Why the Universe is Finite in Size

Contemplating “Of the Nature of Things,” Lucretius (96-55 BCE) questioned the concept of the universe’s infinity when he wrote, “If we should theorize that the whole of space were limited, then it a man ran out to its very last limits and hurled a spear, would you prefer that the spear flew on, or do you think something would block it?”

Writing in an age when most men thought the night sky was gigantic dome made of terra cotta with holes poked in it to let through lamplight from Apollo’s chariot, Lucretius probably thought the former.

Infinity is a big word.

Before the invention of the telescope, man viewed the heavens as slightly out of focus. For millennia, the Milky Way was thought to be a fixed and mysterious cloud in the sky.

It was not until one wintry night in 1609 that Galileo Galilei (1514-1642) first discovered that the Via Lactea was not a cloud at all, but a horizon-spanning nebula of individual stars. His telescope was undoubtedly a purloined knock-off of a Dutch invention, but in an age when philosophers like Giordano Bruno (1548-1600) were burned at the stake for proposing much less – Bruno was made an example of for pushing Copernicus’ heliocentric theory of the Solar System – one must admire Galileo for his courage.

One of Galileo’s biggest fans was German astronomer Johannes Kepler (1571-1630) who advanced the sort of idea that, before the invention of the telescope, might have gotten poor Johann burned at the stake, too. And it is quite simply that the sky, at night, is dark.

Kepler realized that if the universe were infinite it would possess an infinite number of stars, and everywhere you’d look in the sky your line of sight would eventually fall on the surface of another star. This means that, even if the stars are as far as an infinite distance away, in an infinite universe nowhere could you look in the night sky without seeing a blinding sheet of light, inducing Kepler to write in 1610, “If the little disks of 10,000 stars are all fused into one…why do not these suns collectively outdistance our Sun in brilliance?”

With the invention of the telescope came penetrating new heresies, foremost of which was this basic fundamental probing headache: Can the power of God be unlimited in a finite universe?

By 1785, British music teacher William Herschel had mapped enough stars in the Milky Way to announce that the system is lens-shaped, it is enormous, that WE are in it, and that the solar system is heading toward Hercules at an enormous velocity.

This observation echoed Immanuel Kant’s (1724-1804) “Island Universe” idea – Kant was riffing on what Galileo said of the Milky Way – an idea that became a philosophical destination for astronomers through the next century. And concept of a finite Milky Way became the extent of our universe until well into the 20th Century.

Flash forward.

After a survey of the Milky Way using Doppler-shift stellar spectra, a Dutch team led by Jacobus Kapteyn (1851-1922) reported in 1904 that “the stars in the proximity of the Sun could be divided into two distinct streams which moved through each other in opposite directions” or in divergent lanes at different speeds. The pie-shaped wedge Kapteyn produced hinted tantalizingly of a spiral structure.

By 1918, American astronomer Harlow Shapley (1885-1972) located the center of our galaxy in Sagittarius, 30,000 light years away. He estimated it as 100,000 light years across, and containing well over 100 billion stars. More, not only did Shapley realize the existence of more galaxies outside our own, but that the expanse was possibly immeasurable. Shapley was an infinity guy.

Albert Einstein (1879-1955) was on the fence. In “Cosmological Considerations…” (1917), he considered three states of the cosmos: Open, steady state, and closed. Though these states are more easily explained by Friedmann (below), Einstein formulated another important idea:

If the overall density of the universe could be estimated, then its age and “size” could be calculated. But Einstein was clearly not comfortable with an infinite (open) universe, because he realized that little could be KNOWN about an infinite universe because it would ultimately have “zero density.”

“From the standpoint of epistemology,” Einstein wrote later, “it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a closed universe.”

The production of larger, more powerful telescopes led to the discovery of clusters of galaxies outside of our own, which, by 1922, led Swedish astronomer Carl I. Charlier (1862-1934) to revive an old but intriguing philosophical idea: If atoms cluster into stars and planets, and stars and planets cluster into Solar Systems, and Solar Systems cluster into Galaxies, and if Galaxies cluster into Galactic Clusters, then Charlier imagined that it was possible for Clusters of Clusters of Galaxies exist.

Such a continuum of hierarchies reconciles the concept of an infinite universe with a dark night sky, but kills any chance of gaging the overall size of the thing.

If every cluster has an average radius, and if, as you widen your field of view, the distance between clusters increases by a greater degree than the average cluster radius, then, as you consider larger and larger volumes of space, the distance between clusters may multiply to a much greater extent than the radius of each “Island Universe” – so that the darkness of even infinite space might be preserved – a possible solution to Kepler’s antinomy.

