In this section and in Section 2.8 we use mathematical techniques
which are related to the Penrose-Hawking singularity theorems. For
background material, see Penrose [261], Hawking and
Ellis [154], O’Neill [247
], and
Wald [341].
Recall from Section 2.2 that the caustic of the past
light cone of is the set of all points where this
light cone is not an immersed submanifold. A point
is in the caustic if a generator
of the light cone intersects at
an infinitesimally
neighboring generator. In this situation
is said to be
conjugate to
along
. The caustic of the
past light cone of
is also called the “past lightlike
conjugate locus” of
.
The notion of conjugate points is related to the
extremizing properties of lightlike geodesics in the following way.
Let be a past-oriented lightlike geodesic with
. Assume that
is the
first conjugate point along this geodesic. This means that
is in the caustic of the past light cone of
and that
does not meet the caustic at
parameter values between 0 and
. Then a well-known theorem
says that all points
with
cannot be reached from
along a timelike curve arbitrarily close to
, and all points
with
can. For a proof we refer to Hawking and
Ellis [154
], Proposition 4.5.11
and Proposition 4.5.12. It might be helpful to consult
O’Neill [247], Chapter 10,
Proposition 48, in addition.
Here we have considered a past-oriented lightlike
geodesic because this is the situation with relevance to lensing.
Actually, Hawking and Ellis consider the time-reversed situation,
i.e., with future-oriented. Then the result can be
phrased in the following way. A material particle may catch up with
a light ray
after the latter has passed through a
conjugate point and, for particles staying close to
, this is impossible otherwise. The restriction to
particles staying close to
is essential. Particles
“taking a short cut” may very well catch up with a lightlike
geodesic even if the latter is free of conjugate points.
For a discussion of the extremizing property in
the global sense, not restricted to timelike curves close to , we need the notion of cut
points. The precise definition of cut points reads as
follows.
As ususal, let denote the
chronological past of
, i.e., the set of all
that can be reached from
along a
past-pointing timelike curve. In Minkowski spacetime, the boundary
of
is just the past light cone
of
united with
. In an arbitrary spacetime,
this is not true. A lightlike geodesic
that issues from
into the past is always confined to the closure of
, but it need not stay on the boundary. The last
point on
that is on the boundary is by definition [46] the
cut point of
. In other words, it is exactly the part of
beyond the cut point that can be reached from
along a timelike curve. The union of all cut points,
along any past-pointing lightlike geodesic
from
, is called the cut
locus of the past light cone (or the past lightlike cut
locus of
). For the light cone in Figure 24
this is the curve
(actually 2-dimensional) where the two sheets of the light cone
intersect. For the light cone in Figure 25
the cut locus is the
same set plus the swallow-tail point (actually 1-dimensional). For
a detailed discussion of cut points in manifolds with metrics of
Lorentzian signature, see [25
]. For positive
definite metrics, the notion of cut points dates back to
Poincaré [280
] and
Whitehead [350
].
For a generator of the past light
cone of
, the cut point of
does not exist in either of
the two following cases:
Case 2 occurs, e.g., if there is a closed
timelike curve through . More precisely,
Case 2 is
excluded if the past distinguishing
condition is satisfied at
, i.e., if for
the implication
(P1) If, along , the point
is conjugate to
, the cut point of
exists and it comes on or
before
.
(P2) Assume that a point
can be reached from
along two different lightlike geodesics
and
from
. Then the cut point
of
and of
exists and it comes on or
before
.
(P3) If the cut locus of a past
light cone is empty, this past light cone is an embedded
submanifold of .
For proofs see [268]; The proofs can
also be found in or easily deduced from [25
]. Statement (P1) says that
conjugate points and cut points are related by the easily
remembered rule “the cut point comes first”. Statement (P2) says that a
“cut” between two geodesics is indicated by the occurrence of a cut
point. However, it does not say that
exactly at the cut point a second geodesic is met. Such a stronger
statement, which truly justifies the name “cut point”, holds in
globally hyperbolic spacetimes (see Section 3.1). Statement (P3) implies that
the occurrence of transverse self-intersections of a light cone are
always indicated by cut points. Note, however, that transverse
self-intersections of the past light cone of
may occur inside
and, thus, far
away from the cut locus.
Statement (P1) implies that is an immersed submanifold everywhere except at the
cut locus and, of course, at the vertex
. It is known
(see [154
], Proposition 6.3.1)
that
is achronal (i.e., it is impossible to
connect any two of its points by a timelike curve) and thus a
3-dimensional Lipschitz topological submanifold. By a general
theorem of Rademacher (see [113], Theorem
3.6.1), this implies that
is
differentiable almost everywhere, i.e., that the cut locus has
measure zero in
. Note that this argument
does not necessarily imply that the cut locus is a “small” subset
of
. ChruĊciel and Galloway [57] have demonstrated, by
way of example, that an achronal subset
of a spacetime may
fail to be differentiable on a set that is dense in
. So our reasoning so far does not even exclude the
possibility that the cut locus is dense in an open subset of
. This possibility can be excluded in globally
hyperbolic spacetimes where the cut locus is always a closed subset
of
(see Section 3.1). In general, the cut locus
need not be closed as is exemplified by Figure 24
.
In Section 2.8 we investigate the relevance of
cut points (and conjugate points) for multiple imaging.