Null Geodesics, Raychaudhuri Equation, Trapped Surfaces, and Penrose Singularity Theorem ()
1. Introduction
The Penrose singularity theorem (P.S.T.) of 1965 [1] is one of the most important steps in the development of General Relativity theory since its inception by Einstein in 1915, just half a century before. Important for the conception of the theorem, though not cited by its author in its original formulation, was the Raychaudhuri equation (R.E.) [2] for the expansion function associated with the evolution of non-spacelike geodesic congruences (the null case being relevant for the P.S.T.) and its prediction of the existence of caustics or focal points: singularities of the congruences but not of the spacetime itself (points where the expansion diverges and the R.E. looses its validity).
The P.S.T. strongly involves global Lorentzian geometry and topology [3] [4] [5] [6] [7]. An extensive review of the theorem with all of its details can be found in [8]; with a more informative (i.e. less detailed) but equally conceptual version by the same author in [9]. The two crucial new concepts of the theorem are: the definition of spacetime singularity itself as geodesic incompleteness i.e. the impossibility of an affinely parametrized geodesic to extend beyond a finite value of the affine parameter, and therefore the abrupt disappearance of matter or radiation into the “nothing”; and the introduction of the concept of trapped surface: a co-dimension 2 (2-dimensional surface in the case of a 4-dimensional spacetime) spacelike submanifold (compact or not) on which the expansions of both ingoing and outgoing orthogonally emitted geodesic congruences have the same sign (negative for the case of gravitational collapse or positive for the case of matter or radiation creation (big bang)).
It is important to emphasize the well-known fact that all the “classical” singularity theorems, P.S.T., as well as the theorems in [10] and [11], are purely classical in the technical sense i.e. they exclude any reference to quantum mechanics, obviously due to the still missing complete theory of quantum gravity [12]. Some attempt to see a modification of the classical predictions with, however, opposite conclusions are, e.g., those in [13] and [14].
The purpose of the present review is to shortcircuit as much as possible the path from the formulation of the R.E. to the announcement and proof of the P.S.T. with the minimum necessary ingredients from causality theory and global Lorentzian geometry explicitly stated. Of great help in this path have been the already cited reference [8] and the textbook [15]. An extra ingredient is the resume of an attempt of the author [16] to define Feynman propagators associated with the “quantum” evolution of the expansion coefficient corresponding to null geodesic congruences in black holes; in particular in the simplest example: the Schwarzschild-Kruskal-Szekeres (S.K.S.) black hole.
Section 2 is devoted to the R.E.; Section 3 to concepts (definitions and propositions) of causality theory; and Section 4 to the proof of the P.S.T.
2. Null Geodesics Congruences. Raychaudhuri Equation
Let
be a 4-dimensional time-oriented globally hyperbolic spacetime; that is, a connected, Hausdorff, pseudo-Riemannian manifold M with metric
, in local coordinates
,
, given by
. Let
be an open subset in M. A congruence in
is a family (set) of curves in
such that through each point
passes one and only one curve of the family. As a consequence, curves of the congruence do not intersect each other; when this happens i.e. there exists a focal or conjugate point (see below), the congruence breaks down.
Consider a congruence of null geodesics
in
, each affinely parametrized with parameter
and tangent velocity vector field
. The geodesics in the congruence are labelled by a parameter s which allows to define the deviation or separation vector field
. The vector fields
and
satisfy the following set of equations:
(1)
(2)
(3)
and
(4)
where, for arbitrary vectors V and W,
,
, and D is the covariant derivative with respect to the Levi-Civita (L.C.) connection
associated with the metric
.
(1) is the equation that characterizes null geodesics: lightlike tangent vectors.
(2) is the affinely parametrized geodesic equation:
(5)
with
.
(3) expresses the fact that coordinates can be chosen such that the tangent and deviation vectors become orthogonal. In fact,
where in the 6th equality we used (4) and in the 7th equality we used the Leibniz rule and the metric character of the L.C. connection. Then
is constant along each geodesic. It is clear that coordinates can be chosen such that the constant is zero.
