The fame of a physicist

Prof. Stephen Hawking, the famous physicist, died this month.  But why was he famous?

Our astronomer writes:

Stephen Hawking is famous.  His name is known among many people who had no background in physics, much less the high-level and difficult stuff he worked with.  Part of it, no doubt, came from his popular science writings; much of it, from his being confined to a wheelchair and speaking only through a voice synthesizer.  The picture of a man overcoming enormous difficulties to do great things, even if those great things are only vaguely known, is inspirational.  But his standing among scientists is high for other reasons.  I will try to explain some of them; it’s only fair to the memory of a great man.

Physics nowadays is done almost entirely through differential equations.  Strictly speaking, these say how things change only in a tiny region, a short time and a small space (there are rigorous mathematical definitions of what is meant by “short” and “small”).  Newton worked out how to predict the motions of the planets using differential equations.  This contrasts with Kepler’s ellipses, which are geometric paths calculated as wholes.  In a sense, Kepler saw the whole system at once, while Newton (mathematically) could only see a tiny part.  It turns out that sometimes differential equations can be solved to give global solutions, and Newton’s mathematics in a simple situation gives an ellipse.  These are the kind of problems given to students, though in practice they’re rare.  Most of the time one has to set up a computer calculation, whose answer is good only for the specific numbers you put in.

General Relativity, the successor to Newton’s gravity, is expressed in differential equations; ten of them, more complex than Newton dealt with.  It takes time and computing power to work out any solution (other than the few highly symmetric ones known).  So even decades after Einstein presented it, there remained important questions about what could or could not happen; very little of the possible universe had been explored.

Hawking’s first important work applied global methods to GR.  There are ways of saying what is possible without actually solving the equations; and he found that, as long as mass is always positive (in fact, the condition is a bit weaker than that), singularities are inevitable.  They will always happen, though the techniques do not say where or when.  It’s as if one proved that, providing the fuel tank of an airplane is not infinite, it will eventually come to earth.  You need know nothing about the details of the engines, wings, aerodynamics, maneuvers, weather.  (You will need a lemma forbidding air-to-air refueling; but that’s pushing the analogy a bit too far.)

Why are singularities important?  Well, to start working out the solutions of differential equations, we need boundary conditions.  To determine the motion of a guitar string, we need to know where the ends are held still; to work out the motions of the planets, we need to know where they are and how they’re moving at one point in time.  A singularity is an infinity, a dividing-by-zero.  If it’s one of your boundary conditions, you can’t tell what will happen.  So the singularity theorems had the potential to cause the universe to be unpredictable in principle, something that had not happened since Ptolemy set up his epicycles thousands of years ago.  So Hawking’s first theorems threatened the foundations of mathematical physics.

He did much more, some of which I’ll touch on next week.