Fear of Physics (19 page)

Relativity explains this paradox. The muon’s “clock” runs slowly compared to ours, because the muon is traveling near the speed of light. Therefore, in its own frame, the muon does decay on average in a few millionths of a second. However, depending upon how close to the speed of light the muon is traveling during this short time in its frame, perhaps several seconds could elapse in our frame on Earth. Time
really
does slow down for moving objects.

I can’t resist leaving you with another paradox (my favorite!) that indicates how real these effects can be, while also underscoring how personal our notion of reality is. Say you have a brand-new, large American car that you want to show off by driving at a substantial fraction of the speed of light into my garage. Now your car, at rest, is 12 feet long. My garage is also just 12 feet long. If you are moving very fast, however, I will measure your car to be only, say, 8 feet long. Thus, there should be no problem fitting your car in my garage for a brief instant before either you hit the
back wall or I open a door in the back of the garage to let you out. You, on the other hand, view me and my garage to be whizzing past you and thus to you my garage is only 8 feet long, while your car remains 12 feet long. Thus, it is impossible to fit your car in my garage.

The miraculous thing about this paradox is, once again, that both of us are right. I indeed can close my garage door on you and have you in my garage for a moment. You, on the other hand, will feel yourself hit the back wall before I close the door.

Nothing could be more real for either person, but as you can see, reality in this case is in the eye of the beholder. The point is that each person’s
now
is subjective for distant events. The driver insists that
now
the front of his car is touching the back wall of the garage and the back is sticking out the front door, which is still open, while the garage owner insists that
now
the front door is closed and the front of the car has not yet reached the back wall.

If your
now
is not my
now
and your second is not my second, and your inch is not my inch, is there anything left to count on? The answer is yes, and it relates to the new connections between space and time that relativity uncovers. I have already described how once the finite velocity of light is taken into account, space has a timelike character. But these examples push that idea even further. One person’s interval in space, such as the distance between the ends of a train measured at the same instant, can, for another person, also involve an interval in time. The second person will insist that these same measurements were in fact carried out at different times. Put another way, one person’s “space” can be another person’s “time.”

In retrospect, this is not so surprising. The constancy of light connects space and time in a way in which they were not connected
before. In order for a velocity—which is given by a distance traveled in a fixed time—to be measured as the same by two observers in relative motion, both space and time measurements must alter together between the two observers. There
is
an absolute, but it does not involve space separately, nor time separately. It must involve a combination of the two. And it is not hard to find out what this absolute is. The distance traveled by a light ray at a speed
c
for a time
t
is
d
=
ct.
If all other observers are to measure this same speed
c
for the light ray, then their times
t
ʹ and distances
d
ʹ must be such that
d
ʹ =
ct
ʹ. To write it in a fancier way, by squaring these expressions, the quantity
s
²
=
c
²
t
²
–
d
²
=
c
²
t
ʹ
²
–d
ʹ
²
must equal zero and so must be the same for all observers. This is the key that can unlock our picture of space and time in exact analogy to our cave dweller’s leap of insight.

Imagine shadows cast by a ruler on the cave wall. In one case, one may see this:

In another case, the same ruler can cast a shadow that looks like this:

To our poor cavewoman, fixed lengths will not be constant. What gives? We, who do not have to contend with two-dimensional projections but live in three-dimensional space, can see our way around the problem. Looking down on the ruler from above, we can see that in the two different cases the ruler was configured as shown below:

The second time, it had been rotated. We know that such a rotation does not change the ruler’s length but merely changes the “component” of its length that gets projected on the wall. If, for example, there were two different observers viewing shadows projected at right angles, the rotated ruler would have projected lengths as shown:

If the actual length of the ruler is
L,
Pythagoras tells us that
L
²
=
x
²
+
y
²
. Thus, even as the individual lengths,
x
and
y,
change for both observers, this combination will always remain constant. For one observer who measures the
y
direction, the original ruler would have zero extension, while for the other it
would have maximal extension in the
x
direction. The rotated ruler, on the other hand, would have a nonzero
y
extension but smaller
x
extension.

This is remarkably similar to the behavior of space and time for two observers in relative motion. A train moving very fast may appear shorter to me, but it will have some “time” extension—namely, the clocks on either end will not appear synchronized to me if they are to an observer on the train. Most important, the quantity
s
is analogous to the spatial length
L
in the cave example. Recall that
s
was defined as the “space-time” interval
s
²
=
c
²
t
²
–
d
²
. It represents a combination of separate space and time intervals between events, which is always zero between two space-time points lying on the trajectory of any light ray, regardless of the fact that different observers in relative motion will assign different individual values of
d
and
t
to the separate space and time intervals between the points they measure. It turns out that even if these points are not connected by a light ray but represent any two space-time points separated by length
d
and time
t
for one observer, the combination
s
(which need not be zero any longer) will be the same for all observers, again regardless of the fact that the separate length and time intervals measured may vary from observer to observer.

Thus, there
is
something about space and time that is absolute: the quantity
s.
This is for observers in relative motion what
L
is for observers rotated with respect to each other. It is the “space-time” length. The world we live in is thus best described as a four-dimensional space: The three dimensions of space and the “dimension” of time are coupled as closely (although not exactly in the same way) as the
x
and
y
directions were coupled above. And motion gives us different projections of this four-dimensional space on a three-dimensional slice we call
now,
just as rotations
give different projections of a three-dimensional object onto the two-dimensional wall of the cave! Einstein, with his insights based on his insistence on the constancy of light, had the privilege of doing what many of us only dream about. He escaped from the confines of our cave to glimpse for the first time a hidden reality beyond our limited human condition, just as did our cave dweller who discovered that a circle and a rectangle were really reflections of a single object.

 

 

To his credit, Einstein did not stop there. The picture was not yet complete. Again he used light as his guide. All observers moving at constant relative velocities observe a light ray to have the same velocity
c
relative to them, and thus not one of them can prove that it is he who is standing still and the others who are moving. Motion is relative. But what about when they aren’t moving at a constant velocity? What if one of them is accelerating? Will everyone in this case unambiguously agree that the odd one out is accelerating, including this lone observer? To gain some insight into these issues, Einstein considered an experience we have all had. When you are in an elevator, how do you know when and in what direction it has started to move? Well, if it begins to move upward, you momentarily feel heavier; if it moves downward, you momentarily feel lighter. But how do you
know
that it is actually moving, and not that gravity has suddenly gotten stronger?

The answer is, you don’t. There is not a single experiment you can do in a closed elevator that can tell you whether you are accelerating or in a stronger gravitational field. We can make it even simpler. Put the elevator in empty space, with no Earth around.
When the elevator is at rest, or moving at a constant velocity, nothing pulls you toward the floor. If the elevator is accelerating upward, however, the floor must push upward on you with some force to accelerate you along with the elevator. You in the elevator will feel yourself pushing down on the floor. If you have a ball in your hand and let it go, it will “fall” toward the floor. Why? Because if it were initially at rest, it would want to stay that way, by Galileo’s Law. The floor, however, would be accelerating upward, toward it. From your vantage point, which is accelerating upward along with the elevator, the ball would fall down. What’s more, this argument is independent of the mass of the ball. If you had six balls, all with different masses, they would all “fall” with the same acceleration. Again, this is because the floor would actually be accelerating upward toward them all at the same rate.