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\title{Rewriting the exact solutions book (and how we handled the
  cosmological solutions)}
\author{Malcolm A.H. MacCallum\\
School of Mathematical Sciences, \\
Queen Mary, University of London,\\
Mile End Road, LONDON E1 4NS, U.K.}
\date{}
\maketitle

\begin{abstract}
The history of the exact solutions book is recounted and the decisions
made in its evolution discussed. In particular the decisions made for
the second edition, its contents and layout, are described. As an
example, fuller details are given of one of the chapters involving
solutions interpretable as cosmologies.\end{abstract}

\section{Introduction: the genesis of the book}

Talking about the writing process seems a little self-centred, to put
it mildly, but it took up most of my research time for a considerable
period so I feel a seminar on it is justifiable. In the following I
try to preserve the flavour of the talk and in particular I will not
give full references which can be found in the book itself
\cite{SKMHH}: they are listed here by authors and date.

My involvement with the topic started with some hand-written lists of
references collected by George Ellis, which I saw when I was one of his
research students, and which gave classified lists of exact solutions. I
started to collect references myself, on file cards, with the thought
that maybe one day, when I could not think of anything better to do, I
might write a review or book about them. This project seemed suitable
for the end phase of a career.

So I was very pleased when I received a letter from Jena asking for
information and offprints concerning exact solutions. Dietrich Kramer
and Hans Stephani had published a review in 1972 in German (Kramer,
Neugebauer and Stephani 1972), and, I understand, realised it had not
been widely read. They set out to do a fuller version in English, and
had written to a wide circle of workers in the field for information.
I happily offered a copy of my file cards and thought ``Hooray,
somebody else is taking on this thankless task. Good luck to them''.

In 1976 Stephani visited London as part of an ongoing series of
exchanges between Jena and Bill Bonnor's group, and we met, I think
for the first time\footnote{Hans, who sadly died on September 14th,
  2003, became one of my closest colleagues and a good friend.}. It
emerged that the Jena authors wanted a native English speaker to edit
the text, and after some thought I accepted their invitation, later,
of course, becoming a full co-author (I was the primary author of
chapters 8-12 of the original edition and I also assisted the
negotiation of the Western edition by representing the Cambridge
University Press at a meeting with the German publishers in Berlin).

The bibliography for the first edition was kept on file cards with
holes punched around the edges according to a classification scheme
which allowed cards relevant to a particular topic to be extracted
using the holes. The reprints themselves had been diligently collected
and filed with the help of graduate students paid for by the Jena
department. The text was handwritten and given to secretarial staff to
type. It was then exchanged between Jena and London, a process that
took about 6 weeks initially, presumably due to scrutiny by security
on both sides of the Iron Curtain, but speeded up to a week or so by
the end. It has been widely cited, though not always understood. (Some
of the time of the book's authors has been spent as referees correcting the
misapprehensions of authors of papers claiming to have found new
solutions in classes where all solutions are already known, and so
on.)

In the first edition a number of major decisions came from discussions
between the authors. The two main groups were
\begin{enumerate}
\item The overall layout. This was in 5 parts covering
\begin{description}
\item[Part 1] Mathematical background and classifications of solutions
\item[Part 2] Solutions classified by groups of motion
\item[Part 3] Algebraically-special solutions
\item[Part 4] Solutions obtained by other means (almost all solutions
    belonged under Parts 2 or 3)
\item[Part 5] Tables
\end{description}
\item Restrictions on the energy-momenta considered. Those allowed were
\begin{list}{$\bullet$}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} Vacuum
\item{} Electrovac
\item{} Perfect fluid
\item{} ``Pure radiation'' (also called null dust or sometimes null
  fluid).
\end{list}
In particular we excluded
\begin{list}{$\bullet$}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} Combinations of the above cases
\item{} Spinor fields (Dirac, Yang-Mills etc)
\item{} Scalar fields (except, implicitly, massless fields whose energy
  momenta were the same as ``stiff'' fluid)
\item{} Charged fluids.
\end{list}
\end{enumerate}

The latter restrictions could be regarded as laziness or inertia, but
were a good decision in restricting us to classes which could be
treated in a systematic way and to a manageable total amount of work
-- and it leaves something for others to do!

\section{A new edition?}

On the fall of the ``Iron Curtain'', the original publisher, Deutscher
Verlag der Wissenschaften, was sold to H\"uthig who were not
interested in publishing a new edition. As a protective measure Hans
Stephani wrote to the publishers asking about this, because the
contracts we had for the first edition said the rights would revert to
us if no positive response was received within six months, which duly
happened. We then found that the other original publishers, Cambridge
University Press, were keen to do a new version and that their
warehouse was running out of copies of the book (it went out of print
in the late 1990s).

