Space and Time Warps are a commonplace. They are used for rapid
journeys around the galaxy, or for travel through time. But today's
science fiction, is often tomorrow's science fact. So what are the
chances for space and time warps.
The idea that space and time can be curved, or warped, is fairly recent. For more than two thousand years, the axioms of Euc
lidean
geometry, were considered to be self evident. As those of you that were
forced to learn Euclidean geometry at school may remember, one of the
consequences of these axioms is, that the angles of a triangle, add up
to a hundred and 80 degrees.
However, in the last century,
people began to realize that other forms of geometry were possible, in
which the angles of a triangle, need not add up to a hundred and 80
degrees. Consider, for example, the surface of the Earth. The nearest
thing to a straight line on the surface of the Earth, is what is called,
a great circle. These are the shortest paths between two points, so
they are the roots that air lines use. Consider now the triangle on the
surface of the Earth, made up of the equator, the line of 0 degrees
longitude through London, and the line of 90 degrees longtitude east,
through Bangladesh. The two lines of longitude, meet the equator at a
right angle, 90 degrees. The two lines of longitude also meet each other
at the north pole, at a right angle, or 90 degrees. Thus one has a
triangle with three right angles. The angles of this triangle add up to
two hundred and seventy degrees. This is greater than the hundred and
eighty degrees, for a triangle on a flat surface. If one drew a triangle
on a saddle shaped surface, one would find that the angles added up to
less than a hundred and eighty degrees. The surface of the Earth, is
what is called a two dimensional space. That is, you can move on the
surface of the Earth, in two directions at right angles to each other:
you can move north south, or east west. But of course, there is a third
direction at right angles to these two, and that is up or down. That is
to say, the surface of the Earth exists in three-dimensional space. The
three dimensional space is flat. That is to say, it obeys Euclidean
geometry. The angles of a triangle, add up to a hundred and eighty
degrees. However, one could imagine a race of two dimensional creatures,
who could move about on the surface of the Earth, but who couldn't
experience the third direction, of up or down. They wouldn't know about
the flat three-dimensional space, in which the surface of the Earth
lives. For them, space would be curved, and geometry would be
non-Euclidean.
It would be very difficult to design a living being that could exist in only two dimensions.
Food that the creature couldn't digest would have to be spat out the
same way it came in. If there were a passage right the way through, like
we have, the poor animal would fall apart.
So three
dimensions, seems to be the minimum for life. But just as one can think
of two dimensional beings living on the surface of the Earth, so one
could imagine that the three dimensional space in which we live, was the
surface of a sphere, in another dimension that we don't see. If the
sphere were very large, space would be nearly flat, and Euclidean
geometry would be a very good approximation over small distances. But we
would notice that Euclidean geometry broke down, over large distances.
As an illustration of this, imagine a team of painters, adding paint to
the surface of a large ball. As the thickness of the paint layer
increased, the surface area would go up. If the ball were in a flat
three-dimensional space, one could go on adding paint indefinitely, and
the ball would get bigger and bigger. However, if the three-dimensional
space, were really the surface of a sphere in another dimension, its
volume would be large but finite. As one added more layers of paint, the
ball would eventually fill half the space. After that, the painters
would find that they were trapped in a region of ever decreasing size,
and almost the whole of space, was occupied by the ball, and its layers
of paint. So they would know that they were living in a curved space,
and not a flat one.
This example shows that one can not deduce the
geometry of the world from first principles, as the ancient Greeks
thought. Instead, one has to measure the space we live in, and find out
its geometry by experiment. However, although a way to describe curved
spaces, was developed by the German, George Friedrich Riemann, in 1854,
it remained just a piece of mathematics for sixty years. It could
describe curved spaces that existed in the abstract, but there seemed no
reason why the physical space we lived in, should be curved. This came
only in 1915, when Einstein put forward the General Theory of
Relativity.
General Relativity was a major intellectual
revolution that has transformed the way we think about the universe. It
is a theory not only of curved space, but of curved or warped time as
well. Einstein had realized in 1905, that space and time, are intimately
connected with each other. One can describe the location of an event by
four numbers. Three numbers describe the position of the event. They
could be miles north and east of Oxford circus, and height above sea
level. On a larger scale, they could be galactic latitude and longitude,
and distance from the center of the galaxy. The fourth number, is the
time of the event. Thus one can think of space and time together, as a
four-dimensional entity, called space-time. Each point of space-time is
labeled by four numbers, that specify its position in space, and in
time. Combining space and time into space-time in this way would be
rather trivial, if one could disentangle them in a unique way. That is
to say, if there was a unique way of defining the time and position of
each event. However, in a remarkable paper written in 1905, when he was a
clerk in the Swiss patent office, Einstein showed that the time and
position at which one thought an event occurred, depended on how one was
moving. This meant that time and space, were inextricably bound up with
each other. The times that different observers would assign to events
would agree if the observers were not moving relative to each other. But
they would disagree more, the faster their relative speed. So one can
ask, how fast does one need to go, in order that the time for one
observer, should go backwards relative to the time of another observer.
