The Physics of Star Trek (5 page)

So, for Picard, the trajectory of the phaser ray would be curved. What else would Picard
notice? Well, recalling the argument in the first chapter, as long as the inertial dampers
are turned off, he would be thrust back in his seat. In fact, I also noted there that if
Picard was being accelerated forward at the same rate as gravity causes things to
accelerate downward at the Earth's surface, he would feel exactly the same force pushing
him back against his seat that he would feel pushing him down if he were standing on
Earth. In fact, Einstein argued that Picard (or his equivalent in a rising elevator) would
never be able to perform any experiment that could tell the difference between the
reaction force due to his acceleration and the pull of gravity from some nearby heavy
object outside the ship. Because of this, Einstein boldly went where no physicist had gone
before, and reasoned that whatever phenomena an accelerating observer experienced would be
identical to the phenomena an observer in a gravitational field experienced.

Our example implies the following: Since Picard observes the phaser ray bending when he is
accelerating away from it, the ray must also bend in a gravitational field. But if light
rays map out spacetime, then
spacetime
must bend in a gravitational field. Finally, since matter produces a gravitational field,
then
matter must bend spacetime!

Now, you may argue that since light has energy, and mass and energy are related by
Einstein's famous equation, then the fact that light bends in a gravitational field is no
big surpriseand certainly doesn't seem to imply that we have to believe that spacetime
itself need be curved. After all, the paths that matter follows bend too (try throwing a
ball in the air). Galileo could have shown, had he known about such objects, that the
trajectories of baseballs and Pathfinder missiles bend, but he never would have mentioned
curved space.

Well, it turns out that you can calculate how much a light ray should bend if light
behaved the same way a baseball does, and then you can go ahead and measure this bending,
as Sir Arthur Stanley Eddington did in 1919 when he led an expedition to observe the
apparent position of stars on the sky very near the Sun during a solar eclipse.
Remarkably, you would find, as Eddington did, that light bends exactly
twice
as much as Galileo might have predicted if it behaved like a baseball in flat space. As
you may have guessed, this factor of 2 is just what Einstein predicted if spacetime was
curved in the vicinity of the Sun and light (or the planet Mercury, for that matter) was
locally traveling in a straight line in this curved space! Suddenly, Einstein's was a
household name.

Curved space opens up a whole universe of possibilities, if you will excuse the pun.
Suddenly we, and the
Enterprise,
are freed from the shackles of the kind of linear thinking imposed on us in the context of
special relativity, which Q, for one, seemed to so abhor. One can do many things on a
curved manifold which are impossible on a flat one. For example, it is possible to keep
traveling in the same direction and yet return to where you beganpeople who travel around
the world do it all the time.

The central premise of Einstein's general relativity is simple to state in words: the
curvature of spacetime is

directly determined by the distribution of matter and energy contained within it.
Einstein's equations, in fact, provide simply the strict mathematical relation between
curvature on the one hand and matter and energy on the other:

What makes the theory so devilishly difficult to work with is this simple feedback loop:
The curvature of spacetime is determined by the distribution of matter and energy in the
universe, but this distribution is in turn governed by the curvature of space. It is like
the chicken and the egg. Which was there first? Matter acts as the source of curvature,
which in turn determines how matter evolves, which in turn alters the curvature, and so on.

Indeed, this may be perhaps the most important single aspect of general relativity as far
as Star Trek is concerned. The complexity of the theory means that we still have not yet
fully understood all its consequences; therefore we cannot rule out various exotic
possibilities. It is these exotic possibilities that are the grist of Star Trek's mill. In
fact, we shall see that all these possibilities rely on one great unknown that permeates
everything, from wormholes and black holes to time machines.

The first implication of the fact that spacetime need not be flat which will be important
to the adventures of the
Enterprise
is that time itself becomes an even more dynamic quantity than it was in special
relativity. Time can flow at different rates for different observers even if they are not
moving relative to each other. Think of the ticks of a clock as the ticks on a ruler made
of rubber. If I were to stretch or bend the ruler, the spacing between the ticks would
differ from point to point. If this spacing represents the ticks of a clock, then clocks
located in different places can tick at different rates. In general relativity, the only
way to “bend” the ruler is for a gravitational field to be present, which in turn requires
the presence of matter.

To translate this into more pragmatic terms: if I put a heavy iron ball near a clock, it
should change the rate at which the clock ticks. Or more practical still, if I sleep with
my alarm clock tucked next to my body's rest mass, I will be awakened a little later than
I would otherwise, at least as far as the rest of the world is concerned.

A famous experiment done in the physics laboratories at Harvard University in 1960 first
demonstrated that time can depend on where you are. Robert Pound and George Rebka showed
that the frequency of gamma radiation measured at its source, in the basement of the
building, differed from the frequency of the radiation when it was received 74 feet
higher, on the building's roof (with the detectors having been carefully calibrated so
that any observed difference would not be detector-related). The shift was an incredibly
small amount about 1 part in a million billion. If each cycle of the gamma-ray wave is
like the tick of an atomic clock, this experiment implies that a clock in the basement
will appear to be running more slowly than an equivalent atomic clock on the roof. Time
slows on the lower floor because this is closer to the Earth than the roof is, so the
gravitational field, and hence the spacetime curvature, is larger there. As small as this
effect was, it was precisely the value predicted by general relativity, assuming that
spacetime is curved near the Earth.

