1.1. Introduction.

In this paper we investigate the fundamentally simple question of the extent to which the relativistic equations of gravitation determine the motion of ponderable bodies.

Previous attacks on this problem have been based upon gravitational equations in which some specific energy-momentum tensor for matter has been assumed. Such energy-momentum tensors, however, must be regarded as purely temporary and more or less phenomenological devices for representing the structure of matter, and their entry into the equations makes it impossible to determine how far the results obtained are independent of the particular assumption made concerning the constitution of matter.

Actually, the only equations of gravitation which follow without ambiguity from the fundamental assumptions of the general theory of relativity are the equations for empty space, and it is important to know whether they alone are capable of determining the motion of bodies. The answer to this question is not at all obvious. It is possible to find examples in classical physics leading to either answer, yes or no. For instance, in the ordinary Maxwell equations for empty space, in which electrical particles are regarded as point singularities of the field, the motion of these singularities is not determined by the linear field equations. On the other hand, the well-known theory of Helmholtz on the motion of vortices in a non-viscous fluid gives an instance where the motion of line singularities is actually determined by partial differential equations alone, which are there non-linear.

We shall show in this paper that the gravitational equations for empty space are in fact sufficient to determine the motion of matter represented as point singularities of the field. The gravitational equations are non-linear, and, because of the necessary freedom of choice of the coordinate system, are such that four differential relations exist between them so that they form an over-determined system of equations. The overdetermination is responsible for the existence of equations of motion, and the non-linear character for the existence of terms expressing the interaction of moving bodies.

Two essential steps lead to the determination of the motion.

(1) By means of a new method of approximation, specially suited to the treatment of quasi-stationary fields, the gravitational field due to moving particles is determined.

(2) It is shown that for two-dimensional spatial surfaces containing singularities certain surface integral conditions are valid which determine the motion.

In the second part of this paper we actually calculate the first two non-trivial stages of the approximation. In the first of these the equations of motion take the Newtonian form. In the second the equations of motion, which we calculate only for the case of two massive particles, take a more complicated form but do not involve third or higher derivatives with respect to the time.

The method is, in principle, applicable for any order of approximation, the problem reducing to specific integrations at each stage, but we have not proved that higher time derivatives than the second will not ultimately occur in the equations of motion.

In the determination of the field and the equations of motion non-Galilean values at infinity and singularities of the type of dipoles, quadrupoles, and higher poles, must be excluded from the field in order that the solution shall be unique.

It is of significance that our equations of motion do not restrict the motion of the singularities more strongly than the Newtonian equations, but this may be due to our simplifying assumption that matter is represented by singularities, and it is possible that it would not be the case if we could represent matter in terms of a field theory from which singularities were excluded. The representation of matter by means of singularities does not enable the field equations to fix the sign of mass so that, so far as the present theory is concerned, it is only by convention that the interaction between two bodies is always an attraction and not a repulsion. A possible clue as to why the mass must be positive can be expected only from a theory which gives a representation of matter free from singularities.

Our method can be applied to the case when the Maxwell energy-momentum tensor is included in the field equations and, as is shown in part II, it leads to a derivation of the Lorentz force.

In the Maxwell-Lorentz electrodynamics, as also in the earlier approximation method for the solution of the gravitational equations, the problem of determining the field due to moving bodies is solved through the integration of wave equations by retarded potentials. The sign of the flow of time there plays a decisive role since, in a certain sense, the field is expanded in terms of only those waves which proceed towards infinity. In our theory, however, the equations to be solved at each stage of the approximation are not wave equations but merely spatial potential equations. Since such equations as those of the gravitational and of the electromagnetic field are actually invariant under a reversal of the sign of time, it would seem that the method presented here, is the natural one for their solution. Our method, in which the time direction is not distinguished, corresponds to the introduction of standing waves in the wave equation and cannot lead to the conclusion that in the circular motion of two point masses energy is radiated to infinity in the form of waves.

2.1. Extract.

In an article appearing in the American Scientist recently, Edgerton and Britt have given an account of the procedure used for picking the scholarship winners in the "first annual science talent search."

Undoubtedly the procedure of Edgerton and Britt will result in the selection of very clever boys and girls. The validity of the test as a detecter of scientific talent, however, does not rest solely on the calibre of those it selects. The obverse of the picture is equally important. We must know something about the type of candidate the test rejects. We must know not only what the haul is but also what has slipped through the net.

