The Balmer Formula

Leading up to the Formula: 1869 - 1882


In the years after the work of Kirchhoff and Bunsen, the major goal in spectroscopy was to determine the quantitative relationships between the lines in the spectrum of a given element as well as relationships between lines of different substances.
For example, George Johnstone Stoney in 1869 speculated that spectra arose from the internal motions of molecules. However, his mathematical theory was rejected and in 1881, Arthur Schuster concluded:

"Most probably some law hitherto undiscovered exists . . . . "

One year later, Schuster added:

"It is the ambitious object of spectroscopy to study the vibrations of atoms and molecules in order to obtain what information we can about the nature of forces which bind them together . . . But we must not too soon expect the discovery of any grand or very general law, for the constitution of what we call a molecule is no doubt a very complicated one, and the difficulty of the problem is so great that were it not for the primary importance of the result which we may finally hope to obtain, all but the most sanguine might well be discouraged to engage in an inquiry which, even after many years of work, may turn out to have been fruitless. . . . In the meantime, we must welcome with delight even the smallest step in the desired direction."

The Balmer Formula: 1885
On June 25, 1884, Johann Jacob Balmer took a fairly large step forward when he delivered a lecture to the Naturforschende Gesellschaft in Basel. He first represented the wavelengths of the four visible lines of the hydrogen spectrum in terms of a "basic number" h:

Balmer recognized the numerators as the sequence 3², 4², 5², 6² and the denominators as the sequence 3² - 2², 4² - 2², 5² - 2², 6² - 2².
So he wound up with a simple formula which expressed the known wavelengths (l) of the hydrogen spectrum in terms of two integers m and n:

For hydrogen, n = 2. Now allow m to take on the values 3, 4, 5, . . . . Each calculation in turn will yield a wavelength of the visible hydrogen spectrum. He predicted the existence of a fifth line at 3969.65 x 10¯⁷ mm. He was soon informed that this line, as well as additional lines, had already been discovered.

Here are some calculations using Balmer's formula.

At the time, Balmer was nearly 60 years old and taught mathematics and calligraphy at a high school for girls as well as giving classes at the University of Basle. Balmer was very interested in mathematical and physical ratios and was probably thrilled he could express the wavelengths of the hydrogen spectrum using integers.
Balmer was devoted to numerology and was interested in things like how many sheep were in a flock or the number of steps of a Pyramid. He had reconstructed the design of the Temple given in Chapters 40-43 of the Book of Ezekiel in the Bible. How then, you may ask, did he come to select the hydrogen spectrum as a problem to solve?

One day, as it happened, Balmer complained to a friend he had "run out of things to do." The friend replied: "Well, you are interested in numbers, why don't you see what you can make of this set of numbers that come from the spectrum of hydrogen?" (In 1871 Ångström had measured the wavelengths of the four lines in the visible spectrum of the hydrogen atom.)

Balmer published his work in two papers, both published in 1885. The first, titled 'Notiz über die Spektrallinien des Wasserstoff,' is the source of the equation above. He also gives the value of the constant (3645.6 x 10¯⁷ mm.) and discusses its significance:

"One might call this number the fundamental number of hydrogen; and if one should succeed in finding the corresponding fundamental numbers for other chemical elements as well, then one could speculate that there exist between these fundamental numbers and the atomic weights [of the substances] in question certain relations, which could be expressed as some function."

He goes on to discuss how the constant determined the limiting wavelength of the lines described by the Balmer Formula:

"If the formula for n = 2 is correct for all the main lines of the hydrogen spectrum, then it implies that towards the utraviolet end these spectral lines approach the wavelength 3645.6 in closer and closer sequence, but cannot cross this limit; while at the red end [of the spectrum] the C-line [today called H_a] represents the line of longest possible [wavelength]. Only if in addition lines of higher order existed, would further lines arise in the infrared region."

In this second paper, Balmer shows that his formula applies to all 12 of the known lines in the hydrogen spectrum. Many of the experimentally measured values were very, very close to Balmer's values, within 0.1 Å or less. There was at least one line, however, that was about 4 Å off. Balmer expressed doubt about the experimentally measured value, NOT his formula! He also correctly predicted that no lines longer than the 6562 x 10¯⁷ mm. line would be discovered in this series and that the lines converge at 3645.6 x 10¯⁷ mm.
with m = 2, 3, 4, . . . and n = 1, 2, 3, . . . ; but the two constants change in a particular pattern.
By higher order, he means allow n to take on higher values, such as 3, 4, 5, and so on in this manner:

n	m
3	4, 5, 6, 7, . . .
4	5, 6, 7, 8, . . .
5	6, 7, 8, 9, . . .

There is also this one, but I'm not sure if Balmer discussed it:

n	m
1	2, 3, 4, 5, . . .

Before leaving Balmer, several points:

1) Balmer's Formula is entirely empirical. By this I mean that it is not derived from theory. The equation works, but no one knew why. That is, until a certain person.
2) That certain person was born October 7, 1885 in Copenhagen. His name? Niels Henrik David Bohr.
3) Be careful when you read about Balmer's Formula in other books. Often, a form of the formula using frequency rather than wavelength is used.

At first Balmer's formula produced nothing but puzzlement, since no theoretical explanation seemed possible. In 1890 Johannes Robert Rydberg generalized Balmer's formula and showed that it had a wider applicability. He introduced the concept of the wave number v, the reciprocal of the wavelength l, and wrote his formula as

v = 1/l = R (1/n₁² - 1/n₂²)

where n₁ and n₂ are integers and R is now known as the Rydberg constant (value = 10973731.534 m¯¹). Later many other atomic spectral lines were found to be consistent with this formula.
For the lines in the hydrogen spectrum (today called the Balmer series), n₁ = 2 and n₂ takes on the values 3, 4, 5, 6, . . . . If you try the calculations (I don't mind if you do, I can wait.), remember to do one over the answer, so as to recover the wavelength.

