Relativity

Introduction

Relativity is a word that is used in a lot of different contexts to mean a lot of different things; however, in common usage, if you say "relativity," people think "Einstein."

Relativity is the idea that the laws of the universe are the same no matter what direction you are facing, no matter where you are standing, no matter how fast you are moving. Now, to say the laws of the universe are the same does not mean everything looks the same. A person standing in the Mojave desert sees very different things than an astronaut floating in space, or a diver 300 feet under water. But, they all see the same laws of physics and mathematics. Gravity always pulls you towards heavy objects. Objects in motion tend to stay in motion, unless something pushes on them. Electricity can give you a big shock.

Relativity is nearly always presented as a theory about the speed of light and black holes. While this is technically correct, I will take a broader view: I see relativity as the culmination of a 2400 year intellectual struggle to identify and understand mankind's place in the universe. In my opinion, by far the most important consequences of relativity are not the ability of physicists to calculate an extra decimal place or two, but rather the changes in philosophy, our understanding of religion and our relationship to Creation.

For most of western history, people have had their beliefs shaped by our experience on the Earth: the Earth seems to us to be enormously large and completely immovable. We jump up and down, throw large rocks, watch the tides come and go, but the Earth never seems to move. Historically, the stars were thought to be small objects, perhaps painted on a backdrop, but in any case visibly orbiting around the center of the universe, the Earth. Similarly the Sun and the Moon also orbited around the Earth. This universe was thought to be fixed and unchanging, as is only appropriate for a perfect creation of a perfect God. It was only natural for mankind's original idea to be that the Earth was immobile and at the center of this perfect universe.

Meanwhile, Euclid's geometry convinced everyone that mathematics was the most rigorous of all sciences, and therefore must be central to any description of the universe. Euclid, as it turned out, assumed that space was flat.

From about 1500 to 1916, these ideas were slowly examined and found wanting. Eventually, we found what we now consider to be a far more fundamental truth: the universe is a really quite large place, and the Earth is a very tiny little object in the universe. Everything in our universe is constantly in motion - nothing is fixed and unchanging. The Earth moves through the universe on a path determined by the laws of gravity, the very same laws that seem to determine the path of everything else. The Earth is in no way a special object, nor does it occupy a special position. This new understanding, that there seems to be no special place in the universe, no special direction, and no special speed which could be called "at rest," this understanding is called Relativity.

Just to put this into context, our current understanding is that the universe contains about 200 billion galaxies, and each galaxy contains about 200 billion stars. Our particular star, the Sun, seems to be a very average type of star, in no particular way distinguished from the other 40 thousand billion billion stars in the visible universe. We now know of many planets orbiting many stars, and seemingly find new ones on almost a weekly basis. Most of the planets we have found are enormous, imposing giants like Jupiter or even larger, with hundreds or even thousands of times the mass of the Earth - by comparison, the Earth is an all but invisible little rock.

Relativity is the story of mankind learning that we are not the center of creation, at least as far as physics is concerned. This has been a rather upsetting and ego-deflating lesson.

Although this is the story of relativity, it must be emphasized that this story is not complete. At the time of this writing, there is no acceptable quantum theory of gravity. It is widely believed by many, including myself, that before we can build a quantum theory of gravity, first we need a quantum theory of relativity. So, I personally expect that relativity will undergo at least one more major revision, perhaps within my lifetime.

 

Chapter 1: The History of Relativity

Relativity is the culmination of 2400 years of human thought. Originally, we thought that the Earth was at rest at the center of a flat universe that was constructed with a mathematical plan. 500 years ago, we figured out that the Earth was actually moving. 100 years ago we figured out that there is no center to the universe, there is no place in the universe that is at rest, and the universe is not flat. We do continue to delude ourselves into thinking the universe was constructed with a purpose and a mathematical plan.

Today, we live in a world that is constantly changing - 100 years ago there were no airplanes and almost no cars. 50 year ago there were no lasers, no television, nothing we would recognize today a a computer. 25 years ago the Internet was a strange little network occupied by professors and military people. The computers from 5 years ago are doorstops today. We are very accustomed to change; in fact, today people almost don't believe in the idea that things might stay the same.