The answer is given in the following Limerick.
There was a young lady of Wight,
Who traveled much faster than light,
She departed one day,
In a relative way,
And arrived on the previous night.
So all we need for time travel, is a space ship that will go faster
than light. Unfortunately, in the same paper, Einstein showed that the
rocket power needed to accelerate a space ship, got greater and greater,
the nearer it got to the speed of light. So it would take an infinite
amount of power, to accelerate past the speed of light.
Einstein's paper of 1905 seemed to rule out time travel into the past.
It also indicated that space travel to other stars, was going to be a
very slow and tedious business. If one couldn't go faster than light,
the round trip to the nearest star, would take at least eight years, and
to the center of the galaxy, at least eighty thousand years. If the
space ship went very near the speed of light, it might seem to the
people on board, that the trip to the galactic center had taken only a
few years. But that wouldn't be much consolation, if everyone you had
known was dead and forgotten thousands of years ago, when you got back.
That wouldn't be much good for space Westerns. So writers of science
fiction, had to look for ways to get round this difficulty.
In
his 1915 paper, Einstein showed that the effects of gravity could be
described, by supposing that space-time was warped or distorted, by the
matter and energy in it. We can actually observe this warping of
space-time, produced by the mass of the Sun, in the slight bending of
light or radio waves, passing close to the Sun. This causes the apparent
position of the star or radio source, to shift slightly, when the Sun
is between the Earth and the source. The shift is very small, about a
thousandth of a degree, equivalent to a movement of an inch, at a
distance of a mile. Nevertheless, it can be measured with great
accuracy, and it agrees with the predictions of General Relativity. We
have experimental evidence, that space and time are warped. The amount
of warping in our neighbourhood, is very small, because all the
gravitational fields in the solar system, are weak. However, we know
that very strong fields can occur, for example in the Big Bang, or in
black holes. So, can space and time be warped enough, to meet the
demands from science fiction, for things like hyper space drives,
wormholes, or time travel. At first sight, all these seem possible. For
example, in 1948, Kurt Goedel found a solution of the field equations of
General Relativity, which represents a universe in which all the matter
was rotating. In this universe, it would be possible to go off in a
space ship, and come back before you set out. Goedel was at the
Institute of Advanced Study, in Princeton, where Einstein also spent his
last years. He was more famous for proving you couldn't prove
everything that is true, even in such an apparently simple subject as
arithmetic. But what he proved about General Relativity allowing time
travel really upset Einstein, who had thought it wouldn't be possible.
We now know that Goedel's solution couldn't represent the universe in
which we live, because it was not expanding. It also had a fairly large
value for a quantity called the cosmological constant, which is
generally believed to be zero. However, other apparently more reasonable
solutions that allow time travel, have since been found. A particularly
interesting one contains two cosmic strings, moving past each other at a
speed very near to, but slightly less than, the speed of light. Cosmic
strings are a remarkable idea of theoretical physics, which science
fiction writers don't really seem to have caught on to. As their name
suggests, they are like string, in that they have length, but a tiny
cross section. Actually, they are more like rubber bands, because they
are under enormous tension, something like a hundred billion billion
billion tons. A cosmic string attached to the Sun would accelerate it
naught to sixty, in a thirtieth of a second.
Cosmic strings may
sound far-fetched, and pure science fiction, but there are good
scientific reasons to believed they could have formed in the very early
universe, shortly after the Big Bang. Because they are under such great
tension, one might have expected them to accelerate to almost the speed
of light.
What both the Goedel universe, and the fast moving cosmic
string space-time have in common, is that they start out so distorted
and curved, that travel into the past, was always possible. God might
have created such a warped universe, but we have no reason to think that
He did. All the evidence is, that the universe started out in the Big
Bang, without the kind of warping needed, to allow travel into the past.
Since we can't change the way the universe began, the question of
whether time travel is possible, is one of whether we can subsequently
make space-time so warped, that one can go back to the past. I think
this is an important subject for research, but one has to be careful not
to be labeled a crank. If one made a research grant application to work
on time travel, it would be dismissed immediately. No government agency
could afford to be seen to be spending public money, on anything as way
out as time travel. Instead, one has to use technical terms, like
closed time like curves, which are code for time travel. Although this
lecture is partly about time travel, I felt I had to give it the
scientifically more respectable title, Space and Time warps. Yet, it is a
very serious question. Since General Relativity can permit time travel,
does it allow it in our universe? And if not, why not.