So we met some time in 1991 or 1992 (I'm not sure exactly when) and
discussed the idea and agreed to do it.  We then began collecting
references. Having found that about 95\% of the references quoted in
the first edition had come from use of Physics Abstracts to locate
bibliography, we restricted ourselves to that source, using the FIZ
Karlsruhe database, and, later, the ArXiv preprint archive, as well as
anything that came to our attention serendipitously.

The first phase of the work was the bibliographical data collection,
the collection of copies or reprints, for which we again used graduate
student labour, and the formatting of the old references in the
electronic format of the other data. There were many references. In
the end we collected about 4000 new ones (since 1980), to add to the
3000 or so we had studied for the first edition. The cutoff for
systematic collection was 1999 but some later papers are cited: we
leave systematic treatment of the new millenium to our successors. Of
those 4000, Hans Stephani read about 2000 with help from Eddie Herlt,
I read about 900, Dietrich Kramer had about 600 to read and
the rest went to Cornelius Hoenselaers. Some were read by more than
one of us. We had to be brutal in rejecting papers which did not give
first discoveries or which failed to give relevant or complete
solutions, and in omitting methods or commentary of relatively low
utility in our view (which the original authors will not share!).

Some early decisions for the new edition were
\begin{list}{$\bullet$}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} the basic structure was retained
\item{} LaTeX and STN were to be the basic tools
\item{} we would retain the same limitations on energy-momentum, in
  particular
\begin{list}{}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} no massive scalar fields
\item{} no charged fluids
\item{} no mixtures
\item{} no hairy black holes
\end{list}
\item{} the generating techniques chapter would be rewritten completely
  (in fact, some old material was reused and the chapter split in two)
\end{list}

The following new chapters were needed
\begin{list}{[\arabic{enumi}]}
 {\usecounter{enumi}\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} Lie symmetries of differential equations
\item{} Homothetic motions
\item{} Inhomogeneous cosmologies
\item{} Colliding waves
\item{} Special subspaces (including intrinsic symmetry)
\item{} Equivalence problem
\item{} Other generating techniques
\item{} And a catch-all chapter
\end{list}

The decision that the new version would be produced in LaTeX
(some of us used Scientific Word to write it) implied the
production of a LaTeX version of the first edition.  (Unfortunately,
the idea of a Web version, which might have been a serious alternative
were we starting now, did not occur to us then.) Andreas Koutras did a
great deal of the work transforming the old edition into electronic
form and played a crucial role at this phase (during 1995-6): he also
checked many of the solutions in the original book. To
complement this we had to format the references in BibTex format, a
task which I took care of.

In 1996 I spent 3 months in Jena as a G\"astprofessor. During this
time we set up scripts to automate transfer from the FIZ format STN to
BibTeX and devised conventions for our LaTeX input in particular a
cross-referencing scheme which enabled us to label material without
conflicts arising (one part of this being that all the elements in the
first edition retained their original numbers as labels), and started
to use the publisher's macros. It was really only after this that the
bulk of the old material was recast into electronic form. At this
point the new work had largely consisted of rough classification of
new bibliographic material by topic (Stephani did most of this) and
some preliminary reading. From then on there was a great deal of hard
work in writing: for me personally it took most of my research time in
2000-2 as well as significant time before that.

After the rough classification allocating each reference to certain
chapters, we worked by reading and if necessary re-classifying the papers (for
which process we had to devise database maintenance and sharing
procedures, largely administered by Hans Stephani with help from me).
The draft chapters were kept, dated and signed, in a central
repository to which we could each upload as we updated.

Other relevant books appeared before or during our writing and these
somewhat affected our plans in that we were able to refer readers to
them for additional information, especially on interpretation and the
physical significance of the known solutions in some larger
picture. The most important of these were Griffiths' 1991 book on
colliding gravitational waves, Krasinski's 1997 book on inhomogeneous
cosmologies, a veritable tour-de-force which went wider than our book
in considering exotic energy-momentum tensors but was more restrictive
in that the solutions considered had to have a Robertson-Walker limit,
the book on dynamical systems in cosmology edited by Ellis and
Wainwright (1997), which focused mainly on the spatially homogeneous
and related models, Bicak's contribution to the Ehlers Festschrift
(2000), which was selective among solutions but went much further in
interpretation especially of radiative cases, and the book on
gravitational solitons by Belinskii and Verdaguer (2001).