This is all new. Isaac Newton lived in a culture that believed the height of man's knowledge and understanding was reached between the time of Aristotle and Jesus. Newton himself believed that to really understand something, you had to go back to the original greek writings of Aristotle, Euclid, and the Bible. He believed that the modern knowledge of his day was currupted and watered- down versions of the original deeper understandings.

Western culture busied itself with the search for truth. This philosophical orientation led to the assumption that truth existed, and that is was perfect and unchanging. Thus it was an easy step to believe that the Universe and indeed God himself were perfect and unchanging. Eastern philosophy was oriented on the search for what is real. Their presumption was that nothing was fixed and unchanging, that everything has a beginning and an ending. While this search led to many profound understandings, the search for what is real does not seem to lead to the development of science and mathematical logic. This is most curious, as we are now coming to understand that the universe is indeed a place of constant change, that nothing is fixed and unchanging, and that the universe itself has both a beginning and, presumably, an ending.

Our story begins with Aristotle, who compiled a set of ideas about how the world worked. Aristotle lived in Greece from 384 BC to 322 BC. Aristotle was primarily a teacher, and believed the highest calling for a man was to teach. He thought everything was made up of earth, water, air, and fire. He thought things had a natural state - earth-like objects wanted to be at rest, air-like and fire-like objects wanted to rise up. This all apparently seemed intuitively obvious to him. Aristotle taught that heavier objects fall faster than light objects, as a rock falls faster than a feather. At the time, no one actually thought to do experiments and see if any of this was true - in fact, like his teacher Plato, Aristotle thought that the universe was guided by the rules of logic and mathematics, and therefore the laws of the universe could be deduced by logical thought and mathematical reasoning. Aristotle thought that the Earth was immobile at the center of the universe.

Using a set of notes, Aristotle taught classes and basically invented the subjects of logic, physics, astronomy, meteorology, zoology, metaphysics, theology, psychology, politics, economics, and ethics. 250 years after Aristotle's death, his lecture notes were published by Andronicus of Rhodes. For the next 1500 years, the great thinkers of Europe and the Islamic world all traced their roots and primary influences back to Aristotle's teachings.

Euclid of Alexandria lived from 325 BC until about 265 BC. Euclid compiled many theorems and demonstrations that were already known, and carefully ordered them into what we now call an axiomatic system. An axiom is something you take as a given, take on faith as it were. An axiomatic system is a compilation of things you can conclude, demonstrate, or prove based on those axioms. Euclid organized the theorems very carefully, and was able to reduce his assumptions to five axioms.

Euclid's book, The Elements, was and is considered one of the greatest achievements of mankind. The beauty of his work is that once you accept the five axioms, you are led inexorably to his results. This was the first great work of logical reasoning. It has captivated many young thinkers over the ages, and motivated them to try to produce such a work of their own. It also convinced many people that this beauty was an element of the thoughts of God; that it must be the case that the entire universe is governed by a similar set of laws and their logical consequences. Physics is nothing more or less than the search for these laws.

Euclid's five axioms are:

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

The first three of Euclid's axioms are now called axioms of construction, that is, they tell you that you can build certain things. The fourth axiom is now recognized as being of a different nature - the axiom that all right angles are equivalent is the same as assuming that space is the same everywhere, and if you move a right angle around, it stays a right angle.

For 2,000 years it was recognized and agonized over that the first four axioms seem completely obvious and very simple, and the fifth axiom seems by contrast complicated and artificial. Many mathematicians spent substantially their entire careers trying to prove the fifth axiom was a logical consequence of the first four. All such attempts failed, for a simple reason which is now well understood. We now know that the fifth axiom is equivalent to the axiom that space is flat.