Despite the absence of relevant statistics, and despite the contention that the real validity of the tests will not be known until an adequate follow-up study can be made, it is still possible to form a general idea of the type of candidate the test might be likely to favour and the type it might reject by an examination of the test itself. This is not an illogical, unscientific, "theological" procedure. After all, the test was originally devised by a process of reason without the aid of statistics regarding its own performance. It is, therefore, subject to criticism on a like basis.

3.1. Extract 1.

Albert Einstein, whose 70th birthday this month is being noted throughout the civilised world, occupies a position unique among scientists. He has become a legend in his own lifetime. The public adulation of him is so great that he dare not list his telephone number in the directory. When he delivers one of his rare lectures at the Institute for Advanced Study in Princeton, no notice of it may be posted on a bulletin board; the news must be passed around among his colleagues by word of mouth, lest it leak out and the lecture hall be overrun by reporters and curiosity seekers

3.2. Extract 2.

The importance of Einstein's scientific ideas does not reside merely in their great success. Equally powerful has been their psychological effect. At a crucial epoch in the history of science Einstein demonstrated that long-accepted ideas were not in any way sacred. And it was this more than anything else that freed the imaginations of men like Bohr and de Broglie and inspired their daring triumphs in the realm of the quantum. Wherever we look, the physics of the 20th century bears the indelible imprint of Einstein's genius.

4.1. Extract.

A little-known investigation by Sherlock Holmes reveals that the Bard anticipated
wireless, relativity and the atomic bomb. Here Sherlock Holmes explains to Watson Shakespeare's anticipation of relativity:

"Literary research has always attracted me," he continued with a mischievous twinkle in his eye. "For in no other field does the art of detection find such free rein. I am embodying the results of my recent investigations in a little monograph that may have important influence on Shakespearean scholarship. My discoveries suggest the need for a complete revaluation of Shakespeare's works. For they imply that his writings conceal, beneath a cloak of poetry and drama, unparalleled feats of clairvoyance and prognostication."

I was about to protest, but he stilled me with a gesture.

Holmes chuckled and went on: "My own recent discoveries, which are concerned mainly with the Sonnets, bring the strongest possible support to the thesis of the early researchers. Listen, for instance, to the beginning of Sonnet 12:

When I do count the clock that tells the time
And see the brave day sunk in hideous night ....

"It reads like poesy pure and simple. But I shall now point out to you unmistakable indications that these lines refer to Einstein's theory of relativity.

"My dear Holmes!"

"You think it unlikely? Did not Einstein arrive at his theory by rejecting the notion of absolute time? Did he not reason in terms of the behaviour of clocks moving relative to one another? Did he not, in the words of Shakespeare, count the clock that tells the time? In one small line Shakespeare has packed the essence of the theory of relativity."

"Surely it is rather farfetched," I murmured dubiously.

"Oh, yes. One line alone can easily be dismissed as a coincidence. But once we realise that the sonnet refers to the theory of relativity the significance of the second line is immediately clear. How was Einstein's theory tested? Was it not during a total eclipse of the sun, with the brave day sunk in hideous night? If this is still a coincidence it is by now an enormous one."

"Holmes," I laughed. "I am convinced."

5.1. Extract.

This article is about multiple-choice tests; it does not praise them. These tests are now used throughout the country to sift and appraise our youth. Many people are accepting test scores with the blind pseudoscientific faith that afflicts those who cannot see beyond imperfect statistics to the facts the statistics obscure. But how many of those who put their faith in multiple-choice tests have ever examined these tests critically or wondered whether they really do what they are assumed to be doing? How many have weighed the damage these tests do to young minds by rewarding superficiality and penalising depth and intellectual honesty? How many have considered the corrupting effects of these tests on the whole of our educational system?

I am told that a question somewhat like this appeared in a certain test:

Emperor is the name of

(A) a string quartet

(B) a piano concerto

(C) a violin sonata

This seems to be a simple, straightforward question. The average student quickly picks answer B and proceeds untroubled to the next question, perhaps feeling elated at being given so simple a test. But what of the superior student? He knows of the Emperor Concerto of Beethoven, but he also knows of the Emperor Quartet of Haydn; and his knowledge puts him at a disadvantage, for because of it he must pause to weigh the relative merits of A and B while his more fortunate, less well-informed competitors rush ahead.

In this particular case the superior student does not ponder long. Two theories occur to him: the examiner is malicious, or the examiner is ignorant of the Haydn work. If this is the first dubious item that he has encountered on the test, he inclines to the second alternative and chooses answer B with little delay.

