Hiding in the Mirror (6 page)

Thus, the exotic results of Einstein’s
relativity can be understood by analogy to the two-dimensional cave
example. In the latter case, different observations of the same
object appeared inconsistent because each presented a different
two-dimensional projection of the same threedimensional object. In
our universe, different observers in relative motion are simply
presented with different
three-dimensional
slices of an underlying
four-dimensional
universe
where space and time are tied together. Minkowski
called the mathematical combination
d
²
− c
²
t
²
the “spacetime
distance” between the events, to distinguish it from the
threedimensional, purely spatial distances we are used to. Just as
rotations in regular space can change projections, so, too, can
relative motion change the separate time and space intervals
measured by different observers, while the space-time distance is
preserved. Indeed, motion reproduces certain aspects that are
reminiscent of rotations. As Einstein’s train example makes clear,
one man’s space interval can be another man’s time interval. With
this unveiling of what we now call “Minkowski space,”

Minkowski delivered on the promise of his
Cologne lecture. Our Plato’s cave illustration merely makes literal
his metaphorical exclamation that heretofore space by itself and
time by itself would fade away into mere shadows. From 1908 onward,
three-dimensional space and the seemingly distinct and unrelated
one-dimensional progression of time became inextricably linked
together. What had begun with tentative inklings in basement
laboratories filled with compasses and currents had blossomed into a
whole different perspective of our universe to be explored and
understood. This four-dimensional space that we discovered we
occupy, however, differs dramatically from the world that Edward
Abbott envisaged in the plaintive pleas of his
Flatland
hero. The weird relative minus sign between
the spatial part and the time part of space-time distance (remember
that for normal spatial separations, the square of total distance
between two points is the
sum
of the
squares of the individual projections, with no minus signs) changes
everything, so that time and space are tied together in a way that
is quite unlike the way
up
and
sideways
are tied together. We cannot walk into time
as we can apparently walk into space, nor, as far as we yet know,
can we back up. Time travel is so exotic compared to motion in
space that entire movies and (fictional) books have been written to
consider this possibility. The minus sign fundamentally seems to
distinguish between spacetime intervals that are “timelike”
compared to those that are “spacelike.” (Minkowski himself coined
this terminology.)

Physics was thus left at the brink. A fourth
dimension had been discovered, but not the one that Abbott had
imagined. But people most often hear what they want to hear, and
consequently they often tend to interpret the new results of
science in terms that justify their previous expectations. Thus,
the feature that makes Minkowski space special, while profound, was
overshadowed by the newfound freedom of action offered by
Einstein’s special relativity, and the promise of a “fourth
dimension.” But Einstein was not yet finished with space and
time.

What is derived from
experience has only comparative universality,
namely, that which is obtained through induction. We
should therefore
only be able to say that,
so far as hitherto observed, no space has been
found which has more than three dimensions.

—Immanuel Kant,
Critique of
Pure Reason

F
or Kant, space existed
in the mind, as a backdrop for all of our experience. From his
perspective Euclid’s fundamental axioms of geometry were a priori
necessary features of a universe in which thinking beings could
live. Kant felt that these axioms were not derived from experience
or experiment, for if they were, they would merely be provisional,
not absolute.

Well, Kant was correct in at least one respect:
The postulates of Euclid, in particular his famous fifth
postulate—that there is only one line that can be drawn through any
point that does not intersect with (i.e., is parallel with) a given
line—cannot be derived from fundamental principles or from
experience. That is not the case, however, because they are
intrinsic to our existence. It is, rather, because they are not
universally true. On a sphere, for example, lines of longitude are
parallel, but they all meet at the North and South poles.

Such, it seems to me, is the limitation of much
of philosophy: It is often subsumed as empirical knowledge
supplants pure thought. The irony in this statement is that
Einstein’s most significant contribution to human knowledge comes as
close as any major development I know of in the history of physics
to something akin to pure thought. I refer to Einstein’s general
theory of relativity, which he developed in the decade after his
formulation of special relativity in 1905. The term
general
here refers to the fact that special
relativity applied to observers in constant relative motion. What
general relativity did was to extend these considerations to
accelerating observers. Remarkably, in the process, it turned out
to be a new theory of gravity!