When Europe became interested in trying to climb out of the dark ages, the idea of learning and knowledge also became interesting. At this time, education meant learning Greek and Latin, and studying the Bible, Aristotle, and Euclid. It was considered that this was all of knowledge. It was also against this backdrop that the basic ideas of relativity had to evolve - according to Aristotle, the earth was in a special place, and material objects wanted to be at rest in this same special place. According to the interpretations of the Bible at the time, the Earth was a special part of God's creation, the very center and purpose of existence. So, the ideas that things were the same everywhere, that the stars were just like the Sun and the Sun was just like the stars, and that the Earth did not define and occupy the perfect center of creation, but rather was just another part of creation, moving around and subject to the same forces as everything else, these ideas were very controversial and not very welcome. The people who promoted these ideas were similarly controversial and not very welcome.

The Greeks noticed that there were a few "stars" with rather peculiar habits - these stars moved forwards and backwards and then forwards again against the backdrop of the fixed stars. The Greeks called these moving stars "wanderers," or "planets" in Greek. Over the centuries, people continued to observe the planets and chart their courses - it became very fashionable to try to find a way to predict the positions of the planets. For most of this time, the prevailing religious view was that God's creation, the universe, was perfect, and the only perfect shape was the circle, therefore planets must move in circles. It proved impossible to describe the positions of the planets by assuming they moved in circles around the Earth. As observations got better and better, people invented more and more complicated systems of planets moving in circles, which moved on greater circles, which moved on even greater and greater circles. However, although these systems of circles revolving in circles in circles in circles could be made to predict the planets positions reasonably well, the complexity of these artificial systems made a joke of the "perfection" and "simple beauty" of the circle.

Nicolaus Copernicus, a Polish economist and scientist who lived from 1473 to 1543, was the first person to openly challenge the Aristotelian view. In 1513, while in Italy, he published a short paper saying it was the Sun that was at rest at the center of the universe, and everything else including the Earth moved around the Sun. Copernicus said the planets moved in circles around the Sun, as did the Earth. Using this simple system, he was able to predict the orbits of the planets with great precision, but at a great cost: the Earth had to be moved out of the center of the universe. This was a great leap forward, philosophically: the Earth was now considered a moving object, and not a special object at a special place in the universe. This idea was a threat to the established church, since it also effectively removed Earth from being the central object of Creation, and called into question the idea that mankind was God's greatest creation. Copernicus unintentionally started a conflict between religion and science, a conflict which unfortunately maintains to this day.

Tycho Brahe lived and worked in Denmark from his birth in 1546, and died in 1601 in the Czech Republic. On 11 November 1572, in the early evening, he saw a new star in the constellation of Cassiopeia, almost directly overhead. This was remarkable, because the fixed, unchanging stars had changed. This star is now called "Tycho's Supernova." With funding from the King of Denmark, Brahe set up an observatory and made the most accurate observations of the planets up to his time. Brahe was aware of Copernicus' theory, but did not like it. He invented a competing theory, where the Earth was at the center of the universe, and the moon, Sun, and stars revolved around the Earth. In his system, the other planets revolved around the Sun. Thus, in his mind, he melded the most important ideas of the day: the planets all moved in circles, and the Earth was the center of creation again. But, the idea of an unchanging perfect universe was gone forever. And, Brahe set the stage for more and more precise measurements of our universe, and requiring that theories should agree with measurements.

Johannes Kepler was born in Germany in 1571 and lived until 1630. When Kepler was a young man, he was hired by Brahe as a mathematician. Thus Kepler had immediate access to the most precise astronomical observations of the day.

Kepler, like most scientists of his day, was convinced that God had made the Universe according to a mathematical plan, and that mathematics was therefore a strategy for understanding the universe. He spent most of his career calculating the orbits of the planets - for example, it took him nearly 1,000 sheets of paper to calculate the orbit of Mars. It was Kepler who first realized that the planets travel in ellipses, not in circles. Kepler also pointed out that Venus and Mercury are always seen near the Sun, which makes perfect sense if they orbit the Sun, but no sense if they orbit the Earth. Kepler also observed a supernova of his own in 1604, providing more evidence that the universe was a place of change.

