Galileo’s flawed thought experiment and Einstein’s happiest thought
Galileo purportedly demonstrated that objects of different weights fall at the same rate in the famous Leaning Tower of Pisa experiment, which refuted Aristotle’s teachings. The authenticity of this experiment has been debated; whether such an experiment would have yielded the supposed outcome has also been questioned, given the effect of air resistance (see e.g. Adler and Coulter, 1978). But this would not have mattered, because — as often told in conjunction with the Leaning Tower experiment — Galileo reached the same conclusion by the following thought experiment:
Imagine two objects, one lighter than the other, and suppose that the heavier object falls faster than the lighter one, as Aristotle assumed. Now imagine that we connect the two with a rigid rod and then drop the connected objects from the tower. The light object will fall slower than the heavy object, thus it will retard the fall of the heavy object through the rod, and the connected objects should fall slower than the heavy object alone. On the other hand, the connected objects as a whole are heavier than each individual object, thus they together should fall faster than the heavy object alone. This contradiction leads one to conclude that Aristotle’s assumption must be false.
Galileo published this argument in De Motu Antiquora. I quote the English translation — for those who do not read Latin, like me — of the original text at the end of this article. I hope that they make interesting readings, as they are for me, especially the bit about dropping lead balls from the Moon for them to fall back to Earth.
The Leaning Tower experiment — fictional as it may be — and the thought experiment of Galileo have become scientific legends. They symbolize the dawn of modern science, which holds experimentation and logic reasoning over the idealistic philosophy of the School of Athens.
It recently occurred to me, however, that Galileo’s thought experiment, reasonable as it first sounds, is deeply flawed. Galileo accidentally reached the correct conclusion because of an equivalency — unbeknownst to him — that eventually led to Einstein’s conception of the general theory of relativity.
Galileo could not have known the four fundamental forces of nature. Gravity is but one of the four which happens to govern the motion of falling objects. Charged objects in an electric field, in contrast, experience the Coulombic force, which is proportional to the electric charge on the object. The acceleration of the object is thus proportional to its charge-mass ratio.
Galileo’s argument would not have worked for charged objects in an electric field. Objects of different charge-mass ratios will have different rates of moving through the electric field. Combining two objects of different charge-mass ratios will result in an object that move at a different rate than either object, contradicting Galileo’s expectations.
Attentive readers of Galileo’s original text may contend that Galileo was considering two objects of the same genus (species), hence in the case of charged objects he could have been thinking of two objects of the same charge-mass ratio. This, of course, would be a contemporary effort to salvage Galileo’s mistake. Galileo had no way of knowing anything about charge-mass ratio: almost two hundred years would pass before Coulomb’s law was established regarding forces on electric charges. More importantly, the law governing falling objects in the gravitational field is fundamentally different from that governing charged objects in the electric field: without air resistance, any falling object falls at the same rate, regardless of its composition!
The true reason that any object falls at the same rate is the equivalency between its gravitational mass and its inertial mass. According to Newton’s law of gravity, the force on an object is proportional to its mass, while according to Newton’s second law of motion, an object’s acceleration is the force acting on it divided by its mass. One of the universe’s deepest mysteries is that the gravitational mass in the law of gravity is identical to the inertial mass in the second law! As a result, the acceleration of any falling object is equal to the strength of Earth’s gravity. This is why Galileo was right in his conclusion, although he did not know and could not have known the real underlying reason.
The equivalency between the gravitational mass and the inertial mass turned out to be the fundamental principle that led Einstein to the “happiest thought” of his life and to the development of the general theory of relativity. In this way, the imperfect beginning of the scientific study of motion is connected to its ultimate triumph.
Excerpts from the English translation of De Motu Antiquora
From this it is overtly established that Aristotle wants mobiles of the same genus to observe between themselves in the speed of motion the ratio of the sizes that these mobiles have: and he says that very openly in Book IV of the De Caelo, by affirming that a large piece of gold is carried more swiftly than a small one. How ridiculous this opinion is, is clearer than daylight: for who will ever believe that if, for example, two lead balls were released from the sphere of the Moon, one being a hundred times larger than the other, if the larger took an hour to come to Earth, the smaller would use in its motion a space of time of a hundred hours? or, if from a high tower, two stones, one being double the size of the other, were thrown at the same moment, that, when the smaller was at mid-tower, the larger would already have reached the ground? ……
But, in order that we may always make more use of reasons than of examples (for we are seeking the causes of effects, which are not reported by experience), we will bring forth our way of thinking, whose confirmation will result in the downfall of Aristotle’s opinion. We say, then, that mobiles of the same species (let those things be said to be of the same species that are constituted of the same material, such as lead or wood, etc.), though they may differ in size, are however moved with the same swiftness, and a larger stone does not go down more swiftly than a smaller one.
I argue as follows: by proving that mobiles of the same species, of unequal sizes, are carried with the same swiftness.
Let there be two mobiles of the same species, the larger a, and the smaller b; and, if it can be done, as our adversaries hold, let a be moved more swiftly than b. There are then two mobiles one of which is moved more swiftly than the other; hence, according to what has been presupposed, the combination of the two will be moved more slowly than the part, which alone, was moved more swiftly than the other. If then a and b are combined, the combination will be moved more slowly than a alone: but the combination of a and b is larger than a alone: hence, contrary to our adversaries’ view, the larger mobile will be moved more slowly than the smaller; which would certainly be unsuitable. What clearer indication do we require of the falsehood of Aristotle’s opinion? But, I ask, who will not recognize the truth of this on the spot, when he examines it in a pure and simple and natural way? For if we presuppose that the mobiles a and b are equal and that they are very near each other, then, by the consensus of all, they will be moved with equal swiftness: and if we understand that while they are being moved, they are joined, why, I ask, will they double the swiftness of their motion, as Aristotle held, or increase it? Accordingly, let it be sufficiently confirmed that there exists no cause, per se, why mobiles of the same species should be moved with unequal speeds, but there certainly is one why they should be moved with equal speed. But if there were some accidental cause, such as, for example, the shape of the mobile, it must not be classified amongst the causes per se: and moreover, the shape helps or hinders the motion but little, as we shall show in the proper place.
