Hiding in the Mirror (4 page)

Poincaré even discovered in 1905, the same year
that Einstein published his first paper on special relativity, that
the equations of electromagnetism remained unchanged if
measurements of space and time change for different observers in
relative motion in precisely such a way as to reproduce the
“Lorentz contraction”—as he then referred to it—which Lorentz had
earlier proposed to reconcile the negative result of the
MichelsonMorley experiment. Poincaré even demonstrated that the
different observers who synchronize their clocks by light signals
may have different notions of simultaneity.

It is remarkable that in spite of discovering
all of these pieces, Poincaré never fully put the puzzle together.
He remained committed both to the ether and to a dynamic origin for
the contraction of bodies along their direction of motion relative
to the ether. It remained for Einstein to demonstrate that
Maxwell’s equations, when combined with the ideas of Galilean
relativity, provided all that was necessary to resolve the
paradoxes of electrodynamics without additional dynamic hypotheses.
All that one had to do was dispense with the absolute

Henceforth space by itself,
and time by itself, are doomed to fade away
into mere shadows, and only a kind of union of the two
will preserve
an independent
reality.

—Hermann Minkowski

P
hysicists at the turn
of the twentieth century were understandably reluctant to abandon
the security of a sensible worldview that up to that point had
appeared to successfully describe the universe. But ultimately,
once the hidden connections that underlie electromagnetism came
into clear focus, there was no turning back, and the road that
began at “Let there be light” led straight into a fourth dimension.
First, the sensible worldview: If I am running away from you, and
someone behind you throws a ball at you, I would expect that the
ball would appear to be traveling faster relative to you than it
would to me. Common sense similarly suggests that two different
observers in relative motion will measure the same light ray to
travel at different speeds relative to each of them in, say, one
second.

Now the problem: Electromagnetism only makes
sense, in a world where all laboratories measure the same strength
of magnetism and electricity, if the light ray mentioned above
approaches each observer by the same distance in the same time,
even if the observers are moving apart. Now for Einstein’s
solution: If the light ray is to be so measured, then each observer
must use different measures of distance or time. Upon careful
analysis Einstein determined that both measurements must differ.
Specifically, Einstein demonstrated the following implications of
the strange behavior of light, in what we now call Einstein’s
special theory of relativity:

(i)
Remote events that
are simultaneous to one observer will not be simultaneous to
another observer moving with respect to the first.
(ii)
Clocks carried by an observer moving with
respect to another observer will be measured by the latter to be
running slowly.

(iii)
Objects carried
by an observer moving with respect to another observer will be
measured by the latter to be foreshortened along the direction of
their motion.

Einstein arrived at each of these bizarre
conclusions by doing what he called
gedanken,
or “thought experiments,” that get around
the fact that on human scales our perceptions of space and time are
vastly different from what they would be if we could travel at near
light-speed. In this, he followed the spirit of Poincaré’s
thinking. As Poincaré first pointed out, our knowledge of remote
events is always indirect, because remote events are, after all,
remote. We may feel like something we see happening across the room
is happening at the same moment as we see it, so that we are a
“part” of the event, but that is merely an illusion brought about
by the incredibly fast speed of light.

Consider a class photograph. We are accustomed
to thinking that it reflects a single frozen instant in time, when
all of the bright young faces are captured as an enduring memory.
But, strictly speaking, this is not accurate. Just as the different
rows of students are spread out in space, the photograph reflects an
image that is also spread out in time. The light reflected from the
faces of the children in the back row arrives at the camera lens at
the same instant as the light from the faces of the children in the
front row only if the light from the back row began its voyage
slightly earlier. The time difference is imperceptible, perhaps a
billionth of a second or so, but it
isn’t
zero. If each row were separated from the row in front by, say, a
hundred million miles, instead of a few feet, then the students in
the back row could easily have left their seats by the time the
students in the front row had begun to pose for one and the same
photograph. This is because the light from the back row would take
about ten minutes to reach the front row, and would thus reach the
camera at the same time as light emitted from the front row ten
minutes later. In an astronomical context, this is always true.
When we look up at the sky at night, the images of the individual
stars reflect moments spread out by hundreds if not thousands of
years.