---

In 1885, Balmer wrote these prophetic words:

"It appeared to me that hydrogen . . . more than any other substance is destined to open new paths to the knowledge of the structure of matter and its properties. In this respect the numerical relations among the wavelengths of the first four hydrogen spectral lines should attract our attention particularly."

In 1913, Niels Bohr will announce what is now call the Bohr Model of the Atom. He will offer the correct mechanism for the lines in the hydrogen spectrum.

Read More

The Bohr Model of the Atom

On June 19, 1912, Niels Bohr wrote to his brother Harald:

"Perhaps I have found out a little about the structure of atoms."

I. Setting the Stage
The Bohr model of the atom deals specifically with the behavior of electrons in the atom. In constructing his model, Bohr was presented with several problems.
Problem #1: charged electrons moving in an orbit around the nucleus SHOULD radiate energy due to the acceleration of the electron in its orbit. The frequency of the emitted radiation should gradually change as the electron lost energy and spiraled into into the nucleus. Obviously this was not happening, because the spectral lines of a given element were sharply defined and unchanging.
Problem #2: the spectral lines did not show overtones (or harmonics). These are lines where the frequency is double, triple and so on of the fundamental frequency. The lines of the spectrum of each element were scattered about with no apparent pattern, other than the purely empirical formula of Balmer (which dates to 1885). However, no one knew what Balmer's formula meant.
J.J. Tomson's model was constructed with full knowledge of problem #1 above. What Thomson did is to extend the positive charge to the same size as the atom (radius = 10¯⁸ cm.) and allow the electrons to distribute themselves inside. The calculations for his "plum pudding" model, published in 1904, showed that the model did produce electron arrangements that were stable.
However, the Thomson model was conclusively destroyed by Rutherford's 1911 nucleus paper. (In the future -- 1913 and years later -- other discoveries will be made that the Thomson model fails to account for, but the Rutherford model does. Of course, Thomson, Rutherford, Bohr, etc. were not aware of these. There were even efforts in 1914 and 1915 to use the Thomson Model, but these efforts went nowhere.)
The nuclear model of Rutherford's was supported by evidence that could not be refuted. However, if electrons rotated around a nucleus, they would either rip the atom apart or self-destruct. Bohr's answer to the above problems appeared in print for the world to see in July 1913. However, as you can see from Bohr's letter to his brother, the journey to the answer started much earlier.
Bohr wrote out his ideas to date in a memo to Rutherford sometime in June/July 1912. This memo (with one missing page) still exists and if you were a qualified scholar, you could go visit it and read it!! Bohr leaves Manchester right after writing the memo and goes to Copenhagen and is married on August 1, 1912.
The critical part of Bohr's thinking was the making of two assumptions. Bohr himself described these assumptions on page 7 of his famous paper:

"(1) That the dynamical equilibrium of the systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between different stationary states cannot be treated on that basis.
(2) That the latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory."

Making the assumptions at all was a bit shaky, but in September 1913, at a meeting of the British Association for the Advancement of Science, Sir James Jeans remarked:

"The only justification which can be offered for the moment with regard to these hypotheses is the very important one that they work in practice."

---

II. Assumption #1: Electrons Move, yet are Stable
Bohr describes "stationary states" (He never uses the word the modern term "orbit.") on page 5:

"According to the above considerations, we are led to assume that these configurations will correspond to states of the system in which there is no radiation of energy states which consequently will be stationary as long as the system is not disturbed from outside."

As an electron moves in a "stationary state" it emits no radiation whatsoever. This violated a branch of science called electrodynamics (having to do with movement of charged particles and their amount of energy), but the fact is that the atom is stable and DOES NOT emit radiation in the manner predicted. It is this branch which predicts the electron will lose energy and crash into the nucleus (this is the problem #1 mentioned at the top of the file).
In this assumption is some of Bohr's daring nature. While he realized electrodynamics is useless (second part of his sentence), he proposed to use "mechanics" to describe the motion of an electron in its orbit (first part of the sentence). Mechanics deals with things like inertia, momemtum and other features of movement not involving electrical charges. He was willing to throw out well-supported scientific ideas that didn't work, but was also willing to keep other ideas that allowed him to make calculations.
The justification for Bohr deciding to assume mechanics held in the atom, but electrodynamics didn't? The results he got had two features: 1) they concurred with already known results and 2) offered an explanation for why some results were found and not others.

III. Assumption #2 - Incorporation of Planck's Constant
The "latter process" in assumption #2 is described at the end of assumption #1 -- "the passing of the systems between different stationary states"
What Bohr proposed is that the atom will emit (or gain) energy as it moves from one stationary state to another. However, the amounts of energy will not be any old amount, but only certain, fixed values. Those values will be the DIFFERENCES in energy between the stationary states.
Bohr says on p. 7:

"The second assumption is in obvious contrast to the ordinary ideas of electrodynamics but appears to be necessary in order to account for experimental facts."

The experimental facts refered to are the lines in the spectrum of hydrogen.
What Planck had discovered in 1900 was a fundamental limitation on nature. Energy is not emitted or absorbed in a continuous manner, but rather in small packets of energy called quanta. Emission and absorption occured in a DIScontinuous manner. In other words, an atom moved from one energy state to another state in steps. In the mathematical description of this process there occured a new constant of nature, discovered by Planck and named after him.
Planck's constant, symbolized by h, was involved in governing HOW MUCH energy a given quantum had. The amount of energy was directly dependent on the frequency of the radiation according to the following equation:

E = hν

This famous equation was first announced by Planck in 1900.
Bohr argued that Planck's constant should be used to help account for the stability of the atom. This reversed the technique of others who were trying to use atomic models to determine the physical significance of h. Bohr also realized that, if he was correct, his theory should produce a constant with the units of length. This constant would characterize the distance of the electron from the nucleus.
Bohr said on page 2:

The principal difference between the atom-models proposed by Thomson and Rutherford consists in the circumstance the forces acting on the electrons in the atom-model of Thomson allow of certain configurations and motions of the electrons for which the system is in a stable equilibrium; such configurations, however, apparently do not exist for the second atom-model. The nature of the difference in question will perhaps be most clearly seen by noticing that among the quantities characterizing the first atom a quantity appears -- the radius of the positive sphere -- of dimensions of a length and of the same order of magnitude as the linear extension of the atom, while such a length does not appear among the quantities characterizing the second atom, viz. the charges and masses of the electrons and the positive nucleus; nor can it be determined solely by help of the latter quantities.
. . . it seems necessary to introduce in the laws in question a quantity foreign to the classical electrodynamics, i. e. Planck's constant, or as it often is called the elementary quantum of action. By the introduction of this quantity the question of the stable configuration of the electrons in the atoms is essentially changed as this constant is of such dimensions and magnitude that it, together with the mass and charge of the particles, can determine a length of the order of magnitude required.