(4), the deviation vector equation, which states that the parallel transport of the deviation vector
in the direction of the tangent vector k equals the parallel transport of k in the direction of
, amounts to the statement that the deviation vector is Lie transported along the geodesic, and viceversa, the geodesic velocity is Lie transported along the deviation vector. In fact, from the definition of the Lie derivative and the symmetry of the connection,
, and
, and using
,
, while
i.e. the first terms of the r.h.s.’s of
and
are equal; the same holds for the second terms, so
; but
, so
(6)
and then
. (In terms of
and
, Equation (4) is the identity
with
.)
Geodesic deviation equation
(4) can be used to derive the geodesic deviation equation
(7)
where
with
and so
; and
(8)
is the curvature tensor. (7) is a 2nd. order linear ordinary differential Jacobi equation for the deviation vector
(Jacobi field).
Proof of (7): Define the covariant gradient of the velocity field,
(9)
It is easy to proof that it is orthogonal to
:
(10)
Then
(11)
and
(12)
so
(13)
i.e.
is orthogonal to
. (This is consistent with the constancy of
along each geodesic, which implies
, and so
.)
Then
which implies
(14)
i.e.
is also orthogonal to
.
Let us compute the r.h.s. in the expresion for
:
,
where we used the geodesic Equation (2). Then
which is (7) after using the tensor identity
(15)
valid for an arbitrary vector field
.
If
(16)
is the relative velocity between nearby geodesics, then
(17)
is the relative acceleration which, according to the geodesic deviation Equation (7) is given by
(18)
is known as the tidal acceleration between geodesics. According to (18) it is proportional to the curvature.
Since
,
can have a component along
. In fact:
implies
, and the orthogonality of
with
subsists. To isolate
from
we construct the transverse metric, that is, the part of the metric
orthogonal to
. With this aim, we take a null vector field
in a direction such that
. I.e.
(19)
and define the (symmetric) tensor
(20)
The computation of its trace,
, shows that
is the metric of a 2-dimensional surface
to which n and k are orthogonal:
(21)
(22)
The mixed tensor
(23)
acts as a projector operator on
since
(24)
In terms of
, Equation (4) is
(25)
So,
measures the obstruction for the deviation vector
to be parallel transported along the geodesic. Also,
is orthogonal to the velocity vector
i.e. to the geodesics:
, while
since
, but not orthogonal to
since
and
,
are
in general. I.e.
(26)
As a consequence of (26),
and by (25)
have non-vanishing components along n i.e.
is not contained in
. This non-transverse part of
will be later eliminated projecting
with
on this 2-surface.
The transverse part of
is given by
(27)
with covariant derivative along
(28)
where we have used (3), (26), and the covariant derivative of the transverse metric:
(29)
So
also has a non-transverse part given by
. Its projection on
is
(30)
where
is the transverse part of
:
(31)
(30) governs the purely tranverse behavior of the null geodesic congruence.
From (23) and (26),
(32)
In terms of B, n, and k,
is given by
(33)
can be expanded in a part containing its trace
(1 component), its symmetric traceless part
(9 components), and its antisymmetric part
(6 components):
(34)
with
(35)
respectively called the shear tensor, which measures the distortion in shape without change in volume, and the rotation tensor, which measures rotation without change in shape and volume. Also,
, i.e.
(36)
called the expansion scalar, which in what follows will be a fundamental quantity in the theory.
Using (32) and (26), one can easily show that
i.e.
(37)
so that the expansion is nothing but the covariant divergence of the geodesic velocity at each of its points. We also see that
does not depend on the arbitrary choice of the null vector
. From the transversality of
and
, we also have the transversality of
,
, and
:
(38)
The transverse character of
and
implies that
(39)
Raychaudhuri equation for the expansion scalar
From (15), contracting
with
,
where
is the Ricci tensor associated with the curvature tensor
. Multiplying by
one has
; from Leibniz rule
, and choosing
affinely parametrized according to (2), one obtains
(40)
which is the Raychaudhuri equation for the expansion
: 1st. order non-linear differential equation (Riccati equation).
Frobenius theorem
A hypersurface S in
is given by an equation of the form
(41)
where
is a scalar.
A normal vector field to S,
is given by the gradient
(42)
Theorem: A congruence of curves (timelike, spacelike, or null) in
is hypersurface orthogonal if and only if
(43)
where
is the tangent vector to the curve at each of its points. (In our case,
.)