An author was added, Cornelius Hoenselaers joining us to relieve the
pressure on Dietrich Kramer (and eventually the order of the authors
was altered to reflect the overall input of each to the new edition,
including work copied from the first edition).

\section{What became of the old chapters}

Every chapter has been thoroughly revised: in all the chapters
describing the actual solutions, many new solutions were added, and
some old but overlooked solutions and references too.

A few chapters were shortened though the whole book is much longer
(just over 700 pages compared with 400 odd: the cost per page at full
price is about twice the price of the 1980 edition, which is not bad
for inflation over that period!).

Not all solutions are now given explicitly, because the number of
solutions made this impossible. Some are now simply given as
references.

In Part 1, the background section, chapters 1-8 became -- chapters
1-8! However, some material was reordered, and material was added on
fibre bundles, junction conditions, integrability of tetrad methods,
the GHP formalism, homotheties and so on. Two new chapters were added
as described below.

In Part 2, on classification by groups, chapters 9-20 became chapters
11-22. Material was added on homotheties, solutions found by dynamical
systems methods, multipole moments, the factor structure in stationary
axisymmetric solutions, and the boost-rotation case, and the order of
the material on spherically symmetric solutions was changed with some
reordering between chapters. Again there were two new chapters (see
below), and chapter 21 became chapter 24. One problem arising in this
Part was where to allocate all the new work on solutions with a $G_2$
acting on $S_2$ which could be regarded as cylindrical, cosmological
or colliding wave solutions, and which also gave examples of
applications of generating techniques.

In Part 3, on algebraically special solutions, the chapters were
renumbered to run from 26-33 rather than 22-29 and many new solutions
were added.

Part 4, which dealt with other methods, was more extensively
reorganized than other parts. Chapter 30 on generating techniques, as
remarked above, was almost totally rewritten as chapter 34. In chapter
31 (now 35) on special vectors and tensors, collineations were added. Chapter
36 was introduced and chapter 32 became chapter 37.

Finally chapter 33, consisting of tables, was extended.

The one old/new chapter, now chapter 34, is on applications of
generating techniques and is organized with the following main parts
\begin{list}{$\bullet$}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} solutions found by harmonic maps
\item{} solutions found by the HKX method, arising from the
  prolongation for the Ernst equation
\item{} B\"acklund transformations
\item{} the Belinskii-Zakharov method.
\end{list}
Of these, the first and last gave especially large tables of
references to actual applications. The remaining methods are only
briefly described as there was not enough room to expound all them in
full. This is also the place where the bulk of the solutions found
(which in principle is a list as long as one wishes to make it by
applying the known methods) has to be given only in the sense of
references.

\section{What became of the new chapters?}

The new chapters mentioned above were given working numbers 40-46 and
49. I shall describe what happened to them in the order of their
working numbers.

\subsection{Chapter 40}
This chapter concerned the equivalence problem, i.e.\ the problem of
deciding whether two solutions are locally isometric. To describe the
method in full would be a separate lecture, so I will just note the
outline of the resulting chapter. It includes an (almost)
up-to-date review of work on scalar polynomials, limits of
differentiation required, and other research, old and newer, of mine
and of or with {\AA}man, Joly, Paiva, Skea and others.  Other features
are:
\begin{list}{$\bullet$}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item it includes sections from the first edition on scalar
  polynomial invariants and Petrov classification
\item the central section is on the Cartan method and the theory of
  classification algorithms
\item all references to computer algebra programs were omitted
  except acknowledgement to those used, CLASSI and REDUCE, and some references
  to reviews
\item it includes a section on limits of spacetimes.
\end{list}
This became chapter 9.

\subsection{Chapter 41}
This started as a chapter on Lie point symmetries. Extension to
generalized symmetries led on to the Kramer-Neugebauer and Geroch
approches to generating methods, but is also related to some methods
for perfect fluid solutions, Kerr-Schild, and other cases, in
principle. As the strength of the connection with the theory behind
the generating techniques for solutions admitting a $G_1$ or $G_2$
became more apparent, we decided to merge the draft with the relevant
part of the former chapter 30, as rewritten. The new draft then
included a discussion of prolongation (and isovectors, conservation
laws and pseudopotentials), which may now be the most mathematically
demanding part of the book, linearized methods (Kinnersley-Chitre and
the Geroch group), and in brief form, B\"acklund transformations,
Riemann-Hilbert methods, harmonic maps, variational B\"acklund
transformations and Hirota's method. We added other
generating methods (as in an expanded version of the old section
30.6), in particular conformal transformations, and the methods for
perfect fluids due to Racz and Zsigrai, and Krori and Nandy, the
method of Wainwright, Ince and Marshman which gives stiff fluid
solutions from vacua (this had been in chapter 15, with related ideas
of Belinskii and Letelier), and the methods of Ehlers and Buchdahl.
This became chapter 10, the last in Part I.

