Part III
In this part . . .
You’re finally here! This part focuses on Einstein’s great discovery, the special theory of relativity. First, I show you the original theory of relativity, developed by Galileo Galilei, which makes good common sense. (You’re on a train that’s going 100 kph, and you walk from the back to the front of the train at 10 kph. Someone by the railroad tracks will clock you moving at 110 kph.)
Then Einstein came along and changed all of this straightforward stuff. I explain what Einstein meant by relativity and what relativity did to our ideas of time and space. You’ll also see what Einstein’s famous equation, E = mc2, really means.
Chapter 8
In This Chapter
Discovering relative motion
Identifying Galileo’s principle of relativity
Refining the theory of relativity
C ontrary to popular belief, Einstein did not invent relativity. The honor belongs to Galileo Galilei. His ideas shaped the theories that flourished and floundered in the centuries between his own and Einstein’s. And they set the stage for Einstein to establish one of the most revolutionary theories of our time.
In this chapter, I discuss the origins of relativity. Galileo came up with the idea well before Isaac Newton started work on his laws of motion in the 17th century, and Newton used Galileo’s idea of relative motion in his mechanics.
Conducting the First Motion Experiments
As I discuss in Chapter 4, Galileo began his studies on the motion of objects in Pisa in the early 17th century. Lacking a good clock, he timed the swinging of church chandeliers with his pulse. Later, he rigged a simple timer by measuring the water draining out of a wine bottle with a hole in the bottom. He used this clock to time the motion of a falling object so that he could understand how it changed its speed as it fell.
Galileo noticed that as an object fell to the ground, its speed increased by the same amount during equal time intervals. With his experiments, Galileo discovered that the steady increase in speed — or acceleration — of an object falling to the ground is always the same. We call it the acceleration due to gravity and, as Newton taught us, it’s caused by the attraction between the Earth and the object. (Check out Chapter 4 for more on Newton’s theories.)
Experiencing movement on board a ship
Galileo didn’t stop with experiments at rest. He also wanted to know what would happen if he dropped the object while he was moving.
Dropping a ball from the crow’s nest
As the left side of Figure 8-1 shows, if you are in a sailboat moving steadily dead ahead and drop a ball from the crow’s nest, the ball falls straight down and lands at the foot of the mast, not behind. But if a friend is watching your sailboat from shore, the ball seems to move in a curved path, as the right side of Figure 8-1 shows.
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Figure 8-1: The falling ball hits the deck at the foot of the mast (left), but someone looking at the boat from shore sees a curved path (right). |
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Before you drop the ball, your friend on shore sees it being carried along by the ship’s motion. When you drop it, the ball begins to fall but also continues with the horizontal motion that it had when you held it. Like you, your friend sees the ball fall alongside the mast, but the mast and the ball continue moving along with the ship. You see the ball fall straight down, while she sees a curved path.
If you go inside a cabin in the ship and drop the ball there, it again falls straight down to the floor. When you’re inside the cabin traveling on smooth water, you can’t tell if your ship is moving steadily ahead or if it’s docked at port. Everything behaves exactly the same in either case. If you throw a ball straight up, it comes right back down into your hand.
You can even play pool inside the ship. You won’t be any better or worse than you are when you play on your pool table at home. The balls behave in the same way, as long as the ship maintains a constant speed on even seas.
Observing uniform motion
Galileo actually described a thought experiment like the scenario I just described in his bestselling book, Discourses and Mathematical Demonstrations Concerning Two New Sciences Pertaining to Mechanics and Local Motion, familiarly called The Two New Sciences. This is an interesting book. At the time, scholarly books dealing with philosophy or science were written in Latin. Galileo wrote The Two New Sciences in Italian. And he wrote it as a play, a dialogue among three characters. It isn’t dry or difficult; it’s actually entertaining.
In his book, Galileo encourages you to gather some friends and head for the cargo hold of a large ship. He tells you to bring some gnats, flies, and other winged insects, as well as a few fish that you’ll place in a tub. And don’t forget two bottles: one for you to hang upside down so that it drips into the second bottle right underneath.