What Bohr wass pointing out is that Rutherford's model (with its constants of mass and charge) cannot produce a unit of length, but with the introduction of h, such a length constant could be produced. If you write h² / me², you get a value with the units of length and of the proper magnitude. Here are the numbers (modern values):

This length calculation yielded a value which today is named the Bohr radius. Its modern value is 5.292 x 10¯¹¹ m and is symbolized a_o.
Let's stop and review for a moment. What exactly is the Bohr Model of the Atom?
The Bohr model has the following features:

1) There is a nucleus (this was Rutherford's discovery).
2) The electrons move about the nucleus in "stationary states" which are stable, that is, NOT radiating energy.
3) When an electron moves from one state to another, the energy lost or gained is done so ONLY in very specific amounts of energy, not just any old amount.
4) Each line in a spectrum is produced when an electron moves from one stationary state to another.

Today, we call this model an example of a "quantized" atom. The term "quantum" was introduced by Planck to describe a small bundle of energy. So a quantized atom being stimulated is shooting out trillions of quanta (plural) of energy per second.
One way to think about the quanta of energy streaming out is to think of a machine gun shooting out thousands of bullets per second. It SEEMS like a stready, unbroken stream of metal, but it is not. Each bullet is a "quantum."
Another example is water coming out of a hose. It SEEMS like a steady, unbroken stream of water, but we know it is just trillions and trillions of tiny, individual water molecules. The idea is the same with the energy quantum.
Still another example is a stream of gas shooting out a nozzle. It SEEMS like a steady, unbroken stream of gas, but in reality it is trillions and trillonsof individual gas molecules all moving in the same direction. Get the idea now??
Last example. In 1905, Albert Einstein wrote an article in 1905 titled "On a Heuristic Point of View about the Creation and Conversion of Light." In it he uses Planck's idea of a quantum to explain something called the photoelectric effect. He wound up showing that certain equations governing energy behavior are exactly the same as those for a gas in the same volume. By analogy, then, energy could be treated the same way as a gas -- as a collection of particles moving around within the volume. He wrote:

From this we then conclude:
Monochromatic radiation of low density behaves -- as long as Wein's radiation formula is valid -- in a thermodynamic sense, as if it consisted of mutually independent energy quanta . . . .

Two paragraphs later, in referring to "monochromatic radiation," he uses the phrase "discontinuous medium consisting of energy quanta."

---

Feature #4 above is Bohr's explanation of the mechanism for the production of lines in the hydrogen spectrum. The story surrounding this is interesting. At least, I think so.
Bohr was married, as I said, in August, 1912. He and his wife did not honeymoon in Norway as planned, but returned more-or-less immediately to Manchester. The new term was beginning and Bohr was behind schedule in finishing some work that Rutherford had assigned to him. Bohr finished that work and eventually the term ended, ending Bohr's year-long government grant for study.
Bohr even fell behind schedule in writing the paper I am discussing. He wrote a note to Rutherford on November 4, 1912 apologizing for the time he was taking "to finish my paper on the atoms and send it to you." By January 1913, Bohr and his wife were back in Copenhagen where he assumed his new position as assistant to the physics professor. This slowed down the work on the atomic model paper, but it was not abandonded.
The next event in this story (which is not the complete one, only highlights) is a January 31, 1913 letter to Rutherford in which Bohr excluded the "calculation of frequencies corresponding to the lines of the visible spectrum." However, the paper Bohr mailed to Rutherford on March 6, 1913 contained the correct mechanism for the production of the lines of the spectrum. What happened?
The letter shows Bohr HAD been thinking about spectra. It was probably here (in the first week of February 1913) that (according to Bohr's recollection in 1954) he was asked by Hans Marius Hansen (a yound Danish physicist) how Bohr's new atomic model would explain the hydrogen spectrum. Bohr's reply was that he had not seriously considered the issue, believing the answer to be impossibly complex. (Remember that by 1913, several thousand lines, of different elements, were known AND many of these lines exhibited bizarre splittings - called the Zeeman effect -- in magnetic fields. No one, and I mean no one, had any answers for what was going on.)
Hansen disputed Bohr's position and insisted Bohr look up Balmer's work. Bohr did so and soon had the answer for how the lines are produced. In the 1954 interview mentioned above, Bohr said:

"As soon as I saw Balmer's formula, the whole thing was immediately clear to me."

Up to that point, Bohr had not been aware of the Balmer formula. However, it was quickly incorporated into his paper and the final product was shipped off to Rutherford.
So what is the mechanism that makes the lines in the spectrum? On page 11, after discussing the mechanism from p. 8 onward, Bohr said:

" . . . that the lines correspond to a radiation emitted during the passing of the system between two different stationary states."

Later, on p. 14, he wrote:

"We are thus led to assume that the interpretation of the equation (2) is not that the different stationary states correspond to an emission of different numbers of energy-quanta, but that the frequency of the energy emitted during the passing of the system from a state in which no energy is yet radiated out to one of the different stationary states, . . . ."

