The Strangest Man Page 7
For most purposes, the predictions of Einstein’s special theory were extremely similar to the corresponding ones made by Newton’s theory. The two sets of predictions, however, were noticeably different at speeds approaching the speed of light in a vacuum: Einstein claimed that, under these conditions, his theory was more accurate, though it would be several decades before the superiority was convincingly demonstrated by experimenters. In the meantime, Einstein’s reasoning made it possible to amend the description of anything given by Newton’s theory and produce a ‘relativistic’ version – one that agreed with the principles of the special theory of relativity. Two years later, Dirac took up a new hobby, aiming to produce relativistic versions of Newtonian theories – an activity he pursued like an engineer upgrading tried-and-tested designs to ones that perform to a higher specification: ‘There was a sort of general problem one could take, whenever one saw a bit of physics expressed in a non-relativistic form, to transcribe it to make it fit in with special relativity. It was rather like a game, which I indulged in at every opportunity.’43
Einstein’s second theory of relativity applied to all observers, including ones who are accelerating; for example, observers who fall freely under the action of gravity. In this ‘general theory of relativity’, Einstein proposed a geometric picture of gravity, replacing Newton’s concept that an apple and every other mass is subject to a force of gravity by a radically new way of describing the situation. According to Einstein, every mass exists in a curved space-time – roughly analogous to a curved sheet of rubber – and the motion of the mass at every point in space-time is determined by the curvature of space-time at that point. Because the theory is relativistic, information cannot be transmitted faster than light, and all energies contribute to mass (via E = mc2) and therefore to gravity. It turns out that, in the Solar System, where almost all matter has comparatively low density and travels much more slowly than light, the predictions of Einstein’s theory of gravity are in extremely good agreement with Newton’s. But, in some situations, they can be distinguished, and one of the most straightforward ways of doing so involved measuring the bending of starlight by its gravitational attraction to the Sun during a solar eclipse: Einstein’s theory predicted that this deflection would be twice Newton’s value. This was the prediction that Eddington and his colleagues believed they had verified in their solar-eclipse experiments.
It was during one of the early lectures in Broad’s course that Dirac had a revelation about the nature of space and time. Broad was talking about how to calculate the distance between two points. If they lie at the sharpest corners of a right-angled triangle, then every schoolchild knows that the distance between the points (the hypotenuse) is given by Pythagoras’s Theorem: the square of this distance is equal to the sum of the squares of the lengths of the other two sides. In the space-time of the special theory of relativity, things are different: the square of the distance between two points in space-time is equal to the sum of the squares of the spatial lengths minus the square of the time. Dirac later recalled ‘the tremendous impact’ on him of Broad’s writing down that minus sign.44 This dash of chalk on Broad’s blackboard told Dirac that his schoolboy ideas about space and time were wrong. He had assumed that the relationship between space and time could be described using the familiar Euclidean plane geometry, but if that had been true, every sign in the formula for the distance between two points would have been positive. Space and time must be related by a different kind of geometry. Pickering, Dirac’s mathematics teacher at the Merchant Venturers’ School, had already introduced him to the Riemannian geometry that Einstein had used to describe curved space-time. In this way of looking at space and time, the angles of a triangle may not add up to 180 degrees as they do in ordinary Euclidean space. In Einstein’s general theory of relativity, matter and energy are linked with the space and time in which they exist: matter and energy determine how much space-time is curved, and the curvature of space-time dictates how matter and energy move. Thus, Einstein offered a new explanation of why the apple in the tree in Newton’s garden fell: it was not the gravitational pull of the Earth that was responsible but the planet’s curvature of space-time in the region of the apple.45
Inspired by Broad’s lectures, and by Eddington’s semi-popular book Space, Time and Gravitation, Dirac soon taught himself the special and general theories, another early sign of his special talent as a theoretician. The mathematical complexities of Einstein’s general theory so terrified most physicists that they found excuses not to bother with it, whereas Dirac – an engineering undergraduate, not a registered student of physics – studied it voraciously. While other nineteen-year-olds were seeking beauty in the flesh, he sought it in equations.
