# Science Feature: Newton’s Collapsing Universe

Sir Isaac Newton was possibly the greatest scientist who ever lived. But even great men can make mistakes – it’s just that *their* errors are more interesting than other people’s.

Newton’s law of motion (force equals mass times acceleration) is taught in every high school. Newton’s law of gravitation (the inverse-square law) is precise enough to be used by NASA for space mission design and was used by Newton himself to determine the motion of the planets. Given the wide applicability and accuracy of his theory, it was natural that Newton’s questing intelligence would turn to the question of the structure of the universe itself.

Newton was a devout Christian and would have pondered deeply on Genesis, where it is stated:

In the beginning God created the heaven and the earth. And the earth was without form, and void; and darkness was upon the face of the deep.

Newton contemplated the nature of a finite universe filled with a homogeneous substance and realized immediately that this would be unstable, it would soon collapse to a central point: today we call this phenomenon gravitational collapse. So a finite universe could not be stable, but what about an infinite universe? In 1692 Newton wrote to a colleague as follows:

But if matter was evenly diffused through an infinite space, it would never convene into one central mass but some of it might convene into one mass and some into another so as to make an infinite number of great masses scattered at great distances from one another throughout all that infinite space. And thus might the Sun and fixed stars be formed, supposing their matter to be of a lucid nature. [Cambridge, England, December 10th 1692]

Newton must have thought he’d solved it: his theory successfully predicts what you see when you look up at the night sky and also demonstrates that the universe, like God, is infinite. Another triumph! But infinity is tricky and even the best guesses of a genius can sometimes mislead. Using simple high school math let’s put Newton right.

Imagine a *finite* sphere of radius **r** meters filled with God’s homogeneous substance of constant density, initially at rest. Consider a test particle on the outer surface of this sphere, distance **r** from the center. Newton’s law of gravitation tells us that it feels a central force as if all the mass of the sphere were concentrated at the center so the test particle will start infalling with a certain acceleration.

Because the sphere’s mass increases as the cube of the radius while the gravitational attraction at its boundary drops off as only the inverse-square we get our first interesting result. The initial acceleration at the boundary of a sphere undergoing gravitational collapse is simply proportional to the sphere’s radius.

If we then work out how long it takes for the sphere’s surface to collapse to its central point we get a surprise. The collapse time is exactly the same regardless of the original radius. The increase in acceleration for a larger sphere precisely compensates for the extra distance it has to fall. This has a startling consequence: if we let the radius tend to infinity to model infinite space, the infinite sphere still collapses in finite time. How long is that? Putting in the density numbers for our universe, an initially static infinite Newtonian universe takes around twenty billion years – from the time God first created it – to collapse to a uniform state of infinite density.

You might think that there is a flaw in this argument. Only an observer in the “center” of the “infinite sphere” would see everything falling towards them. But you would be wrong. Consider an observer **O** at a distance **x** meters from the “center” and consider a point **P**, **y** meters further out from the center and a point **Q**, **y** meters closer in to the center.

From **O**’s point of view ‘further out’ **P** is accelerating towards her with an acceleration **k(x + y) – kx = ky**, while ‘nearer the center’ **Q** is *also* accelerating towards her with an acceleration **kx – k(x-y) = ky**. In fact every observer sees the whole universe falling directly into her: there is no scope for the kind of clumping into separate suns and stars which Newton thought would occur.

Just one last surprising result: in Newton’s theory there is no maximum velocity, but we can ask how big a *finite universe* which looked like ours would have to be, in order to finish its collapse as a black hole. It turns out to be around 150 billion light years. This radius is called the Schwarzschild radius and is the size of a black hole with the mass-energy density of our universe: remarkably, the formula is the same both in Newton’s theory and in General Relativity.

Somehow even when Newton was wrong, he got to be right!

**Further Reading**

1. The Inflationary Universe, Alan Guth, (1998), Appendix B.

2. Is the Big Bang a black hole?

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