Proof:
) The congruence is orthogonal to S if
i.e.
, where the scalar
is constant on S. Then
. Since
is a 1-form, then
and
from
. Since
is a scalar,
and so
. Then,
(44)
with
, is the equation obeyed by the tangent vectors
to a congruence of curves which are orthogonal to a family of hypersurfaces S’s. From (44) one obtains (43).
) Exercise.
Remarks: i) The fact that a congruence of curves is hypersurface orthogonal is determined only from the knowledge of the tangent vector field
. ii) Neither the normalization of
nor the geodesic equation were used in the proof of the theorem. Then it is valid for arbitrary curves (geodesics or non-geodesics, timelike, null, or spacelike). iii) By definition, any 4-vector
orthogonal to
i.e. such that
is tangent to the hypersurface S. Since
,
is also tangent to S i.e.
is both orthogonal and tangent to the hypersurface S. So, a hypersurface orthogonal null geodesic congruence is part of S. The geodesics are called the generators of S. Also, from (21),
is orthogonal to the 2-dimensional surface
.
Corollary: A hypersurface orthogonal congruence of null geodesics has no rotation, and viceversa. (The result is also valid for timelike geodesics.)
Proof:
)
(45)
) Let
; one can then prove that
for some function
(exercise).
So, the Raychaudhuri equation for the expansion scalar of a hypersurface orthogonal null geodesic congruence is
(46)
Geodesic focussing
If the null convergence condition (N.C.C.):
(47)
holds, which by Einstein’s equations
is related to the condition on the energy-momentum tensor
:
(48)
then
(49)
meaning that the expansion decreases during the congruence evolution: if at
,
(initially divergent congruence), then the congruence will diverge less rapidly in the future (
); if at
,
(initially convergent congruence), then the congruence will converge more rapidly in the future (
). So, for hypersurface orthogonal null geodesic congruences, if the N.C.C. holds, gravity is atractive, i.e. geodesics are focussed (focussing theorem).
For vacuum solutions,
, and then
(50)
So, from (39),
(51)
with equality if and only if the shear vanishes.
The integration of (51) is
(52)
where
is the expansion at the hypersurface S (we choosed
). If
,
and so
and the congruence breaks down since it converges to a point p: focal point or conjugate point to S ( [8] [17]; see also M. Basquens, Singularity theorems, Universitat Politécnica de Catalunya, 2016) when
. (
if and only if
and/or
do not vanish.) This focussing process is ilustrated in Figure 1.
It should be stressed that p is not a singularity of the spacetime, but a singularity of the congruence, i.e. a point where the Raychaudhuri equation loses its validity.
Interpretation of Θ
If we call
the parameter of the curves with tangents
(the null vector field defined by (19)), each constant value of
defines a hypersurface
in M. Let
(fixed
and
) be an infinitesimal (2-dimensional) surface element, contained in the intersection
of the hypersurfaces
and
. So, in particular,
. If
is the area of
, then
is the fractional rate of change of
along the geodesics, i.e.
(53)
Proof: Let
,
be coordinates on
. Together with
and
one has a 4-dimensional coordinate system
with
and tangent vectors to
,
,
,
. Then, the metric on
is
(54)
In fact, on
,
, i.e.
and so
. The infinitesimal area element is
with
. Let
be the inverse metric of
, i.e.
; then
. It is easy to see that the
’s vectors are Lie transported along
:
(55)
Figure 1. Focussing of hypersurface orthogonal null geodesics congruence.
with the necessary differentiability. Since with a symmetric connection (
), in the Lie derivative
can be replaced by D, one has
(56)
i.e.
. From (54) and (56) one can prove that
(57)
Then
(58)
where we used
(59)
which can be proved contracting
with
, using (20), and the fact that
. This of course implies that
.
Along the geodesics,
are constant, and so the change in
comes only from the change in
; so
(60)
(for non-singular matrices N,
. Together with (58), we obtain (53).
Feynman propagators for null geodesic congruences
In terms of the function
defined by ( [18] [19])
(61)
the Raychaudhuri Equation (40) becomes
(62)
with
(63)
which is nothing but the equation of a classical 1-dimensional harmonic oscillator with
(“time”)-dependent frequency
. After Hill ( [20]), (62) is known as a “Hill-type” equation. If at
the congruence has a focal point, i.e.
as
, then
must be a zero of
if
is finite.
