A BEEMIAN SAMPLER: 1966-2002

25

8. A Sampler of Reactions to Global Lorentzian Geometry

The first author has sometimes been tempted to sit down with a group of long-

time associates at a conference and ask them what aspects of the monograph Global

Lorentzian Geometry they found particularly helpful. However, he has never

done so. Nonetheless, to provide a lighter touch to the ending of his presentation

at the Beemfest, a sample of reactions was provided which have randomly reached

his attention over the years, and which prove that it is hard to predict what aspect

of a publication or book will be found to be of most interest by the readership.

The first thing which has amused us over the years is that around 1981, a

reviewer of the First Edition commented very enthusiastically on what a wonderful

proof had been given in Proposition 2.6 (which became Proposition 3.10 in the

Second Edition) of the basic fact that a compact space-time contains a closed

timelike curve and hence fails to be chronological. The first author, of course,

enjoyed the proof that Beem drafted for this result, using an open cover of future

chronological sets, but could hardly have imagined that such a standard type of

topological proof would have been greeted with such enthusiasm.

A second thing which caused us to smile was that a rather routine lemma on

totally vicious space-times (Lemma 3.2 (b) in the First Edition, which became

improved Lemma 4.2 in the Second Edition) inspired publications on two

different

continents offering improvements-first,

T.

Ikawa and

H.

Nakagawa

[66]

in Japan,

and then

B.

Wegner

[91]

in Germany, commenting on their earlier paper. This

Lemma was chosen for the text solely because it provided an example of a causality

condition which had been given by the General Relativists themselves, for which

the Lorentzian. distance function always took on infinite values. Hence, it was

selected to highlight that this possibility did indeed occur for things which had

been formulated by the physicists.

At around that time, this same Professor

B.

Wegner at TU Berlin had several

Ph.D. students who wrote interesting dissertations on the differential geometry of

space-times. In due course of time, we received copies of their works, and the first

author was flattered yet chagrined after receiving one of these dissertations to read

in it that if Beem and Ehrlich had studied two--dimensional space-times in Section

2.4 of the First Edition, then these things were legitimate objects of study. (This

was before physicists began writing about 1

+

1, 2

+

1, and 3

+

1 space-times as

a common notation, so one supposes that this researcher felt the need to justify

working in 2 dimensions rather than 4.)

For Figure 2.3 in the First Edition (which became Figure 3.3 in the Second

Edition), as decided non--experts in General Relativity, we just selected from the

many causality conditions considered in relativity those which seemed most appro-

priate and would fit on the table. We started with global hyperbolicity and ended

with chronology. (The first author was not keen on figures anyway, but Professor

Beem had told him that physicists enjoy figures, so he gave Beem his blessing to

produce some for the First Edition; Beem drew them by hand with Indian ink for

the First Edition, while the typists typed the accompanying captions.) We termed

this figure ourselves the "hierarchy" of causality conditions. When the first author

relocated to Florida and several years later began the study of the geometric prop-

erties of gravitational plane waves with Professor G. Emch, he found that Emch

was referring to these conditions, which we haphazardly selected, as "the ladder

of causality," and Emch told the first author that he wished to determine

exactly