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## James Jeans and our finite universe(?)

“Leave only three wasps alive in the whole of Europe and the air of Europe will still be more crowded with wasps than space is with stars, at any rate in those parts of the universe with which we are acquainted.”

I love a good illustration.

For whatever reason, I’m drawn to old popular-level science books. I just finished reading “The Stars in Their Courses” by James Jeans, first published in 1931. Jeans is best known in my field for the “Jeans length”. Suppose a cloud of gas is trying to collapse under its own gravity, but is being held back by gas pressure. Jeans showed that there is a critical length scale, such that if the object is smaller than the Jeans length then pressure wins and the cloud is stable, but if it is larger then gravity wins and collapse ensues.

Jeans gives an overview of all of the astronomy of his day. It’s mostly familiar material, of course; the interesting bit is the glimpse inside the mind of the great scientist. Here’s a neat illustration:

“If we could take an ordinary shilling out of our pocket, and heat it up to the temperature of the sun’s centre [40 million kelvin], its heat would shrivel up every living thing within thousands of miles of it.”

Repeating this calculation, I think Jeans is reasoning as follows. A shilling is about 5 grams of copper (specific heat capacity 0.385 J/gram/kelvin), and so at 40,000,000 K we have about $8 \times 10^7$ J of energy. This is ‘only’ 20 kg of TNT – most bombs are at least a tonne of TNT equivalent, and they don’t do miles of damage. That much energy could raise the temperature of the surrounding air to boiling point for about a 10 metre radius. Not too promising. However, the coin will be emitting thermal radiation at x-ray wavelengths. A lethal dose of x-rays is about 5 J/kg, so our coin has enough energy to kill about 100,000 people. One must factor in the fraction of energy emitted horizontally, the fraction absorbed by biological material, the cooling of the coin, etc, but certainly it’s a very dangerous coin.

Jeans’ views on cosmology are very revealing. He is writing within 5 years of the discovery of the expansion of the universe by Lemaitre (first!) and Hubble. Jeans says:

“… we cannot say anything with certainty on the age of the universe until we know the truth as to the apparent recession of the nebulae. If these prove to be real, it will become necessary to pack all the events of astronomy, somehow or other, into a past of a few thousand millions of years. At present the whole general evidence of astronomy seems to cry out in protest against so short a past; it hardly seems possible to account for the present arrangement of the stars if their lives have been as short as this. I think it most likely that the apparent recessions of the nebulae will prove to be spurious, in which case the arrangement of the stars points to a past of millions of millions of years.”

With hindsight, we know that Jeans was wrong. The recessions of the nebulae are not spurious (though Hubble’s estimate of the age of the universe was in error), and indicate an age for the universe of around 13.7 billion (thousand million) years. So what are these “arrangements of the stars” that demand a longer timescale? I’m not sure what Jeans is referring to. My first guess is that the arrangement of stars into stable (relaxed, virialised) disk galaxies via two body interactions implies a timescale much longer than the age of the universe. Ideas? How good were estimates of the lifetimes of stars in the early 1930’s?

What shape is our universe?

“… modern astronomy regards the universe as a finite closed space, as finite as the surface of the earth, and if [the astronomer] is not yet acquainted with the whole universe, he has good reason to hope that he will be before very long. We are beginning to think of the universe as … something enormously big, but nevertheless not infinitely big; something whose limits we can fix; something capable of being imagined and studies as a single complete whole. … If [Einstein’s] theory of relativity is true, space cannot go on for ever; it must bend back on itself like the surface of the earth.”

This sounds strange to modern cosmologists. We think that there are three geometries – flat, hyperbolic and spherical – and the geometry of the universe depends on the amount of matter-energy that it contains. Why does Jeans only consider the spherical geometry? Again, I’m guessing … I can think of two reasons. It might be that, writing in 1931, only the spherical case was known. The derivation of the full Robertson-Walker metric, with its three geometry cases, was in 1935. In (the 1931 translation of) Lemaitre’s 1927 paper, he begins by saying:

According to the theory of relativity, a homogeneous universe may exist such that all positions in space are completely equivalent; there is no centre of gravity. The radius of space R is constant … straight lines starting from a point come back to their origin after having travelled a path of length pi R; the volume of space has a finite value pi^2 R^3 … Two solutions have been proposed [that of de Sitter and that of Einstein].

There is no mention of the other cases.

There is another option, going back to Einstein and highlighted by Misner, Thorne and Wheeler (pg 704):

Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space-bounded, universe:

1. From the standpoint of the theory of relativity, the condition for a closed surface is very much simpler than the corresponding boundary condition at infinity of the quasi-Euclidean structure of the universe.

