10.8. Three Problems for Newton#
The successes of Newton’s model for gravitation were many and astounding. It’s sometimes said that the Enlightenment was a direct result of his success of the naturalistic approach to explaining the world. Here are some of what his theory of gravitation demonstrated:
He showed that the inverse-square rule for gravitation explained Kepler’s Laws, that they would accommodate circular, elliptical, and parabolic orbits. Famously, Hayley’s Comet was discovered and the predictions that Newton’s friend made were based on Newton’s rules. He simply assumed that the comet’s path was elliptical (but squashed) around the Sun as its focus and could then use Newtons’ Gravitation law. He was right…Halley’s Comet’s path takes it all the way past Neptune before it starts coming back towards the Sun. It’s a 76 year round trip.
The Earth’s axis wobbles a tiny bit just like a top and Newton explained that, the “precision of the equinoxes.”
He explained the twice-a-day, high and low tides as a feature of the Moon’s attraction for the ocean water closest to it as opposed to the water on the other side of the Earth from the Moon.
Earth should not be a perfect sphere since it’s not an absolutely rigid mass. Because it rotates material closest to the axis through the poles (near the poles) would feel a different centripetal force due to the material inside of its radius from material furthest away from the axis of rotation (equator). So there should be a measurable difference in the gravitational attraction at different longitudes and this stimulated heroic teams of explorers who traveled very far north with pendulums to make measurements of \(g\) everywhere they could. Newton’s explanation worked.
And of course his model explained all of the observed orbital motions of the known planets, a concept that was not even thought possible, or even desirable while Newton was a child. He determined the relative masses of the planets and the Sun.
Of course in addition, he had other unparalleled (including to this day) achievements:
He properly conceived of the idea of momentum and completely described motion and dynamics.
He correctly conceived of the theory of colors as mixing together to make white, in contradiction to the prevailing views led by Descartes.
He invented and the pioneered the use of calculus.
By the time he died in 1726:
Magic and superstition as explanations for nature were gone.
Subservience to Aristotle was gone.
Everyone believed that…everything could be known.
The very essence of the Enlightenment period in western history had begun.
But there were issues that were more philosophical that required his attention. Here are three.
10.8.1. Problem 1: Action At A Distance#
Two distinct camps developed in physics. While a dominant belief in naturalism now reigned in Europe, the French followed the lead of Descartes while the British remained loyal to Newton. But everyone agreed that the actual mechanism of gravity was problematic. In a letter he wrote:
“It is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact…that one body may act upon another at a distance through a vacuum, without the mediation of anything else…is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.”
Everyone agreed on the facts, but two opinions emerged about proper process. The Continental view was that until you can enunciate the mechanism of gravity and then reason deductively from that, you’re not doing acceptable science. By contrast, there was the British view—and the one that we all follow today—that what’s important is that if it works, that’s good enough.
The idea of "Action at a Distance" for any physical phenomenon bothered everyone and would today still be anathema.
But while bothered…Newton was undeterred. In many ways, Newton first differentiated the contrast between why a phenomenon occurs and how it occurs…and only an answer to how can be a mathematical model. He famously said: “I feign no hypotheses.” which even is its own Latin catch-phrase, “hypotheses non fingo” (Google it!). That’s his throw-down to the Cartesians.
No guessing, Newton's modeling.
The bar to making progress that Descartes set up (the “Continental view”) is too high. One should “hypothesize” (I’d say model-build) and deduce empirical observables, test them and then refine your model. Then you’ve turned science into a Process that improves on its conclusions. Eventually—and gravity is a good example—one might find an acceptable why…but until that, how is good enough and makes progress possible. Descartes’ crowd could never get started and so he and his French followers were left in the scientific dust.
10.8.2. Problem 2: Stability of the Universe—Newton’s Cosmology#
Newton wasn’t shy about how to apply his model. As described above, there were plenty of terrestrial and astronomical applications that were predicted and tested positive. But what about the whole enchilada? What about the whole Universe?
He recognized quickly that he faced a puzzle even more prickly than Action at a Distance. He couldn’t explain the improbability of why we’re here at all. Here’s the problem, which I can form as a question:
Is the Universe finite or infinite? His theory seems to suggest that the Universe must be infinite, with an infinite number of stars (all anyone knew about were planets and stars…no galaxies). Imagine this enormous space filled with stars, each of which is attracting every other object in it, and is in turn being attracted by every other object. The left hand picture below is a cartoon of such a situation. That one star on the right is being pulled on by everyone…and the fact that the Gravitation law varies like \(1/R^2\) means that there is an influence from all objects, all the way out to infinity. But it’s in balance since there are the same number (infinite!) of stars all around it and they are evenly distributed in all directions. So, great.
If the Universe had an edge, then the right hand figure would crudely be the story. Notice now that our target star is being pulled to the left and there’s no balancing set of forces to the right. That should start our star accelerating which would then pull on other stars differently as it moves and they’d start to accelerate—the end result would be a huge collapse of everything on top of itself. Yet: we’re here and so this hasn’t happened. Therefore the Universe is infinite. That’s the argument, but it’s flawed…or at least highly improbable.