Galileo Galilei lived in Italy from 1564 to 1642. Galileo studied the works of Copernicus, and was a great believer his views. In 1609, Galileo heard of a Dutch invention, the spyglass, and quickly made his own version, the first telescope. With his telescope, he discovered the moons of Jupiter, and he saw that Venus had phases like the moon. This proved that Venus orbited the sun, not the Earth. Galileo also noticed that the moons of Jupiter orbited with a fixed period, just as our moon makes a complete revolution around the Earth every 28 days. However, Galileo noticed that a few months later, his predictions of the orbits of Jupiter's moons were off by ten to fifteen minutes. Galileo correctly realized that this was because when the Earth is on the far side of the Sun from Jupiter, the light from the moons takes extra time to reach us over the extra distance. From this effect, Galileo was able to make a quite good estimate of the speed of light. The idea that light traveled at a finite speed, not instantaneously, was very new.

At the time, the views of Copernicus were quite controversial. Galileo had the habit of not only supporting these views, but trying to make his opponents look like fools, a habit which did not serve him particularly well. In 1616, Galileo was subjected to the Inquisition, and given a secret warning to recant his Copernican views. In 1632, Galileo published his famous Dialogue concerning the two greatest world systems, which only got him into further trouble, first for continuing to support Copernicus' views, and second for continuing to try to make his opponents look like fools. Galileo was summoned to Rome, where he was found to be "vehemently suspected of heresy", and forced to recant his Copernican views and sentenced to house arrest for life. He is reputed to have muttered under his breath as he left the Inquisition, "Never the less, it still moves."

Galileo set the stage for modern physics by noticing that all things fall at the same rate without regard for their composition. Aristotle had taught that heavier things fall faster. Galileo realized that by rolling things down a ramp, he could slow the effects of gravity and make more careful measurements. In this fashion Galileo collected the data that Newton would use to create his great theory of gravity.

Sir Isaac Newton lived in England from 1643 to 1717. Newton took Galileo's work and put it into a mathematical form. Newton, as one of his axioms, said objects at rest tend to stay at rest, and objects in motion tend to stay in motion. This is in complete disagreement with Aristotle's view that moving objects tend towards their natural state of being at rest on the Earth. Newton codified Galileo's views on systems in mathematics - the idea that the laws of physics are the same no matter what direction you face, no matter where you stand, no matter how fast you move. In codifying these ideas, Newton made a distinction between moving at a constant velocity, and accelerating. Newton claimed that the laws of physics were valid so long as you were moving at a constant velocity, but not if you were accelerating. These ideas are today called "Galilean Relativity." Although Newton undoubtedly knew of the speed of light, he did not think it was in any way special, and he thought that you could go as fast as you like, if you had the means to propel yourself.

Newton once said, "If I have seen far, it is because I have stood on the shoulders of giants." Here we have seen something of the giants whose shoulders he used. The mathematical abilities and physical intuition of Newton truly stand out in history and mark him as perhaps the most prominent physicist who ever lived. However, just as he indicated, much of the philosophical "heavy lifting" had already been done. The notion that to stand still on the Earth was to be perfectly at rest in the precise center of God's perfect, unchanging creation was painful to give up. It took at least four noteworthy geniuses 150 years to set an appropriate stage for Newton's great work.

Johann Carl Friedrich Gauss lived in Germany from 1777 until 1855. He was one of the greatest mathematicians who ever lived, and made many important contributions to physics, also. Gauss considered non-Euclidean geometry, also called curved spaces, and worked out much of the math, but never published anything on the topic. He was afraid of the backlash from his church and the community. However, he though a lot about these issues, and at one point actually measured the angles in a triangle about 5 miles on a side in an attempt to determine if space was flat or curved. Today, we think Gauss and his student Riemann came very close to developing General Relativity.

Michael Faraday lived in England from 1791 until 1867. By today's standards, Faraday had little in the way of formal education or mathematical training or abilities, which makes his scientific accomplishment all the more impressive. When Faraday was young, he was apprenticed to a book binder, and he used all his spare time educating himself by reading the scientific books laying around. Later, Faraday got himself a job as an assistant in a lab where he worked ceaselessly. Faraday almost single-handedly worked out the laws of the relationship between electricity and magnetism. Today, it's Faraday's work that tells us how to make electric motors and generators, the basic foundation of our entire technological civilization.

