D. How far apart were those events?
How did Einstein conclude that space and time are related in this way? We can get a better idea by switching things around. We've been looking at how events in different places can either occur at the same time or at different times in different frames of reference. Now, let's take another look at the earlier events of your picking up and putting down your phone, which occurred at different times, but in the same or different places according to the frame of reference. Here is how these events look on a diagram in your frame of reference:
The same two events, as the highway patrol officer parked in his car by the roadside would have seen them, did not occur at the same place:
The slight difference in position simply corresponds to how far your vehicle traveled down the road while you were talking on the phone. (Again, we exaggerate your speed to make the effect visible on the diagram.)
Suppose that, instead of traveling at highway speeds of a few dozen kilometers per hour, you had traveled as fast as an electron in an electronic device, about 42,000 kilometers per second. Your car wouldn't be able to reach that speed with just the power of its engine, but if it could, the patrol officer would have seen the events somewhat like this:
No exaggeration this time. The officer would have found, not surprisingly, that your vehicle had moved even farther down the road-in fact, much farther in this case-while you were on the phone.
Suppose that, instead of traveling at electronic-device electron speed, your vehicle had gone as fast as an electron emitted from a radioactive substance. Electrons in this situation often move much faster. A typical speed for an electron emitted by a decaying copper-64 nucleus is 270,000 kilometers per second. If you had moved that fast, the patrol officer would find:
As you move at faster and faster speeds, the distance between the paces where you pick your phone up and where you put it down again, as seen by someone parked by the side of the road, is greater and greater. But because of the way space and time are related, the distance between these events in the patrol officer's reference frame grows much more quickly with your speed that our ordinary experience would immediately suggest. For a phone conversation lasting five minutes in your own frame of reference, the distances at different vehicle speeds are as follows:
|vehicle speed||distance between events|
|88.5 km/hour (0.0246 km/sec)||7.38 km|
|42,000 km/sec||12.7 million km|
|270,000 km/sec||186 million km|
Note how the distance between events increases faster than the speed does. While 270,000 kilometers per second is less than seven times faster than 42,000 kilometers per second, 186 million kilometers is over 14 times further than 12.7 million kilometers.
This is rather different from what we would ordinarily expect. In our usual experience, if you want to cover twice as much distance in a given amount of time, you have to go twice the speed. But the same relationship between space and time that makes events' time separation depend on velocity also affects how their space separation depends on velocity. The greater the velocity difference is between reference frames, the less conventional this dependence is.
From our conventional experience, we might reasonable expect that in order to put an infinite distance in finite time between our picking a phone up and putting it down, we'd have to travel infinitely fast. As it turns out, though, there's a finite speed that would serve the purpose.
|vehicle speed||distance between events|
This particular finite speed turns out to act as though it were infinite in another way. If you ran after something that was traveling infinitely fast, you could never overtake it. Infinite speed in one frame of reference would also be infinite in any other frame of reference. That, at least, is what we would expect if space and time weren't interrelated as the Minkowski diagram illustrates. But in fact, it is the finite speed of 299,792.458 kilometers per second that behaves this way. Start running after something that's traveling 299.792.458 kilometers per second, and you'll find that even in your new frame of reference, it's still pulling away from you at 299.792.458 kilometers per second-not at almost 299.792.458 kilometers per second, but at exactly 299,792.458 kilometers per second. Even if you were to take off at very close to that speed-say, 299,792.457 kilometers per second-you'd be no better off. You wouldn't reduce the speed difference to one meter per second. Because of how time itself changes between reference frames, in your new reference frame the object you were pursuing would still be getting away from you at exactly the same 299,792.458 kilometers per second that it was traveling in your original frame of reference.
299,792.458 kilometers per second, as it happens, is how fast light travels through a vacuum. (.....continued)