The second implication of curved space is perhaps even more exciting as far as space
travel is concerned. If space is curved, then a straight line need not be the shortest
distance between two points. Here's an example. Consider a circle on a piece of paper.
Normally, the shortest distance between two points A and B located on opposite sides

of the circle is given by the line connecting them through the center of the circle:

If, instead, one were to travel around the circle to get from A to B, the journey would be
about
1 1/2
times as long. However, let me draw this circle on a rubber sheet, and distort the central
region:

Now, when viewed in our three-dimensional perspective, it is clear that the journey from A
to B taken through the center of the region will be much longer than that taken by going
around the circle. Note that if we took a snapshot of this from above, so we would have
only a two-dimensional perspective, the line from A to B through the center would look
like a straight line. More relevant perhaps, if a tiny bug (or two-dimensional beings, of
the type encountered by the
Enterprise)
were to follow the trajectory from A to B through the center by crawling along the surface
of the sheet, this trajectory would appear to be straight. The bug would be amazed to find
that the straight line through the center between A and B was no longer the shortest
distance between these two points. If the bug were intelligent, it would be forced to the
conclusion that the two-dimensional space it lived in was curved. Only by viewing the
embedding of this sheet in the underlying three-dimensional space can we observe the
curvature directly.

Now, remember that we live within a four-dimensional spacetime that can be curved, and we
can no more perceive the curvature of this space directly than the bug crawling on the
surface of the sheet can detect the curvature of the sheet. I think you know where I am
heading: If, in curved space, the shortest distance between two points need not be a
straight line, then it might be possible to traverse what appears
along the line of sight
to be a huge distance, by finding instead a shorter route through curved spacetime.

These properties I have described are the stuff that Star Trek dreams are made of. Of
course, the question is: How many of these dreams may one day come true?

WORMHOLES: FACT AND FANCY: The Bajoran wormhole in
Deep Space Nine
is perhaps the most famous wormhole in Star Trek, although there have been plenty of
others, including the dangerous wormhole that Scotty could create by imbalancing the
matter-antimatter mix in the
Enterprise's
warp drive; the unstable Barzan wormhole, through which a Ferengi ship was lost in the
Next Generation
episode "The

Price"; and the temporal wormhole that the
Voyager
encountered in its effort to get back home from the far edge of the galaxy.

The idea that gives rise to wormholes is exactly the one I just described. If spacetime is
curved, then perhaps there are different ways of connecting two points so that the
distance between them is much shorter than that which would be measured by traveling in a
“straight line” through curved space. Because curved-space phenomena in four dimensions
are impossible to visualize, we once again resort to a two-dimensional rubber sheet, whose
curvature we can observe by embedding it in three-dimensional space.

If the sheet is curved on large scales, one might imagine that it looks something like
this:

Clearly, if we were to poke a pencil down at A and stretch the sheet until we touched B,
and then sewed together the two parts of the sheet, like so:

we would create a path from A to B that was far shorter than the path leading around the
sheet from one point to another. Notice also that the sheet appears flat near A and also
near B. The curvature that brings these two points close enough together to warrant
joining them by a tunnel is due to the global bending of the sheet over large distances. A
little bug (even an intelligent one) at A, confined to crawl on the sheet, would have no
idea that B was as “close” as it was, even if it could do some local experiments around A
to check for a curvature of the sheet.

As you have no doubt surmised, the tunnel connecting A and B in this figure is a
two-dimensional analogue of a three-dimensional wormhole, which could, in principle,
connect distant regions of space-time. As exciting as this possibility is, there are
several deceptive aspects of the picture which I want to bring to your attention. In the
first place, even though the rubber sheet is shown embedded in a three-dimensional space
in order for us to “see” the curvature of the sheet, the curved sheet can exist without
the three-dimensional space around it needing to exist. Thus, while a wormhole could exist
joining A and B, there is no sense in which A and B are “close”
without
the wormhole being present. It is not as if one is free to leave the rubber sheet and move
from A to B through the three-dimensional space in which the sheet is embedded. If the
three-dimensional space is not there, the rubber sheet is all there is to the universe.

Thus, imagine that you were part of an infinitely advanced civilization (but not as
advanced as the omnipotent Q beings, who seem to transcend the laws of physics) that had
the power to build wormholes in space. Your wormhole building device would effectively be
like the pencil in the example I just gave. If you had the power to produce huge local
curvatures in space, you would have to poke around blindly in the hope that somehow you
could connect two regions of space that, until the instant a wormhole was established,
would remain very distant from each other. In no way whatsoever would these two regions be
close together until the wormhole produced a bridge. The bridge-building process
itself
is what changes the global nature of spacetime.

Because of this, making a wormhole is not to be taken lightly. When Premier Bhavani of
Barzan visited the
Enterprise
to auction off the rights to the Barzan wormhole, she exclaimed, “Before you is the first
and only stable wormhole known to exist!” Alas, it wasn't stable; indeed, the only
wormholes whose mathematical existence has been consistently established in the context of
general relativity are transitory. Such wormholes are created as two microscopic
“singularities” regions of spacetime where, the curvature becomes infinitely sharp find
each other and momentarily join. However, in a time shorter than the time it would take a
space traveler to pass through such a worm-hole, it closes up, leaving once again two
disconnected singularities. The unfortunate explorer would be crushed to bits in one
singularity or the other before being able to complete the voyage through the wormhole.

The problem of how to keep the mouth of a wormhole open has been hideously difficult to
resolve in mathematical detail, but is quite easily stated in physical terms: Gravity
sucks! Any kind of normal matter or energy will tend to collapse under its own
gravitational attraction unless something else stops it. Similarly, the mouth of a
wormhole will pinch off in nothing flat under normal circumstances.

So, the trick is to get rid of the normal circumstances. In recent years, the Caltech
physicist Kip Thorne, among others, has argued that the only way to keep wormholes open is
to thread them with “exotic material.” By this is meant material that will be measured, at
least by certain observers, to have “negative” energy. As you might expect (although naive
expectations are notoriously suspect in general relativity), such material would tend to
“blow” not “suck,” as far as gravity is concerned.