# A Brief Intermission for Large and Small

We are about to enter the world of the very large and the very small.  The universe is enormous; in fact, we don’t know whether it is infinite or has a boundary and probably never will because of the limits of how far we can see.  As best we can determine the universe is about 13.7 billion years old.  This puts an absolute limit on how far we can see: 13.7 billion light years.  A light year is the distance light travels in one year.  Light coming from any objects farther than 13.7 billion light years is undetectable simply because it hasn’t had time to get here yet.

A simulated view of the entire observable universe, approximately 14 billion light-years across. (Photo credit: Wikipedia)

In the terms we are going to speak, 13 billion is a small number.  Rather than write “a thousand billion” or “a billion billion” it helps to have some sense of the magnitude of those numbers by writing them in terms we can appreciate.  Just how big is a thousand billion?  Written out it would be 1,000,000,000,000.  A billion billion is 1,000,000,000,000,000,000.  Numbers like this convey more of a sense of the size of the number than the words “billion billion” but they are cumbersome to write.  Instead of writing numbers in this way, science uses scientific notation to express both very large and very small numbers.  Scientific notation writes numbers as powers of 10.  Ten to the first power is written as 101 and means 10 x 1.  Ten to the second power is 102 meaning 10 x 10 or 100.  Ten to the third power is 103 for 10 x 10 x 10 or 1,000.  One million (1,000,000) is written as 106.  As you can see, the number of zeroes that follows the one is the same as the power to which 10 is raised, three in the case of 103 (1,000) and six in the case of 106 (1,000,000).  In this way we can know that 1024 is a really big number and that it is bigger by 100 times than 1022 without having to count zeroes to see which number has more, or figuring out how many billion trillion that number is.

When it comes to very small numbers, scientific notation still raises 10 to a given power, but this time makes a fraction, such as 1/102 to represent 1/100 or .01.  However, to avoid 1/10, the number is written 10-2, the minus sign indicating that this is a fraction with 102 in the denominator.  Thus, one one-thousandth (1/1000) is written 10-3.  As with large numbers we can readily see that 10-6 is a very small number, one one-millionth (1/1,000,000).

So to give some perspective about how far 13.7 billion light years is, we can convert that to miles.  Light travels about 186,000 miles per second.  In a year there are 60 seconds/minute x 60 minutes/hour x 24 hours/day x 365 days per year, or 31,536,000 seconds.  For every second light travels 186,000 miles so a light year is 31,536,000 x 186,000 = 5.87 x 1012 miles.  Now multiply that by 13.7 billion (13.7 x 109) and you get an idea of how far away the “end of the universe” is: 80 x 1021 miles, or 80 followed by 21 zeroes.

While this post might remind you too much of high school, all you need to remember is that if ten is raised to a positive power (2, 5, 24, etc.) it’s a big number, while if ten is raised to a negative number (-2, -5, -24, etc.) it’s a small number and gets smaller as the power gets bigger in absolute terms.

With this technical background we can now start to talk about how the steady state proponents were finally vanquished by the big bang backers.

# Lemaitre Sheds Light and Creates Conflict

Georges Lemaitre was born in 1894.  He began studying engineering but, like Friedmann, his studies were interrupted by World War I.  In the trenches he observed first-hand the effects of German mustard gas and won the Croix de Guerre.  After the war he returned to his studies but switched to theoretical physics.  He also enrolled in the seminary and was ordained a priest in 1923.  For the remainder of his life he pursued two careers, physics and the priesthood, saying “There were two ways of arriving at the truth.  I decided to follow them both.”

Georges Lemaître is credited with proposing the Big Bang theory of the origin of the universe in 1927. (Photo credit: Wikipedia)

In 1923 after spending two years in Cambridge with Arthur Eddington, Lemaitre returned to Belgium and began his own cosmological quest for truth.  He adopted Einstein’s general relativity but, like Friedmann, rejected the notion of the cosmological constant.  Without knowing anything about Friedmann’s work, Lemaitre resurrected the expanding universe model.  Unlike Friedmann who was a mathematician, interested mainly in the numbers of the theory, Lemaitre wanted to understand the reality behind the numbers.  If the cosmos were expanding, Lemaitre decided to run the clock backwards.  An expanding universe implied that things were closer together yesterday, closer still 100 years ago and still closer 1 million years ago.  Run the clock backward enough and the inescapable conclusion was that everything was together at one point.

