Abstract
In this thesis, we study phenomena related to trapping of light in stationary spacetimes.
After a general introduction to Mathematical General Relativity, we first prove a
uniqueness result for “quasilocal photon spheres” and static horizons in asymptoti-
cally flat so-called pseudo-electrostatic systems. Our result implies that an asymptot-
ically Reissner–Nordström electrostatic system of arbitrary dimension which contains
a “subextremal” photon sphere is a Reissner–Nordström manifold.
We define pseudo-electrostatic systems as a generalization of electrostatic systems
by replacing one of the dimensionally reduced Einstein equations for electrovacuum
with an inequality for the scalar curvature. Furthermore, we define “quasilocal pho-
ton spheres” in (pseudo-)electrostatic systems by equalities relating their intrinsic and
extrinsic geometry; this notion generalizes (as we will show) photon spheres in electro-
static systems. We need to postulate a subextremality condition (an inequality relating
mean curvature and scalar curvature of quasilocal photon spheres) as an assumption
of our theorem, since the conclusion of the theorem does not hold, for example, for
superextremal (|q| > m) Reissner–Nordström manifolds; and these manifolds may con-
tain superextremal (in the sense of the quasilocal inequality) photon spheres.
The methods used in the proof of this theorem go back to the classical black hole
uniqueness proofs of Bunting and Masood-ul-Alam [6] and Ruback [55], which rely
on an application of the rigidity case of the positive mass theorem. These techniques
are combined with newer ideas developed by Cederbaum–Galloway [12] and Ceder-
baum [8], which allow (by gluing suitably constructed “necks” to the photon sphere
inner boundary of the (pseudo-)electrostatic system up to a static horizon) to reduce
the (quasilocal) photon sphere case to the static horizon case. Our theorem extends to
the much weaker case of pseudo-electrostatic systems due to the realization that the
equation for the Ricci tensor is not necessary for this type of proof but can be replaced
by an inequality for the scalar curvature. Generalizing from the higher-dimensional
vacuum case treated in [8] and the 3 + 1-dimensional electrovacuum case from [12]
(which are both covered by our result) to the higher-dimensional electrovacuum case
also requires a different strategy for the calculations that prove quasilocal properties of
photon spheres, a new choice of the “mass” and “charge” of the glued-in necks, the usage
of a different partial differential equation in the last step of the proof, and adjustment
of many calculations (for example to prove regularity statements) along the way.
In the second part of this thesis, we leave the static setting and investigate trapped
light in the Kerr spacetime. Studying the photon region in this paradigmatic example
of a stationary but not static spacetime is an important step towards a better understanding of the structure of trapped light in stationary spacetimes. We give a new
and (compared to [20]) more direct proof that the photon region in the Kerr spacetime
can be naturally understood as a submanifold of the phase space and has topology
SO(3) × R2. We first prove rigorously that the photons of constant trapped coordinate
radius, which are explicitly given in [64], are the only trapped photons in the Kerr
spacetime in the sense that they stay away both from the horizon and from spatial
infinity. We then proceed to describe the photon region as a zero set of a smooth map
from the Kerr phase space to R3, using the characterization of photons of constant
radius via their constants of motion from [64], and use the implicit function theorem
to show that this set is a submanifold of the phase space. Finally, we show that this
submanifold is R2 times a closed 3-dimensional manifold. By explicitly calculating the
first fundamental group of this 3-manifold (using the Seifert–van Kampen theorem) as
Z2, and by then appealing to the elliptization conjecture, we conclude that the manifold
in question is SO(3), so that the Kerr photon region in the phase space is topologically
SO(3) × R2.