Before the ship sails, he asks you to carefully observe how the insects fly, how the fish swim, and how the water drips into the bottle below. Galileo also wants you to throw a ball to your friends and to look at its motion.
When the ship sets sail and is moving smoothly, not tossing or lurching, you’ll find out that there is no change in the way the insects fly, the fish swim, the water drips, and the ball moves. You won’t be able to tell if the ship is moving by watching any of these events. Galileo writes that this is because the ship’s motion is shared by all the things in it, including the air.
According to Galileo,
Uniform motion (motion along a straight line at constant speed) cannot be discovered without a reference point.
Bringing his ideas to Earth
In The Two New Sciences, Galileo also discussed the problem of the behavior of falling objects on the moving Earth. If the Earth is moving, why does a ball thrown straight up into the air fall back to the same spot from which it was thrown? Many of Galileo’s contemporaries used this argument to deny that the Earth moved.
Based on his thought experiments, Galileo concluded that if the Earth were moving, a ball would fall to the ground exactly as it would appear to do if the Earth weren’t moving. Galileo said that you can’t tell whether the Earth is moving by watching an object fall, just as you can’t tell whether a ship is moving by watching an insect fly.
Finding uniform motion on a train
Einstein liked to use trains to illustrate his own theory of relativity. Trains were the main means of transportation during his time, and Einstein was very familiar with them. Trains offered a smoother ride than horse-drawn carriages. They offered one of the few ways people had to experience what Galileo called uniform motion: motion along a straight line at a steady speed.
Establishing the Principle of Relativity
According to Galileo, you can’t distinguish steady motion in a straight line from rest. If you can’t distinguish the two, they are the same. Uniform motion (as Galileo called it) and rest are the same thing.
That concept may sound a bit strange. You may not know when you’re moving, but you certainly know when you’re not moving. When you’re sitting at home watching television, you aren’t going anywhere. Or are you?
Let’s say some astronauts are on their way to Mars. They have a powerful telescope and, for some reason, become interested in you. Through their telescope they see you sitting by the window in your living room, moving around the sun (along with the Earth) at 30 kilometers per second (kps). And that’s without taking into account the motion of the sun and the entire solar system moving around the Milky Way galaxy, or the galaxy’s motion within the Local Galactic Cluster.
Understanding that motion is relative
We usually refer to motion in everyday situations relative to the Earth. Earth is our reference point, or reference frame (as physicists say). Relative to the Earth, you are clearly at rest while you watch television in your home. At the same time, you are moving at 30 kps relative to the sun.
Imagine astronauts of the future, traveling in interplanetary space, away from any solar system. If they encounter another ship returning from an expedition, heading in the opposite direction, the astronauts won’t be able to determine how fast the other ship is moving or how fast they are moving without checking their instruments, calibrated to read speeds relative to the Earth.
Are we there yet?
Sitting in an airplane, anxiously waiting to take off, you may be fooled into thinking that you are finally moving only to discover that the airplane taxiing next to yours is moving in the opposite direction. Unless you look outside and see the ground or the terminal, you can’t tell who is moving.
You’ve also probably experienced a momentary confusion when driving on a multilane road during a traffic jam. The slow motion of the cars around you may make you look carefully to figure out who is moving.
Riding the bullet train
Consider another example. The bullet train between Madrid and Seville reaches speeds of up to 300 kilometers per hour (kph), or 188 mph, taking about two hours to travel the 470-km (293-mi) route through Spain’s countryside. When the train travels at 230 kph, your camera is at rest relative to the train but moves with you and all the passengers at 230 kph relative to the ground.
A train attendant pushing a food cart toward the front of the train at 2 kph while the train travels at 230 kph has a speed relative to the ground of 232 kph. (His speed relative to the train is, of course, 2 kph.) If he boards the rear door of the last car and steps down at the end of the trip from the front door of the first car, he shaves off a few minutes of travel time.
If the train attendant walks toward the rear of the train at 3 kph to get some decaf for a passenger, his speed relative to the ground is 227 kph. What is the train attendant’s real speed? Is it 3 kph or 227 kph? Both. It depends on the reference frame. As long as you’re careful in its description, each is equally valid. There are no fixed or absolute speeds.
