Outer Limits of Reason (44 page)

With the notion of simultaneity destroyed, there is going to be a problem with causality. If you cannot determine what came first, then you will not be able to determine what caused what. How are we to understand the laws of the universe when our very notion of causality is problematic?

Mass-Energy Equivalence

It would be morally reprehensible of us to leave special relativity without mentioning the world's most famous equation:

E
=
mc
²
.

That is, energy (
E
) can be exchanged for mass (
m
) multiplied by the square of the speed of light (
c
²
). This equation is a direct consequence of special relativity, and it describes how energy and matter can be converted into each other.

Before we begin to understand this, we need to be reminded what mass is. Usually the mass of an object tells us how much matter the object has. Given two balls of the same size where one is made of steel and the other of cork, the ball of steel has more mass. Physicists have two ways of measuring mass: gravitational mass and inertial mass.

Gravitational mass is basically how much the object weighs—that is, how much force gravity exerts on the ball. Obviously, the steel ball weighs more than the cork ball. Such a measurement depends on where you measure it. The same ball weighs slightly more near the Dead Sea than on top of Mount Everest. (I clarify why this is true shortly.) So gravitational mass is relative to how it is measured and is not absolute.

Inertial mass is how much the object resists pressure. In other words, if the ball is pushed, how much will it move? If you push the steel and cork balls with the same amount of force, the cork ball will move at a faster rate. Special relativity shows us that measuring how fast things move is a relative process. That is, it depends on the rate of speed of the observer. We conclude that inertial mass is also not absolute.

These two ways of measuring mass give you the same answer. (In fact, this idea is central in our forthcoming discussion of general relativity.) The fact that both ways of defining mass are subject to relativity shows that the very nature of matter is also relative. Not only are space and time relative terms, but so is mass.

Now to explain the mass-energy equivalence. It would take us too far afield to actually give details of Einstein's famous equation, but we can at least get an intuition as to why mass and energy can be converted into each other. Imagine taking a mass and applying a huge force to the mass while it is in outer space. By Newton's law (force = mass times acceleration or
F
=
ma
), the mass will accelerate in comparison to the force applied to it. Notice we are talking about acceleration and not about velocity. That means the mass will go faster and faster and there is seemingly nothing to stop it. But remember that special relativity teaches us that nothing can go faster than the speed of light. To ensure that the object never goes faster than the speed of light, the mass of the object increases as it goes along. An increasing mass will guarantee that the object slows down. We have successfully converted force—that is, energy—into mass. In contrast, the process that occurs inside a nuclear reactor is an example of mass turning into energy.

As a result, every object that is moving has more mass than when it is stationary. This extra mass might be imperceptible when we are dealing with speeds that are minuscule compared to the speed of light. Nevertheless a moving object has more mass. Similarly, an object with more energy has more mass. So an electric iron turned on weighs more than one that is turned off.

This equivalence is the basis of nuclear energy and nuclear bombs. The fact that
c—
and hence
c
²
—is such a huge number shows that small amounts of mass can be converted into enormous amounts of energy. This is where atom bombs and nuclear reactors get their immense power. The mass-energy equivalence is actually the basis of a lot more: within the sun, there are constant nuclear reactions that are converting mass into energy. This energy comes to Earth and gives us life.

General Relativity

Throughout our discussion of special relativity, we have always restricted ourselves to being an observer moving at a constant speed and not making any turns. Let us drop this restriction. In contrast to people in a car moving at a constant speed, passengers in a car speeding up will feel as if they are going into the seat and when the car makes a sharp right turn, they will feel themselves pushed to the left. This is different from the smoothness of Galileo's ship, discussed above.

To understand acceleration, Einstein posed the following thought experiment. Imagine a child closed up in a box. If she drops a ball, the ball will fall to the floor.

There are two ways to explain why the ball falls, as in
figure 7.28
.
³⁸
One possibility is that the box is on the Earth and that gravity is pulling the ball. The other possibility is that the box is in outer space and the spaceship is accelerating. In that case, the ball will fall to the floor for the same reason that an astronaut feels pulled into the seat when blasting off. This is similar to the feeling of a force when you accelerate in your car. Just as the scientist stuck inside Galileo's ship cannot determine if the ship is moving or still, so too will the child in the box not be able to tell if gravity or acceleration is affecting the ball. Einstein concluded that there is really no way to tell the difference between gravity and acceleration. This leads us to his general principle of relativity.