Yet even in this simple case he suffers a penalty far exceeding the slight loss of time. For he has been led to call into question both the good will and the competence of the examiner; and this subjects him to a psychological handicap, the severity of which will depend upon how faulty or impeccable is the rest of the test. It is no longer possible for him to skim innocently ahead. Instead, he must proceed warily and dubiously, ever alert for intentional and unintentional pitfalls. And whenever he comes to a question for which he, with his superior ability, sees more than one reasonable answer, he must stop to evaluate afresh the degrees of malice and incompetence of the examiner. Such a test becomes for the superior student a highly subjective exercise in applied psychology - and, if he is sensitive, an agonising one.

6.1. Extract.

I am, of course, not alone in criticising current test procedures. Among the criticisms of tests that I have been making over the years is that even the best multiple-choice tests penalise depth, subtlety, creativity, intellectual honesty, and superior knowledge. I have explained how they do this, and have shown that arguments used by important testers in rebuttal have in fact been tantamount to admissions that the charge is valid.

If the charge is valid, multiple-choice tests have a defect of major proportions, and their widespread use has grave educational and national consequences. This is surely something that we dare not ignore or even treat lightly. Yet there has hitherto seemed to be considerable reluctance in many psychometric circles to face this and related issues squarely. Indeed, when my various criticisms of multiple-choice tests appeared in print they evoked an understandable but nonetheless unfortunate defensive reaction from a number of psychometrists (though there were notable exceptions). If the charge were false, the obvious strategy for the psychometrists would have been to seek to demonstrate the fact by the statistical methods that they find convincing. Yet it is almost a decade now since I brought the charge to the attention of Educational Testing Service, and in all that time they have produced no evidence, statistical or otherwise, that refutes it despite their unrivalled opportunity to make experiments using their own multiple-choice tests, which are certainly among the very best.

The important fact, then, is not just that there does not happen to be statistical or other evidence to refute the charge but that, had the charge been refutable, there ought to have been such evidence by now. Because there is not, the charge that multiple-choice tests penalise depth, subtlety, creativity, intellectual honesty, and superior knowledge must be held to prevail not only on its own merits but also by default. And this leaves a crucial question that we must all face: What is to be done about the matter?

7.1. Extract 1.

There is no generally satisfactory method of evaluating human abilities and capabilities, though occasionally it can be done individually with remarkable prescience. Rough, superficial evaluations are of course possible, and they can be made on a mass production basis. But the detection and evaluation of other than superficial ability is inevitably an art demanding insight, taste, and knowledge. Current attempts to reduce it to a science and then mechanise it are not only dangerous but in a profound sense unscientific. These are hard words, and it would be too much to expect psychometrists to find them pleasing. Yet the psychometrists' defensive reaction to past criticism has been unfortunate. By it the psychometrists have not only prevented a significant confrontation of issues; they have unwittingly advertised their own insecurity.

7.2. Extract 2.

Most of the weaknesses of psychometrists' current methods have to do with the manner in which they use statistics. They seem unaware of the fallacies in their procedures. To illustrate some of these, consider the following skeleton of a multiple-choice question. One is required, essentially, to say which one of the words, if any, in the following sentence makes it defective and should thus be changed:

Among them Tom and Dick could not find enough money.

It does not take long to pick among on the ground that Tom and Dick are two persons and the preposition should therefore be between. On a pretest of this question the statistics would strongly support this as the key.

But if one pauses just a little to think - and has the ability to do so - one concludes that the question is ambiguous. For Tom and Dick might have been holding up a large group at gunpoint and "among them" could not find enough money.

Would this ambiguity show up in the statistics? Think carefully before you answer. There is a trap in all of this. I have tried this question verbally on various groups. Some people, unaware that among and between mean different things, see nothing wrong with the sentence. The question discriminates excellently between these people (call them Group I - for ignorant) and those who say among should be between (call them Group K - for knowledgeable.) Only a small percentage of the people seem to see the alternative meaning fairly quickly and to conclude therefore that the question is ambiguous (call them Group D - for deep). Since Group D is small, it has no great effect on the pretest statistics. These statistics, therefore, will not reveal to the test-maker that the question is defective.

8.1. Extract 1.

Collaborating with Einstein was an unforgettable experience. In 1937, the Polish physicist Leopold Infeld and I asked if we could work with him. He was pleased with the proposal, since he had an idea about gravitation waiting to be worked out in detail. Thus, we got to know not merely the man and the friend, but also the professional.