(62) is the Euler-Lagrange equation of the
-dependent Lagrangian
(64)
For a suitable domain of definition of
, (62) admits a solution
subject to the boundary conditions
and
with, e.g.
. (Notice that if
, then
since
and
.)
It is well known ( [21] [22]) that a classical Lagrangian of the form
(65)
has associated with it a Feynman propagator
given by the path integral
(66)
(
) where, formally,
(67)
The result is
(68)
with
and
solution of
(69)
with
,
, and
solution of
(70)
with
and
.
Since (64) and (65) (and therefore (62) and (69)) have the same form, the path integral
(71)
with
(72)
and
solution of (70) with t’s replaced by
‘s, can be formally considered the Feynman propagator describing the “quantum” flow of the geodesic congruence from
to
or, equivalently, the quantity which includes all admisible “quantum” fluctuations of the expansion
from
(corresponding to
) to
(corresponding to
). The quotation marks in “quantum” is due to the still missing existence of a final theory of quantum gravity ( [12]).
As a concrete example ( [16]), we consider the evolution of the ingoing radial null geodesic congruences within the black hole (B.H.) region of the Schwarzschild-Kruskal-Szekeres spacetime, from the future (
) and past (
) horizons to the future singularity at
(
). In both cases the affine parameter is
( [15]). In this case, since
, the Raychaudhuri equation reduces to
, the focal points are at the singularity, and the propagators result
(73)
where M is the energy of the B.H.,
with
and
, i.e.
at
,
at
, and
. Even if
and therefore
, finite initial values for
guarantee finite values for the propagators, which suggests that the introduction of a quantum description should smooth or even disappear the singularities of the classical theory ( [13]) (See Figure 2).
3. Elements of Causality Theory
Let
be a 4-dimensional pseudo-Riemannian (Lorentzian) manifold.
1. A vector field X over M is causal if
is nonspacelike (temporal or null)
.
2. A curve
is causal if its tangent vector is nonspacelike at each of its points.
3. A Lorentzian manifold
is temporally orientable if it admits a smooth global temporal vector field T, locally
,
, called a temporal orientation. The triplet
is called a Lorentz oriented manifold.
defines the opposite temporal orientation. A non-zero causal vector field X is future (past) directed if
(
).
Figure 2. Future directed null ingoing geodesic propagators: (a):
, (b):
.
4. It can be shown that any compact without boundary time oriented spacetime contains closed timelike curves ( [6]). To avoid such causality violation situations, all spacetimes considered here are non-compact.
5. Two spacetimes
and
are isometric if there exists a diffeomorphism
such that
, where
denotes the pull-back operation.
Extendibility
A spacetime
is extendible if it is isometric to a proper subset of a spacetime
.
is called an extension of
. For example, the Schwarzschild-Kruskal-Szekeres (SKS) spacetime is an extension (the maximal extension) of the Schwarzschild spacetime
with the isometry generated by the inclusion
.
6. Let
be a future directed causal curve in M.
is a future final point of
if for any open neighborhood
of p there exists
such that
,
.
is future inextendible if it has no future final point. Example: Let
,
. It is clear that
is a future final point of
. So
is not future inextendible; that is,
is future extendible. Instead, let
,
.
has no future final point since
has been eliminated from the spacetime; so,
is future inextendible.
Changing “future” by “past” the previous definitions pass to past directed causal curves, and one has the analogous concepts of past final point and past inextendible causal curves.
A causal curve is inextendible if it is both future and past inextendible, i.e. it has neither a future nor a past final point. Geodesic completeness
7. A geodesic is complete if for it there exists an affine parameter
which extends from
to
.
A spacetime is geodesically complete if all its causal (timelike or null) inextendible geodesics are complete.
A spacetime is singular if: i) as a spacetime it is inextendible, and ii) it is geodesically incomplete. So, an inextendible spacetime is singular if it has at least one causal inextendible geodesic which does not admit an affine parameter
extending from
to
.
Examples: i)
is not singular. ii) SKS is singular since it has causal geodesics which die at
at a finite value of their affine parameter.