2. The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contains, to a first approximation, in the equations of the theory of relativity; But this idea of Mach’s corresponds only to a finite universe, bounded in space, and not to a quasi-Euclidean, infinite universe”. [Einstein (1950)]

If Jeans followed Einstein on this, the flat and open geometries are not ignored but discarded as unphysical. Note also, with MTW, that flat and curved geometries do not determine the topology of the universe. For example, a geometrically flat universe could have the topology of a torus (donut), and thus be finite.

The question now is: why don’t we side with Einstein? I don’t know of any modern cosmologist who argues that the universe must be spherical (or at least finitely large) for these reasons. I’ll let John Wheeler elaborate on why he believed (in 1976) that the universe is closed:

Einstein long ago, of course, was led into general relativity not least by his idea—going back to Ernst Mach—that the inertia of one particle here and now arises from its interaction with other particles elsewhere in the universe. Later on, in his famous book, The Meaning of Relativity, p. 150, he talks about his reasons for still believing in a closed universe. Closure would mean a ﬁnite number of particles in the universe for given particle to interact with.

Today, of course, there is another reason that one has to think of a closed universe instead of an open universe: there is no natural way to deﬁne the boundary conditions for an open universe. One might at ﬁrst think the most natural boundary condition for the universe is asymptotic ﬂatness. But in a universe that goes asymptotically ﬂat one has no way to deﬁne what ﬂat is in the framework of a modern quantum theory. The metric is really oscillating and ﬂuctuating everywhere. No matter how great the distance to which one goes one never comes to a distance so great that space becomes ﬂat. Therefore “asymptotic ﬂatness” is a physically impossible boundary condition. No alternative boundary condition has ever been proposed for an open universe that does not run into the same diﬃculty. Closure is the only boundary condition we know that is at the same time mathematically well-deﬁned and physically reasonable.

But still we have to recognize that the universe is not something that we necessarily can be conﬁdent in making theories about. …

But I do believe it is worthwhile remembering back to 1953. That was the time when it looked as if one were in great diﬃculty with Einstein’s idea of a universe with its expansion slowing down with time. The astrophysical evidence at the time pointed to an expansion of the universe that speeded up with time. Imaginative investigators put forward all kinds of theories like “the theory of continuous creation” and “the steady state universe”. Every one of those theories meant giving up Einstein’s simple ideas. In the end it turned out that they were all wrong—that Einstein’s original idea was correct that the expansion of the universe is slowing down. The trouble was only that the astrophysical data on distances to the other galaxies had been wrong by a factor of six.

That is the story of one diﬃculty with the idea of a closed universe—a diﬃculty that turned out not to be a diﬃculty. Well, today what is the diﬃculty with the idea of the closed universe? It’s primarily that we do not see enough matter around; we appear to be short by a factor of something like 30 from having enough matter to curve up the universe into closure. However, today our colleagues in the world of astrophysics are beginning to tell us that there is much more matter in the universe than we had realized a few years ago. They ﬁnd evidence that typical galaxies weigh somewhere between 3 and 20 times as much as one had ﬁrst believed.

We now think that the expansion of the universe is accelerating, so we’re back to 1953 again. But Einstein’s simplicity is not completely lost. According to the latest data, there is enough mass-energy to close the universe, but not from matter alone. We need dark energy, which looks a lot like the “continuous creation” fields of the steady state model. This dark energy ensures that the universe does not recollapse. However, the data is not precise enough to definitively rule out flat and open geometries. Should we side with Einstein and a priori favour a closed geometry?

EInstein also raises the issue of actual infinities in the universe. The absurdities of such infinities should not be underestimated, and there are those who argue that there cannot be such in such infinities in reality. For example, George Ellis quotes Hilbert with approval:

Any claims of actual existence of physical inﬁnities in the real universe should be treated with great caution (c.f. [3], section 9.3.2), as emphasized by David Hilbert long ago ([8], p. 151):

“Our principal result is that the inﬁnite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the inﬁnite to play is solely that of an idea . . . which transcends all experience and which completes the concrete as a totality . . .”

That’s a topic for another day.

### 4 Responses

1. Very well written! Love it!

2. […] read quite a few astronomy books in my time, and this is one of the best. The problem with a lot of these books is […]

3. Most interesting post. Although i have heard of Jeans’s paper on steady-state cosmology, I have never seen it – can you give a reference? I have recently found a similar model by Einstein (never published) and suspect it predates Jeans!
Cormac O’Raifeartaigh

4. Most interesting post. Although i have heard of Jeans’s paper