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
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
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.
would be very difficult to design a living being that could exist in only two
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
Space and Time Warps
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,
traveled much faster than light,
She departed one day,
In a relative
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,
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
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
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
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.
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
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
Since we haven't cracked time travel yet, I have run out
научной фантастике мы довольно часто встречаемся с кривизной
пространства и времени. Именно этот феномен используются для мгновенных
путешествий вокруг Галактики, или для путешествий во времени. Но, как мы
знаем, сегодняшняя научная фантастика, зачастую, это завтрашняя наука с
её фактами, фиксирующими изменения кривизны пространства и времени.
Идея, что пространство и время могут быть искривлены или испещрены
кавернами, возникла относительно недавно. Больше двух тысяч лет аксиомы
Евклидовой геометрии, как полагали, были сами по себе очевидны. И те из
вас, кто был вынужден изучать Евклидову геометрию в школе, должен
помнить одно из следствий этих аксиом, что сумма углов в треугольнике
составляет 180 градусов.
Однако, в прошлом столетии, люди начали понимать, что возможны
и другие формы геометрии, в которых углы треугольника, не должны
составлять в сумме 180 градусов. Рассмотрим, например, поверхность
Земли, на которой наиболее приближен к прямой линии большой, как его
называют, круг, [который состоит из]
кратчайших расстояний между двумя точками - пунктами сообщения
воздушных линий. Рассмотрим теперь треугольник на поверхности Земли,
образованный линией экватора в 0 градусов долготы, проходящей через
Лондон , и линии в 90 градусов восточной
долготы, проходящей через Бангладеш.
Две линии долготы встречаются на экваторе под прямым углом в 90
градусов. Две линии долготы также встречают друг с другом под прямым
углом в 90 градусов. Таким образом, имеется треугольник с тремя правильными
углами. Углы этого треугольника составляют в сумме двести семьдесят
градусов. Это больше чем 180 градусов для треугольника на плоской
Если [треугольной] поверхности
[Земли] придать форму седла, потянув
[за одну из сторон], то окажется, что углы
[такого треугольника] в сумме
составляют менее 180 градусов. И поверхность Земли
представляет то, что называется двухмерным пространством.
Таким образом, мы можем перемещаться по поверхности Земли,
, в двух
направлениях под прямым углом друг к другу: на север-юг или на
восток-запад. Но конечно, есть ещё и третье направление - под прямым углом
к этим двум, - это вверх и вниз. Другими словами, поверхность Земли
существует в трехмерном пространстве. Трехмерное пространство - плоское,
т.е. отвечает Евклидовой геометрии, в которой углы треугольника
составляют в целом 180 градусов. Однако, можно представить гонку
существ, которые могли бы перемещаться по поверхности Земли, но которые
не могли бы перемещаться в третьем направлении - в направлении
вверх-вниз. Они не знали бы о плоском трехмерном пространстве, в котором
существует поверхность Земли. Для них это пространство было бы изогнуто,
и геометрия не была бы Евклидовой.
Было бы очень трудно проектировать живое существо, которое могло
существовать только в двух измерениях.
Пища, которую существо не могло переварить, выплёвывалась бы тем же
самым путем, каким она в него вошла. Если был бы прямой путь через тело
животного, то бедное животное развалилось бы...
left ear &
right Christian IX ear -
- [Absolute ear for musical
signals in range///]
Так три измерения, кажется, минимум для жизни. Но так же, как можно
думать о двух размерных существах, живущих на поверхность Земли, таким
образом можно было предположить, что трехмерное место, в котором мы
живем, было поверхностью сферы, в другом измерении, которое мы не видим.
Если бы сфера была очень большой, космический была бы почти плоская, и
Евклидова геометрия, то было бы очень хорошее приближение по маленьким
расстояниям. Но мы заметили бы, что Евклидова геометрия сломалась, по
большим расстояниям. Как иллюстрация этого, вообразите команду
живописцев, добавляя краску на поверхность большого шара. Поскольку
толщина слоя краски увеличилась, поверхностная область повысится. Если
бы шар был в плоском трехмерном месте, то можно было бы пойти при
добавляющей краске неопределенно, и шар станет больше и большим. Однако,
если бы трехмерное место, была действительно поверхность сферы в другом
измерении, то его объем был бы большим, но конечным. Поскольку каждый
добавил больше слоев краски, шар в конечном счете заполнит половину
места. После этого живописцы нашли бы, что они были пойманы в ловушку в
области когда-либо уменьшающегося размера, и почти всем месте, был занят
шаром, и его слоями краски. Таким образом они знали бы, что они жили в
кривом месте, и не плоском.
с. ш. 0°00′05.31″ в. д.)
углубляясь в Землю, её земную кору, по плоскостям сторон треугольника.
двумерные существа также могут находиться в состоянии покоя.
There was a young lady of Wight,
traveled much faster than light,
She departed one day,
In a relative
And arrived on the previous night.
night was Yesterday -
Мысль стремится в
порыве всю память объять
пространство со временем вспять,
В Юрский Парк, к
трилобитам в мгновение ока,
Превращая луч света в
мой синопсис тока,
Что находит забытые
Там, где кто-то
оставил один мокрый след.
Curvature of Time (and
- Curvature of the Earth
extraterrestrial life 'in face of' superintelligent
gala-populations with own religious multipackets'programs (Christ-Maria-Mohammed-Budda-Jowa-Ilja-StGerman,
etc.\КВН) for us, primitive Earth citizens.
super-electrocurrent-pipes (synopsis-pipes) in the Sun Galaxy (in Cosmos
Какой каркас могу увидеть
Arecibo-Botschaft & Radiowellen-Signals
not like pay bills - joint Russian & German Phobos project:
Recalculation of orbital trajectories
список потерь РосКосмоса в
миллиардах? (анализ причин)
Recalculation of Orbital Trajectories