2013 April 8 - May 27
A New Approach to Experimental History of Science
Part 5: An Example (continued)
"How about that? Mr. Galileo was correct in his findings."
Certainly a beautiful demonstration for students and the public: so the astronauts intended it to be, and so it is. As a repetition of the "Tower of Pisa" experiment, somewhat anachronical, even triumphalistic: a celebration of the heroic scientific genius of the Renaissance, the rational spirit which enabled humanity to reach the Moon. A realisation, too, of Galileo's own fantasy:
"SALVIATI: We have already seen that the difference of speed between bodies of different specific gravities is most marked in those media which are the most resistant: thus, in a medium of quicksilver, gold not merely sinks to the bottom more rapidly than lead but it is the only substance that will descend at all; all other metals and stones rise to the surface and float. On the other hand the variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed.
"SIMPLICIO: This is a remarkable statement, Salviati. But I shall never believe that even in a vacuum, if motion in such a place were possible, a lock of wool and a bit of lead can fall with the same velocity!"
Poor Simplicio: wrong again. Or was he?
One cannot read Galileo's Two New Sciences without noticing his constant reference to experiments beyond the reach of Seventeenth Century technology, his profound awareness of the limitations and errors intrinsic in experiment:
"SIMPLICIO: Your discussion is really admirable; yet I do not find it easy to believe that a bird-shot falls as swiftly as a cannon ball.
"SALVIATI: Why not say a grain of sand as rapidly as a grindstone? ... Aristotle says that 'an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.' I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths; now you would not hide behind these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one ...
"SIMPLICIO: Perhaps the result would be different if the fall took place not from a few cubits but from some thousands of cubits.
"SALVIATI: If this were what Aristotle meant, you would burden him with another error which would amount to a falsehood; because, since there is no such sheer height available on earth, it is clear that Aristotle could not have made the experiment; yet he wishes to give us the impression of his having performed it when he speaks of such an effect as one which we see."
A truly cenochronical re-creation of Galileo's work -- a re-creation in which one becomes a genuine participant in the dialogue -- would perhaps begin by attempting to perform the experiments which he could only imagine. Dropping hammers and falcon-feathers on the moon is a step in that direction, but only stands symbolically for the Space Age's grand cenochonic replication of the "Tower of Pisa" experiment: the lunar laser-ranging project.
Both the Apollo expeditions and the unmanned Lunokhod probes left corner-cube prisms (in effect, mirrors) on the Moon, so that the exact value of the Earth-Moon distance could be measured (assuming the constancy of the speed of light, Simplicio would point out). The lasers of West Texas, shining onto the lunar surface, permit accuracies of a centimetre or better -- Galileo's "finger-breadth". Since the Earth and Moon are continually falling toward their common centre of gravity with the Sun -- such is the meaning of "orbit" -- they are gigantic test-bodies, exactly like Galileo's. Thus is the equivalence principle tested and verified (to one part in 1013) by modern cenochronic historians of science, though they may not think of themselves as such.
The work of Robert Reasenberg and his collaborators, known as POEM, the Principle Of Equivalence Measurement experiment at Harvard-Smithsonian Center for Astrophysics, is conceptually even closer to Galileo's Tower of Pisa (and has produced talks with amusing titles: Checking the Equivalence Principle While Riding on a Pogo Stick and Testing the Equivalence Principle on a Trampoline). The space-based version, SR-POEM, faces the new technical challenges of conducting Galileo's experiment in space (aboard a sounding rocket). SR-POEM is designed to measure the position of two falling 900-gramme dumbells for 40 seconds. The experimenters believe that they will be able to achieve accuracy of one part in 1016.
But what if one of the dumbells were to fall upwards?
Few if any modern synopses of Two New Sciences include one of Simplicio's more curious objections:
"SIMPLICIO: But if we find that air has levity instead of gravity what then shall we say of the foregoing discussion which, in other respects, is very clever?
"SALVIATI: I should say that it was empty, vain, and trifling. But can you doubt that air has weight when you have the clear testimony of Aristotle affirming that all the elements have weight including air, and excepting only fire? As evidence of this he cites the fact that a leather bottle weighs more when inflated than when collapsed.
"SIMPLICIO: I am inclined to believe that the increase of weight observed in the inflated leather bottle or bladder arises, not from the gravity of the air, but from the many thick vapors mingled with it in these lower regions. To this I would attribute the increase of weight in the leather bottle.
