## The Net Advance of Physics RETRO:Weblog 2013 April 8 - May 27 A New Approach to Experimental History of Science Part 4: An Example

Galileo, according to his perhaps-unreliable biographer Viviani, timed the period of the chandeliers in Pisa Cathedral, and dropped weights from the Leaning Tower.

2013 April 29:

The principle of equivalence -- in its "weak" version -- was announced by Galileo in 1638: heavy stones and light fall with the same acceleration, as would all bodies in the absence of air resistance. The principle could be tested by dropping heavy objects, by rolling balls down inclined planes, by observing the regular swing of pendula ...

Chandelier in Pisa Cathedral. Photo by Katarina Jankovic.

This was a cutting-edge research-area throughout the Seventeenth Century, as gravitational theory developed into its Cartesian and then its Newtonian forms. Newton himself conducted an ingenious experiment in the Galilean tradition:[ Principia III, vi, 6]:

"It has been, now of a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to the earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums.

"I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal : I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations. And therefore the quantity of matter in the gold (by Cor. 1 and 0, Prop. XXIV, Book II) was to the quantity of matter in the wood as the action of the motive force upon all the gold to the action of the same upon all the wood ; that is, as the weight of the one to the weight of the other : and the like happened in the other bodies. By these experiments, in bodies of the same weight, I could manifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been."

From this experiment Newton derived two much stronger statements of the equivalence principle than Galileo's:

"Corollary 1. Hence the weights of bodies do not depend upon their forms and textures ; for if the weights could be altered with the forms, they would be greater or less, according to the variety of forms, in equal matter ; altogether against experience.

"Corollary 2. Universally, all bodies about the earth gravitate towards the earth ; and the weights of all, at equal distances from the earth's centre, are as the quantities of matter which they severally contain. This is the quality of all bodies within the reach of our experiments ; and therefore (by Rule III) to be affirmed of all bodies whatsoever. If the æther, or any other body, were either altogether void of gravity, or were to gravitate less in proportion to its quantity of matter, then, because (according to Aristotle, Des Cartes, and others) there is no difference betwixt that and other bodies but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter ; and, on the other hand, the heaviest bodies, acquiring the first form of that body, might by degrees quite lose their gravity. And therefore the weights would depend upon the forms of bodies, and with those forms might be changed : contrary to what was proved in the preceding Corollary."

I had expected to find without difficulty a short video showing that simple pendula with different masses have the same period. Amazingly, very few (excluding simulations) seem to be available; most of those I found, like this one, are obviously student lab-reports. [1 min]

Newton's version of the pendulum experiment typifies how scientific replication usually works in the context of an ongoing research-programme. One reads an account of some fairly recent experiment (in Newton's day meant "recent" meant less than a century old; today it might mean weeks) but does not attempt to exactly copy it as given in the literature; rather, one designs a related experiment testing the predictions of the first experiment by a slightly different method. Newton did not travel to Pisa, or even to a domed building in England, and measure the periods of chandeliers; he constructed small pendula of a standard length but bobs of various weights and compositions. Had this experiment yielded a wide variety of different periods, the validity of Galileo's observations would have been called into question, and a visit to a cathedral might have been warranted (to investigate the possibility that only very long pendula have a mass-independent period). Instead, of course, Newton's results agreed with what Galileo might have predicted, and also added some additional information.

Although the intention here was clearly not to conduct an historical investigation, one might nevertheless describe such replications as cenochronic: one is checking, and also extending, the work of a contemporary researcher.

In modern retellings, the equivalence principle of Galileo and Newton is usually given as:

1. Weak Form: weight is proportional to quantity of matter; the "mass" that appears in the law of gravity is the same as the "mass" in the law of inertia. If mass is the ratio of force to acceleration and weight (i.e. gravitational force) is the product of mass and gravitational field, then:
2. Stronger Form: the gravitational field is the acceleration due to gravity. Thus the gravitational response of an object does not depend on its mass or any other physical properties (Newton's "forms and textures"), but only on its location in the field (and perhaps its velocity).

The gravitational field is acceleration; note the copula. In all of the 1700s -- in all of the 1800s -- noöne seems to have thought to reverse the sentence and say that acceleration is a gravitational field. Noöne, that is, except one mathematician of medium repute, far better known as an author of children's novels ... not so much of this novel, however. Sylvie and Bruno takes place in at least three different universes, parallel yet intersecting in non-Euclidean fashion; the part which concerns us here plays out in ours, among the familiar rituals of a Victorian country-house.

