The Penrose–Hawking singularity theorems
are a set of results in general relativity
that attempt to answer the question of when
gravitation produces singularities.
== Singularity ==
A singularity in solutions of the Einstein
field equations 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-holes,
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 (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: it
guarantees that the (classical) Big Bang 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, obeys this condition. During
inflation, 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, 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, the
Reissner–Nordström metric, the Kerr metric
and the Kerr–Newman metric 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 to form a black
hole, if the star is spinning and thus possesses
some angular momentum, maybe the centrifugal
force 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 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 and its
topology. The Bonnet–Myers theorem states
that a complete Riemannian manifold that has
Ricci curvature everywhere greater than a
certain positive constant must be compact.
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 geodesics 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 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 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 boundaries 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 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
cube 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 isn't significant, because
the outgoing light rays for any sphere inside
the horizon of a black hole 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
geodesics, 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, which describes the divergence
θ
{\displaystyle \theta }
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
θ
˙
=
−
σ
a
b
σ
a
b
−
1
3
θ
2
−
E
[
X
→
]
a
a
{\displaystyle {\dot {\theta }}=-\sigma _{ab}\sigma
^{ab}-{\frac {1}{3}}\theta ^{2}-{E[{\vec {X}}]^{a}}_{a}}
where
σ
a
b
{\displaystyle \sigma _{ab}}
is the shear tensor of the congruence and
E
[
X
→
]
a
a
=
R
m
n
X
m
X
n
{\displaystyle {E[{\vec {X}}]^{a}}_{a}=R_{mn}\,X^{m}\,X^{n}}
is also known as the Raychaudhuri scalar (see
the congruence page for details). The key
point is that
E
[
X
→
]
a
a
{\displaystyle {E[{\vec {X}}]^{a}}_{a}}
will be non-negative provided that the Einstein
field equations 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:
Suppose we have a spacetime that is globally
hyperbolic, and two points
p
{\displaystyle p}
and
q
{\displaystyle q}
that can be connected by a timelike or null
curve. Then there exists a geodesic of maximal
length connecting
p
{\displaystyle p}
and
q
{\displaystyle q}
. Call this geodesic
γ
{\displaystyle \gamma }
.
The geodesic
γ
{\displaystyle \gamma }
can be varied to a longer curve if another
geodesic from
p
{\displaystyle p}
intersects
γ
{\displaystyle \gamma }
at another point, called a conjugate point.
From the focusing theorem, we know that all
geodesics from
p
{\displaystyle p}
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, there are
several versions of the Penrose–Hawking
singularity theorem. Most versions state,
roughly, that if there is a trapped null surface
and the energy density is nonnegative, then
there exist geodesics 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:
AssumeThe null energy condition holds.
We have a noncompact connected Cauchy surface.
We have a closed trapped null surface
T
{\displaystyle {\mathcal {T}}}
.Then, we either have null geodesic incompleteness,
or closed timelike curves.
Sketch of proof: Proof by contradiction. The
boundary of the future of
T
{\displaystyle {\mathcal {T}}}
,
J
˙
(
T
)
{\displaystyle {\dot {J}}({\mathcal {T}})}
is generated by null geodesic segments originating
from
T
{\displaystyle {\mathcal {T}}}
with tangent vectors orthogonal to it. Being
a trapped null surface, by the null Raychaudhuri
equation, both families of null rays emanating
from
T
{\displaystyle {\mathcal {T}}}
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
J
˙
(
T
)
{\displaystyle {\dot {J}}({\mathcal {T}})}
have to terminate, however, 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
T
{\displaystyle {\mathcal {T}}}
, a contradiction. But as
T
{\displaystyle {\mathcal {T}}}
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,
J
˙
(
T
)
{\displaystyle {\dot {J}}({\mathcal {T}})}
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
J
˙
(
T
)
{\displaystyle {\dot {J}}({\mathcal {T}})}
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
J
˙
(
T
)
{\displaystyle {\dot {J}}({\mathcal {T}})}
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, it is possible for
E
[
X
→
]
a
a
{\displaystyle {E[{\vec {X}}]^{a}}_{a}}
to be negative even if the Null Energy Condition
holds.
== Notes