*
Broad was sceptical of the contribution philosophy can make to advance the understanding of the natural world (he called it ‘aimless wandering in a circle’), but his lectures persuaded Dirac that the subject was worth pursuing. One text he took out of the library was John Stuart Mill’s A System of Logic, which the young Einstein had studied some fifteen years before.46 Mill had been the nineteenth century’s pre-eminent British philosopher, the most cogent voice of empiricism, the belief that human beings should ground every concept in verifiable experience.47 His approach to ethics was largely utilitarian, believing that the ultimate good is one that brings the most happiness to the greatest number of people and that the rightness of any human action should be judged according to its contribution to public happiness. Mill was influenced by other empiricists, notably by his friend Auguste Comte, the French pioneer of the positivist belief that all true knowledge is scientific, including knowledge about ‘sociology’, a word that Comte coined. Mill had no time for the Kantian ‘intuitionist’ view that some truths are so exalted that they transcend experience: he dismissed as meaningless many unverifiable statements made by bishops, politicians and others he regarded as airy-fairy moralists. Mill’s views and his feet-on-theground public spiritedness were enormously influential among Victorians and have become the essence of the liberal English consensus. He influenced Dirac, and many others, more than they knew.
A System of Logic, published in 1843, is a plain-spoken if laborious account of how empiricism can shape every aspect of human life.48 The book features Mill’s agenda for science, which assumes that there is an underlying ‘uniformity of nature’. The aim of scientists should be to explain more and more observations in terms of fewer and fewer laws, every one of them grounded in experience and induced from it. For Mill, the agreement between an experimental measurement and a corresponding theoretical prediction does not imply that the theory is correct, as there may well be many other theories that give equally good agreement. He argued that scientists have the never-ending task of finding theories that are in ever-better agreement with empirical observations.
In a memoir he wrote in his seventies, Dirac said he gave ‘a lot of thought’ to philosophy, trying to understand what it could contribute to physics. He recalled that he read A System of Logic ‘all through’, which we can safely interpret to mean that he read and pondered almost every word of it, his usual practice.49 Although he found it ‘pretty dull’, it introduced to him the important idea that the disparate scientific observations and theories he had learned about had an underlying unity. Furthermore, science should seek to describe this unity using the fewest possible laws of nature, each of them formulated in the simplest possible way. Although this probably influenced the thinking of the young Dirac, he concluded that philosophy was not an effective way of finding out what makes nature tick. Rather, as he put it in an interview in 1963, ‘it’s just a way of talking about discoveries which have already been made’.50
The best way of understanding nature’s regularities, he was coming to believe, was through mathematics. Dirac’s lecturers in the engineering classes had drummed into him that mathematical rigour is unimportant; mathematics is simply a tool to obtain useful answers that are correct or, at least, accurate e
nough for the purpose in hand. One exponent of this pragmatic approach to the mathematics of engineering was Oliver Heaviside, an acid-tongued recluse who had invented a battery of powerful techniques that made it easy to study the effects of passing pulses of electric current through electrical circuits. No one quite understood why these methods worked, but he didn’t care: what mattered to him was that they gave correct results, with a speed more rigorous methods could not match and without generating inconsistencies with other parts of mathematics. Engineers prized Heaviside’s methods for their usefulness, but mathematicians mocked them for their lack of rigour. Heaviside had no time for pedantry (‘Shall I refuse my dinner because I do not understand digestion?’51) and rejected the attacks of his detested opponents. He even entitled his autobiography after them: Wicked People I Have Known.52
Dirac studied Heaviside’s techniques and later remarked that there was ‘some sort of magic’ about them.53 Another of the engineers’ clever tricks that impressed Dirac concerned the calculation of the stresses exerted on materials; for example, by a gymnast balancing on a beam. Engineers routinely calculate these stresses using special diagrams that generate correct answers much more quickly than the mathematicians’ rigorous techniques. In his classes, Dirac used this method to represent stresses in this way and saw its power; within a few years, he would use similar techniques in a different context, to understand atoms.54
One of the lessons he learned in his engineering classes was the value of approximate theories. In order to describe how something works, it is essential to take into account the quantities that do most to affect its behaviour and to single out the quantities unimportant enough to be ignored. David Robertson taught Dirac a lesson he later regarded as crucial: even approximate theories can have mathematical beauty. So, when Dirac studied electrical circuits, the stresses on revolving shafts in engines and the windings of the rotors in electric dynamos, he was aware that the underlying theories had, like Einstein’s general theory of relativity, a mathematical beauty.
