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-Nordstroem 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.
General
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