Perhaps influenced by his theological training, Lemaitre realized that general relativity implied a moment of creation.  He concluded that the universe began in a relatively small, compact region that suddenly expanded and evolved into what we observe today.  He refined his theory into what he called the primeval atom that contained all of the matter that eventually became the stars and planets.  Though a moment of creation was central to his theory, Lemaitre was interested in the evolution of the universe from the primeval atom to the stars and galaxies.

Lemaitre published his theory and was met with the same deafening silence that greeted Friedmann.  To make matters worse, Lemaitre also had a run-in with Einstein who rebuffed him, saying that his mathematics were correct but his “physics is abominable.”  Einstein had thus been offered two chances to accept an alternative to the steady state view of the universe and rejected both.  As the world authority on cosmology, Einstein’s words had the force of law.  It is ironic that, having challenged authority in his early career Einstein had now become the authority behind whom virtually all scientists fell into line.  It probably didn’t help Lemaitre that he was a priest and his theory smacked of a Creator.  Though it had been nearly four centuries since Galileo was forced to confess, the wounds science felt from religion were still tender.

The truth is, both theories were appealing and both had flaws.  The flaw in the steady state theory was the cosmological constant, which, as we have seen, is nothing but a fudge factor to make the theory conform to the accepted view of how things are.  The flaw in the nascent big bang theory (it still had not been thus named) was that there was no evidence to support the theory of a sudden explosion, other than the logic behind an expanding universe.  For that matter, though, there was no evidence to support a steady state model other than the belief that this is how things are.  The theorists needed evidence to support their various theories so they turned to the experimental physicists, the astronomers.

# Friedmann’s Models of the Universe

Alexander Friedmann was a brilliant mathematician who also had a penchant for science and technology.  After enduring both the First World War and the Russian revolution in 1917, Friedmann was eventually introduced to Einstein’s Theory of General Relativity.  It may have been a combination of his delayed exposure to the theory and Russia’s relative isolation from the rest of the world that allowed Friedmann to ignore Einstein’s (and most other physicists’) view of the universe as static and formulate an entirely new and radical approach.

While Einstein started with the assumption that the universe is static and introduced the cosmological constant to counter the effect of gravity, which, under his view, would eventually lead to the collapse of the universe, Friedmann ignored the cosmological constant and then looked at general relativity to see what kind of universe it predicted.  First of all, Friedmann’s model was one of a dynamic universe, one that had started with an initial expansion (the term Big Bang wouldn’t come along for some 30 years).  This initial expansion led to three possible results:

friedmann models (Photo credit: Wikipedia)

First, if the initial expansion wasn’t great enough, gravity would eventually pull the universe back in on itself.  Like a ball thrown upward, the universe would at first move quickly, then slow to a complete and brief (in cosmological terms) time, and then begin to contract ever faster.  Friedmann envisioned it then expanding again, endlessly like a bouncing ball.

Secondly, if the initial expansion was great enough, the universe would continue to expand infinitely.

The third view was a middle ground in which the initial expansion was enough so that the universe would continue to expand, though slower and slower, never quite stopping.  It is like the problem of the rabbit and the lettuce.  Each second a rabbit moves one-half of the remaining distance between himself and a piece of lettuce.  For example, the rabbit starts out four feet from the lettuce.  In the first second he moves two feet closer.  The second second he moves one foot closer.  The third second he moves six inches closer and so forth.  Does he ever reach the lettuce?  The answer is, no, he never covers the remaining half-distance to the lettuce.  Similarly Friedmann’s third model never quite reaches a point where gravity overcomes expansion, though the rate of expansion continually slows until, like the rabbit, the universe is creeping forward.

Friedmann thus proposed a model of the universe based on general relativity that did not match the model proposed by the creator of the Theory of General Relativity, Albert Einstein.  Although Einstein would admit that Friedmann’s view was mathematically correct based on general relativity, he claimed it was scientifically irrelevant because the universe was static.