The intensity and depth of his concentration were fantastic. When battling a recalcitrant problem, he worried it as an animal worries its prey. Often, when we found ourselves up against a seemingly insuperable difficulty, he would stand up, put his pipe on the table, and say in his quaint English, "I will a little tink" (he could not pronounce "th"). Then, he would pace up and down, twirling a lock of his long greying hair around his forefinger.

A dreamy, faraway yet inward look would come over his face. There was no appearance of concentration, no furrowing of his brow - only a placid inner communion. The minutes would pass, and then suddenly Einstein would stop pacing as his face relaxed into a gentle smile. He has found the solution to the problem. Sometimes it was so simple that Infeld and I could have kicked ourselves for not having thought of it. But the magic had been performed invisibly in the depths of Einstein's mind, by a process we could not fathom.

8.2. Extract 2.

Einstein was an accomplished amateur musician. We used at play duets; he at the violin, I at the piano. One day he surprised me by saying that Mozart was the greatest composer of all. Beethoven, he said, "created" his music but the music of Mozart was of such purity and beauty that one felt he merely "found" it - that it had always existed as part of the inner beauty of the Universe, waiting to be revealed.

It was this very Mozartian simplicity that most characterised Einstein's methods. His 1905 theory of relativity, for example, was built on two simple assumptions. One is the so-called principle of relativity, which means, roughly speaking, that we cannot tell whether we are at rest or moving smoothly. The other assumption is that the speed of light is the same, no matter what the speed of the object that produces it. You can see how reasonable this is if you think of agitating a stick in a lake to create waves. Whether you wiggle the stick from a stationary pier, or from a rushing speedboat, the waves once generated are on their own, and their speed has nothing to do with that of the stick.

Each of these assumptions, by itself, was so plausible as to seem primitively obvious. But together they were in such violent conflict that a lesser man would have dropped one or the other and fled in panic. Einstein daringly kept both - and by doing so he revolutionised physics. For he demonstrated that they could after all, exist peacefully side by side, provided we give up cherished beliefs about the nature of time.

8.3. Extract 3.

Science is like a house of cards, with concepts like time and space at the lowest level. Tampering with time brought most of the house tumbling down, and it was this made Einstein's work so important - and so controversial. At a conference in Princeton in honour of his 70th birthday, one of the speakers, a Nobel prize winner, tried to convey the magical quality of Einstein's achievement. Words failed him, and with a shrug of helplessness he pointed to his wrist-watch, and said in tones of awed amazement, "It all came from this." His very ineloquence made this the most eloquent tribute I have heard to Einstein's genius.

8.4. Extract 4.

We think of Einstein as one concerned only with the deepest aspects of science. But he saw scientific principles in every day things to which most of us would give barely a second thought. He once asked me if I had ever wondered why a man's feet will sink into either dry or completely submerged sand, while sand that is merely damp provides a firm surface. When I could not answer, he offered a simple explanation. It depends, he pointed out, on surface tension, the elastic-skin effect of a liquid surface. This is what holds a drop together, or causes two small raindrops on a window pane to pull into one big drop the moment their surfaces touch.

When sand is damp, Einstein explained, there are tiny amounts of water between the grains. The surface tensions of these tiny amounts of water pull all the grains together, and friction then makes them hard to budge. When the sand is dry, there is obviously no water between grains. If the sand is fully immersed, there is water between grains, but there is no water surface between them to pull them together. This is not as important as relativity; yet as his youthful question, about running abreast of a light wave showed, there is no telling what seeming trifle will lead an Einstein to a major discovery. And, the puzzle of the sand gives us an inkling of the power and elegance of Einstein's mind.

9.1. Review.

The April 1960 Baker Street Journal included a contribution by Banesh Hoffman, a professor of mathematics at Queens College, NY. "Sherlock, Shakespeare and the Bomb" - and isn't that a lot to get into one pastiche - closes with the note that this was the second appearance of the story as it was "Enlarged from The Scientific American, April 1951."

Hoffman's pastiche had a third reprinting and that leads us to our 50 Years Ago piece. The February 1966 Ellery Queen Mystery Magazine contained, along with mysteries by Agatha Christie and Edward Hoch, "Sherlock, Shakespeare, and the Bomb." Banesh Hoffman, BSI (1963: "The Dynamics of an Asteroid") sent a reprinted excerpt from the magazine containing the story to a friend. It is inscribed "To Tom Mahoney, With all good wishes, from Banesh Hoffman" and is held in The Sherlock Holmes Collections.