Put another way, if the distances between each cluster become greater like the reciprocals of an infinite series, then the equation can converge to a real number – like the binomial series, or the series for the natural exponent.

Naturally, Charlier came down on the side of an infinite universe.

By the 1920s, everybody was piling on.

During a spectrographic survey of galaxies in 1920-22, Vesto M. Slipher (1875-1969) of Flagstaff’s Lowell Observatory measured the “radial velocities” of over 400 galaxies and concluded that “only 5 from a list of 41 spirals appeared to be approaching” our system. The rest – as shown by their red-shifted spectra – were flying away, into space.

Slipher’s findings meant the obvious. If some constant of expansion could be discovered, and agreed on, then it would constitute a relatively accurate way of measuring distances to distant galaxies. The problem was that recession velocities don’t readily show consistency. They were all over the place.

But the best possible explanation for it arose that year.

In 1922, while watching fireworks, Georges Lemaitre (1894-1966) a Belgian, and Catholic Priest in his day job, considered that galactic red-shifts were the remnants of a hyper-explosion which brought the material universe into being, the product of “a giant lump of material – a sort of superatom – that was unstable, exploded and condensed as galaxies.” Lemaitre privately called this his “fireworks theory.”

Here’s what this means. Light takes time to cross space. The farther the distance away, the longer it takes light to cross, and the older is the thing you’re looking at. In short, the farther away an object is, the farther back in time you’re looking. Objects farthest out are also “closer” to the initial time of expansion.

If the universe did originate from a single “superatom,” then its speed of expansion might be considered very great, close even to the speed of light. Over a time scale that could be measured in eons, the expansion of space (relativists insist), gradually slows down, so that in our own little corner of the universe the speed of expansion is practically zero. Here comes the time thing again: The farther out you image, the closer you get to the moment of creation; the closer you get to the moment of creation, the faster will you measure the universe’s fiery debris flying away. Galactic recession explained.

That same year, 1922, Aleksandr Friedmann, a Russian meteorologist (1888-1925), went further when he realized that the universe could fall into one of three states: Open, steady and closed. If the universe had not enough mass to stop its expansion, it would keep on expanding forever as an “open” universe. If the mass of the universe is just enough to overcome and keep further expansion at zero, the universe is “steady state” – but, since nature tends to botch such tightrope acts, this is unlikely to be the case. If the universe had enough mass to stop and pull itself inward, then it’s said to be “closed.” And if closed, the universe will fall back inward, pull itself into another little but massive singularity, and blow itself into the smithereens of another fresh creation. This last is what is meant by an oscillatory universe.

By 1928, Jan Oort (1900-92), a Dutch astronomer who studied under Kapteyn, confirmed that the Milky Way was a vast spiral galaxy, around which the Solar System takes about 225 million years make one revolution around the center. 225 million years is either an interesting coincidence, or of significant interest, as it is roughly the period between mass extinctions on the Earth.

I’ll leave it to the reader to use Shapley’s orbital radius of 30,000 light years and Oort’s 225 million year Solar System orbit to calculate our speed around the galaxy.

But Oort realized that the Milky Way wasn’t massive enough to hold itself together. It has matter missing. This is the Dark Matter astronomers are so eager to find. But at least Oort got a zone of comets named after him.

[An Italian/Chinese team of physicists recently claimed to have detected Dark Matter via a pool of sodium iodide a mile under Italy’s Gran Sasso Mountain. But I’ll leave it to the reader to pull up the 4-17-08 New York Times (p. A16) story to read further.]

Finally, after a survey of 46 nebulae, Edwin Hubble (1889-1953) measured their emission lines red-shifted pretty much in one direction, and announced his best estimate in 1929. He was on the right track, but his first number is far too high: 500 km/s per megaparsec. And it was a number that had Einstein kvetching:

In a 1945 revision of “Cosmological Considerations,” Einstein wrote:

“From the measured value of h [the Hubble constant] we get for the time of existence of the world up to now 1.5 [billion] years. This age is about the same as that which one has obtained from the disintegration of uranium for the firm crust of the earth. This is a paradoxical result, which for more than one reason has aroused doubts as to the validity of the theory…. In this case I see no reasonable solution.”

Compared to his other literary voice, this is sheer literary apoplexy. The Earth itself is now estimated to be between 4.5 and 4.8 billion years old. So how can the Hubble constant lead to an age that’s three times younger than our Solar System?

