From Eternity to Here (14 page)

Advances in science are rarely straightforward, and the correct way to interpret the Michelson-Morley results was not obvious. Perhaps the aether is dragged along with the Earth, so that our relative velocity remains small. After some furious back-and-forth theorizing, physicists hit upon what we now regard to be the right answer: The speed of light is simply a universal invariant. Everyone measures light to be moving at the same speed, independent of the motion of the experimenter.
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Indeed, the entire content of special relativity boils down to these two principles:

• No local experiment can distinguish between observers moving at constant velocities.

• The speed of light is the same to all observers.

When we use the phrase
the speed of light
, we are implicitly assuming that it’s the speed of light through empty space that we’re talking about. It’s perfectly easy to make light move at some other speed, just by introducing a transparent medium—light moves more slowly through glass or water than it does through empty space, but that doesn’t tell us anything profound about the laws of physics. Indeed, “light” is not all that important in this game. What’s important is that there exists some unique preferred velocity through spacetime. It just so happens that light moves at that speed when it’s traveling through empty space—but the existence of a speed limit is what matters, not that light is able to go that fast.

We should appreciate how astonishing all this is. Say you’re in your spaceship, and a friend in a faraway spaceship is beaming a flashlight at you. You measure the velocity of the light from the flashlight, and the answer is 300,000 kilometers per second. Now you fire your rockets and accelerate toward your friend, until your relative velocity is 200,000 kilometers per second. You again measure the speed of the light coming from the flashlight, and the answer is: 300,000 kilometers per second. That seems crazy; anyone in their right mind should have expected it to be 500,000 kilometers per second. What’s going on?

The answer, according to special relativity, is that it’s not the speed of light that depends on your reference frame—it’s your notion of a “kilometer” and a “second.” If a meterstick passes by us at high velocity, it undergoes “length contraction”—it appears shorter than the meterstick that is sitting at rest in our reference frame. Likewise, if a clock moves by us at high velocity, it undergoes “time dilation”—it appears to be ticking more slowly than the clock that is sitting at rest. Together, these phenomena precisely compensate for any relative motion, so that everyone measures exactly the same speed of light.
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The invariance of the speed of light carries with it an important corollary: Nothing can move faster than light. The proof is simple enough; imagine being in a rocket that tries to race against the light being emitted by a flashlight. At first the rocket is stationary (say, in our reference frame), and the light is passing it at 300,000 kilometers per second. But then the rocket accelerates with all its might, attaining a tremendous velocity. When the crew in the rocket checks the light from the (now distant) flashlight, they see that it is passing them by at—300,000 kilometers per second. No matter what they do, how hard they accelerate or for how long, the light is always moving faster, and always moving faster by the same amount.
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(From their point of view, that is. From the perspective of an external observer, they appear to be moving closer and closer to the speed of light, but they never reach it.)

However, while length contraction and time dilation are perfectly legitimate ways to think about special relativity, they can also get pretty confusing. When we think about the “length” of some physical object, we need to measure the distance between one end of it and the other, but implicitly we need to do so
at the same time
. (You can’t make yourself taller by putting a mark on the wall by your feet, climbing a ladder, putting another mark by your head, and proclaiming the distance between the marks to be your height.) But the entire spirit of special relativity tells us to avoid making statements about separated events happening at the same time. So let’s tackle the problem from a different angle: by taking “spacetime” seriously.

SPACETIME

Back to the spaceship with us. This time, however, instead of being limited to performing experiments inside the sealed ship, we have access to a small fleet of robot probes with their own rockets and navigation computers, which we can program to go on journeys and come back as we please. And each one of them is equipped with a very accurate atomic clock. We begin by carefully synchronizing these clocks with the one on our main shipboard computer, and verifying that they all agree and keep very precise time.

Then we send out some of our probes to zip away from us for a while and eventually come back. When they return, we notice something right away: The clocks on the probe ships no longer agree with the shipboard computer. Because this is a thought experiment, we can rest assured that the difference is not due to cosmic rays or faulty programming or tampering by mischievous aliens—the probes really did experience a different amount of time than we did.

Happily, there is an explanation for this unusual phenomenon. The time that clocks experience isn’t some absolute feature of the universe, out there to be measured once and for all, like the yard lines on a football field. Instead, the time measured by a clock depends on the particular trajectory that the clock takes, much like the total distance covered by a runner depends on their path. If, instead of sending out robot probes equipped with clocks from a spaceship, we had sent out robots on wheels equipped with odometers from a base located on the ground, nobody would be surprised that different robots returned with different odometer readings. The lesson is that clocks are kind of like odometers, keeping track of some measure of distance traveled (through time or through space) along a particular path.