Closely
related to time travel, is the ability to travel rapidly from one
position in space, to another. As I said earlier, Einstein showed that
it would take an infinite amount of rocket power, to accelerate a space
ship to beyond the speed of light. So the only way to get from one side
of the galaxy to the other, in a reasonable time, would seem to be if we
could warp space-time so much, that we created a little tube or
wormhole. This could connect the two sides of the galaxy, and act as a
short cut, to get from one to the other and back while your friends were
still alive. Such wormholes have been seriously suggested, as being
within the capabilities of a future civilization. But if you can travel
from one side of the galaxy, to the other, in a week or two, you could
go back through another wormhole, and arrive back before you set out.
You could even manage to travel back in time with a single wormhole, if
its two ends were moving relative to each other.
One can show
that to create a wormhole, one needs to warp space-time in the opposite
way, to that in which normal matter warps it. Ordinary matter curves
space-time back on itself, like the surface of the Earth.
However, to create a wormhole, one needs matter that warps space-time in
the opposite way, like the surface of a saddle. The same is true of any
other way of warping space-time to allow travel to the past, if the
universe didn't begin so warped, that it allowed time travel. What one
would need, would be matter with negative mass, and negative energy
density, to make space-time warp in the way required.
Energy is
rather like money. If you have a positive bank balance, you can
distribute it in various ways. But according to the classical laws that
were believed until quite recently, you weren't allowed to have an
energy overdraft. So these classical laws would have ruled out us being
able to warp the universe, in the way required to allow time travel.
However, the classical laws were overthrown by Quantum Theory, which is
the other great revolution in our picture of the universe, apart from
General Relativity. Quantum Theory is more relaxed, and allows you to
have an overdraft on one or two accounts. If only the banks were as
accommodating. In other words, Quantum Theory allows the energy density
to be negative in some places, provided it is positive in others.
The reason Quantum Theory can allow the energy density to be negative, is that it is based on the Uncertainty Principle.
This says that certain quantities, like the position and speed of a
particle, can't both have well defined values. The more accurately the
position of a particle is defined, the greater is the uncertainty in its
speed, and vice versa. The uncertainty principle also applies to
fields, like the electro-magnetic field, or the gravitational field. It
implies that these fields can't be exactly zeroed, even in what we think
of as empty space. For if they were exactly zero, their values would
have both a well-defined position at zero, and a well-defined speed,
which was also zero. This would be a violation of the uncertainty
principle. Instead, the fields would have to have a certain minimum
amount of fluctuations. One can interpret these so called vacuum
fluctuations, as pairs of particles and anti particles, that suddenly
appear together, move apart, and then come back together again, and
annihilate each other. These particle anti particle pairs, are said to
be virtual, because one can not measure them directly with a particle
detector. However, one can observe their effects indirectly. One way of
doing this, is by what is called the Casimir effect. One has two
parallel metal plates, a short distance apart. The plates act like
mirrors for the virtual particles and anti particles. This means that
the region between the plates, is a bit like an organ pipe, and will
only admit light waves of certain resonant frequencies. The result is
that there are slightly fewer vacuum fluctuations, or virtual particles,
between the plates, than outside them, where vacuum fluctuations can
have any wavelength. The reduction in the number of virtual particles
between the plates means that they don't hit the plates so often, and
thus don't exert as much pressure on the plates, as the virtual
particles outside. There is thus a slight force pushing the plates
together. This force has been measured experimentally. So virtual
particles actually exist, and produce real effects.
Because there
are fewer virtual particles, or vacuum fluctuations, between the plates,
they have a lower energy density, than in the region outside. But the
energy density of empty space far away from the plates, must be zero.
Otherwise it would warp space-time, and the universe wouldn't be nearly
flat. So the energy density in the region between the plates, must be
negative.
We thus have experimental evidence from the bending
of light, that space-time is curved, and confirmation from the Casimir
effect, that we can warp it in the negative direction. So it might seem
possible, that as we advance in science and technology, we might be able
to construct a wormhole, or warp space and time in some other way, so
as to be able to travel into our past. If this were the case, it would
raise a whole host of questions and problems. One of these is, if
sometime in the future, we learn to travel in time, why hasn't someone
come back from the future, to tell us how to do it.
Even if
there were sound reasons for keeping us in ignorance, human nature being
what it is, it is difficult to believe that someone wouldn't show off,
and tell us poor benighted peasants, the secret of time travel. Of
course, some people would claim that we have been visited from the
future. They would say that UFO's come from the future, and that
governments are engaged in a gigantic conspiracy to cover them up, and
keep for themselves, the scientific knowledge that these visitors bring.