It was probably Dirac’s reflections on Einstein’s theory that first led him to believe that the goal of theoretical physicists should be to find equations that describe the natural world, but his studies of engineering were the source of a proviso: that the fundamental equations of Nature are only approximations.55 It was the job of scientists to find ever-better approximations to the truth, which always lies tantalisingly beyond their reach.
Apart from the embarrassing report Dirac had been given in Rugby, his record during his degree was almost flawless: only once in three years did he fail to top his class in every subject (the spoilsport was the assessor of a Strength of Materials course who ranked him second). 56 But it was clear that his real talents were in theoretical subjects and mathematics. Early in 1921, within a few months of completing the degree, his father suggested that he set his sights on studying at Cambridge.57 Early in February, Charles wrote to St John’s College, almost certainly acting on the advice of Ronald Hassé, head of Bristol University’s mathematics department and a member of Cambridge University’s network of talent-spotters. Hassé was a graduate and research student of the college, notable as the first person in Cambridge to speak of Einstein’s ‘theory of relativity’.58
Charles enquired whether the college would let him have details of ‘any open scholarship in mechanical science or mathematics’ that his son could apply for.59 The college responded swiftly and arranged for Dirac to make his trip to Cambridge in June 1921, to sit the college’s entrance examination.60 Dirac’s application to the college, made when he had just turned nineteen, is the earliest extant example of his adult handwriting. It shows that he wrote with the precision and clarity of a calligrapher, each letter standing upright with some of the capitals decorated unobtrusively with a tiny curlicue.61
Dirac passed the entrance examination handsomely, winning an annual scholarship of £70, which was disappointingly short of the minimum of £200 a year that he needed to live in Cambridge.62 Charles argued that it was ‘out of the question’ to give his son the additional money as he earned only £420 a year and had no other income, neglecting to mention his lucrative private tuition. Bristol council refused to help because Charles and Paul had become British citizens only two years before and were therefore ineligible for financial assistance. Disappointed, Charles later wrote to Cambridge asking to be kept informed if any other opportunities should arise for his son. He concluded, ‘I am sorry to trouble you, but I believe the boy has an exceptionable [sic] head for mathematics and I am trying to do my best for him.’63 When an official at St John’s College offered tactfully to advise him further if he would provide more information about his family’s finances, Charles did not reply.64
Although Paul’s Cambridge application had stalled, by July he had a first-class honours degree in engineering, a qualification that he and his father hoped would all but guarantee him employment. However, his graduation coincided with the worst depression in the UK since the industrial revolution: unemployment soared to two million. To every job application, Dirac drew a blank. Thus, the most talented graduate Bristol had ever produced found himself unemployed. But this turned out to be a stroke of luck.
Notes - Chapter three
1 Stone and Wells (1920: 371–2).
2 Bristol Times and Mirror, 12 November 1918, p. 3.
3 ‘Recollections of Bristol University’, Dirac Papers, 2/16/3 (FSU).
4 Lyes (n.d.: 29). At the Dolphin Street picture house, for example, Fatty Arbuckle starred in The Butcher Boy.
5 Quoted in Sinclair (1986).
6 Dirac Papers, 2/16/3 (FSU).
7 The list of textbooks that Dirac studied as an engineering student is in Dirac Papers, 1/10/13 and
1/12/1 (FSU).
8 BRISTU, papers of Charles Frank. ‘[N]ot the faintest idea’ is the testimony of Mr S.Holmes, a lecturer in electrical engineering, given to G. H. Rawcliffe, who, in turn, passed it to Charles Frank on 3 May 1973.
9 Papers of Sir Charles Frank, BRISTU. ‘Even as an engineering student, he spent much time reading in the Physics Library,’ wrote Frank in a note in 1973.