We are accustomed to this phenomenon in a
reverse context because of the fact that sound travels much more
slowly than light. When we see lightning strike in the distance,
and we hear the thunder clap many seconds later, we know that they
relate to one and the same event, even though we experience its
different aspects at different times. It is equally true however,
that things we experience in a single instant can reflect not one
event, but many separate ones.

Einstein imagined a scenario where this would
be explicit. Picture, for example, a train so long that light from
one end of it would take several seconds to reach the other. Now
picture that you are in the middle of the train. Now picture,
finally, another implausible series of events: Lightning strikes
both ends of the train at exactly the same instant. How do you know
that the two lightning bolts hit either end of the train at the
same time? Simple: You see the two flashes in your car at the same
instant. Since you are in the middle of the train, you know that,
even accounting for the fact that it has taken some time for the
images to reach you, since the time for both images to reach you is
the same, the flashes must have been simultaneous.

Now, what about someone on the ground whom you
see directly opposite you at exactly the instant when the lightning
bolts struck the ends of the train (not later, when you actually
see the flashes!)? What would she see (assuming the flashes were
bright enough for her to see them as well)?

Well, since you are moving with respect to her,
by the time you see the flashes she must now be closer to one end of
the train than the other.

Thus, the light from one of the flashes must
have passed her location before it made its way to you. Hence, she
will see one of the flashes before the other. But since she was
opposite you when the lightning hit either end of the train, and
was thus also midway between the flashes, and since she sees one
before the other, she must infer that one of the flashes hit before
the other.

What is wrong with this picture? Well, in a
sensible universe the person on the ground would indeed see one
flash before the other, and the person on the train would see both
flashes at the same time. But the person on the train, whom the
person on the ground would see moving toward one flash and away from
the other, would also be able to (if she had the proper apparatus)
measure that the light ray from the side of the train that she was
moving toward would be traveling relative to her faster than the
other light ray, which she would be moving away from. Thus,
although she saw both flashes at the same time, she would indeed be
able to infer from her measurements that one event had to have
occurred before the other in order for her to experience them
simultaneously, in agreement with the assessment of the person on
the ground.

But the universe isn’t so sensible. Maxwell,
Einstein, and experiment all tell us that both observers will
measure the speed of both light rays relative to themselves to be
exactly the same, and, as a result, each observer is forced to a
different conclusion about the simultaneity of the two events. It
is important here not to think that one observer is right, and one
is wrong. They are both right. There is not a single experiment
either person can do to change her own perception of the events or
to prove the other person wrong. If they could, then one of them
would be able to prove that she was at rest while the other person
was moving. But that is the whole point. There is no absolute rest
frame with respect to the speed of light. All observers are
equivalent. So that means that whether or not distant events are
simultaneous depends upon who is doing the observing. There is no
absolute “now.” “Now” means something unambiguous only right where
you are. Anything you conclude about “now” elsewhere is simply an
inference, and it is unique to you. To put another way, “now” is
relative.