All I can say is, that if governments were hiding something, they are
doing a pretty poor job, of extracting useful information from the
aliens. I'm pretty skeptical of conspiracy theories, believing the cock
up theory is more likely. The reports of sightings of UFO's can't all be
caused by extra terrestrials, because they are mutually contradictory.
But once you admit that some are mistakes, or hallucinations, isn't it
more probable that they all are, than that we are being visited by
people from the future, or the other side of the galaxy? If they really
want to colonize the Earth, or warn us of some danger, they are being
pretty ineffective.
A possible way to reconcile time travel, with
the fact that we don't seem to have had any visitors from the future,
would be to say that it can occur only in the future. In this view, one
would say space-time in our past was fixed, because we have observed it,
and seen that it is not warped enough, to allow travel into the past.
On the other hand, the future is open. So we might be able to warp it
enough, to allow time travel. But because we can warp space-time only in
the future, we wouldn't be able to travel back to the present time, or
earlier.
This picture would explain why we haven't been over run by tourists from the future.
But it would still leave plenty of paradoxes. Suppose it were possible
to go off in a rocket ship, and come back before you set off. What would
stop you blowing up the rocket on its launch pad, or otherwise
preventing you from setting out in the first place. There are other
versions of this paradox, like going back, and killing your parents
before you were born, but they are essentially equivalent. There seem to
be two possible resolutions.
One is what I shall call, the
consistent histories approach. It says that one has to find a consistent
solution of the equations of physics, even if space-time is so warped,
that it is possible to travel into the past. On this view, you couldn't
set out on the rocket ship to travel into the past, unless you had
already come back, and failed to blow up the launch pad. It is a
consistent picture, but it would imply that we were completely
determined: we couldn't change our minds. So much for free will. The
other possibility is what I call, the alternative histories approach. It
has been championed by the physicist David Deutsch, and it seems to
have been what Stephen Spielberg had in mind when he filmed, Back to the
Future.
In this view, in one alternative history, there would
not have been any return from the future, before the rocket set off, and
so no possibility of it being blown up. But when the traveler returns
from the future, he enters another alternative history. In this, the
human race makes a tremendous effort to build a space ship, but just
before it is due to be launched, a similar space ship appears from the
other side of the galaxy, and destroys it.
David Deutsch claims
support for the alternative histories approach, from the sum over
histories concept, introduced by the physicist, Richard Feinman, who
died a few years ago. The idea is that according to Quantum Theory, the
universe doesn't have just a unique single history.
Instead,
the universe has every single possible history,each with its own
probability. There must be a possible history in which there is a
lasting peace in the Middle East, though maybe the probability is low.
In some histories space-time will be so warped, that objects like
rockets will be able to travel into their pasts. But each history is
complete and self contained, describing not only the curved space-time,
but also the objects in it. So a rocket can not transfer to another
alternative history, when it comes round again. It is still in the same
history, which has to be self consistent. Thus, despite what Deutsch
claims, I think the sum over histories idea, supports the consistent
histories hypothesis, rather than the alternative histories idea.
It thus seems that we are stuck with the consistent histories picture.
However, this need not involve problems with determinism or free will,
if the probabilities are very small, for histories in which space-time
is so warped, that time travel is possible over a macroscopic region.
This is what I call, the Chronology Protection Conjecture: the laws of
physics conspire to prevent time travel, on a macroscopic scale.
It seems that what happens, is that when space-time gets warped almost
enough to allow travel into the past, virtual particles can almost
become real particles, following closed trajectories. The density of the
virtual particles, and their energy, become very large. This means that
the probability of these histories is very low. Thus it seems there may
be a Chronology Protection Agency at work, making the world safe for
historians. But this subject of space and time warps is still in its
infancy. According to string theory, which is our best hope of uniting
General Relativity and Quantum Theory, into a Theory of Everything,
space-time ought to have ten dimensions, not just the four that we
experience. The idea is that six of these ten dimensions are curled up
into a space so small, that we don't notice them. On the other hand, the
remaining four directions are fairly flat, and are what we call
space-time. If this picture is correct, it might be possible to arrange
that the four flat directions got mixed up with the six highly curved or
warped directions. What this would give rise to, we don't yet know. But
it opens exciting possibilities.
The conclusion of this
lecture is that rapid space-travel, or travel back in time, can't be
ruled out, according to our present understanding. They would cause
great logical problems, so let's hope there's a Chronology Protection
Law, to prevent people going back, and killing our parents. But science
fiction fans need not lose heart. There's hope in string theory.
Well, a neutrino (denoted by the Greek letter ν ) is a "ghostly" elementary particle with some important properties... 
Black holes can be used to travel into the future only. So far as we know, our universe prohibits traveling into the past.
