Hiding in the Mirror (5 page)

A similar relativity occurs for the slowing of
clocks. If I am moving very fast relative to you, you will measure
my clocks to be running slowly. I will appear to you to age more
slowly if you watch me recede into the distance. But I will in turn
measure your clocks to be running slowly as, well, you will appear
to me to age more slowly.

At this point a conventional reaction to the
implications of relativity is to throw up one’s hands and decide
that the world has no order in it whatsoever and that there exist
no absolutes. Everything is relative, so anything goes! Indeed,
this was the reaction of many artists and writers in the early part
of the twentieth century to the results of relativity, as I shall
soon discuss. But even if it feels justified, this is not the
correct response. Hermann Minkowski had been one of Einstein’s
mathematics teachers in Zurich—in fact, one of the few whose
lectures Einstein actually enjoyed. In 1902 Minkowski moved to the
University of Göttingen, where one of the most renowned
mathematicians of his time, David Hilbert, was located.
Interestingly, Hilbert would later help Einstein provide the
mathematical tools that would change our picture of space and time
in profoundly new ways. But well before that, the first Göttingen
mathematician to have had such an impact was Minkowski.

In 1908 in Cologne Minkowski gave a lecture
entitled “Space and Time,” which created a tremendous stir and has
since been recognized as a watershed moment in our understanding of
physical reality. The epigraph of this chapter is from that
lecture, which began with words that are both enticing and
particularly significant for a mathematician to have uttered: “The
views of space and time which I wish to lay before you have sprung
from the soil of experimental physics, and therein lies their
strength. They are radical.” In his speech, and in the more
technical paper that accompanied it, Minkowski delivered exactly
what he had promised. By the time he was finished, space and time
could no longer sensibly be individually discussed, and only a
union of the two, which we now call space-time, was understood to
retain any independent reality.

The seeds of Minkowski’s realization lie in the
example I presented involving Einstein’s long train. Recall that
simultaneous lightning bolts for an observer on a moving train
provide an ideal method for her to measure the length of the train.
She merely has to later disembark, return to the scene of the
lightning strikes, and measure the distance between the scorch
marks on the tracks.

Now, also recall that an observer on the ground
will contest this measurement, arguing that the two lightning bolts
were not simultaneous and therefore the scorch marks represent
events that happened at two different times at either end of a
moving train. Thus, the distance between the scorch marks must
represent a larger distance than the true length of the train. Let
us then consider what this implies by thinking in terms of what,
precisely, is meant by a measurement. The observer on the train
measures an interval in space. That is, after all, what a
measurement of distance is. For the observer on the ground,
however, this same measurement involves an interval in space
and
time.Seen from this perspective,
perhaps it is not surprising that the individual distance and time
difference measurements for the two observers differ. To visualize
this a little more dramatically, let us imagine two observers in
Plato’s cave. One of them sees the following shadow on the cave
wall, in the morning:

Later in the day, the other observer sees this
one:

Has the person whose shadow they have seen at
different times of day changed in height? No, of course not.
Rather, the sun is higher in the sky, and the length of the shadow
on the back wall of the cave will change accordingly.

Let’s simplify the issue. Imagine the cave
dwellers are viewing the shadow of a transparent ruler:

Now suddenly the shadow changes:

The shadow-ruler has inexplicably changed in
length. How was this possible? Simple: The original ruler was
rotated with respect to the light source. As seen from above, the
two different situations appear as follows:

  The length of the original ruler
has certainly not changed by this rotation, but the
projection
of this three-dimensional object onto the
twodimensional wall at the back of the cave has. Physicists in this
cave-dwelling society may initially be baffled by the fact that the
lengths of shadowobjects are apparently not absolute. But
eventually someone would intuit that the objects being observed
behave as two-dimensional projections of three-dimensional objects
that can be rotated perpendicular to the wall. Mathematically,
there is a quantity that is absolute and doesn’t change under such
rotations—namely, the length of the original ruler. If this ruler
has a length
L,
while the length of the
shadow-ruler (i.e., the projection of
L
on
the cave wall) is
X
, then a cave
mathematician, who, for the sake of argument we might call
Pythagoras, might suggest that there is a quantity,
L,
whose value does not change, and that is given by
the relation
L
²
=
X
²
+
Y
²
, where
X
is the projection of
the ruler on the cave wall and
Y
is the
projection of
L
perpendicular to the cave
wall:

By now you don’t have to be Einstein to see
where we are heading. What Hermann Minkowski realized is that there
is a similarity (but
just
a similarity)
between this scenario and what occurs, according to relativity, for
observers in relative motion measuring the same object. Recall that
the speed of light in empty space,
c
, is
measured to be the same by all observers. Say one observer measures
the distance traveled by a light ray in some time
t
to have a value
d
. Since
distance traveled is determined by the speed of the light ray times
the time it travels, this observer thus finds
d
=
ct
. Any other observer
moving with respect to this observer may in general measure a
different length
d
' and time
t
'
,
but they must find
d
' =
ct
' if they
are to determine the same speed relative to them for this light
ray. Thus, at least for a light ray, different observers in
relative motion will measure distances and times such that the
combination
d
²
−
c
²
t
²
=
d
'
²
−
c
²
t
'
²
= 0 for any light ray. While this is manifestly true
for a light ray, it turns out that this combination will be
measured to be the same by all observers for any two “space-time”
events measured to be separated by a distance
d
and time
t
for any one of
them, so that
d
²
− c
²
t
²
= d '
²
−c
²
t
'
²
for all events separated in space and time even
if the combination is not zero (i.e., the two distances and times
are not for points connected by the trajectory of a light ray).
This will be true even though the separate observers will in
general arrive at different separate measurements of
d
and
t
. Minkowski realized
that this particular combination of distance and time, which
Einstein recognized remains invariant between observers in relative
motion, is strikingly analogous to the way the different length
projections of a ruler can be combined to always produce the same
value—namely the length of the ruler itself—regardless of its
orientation. Except for the weird minus sign (i.e.,
d
²
− c
²
t
²
instead of 
d
²
+ c
²
t
²
), which we will
discuss shortly, the combination is the same.