy condition
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically-phrased mathem ...
holds and the geodesic congruence is timelike.
When these hold, the divergence becomes infinite at some finite value of the affine parameter. Thus all geodesics leaving a point will eventually reconverge after a finite time, provided the appropriate energy condition holds, a result also known as the focusing theorem.
This is relevant for singularities thanks to the following argument:
# Suppose we have a spacetime that is
globally hyperbolic, and two points
and
that can be connected by a
timelike
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 differen ...
or
null curve
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Introduction
In modern physics (especially general relativity) spacetime is represented by a Lorentzian m ...
. Then there exists a geodesic of maximal length connecting
and
. Call this geodesic
.
# The geodesic
can be varied to a longer curve if another geodesic from
intersects
at another point, called a conjugate point.
# From the focusing theorem, we know that all geodesics from
have conjugate points at finite values of the affine parameter. In particular, this is true for the geodesic of maximal length. But this is a contradictionone can therefore conclude that the spacetime is geodesically incomplete.
In
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 ...
, there are several versions of the Penrose–Hawking singularity theorem. Most versions state, roughly, that if there is a
trapped null surface
Closed trapped surfaces are a concept used in black hole solutions of general relativity which describe the inner region of an event horizon. Roger Penrose defined the notion of closed trapped surfaces in 1965. A trapped surface is one where light ...
and the
energy density
In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or .
Often only the ''useful'' or extract ...
is nonnegative, then there exist
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 connection. ...
s of finite length that cannot be extended.
These theorems, strictly speaking, prove that there is at least one non-spacelike geodesic that is only finitely extendible into the past but there are cases in which the conditions of these theorems obtain in such a way that all past-directed spacetime paths terminate at a singularity.
Versions
There are many versions. Here is the null version:
: Assume
# The
null energy condition holds.
# We have a noncompact connected
Cauchy surface In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as ...
.
# We have a closed
trapped null surface
Closed trapped surfaces are a concept used in black hole solutions of general relativity which describe the inner region of an event horizon. Roger Penrose defined the notion of closed trapped surfaces in 1965. A trapped surface is one where light ...
.
: Then, we either have null geodesic incompleteness, or
closed timelike curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Sto ...
s.
:: ''Sketch of proof'': Proof by contradiction. The boundary of the future of
,
is generated by null geodesic segments originating from
with tangent vectors orthogonal to it. Being a trapped null surface, by the null
Raychaudhuri equation
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.
The equation is important as a fundamental lemma for the Penrose–Hawking singularity the ...
, both families of null rays emanating from
will encounter caustics. (A caustic by itself is unproblematic. For instance, the boundary of the future of two spacelike separated points is the union of two future light cones with the interior parts of the intersection removed. Caustics occur where the light cones intersect, but no singularity lies there.) The null geodesics generating
have to terminate, however, i.e. reach their future endpoints at or before the caustics. Otherwise, we can take two null geodesic segmentschanging at the causticand then deform them slightly to get a timelike curve connecting a point on the boundary to a point on
, a contradiction. But as
is compact, given a continuous affine parameterization of the geodesic generators, there exists a lower bound to the absolute value of the expansion parameter. So, we know caustics will develop for every generator before a uniform bound in the affine parameter has elapsed. As a result,
has to be compact. Either we have closed timelike curves, or we can construct a congruence by timelike curves, and every single one of them has to intersect the noncompact Cauchy surface exactly once. Consider all such timelike curves passing through
and look at their image on the Cauchy surface. Being a continuous map, the image also has to be compact. Being a
timelike congruence In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Oft ...
, the timelike curves can't intersect, and so, the map is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. If the Cauchy surface were noncompact, then the image has a boundary. We're assuming spacetime comes in one connected piece. But
is compact and boundariless because the boundary of a boundary is empty. A continuous injective map can't create a boundary, giving us our contradiction.
:: ''Loopholes'': If closed timelike curves exist, then timelike curves don't have to intersect the ''partial'' Cauchy surface. If the Cauchy surface were compact, i.e. space is compact, the null geodesic generators of the boundary can intersect everywhere because they can intersect on the other side of space.
Other versions of the theorem involving the weak or strong energy condition also exist.
Modified gravity
In modified gravity, the Einstein field equations do not hold and so these singularities do not necessarily arise. For example, in
Infinite Derivative Gravity
Infinite derivative gravity is a theory of gravity which attempts to remove cosmological and black hole singularities by adding extra terms to the Einstein–Hilbert action, which weaken gravity at short distances.
History
In 1987, Krasnikov con ...
, it is possible for
to be negative even if the Null Energy Condition holds.
Notes
References
* The classic reference.
*
*
*. Also available as
*Kalvakota, Vaibhav R. (2021),
A discussion on Geometry and General Relativity
*See also for a relevant chapter from ''The Large Scale Structure of Space Time''.
*Witten, Edward (2020), "Light Rays, Singularities, and All That", ''Rev. Mod. Phys.'' 92, 45004 (2020), https://doi.org/10.1103/RevModPhys.92.045004 . Also available as https://arxiv.org/abs/1901.03928. An excellent pedagogical review of causal properties 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 ...
.
{{DEFAULTSORT:Penrose-Hawking Singularity Theorems
General relativity
Mathematical methods in general relativity
Theorems in general relativity