Spacetime geometry from graviton condensation

In this thesis we introduce a novel approach viewing spacetime
geometry as an emergent phenomenon based on the condensation of a
large number of quanta on a distinguished flat background. We
advertise this idea with regard to investigations of spacetime
singularities within a quantum field theoretical framework and
semiclassical considerations of black holes. Given that in any
physical theory apart from General Relativity the metric background
is determined in advance, singularities are only associated with
observables and can either be removed by renormalization techniques
or are otherwise regarded as unphysical. The appearance of
singularities in the spacetime structure itself, however, is
pathological. The prediction of said singularities in the sense of
geodesic incompleteness culminated in the famous singularity
theorems established by Hawking and Penrose. Though these theorems
are based on rather general assumptions we argue their physical
relevance. Using the example of a black hole we show that any
classical detector theory breaks down far before geodesic
incompleteness can set in. Apart from that, we point out that the
employment of point particles as diagnostic tools for spacetime
anomalies is an oversimplification that is no longer valid in high
curvature regimes. In view of these results the question arises to
what extent quantum objects are affected by spacetime
singularities. Based on the definition of geodesic incompleteness
customized for quantum mechanical test particles we collect ideas
for completeness concepts in dynamical spacetimes. As it turns out,
a further development of these ideas has shown that Schwarzschild
black holes, in particular, allow for a evolution of quantum probes
that is well-defined all over. This fact, however, must not
distract from such semiclassical considerations being accompanied
by many so far unresolved paradoxes. We are therefore compelled to
take steps towards a full quantum resolution of geometrical
backgrounds. First steps towards such a microscopic description are
made by means of a non-relativistic scalar toy model mimicking
properties of General Relativity. In particular, we model black
holes as quantum bound states of a large number N of soft quanta
subject to a strong collective potential. Operating at the verge of
a quantum phase transition perturbation theory naturally breaks
down and a numerical analysis of the model becomes inevitable.
Though indicating 1/N corrections as advertised in the underlying
so-called Quantum-N portrait relevant for a possible purification
of Hawking radiation and henceforth a resolution of the
long-standing information paradox we recognize that such a
non-relativistic model is simply not capable of capturing all
relevant requirements of a proper black hole treatment. We
therefore seek a relativistic framework mapping spacetime geometry
to large-N quantum bound states. Given a non-trivial vacuum
structure supporting graviton condensation this is achieved via
in-medium modifications that can be linked to a collective binding
potential. Viewing Minkowski spacetime as fundamental, the
classical notion of any other spacetime geometry is recovered in
the limit of an infinite constituent number of the corresponding
bound state living on Minkowski. This construction works in analogy
to the description of hadrons in quantum chromodynamics and, in
particular, also uses non-perturbative methods like the auxiliary
current description and the operator product expansion.
Concentrating on black holes we develop a bound state description
in accordance with the isometries of Schwarzschild spacetime.
Subsequently, expressions for the constituent number density and
the energy density are reviewed. With their help, it can be
concluded that the mass of a black hole at parton level is
proportional to its constituent number. Going beyond this level we
then consider the scattering of a massless scalar particle off a
black hole. Using previous results we can explicitly show that the
constituent distribution represents an observable and therefore
might ultimately be measured in experiments to confirm our
approach. We furthermore suggest how the formation of black holes
or Hawking radiation can be understood within this framework. After
all, the gained insights, capable of depriving their mysteries,
highlights the dubiety of treating black holes by means of
classical tools. Since our setup allows to view other,
non-black-hole geometries, as bound states as well, we point out
that our formalism could also shed light on the cosmological
constant problem by computing the vacuum energy in a de Sitter
state. In addition, we point our that even quantum chromodynamics,
and, in fact, any theory comprising bound states, can profit from
our formalism. Apart from this, we discuss an alternative proposal
describing classical solutions in terms of coherent states in the
limit of an infinite occupation number of so-called corpuscles.
Here, we will focus on the coherent state description of Anti-de
Sitter spacetime. Since most parts of this thesis are directed to
find a constituent description of black holes we will exclude this
corpuscular description from the main part and introduce it in the
appendix.