Suppose the Universe is infinite and this incredibly delicate balance is at work. A butterfly could cause the whole thing to collapse, much less Jupiter orbiting the Sun. That is, the nature of his Gravitational law is such that the delicate balance that holds everything just right…has to be absolutely perfect. That seems improbable.
Newton had a famous correspondence with the leading theologian in Britain, Richard Bentley in 1692. Bentley was erudite and familiar with science and Newton took him seriously. He wrote to the reverend:
“As to your first query, it seems to me that if the matter of our Sun and planets and all the matter in the universe were evenly scattered throughout all the heavens, and every particle had an innate gravity toward all the rest, and the whole space throughout which this matter was scattered was but finite, the matter on the outside of the space would, by its gravity, tend toward all the matter on the inside, and by consequence, fall down into the middle of the whole space and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space, it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one to another throughout all that infinite space. And thus might the Sun and fixed stars be formed, supposing the matter were of a lucid nature. But how the matter should divide itself into two sorts, and that part of it which is fit to compose a shining body should fall down into one mass and make a Sun and the rest which is fit to compose an opaque body should coalesce, not into one great body, like the shining matter, but into many little ones; or if the Sun at first were an opaque body like the planets, or the planets lucid bodies like the Sun, how he alone would be changed into a shining body whilst all they continue opaque, or all they be changed into opaque ones whilst he remains unchanged, I do not think explicable by mere natural causes, but am forced to ascribe it to the counsel and contrivance of a voluntary Agent.”
And again,
“The reason why matter evenly scattered through a finite space would convene in the midst you conceive the same with me, but that there should be a central particle so accurately placed in the middle as to be always equally attracted on all sides, and thereby continue without motion, seems to me a supposition as fully as hard as to make the sharpest needle stand upright on its point upon a looking glass. For if the very mathematical center of the central particle be not accurately in the very mathematical center of the attractive power of the whole mass, the particle will not be attracted equally on both sides. And much harder it is to suppose all the particles in an infinite space should be so accurately poised one among another as to stand still in a perfect equilibrium. For I reckon this as hard as to make, not one needle only, but an infinite number of them (so many as there are particles in an infinite space) stand accurately poised upon their points. Yet I grant it possible, at least by a divine power; and if they were once to be placed, I agree with you that they would continue in that posture without motion forever, unless put into new motion by the same power. When, therefore. I said that matter evenly spread through all space would convene by its gravity into one or more great masses, I understand it of matter not resting in an accurate poise.”
“… a mathematician will tell you that if a body stood in equilibrio between any two equal and contrary attracting infinite forces, and if to either of these forces you add any new finite attracting force, that new force, howsoever little, will destroy their equilibrium and put the body into the same motion into which it would put it were those two contrary equal forces but finite or even none at all; so that in this case the two equal infinities, by the addition of a finite to either of them, become unequal in our ways of reckoning; and after these ways we must reckon, if from the considerations of infinities we would always draw true conclusions.”
So Newton’s solution to this delicate balance was an appeal to God. God’s job is holding everything in place. Today we don’t do that in science.
The stability of the universe in much this same guise will come back and vex Albert Einstein and lead him into an intellectual cul-de-sac where turning around was his single, humiliating choice.
10.8.3. Problem 3: Absolute Space and Time#
Newton’s mechanics led to big questions that required speculation about space and time…that is, Space and Time! He asked himself questions like this (although not this particular one).
Suppose the universe consists of only four objects: you, your friend, a rope, and a knife. You and your friend are connected by a rope. Are you stationary or are you rotating around the center point of the rope? Remember, the universe is empty but for you two. How could you tell?
This is sticky matter of relative motion. If I were to ask if you were moving linearly with respect to your friend, you could tell me that because you’d see your friend approach, pass you, and recede. Which would be really sad. (By the way, your friend would see exactly the same thing except in the other direction.) So you might not agree about who is moving and who is stationary, but you’d have no trouble believing that relative motion exists between you.
But rotation is a different matter and this question is specifically about an accelerated “frame of reference” since in order to rotate about that center point, a centripetal force through the rope would be required and so an acceleration is at work. Well, one of you has a knife and if you cut the rope one of two things might happen. Nothing! In which case you’d conclude that you were not rotating because the other thing that might happen could be that you’d immediately begin to separate meaning that: you had each been orbiting the center and that when the rope no longer connected you, you’d start straight line motion in accordance with Newton’s First law. And you’d become very lonely.
The question is…if you are rotating in this situation, with respect to what are you rotating? Newton felt that he needed an absolute measure for inertia and acceleration and he chose Space, with a capital S. Take everything out of the universe and space will still be there acting as an absolute coordinate system. All motion, constant velocity and accelerated, can be described mathematically with respect to this absolute coordinate system. To Newton, space was a thing, almost like a physical substance. Newton also insisted that there is an absolute clock…absolute Time, with a capital T.
Needless to say, there was also the Continental point of view, championed by Leibnitz who said that space was defined by the relative positions of things. Take away the things and there is no space…it’s completely relative…to stuff.
This argument is going to come back to haunt us a few more times before we reach the 21st century! But the important thing is that nobody talked like this; nobody theorized scientifically about the universe before Newton.