Georg Riemann  lived in Germany from 1826 until 1866. He was Gauss' last and most famous student. To finish his doctoral degree,  in 1854 Riemann was required to give a lecture. Gauss asked him to lecture on geometry. In this single lecture, Riemann laid almost all of the mathematical foundation for Einstein's General Relativity. Riemann went on to become an important mathematician, but did little work on curved spaces after this one lecture. Riemann was not a Catholic, and therefore was completely unconcerned about any backlash his lecture might generate.

James Clerk Maxwell lived in England from 1831 until 1879. Maxwell took the works of several other physicists including Coulomb, Ampere, Gauss, and especially Faraday, and brought them together into one set of equations that described all electric and magnetic phenomenon. Maxwell's equations are the first, and by far the most successful example of what we now call a Unified Field Theory. Prior to Maxwell it was thought that electricity, magnetism, and light were all completely unrelated. Maxwell showed that these phenomenon are all different aspects of the same thing.

Maxwell's equations made a very peculiar prediction: with his equations, he was able to calculate the speed of light. This is strange because up until that time it was thought that velocities simply added: if you were on a train moving at 100 miles per hour, and you turned on a flashlight pointing forwards, the light from that flashlight would go 100 miles per hour faster than light from a flashlight that was not moving. So, according to the physics of Newton, a theory should not predict the speed of light, since that speed should depend on how fast the observer and the source were moving. No one really knew quite what to make of this, but the longer people thought about it the more it bothered them.

At this time, people thought that since light is a wave, there must be something that is waving. Waves in the ocean are moving water. So light waves must be moving something. This something was given the name the Luminiferous Ether. From about 1865 until about 1920 people searched diligently for this Ether, but no one could ever find any. Today, we simply accept that light propagates, and we don't worry about what is moving. We also know today that light travels in our universe at an absolute velocity - no matter how you produce it, no matter how you are moving when you see it, you always see light moving at the same speed. This is the single most important point of Einstein's relativity. This is also the reason why Maxwell's equations were able to predict a particular speed for light - Maxwell's equations turned out to be fully compatible with Einstein's relativity, even though Maxwell wrote his equations down 40 years before Einstein developed Special Relativity.

Maxwell did another thing which I think in the long run will prove important. Maxwell knew of the rings of Saturn, and wondered what they were. He was able to show rather quickly that the rings could not be a solid disk, because the gravitational forces on a large solid disk would either tear the ring apart or make it crash into Saturn. Next he considered the idea that the rings were liquid, but again he was able to show that liquid rings could not orbit Saturn in a stable fashion. Lastly, he considered the possibility that the rings were made of dust, that is uncounted trillions of tiny individual particles. This system turned out to be stable, and today we know that Saturn's rings are indeed made up of tiny little ice chips, grains of sand, and small rocks. This was one of the first times that it was shown that continuous systems can have stability problems, but quantized systems can work in the same circumstances. Today, all of our theories suffer from stability problems brought on by the continuum, and a need for a theory of quantized space and time is becoming more and more apparent.

Ernst Mach was an Austrian physicist and philosopher who lived from 1838 until 1916. Mach did important work on sound, so the speed of sound is called Mach 1. He also criticized the existing physical theories of his day on several grounds. Mach said that since all we ever know of the universe comes to us through our senses, our theories should speak only of things which can be observed and measured. Mach also asserted that all phenomenon in our universe must have causes from within our universe. He proposed Mach's principle, which is that inertia, the tendency of a body in motion to stay in motion and to resist accelerations, must be a result of interactions with the other matter in the universe.