Friedmann was eventually proven correct, but he did not live to see himself vindicated as Einstein had done with his theory.  Friedmann died of a serious illness, probably typhoid fever, in a delirium, reportedly lecturing to an imaginary audience.  Part of Friedmann’s problem was that his notion was too radical.  He suffered the same fate as Copernicus in that the scientific world simply wasn’t ready for his view of the universe.  Another problem was that he locked horns with Einstein himself, the world’s foremost cosmologist at the time.  Finally, Friedmann, a mathematician, was not an astronomer and was therefore an outsider to the cosmological world.  Though Friedmann’s papers were published during his lifetime they received almost no notice and would not until they were rediscovered by a Belgian scientist, Georges Lemaitre.

# The Cosmological Consequences of General Relativity

In 1917 Einstein published a paper entitled Cosmological Considerations of the General Theory of Relativity. The title is significant.  Unlike Galileo, Copernicus, Kepler, Brahe and Newton, Einstein was concerned with the entire universe, not just the solar system.  Calculating the orbit of Mercury to predict its movement around the sun was hard enough, but Einstein had the audacity to attempt to predict the movement of stars and galaxies.  In order to do this he had to make an assumption.  His assumption, known as the cosmological principle, is that the universe is isotropic and homogeneous.  This means that the universe looks pretty much the same in every direction, and that it looks that way from whatever vantage point in the universe you have.  This means that we do not occupy a special place in the universe.

When he applied his theory to the universe, the result was unsettling to say the least.  General relativity predicted that the universe was destined to crash in on itself as a result of gravity’s relentless pull on the stars and galaxies.  Newton’s theory had predicted the same thing.  In the early 20th Century, the view of the universe was that it was stable, that it had always existed more or less as it is today.  But Einstein’s prediction was that things would start creeping slowly toward each other.  The creep would turn into a frantic dash as stars and galaxies collided.

Anxious to make general relativity consistent with the observed and accepted notion of what the universe is, Einstein realized that if he introduced a constant, which he called the cosmological constant into general relativity, he could keep the result from being a huge crunch at some point in the future.  The cosmological constant was a sort of anti-gravity that repelled matter, offsetting the natural attraction of gravity just enough to prevent collapse.  It was, in essence, a fudge factor similar to that found by proponents of Newton’s theory of gravity who were unwilling to accept general relativity.  Even Einstein was somewhat ashamed of the need for this fudge factor, saying that it was “detrimental to the formal beauty of the theory [general relativity].”

Most scientists were content to accept the cosmological constant as a “refinement” to general relativity because it made the theory fit with their notion of the universe.  Remember we began with the adage that things are what they are and if the theory doesn’t explain them we better get a new theory.  No one wanted to jettison general relativity after having just adopted it.

No one, that is, except Alexander Friedmann, a Russian scientist born in 1888.  Friedmann would tackle orthodoxy head on and come up with a radically different view of the universe, one that would not be confirmed for decades.

# Einstein’s Victory

Einstein needed to be able to predict a result that hadn’t been observed yet.  Only in this way could he definitively show the superiority of his theory over Newton’s.  Think of it this way:  Suppose you have been talking to two investment advisers, Fred and Barney.  Fred shows you his calculations that predicted yesterday what the stock market did today.  Barney shows you his calculations today that predict what the market will do tomorrow.  Tomorrow the market performs exactly as Barney predicted.  Whom do you believe?  Fred could have manipulated his theory to match today’s results because he knew the result before he gave you his prediction.  But Barney couldn’t have known what the market would do today except through his theory.

Einstein had long pondered the relationship between light and gravity.  Since the General Theory of Relativity defines gravity as a warp in the fabric of spacetime and light travels through spacetime, under Einstein’s theory light should be affected by a massive body such as the sun.  Einstein theorized that light from a star behind the sun would be bent, resulting in an apparent shift in its position.  The problem was, you can’t see any stars when the sun is shining.  Einstein realized that during a total solar eclipse stars are visible in the daytime sky so he looked for an opportunity and an astronomer (remember, Einstein was a theoretical physicist — he didn’t actually conduct experiments) to observe stars during a solar eclipse.  He found both a partner in Arthur Eddington and an opportunity in the Southern Hemisphere in 1919.  Eddington and one team went to the island of Principe off West Africa and another team went to Brazil, hoping that if clouds obscured the eclipse in one location, the other location would be clear.

