Penrose-Hawking singularity theorems

# Penrose-Hawking singularity theorems

Overview
The Penrose–Hawking singularity theorems are a set of results in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

which attempt to answer the question of when gravitation produces singularities
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...

.
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Encyclopedia
The Penrose–Hawking singularity theorems are a set of results in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

which attempt to answer the question of when gravitation produces singularities
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...

.

A singularity in solutions of the Einstein field equations
Solutions of the Einstein field equations
Where appropriate, this article will use the abstract index notation.Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations of general relativity. Solving the field equations actually gives Lorentz metrics...

is one of two things:
1. a situation where matter is forced to be compressed to a point (a space-like singularity)
2. a situation where certain light rays come from a region with infinite curvature (time-like singularity)

Space-like singularities are a feature of non-rotating uncharged black-hole
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

s, while time-like singularities are those that occur in charged or rotating black hole exact solutions. Both of them have the following property:
geodesic incompleteness: Some light-paths or particle-paths cannot be extended beyond a certain proper-time or affine-parameter (affine parameter is the null analog of proper time).

It is still an open question whether time-like singularities ever occur in the interior of real charged or rotating black holes, or whether they are artifacts of high symmetry and turn into spacelike singularities when realistic perturbations are added.

The Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

theorem guarantees that some sort of geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

incompleteness occurs inside any black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

, whenever matter satisfies reasonable energy condition
Energy condition
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is one of various alternative conditions which can be applied to the matter content of the theory, when it is either not possible or desirable to specify this content explicitly...

s. (It does not hold for matter described by a super-field, i.e., the Dirac field) 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: in Hawking's original formulation, it guaranteed that the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

has infinite density. Hawking later revised his position in A Brief History of Time
A Brief History of Time
A Brief History of Time is a popular science book written by renown physicist Stephen Hawking and first published by the Bantam Dell Publishing Group in 1988. It became a best-seller and has sold more than 10 million copies...

(1988) where he stated "There was in fact no singularity at the beginning of the universe" (p50). This revision followed from quantum mechanics, in which general relativity must break down at times less than the Planck time. Hence general relativity cannot be used to show a singularity.

Penrose's theorem is more restricted, it only holds when matter obeys a stronger energy condition, called the dominant energy condition, which means that the energy is bigger than the pressure. All ordinary matter, with the exception of a vacuum expectation value of a scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

, obeys this condition. During inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...

, the universe violates the stronger dominant energy condition (but not the weak energy condition), and inflationary cosmologies avoid the initial big-bang singularity, rounding them out to a smooth beginning.

## Interpretation and significance

In general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It 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 space-time stops being a manifold. Singularities can be found in all the black-hole spacetimes, the Schwarzschild metric
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

, the Reissner–Nordström metric and the Kerr metric
Kerr metric
The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...

, and in all cosmological solutions which don't 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
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

to form a black hole, if the star is spinning and thus possesses some angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

, maybe the centrifugal force
Centrifugal force
Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

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
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

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 (see No-hair theorem). The part inside the event horizon necessarily has a singularity somewhere. The proof is somewhat constructive— it 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, 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 mathematics, there is a deep connection between the curvature of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

and its topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

. The Bonnet–Myers theorem states that a complete Riemannian manifold which has Ricci curvature
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

everywhere greater than a certain positive constant must be compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

. The condition of positive Ricci curvature is most conveniently stated in the following way: for every geodesic there is a nearby initially parallel geodesic which will bend toward it when extended, and the two will intersect at some finite length.

When two nearby parallel geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

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
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

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.

Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

argued analogously in relativity. If null geodesics, 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
Congruence (general relativity)
In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime...

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
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

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 which is eventually generated by new light rays which 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
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

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:
1. An energy condition on the matter,
2. A condition on the global structure of spacetime,
3. 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
Raychaudhuri equation
In general relativity, the Raychaudhuri equation, or Landau-Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter....

, which describes the divergence of a congruence
Congruence (general relativity)
In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime...

(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 (see the congruence
Congruence (general relativity)
In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime...

page for details). The key point is that will be non-negative provided that the Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

hold and
• the null energy condition holds and the geodesic congruence is null, or
• the strong energy condition 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
1. Suppose we have a spacetime that is globally hyperbolic
Globally hyperbolic
In mathematical physics, a spacetime manifold is globally hyperbolic if it satisfies a condition related to its causal structure. This is relevant to Einstein's theory of general relativity, and potentially to other metric gravitational theories....

, and two points and that can be connected by a timelike or null curve. Then there exists a geodesic of maximal length connecting and . Call this geodesic .
2. The geodesic can be varied to a longer curve if another geodesic from intersects at another point, called a conjugate point.
3. 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 contradiction – one can therefore conclude that the spacetime is geodesically incomplete.

In general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It 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
Trapped null surface
A trapped null surface is a set of points defined in the context of general relativity as a closed surface on which outward-pointing light rays are actually converging ....

and the energy density
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

is nonnegative, then there exist geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s of finite length which can't 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
1. The null energy condition holds.
2. We have a noncompact connected Cauchy surface
Cauchy surface
Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future uniquely....

.
3. We have a closed trapped null surface
Trapped null surface
A trapped null surface is a set of points defined in the context of general relativity as a closed surface on which outward-pointing light rays are actually converging ....

.
Then, we either have null geodesic incompleteness, or closed timelike curve
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...

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
Raychaudhuri equation
In general relativity, the Raychaudhuri equation, or Landau-Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter....

, 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.) However, the null geodesics generating have to terminate, i.e. reach their future endpoints at or before the caustics. Otherwise, we can take two null geodesic segments — changing at the caustic — and 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, the timelike curves can't intersect, and so, the map is injective. 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.