For example, if you whirl a bucket full of water about your head, the surface of the water will assume a curved shape. But, how does the water know to do that? How does the bucket know that it is accelerating in a circle, and you are standing still? From the bucket's point of view, it seems equally reasonable to assert that the bucket is just hanging out, and you are whirling around the bucket. Mach's answer was that the key difference between you and the bucket is that you see the distant stars in the sky standing still, and the bucket sees the stars whirling about. Therefore, the surface of the water curves due to some interaction between the water and the distant stars. Mach therefore claimed that if there was nothing in the universe but you and the bucket, and you whirled the bucket about your head, the water would not curve. Mach further claimed that if you could spin the entire universe about the bucket, the water surface would curve. Unfortunately, we do not currently know how to test either of these very interesting ideas.

There is a logical problem with Newton's physical explanation of the water curving. We say the water surface curves because the bucket is accelerating due to an external force - you're pulling on the bucket handle. But, you may ask, how does the water know it's being pulled? According to Newton's theories, the answer is that the water knows its accelerating because its surface is curving, which is a sure sign of acceleration and forces. So, in the end, all we can really say is that the water surface is curving because it is curving. Not very satisfying. While physics can tell us with enormous accuracy precisely how much the water surface will curve, physics is an almost total failure at telling us why it will curve. Mach did not like this in the slightest.

The idea that things which happen in the universe must have causes from within the universe seems so obvious as to be tautological. However, in spite of this, many of our natural assumptions and theories fail this criteria. For example, we consider inertia and electric charge to be properties of material objects with names but no causes. Perhaps more critically, we all know that time marches forwards inexorably, but we consider this most pervasive of effects to be without cause - it just is. Of course, the idea that everything has a cause is just an idea, and may be wrong. Or it may truly be that some things were simply chosen by God, so that their cause is not within our universe. However, it seems to me and most scientists that we should continue to try to explain what happens in our universe strictly in terms of things which are already in the universe until we're quite clear that this approach simply isn't working.

Einstein cited Mach as one of his primary inspirations. As perhaps you can tell, I like Mach a lot too.

Hendrik Lorentz was a Dutch physicist who lived from 1853 until 1928. Lorentz worked out the basic mathematics of Special Relativity, which is now called the Lorentz transform. Einstein used the formulas invented by Lorentz to develop his theories.

Jules Henri Poincaré was a French physicist who lived from 1854 to 1912. His theories of coordinate transformations were also instrumental in the development of Special Relativity. Today, we consider that Special Relativity was simultaneously discovered by Lorentz, Poincaré, and Einstein.

Albert Einstein was born on March 14th, 1879 in Ulm, Germany and died on April 18th, 1955 in Princeton, USA. Einstein worked only in the field of theoretical physics because of "my disposition for abstract and mathematical thought, and my lack of imagination and practical ability."

Einstein created two theories of relativity, which he called the Special Theory of Relativity, and the General Theory of Relativity. Einstein said he started working on his theory of relativity when he was 16. He had just learned about Maxwell's equations and their predictions of the speed of light. Einstein liked to perform what he called "thought experiments", in his native German "Gedankenexperiments." Here are two he thought up when he was 16. First, he imagined himself holding a mirror in his hand at arms length, and looking at his own reflection. Then, he imagined starting to run faster and faster, until he was running at the speed of light. Would he be able to see his own reflection? Second, he imagined a light ray zooming past him, and he ran to catch up with it. What would the light ray look like when he was running right alongside of it?

Ten years later, in 1905, he figured out the answers to these two questions. You cannot run at the speed of light, and at any lesser speed, you simply see your own reflection. If you could run at the speed of light, the light ray would look like an electric field which changes in space but not in time. This is impossible according to Maxwell's equations, but you cannot run at the speed of light so you can never see such a thing.

We have seen that originally, people thought that the Earth was precisely at rest precisely in the center of the universe. This belief, of course, would mean that we would have a very hard time figuring out what the laws of physics would look like if we were somewhere far from the center of the universe traveling at some high rate of speed. From about 1500 until about 1680, several very smart people figured out what we now call Galilean Relativity, which is that the laws of physics are the same for anyone anywhere, traveling at any speed, so long as they were traveling at a constant speed and not accelerating. Einstein's first contribution to relativity was to add to this, The speed of light is a constant, and does not depend on where you are or how fast you are moving. Once he understood this very counter-intuitive fact, he was able to quickly work out the laws of Special Relativity, which he published in 1905. Maxwell's equations were already compatible with Special Relativity, however Newton's equations were not. So, Einstein had to reformulate Newtonian Mechanics so as to be consistent with Special Relativity.