So, in other words, each line is equal to the DIFFERENCE in energy as an electron moves between two stationary states. And, if you read Bohr's paper, at the bottom of page 8, you will see two equations where he writes W_τ2 minus W_τ1. This subtraction, of course, yields an answers which we call a "difference."
This is also the reason why the harmonics (see back to problem #2) were not observed. Harmonics are observed when an object (such as an electron) resonates at a specific frequency, called the fundamental. Overtones of twice the frequency, three times the frequency and so on are also observed. Here the frequency of the line in the spectrum had to do with energy gained or lost as the electron moved from one stationary state to another.The line is an energy DIFFERENCE!! The fundamental frequency of the electron was not involved at all with the lines!! No one until Bohr, not even Planck or Einstein, had thought to challenge the idea that spectral lines were produced by electrons emitting the specific frequenceies of each line.
By the way, exactly how an electron moves between two stationary states is never discussed by Bohr. How does the electron know to go to another stationary state of a specific energy amount? Why that amount and not some other amount? How does the electron know to stop at the right energy amount? Maybe Bo knows, but Bohr certainly didn't.
Bohr sent a second draft to Rutherford about two weeks after the first since he had "found it necessary to introduce some small alterations and additions." He then traveled to Manchester to persuade Rutherford that the paper would harmed greatly by any reduction in length. On May 10, Bohr returned the final, corrected proof to Rutherford and this became the published version.
In August 1914 (slightly more that a year after Bohr's paper was published), Rutherford said:

"N. Bohr has faced the difficulties [of atomic structure] by bringing in the idea of the quantum. At all events there is something going on which is inexplicable by the older mechanics."

Of course, Rutherford was completely correct. The "older mechanics" had to give way to the "quantum mechanics." Bohr would lead a revolution in thinking during the 1920s and 1930s that continues to be profitably mined even today.
At age eighty, J.J. Thomson, stooped by age, but still sharp of mind, wrote:

"At the end of 1913 Niels Bohr published the first of a series of researches on spectra, which it is not too much to say have in some departments of spectroscopy changed chaos into order, and which were, I think, the most valuable contributions which quantum theory has ever made to physical science."

---

Read More

Expanding Universe

 Expanding Universe

I: Introduction

In 1929 an astronomer named Edwin Hubble announced a remarkable observation that changed our view of the world more than almost any other single discovery made this century. In every direction he looked, every galaxy in the sky was moving away from us. The nearby ones were moving relatively slowly, but the farther away a galaxy was the faster it was heading out. What can account for our great unpopularity? Is our galaxy somehow different from all others? It turns out that there is another, arguably simpler explanation that is well supported by many other observations. It is that the entire universe itself is expanding! As I will explain below, this expansion means not only that we should see every other galaxy moving away from us, but that observers in another galaxy should see exactly the same thing. In a uniform expanding universe, every observer sees herself at the center of the expansion, with everything else moving outwards from her.

This statement forms the basis of our current theories of the structure and history of the universe. The study of the overall structure of the universe is called cosmology. The theory that has come to dominate cosmology since Hubble's observations goes by several names, but is most commonly known as the big bang model. (As I will explain later, this name is somewhat misleading, but owing to its widespread acceptance I will continue to use it.)

This paper describes the big bang model. Section II describes what it means to say the universe is expanding, and subsequent sections address some questions that commonly arise in connection with the model.

In Section III I discuss whether the universe is infinite or finite. While we don't yet know the answer to this question, Einstein's general theory of relativity predicts that finite universes contain a larger density of matter than infinite ones, so by measuring the density in the universe we should hopefully be able to make the determination. I conclude this section by describing what it would mean for the universe to be infinite or finite.

In Section IV I talk about the origin and history of the universe. As the universe expands and galaxies move apart from each other the average density is decreasing. If we extrapolate the expansion backwards we conclude that there was a time roughly 10-15 billion years ago when the density was nearly infinite. In this section I briefly outline the history of the universe from that time to the present.

In Section V I continue the story, describing what relativity theory predicts will happen to the universe in the future. The two possibilities are that the universe will continue to expand forever or that it will eventually slow down and begin contracting. The theory tells us that which one will happen depends on whether or not the average density of the universe exceeds a certain value, the same value that determines whether the universe is infinite or finite. Relativity predicts that an infinite universe will continue to expand forever, whereas a finite universe will expand for a finite time and then contract. I conclude by describing these two scenarios.

The paper is followed by a series of endnotes that discuss other issues, including evidence for the big bang model as well as possible problems with it and the proposed solutions. It is not necessary to read the endnotes to understand the rest of the paper. 

 

II: The Expanding Universe: An Overview

Simple analogies can clarify what it means for the universe to expand, but they can also be misleading. I will make heavy use of one analogy, attempting to point out its shortcomings as I proceed. Think of the universe as a rubber sheet being stretched out. (If you are comfortable with visualization in three dimensions you can imagine a raisin cake expanding instead, but for the purpose of illustration I will stick with the two dimensional case.) Now imagine that there are thumbtacks stuck into the rubber at various points representing galaxies. (In the raisin cake analogy these would be the raisins.) As the rubber (the universe) is stretched (expands), the thumbtacks (galaxies) all get farther apart. Note that I haven't said anything yet about how big the rubber sheet is. For all we know it might be infinite. (This point will be addressed in a later section.) What I mean when I talk about expansion is that the rubber is being stretched out, causing the distances between the thumbtacks to increase.

To see what this expansion should look like to us, imagine an observer sitting on one of the thumbtacks. This observer imagines himself to be at rest and measures all movement relative to his thumbtack (galaxy). Since the distance between any two thumbtacks is increasing, it will appear to him that all the other ones are moving away from him. How fast will another thumbtack appear to move? That depends in part on how fast the rubber sheet is being stretched out, i.e., how fast the universe is expanding. In addition, however, the apparent speed of the other thumbtacks is also dependent on their positions relative to the observer. The nearby thumbtacks will appear to be moving away very slowly, whereas the distant ones will appear to be moving away much faster. To see why this is so, suppose the rubber sheet doubles in size in one second. 

