This book explores the role of singularities in general relativity (GR): The theory predicts
that when a sufficient large mass collapses no known force is able to stop it until all mass
is concentrated at a point. The question arises whether an acceptable physical theory should
have a singularity not even a coordinate singularity. The appearance of a singularity shows
the limitations of the theory. In GR this limitation is the strong gravitational force acting
near and at a super-massive concentration of a central mass. First a historical overview is
given on former attempts to extend GR (which includes Einstein himself) all with distinct
motivations. It will be shown that the only possible algebraic extension is to introduce
pseudo-complex (pc) coordinates otherwise for weak gravitational fields non-physical ghost
solutions appear. Thus the need to use pc-variables. We will see that the theory contains a
minimal length with important consequences. After that the pc-GR is formulated and compared
to the former attempts. A new variational principle is introduced which requires in the
Einstein equations an additional contribution. Alternatively the standard variational
principle can be applied but one has to introduce a constraint with the same former results.
The additional contribution will be associated to vacuum fluctuation whose dependence on the
radial distance can be approximately obtained using semi-classical Quantum Mechanics. The main
point is that pc-GR predicts that mass not only curves the space but also changes the vacuum
structure of the space itself. In the following chapters the minimal length will be set to
zero due to its smallness. Nevertheless the pc-GR will keep a remnant of the pc-description
namely that the appearance of a term which we may call dark energy is inevitable. The first
application will be discussed in chapter 3 namely solutions of central mass distributions. For
a non-rotating massive object it is the pc-Schwarzschild solution for a rotating massive
object the pc-Kerr solution and for a charged massive object it will be the Reissner-Nordström
solution. This chapter serves to become familiar on how to resolve problems in pc-GR and on how
to interpret the results. One of the main consequences is that we can eliminate the event
horizon and thus there will be no black holes. The huge massive objects in the center of nearly
any galaxy and the so-called galactic black holes are within pc-GR still there but with the
absence of an event horizon! Chapter 4 gives another application of the theory namely the
Robertson-Walker solution which we use to model different outcomes of the evolution of the
universe. Finally the capability of this theory to predict new phenomena is illustrated.
