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Space and Time Warps
This lecture is the intellectual property of Professor S.W. Hawking.
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In science fiction, 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 Euclidean
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.
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Space and Time Warps Cont...
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.
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Space and Time Warps Cont...
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.
Since we haven't cracked time travel yet, I have run out of time.
Thank you for listening.
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