\subsection{Chapter 42}
This new chapter on special subspaces (mostly hypersurfaces) contains
a general introduction, and sections on solutions with flat slices
(as discussed by Stephani, Wolf and others), sections with conformally
flat slices, and other intrinsic symmetries. It became chapter 36.

\subsection{Chapter 43}
This was the catch-all chapter, but in the end all papers assigned to
this chapter because they did not apparently relate to any existing
topic were reassigned elsewhere.

\subsection{Chapter 44}
This was to be a  chapter on homotheties. It covered homothety groups --
the general theorems and their enumeration, the possible metric forms
with homothety (not yet solutions), and a survey of solutions. These
parts were in the end added to the former chapters 8 and 9 (now 8 and
11).

\subsection{Chapter 45}
This chapter was on inhomogeneous cosmologies. It was one of the
chapters I was responsible for and I describe below at greater length
how it developed as an example of the writing process. It became
chapter 23.

\subsection{Chapter 46}
This chapter on colliding waves involved a major writing effort since
there had been many papers and a book (Griffiths, 1991). It includes
\begin{list}{$\bullet$}{\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
\item{} an expanded version of the introduction from the old chapter 15
\item{} the relevant forms of the vacuum field equations
\item{} vacuum cases with collinear polarization
\item{} vacuum cases with non-collinear polarization
\item{} Einstein-Maxwell cases
\item{} stiff fluids and pure radiation
\end{list}
In the last section mentioned it was necessary to revisit a 1980s
controversy between Chandrasekhar and Xanthopoulos, Feinstein,
Senovilla and myself, and a number of other authors, about the
significance of the fact that the gravitational field equations do not
uniquely specifiy the energy-momentum in the interaction
region. Our wording here was chosen rather carefully! It became
chapter 25.

\subsection{Chapter 49}
This chapter was to be on generating methods other than those for
solutions with a $G_1$ or $G_2$ and included the old section 30.6. It
was incorporated in other chapters.

\section{The writing of the chapter on inhomogeneous cosmologies}
This was the last chapter to be written. The first cut was that it
should be ordered by group types in order, i.e.\ solutions with $H_3$,
$G_2$ (only), $H_2$, $G_1$ and $H_1$, where $H_n$ denotes a homothety
group and necessarily contains a $G_{n-1}$. After some discussion, it
was decided the chapter should only include perfect fluid solutions
and that the related vacua should appear elsewhere. Then, recognizing
that not all such fluid solutions can reasonably be considered
cosmological, the chapter was retitled ``Inhomogeneous perfect fluid
solutions with symmetry''.

It was decided the chapter should include the general forms of the
solutions obtained by the Wainwright, Ince and Marshman method which
generates solutions with stiff fluid; in the old edition only the
specializations with a $G_3$ had been given in full. Now the detailed
$H_3$ or $G_2$ on $S_3$ solutions are given.

Many papers giving stiff fluid solutions obtained from vacua or
obtainable in that way had appeared. The applications of the
Wainwright, Ince and Marshman method itself were listed in what became
chapter 10, while others which could have been so obtained were given,
but only as references, in chapter 23.

Quite a few solutions found are definitely unphysical e.g.\ they
violate energy conditions. These were ignored. Many more, almost all,
are non-barotropic fluids (i.e.\ one cannot write $p=p(\mu)$): these
were kept because we know so few inhomogeneous fluid models that even these
may be useful in cosmology.

There were classifications of possible solutions. Wainwright (1979)
gave a very general scheme but almost all the cases described were
unpopulated by actual solutions. In 1981 he gave a scheme for the more
restricted class with a commuting $G_2$ and orthogonal flow: this has
been widely used, its classes are populated (indeed it covers all
explicitly solutions known to belong in this chapter except Kramer's)
and it was used in the final writeup. A similar scheme for non-Abelian
$G_2$ was devised by Sintes but few such solutions are known.

Some papers considered contained, or appeared to contain, spatially
homogeneous solutions. These were read first because they were needed
for chapters 11 and 12 (new numbers).

In total there were about 165 papers to read for this chapter. About
20 of these were eliminated early on, mainly because of
misclassification or because the energy-momentum tensor did not belong
or there was no solution, only equations.