Nothing strange or new here. Galileo’s relativity agrees with our common sense. However, Einstein will disagree with what we are saying. His relativity will be strange, as I show you in the next chapter.
Stating Galileo’s principle of relativity
According to Galileo, then, all motion along a straight line at a constant speed is relative. It’s the same as saying that you don’t have fixed, absolute speeds, that you can’t distinguish between rest and uniform motion.
Galileo believed that no experiment in mechanics could reveal whether you are in uniform motion or at rest; all mechanics experiments work the same way regardless of your motion. This is the Galilean principle of relativity. Simply put, it says
The laws of mechanics are the same in all frames of reference in uniform motion.
Not being able to distinguish uniform motion from rest means that all reference frames are equivalent. No reference frame is special or absolute, as scientists call it. There is no absolute standard of rest, and uniform motion has to be always referred to a frame of reference. Uniform motion is relative.
If you measure the speed of your boat at 30 kph relative to the water you’re navigating in, someone else may measure it to be 20 kph relative to shore. And someone else on another boat wanting to catch up to you may measure it to be 5 kph relative to his boat. Which is correct? All are. It depends on the reference frame.
Creating Another Relativity
For Galileo, physics was mechanics. In reality, in Galileo’s time, not much was known about mechanics. And what was known had been discovered by him, like the motion of falling objects and the ideas of uniform motion and accelerated motion. Newton came along and developed all of mechanics. In doing so, he built a system of the world, a way to look at the universe that worked like clockwork.
In Newton’s mechanical universe, objects move in predictable ways that can be calculated precisely using the three laws of motion and the universal law of gravitation (see Chapter 4). In Newton’s world, space and time are fixed, absolute. A sailor walking on the deck along the centerline toward the bow of his boat at 4 kph while his boat sails at 20 kph relative to land can easily figure out that he is moving at 24 kph past the land. When he returns toward the stern at the same speed, he moves at 16 kph past the land.
Newton adopted Galileo’s principle of relativity and used it when he developed his mechanics. He even stated it clearly in his Principia, the masterpiece he wrote to describe his mechanics. It was clear to Newton that uniform motion made no difference to the laws of mechanics.
Integrating the laws of motion with the speed of light
All was going well for Galileo’s relativity until James Clerk Maxwell came up with his theory of electromagnetism in the 19th century. His four equations (which I discuss in Chapter 6) taught us that light is an electromagnetic wave. But if it’s a wave, it needs some sort of substance to move through. Sound waves move through the air or water, or even through solids.
The mysterious substance that light supposedly traveled through was called the ether. The commonly held belief after Maxwell developed his theory was that light traveled like a wave at about 300,000 km (186,000 mi) per second through the ether. (By 1882, the measured value of the speed of light was very close to the modern value of 299,792.458 kps or 186,282.397 mps.)
If light traveled through the ether, the Earth did as well. And if Earth were moving through the ether, the speed of light would change. When the Earth moved in the same direction as the ether, it would gain some ground on the light beam, and you would measure a smaller value. When the Earth moved in the opposite direction in its orbit around the sun, you’d measure a larger value for the speed of light, because you’d be losing ground.
At the end of the 19th century, two physicists at what is today called Case Western Reserve University, Albert Michelson and Edward Morley, set up a delicate experiment to measure the change in the speed of light as the Earth moved through the ether. I discuss this experiment in detail in Chapter 9. But strangely enough, the experiment measured the same speed for light no matter how the Earth moved.
That result was difficult to understand. It was like trying to measure from shore the speed of a boat that’s moving in the water at 20 kph and coming up with the same speed whether it goes upstream or downstream. The water is going to carry the boat faster when it travels downstream.
Didn’t light follow the principle of relativity? Perhaps you can’t add up the speeds of the Earth and light, like you do with boats and trains. The results of the Michelson–Morley experiment created a new problem for scientists.
