A spacetime is called
asymptotically simple and empty if
there is a strongly causal spacetime
with the
following properties:
(S1) is an open
submanifold of
with a non-empty boundary
.
(S2) There is a smooth function
such that
,
,
everywhere on
and
on
.
(S3) Every inextendible lightlike
geodesic in has past and future end-point on
.
(S4) There is a neighborhood of
such that the Ricci tensor of
vanishes on
.
Asymptotically simple and empty spacetimes are mathematical models of transparent uncharged gravitating bodies that are isolated from all other gravitational sources. In view of lensing, the transparency condition (S3) is particularly important.
We now summarize some well-known facts about
asymptotically simple and empty spacetimes
(cf. again [154], p. 222,
and [117], Section 2.3). Every
asymptotically simple and empty spacetime is globally hyperbolic.
is a
-lightlike hypersurface of
. It has two connected components, denoted
and
. Each lightlike geodesic in
has past end-point on
and future
end-point on
. Geroch [134] gave a proof
that every Cauchy surface
of an asymptotically simple
and empty spacetime has topology
and that
has topology
. The original
proof, which is repeated in [154
], is incomplete. A
complete proof that
must be contractible and that
has topology
was given by
Newman and Clarke [238
] (cf. [237
]); the stronger
statement that
must have topology
needs the assumption that the Poincaré conjecture is
true (i.e., that every compact and simply connected 3-manifold is a
3-sphere). In [238
] the authors
believed that the Poincaré conjecture was proven, but the proof
they are refering to was actually based on an error. If the most
recent proof of the Poincaré conjecture by Perelman [263]
(cf. [346])
turns out to be correct, this settles the matter.
As is a lightlike hypersurface
in
, it is in particular a wave front in the sense of
Section 2.2. The generators of
are the integral curves of the gradient of
. The generators of
can be interpreted
as the “worldlines” of light sources at infinity that send light
into
. The generators of
can be interpreted
as the “worldlines” of observers at infinity that receive light
from
. This interpretation is justified by the observation
that each generator of
is the limit curve for a
sequence of timelike curves in
.
For an observation event inside
and light sources at
infinity, lensing can be investigated in terms of the exact lens
map (recall Section 2.1), with the role of the source
surface
played by
. (For the mathematical
properties of the lens map it is rather irrelevant whether the
source surface is timelike, lightlike or even spacelike. What
matters is that the arriving light rays meet the source surface
transversely.) In this case the lens map is a map
, namely from the celestial sphere of the observer to
the set of all generators of
. One can construct it in two
steps: First determine the intersection of the past light cone of
with
, then project along the
generators. The intersections of light cones with
(“light cone cuts of null infinity”) have been
studied in [189, 188
].
One can assign a mapping degree (= Brouwer degree
= winding number) to the lens map and prove
that it must be
[270
]. (The proof is
based on ideas of [238, 237]. Earlier
proofs of similar statements - [188], Lemma 1,
and [268], Theorem
6 - are incorrect, as outlined in [270
].) Based on this
result, the following odd-number theorem can be proven for observer
and light source inside
[270]: Fix a
point
and a timelike curve
in an
asymptotically simple and empty spacetime
. Assume that the image of
is a closed subset of
and that
meets neither the point
nor the caustic of
the past light cone of
. Then the number of
past-pointing lightlike geodesics from
to
in
is finite and odd. The same result can
be proven with the help of Morse theory (see Section 3.3).
We will now give an argument to the effect that
in an asymptotically simple and empty spacetime the non-occurrence
of multiple imaging is rather exceptional. The argument starts from
a standard result that is used in the Penrose-Hawking singularity
theorems. This standard result, given as Proposition 4.4.5
in [154], says that along a
lightlike geodesic that starts at a point there must be a point conjugate to
, provided that
The last assumption is certainly true in an asymptotically simple and empty spacetime because there all lightlike geodesics are complete. Hence, the generic condition and the weak energy condition guarantee that every past light cone must have a caustic point. We know from Section 3.1 that this implies multiple imaging for every observer. In other words, the only asymptotically simple and empty spacetimes in which multiple imaging does not occur are non-generic cases (like Minkowski spacetime) and cases where the gravitating bodies have negative energy.
The result that, under the aforementioned
conditions, light cones in an asymptotically simple and empty
spacetime must have caustic points is due to [165]. This paper
investigates the past light cones of points on and their caustics. These light cones are the
generalizations, to an arbitrary asymptotically simple and empty
spacetime, of the lightlike hyperplanes in Minkowski spacetime.
With their help, the eikonal equation (Hamilton-Jacobi equation)
in an asymptotically simple and empty
spacetime can be studied in analogy to Minkowski
spacetime [126, 125
]. In Minkowski
spacetime the lightlike hyperplanes are associated with a
two-parameter family of solutions to the eikonal equation. In the
terminology of classical mechanics such a family is called a complete integral. Knowing a complete integral allows
constructing all solutions to the Hamilton-Jacobi equation. In an
asymptotically simple and empty spacetime the past light cones of
points on
give us, again, a complete integral for
the eikonal equation, but now in a generalized sense, allowing for
caustics. These past light cones are wave fronts, in the sense of
Section 2.2, and cannot be represented as
surfaces
near caustic points. The way in which
all other wave fronts can be determined from knowledge of this
distinguished family of wave fronts is detailed in [125]. The
distinguished family of wave fronts gives a natural choice for the
space of trial maps in the Frittelli-Newman variational principle
which was discussed in Section 2.9.