The thumbtack that began one foot away from you is two feet away, meaning it appears to have moved by a foot. Its apparent velocity is therefore 1 foot per second. In the same time the thumbtack that started out three feet away also ends up twice as far away (six feet), but this means that it appears to have moved away at three times the speed of the first thumbtack (three feet per second). In terms of the expanding universe, this means that not only will every galaxy appear to be moving away from us, but the speed with which it does so will be directly proportional to its distance from us. A galaxy that is four million light years away will have twice the apparent velocity of one that is two million light years away.

This pattern is precisely what Hubble observed. Not only did he see that all distant galaxies are moving away from us and that the more distant ones are moving away more rapidly, but he found that the rate at which they were receding from us was proportional to their distance from us. In short, his observations exactly matched what we just predicted for an expanding universe. This proportionality is known as Hubble's Law.¹

A problem arises when we consider an expanding universe. Suppose everything in the universe were to double in size. The distances between galaxies would double, the size of the Earth would double, the size of all our meter sticks would double, and so on. It would seem to an observer (who will also have doubled in size) as if nothing had happened at all. So what do we mean by saying the universe expands?

In fact, not everything grows as the universe expands. In the example of the rubber sheet, the distance between thumbtacks keeps increasing but the thumbtacks themselves remain the same size. Similarly, while distant galaxies are pulled away from each other by the expansion, smaller objects like meter sticks, people, and the galaxies themselves are held together by forces that prevent them from expanding. So we expect that billions of years from now galaxies will still be roughly the same size they are today, but the distances between them will on average be much larger.

III: Infinite or Finite

People have wondered for millennia whether the universe is limited in size or goes on forever. Fortunately we now have modern science to step in and supply us with the answer, which is that we don't know.

We believe that the universe is governed by Einstein's theory of general relativity, which among other things addresses such matters as the overall structure of the universe. In the early 1920s Alexander Friedmann showed that using one assumption (which I discuss below), the equations of general relativity can be solved to show that a finite universe must have a larger density of matter and energy inside it than an infinite universe would have.²  There is a certain critical density that determines the overall structure of the universe. If the density of the universe is lower than this value, the universe must be infinite, whereas a greater density would indicate a finite universe. These two cases are referred to as an open and closed universe respectively.³

The critical density is about 10^-29 g/cm³, which is equivalent to about five hydrogen atoms per cubic meter.⁴  This may not seem like a lot; by comparison the density of water is roughly 1 g/cm³ or about 500 billion billion billion hydrogen atoms per cubic meter. However, we live in a very dense part of the universe. Most of the universe is made up of intergalactic space, for which a density as low as the critical density is plausible.

Aside from the theory of relativity itself, Friedmann's only other assumption in deriving his results was that on average the density of the universe was the same everywhere. This doesn't mean that every place in the universe is exactly the same. I already mentioned that the Earth is much more dense than space. However, if I measure the average density in our galaxy it will be about the same as the average density in any other galaxy, and the number of galaxies per unit volume should be roughly the same in different parts of the universe. This assumption matches all our observations to date. Individual galaxies differ from one another in some of their specific properties, but on average their properties don't appear to change from one region of the sky to another. Nonetheless, the idea that the universe is roughly the same everywhere—a property known as homogeneity—is still an assumption. We can probably only see a tiny fraction of the universe and we have no guarantee that the parts we cannot see look like the parts we can. Lacking any evidence to the contrary, however, we will assume that the property of homogeneity holds true.

So we should be able to answer the question of the universe being infinite or finite by measuring the density of everything around us and seeing whether it is above or below the critical value. This is true in principle, and measuring the average density of the universe is a very active field of research right now. The problem is that the measured density turns out to be pretty close to the critical density. Right now the evidence seems to favor an infinite universe, but it is not yet conclusive.

To recap, one of the assumptions of the standard big bang model is that the universe is more or less homogeneous—the same everywhere. As far as we can see, which is billions of light years in every direction, this assumption appears to be correct. Under this assumption general relativity says that whether the universe is infinite or finite depends on its density. Measurements of that density reveal that it is close to the critical value. Right now the data seem to point more towards an open (infinite) universe, but new data coming in the next 10-20 years should resolve the question much more definitively.

Given our uncertainty about this question, I will say a few things about what it would mean if the universe is infinite or finite and how those two possibilities relate to the idea of the universe expanding.

An infinite universe is in some ways easier to imagine than a finite one. Since the universe is supposed to be everything that exists, it seems intuitive that it should go on forever. Of course an infinite universe is impossible to picture, but we can get at what it means by saying that no matter how far you go there will always be more space and galaxies. It is hard, however, to reconcile this picture with the idea that the 

universe is expanding. If it's already infinite, how can it expand?

To see how, remember that by expansion we mean that the distance between galaxies is increasing. Suppose right now there is a galaxy every million light years or so. After a long enough time this infinite grid of galaxies will stretch out so that there is a galaxy every two million light years. The total size of the universe hasn't changed—it's still infinite—but the volume of space containing any particular group of galaxies has grown because the separation between the galaxies is now larger.⁵

What about a finite universe? This phrase sounds like a contradiction because if the universe ends somewhere then we would naturally want to know what was beyond it, and since the universe includes everything, whatever is beyond that edge should still be called part of the universe. The resolution of this paradox is that even if the universe is finite, it still doesn't have an edge. If I head off in one direction and resolve to keep going until I find the end of the universe, I eventually find myself right back where I started. A finite universe is periodic, meaning that if you go far enough in any direction you come back to where you started.

Trying to picture a closed (finite) universe is in some ways even harder than trying to picture an open (infinite) universe because it is easy to mislead yourself. For example, people often compare a two-dimensional closed universe to the surface of a balloon. This analogy is helpful because such a surface has the property of being periodic in all directions, and it is easy to picture the expansion of such a universe by imagining the balloon being blown up. In fact, this analogy is like the rubber sheet analogy I used before, except now the sheet has been wrapped up to form a sphere. The problem is that this picture immediately leads to the question of what is inside the balloon.