10 The college had classes on Saturday mornings as well as during weekdays (as was traditional, Wednesday afternoons were usually free for sporting activities). Information on Dirac at the Merchant Venturers’ College is in the college’s Year Books (BRISTRO 40659/1). Dirac’s student number was 1429.
11 Letter to Dirac from Wiltshire, 4 May 1952, Dirac Papers, 2/4/7 (FSU). The first two names of Wiltshire, known to most people as Charlie, were Herbert Charles.
12 Dirac Papers, 2/16/3 (FSU).
13 Dirac Papers, 2/16/3 (FSU).
14 Interview with Leslie Warne, 30 November 2004.
15 Records of the Merchant Venturers’ Technical College, BRISTRO.
16 The photograph shows the visit of the University Engineering Society’s visit to Messrs. Douglas’ Works, Kingswood, 11 March 1919, Dirac Papers, 1/10/13 (FSU).
17 ‘Miscellaneous collection, FH Dirac’, September 1915, Dirac Papers, 1/2/2 (FSU).
18 Testimony to Richard Dalitz by E. B. Cook, who
taught with Charles from 1918 to 1925.
19 Testimony to Richard Dalitz by W. H. Bullock, who joined the Cotham Road School staff in 1925 and was later Charles’s successor as Head of the French Department.
20 Charles Dirac’s letter is reproduced in Michelet (1988: 93).
21 See Charles Dirac’s Certificate of Naturalization, Dirac Papers, 1/1/3 (FSU). The papers concerning Charles Dirac’s application for British citizenship are in UKNATARCHI HO/144/1509/374920.
22 Interview with Mary Dirac, 21 February 2003.
23 Interview with Dirac, AHQP, 1 April 1962, p. 6.
24 Letter to Dirac from Wiltshire, 10 February 1952, Dirac Papers, 2/4/7 (FSU).
25 Dirac (1977: 110).
26 Sponsel (2002: 463).
27 Dirac (1977: 110).
28 Five shillings (25 pence) secured a copy of Easy
Lessons in Einstein by Dr E. L. Slosson, a guinea (£1.05) The Reign of Relativity by Viscount Haldane.
29 Eddington (1918: 35–9).
30 Dirac Papers, 1/10/14 (FSU).
31 Testimonies of Dr J. L. Griffin, Dr Leslie Roy Phillips and E. G. Armstead, provided to Richard Dalitz.
32 Letter to Dirac from his mother, undated but written at the beginning of his sojourn in Rugby, c. 1 August 1920, Dirac Papers, 1/3/1 (FSU).
33 Rugby and Kineton Advertiser, 20 August 1920.
34 Letters to Dirac from his mother, August and September 1920, especially 30 August and 15 September (FSU).
35 Interview with Dirac, AHQP, 1 April 1962, p. 7.
36 Letter from G. H. Rawcliffe, Professor of Electrical Engineering at Bristol to Professor Frank on 3 May 1973. BRISTU, archive of Charles Frank.
37 Broad (1923: 3).
38 Interview with Dirac, AHQP, 1 April 1962, p. 4 and 7.
39 Schilpp (1959: 54–5). I have replaced Broad’s archaic term ‘latches’ with ‘laces’.
40 Broad (1923: 154). This book is based on the course of lectures that Broad gave to Dirac and his colleagues. Broad prepared all his lectures meticulously and wrote them out in advance, making it easy for him to publish them. What we read in this book is therefore likely to be the material that Broad presented to Dirac.
41 Broad (1923: 486).
42 Broad (1923: 31).
43 Dirac (1977: 120).
44 Dirac (1977: 111).
45 Schultz (2003: Chapters 18 and 19).
46 Galison (2003: 238).
47 Skorupski (1988).
48 Mill (1892). His most important comments about the nature of science are in Book 2 and in Book 3 (Chapter 21).
49 Dirac (1977: 111).
50 Interview with Dirac, AHQP, 6 May 1963, p. 6.
51 See http://www.uh.edu/engines/epi426.htm (accessed 27 May 2008).