Special Relativity is the idea that the laws of physics are the same everywhere in the universe, no matter where you are, no matter how fast you are moving, so long as you are not accelerating, and one of the laws of physics is that light always travels at the same speed.

Very shortly after publishing Special Relativity, Einstein realized that the theory did not actually apply anywhere in our universe. The reason is very simple to understand: we live in a universe filled with thousands of billions of billions of planets and stars. No matter where you go in the universe, you are being pulling in some direction by gravity. So, Einstein realized, there is actually no such thing as a place or a person who is not accelerating, because there is no such thing as a place or a person who is not being pulled on by gravity. Einstein mulled over this idea for another 11 years - one might almost think he was a bit slow.

In 1907, two years after he had published Special Relativity, Einstein walked home from his job as a patent examiner in the Swiss Patent office, and said to his wife, "I had today the happiest thought of my entire life. I realized that a man freely falling in an elevator does not feel his own weight." Einstein had discovered a new axiom, that gravitational mass was equivalent to inertial mass. This was already known as a measured fact, but was not understood. Einstein decided to elevate this fact from a curious coincidence to a principle, which he called the principle of equivalence. What Einstein had decided was that you could not tell the difference between the force of gravity and some other kind of force. For example, if you were in an elevator with the doors closed and it was freely falling, you could not tell if you were close to the Earth or floating in space very far away from any other mass. Alternatively, if the elevator were sitting on the ground, you would feel your weight, but you could not tell if you were sitting on the Earth, or if the elevator were being pushed by a rocket motor with an acceleration of precisely one g, the acceleration of gravity on the Earth.

What Einstein had decided was that the closest thing there was to an inertial frame, that is moving at a constant velocity without acceleration, was to be freely falling. However, there are limits to this. Imagine you had an elevator which was 8,000 miles wide, and it was falling near the Earth. The Earth itself is only about 8,000 miles across, so clearly gravity will pull you straight down in the middle of the elevator, but at each end of the elevator you'll be pulled towards the middle. So, there's a limit to how big the elevator can be and still represent a freely falling frame. Also, after a little while, the elevator will hit the Earth - this will be a big clue to the people inside that something has changed. So, there's a limit to how long this situation can last and still represent a freely falling frame.


A very large elevator freely falling towards the Earth

Einstein realized that this was a lot like a curved space. If you're on a ship on the ocean, it looks like the Earth is flat. But, if there's another nearby ship and it sails away, after a time the ship seems to sink below the horizon. So, there's a limit to how big an area you can look at on the Earth and convince yourself that the Earth is flat.

Einstein spent much of the next 8 years learning the math invented by Riemann (invented in one week of work for a single afternoon lecture, which gives you an idea of their relative abilities at mathematics), and finally was able in 1916 to publish his General Theory of Relativity, which says that there is no center to the universe and nothing is at rest. The only places in the universe that seem to be moving at a constant velocity are small areas that only last for a short time.

Is this the end of the story of Relativity? Well, yes and no. This brings us up to our best current understanding. However, many people are dissatisfied with the current situation, as was Einstein himself. There are a lot of pretty strong hints that we're still missing several important ideas. Einstein noted that gravity waves travel at the same speed as light waves. He could not believe this was a coincidence, and felt strongly that this indicated there was a link between gravity and electro-magnetism. It is widely believed that we should somehow be able to make a quantum theory of gravity, but non-stop efforts from 1930 until 2004 have failed to produce a working theory. We have managed to prove that the techniques we currently use to build quantum field theories will not work for gravity, and we have no clue what techniques will work. All we're really certain of is that we have a lot left to learn.

Further Reading

Considerably more detailed biographical sketches on these scientists can be found at the The MacTutor History of Mathematics web page - http://www-groups.dcs.st-and.ac.uk/~history/.

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Copyright © 2002 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.