I knew that some important work had been done by Senovilla and his
students, by Mars in his thesis (of which I was an examiner) and
later, in Vera's thesis, by Carot and his colleagues and in Sintes'
thesis, and earlier by Koutras in finding solutions with a homothety
in the first edition whose symmetry had not been noted. I started with
these references. The solutions were checked, using CLASSI and REDUCE.
This led to a subdivision into those solutions with diagonal Abelian $G_2$
form, non-diagonal solutions with orthogonal transitivity, and
solutions without orthogonal transitivity. No section for the
non-Abelian case of $G_2$ was needed because all explicitly known
solutions with such a group have a $G_3$ on $V_2$.

Reading the references just picked out suggested that the main methods
for finding solutions had been: assuming separability of the metric
coefficients, whether in comoving or non-comoving frames; assuming
additional conformal symmetries; and, for the $H_3$ cases, dynamical
systems methods.

Putting these ideas together led to an outline as follows.

\setcounter{section}{23}
\subsection{$H_3$ on $S_3$}
Including the Wainwright, Ince and Marshman solutions
\subsection{$H_3$ on $T_3$}
Including solutions due to Uggla
\subsection{$G_2I$ on $S_2$}
\subsubsection{Diagonal cases}
Separable in comoving coordinates (Senovilla and Ruiz)\\
Partially separable (Mars, Roy and Prasad)\\
Separable in non-comoving coordinates (Vera)\\
Conformal motions (Carot et al, Mars and Wolf)
\subsubsection{Non-diagonal cases, orthogonally-transitive}
Including work of Mars and Senovilla
\subsubsection{Non-orthogonally-transitive cases}
Including work of Wils and of van den Bergh. All of these are stiff fluid
solutions but are not obtainable by the Wainwright et al method:
instead they are found by separability assumptions.
\subsection{Solutions with a maximal $H_2$ or smaller groups}
In the final event, this subsection was handed over to Hans Stephani
but there proved to be very little to say beyond references to special
cases treated elsewhere.
\vspace{10pt}

\setcounter{section}{5}
The next steps were to put the Senovilla and Ruiz classes into as
simple a form as possible (which was not trivial) and similarly the
Roy and Prasad solutions. Then I tackled the Vera, and Senovilla and
Vera, solutions: these have a subclassification based on the numbers
of independent functions of each of the two remaining variables. Only
a few of the possible cases have been completely integrated. Next
came a similar study of the Mars and Senovilla cases. Throughout this
phase it was important to keep a lookout for duplication of known
solutions (Allnutt's type D solution is one which recurs) and for
additional homothetic motions.

Some of the main problems are outlined in the following paragraphs.

Uggla's 1992 solutions with an $H_3$ all have a $G_2$. Where do they
fit among the $G_2$ solutions? This was not easy to work out due to
the very complicated form in which the solutions naturally arose from
Uggla's dynamical systems approach -- they have numerous interrelated
constant parameters. Eventually they turned out to be special cases of
the Senovilla-Ruiz class (which had been given in a simpler form)
together with one special subclass overlooked by Senovilla and Ruiz.
For example, the solution given in the first column of table 1 of
Uggla (1992) turns out to be a special case of the Senovilla and Ruiz
class labelled as $2m=1+n$. (In the talk I showed the details of these
solutions.)

The second set that were hard to check were the cases with conformal
motion. This is because the metric forms obtained had been derived
from the allowable coordinate forms for the conformal symmetry groups
under different assumptions hard to reconcile. In particular, how did
the assumptions of the two main papers intersect? It turned out that
two classes of solutions were duplicated and/or generalized one
another and/or differed by the interchange of sinh and cosh or similar
trivial changes. One class in the Carot et al paper is also in that of
Vera. An example given in the talk is that the solution of
equation (140) of Carot, Coley and Sintes (1996) is conformal to a
2+2 decomposable solution (like the ``Bertotti-Robinson'' metric, more
correctly ascribed to Levi-Civita): this implies it admits a $C_6$
rather than the $C_3$ assumed.

The final example of problems was the many other papers with special
cases (mostly older papers): here the effort was to identify them and
fit them into place in the main classes. Examples are given in the
chapter but a number more papers are not cited because they amount to
rediscovery of (usually, special cases of) already known metrics.

\section{Conclusion}
I hope that by describing the processes we followed to plan and write the
new edition of the book, I have given some insight into the process of
producing such a work, and into the things people entering this
field should consider before claiming to have found a new solution.

\begin{thebibliography}{99}
\bibitem{SKMHH} H. Stephani, D. Kramer. M.A.H. MacCallum, C.A. Hoenselaers
  and E. Herlt, {\em Exact solutions of Einstein's field equations}
  (Cambridge University Press, Cambridge, 2003).
\end{thebibliography}
\end{document}