11. Relativity

A dialogue

The student’s words are in plain text. My responses are in italics.

What is time?

I don’t know.

Someone told me it was the fourth dimension.

That’s just a physicist’s way of confusing you. It’s true, but it is a much less deep statement than you would guess.

Time moves. I know that. But I don’t understand it. What does it mean that time moves?

I don’t know.

Does time ever slow down?

Yes.

How can you say that when you don’t know what the motion of time is?

Because we can measure relative rates. We can make time slow down in the laboratory. We see it in the stars.

Can we travel in time?

Sure. We’re doing it right now. We are both going forward in time.

I meant, can we go back in time?

Nothing in physics prevents that. But I don’t believe we can.

Believe? I thought we were discussing physics.

Nothing in physics prevents going backwards in time. But backward time travel violates my belief in my own free will.

What determines the direction of time?

Some people will tell you it is determined by entropy. But that is controversial, and not proven. There is no way to test the idea, so it too is more a belief than it is solid physics.

Nothing about time is obvious. Yet, given the mysterious nature of time, you may be surprised at some of the things we do know about it. For example, we know that if two twins are exactly the same age and one travels while the other stays at home, then when they are brought back together, the moving twin will have experienced less time than the other twin!

There is nothing odder about time than that. Yet Albert Einstein gave us a formula that tells us precisely how much less time the moving twin experienced. And that fact has been experimentally verified with very sensitive clocks flown on airplanes. Even radioactive atoms, when they move, experience less time than those that are stationary. That fact is verified every day at accelerator laboratories where such atoms are sent near the speed of light, and physicists note that their radioactive decays slow down.

The nature of time (and space) is at the heart of the theory of relativity. That’s what this chapter is about. Einstein created the theory of relativity in the early 1900s. The theory of relativity consists of two parts. The first is called “the special theory of relativity” and it has to do with the nature of time, space, energy, and momentum. It was in this work, published in 1905, that Einstein presented his famous equation E = mc². The second part was published in 1916 and is called the “General Theory of Relativity.” It is really a theory of gravity. It “explains” all of gravity as due to a bending of space and time. This theory is needed to understand some of the recent discoveries in cosmology about the nature of the Universe.

This chapter departs a bit from my previous philosophy. Future presidents don’t really need to know about the theory of relativity. It is important, however, to physicists, to philosophers, to those who plan trips to other planets, and to anyone who wants to have their mind stretched beyond what this course has already done.

Events – and “the fourth dimension”

Time is often called the “fourth dimension.” That turns out to be a useful definition, not an observable fact. And it is not something super mysterious or deeply abstract. In fact, when used in that way, the word “dimension” is being used in a very technical and narrow way: the “dimension” of a quantity is the number of different numbers you need to describe it.

Suppose you wish to specify a location on the Earth. You can do that with three coordinates, such as latitude, longitude, and altitude. Or you could use a system with x, y, and z. The key thing is that you need only three numbers. Any two objects that have the same set of three numbers must be at the same location. In math, we say that a location is a 3-dimensional number. That’s all that the fancy word dimension actually means. Space is three dimensional.

If you want to specify an event, rather than a location, then it is sufficient to give the location and the time of the event. Suppose I were to tell you that there is an event at my house at 8 pm tonight. Then there is no confusion; you might not know what is going to happen, but you have located it in both time and space. The event can have a name, such as “Elizabeth’s birthday party” or “Melinda goes to bed.” But to be unique (Elizabeth has a birthday each year, and Melinda goes to bed almost every night) you also specify the time. Events are specified by four numbers. So we say that events are four dimensional. That’s not deep. It is trivial. That’s is the entire meaning of saying that time is the fourth dimension.

That is not what is interesting about time. What is interesting is that the amount of time can change depending on the velocity that an object is moving in the three dimensions of space. That idea is deep, and requires some explanation.

Time dilation

I described in the opening of this chapter how two twins can experience different amounts of time. That seems to violate common sense. How can it be true? The answer is that the effect is very small unless the velocity is very fast--that’s why you never notice it. Common sense is based on experience, and that kind of time dilation is not part of our normal lives, so it violates our common sense. It also violated the common sense of ancient men to think that the sun has a million times the volume of the earth. Does that violate your common sense? Sometimes, all it takes to incorporate something into your common sense is to hear it many times, or to gain familiarity with it. Maybe after you have read this chapter, time dilation will start to become part of your common sense.

Time dilation is so small that it’s difficult even to measure unless the velocity is near the speed of light. For airplanes moving at 675 mi/h, the effect is about 5 x10^–13. That means that if you traveled at this speed for one day, you would lose 43 nanoseconds.[1] (That is the time it would take light to travel about 43 ft.) If you fly for a year, you will experience 16 microseconds less time than your twin who doesn’t travel.

This small effect becomes large if the velocity approaches the speed of light. At 60% of the speed of light, the time dilation factor is 0.8! Let me show you how to do the calculation yourself. Suppose one object is moving at a velocity v. Let the speed of light be called c. In science fiction, the ratio of v to c is called the “lightspeed.” If you are moving at 60% the speed of light, your lightspeed is 0.6. In physics, we usually call the lightspeed “beta” and use the Greek letter b (which looks like a B with a tail).

b = v/c = light speed

Einstein gave an exact formula for calculating this. Although the term is not usually given a name, I like to call it the Einstein factor. Time will slow down by

The Einstein Factor:

You don’t have to memorize this, but you might want to anyway, because then you can do real relativistic calculations.

Let’s get back to our example. If the light speed β = 0.6, then the equation gives the Einstein factor to be

If a man named John stays at home, and his fraternal twin Mary travels at 0.6 lightspeed, then her time will go slower at a rate that is only 0.8 that of John’s time. If John ages 1 year, then Mary will age only 0.8 years. When Mary returns, and they compare ages, John will be 0.2 years older than Mary (i.e. a little more than 2 months older). Yet they are twins, born at the same time.

The effect gets much more dramatic as Mary’s velocity increases. Suppose she travels at light speed 0.99999, i.e. at 99.999% the speed of light. If you plug that into the time equation, you’ll find Mary’s time progresses at a rate only 0.0045 the rate of John’s time. If John ages 1 year, Mary will age 0.0045 years. To convert that to days, multiply by the number of days in a year: 0.0045 years x 365 days/yr = 1.6 days.

Not only that, but she will experience only 1.6 days, while John experiences a full year. If they began as freshmen, Mary will still be a freshman, but John will be a sophomore.

The fastest that any astronaut has ever traveled is approximately Earth escape velocity, about 11 km/s. This is equivalent to light speed β = 0.0037. Plug this into the time equation (use a calculator) and you’ll find that the astronaut time goes at a rate that is 0.99999933 slower than Earth time. That isn’t a big change (since the number is so close to 1). If he travels for 1 year (that is, 365 days x 24 h x 3600 seconds per hour = 3.16x10⁷ s), then he will experience 0.02 seconds less than if he stayed at home. That’s not enough for him to notice unless he is carrying a very accurate clock.

We have sent radioactive atoms at velocities close to the speed of light, and their radioactivity does slow down, by exactly the predicted factor.