If clocks are kind of like odometers, then time is kind of like space. Remember that even before special relativity, if we believed in absolute space and time à la Isaac Newton, there was nothing stopping us from combining them into one entity called “spacetime.” It was still necessary to give four numbers (three to locate a position in space, and one time) to specify an event in the universe. But in a Newtonian world, space and time had completely separate identities. Given two distinct events, such as “leaving the house Monday morning” and “arriving at work later that same morning,” we could separately (and uniquely, without fear of ambiguity) talk about the distance between them and the time elapsed between them. Special relativity says that this is not right. There are not two different things, “distance in space” measured by odometers and “duration in time” measured by clocks. There is only one thing, the
interval in spacetime
between two events, which corresponds to an ordinary distance when it is mostly through space and to a duration measured by clocks when it is mostly through time.

What decides “mostly”? The speed of light. Velocity is measured in kilometers per second, or in some other units of distance per time; hence, having some special speed as part of the laws of nature provides a way to translate between space and time. When you travel more slowly than the speed of light, you are moving mostly through time; if you were to travel faster than light (which you aren’t about to do), you would be moving mostly through space.

Let’s try to flesh out some of the details. Examining the clocks on our probe ships closely, we realize that all of the traveling clocks are different in a similar way: They read
shorter
times than the one that was stationary. That is striking, as we were comforting ourselves with the idea that time is kind of like space, and the clocks were reflecting a distance traveled through spacetime. But in the case of good old ordinary space, moving around willy-nilly always makes a journey longer; a straight line is the shortest distance between two points in space. If our clocks are telling us the truth (and they are), it would appear that unaccelerated motion—a straight line through spacetime, if you like—is the path of
longest
time between two events.

Figure 12:
Time elapsed on trajectories that go out and come back is less than that elapsed according to clocks that stay behind.

Well, what did you expect? Time is kind of like space, but it’s obviously not completely indistinguishable from space in every way. (No one is in any danger of getting confused by some driving directions and making a left turn into yesterday.) Putting aside for the moment issues of entropy and the arrow of time, we have just uncovered the fundamental feature that distinguishes time from space: Extraneous motion
decreases
the time elapsed between two events in spacetime, whereas it
increases
the distance traveled between two points in space.

If we want to move between two points in space, we can make the distance we actually travel as long as we wish, by taking some crazy winding path (or just by walking in circles an arbitrary number of times before continuing on our way). But consider traveling between two events in spacetime—particular points in space, at particular moments in time. If we move on a “straight line”—an unaccelerated trajectory, moving at constant velocity all the while—we will experience the longest duration possible. So if we do the opposite, zipping all over the place as fast as we can, but taking care to reach our destination at the appointed time, we will experience a shorter duration. If we zipped around at precisely the speed of light, we would never experience any duration at all, no matter how we traveled. We can’t do exactly that, but we can come as close as we wish.
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That’s the precise sense in which “time is kind of like space”—spacetime is a generalization of the concept of space, with time playing the role of one of the dimensions of spacetime, albeit one with a slightly different flavor than the spatial dimensions. None of this is familiar to us from our everyday experience, because we tend to move much more slowly than the speed of light. Moving much more slowly than light is like being a running back who only marched precisely up the football field, never swerving left or right. To a player like that, “distance traveled” would be identical to “number of yards gained,” and there would be no ambiguity. That’s what time is like in our everyday experience; because we and all of our friends move much more slowly than the speed of light, we naturally assume that time is a universal feature of the universe, rather than a measure of the spacetime interval along our particular trajectories.

STAYING INSIDE YOUR LIGHT CONE

One way of coming to terms with the workings of spacetime according to special relativity is to make a map: draw a picture of space and time, indicating where we are allowed to go. Let’s warm up by drawing a picture of Newtonian spacetime. Because Newtonian space and time are absolute, we can uniquely define “moments of constant time” on our map. We can take the four dimensions of space and time and slice them into a set of three-dimensional copies of space at constant time, as shown in Figure 13. (We’re actually only able to show two-dimensional slices on the figure; use your imagination to interpret each slice as representing three-dimensional space.) Crucially, everyone agrees on the difference between space and time; we’re not making any arbitrary choices.

Figure 13:
Newtonian space and time. The universe is sliced into moments of constant time, which unambiguously separate time into past and future. World lines of real objects can never double back across a moment of time more than once.

Every Newtonian object (a person, an atom, a rocket ship) defines a world line—the path the object takes through spacetime. (Even if you sit perfectly still, you still move through spacetime; you’re aging, aren’t you?
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) And those world lines obey a very strict rule: Once they pass through one moment of time, they can never double backward in time to pass through the same moment again. You can move as fast as you like—you can be here one instant, and a billion light-years away 1 second later—but you have to keep moving forward in time, your world line intersecting each moment precisely once.