(H1) The past light cone of any
event , together with the vertex
, is closed in
.
(H2) The cut locus of the past
light cone of is closed in
.
(H3) Let be in the cut locus
of the past light cone of
but not in the conjugate
locus (= caustic). Then
can be reached from
along two different lightlike geodesics. The past
light cone of
has a transverse self-intersection at
.
(H4) The past light cone of is an embedded submanifold if and only if its cut
locus is empty.
Analogous results hold, of course, for the
future light cone, but the past version is the one that has
relevance for lensing. For proofs of these statements
see [25],
Propositions 9.35 and 9.29 and Theorem 9.15,
and [268
],
Propositions 13, 14, and 15. According to
Statement (H3), a “cut point” indicates a “cut” of
two lightlike geodesics. For geodesics in Riemannian manifolds
(i.e., in the positive definite case), an analogous statement holds
if the Riemannian metric is complete and is known as Poincaré theorem [280, 350]. It was
this theorem that motivated the name “cut point”. Note that
Statement (H3) is not true without the assumption
that
is not in the caustic. This is exemplified by the
swallow-tail point in Figure 25
. However, as points
in the caustic of the past light cone of
can be reached from
along two “infinitesimally close” lightlike
geodesics, the name “cut point” may be considered as justified also
in this case.
In addition to Statemens (H1) and (H2) one would
like to know whether in globally hyperbolic spactimes the caustic
of the past light cone of (also known as the past
lightlike conjugate locus of
) is closed. This question is
closely related to the question of whether in a complete Riemannian
manifold the conjugate locus of a point is closed. For both
questions, the answer was widely believed to be ‘yes’ although
actually it is ‘no’. To the surprise of many, Margerin [215]
constructed Riemannian metrics on the 2-sphere such that the
conjugate locus of a point is not closed. Taking the product of
such a Riemannian manifold with 2-dimensional Minkowski space gives
a globally hyperbolic spacetime in which the caustic of the past
light cone of an event is not closed.
In Section 2.8 we gave criteria for the number
of past-oriented lightlike geodesics from a point (observation event) to a timelike curve
(worldline of a light source) in an arbitrary
spacetime. With Statements (H1), (H2), (H3), and (H4) at hand,
the following stronger criteria can be given.
Let be globally hyperbolic, fix
a point
and an inextendible timelike curve
in
. Then the following is true:
(H5) Assume that enters into the chronological past
of
. Then there is a past-oriented
lightlike geodesic
from
to
that is completely contained in the boundary of
. This geodesic does not pass through a cut point or
through a conjugate point before arriving at
.
(H6) Assume that can be reached from
along a
past-oriented lightlike geodesic that passes through a conjugate
point or through a cut point before arriving at
. Then
can be reached from
along a second past-oriented lightlike geodesic.
Statement (H5) was proven in [327] with the help of
Morse theory. For a more elementary proof see [268
],
Proposition 16. Statement (H5) gives a characterization of the
primary image in globally hyperbolic
spacetimes. (The primary image is the one that shows the light
source at an older age than all other images.) The condition of
entering into the chronological past of
is necessary to exclude the case that
sees no image of
. Statement (H5) implies
that there is a unique primary image unless
passes through the cut locus of the past light cone
of
. The primary image has even parity. If the weak
energy condition is satisfied, the focusing theorem implies that
the primary image has magnification factor
, i.e., that it appears brighter than a source of the
same luminosity at the same affine distance and at the same
redshift in Minkowski spacetime (recall Sections 2.4 and 2.6, in particular
Equation (46
)).
For a proof of Statement (H6)
see [268],
Proposition 17. Statement (H6) says that in a globally hyperbolic
spacetime the occurrence of cut points is necessary and sufficient
for multiple imaging, and so is the occurrence of conjugate points.