Edmund C. Tarbell: Arrangement in Pink and Gray (Afternoon Tea)
(Worcester Art Museum)

Here is Lewis Carroll, trying his best to write like Austen or Trollope as he documents the bumpy course of true love among the gentry ... Naturally, there are some distractions besides the usual ones:

" 'How convenient it would be,' Lady Muriel laughingly remarked, à propos of my having insisted on saving her the trouble of carrying a cup of tea across the room to the Earl, 'if cups of tea had no weight at all ! Then perhaps ladies would sometimes be permitted to carry them for short distances !'

" 'One can easily imagine a situation,' said Arthur, 'where things would necessarily have no weight, relatively to each other, though each would have its usual weight, looked at by itself.'

" 'Some desperate paradox !' said the Earl. 'Tell us how it could be. We shall never guess it.'

" 'Well, suppose this house, just as it is, placed a few billion miles above a planet, and with nothing else near enough to disturb it : of course it falls to the planet ?'

"The Earl nodded. 'Of course -- though it might take some centuries to do it.'

" 'And is five-o'clock-tea to be going on all the while ? ' said Lady Muriel.

" 'That, and other things,' said Arthur. 'The inhabitants would live their lives, grow up and die, and still the house would be falling, falling, falling ! But now as to the relative weight of things. Nothing can be heavy, you know, except by trying to fall, and being prevented from doing so. You all grant that ? '

"We all granted that.

" 'Well, now, if I take this book, and hold it out at arm's length, of course I feel its weight. It is trying to fall, and I prevent it. And, if I let go, it falls to the floor. But, if we were all falling together, it couldn't be trying to fall any quicker, you know : for, if I let go, what more could it do than fall ? And, as my hand would be falling too -- at the same rate -- it would never leave it, for that would be to get ahead of it in the race. And it could never overtake the falling floor ! '

" 'I see it clearly,' said Lady Muriel. 'But it makes one dizzy to think of such things ! How can you make us do it ?'

" 'There is a more curious idea yet,' I ventured to say. 'Suppose a cord fastened to the house, from below, and pulled down by some one on the planet. Then of course the house goes faster than its natural rate of falling : but the furniture -- with our noble selves -- would go on falling at their old pace, and would therefore be left behind.'

" 'Practically, we should rise to the ceiling,' said the Earl. 'The inevitable result of which would be concussion of brain.'

" 'To avoid that,' said Arthur, 'let us have the furniture fixed to the floor, and ourselves tied down to the furniture. Then the five-o'clock-tea could go on in peace.'

" 'With one little drawback!' Lady Muriel gaily interrupted. 'We should take the cups down with us : but what about the tea?'

" 'I had forgotten the tea,' Arthur confessed. 'That, no doubt, would rise to the ceiling- unless you chose to drink it on the way !'

" 'Which, I think, is quite nonsense enough for one while !' said the Earl. 'What news does this gentleman bring us from the great world of London ?'

"... [T]he conversation ... now took a more conventional tone ..."

A more conventional tone indeed. Carroll/Dodgson in these paragraphs had done better and deeper physics than Ernst Mach and his entire over-rated school, but noöne in the scientific world seems to have paid the least attention! (It's hard to blame them too much, though -- reading Sylvie and Bruno can be an ordeal: " 'It are gone !' Bruno solemnly replied ... 'Oo couldn't touch it, oo know. If oo walked at it, oo'd go right froo!'")

Not Bruno but Albert.

Einstein, therefore, had to come up with it all over again, starting from scratch in 1907: [ Relativity: the Special and General Theory, (1920), Chapter XX]:

"THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS AS AN ARGUMENT FOR THE GENERAL POSTULATE OF RELATIVITY

"We imagine a large portion of empty space, so far removed from stars and other appreciable masses, that we have before us approximately the conditions required by the fundamental law of Galilei. It is then possible to choose a Galileian reference-body for this part of space (world), relative to which points at rest remain at rest and points in motion continue permanently in uniform rectilinear motion. As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer. He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room.

