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Herewith the first three chapters of a
projected six-chapter book on general
relativity and cosmology. The only
prerequisites are calculus and that elusive
quality that is usually called ``mathematical
maturity'' but is better described as a
spirit of playfulness---a willingness to
tinker with ideas, take them apart and ask
what makes them work.
I have striven for the clarity that can come
only from mathematical precision. A one-form
is defined as a section of the cotangent
bundle, not by the transformation rules it
satisfies. The emphasis is on coordinate-
free and basis-free reasoning. There is no
debauch of indices because there are
no indices.
In selecting material for inclusion, my motto
has been: ``All the mathematics that is
necessary, and not a jot more''. Functors
play a central role in this book, because
general relativity is fraught with
natural equivalences, and a precise
description of natural equivalence requires
functors or their equivalent. But because
there is no need for the generalized
formalism of abstract category theory, the
functors in this book are defined only on the
concrete category of vector spaces.
This book is mostly self-contained. The only
facts I have quoted without proof are the
standard existence and uniqueness theorems from
the theory of differential equations and an
occasional lemma about extendibility of smooth
functions from closed sets to open neighborhoods.
Because this book is aimed at beginning
students, I have included a lot of detail
that will strike more sophisticated readers
as pedantic. This is particularly so when it
comes to defining the natural equivalences
that, as mentioned above, are at the heart of
the theory. It's important for students to
see that this level of precision is necessary
even when it is trivial.
The in-text exercises are mostly very easy. Their
purpose is not to challenge the student, but to
slow him down, insuring that he has understood
one definition before going on to the next.
The six projected chapters are:
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