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% Preface for the book in preparation:
% "Understanding Spacetime Splittings and Their Relationships"
% by Robert T Jantzen, Paolo Carini, and Donato Bini
\beginpreface
I was a sophomore in John Wheeler's modern physics course
at Princeton University in 1972
when the proofs for some chapters of {\it Gravitation\/} began
showing up. This book and its authors (Misner, Thorne and
Wheeler) have helped to shape generations of relativists
including myself and have done much to establish the use of more
modern mathematical notation and style as commonplace in the
field and in neighboring areas. In particular it is the one
textbook which presents the $3+1$ approach to general relativity
to a wide audience.
However, an alternative ``$1+3$" approach to the splitting of
spacetime into space plus time also existed and was most readily found
in the text {\it The Classical Theory of Fields} by Landau and
Lifshitz, but few versed in the $3+1$ school took the time to
understand this alternative in terms of the same beautiful
language that made the $3+1$ approach such a powerful tool in
gravitational physics. The culprit is of course quantum gravity
which was always lurking in the background as one of the prime
motivations for using the $3+1$ approach. In astrophysical
applications the superiority of either approach is not obvious
and it seems clear that both approaches together can reveal
complementary features of a physical problem, shedding more
insight than either one alone. Indeed the $1+3$ approach
underlies much of what is done in the post-Newtonian
approximation of general relativity, although it is not generally
recognized,
and provides the variable decomposition for the stationary
exact solution industry,
while in the cosmological context it appears that
gauge invariant perturbation theory for
Friedmann-Robertson-Walker cosmological models may even be
simpler in this approach.
Unfortunately the $1+3$ approach, which apparently flourished in
the fifties before the $3+1$ school took over, suffered a
near fatal public relations blackout in the intervening decades
preceding the nineties.
This has left many of us who have entered the field since the
seventies almost ``completely unaware" of much of this formalism
beyond recognition of it from limited exposure to it in {\it The
Classical Theory of Fields\/} or in the closely related
congruence approach systematized for applications in cosmology by
Ehlers and more widely known through the work of Hawking and
Ellis. When Paolo Carini as a student of Remo Ruffini in Rome
in 1989
got
me interested in trying to explain what was really underlying the
peculiar coordinate decomposition of the electromagnetic field
found in an exercise in the Landau-Lifshitz text and its
relationship to the $3+1$ and $1+3$ approaches, we were drawn
into an examination of the very foundations of the splitting
formalisms independent of electromagnetism. We were also
``completely unaware" of the work in the fifties by M\o ller,
Zel'manov, and Cattaneo on the $1+3$ point of view. Driven by our
ignorance of these matters and the lack of a clear discussion of
their mathematical structure %foundations
in the literature, we embarked on a study in which we learned a
great deal not only about the larger framework in which these
questions fit together, but also about the work that had been
done and effectively buried in the past. Later joined by Donato
Bini, we went on to examine some new aspects of the application
of splitting formalisms to questions in general relativity. It
seems not only valuable to share with the larger community what
we have understood in a notation that easily allows comparison of
the different approaches but also appropriate that some of the
historical roots of the subject be recalled and not forgotten.
Meanwhile during the decade of the nineties the $3+1$ approach
using the more modern point of view of congruences has enjoyed
increasing visibility through many articles by Ellis and
collaborators, and the need for understanding the relationships
among various approaches has only increased.
In this spirit the following monograph
has grown to its present form. Certainly it is not meant to be
exhaustive, especially where the $3+1$ approach is concerned
given the enormous amount of existing literature which deals with
that case alone. Rather it is meant to reveal more clearly the
relationships between the various approaches in an effort to
break down the artificial barriers which divide them. If this
attempt is only partially successful, our efforts will have been
well spent.
The International Center for Relativistic Astrophysics at the
University of Rome and its director Remo Ruffini are especially
thanked for suggesting and supporting this project.
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