"SALVIATI: I would not have you say this, and much less attribute it to Aristotle; because, if speaking of the elements, he wished to persuade me by experiment that air has weight and were to say to me: "Take a leather bottle, fill it with heavy vapors and observe how its weight increases," I would reply that the bottle would weigh still more if filled with bran; and would then add that this merely proves that bran and thick vapors are heavy, but in regard to air I should still remain in the same doubt as before. However, the experiment of Aristotle is good and the proposition is true. But I cannot say as much of a certain other consideration, taken at face value; this consideration was offered by a philosopher whose name slips me; but I know I have read his argument which is that air exhibits greater gravity than levity, because it carries heavy bodies downward more easily than it does light ones upward.
"SAGREDO: Fine indeed! So according to this theory air is much heavier than water, since all heavy bodies are carried downward more easily through air than through water, and all light bodies buoyed up more easily through water than through air; further there is an infinite number of heavy bodies which fall through air but ascend in water and there is an infinite number of substances which rise in water and fall in air ...
"SALVIATI: The experiment with the inflated leather bottle of Aristotle proves conclusively that air possesses positive gravity and not, as some have believed, levity, a property possessed possibly by no substance whatever; for if air did possess this quality of absolute and positive levity, it should on compression exhibit greater levity and, hence, a greater tendency to rise; but experiment shows precisely the opposite."
The problem of negative mass -- "levity" or antigravity -- is one of the most vexing in classical physics. The principle of equivalence states in mathematical language that ma = mg and therefore that a=g. If the equivalence principle should be wrong, we should write not m but minertial and mpassive for the inertial and gravitational masses.
Why the subscript "passive"? As Newton knew though Galileo did not, the g due to a point source of mass M is proportional to M / r2. There is no obvious reason that this "active" gravitational mass -- the source of gravity -- must necessarily be the same as the mass which passively responds to the field.
So when a point particle with inertial mass minertial and passive gravitational-mass mpassive interacts with another point-mass with active gravitational-mass Mactive, the actual relationship between acceleration and gravity is
where G is Newton's constant and the minus sign is required to ensure that, when all masses are positive, the resulting acceleration is an attraction.
To turn the attraction into a repulsion requires only that one of the masses in the acceleration formula (or all three) be negative. If such a breakdown of the weak equivalence-principle were to occur, the stronger forms would face difficulties as well; the reader might enjoy trying to imagine how objects dropped in a "falling house" could seem to fall towards the ceiling.
The long history of negative mass need not concern us here, except in one way: some of the most sophisticated modern replications of Galileo's experiment are attempts to determine whether the three types of mass -- evidently all equal and positive in ordinary matter -- might differ, or even become negative, in antimatter. According to the famous geometrical interpretation advocated by Feynman, antiparticles are normal particles moving backward in time. This suggests the thought experiment of "dropping", that is releasing, an anti-apple. If one films the result and then plays the film backward, would one not then appear to be seeing an ordinary apple falling to the ground? And if so, does not this mean that the unreversed film, showing the actual trajectory of the anti-apple, would show it rising?
There are flaws in this argument, but the situation is confusing enough to certainly merit experimental testing, and such testing has been, and is, taking place. Among the best-funded of such experiments is CERN's ALPHA collaboration, which has very recently announced that the ratio between the inertial and passive-gravitational masses of antihydrogen is no less than 1/75. [Nature Communications 4, 1785 (2013)]. This may not sound like very good news for the equivalence principle; however, it may merely reflect the enormous difficulty of the experiment, and future replications will yield ratios approaching 1.
Let us return, however, to the main thrust of this essay. I contend that such research is not merely physics, but also history of science. Rather than viewing the intellectual development of humanity as a trajectory leading inexorably to the present (the anachronic point-of-view) or as a series of discrete and ultimately unknowable "lost worlds" (the diachronic point-of-view), a cenochronic historian sees it as an interconnected whole, with all past and future stages subtly united. No past idea or discovery is truly complete; each can always give birth to new continuations, not only those which happened to occur in chronological proximity.
Next Monday, in the penultimate installment of A New Approach to Experimental History of Science, we will examine this point more closely. For now, though, farewell to Galileo Galilei, our companion for the last four weeks!
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