"Sherlock, Shakespeare and the Bomb" is the story of Holmes's and Watson's annual Christmas reunion in Sussex. It must have been a wonderful meeting, as Dr Watson writes that both he and Holmes have "become more mellow, and a growing humour and warm sentiment have enriched our latter years." Holmes's "special intellectual treat" in which he recounts "some fictitious case, told with such a wealth of corroborative detail that one would almost believe it true" were annual highlights for Watson. But that year was the exception; there was no case to discuss. Instead, he had "something just as good. Better, indeed, since it rests on documentary evidence." What follows is Holmes's discussion of his "literary research" which he feels will "suggest the need for a complete revaluation of Shakespeare's works. For they imply that his writings conceal, beneath a cloak of poetry and drama, unparalleled feats of clairvoyance and prognostication."

In this thrice reprinted essay, Holmes presents his friend with his belief that Shakespeare's The Tempest contained passages that foretold the invention of "the modern miracle of wireless [radio]", two sonnets predicting Einstein's theory of relativity, and one sonnet "that describes the present atomic situation so aptly that every line of it can be understood as a message for today." When questioned by Watson about Shakespeare's failure to declare himself a seer, Holmes responds that "It is one of the strongest points in favour of my thesis. Had Shakespeare couched his prophetic utterances in forthright terms, would they have survived? ... Would he have been believed, or would he rather have been dismissed as a harmless lunatic?"

As Holmes is wont to do with Watson, he presents his case for his interpretations. By the time he references the Bikini bomb tests, the "oblique reference to Los Alamos," and quantum theory, Watson states "I must defer to your superior knowledge," falling far short of his usual "When I hear you give your reasons ... the things always appears to me to be so ridiculously simple that I could easily do it myself." (SCAN)

Most of us would be in Watson's shoes when deferring to the superior knowledge possessed by not only Holmes, but of Banesh Hoffman. In the April 1960 BSJ he's listed as a professor of mathematics at Queens College, but there was far more to the man.

Banesh Hoffman was born in Richmond, Britain on 6 September 1905 to a Russian mother and Polish father. He studied mathematics and theoretical physics at Merton College, Oxford University where he received his Bachelor of Art with first class honours in 1929. By 1930 he was living in the United States and earned his doctorate at Princeton University in 1932. He worked as an instructor at the University of Rochester and joined Princeton's Institute for Advanced Study in 1935, where he collaborated with Albert Einstein on the paper "Gravitational Equations and the Problem of Motion." Hoffman joined the Mathematics Department at Queens College, New York in 1937 and was also a visiting professor at Harvard and at Kings College, University of London. In 1947 he authored The Strange Story of the Quantum. He retired from full time teaching in the 1960s and wrote The Tyranny of Testing in which he argued that standardised testing was a superficial means to measure knowledge. He taught one course a semester until he fully retired in the late 1970s. Hoffmann co-authored, with Einstein's secretary Helen Dukas, the book Albert Einstein: Creator and Rebel in 1972.

Dr Hoffman's wife, the former Doris Goodday, wrote that among her husband's many accomplishments he was an amateur pianist and would play duets with Einstein, an amateur violinist. Dr Hoffman died in Queens on 5 August 1986. The Banesh Hoffman Memorial Award is given at CUNY Queens College.

Peter Calamai, BSI is an adjunct research professor at Ottawa's Carleton University and served as national science reporter for The Toronto Star for ten years. When asked to call on his scientific knowledge in regard to Hoffman's writings, Calamai in turn contacted his friend the esteemed Cliff Will. In addition to his book Theory and Experiment in Gravitational Physics, Will also authored Was Einstein Right?: Putting General Relativity to the Test in 1993. Peter wrote that Will "says the science in Hoffman's Sherlockian [piece] is bang on, even if some of the Shakespearean connections are far-fetched. Only someone intimately familiar with the ins and outs of general relativity could have made these connections."

At the end of the essay Holmes extends his hope that his Christmas story was sombre but hopes that Watson enjoyed it. "Indeed, I perceive that you have [enjoyed it] for you have let your breakfast get cold. Have some warm eggs, Watson." The last line in "Sherlock, Shakespeare and the Bomb" displays Hoffman's "certain unexpected vein of pawky humour" (VALL) with his Shakespearean knowledge when Watson responds "Thank you, Holmes. But no more bacon."