Eddington’s photograph of a solar eclipse, which confirmed Einstein’s theory that light “bends”. (Photo credit: Wikipedia)

In the end both teams were successful in observing the eclipse and taking what photographs needed to be taken.  After developing the plates, making measurements and calculating margins of error, Einstein’s predictions were within the margin of error while the results predicted by Newton’s laws were far too low.  Both Eddington and Einstein became virtual rock stars in science.  Eddington became inextricably connected to general relativity.  At one point someone said to him that he was one of three people in the world who understood the General Theory of Relativity.  Eddington was silent for several seconds and finally the person urged him not to be so modest.  “On the contrary,” Eddington replied, “I am trying to think who the third person is.”

The vindication of Einstein’s theory required a paradigm shift, a complete and sudden alteration in the way science viewed the universe.  Science usually moves incrementally, in small changes.  It isn’t often that Saul becomes Paul on the road to Damascus.  When a paradigm shift such as this occurs it usually isn’t generally accepted.  Older scientists who have devoted their professional lives to a particular view of things are reluctant to change and often it requires a dying off of the older generation before a new theory is completely accepted.  Such was the case with the General Theory of Relativity.  Yet his theories have been proven correct.  Einstein was once asked by a student how he would have felt if the universe had ultimately turned out to be different than his theory predicted.  With tongue firmly in cheek he replied, “Then I would feel sorry for the Good Lord.  The theory is correct anyway.”

# A New Theory of Gravity

Induced spacetime curvature (Photo credit: Wikipedia)

Remember the example of traveling in a train at near the speed of light past a friend on a station platform.  The friend sees time as slowing down and you as being thinner.  Both space and time have changed in his perception.  Remember also that the theory of special relativity combines both space and time into one, spacetime.   What is this spacetime?

Mathematics is often called the language of science because many physical phenomena can best be expressed mathematically.  For those of us who have trouble with high school algebra and geometry, describing scientific theories can be daunting and understanding them well-nigh impossible.  We have to resort to crude analogies like the train traveling past the station.  But using another everyday example lets us at least visualize spacetime.

Imagine a trampoline stretched taut in its frame.  The surface is perfectly flat.  This represents spacetime in the absence of any matter.  If you roll a BB across the trampoline it will go in a straight line from one side to the other.  Matter is composed of mass so let’s see what happens when mass is introduced into spacetime.

Now imagine placing a bowling ball in the middle of the trampoline.  Spacetime becomes deformed as the trampoline sags to support the ball.  If you now roll a tennis ball across the trampoline it will follow what appears to be a curved path caused by the indentation from the bowling ball.  In fact, if you roll the tennis ball at just the right speed it will circle the bowling ball like a roulette ball circles the spinning wheel.

The bowling ball can be thought of as a star, our sun for example.  The tennis ball is a planet orbiting that star, our earth or one of the other planets.  The tennis ball makes its own slight indentation in the trampoline, just as the Earth makes its own indentation in spacetime.  We can imagine a marble circling the tennis ball, following that indentation.  In the same way the moon orbits the Earth.

This three-dimensional image of spacetime soon breaks down because friction of the tennis ball on the trampoline and air resistance slow the tennis ball and it drops into the indentation.  However, that in itself is instructive because that is the ultimate fate of Earth: eventually the earth will spiral into the sun just like the tennis ball.

Einstein’s view of gravity is thus fundamentally different from Newton’s.  Newton believed that gravity is a natural attraction of bodies for each other that is caused by a property inherent in mass.  Einstein postulated that gravity is caused by a deformation of spacetime, which in turn is caused by the body.  There is nothing inherent in mass that creates gravity.  Instead it is the nature of spacetime that causes gravity.

