Stephen Hawking - Space & Time Warps

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Space and Time Warps

This lecture is the intellectual property of Professor S.W. Hawking. You may not reproduce, edit

or distribute this document in anyway for monetary advantage.

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.

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.

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

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摘要：

SpaceandTimeWarpsThislectureistheintellectualpropertyofProfessorS.W.Hawking.Youmaynotreproduce,editordistributethisdocumentinanywayformonetaryadvantage.Insciencefiction,spaceandtimewarpsareacommonplace.Theyareusedforrapidjourneysaroundthegalaxy,orfortravelthroughtime.Buttoday'ssciencefiction,isoften...

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Stephen Hawking - Space & Time Warps

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