This question comes from taking the analogy too literally. Nothing in general relativity says that a two-dimensional closed universe would have to exist as a sphere inside a three-dimensional space; the theory only says that such a universe would have certain properties (e.g. periodicity) in common with such a sphere. For this reason I think it is useful to keep the balloon in mind as a convenient analogy but it is ultimately best to think of the closed universe as a three-dimensional space with the strange property that things which go off to the right eventually come back again from the left.

What does expansion mean in a closed universe? Since this universe has a finite size, it makes sense to talk about that size increasing. Again suppose that there is now a galaxy every million light years. Suppose also that if I were to head off in a straight line I would travel 100 billion light years before coming back to where I started, passing about 100,000 galaxies on the way. If I take the same journey billions of years later, the number of galaxies won't have changed but the distances between them will have doubled, so the total distance for the round trip will now be 200 billion light years.⁶

IV: The Big Bang & the History of the Universe

At the beginning of this century, physicists generally had a strong bias toward the idea that the universe was essentially unchanging. Local phenomena would of course change from minute to minute, and stars and galaxies might be born and die, but taken as a whole the universe was assumed to be more or less the same now as it had been billions or trillions of years ago, with no beginning or end. Einstein, disturbed that his theory of general relativity seemed to be inconsistent with a static universe, tried to modify the equations of the theory. When Hubble's observations showed that the universe was indeed expanding, Einstein retracted this modification and called it the biggest blunder of his life.

Given that the universe is growing, the question of whether the expansion started at some point in the past inevitably arises. Our current theories say the expansion did have a beginning. This section discusses why we believe this and what it means to even say so. It also contains a brief outline of the history of the universe from that beginning to the present day.

The Big Bang

To see what it means to say the universe had a beginning, consider a group of galaxies chosen at random throughout the universe. The illustration below shows five galaxies as they appear now and as they would have appeared at several times in the distant past. At some point in the past (about 6-10 billion years ago), all of these galaxies would have been half as far apart as they are now. At an earlier time they would have been half as far apart as that, and so on. If you extrapolate this process backwards you eventually come to a time in the past when the galaxies would have been right on top of one another. Put another way, the density of matter (or energy) in the universe was higher at earlier times, and extrapolating this process backwards we come eventually to a time when that density would have been infinite. This moment of infinite density is called the big bang

Having defined the moment of the big bang in this way—the time when all distances between objects were zero—I am not going to talk about that time. A point of infinite density, known in physics as a "singularity," makes no sense. Moreover, our current theories do not predict that such a moment occurred in the past. Our best physical theories, including general relativity and quantum mechanics, stop working when we try to describe matter that is almost infinitely dense. That word "almost" is important. The theories don't simply break down at the instant of the big bang singularity; rather, they break down a short time afterwards when the density has a certain value called the Planck density.

The Planck density, which is the highest density we can hope to describe with our current physics, is over 10⁹³ g/cm³, which corresponds to roughly 100 billion galaxies squeezed into a space the size of an atomic nucleus. For virtually any application we can imagine this limitation of our theories is completely irrelevant, but it means we can't describe the universe immediately after the big bang. We can only say that our current model of the universe begins when the density was somewhere below the Planck density and we can say virtually nothing about what the universe was like before that. We therefore take as our initial condition a universe at or just below the Planck density, and any questions about the instant of the big bang itself are eliminated from consideration.

Is this a cop-out? It certainly is. Physicists have not given up on understanding what happened before this time, but we admit that right now we have no theory to describe it. Many people are working to develop such a theory, but until that happens we are left having to start our description of the universe when the density was large but still finite.

Once we impose this limitation on ourselves, our picture of the universe works equally well for an infinite or a finite universe. If the universe is finite then it may very well have been extremely small at the moment when the density was at the Planck level. If the universe is infinite then it was also infinite at that early time. The density was enormous and the distances between particles vanishingly small, but that dense mass of particles went on forever.

The History of the Universe

Describing the history of the universe is obviously a fairly large task, so I will content myself with mentioning a few highlights. For a very good description of much of the early history I recommend the book The First Three Minutes by Steven Weinberg.

At the moment when the density of matter equaled the Planck constant, the universe consisted of a hot soup of elementary particles. When I say this medium was hot that means that the particles, on average, had very high energies. All of the fundamental particles such as quarks, electrons, and photons were present. At present these particles are mostly combined into larger units such as atoms, molecules, penguins, and so on, but at the extremely high temperatures of the early universe they remained separate. If several particles were to have combined into a more complicated structure such as an atom they would have been instantly ripped apart in collisions with the high energy particles flying around everywhere. As the universe expanded, the density and temperature of this mixture decreased. After a small fraction of a second the quarks combined into protons and neutrons in a process called baryogenesis. A few minutes later the protons and neutrons combined into atomic nuclei in a process referred to as nucleosynthesis. Hundreds of thousands of years later these protons and neutrons combined with electrons to form atoms. This last process is called recombination (despite the fact that particles had presumably never been bound into atoms before).

In the period of recombination the universe was still almost perfectly homogeneous, meaning that the density was the same everywhere. While the density still is the same everywhere when averaged over huge regions of space, it certainly varies locally. The density of the Earth is vastly larger than the density of interstellar space, which is in turn much greater than the density of intergalactic space. In contrast, the difference in density between the most and least dense regions at the time of recombination was about one part in 100,000. Between then and now the clumping of matter into galaxies, stars, etc. took place.

The mechanism by which this clumping occurred is fairly simple, although its details continue to be studied and debated. At the time of recombination the universe consisted of a nearly uniform hot gas with regions very slightly denser than the average and others very slightly less dense. If the density had been exactly the same everywhere then it would have always stayed that way. However, a region slightly denser than the surrounding gas would have a stronger gravitational attraction, and mass would tend to flow into it. This process would make this region even denser, causing it to attract matter even more strongly. In this way the almost uniformly dense universe gradually became less and less uniform, resulting in the dense clumps of matter we see around us now. On a fairly large scale these clumps make up galaxies, and matter that clumped on a smaller scale makes up the stars inside those galaxies. A very small portion formed into smaller objects orbiting around those stars and a small portion of that matter formed into people reading physics papers on the Internet.