Domains of dependence. Cauchy surfaces. Cauchy horizons
8. Let
be a spacetime.
is a partial Cauchy surface if
is an hypersurface of M such that no pair of points
,
, can be joined by a causal curve in M (then in particular in
).
The future domain of dependence of
,
, is the set of points
such that any past inextendible causal curve
which passes through p intersects
. In particular
.
The past domain of dependence of
,
, is the set of points
such that any future inextendible causal curve
which passes through q intersects
. In particular
. So,
.
Since
is acausal (see below), the number of intersections of
and
with
is 1 i.e.
.
The domain of dependence of
,
, is the union of
and
i.e.
(74)
Example: Let
in
be given by the positive x-axis i.e.
.
is closed in
since
is open. Also,
is achronal (see below 15.) and spacelike i.e. acausal: no two of its points can be joined by a causal curve. Therefore
is a partial Cauchy surface
,
,
with
and
. Also,
; clearly,
. If instead,
, then
.
A partial Cauchy surface
in M is a Cauchy surface if
. In this case the spacetime is called globally hyperbolic.
If
is a partial Cauchy surface but not necessarily a Cauchy surface, the future (past) boundary of
,
is called a future (past) Cauchy horizon of
. In the previous example in
,
; one has
and
.
Chronological and causal conditions
9. A time oriented spacetime satisfies the chronological condition if it has no closed future (past) directed timelike curves; (in particular geodesics). Such spacetime is said to be chronological.
A time oriented spacetime satisfies the causality condition if it has no closed future (past) directed causal curves; in particular geodesics. Such a spacetime is said to be causal.
So, a causal spacetime is a chronological spacetime, but not necessarily the other way around.
10. A spacetime M is strongly causal if
and
(open neighborhood of p),
such that if
and
is a causal curve, then
. Since
is arbitrarily small, taking
, the unique causal curve
is the trivial one
So, in a strongly causal spacetime there are no nontrivial closed causal curves.
So, a strongly causal spacetime is a causal spacetime.
11. Proposition: A globally hyperbolic spacetime has no closed causal curves.
Proof: Let
with local coordinates
be a closed causal curve in the spacetime.
should be periodic i.e.
or, equivalently,
. So,
would be an inextendible causal curve intersecting the Cauchy surface
-many times, in contradiction with the definition 8.
It can be shown that global hyperbolicity implies strong causality ( [7]: p. 11) and so we have the chain of implications:
(75)
Chronological and causal futures and pasts; horisms
Let
.
The chronological future (past) of p is given by the set
(76)
One writes
.
The causal future (past) of p is given by the set
(77)
One writes
.
Clearly,
(78)
but not viceversa.
The set of future (past) horisms of p is given by
(79)
If N is a subset of M, the corresponding quantities are defined by:
(80)
For any
,
since the constant curve
for any
is null and so causal (a point).
12. It can be proved ( [5]):
i)
is open (rough argument: a sufficient small deformation of a timelike curve remains timelike)
ii)
is open (union of open sets is open)
iii)
iv)
. (
is not necessarily closed; if
then
is closed i.e.
)
v)
vi)
13. Theorem: If a spacetime M is strongly causal and
,
is compact, then M is globally hyperbolic.
Proof: [8]: p. 734.
14. A spacetime is stably causal if it exists a global function
such that
is timelike (i.e.
). (
is the vector field associated with the 1-form dt by g; locally
.) t is called a time function. A stably causal spacetime is time orientable. The preimage of
by t,
, is called a level set of t and is denoted by
. If the domain of dependence of each of the level sets is M i.e.
,
, then all level sets are homeomorphic to each other, each level set is a Cauchy surface, the spacetime is globally hyperbolic and, topologically,
for any
. So,
(81)
but not the other way around.
Proposition: A stably causal spacetime satisfies the chronological condition.
Proof: t increases in the direction of
, so t increases in the direction of any forward directed timelike curve. So, there is no closed forward directed timelike curve.
Theorem: A stably causal spacetime is strongly causal.
Proof: [7]: p. 11; [17]: p. 199.
One has the chain of implications:
(82)
Achronal and future sets, achronal boundaries, edges
15. An achronal set
is a subset of M such that no two of its points can be joined by a timelike curve. (Notice that if a set N is acausal, then it is achronal, but not the other way around.)
i) Proposition:
is achronal
.