"To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a 'being' (what kind of a being is immaterial to us) begins pulling at this with a constant force. The chest together with the observer then begin to move 'upwards' with a uniformly accelerated motion. In course of time their velocity will reach unheard-of values provided that we are viewing all this from another reference-body which is not being pulled with a rope.

"But how does the man in the chest regard the process ? The acceleration of the chest will be transmitted to him by the reaction of the floor of the chest. He must therefore take up this pressure by means of his legs if he does not wish to be laid out full length on the floor. He is then standing in the chest in exactly the same way as anyone stands in a room of a house on our earth. If he release a body which he previously had in his hand, the acceleration of the chest will no longer be transmitted to this body, and for this reason the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that the acceleration of the body towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment.

"Relying on his knowledge of the gravitational field (as it was discussed in the preceding section), the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time. Of course he will be puzzled for a moment as to why the chest does not fall, in this gravitational field. Just then, however, he discovers the hook in the middle of the lid of the chest and the rope which is attached to it, and he consequently comes to the conclusion that the chest is suspended at rest in the gravitational field.

Owen Dunn: "A Waterloo & City Line train being lifted by crane," ©2006.

"Ought we to smile at the man and say that he errs in his conclusion ? I do not believe we ought to if we wish to remain consistent ; we must rather admit that his mode of grasping the situation violates neither reason nor known mechanical laws. Even though it is being accelerated with respect to the 'Galileian space' first considered, we can nevertheless regard the chest as being at rest. We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity ...

" ... Suppose that the man in the chest fixes a rope to the inner side of the lid, and that he attaches a body to the free end of the rope. The result of this will be to stretch the rope so that it will hang 'vertically' downwards. If we ask for an opinion of the cause of tension in the rope, the man in the chest will say : 'The suspended body experiences a downward force in the gravitational field, and this is neutralised by the tension of the rope ; what determines the magnitude of the tension of the rope is the gravitational mass of the suspended body.'

"On the other hand, an observer who is poised freely in space will interpret the condition of things thus : 'The rope must perforce take part in the accelerated motion of the chest, and it transmits this motion to the body attached to it. The tension of the rope is just large enough to effect the acceleration of the body. That which determines the magnitude of the tension of the rope is the inertial mass of the body.' Guided by this example, we see that our extension of the principle of relativity implies the necessity of the law of the equality of inertial and gravitational mass. Thus we have obtained a physical interpretation of this law."

Einstein in his Berlin years [Scientific Monthly 10, 418 (1920)]

Obviously both Einstein and Dodgson were conducting thought experiments; nonetheless, it would be correct to say that they were (mentally) re-creating the work of Galileo and of Newton, and doing so at a very high level. It should be emphasised that the equivalence principle, in its Newtonian form, was no longer a research topic in the 1880s or the 1900s; it was standard, textbook material. Although there were numerous real-world experiments descended from Galileo's and conducted by researchers, their purpose was not to shed light upon the nature of mass or gravity, but to use the established results to (for example) "weigh the Earth", add more digits to the constant G, or locate buried geological structures by their gravitational signatures. In such experiments and surveys, the equivalence principle was taken for granted, as something long known. In a sense, then, these could be described as "anachronic" replications.

In devising the thought experiments which led to general relativity, by contrast, Dodgson and Einstein asked difficult questions about the relationship between acceleration, mass, and gravity -- the same questions that Newton had asked, and that Galileo had asked to the extent that his vocabulary permitted. For Dodgson and Einstein, Newton and Galileo were not departed ancestors (to be venerated or forgotten) but contemporaries across time. By (mentally) repeating the classic falling-body experiments, a new insight was obtained, one already partly latent in the older work: acceleration and gravity cannot be distinguished at all. This is a cenochronic insight.

Readers of this blog-post may also wish to read " Einstein's Pathway to the Equivalence Principle" by Galina Weinstein. Next week we will see how cenochronic re-creations of Galileo's experiment in the laboratory have become an ongoing part of modern gravitational research. But for now ---

"'How perfectly isochronous!' the Professor exclaimed with enthusiasm. He had his watch in his hand, and was carefully counting Bruno's oscillations. 'He measures time quite as accurately as a pendulum!'

" 'Yet even pendulums,' the good-natured young soldier observed, as he carefully released his hand from Bruno's grasp, 'are not a joy for ever! Come, that's enough for one bout, little man!'"

[Sylvie and Bruno, Volume I, chapter 18]