Postulate 3:
  All observers will observe the same laws of motion.

This means that the laws of physics must be the same regardless of whether the person is feeling gravity or acceleration.

The length contraction and the time dilation of special relativity also occur in general relativity. A person accelerating or decelerating will perceive light traveling at a constant speed, so their measuring rods and clocks must also change. However, the length contraction and the time dilation will no longer be constant. As the traveler speeds up, the stationary observer will observe the moving measuring rods shrinking and the clocks slowing down. In contrast, when the traveler decelerates, the stationary observer will see the measuring rods get larger and the clocks speed up. There is, however, another aspect of general relativity: since acceleration and gravity are the same, measuring rods and clocks will also be perceived to change near any great mass. A spaceship traveling near the sun or near a black hole will shrink and its clocks will go slower. The limit of this is that if a traveler actually entered a black hole with a watch that miraculously survived, it will not move. Of course, the traveler will not notice it either.

As we saw with
figure 7.26
, space and time are not separate entities. Rather, spacetime is a unified four-dimensional arena where all motion and the laws of physics take place. General relativity makes this spacetime much more interesting. Rather than just being a flat four-dimensional arena, it now has curves and bends. Mass (or equivalently, energy) curve and bend the arena. It is very hard to envision a four-dimensional space and it is even harder to imagine such a space curved. A helpful way to wrap your mind about this concept is to think of a two-dimensional flat rubber sheet. When weights are placed on the sheet, it curves around it as in
figure 7.29
. The mass curves the very fabric of spacetime and an object near the mass will tend to come toward it. So mass curves spacetime and the curves of spacetime affect the mass. The two spheres have curved spacetime and will attract each other. We call this gravity.

This is the general-relativity explanation of gravity. Acceleration can also be understood from this point of view. An object going at a constant pace is equivalent to a straight line in spacetime. When an object accelerates, its path curves and veers away from a straight line.

Five years after Einstein formulated general relativity it was experimentally confirmed in a famous experiment with a solar eclipse. An astronomer named Arthur Stanley Eddington (1882–1944) traveled to the island of Principe off the west coast of Africa to be there on May 29, 1919. There was going to be a total solar eclipse in the southern hemisphere and he wanted to witness this event. A solar eclipse is when the moon passes between the Earth and the sun, blocking the solar rays from hitting the Earth. Eddington calculated the position of the sun when the eclipse would happen and measured the distance between two stars that were near opposite sides of the sun, as in the top part of
figure 7.30
. He then waited for the sun to come between the stars. At that point, since the sun was so bright, the stars were not visible. Then the moon came between the Earth and the sun for 410 seconds. During this time, the sunlight was blocked off and the stars became visible again. Eddington was able to measure the distance between the two stars. As Einstein predicted, the stars seemed farther apart. The sun's gravitational pull actually tugged the light rays from the two stars. With those rays being pulled around the sun, the stars actually looked as if they were farther apart than before the sun came between the stars. Light is also affected by gravity because gravity is the curving of space. Eddington's results were broadcast all over the world: general relativity had been confirmed.

Question: Exactly how far apart are the two stars? Answer: It depends on when you are looking at them. The rays of light are curved because spacetime itself is curved. That is why the light is being pulled. You might protest that the positions of the two stars are fixed regardless of what it looks like when the sun passes between them. A similar phenomenon happens when you see a straw bend as water is poured into a glass. The straw seems to bend, but in fact, it is only an optical illusion. Such a protest would be slightly misguided. It's not only that the stars
look
farther apart. Rather, they
are
farther apart. If you wanted to touch the bent straw with your finger, you would not put your finger where you see the straw. You would take the illusion into account and put your finger where it should be. In contrast, if you wanted to travel to one of those two stars, you must take into account the curvature of spacetime. If your spaceship will pass near the sun, you must take its gravitational pull into account. The curvature of spacetime is not an illusion.