Expressing the idea of contraction
The ether experiment of Michelson and Morley was very carefully done, and scientists were puzzled by its results. After the results were published, two physicists came up with an explanation. And by coincidence, they did so independently of one another. The explanation wasn’t what anybody was expecting.
George FitzGerald at Trinity College in Dublin, Ireland and Hendrik Antoon Lorentz of the University of Leyden in the Netherlands said at about the same time that the reason the speed of light is independent from the motion of the Earth is that objects shrink when they move, and the shrinking is along the direction of motion. The faster the object moves, the more it shrinks. What’s more, the shrinking is exactly in the amount needed to keep the speed of light measurements unchanged.
That was some strange idea. Objects shrink when they move?
Lorentz provided an explanation for it. He had developed a theory of matter based on electrons. According to Lorentz, all matter is composed of electrons, and properties such as elasticity or hardness are due to the way these electrons interact. When a body moves through the ether, these electrons flatten out, producing an overall reduction in the size of the object.
Lorentz modified the simple rule in Galileo’s principle of relativity and came out with an equation that could be used to compute the length reduction with speed. We know this equation today as the Lorentz–FitzGerald contraction.
In the end, the length contraction idea of Lorentz and FitzGerald wasn’t taken seriously. It was too much of a coincidence that the shortening of lengths was what was needed to make the Michelson–Morley experiment work.
Lorentz published his paper in 1895. That year, the 16-year-old Einstein was thinking about what he would see if he could travel at the speed of light. Ten years later, he provided the final explanation for this puzzle. Einstein used the Lorentz–FitzGerald equation in his theory of relativity, but with a different interpretation. I cover Einstein’s version in Chapter 9.
Identifying the Man Who Almost Discovered Relativity
Jules Henri Poincaré in France had a different approach to explain the strange results of the Michelson–Morley experiment. He sympathized with Lorentz but was unhappy with introducing the idea of length contraction only to explain the result of the experiment.
Thinking about elastic time
Poincaré was unhappy that Lorentz had abandoned the principle of relativity. The laws of physics appeared to be the same in all frames of reference, which told him there must be some more general principle of relativity. He asked Lorentz to work on an extension to his length contraction equation.
Lorentz went back to the drawing board. He came back with a new set of transformation equations that included his length contraction but added time dilation to the mix. What his new theory said was that if you’re moving relative to the Earth, for example, not only do objects change in length, but your clock ticks at a different rate. Time has a sort of elastic property that stretches and contracts depending on how you move.
Of course, no one — not even Lorentz himself — thought that these equations applied to the real world. They just helped in the calculations; they were a math tool with no connection to reality.
If the Lorentz transformations were correct (mathematically), how do you measure time? Poincaré suggested that clocks in different moving frames could be synchronized with light signals. But he said that these clocks don’t show “true time,” and some clocks are slower and some faster. The only true time is the one measured relative to the ether.
Expressing an unrealized hope
At the International Congress of Arts and Sciences in St. Louis in 1904, Poincaré delivered an invited lecture where he gave a clear and simple description of the principle of relativity. It was an extension of Galileo’s principle, to include all the laws of physics. He said that the laws of physical phenomena should be the same whether you are in uniform motion or at rest so that you couldn’t have any way of knowing if you are moving or not.
As I show you in Chapter 9, Poincaré’s version of the principle of relativity is essentially the same as Einstein’s. And Poincaré stated it one year before Einstein’s own relativity paper was published. It’s surprising that Poincaré didn’t take the last step and discover the correct version of relativity.
In his lecture, Poincaré mentioned the experiments of Michelson and Morley, who had “pushed precision to its last limits.” He also included Lorentz’s transformations, which gave the formula to change length and time so that the speed of light remains the same when you measure it while in motion through the ether.
Poincaré said that the principle of relativity needed to be explained, so that we can perhaps build a whole new mechanics where the speed of light can’t be exceeded. Poincaré, the experienced and respected scientist, who had made original and significant contributions in mathematics and physics (and continued to do so), ended his lecture by saying that this new mechanics was an “unrealized hope and conjecture.”
Just a year later, the 26-year-old inexperienced and unknown Albert Einstein would make this unrealized hope a reality.