That is not to say, however, that Einstein’s
general relativity was motivated by mathematical concerns alone,
either the beauty of tensor algebra, which made his theory
calculable, or that of Riemannian geometry, which Einstein had to
master in order to ultimately describe curved space. Far from it.
The origins of Einstein’s general theory of relativity stem from
the same type of thought experiments involving physical phenomena
that led to the special theory. In this case they came about as
Einstein was pondering Newton’s law of gravity, electromagnetism,
and special relativity in 1907, a year in which he later stated he
had had “the happiest thought of his life.”

We have already seen how the relationship
between electricity and magnetism implies that what one observer
measures as a magnetic force, another could measure as an electric
force. This “observer-dependence” of electromagnetism played a key
role in the development of special relativity and the unification of
space and time into space-time. Perhaps not surprisingly, a similar
notion played a central role in Einstein’s thinking when, in 1907,
while considering Newton’s gravity, he suddenly realized that it,
too, was observer dependent.

He reasoned as follows: An observer who is
free-falling in a gravitational field—like someone who jumps out of
a plane—feels no gravitational forces at all. For this observer,
the gravitational field is undetectable (at least until the rude
awakening, followed by a quick demise, upon later hitting the
ground). Ignoring any effects of air resistance, an object
“dropped” from such an observer’s hand would fall at the same rate
of acceleration as that of the observer, so it would remain at rest
relative to the observer. For all intents and purposes, gravity
wouldn’t exist for this individual. In this regard, as would be
equally true for Galileo’s observer moving at a constant speed in
the absence of gravity, such a free-falling observer would have
every right to consider herself at rest, because all objects at
rest in her frame would remain at rest if no other
(nongravitational) force was applied to them.

In this sense gravity, like electricity or
magnetism, seems to exist truly in the eye of the beholder. But
this picture is true only if all objects fall at the same rate. If
a single object accelerated at a different rate from all other
objects in a gravitational field, the whole notion that gravity
might be invisible would fall apart. A free-falling observer would
see this object as accelerating relative to her, and thus would be
able to conclude that some external force was acting upon it.

This idea—all objects fall at the same rate due
to gravity, independent of their composition—Einstein labeled the
Equivalence Principle, and it was central to his development of
general relativity. Only if it remained true could gravity arise as
an accident of one’s circumstances, just as the electric force that
one might experience could actually be due to a distant, changing
magnetic field.

While a violation of the equivalence principle
would put an end to any chance of “replacing” gravity with
something more fundamental and less observer-dependent, it is not
obvious from this example what one might actually replace it with.
Once again, Einstein provided a thought experiment that showed the
way. If falling in a gravitational field can get rid of any
observable effects of gravity, accelerating in the absence of one
can create the appearance of a gravitational field. Consider the
following famous example. Say, for some inexplicable reason, you
are in an elevator deep in space. As everyone who has ever been in
an elevator has experienced, when it first starts to accelerate
upward, you feel slightly heavier; namely, you feel a greater force
exerted by the floor on your feet. If you were in outer space, where
you would otherwise feel weightless, and the elevator you were in
started to accelerate upward, you would feel a similar force
pushing you down against the floor.

Einstein reasoned that, if the equivalence
principle was indeed true, then there is no experiment you could
perform in the elevator that could distinguish between whether that
elevator was accelerating upward in the absence of a gravitational
field, or whether it was at rest in a gravitational field, where the
force the observer would feel pushing her down against the floor
would be due to gravity.

So far so good. Now, imagine what would happen
if the observer in the accelerating elevator were to shine a laser
beam from one side of the elevator to the other. Since, during the
time the light beam was crossing the elevator, the elevator’s
upward speed would have increased, this would mean that the light
ray, which is traveling in a straight line relative to an observer
at rest
outside
the elevator, would end up
hitting the far side of the elevator somewhat below the height
where it was emitted, relative to the floor of the elevator

.