V: The End (?) of the Universe

Hubble's observation that the universe is expanding suggested more generally that the universe is changing with time. As in most subjects, we know more about the past than we do about the future, but if we assume that our current physical theories are correct then we can predict a great deal about the future of our universe. Is the universe going to exist forever or will it someday come to an end as it began? Put another way, will the expansion of the universe continue forever? If the universe keeps on expanding it will presumably continue to exist for an infinitely long time. On the other hand, if the expansion ever stops, then the universe will contract until it once again reaches the Planck density (and after that we have no idea what it will do). In what follows I will explain what determines which of these scenarios is going to occur and say more about what each of them means.

We know from general relativity that expansion of the universe is slowed down by the mutual gravity of all the matter inside it. Whether or not the expansion will continue forever depends on whether or not there is enough matter in the universe to reverse it. If the density of matter in the universe is less than a certain critical value, then the universe will never stop expanding. If, on the other hand, the density of matter is greater than the critical value, then the pull of gravity will eventually be strong enough to stop the expansion and the universe will begin contracting. In Section III we saw that whether or not the universe is finite or infinite depends on whether the density of matter is above or below a critical value. That value turns out to be exactly the same as the critical value that determines whether or not the expansion will reverse. In other words, general relativity says that an open (infinite) universe will expand forever and a closed (finite) universe will eventually recollapse.⁸

If the universe expands forever, the clusters of galaxies in it will move farther and farther apart. Eventually each galaxy cluster will be alone in a vast empty space. The stars will burn out their fuel and collapse, leaving nothing but cold rocks behind. Eventually these will disintegrate as well. This whole process will take an unimaginably long time but it will occur eventually, and the universe will thereafter consist of nothing but loosely spread out elementary particles. All of the energy in the universe will then be distributed in a more or less uniform way at some extremely low temperature, and as the universe continues to expand this temperature will fall and the universe will become ever more empty and cold. This scenario is sometimes referred to as the heat death of the universe.

On the other hand, if the universe has a high enough density, then the galaxies will eventually start moving back towards each other. Once they are close enough together all galaxies and stars will collapse, until at some point the universe will once again consist of nothing but densely packed, highly energetic particles. 

Eventually all matter will be compressed to the Planck density, the density at which our current theories fail. Lacking a theory for such densities, we cannot predict what will happen then. One possibility is that the universe will bounce back—indeed, perhaps it has been in a cycle of expanding and contracting forever. Then again perhaps the universe will simply annihilate itself and cease to exist. Determining which of these possibilities would occur will require the development of a theory of physics at extremely high densities.

More than any other time in history, mankind faces a crossroads. One path leads to despair and utter hopelessness. The other, to total extinction. Let us pray we have the wisdom to choose correctly.

-Woody Allen

Endnote I: The Evolution of the Critical Density

As the universe expands, the density of matter inside it decreases. Yet relativity says that the questions of whether the universe is infinite or finite (section III) and of its ultimate fate (section V) depend on its density. Suppose the density of the universe is greater than 10^-29 g/cm³, meaning the universe is finite. What happens when the expansion of the universe causes the density to drop below that value? The answer is that the critical density changes with time, so that by the time our universe has dropped below that particular value the critical density itself will be lower still. (In fact the critical density drops faster than the actual density, so that if our current density is twice the critical density it will at later times be four times it, and so forth.) In other words, if we are currently above the critical density we will always continue to be so. Whether the universe is open or closed does not change with time.

Endnote II: Evidence for the Big Bang Model

Many observations provide evidence for the big bang model as we have outlined it. One is Hubble's observation of the expansion of the universe. We have measured distances and recession speeds for thousands of galaxies and other objects and they all match Hubble's law as accurately as we can measure them.⁹  These measurements provide very strong evidence that the universe is expanding. Nonetheless, when these data became known early in this century physicists were generally reluctant to abandon the idea that the universe is unchanging. This reluctance led to the development of so-called steady-state models of the universe that tried to reconcile Hubble's law with an eternally unchanging universe.

The steady-state models were dealt their death blow with the second great piece of observational evidence for the big bang model, namely, the discovery of the microwave radiation left over from the early universe. Prior to recombination, the universe consisted of a uniform hot mixture of particles. Such a mixture emits a recognizable spectrum of radiation that, if emitted then, should still be around today. Moreover, since that mixture filled the entire universe, that radiation should have been emitted everywhere in all directions, and should thus fill all of space. In 1964 Arno Penzias and Robert Wilson discovered microwaves coming from all directions in the sky, with exactly the spectrum predicted by the theory. (The spectrum of radiation is a description of the intensity of the radiation at different frequencies.) Almost immediately after this discovery, the steady state theories were abandoned and big bang cosmology became nearly universally accepted.¹⁰

Another prediction of the big bang model concerns the relative abundances of certain light elements. According to the model, the universe started with only elementary particles that eventually formed into atomic nuclei. A hydrogen nucleus is simply a single proton, so hydrogen was the first atomic nucleus to appear . Some of the protons eventually combined with other protons and/or neutrons to form other light elements such as deuterium, helium, and lithium. The laws governing nuclear physics are fairly well understood, so physicists have been able to work out the proportions of these different elements that should have been produced. Those proportions closely match what we observe in the universe today.

Endnote III: Problems and Lingering Questions

Despite the successful predictions of the big bang model, many people find the model problematic. The problems involve assumptions that must be made for the model to work and certain predictions of the theory that don't match our observations.