Newton’s theories had served science well for nearly 400 years.  It wasn’t enough for Einstein’s theory to predict the same thing as Newton’s.  Einstein’s theory had to predict an observable event not predicted by Newton’s theory.  Only in this way could Einstein’s theory be shown superior to Newton’s.  This is how most theories are “proven.”  In practice a theory can’t be proven right; it can only be proven wrong in the sense that it predicts something that is not observed or fails to predict an observation.  Einstein’s problem was that for all situations on earth, including the orbit of the moon, both theories predicted the same thing with the same accuracy.  Einstein’s theory predicted a difference in an extreme gravitational field.  The only problem was finding such a situation where the two predictions could be observed and compared.

He finally found such a situation in the orbit of Mercury around the sun.  Mercury’s orbit had long been observed as rotating.  Mercury’s orbit is elliptical like the other planets’ but the ellipse itself rotates around the sun by 574 arcseconds per century.  An arcsecond is 1/3,600 degrees.  It takes Mercury over one million orbits of the sun and 200,000 years to return to the same orbit.  Newton’s theory accounted for only 531 of the 574 arcseconds but Einstein’s theory account for exactly 574 arcseconds.

This was enough to put Newton’s theory in doubt but not enough for most scientists.  Someone discovered that if, instead of r2 in Newton’s formula for gravitational attraction there was r2.00000016 the two theories made the same prediction.  This was just number fudging but it shows the lengths to which scientists will go to salvage pet theories.  Einstein realized he would need something even more spectacular to win the controversy.

# That’s Heavy

In an earlier post we saw that Newton developed three significant theories, any one of which would probably have earned him a Nobel Prize but for two small details: The Nobel Prize didn’t exist when Newton was alive and the rules of the Nobel Prize prevent posthumous awards.  In any event, his theory of gravity governed the cosmos for over three centuries.  The story of Newton developing his theory while sitting under an apple tree and having an apple fall and hit him on the head is probably apocryphal, but there is little doubt that he was inspired by watching apples fall.

Newton’s theory of gravity simply says that any two bodies attract each other with a force that is proportional to their two masses and inversely proportional to the square of the distance between them.  As millions of high school physics students know, this formula is expressed as

F = (G x m1 x m2)/r2

Here F represents the gravitational force, G is the gravitational constant which is necessary to make the force of gravity somewhat congruent with other natural forces such as magnetism, m represents the masses of the two objects in question and r is the distance between the two objects.  This formula explained why planets remain in orbit around the sun, why apples drop from trees and even why astronauts feel weightlessness in space.  It coalesced all the concepts of Kepler, Copernicus and Galileo about the solar system.  It seemed complete, though even Newton himself felt it wasn’t the final answer.  He said “I seem to myself to have been a little boy playing on the seashore and diverting myself now and then in finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay undiscovered before me.”

The problem with Newton’s theory is it doesn’t explain how gravity works.  We say that it is an attraction between two bodies, a force that pulls them together, but how is that force transmitted through space?  For example, consider a golfer addressing a golf ball on the tee.  He swings the club, which strikes the ball and sends it into a graceful arc, up, up, until gravity, that relentless force that cannot be defied, pulls it back to earth.  At least that’s how it works in my mind when I swing a club.  Reality is uglier.  If we consider the source of the forces acting on the golf ball, it initially moves due to the impact of the club head.  That impact is the product of the momentum of the club head, which is caused by the synchronous contraction of the golfer’s muscles, which in turn is caused by complex chemical reactions that contract and relax the muscles.  Those chemical reactions are caused by the conversion of food the golfer ingests.  All of this is complex but completely understood by science.

On the other hand, what is this mysterious force of gravity?  All we know about it is that it can be measured by Newton’s equation, but that doesn’t tell us how it arises.  We generally pass that off as a natural property of mass.  But why should mass have this property?  And, more importantly, how is it transmitted through space?

Einstein’s special theory of relativity led him to the conclusion that gravity is caused by a deformation of the very fabric of spacetime.  This deformation causes bodies to follow the curvature of spacetime.  This notion sounds crazy and indeed Einstein himself doubted himself.  Yet he could not bring himself to abandon the theory.  He spent eight years, from 1907 to 1915, trying to refine this vague notion of curvature of spacetime into a formal theory explainable by a mathematical formulation.

In the next post we’ll try to describe this whole curvature of spacetime and how it creates gravity concept.