Stephen Hawking was one of the most gifted and most famous scientists of the last fifty years. His science demonstrated a blend of technical ability and intuition. Hawking’s best-known results concern black holes. His earliest work was on singularities in general relativity, what became known as the Hawking-Penrose theorems. His discovery of Hawking radiation was a landmark result that fundamentally changed our understanding of black holes. Hawking had a remarkable life story, some of which was represented in the movie *The Theory of Everything*. Hawking had a playful spirit, which served him well and helped him to connect with the general public. It also endeared him to those who saw his appearances on television shows such as *Star Trek: The Next Generation*, *The Simpsons*, *Futurama*, *The Big Bang Theory, *and even *Last Week Tonight.*

MathSciNet has 148 publications for Stephen Hawking, which are cited by over 2000 different authors. Outside academia, Hawking’s most famous book is *A Brief History of Time*. Among physicists and mathematicians, his book with Ellis, *The Large Scale Structure of Space-time* is a true classic, and still a great place to learn the mathematics of general relativity.

Of his eight earliest papers in MathSciNet, six of them have “singularities” as part of the title. Thirty-five later papers have “black holes” in the title. Clearly this was a theme in his work. Hawking’s *Comm. Math. Phys.* paper on radiation of black holes is:

MR0381625

Hawking, S. W.

Particle creation by black holes.

Comm. Math. Phys. 43 (1975), no. 3, 199–220.

The heart of the paper is a rather serious calculation. But if you look at the paper, you see that it is quite dominated by words, not formulas. (The same is true of the shorter, earlier paper in *Nature*.)

As mentioned above, Hawking could be playful. He clearly enjoyed interacting with people. When I was a post-doc in Roger Penrose’s group at Oxford, some members of the group organized a conference at Durham University. Hawking was there for a couple of days. One night, he invited Penrose and some others out for drinks in a local pub. Hawking didn’t drink, but bought drinks for others. Most of the conversation was about the mathematics and physics that was being presented at the conference. But later, Hawking started posing problems — and was goading the more mathematical members of the group. I remember one problem in particular. Consider $x^{x^{x^\cdots}}$. For what positive values of $x$ does this expression converge? The “obvious” answer to a mathematician who has had a couple of beers is $0<x \le 1$. However, that is not quite right as $x$ can actually be a little larger than $1$. For some stupid reason, we were able to get up to $x=\sqrt{2}$. (It is stupid, because once you understand the problem, you shouldn’t be thinking of square roots.) Hawking prodded us, and plied us with more drinks. Eventually we realized that $e$ had to be involved, which led us to $0 < x \le e^{1/e}$, but we couldn’t prove it — even with more beers. The problem goes way back, having been considered by Euler. The earliest published paper on it that I know of is

MR1578416

Eisenstein, G.;

Entwicklung von $\alpha^{\alpha^{\alpha^\cdots}}$. (German)

J. Reine Angew. Math. 28 (1844), 49–52.

A more recent account is in

MR2091543

Anderson, Joel(1-PAS)

Iterated exponentials.

Amer. Math. Monthly 111 (2004), no. 8, 668–679.

Somehow, Hawking had this at his fingertips.

Several obituaries of Hawking have been published in high profile publications: The New York Times, BBC, and The Washington Post. Two that struck me are the obituary in The Guardian written by Roger Penrose, which is quite forthright, and the announcement from his research group at Cambridge University.

**MR0381625**

Hawking, S. W.

Particle creation by black holes.

*Comm. Math. Phys.* **43** (1975), no. 3, 199–220.

83.53

The author demonstrates that black holes are not completely black, but emit thermal radiation with a characteristic temperature of about $10^{-6}(M_\odot/M)^\circ K$ for a Schwarzschild object of mass $M$. This result had been conjectured previously on thermodynamic grounds.

The production of radiation by black holes is a quantum phenomenon caused by the disturbance of the vacuum state by the gravitational field of a collapsing massive object. The nature of the radiation is deduced by studying the effect of the collapse on the normal modes of a massless scalar field that is initially free to propagate through the centre of the collapsing matter and out again. An explicit calculation of the Bogoljubov transformation between the initial (undisturbed) and final states is given under the assumption that null rays are to be treated as in geometrical optics. Only particle states in the asymptotic region (where they are well defined) are discussed. The result is a Planck radiation spectrum.

Some conjecture is given about back-reaction on the metric from the particle production. This should cause the horizon area to shrink (in contrast to the classical case), possibly terminating in a naked singularity.

{For errata to the bibliographic data of the original MR item see E 52 9960 Errata and Addenda in the paper version. See MR0389129}

Reviewed by P. C. W. Davies

**MR0424186**

Hawking, S. W.; Ellis, G. F. R.

**The large scale structure of space-time**.

Cambridge Monographs on Mathematical Physics, No. 1. *Cambridge University Press, London-New York*, 1973. xi+391 pp.

83.58

Despite its imposing title, this book is a text on general relativity with a very mathematical orientation. It is an excellent introduction to the subject for a mathematician interested in relativity, because it is much more rigorous and uses a language much more familiar to the mathematician than that found in the usual texts. The thrust of the book is toward proving the “singularity theorems”, stating that large classes of solutions of Einstein’s equations with reasonable equations of state reach a singularity (of some sort) in a finite time. To deal with these theorems the authors introduce mathematical machinery for those not familiar with the tools they use, and discuss general relativity in this context, arriving at a description of several exact solutions of Einstein’s equations. This discussion takes up half of the book, and the remaining half is dedicated to the discussion of more specialized mathematical tools and to proving the singularity theorems for collapsing stars, and finally for the universe. The book ends with a short discussion of the meaning of singularities.

Reviewed by Michael P. Ryan Jr.