The success of the big bang model required the assumption that the universe was almost exactly homogeneous (the same everywhere) at early times. If the universe had been slightly less homogeneous initially, it would look very different now, whereas if it had been perfectly homogeneous then structures such as galaxies could never have formed. Another necessary assumption is that the expansion began simultaneously throughout a very large and possibly infinite universe.

The big bang model also requires the density of matter in the early universe to have been extremely close to the critical density. If it had been too high, the universe would have recollapsed before any structure had time to form, while if it had started out too low galaxies could not have formed. I noted in endnote I that over time the universe tends to move away from the critical density. It turns out that if the universe had initially been above or below the critical density by more than one part in 10⁵⁵, life as we know it could not have arisen!

These objections, while they make the theory seem strange, can be dismissed by saying that the universe just happened to start that way. Since the big bang model says nothing about how the universe got here in the first place, we have to assume some initial conditions. We are free to assume that for whatever reason the 

universe started out in exactly the way it had to in order to produce galaxies, stars, and ultimately you.

There is, however, another class of problems with the big bang model that cannot be explained away so easily. These problems have to do with exotic objects that should have been formed when the universe was extremely hot and dense. Our current theories predict that many different kinds of particles would have been created at those temperatures that could not be created today. Some of them would have decayed by now into normal matter and thus we would not expect to see them now, but others—called relic particles— would be expected to be stable enough to still be present in large quantities and easily detectable. These particles—which I won't describe in detail—include magnetic monopoles, gravitinos, axions, and even stranger beasts such as hedgehogs, cosmic strings, and domain walls. (The last two aren't particles but large objects, but the basic idea is the same.) The fact that we don't see any of them now cannot be explained by the standard big bang model. Moreover, some of these particles, if they had been around at the time of nucleosynthesis, would spoil our successful predictions of the relative abundances of light elements (see endnote II).

Physicists have tried for decades to formulate theories that could eliminate both the questionable assumptions and the problematic particles associated with the standard big bang model. Currently the only plausible candidate is a theory called inflationary cosmology, which is widely accepted by most cosmologists to be a necessary modification of the big bang model. This theory says that there was a period of very rapid expansion in the first fraction of a second after the big bang, or more precisely, after the density fell below the Planck level. A detailed explanation of why this happened or how it resolves all the problems cited above would be beyond the scope of this paper. I simply note that this rapid expansion period would have caused the universe to become almost perfectly homogeneous and almost exactly at the critical density regardless of how it started out. It would also get rid of all unwanted relic particles while still allowing for the creation of the ordinary particles that make up the universe today.

Finally I should mention the last great failing of the big bang model. Even when supplemented by inflation, big bang cosmology cannot explain why the universe is here in the first place. Inflation greatly reduces the number of assumptions you have to make about the origin of the universe. In fact some versions of inflationary cosmology suggest that the universe had no beginning but has existed forever. But whether the universe has existed forever or for only 10-15 billion years, the question of why it exists at all remains a mystery. Even if we could eventually come up with a set of laws that explained how the universe came into being, as some people are currently trying to do, the mystery of why those laws should exist would remain. That mystery will perhaps remain forever beyond the ability of science to explain.

Footnotes

(Clicking on the footnotes in the text will cause them to appear in a separate window, but they are reproduced here as well for the benefit of anyone printing out the paper.)

1. If you know something about the theory of relativity it may occur to you that Hubble's law seems to predict that very distant objects will recede from us faster than light, whereas Einstein's special theory of relativity predicts that nothing can move faster than light. For readers who are familiar with special relativity I can note that an observer in an expanding universe is not in an inertial reference frame, and therefore the laws of special relativity do not apply. They will still be good approximations for measurements of nearby objects, but not for very distant ones. For readers not familiar with special relativity I will simply note that Hubble's law is correct and that the explanation of why this is possible requires more relativity theory than I can explain in this footnote.

2. Actually saying "matter and energy" is redundant, because according to relativity theory matter is just another form of energy, with the amount of energy corresponding to a given mass being given by the famous equation E=mc². So from now on when I say "density of matter" I will be including all other forms of energy, such as electromagnetic radiation.

3. If the density has exactly this critical value then the universe is also infinite, but in this case it is called "flat" rather than "open."

4. Actually the value of the critical density changes with time. For a discussion of this issue see Endnote I

5. This picture of a uniform grid of galaxies is only a rough description. For example, many galaxies clump together in large groups called clusters. These clusters are held together by the mutual gravitational attraction of the galaxies so they don't grow as the universe expands. In such cases it is the distance between clusters of galaxies that grows in the way I've described.

6. The rather fanciful journey I'm suggesting is unrealistic in several ways. First of all I'm assuming that I could travel so quickly that the universe wouldn't grow much while I was making the trip. In fact even a light beam can't travel that fast and nothing can travel faster than a light beam. I also assumed for the purpose of illustration that galaxies wouldn't be created or destroyed in such a long time.

7. I'm being unrealistic when I talk about the distances between galaxies at these early times. Galaxies did not form until many millions of years after the big bang. The very early universe consisted of a dense mass of particles and the expansion of the universe at this time consisted of the distances between these particles increasing.

8. These conclusions about the future of the universe depend on an assumption that the universe is made up of ordinary matter. Recent observations suggest that the universe may instead be largely made up of a poorly understood form of matter that repels rather than attracts—a kind of antigravity. If these observations are confirmed and the universe does contain such matter, then the expansion will continue forever regardless of whether the universe is infinite or finite.

9. Actually this isn't true for nearby galaxies. Having nothing to do with the expansion of the universe, galaxies have their own velocities relative to each other, known as peculiar velocities. For nearby galaxies these peculiar velocities dominate and the galaxies may be moving towards or away from us. For distant galaxies, however, the recession rate due to the expansion of the universe is so great that the peculiar velocity makes no noticeable difference.

10. The discovery of the microwave background radiation by Penzias and Wilson was a remarkable example of serendipity in science. They were doing an unrelated experiment and found that their detectors were picking up a background signal coming from all directions. It wasn't until they discussed this finding with a colleague that they understood the significance of the discovery.

Read More