Proof:
) Suppose that
;
and
, which implies that
such that
i.e.
a forward directed timelike curve
, which is a contradiction with the achronality of N.
) Let
;
: if
i.e.
such that
is a forward directed timelike curve.
(In words, if a subset of M is achronal, the subset and its chronological future and past are disjoint; and viceversa.)
ii) An achronal boundary is a subset of M of the form
(or
) for some subset
.
iii) Lemma: Let N be a subset of M. If
.
Proof: Let
;
and so
is an open neighborhood of p. Since
and
.
.
iv) Proposition: An achronal boundary is achronal.
Proof: Let
with
. Because of the Lemma,
.
, what is a contradiction since
is open. Then
and
is achronal.
v) It can also be shown that an achronal boundary
is a closed, continuous hypersurface of M.
vi) Let
be an achronal subset of M.
is an edge point of N if any
(open neighborhood of p) contains a timelike curve
from
to
that does not meet N. (
is the chronological future (past) of p within
.) We denote
(83)
It is clear that
. If
one says that N is edgeless. If
,
,
, which amounts to
, i.e. N is closed.
vii) Proposition: An achronal boundary is edgeless.
Proof: Let N be a subset of M;
is an associated boundary (we could choose
). Let
. Let
be any timelike curve from
to
i.e.
with
and
.
. Then
is edgeless.
viii) Let N be a subset of M. N is a future set if
i.e. N is enough big to contain its chronological future.
ix) Proposition: Let N be a subset of M with with
. If N is achronal
is not a future set.
Proof: N achronal
and N future set
.
, which is a contradiction.
x) The above proposition is equivalent to: If N is a future set
can not be achronal (unless
).
xi) Proposition: For any
,
is a future set, i.e.
.
Proof: [8]: p. 731:
.
The same occurs for causal sets: since
(same Ref.),
is a future set, i.e.
.
xii) Proposition:
,
is achronal.
Proof:
, and
is achronal.
Trapped surfaces
16. A future (past) trapped surface S in M is a 2-dimensional spacelike submanifold of M such that for both outgoing (+) and ingoing (−) forward directed null geodesic curves emitted orthogonally from S, the expansions
are both negative (positive) on S.
A closed trapped surface is a compact without boundary trapped surface.
17. Facts [8]:
i) If
a forward directed causal curve from p to q but
a forward directed timelike curve from p to q,
every forward directed causal curve joining p to q must be a null geodesic segment (prop. 2.13, p. 729).
ii) Given a causal curve
from p to q, (a): there is no neighborhood of
containing a timelike curve from p to q
(b):
is a null geodesic segment from p to q without any point conjugate to p between p and q (prop. 2.14, p. 729). In particular − (b)
− (a): If
is a null geodesic segment from p to q with a focal point conjugate to p between p and q,
open neighborhood of
which contains a timelike curve from p to q.
iii) Given a causal curve
from a spacelike surface S to q, (a): there is no neighborhood of
containing a timelike curve from S to q
(b):
is a null geodesic segment orthogonal to S to q without any point focal to S between S and q (prop. 2.14, p. 729). In particular, − (b)
− (a): If
is a null geodesic segment emanating orthogonally from S to q with a focal point to S between S and q,
open neighborhood of
which contains a timelike curve from S to q.
iv) If
and
,
lies on a future directed null geodesic segment from N.
18. Proposition: Let
be a closed trapped surface and assume that the N.C.C. holds, i.e.
,
null vectors
.
(a):
is compact, or (b): the spacetime is null geodesically incomplete to the future (past) ( [8]: prop. 4.1, p. 780). (The disjunction connector “or” is exclusive.)