It is also important not to think that any
sense of “now” is therefore completely arbitrary. It is just as
constrained after Einstein’s
gedanken
experiment as it was before. Each observer can base a consistent
reality on what she sees, and she can count on the fact that events
never precede causes, and so on, even if it turns out that for one
observer one event may happen before another, while for another
observer precisely the opposite may be true. It turns out that the
mathematics of relativity happily only allow this reversal in
temporal ordering for events witnessed by different observers
whenever the events are sufficiently remote in space and close in
time, so that one event cannot have been the cause of the other.
Put another way, if a signal can travel between the events in the
time between them, then all observers will end up agreeing about
which happened first, even if the observers might disagree about how
much time had elapsed between them. But just in case you were
beginning to think things might be sensible after all, consider the
following: The same type of reasoning that led Einstein to
recognize that simultaneity was relative led him to recognize that
measures of length and time themselves were also relative. For
example, let us return to our train example. When the lightning
struck simultaneously (for the observer on the train), let us say
it scorched the tracks at the same time. Thus, that observer can
come back later and measure the distance between the scorch marks
to determine the length of the train. But the observer who was on
the ground at the time will call foul. She will insist that because
one lightning bolt hit before the other, and during the time
between the two events, the train was moving, that, the scorch
marks on the ground represent a distance that is
longer
than the actual size of the moving train. In
short, the observer on the ground who sees the train moving past
will insist that the train is
shorter
than
will an observer on the train, who is at rest with respect to it.
So far so good. Moving objects are measured to be contracted along
their direction of motion. In fact, this contraction is precisely
that calculated earlier by Lorentz (which was dubbed the “Lorentz
contraction” by Poincaré) when he tried to make sense of the
Michelson-Morley experiment. But here the resemblance ends. In
Lorentz’s worldview, where there was an ether and a universal rest
frame, moving objects could be contracted relative to those
standing still. But in Einstein’s universe, which happens to be the
one we live in, all motion is relative. There is no universal rest
frame and no ether. So, for a person on the moving train, it is the
person on the ground who can be said to be moving past, in the
opposite direction. And exactly the same type of reasoning as given
above will convince you that the person on the train will measure
the lengths of objects at rest with respect to the person on the
ground to be shorter than will the person on the ground!

Thus, each observer will measure the length of
objects at rest in the other person’s frame of reference to be
shorter. The Lorentz contraction is not absolute; it is
relative.

Once again, the relative nature of the Lorentz
contraction should not lull you into assuming that it is not real.
It is as real as the nose on my face, whose size will, of course,
depend upon who is viewing it. This is illustrated by my very
favorite paradox from relativity. Thankfully for you, it is the
last one I will attack your brain with here.

Say I have a fast sports car—a
really
fast one, which can travel at a large
fraction of the speed of light, where the mysterious effects of
relativity become more apparent. After all, if you consider the
gedanken
experiments I have discussed
above, clearly the discrepancies about length and time between
observers are related to how far the train could have traveled
during the time the light rays crossed it. To have observable
effects, one needs either very large trains or very fast ones.
Well, say I am moving past you at a very large fraction of the
speed of light. My car will therefore be measured by you to be
shorter than I will measure it to be. Now, say you have a garage
with two doors, one at either end, into which I am driving. If my
car is ten feet long to me, say it would be measured to be six feet
long by you. Say your garage is eight feet in depth. Then, for you
it should certainly be possible to quickly close the front door of
your garage after my car has entered and continues speeding along,
completely enclosing it within the garage. You would then hopefully
run very quickly to open the door at the rear of the garage so that
my speeding car would not run into it. Relativity tells us that
this is certainly possible, at least in principle. But now there is
a problem. In my reference frame, it is your moving garage that is
shorter. To me, it appears to be only five feet long, and there is
no way that my car will fit within it!

Am I doomed to crash? Well, if I do hit a door,
both observers would have to agree that such an event happened.
(After all, they can come back together afterward and see the
tangled mess, if both people are still alive.) So, if one observer
sees me making it through the garage safely, then I must have done
so. Rather, I will insist that my car and I were never entirely
within the garage, because I will measure the order in time of the
remote events, including opening and closing the garage doors, to
be different than will the observer on the ground. I will insist
that, for example, the rear door of the garage was opened before
its front door was closed. Thus, as I sped through, the front of my
car exited the back of the garage before its rear end passed
through the front of the garage. The point is that each observer’s
reality is real. For you, my car was completely inside the garage.
For me, it never was. There is no experiment you can perform that
will prove me wrong, and vice versa. At the same time, it is clear
that the contraction, while real, is still very much in the eye of
the beholder. Or, as Einstein would say, measurements of length are
relative.