Now, if gravity is to produce effects that are
completely equivalent to those we would measure in an accelerating
system this would mean that if I shined a laser beam in an elevator
at rest in a gravitational field (say, on Earth), I would also see
the light ray’s trajectory bend downward. (Of course, the effect
would be very small, but since we are doing a thought experiment
here, we are free to imagine an arbitrarily accurate measuring
device.)

But, special relativity tells us that light
rays move at constant speed in straight lines. How can we reconcile
this behavior with what you would measure in the elevator? Well,
one way to go in a straight line and also travel in a curve is to
travel on a straight line on a curved surface. This realization led
Einstein on a long mental journey in the course of which he was
drawn to the inescapable conclusion that space and time are not
only coupled together, but are also themselves dramatically
different than we perceive them to be. Space, and to some extent
time, can be curved in the presence of mass or energy. The result
was perhaps the most dramatic reformulation of our understanding of
the underlying nature of the physical universe in the history of
science.

Einstein’s journey was replete with false
starts and dead ends, and the slowly dawning acceptance that
mathematical concepts that he had vaguely been exposed to while a
student might actually be useful for understanding the nature of
gravity. In 1912 Einstein finally realized that the mathematics of
Gauss, and then Riemann, which described the geometry of curved
surfaces and ultimately curved spaces, held the key to unlocking
the puzzle he had been wrestling with all those years. By November
1915, after almost having been scooped by the best mathematician of
that generation, David Hilbert, Einstein unveiled the final form of
his “gravitational field equations.” Einstein’s equations, as we
usually call them, provide a relation between the energy and
momentum of objects moving within space and the possible curvature
of that space. There are at least two fascinating and unexpected
facets of this relation. First, it turns out to be completely
independent of whatever system of coordinates one might use to
describe the position of objects within the curved space. Second,
and true to the spirit of special relativity—which by tying
together space and time also turned out to tie together mass and
energy—energy becomes the source of gravity. In general relativity,
however, such energy influences the very geometry of space itself—a
fact that makes general relativity almost infinitely more complex
and fascinating than Newton’s earlier law of gravitation. This is
because the energy associated with a gravitational field, and hence
with the curvature of space, in turn affects that curvature.

In the jargon of mathematicians, general
relativity is a “nonlinear” theory. While technically speaking this
means that it is difficult to solve the relevant equations, in
physical terms it means that the distribution of mass and energy in
space determines the strength of the gravitational field at any
point, which in turn determines the curvature of space at any
point, which in turn determines subsequent distribution of masses
and energy, which in turn determines the curvature of space, and so
on. Nevertheless, in spite of the difficulty of dealing with these
equations, the single fact that affected Einstein during that
fateful November in 1915 more deeply than perhaps any other
discovery he had made in his lifetime was the realization that the
mathematical theory he had just proposed explained an obscure but
mysterious astronomical observation about the orbit of Mercury
around the sun. One of the most successful and stunning predictions
of Newton’s law of gravity is that the orbit of planets around a
central body such as the Sun should be described by mathematical
curves called ellipses. That the planetary motions were not perfect
circles had first been discovered, somewhat to his dismay, by
Johannes Kepler, and in short order Newton proved that his
universal law universally implied elliptical orbits. Nevertheless,
in 1859 the French astronomer Urbain Jean Joseph Le Verrier
discovered that the orbit of Mercury was anomalous. Instead of
returning exactly to its initial position after each orbit, the
planet advanced slightly, so that rather than forming perfect
ellipses, the orbits traced a figure that was more like a spiral,
with the axis of each successive orbit being slightly shifted
compared to the one before it, as shown in an exaggerated view
below:

This “precession” was extremely small,
measuring only about 1/100 of a degree per century. Nevertheless,
in physics, as in horseshoes, being merely close is not good
enough; if Newton was correct, there should be no such precession.
Barring the presence of some new, undiscovered massive body nearby
exerting a gravitational pull on Mercury, the only way such a
precession could be explained was to slightly alter the nature of
Newtonian gravity.