Proof: (We prove the Prop. “to the future”; the proof “to the past” is analogous.) Let us assume that the spacetime is null geodesically complete to the future, i.e. we assume − (b). Since S is a future trapped surface,
the expansions
of both null geodesic congruences emanating orthogonally from S are negative on S. Since S is compact, the maximum and the minimum of
are atained on S. Let
be the maximum value of both
on S (
). Since the N.C.C. holds and the spacetime is null geodesically complete, then the Raychaudhuri equation implies that there is a focal point at a finite value of the affine parameter
. Let K be the subset of M containing all these null geodesics in both orthogonal congruences from S up to
included. By construction, K is compact and closed. Given that
(since if
lies in a forward directed null geodesic segment from S (by 17.iv)), to see that
is compact it is enough to show that
is closed (since any closed set in a compact set is compact). Let
be a sequence of points in
converging to p, i.e.
as
. By construction,
. By the closedness of K,
. So, it is enough to prove that
. If
, what
that some
which is imposible since
and
.
i.e.
is closed and
compact.
Note: We have proved that -(b)
(a). This amounts to -(a)
(b), i.e.
If
is not compact
the spacetime is null geodesically incomplete to the future.
19. Proposition: Let
and assume that the N.C.C. holds. If the expansion
of the forward directed null geodesic congruence emanating from p becomes <0 along any geodesic of the congruence,
(a):
is compact, or (b): the spacetime is null geodesically incomplete ( [8]: prop. 4.2, p. 780). (The disjunction connector “or” is exclusive.)
Proof: Assume that the spacetime is null geodesically complete, i.e. we assume −(b). If
at some point in each null geodesic from p and the N.C.C. holds, by geodesic completeness we know that there will be a conjugate point to p along each geodesic before or at the finite value
of the affine parameter.
, as in the previous Proposition, there exists a compact set K and, with analogous following steps, one arrives at the conclusion that
is compact.
Note: We proved that −(b)
(a). This is equivalent to −(a)
(b), i.e.
If
is not compact
the spacetime is null geodesically incomplete to the future.
20. A non-empty achronal subset N of M is called future (past) trapped if
is compact.
A proper achronal boundary is the boundary of a future set.
21. Proposition: Let N be a subset of M, and let
be closed, i.e.
.
: i)
; ii)
is boundaryless; iii)
is achronal; iv)
is a proper achronal boundary.
Proof:
i)
.
ii)
.
iii)
, and
is achronal.
iv)
is a future set.
4. Penrose Singularity Theorem
Theorem: Let M be a spacetime. Assume the N.C.C. holds. If there exists a non-compact Cauchy surface
and a closed trapped surface S, then the spacetime is null geodesically incomplete.
Notes: i) Recall that
,
. ii) The three conditions stated in the theorem are respectively known as the curvature condition, the causality condition, and the initial/boundary condition. iii) Since by 14.,
, M connected
connected.
Proof: i) Suppose the spacetime is null geodesically complete. Since S is a closed trapped surface (future or past) and the N.C.C. holds, then by Prop. 18,
is compact. By Prop. 15.iv and 15.xii,
is achronal; on the other hand, by Prop. 21.i,
, and by Prop. 21.iii,
is achronal. Moreover, from Prop. 21.iv,
is a proper achronal boundary.
ii) Consider a timelike geodesic congruence in the spacetime. Since
is a Cauchy surface i.e. the spacetime is globally hyperbolic, then every curve of the congruence intersects
exactly once, and since
is achronal, then every curve of the congruence intersects
at most once. We take the timelike geodesic congruence such that each of its curves intersects
exactly once.
iii) Define the map
which transports the points in
to
along the curves of the timelike geodesic congruence. Since f is continuous (because of the continuity of the congruence) and 1-1, then
is homeomorphic to its image T on
(
), then T is compact since
is compact. But
(or T), being a proper achronal boundary, is an embedded 3-dimensional submanifold (and therefore open and without boundary) of
( [8]: prop. 2.16, p. 732). Then
is an open subset of
. Since
is Hausdorff and, by Prop. 18,
is compact, then
must also be closed ( [23]: Thm. (3.6)). On the other hand, since
is connected, then
or
. But
because M is null geodesically complete; then
, what is a contradiction since
is non-compact. So,
can not be compact. The contradiction comes from the assumption that the spacetime is null geodesically complete. So, M is null geodesically incomplete.
Acknowledgements
The author thanks for hospitality to the Instituto de Astronomía y Física del Espacio (IAFE) of the Universidad de Buenos Aires and CONICET, Argentina, where part of this work was done; and to Oscar Brauer at the University of Leeds, U.K., for the drawing of Figure 1 and Figure 2.