The Hubble Constant isn’t a constant, it’s an estimate, because galactic recession isn’t constant. Some are flying parallel to us, and a few, like the Andromeda 2. 4 million light years away, are heading right toward us. In the case of Andromeda, at 35 km/s. (I’ll let you do the math.) But the numbers of receding galaxies tend to cluster along a line in the Hubble diagram enough to plot a single line through them. The art of the science is figure out just where to put the line.

To date, we have a better fit. But I must admit to have pulled the current Hubble constant off Wikipedia, and it is:

71 km/s per Mpc.

What does it mean?

Mpc means mega-parsec. Mega- is a prefix for million. A parsec is the distance that light travels in 3.26 years, so a megaparsec the distance light travels in 3.26 million years.

The Hubble constant means that for every megaparsec, or for every million parsecs distant, a will be red-shifted as moving another 71 kilometers per second FASTER.

The simplest model is the one proposed in 1935 by British cosmologist Edward A. Milne (1896-1950) who proposed a simple spherical universe expanding from a single point. (Contemporary theory requires no actual center of expansion, but holds that the entire universe occupies the surface, or membrane, of a four dimensional hyper-sphere. But as a former teacher I know that it’s essential to teach the simplest idea first, no matter how jerry-rigged.)

Now let’s take another look at the Hubble constant: For every 3.26 million light years we peer into the past, the nebulae are traveling 71 km/sec faster.

Now, turn this idea around: For every 3.26 million years in the future, the nebulae have slowed down by 71 km/sec.

So if we treat the megaparsec as a TIME, the Hubble constant becomes a DECELERATION.

Change 71 km/sec into 71,000 m/s, and convert Mpc, or 3.26 million years into seconds, then divide:

a = -71 x 1000 (m/s) / (3.26 x 1,000,000 years x 31536000 s/year)

The negative sign means deceleration; and I’m using “a” for acceleration so as not to confuse my interpretation with the real symbol for the Hubble constant, h. What we get is the more easily portable

a = -.00000000061 m/ss

Think of the Big Bang as expanding gases exploding from a single point at an initial speed of the speed of light, c = 300,000,000 m/s. Using the standard equation of motion

(v – c) = at

Consider the final speed of expansion, v, to be so low that it’s practically nothing. And solve for t

-c/a = t

And congratulations. You’ve just calculated the age of the universe. (That is, after you’ve converted from seconds back to years. Note also that the negative signs cancel out.) Not only is it close to the currently accepted answer, but it differs from what Einstein would’ve calculated by 2/3rds, that is, if he had the correct number in the first place.

What’s more, by integrating the standard equation of motion with respect to time, you can calculate the radius of Milne’s universal sphere. (This printer can’t handle subscripts, but any standard physics textbook will show the required equation.)

There’s only one problem with this.

It’s only a billion and a quarter years older than our galaxy, and gets a radius one-third of the visual field (about 18 billion light years), neither of which makes much sense.

Einstein voiced his reservations when he wrote in the 1945 “Cosmological Considerations”:

“An infinite universe is possible only if the mean density of matter in the universe vanishes. Although such an assumption is logically possible, it is less probable than the assumption that there is a finite mean density of matter in the universe.”

Immanuel Kant argued that labeling something as tangible and complete as the universe with an intangible concept like infinity is an unproductive contradiction. A bit like asking if a rock possesses consciousness. Kant liked his universe tidy, finite, and with a definite beginning.

Einstein channels Kepler:

“Why is space not so filled with radiation as to make the nocturnal sky look like a glowing surface?”

But it does glow. Though coolly. At about 3 Kelvin.

What the two Bell Telephone lab scientists Arno Penzias and Robert Wilson picked up in 1965 was the low hiss of microwave radiation: That 3 Kelvin glow from everywhere in the sky. Instead of trying to eliminate it (which was their job), they studied it and realized it might be the afterglow from the first seconds of creation. It was temperature to which the universe has cooled in the 13.767 billion years of its estimated age.

It is a remnant of the Big Bang.

Or is it evidence of something else? Does that spooky 3 Kelvin glow originate from somewhere else? From somewhere out there. From somewhere beyond?

As to the question of an infinite universe, perhaps we’re asking the wrong question. Perhaps we will be no more aware of the true nature of the universe than an ant would be of the shape of the Earth.

But if you twist my arm about it, I will guarantee with absolute certainty that there’s a solid 50/50 chance that the universe is finite.

It’s space that’s infinite.