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To describe how quantum theory shapes time and space, it is helpful to introduce
the idea of imaginary time. Imaginary time sounds like something from science
fiction, but it is a well-defined mathematical concept: time measured in what
are called imaginary numbers. One can think of ordinary real numbers such as 1,
2, -3.5, and so on as corresponding to positions on a line stretching from left
to right: zero in the middle, positive real numbers on the right, and negative
real numbers on the left.

Imaginary numbers can then be represented as corresponding to positions on a
vertical line: zero is again in the middle, positive imaginary numbers plotted
upward, and negative imaginary numbers plotted downward. Thus imaginary numbers
can be thought of as a new kind of number at right angles to ordinary real
numbers. Because they are a mathematical construct, they don't need a physical
realization; one can't have an imaginary number of oranges or an imaginary
credit card bill.

One might think this means that imaginary numbers are just a mathematical game
having nothing to do with the real world. From the viewpoint of positivist
philosophy, however, one cannot determine what is real. All one can do is find
which mathematical models describe the universe we live in. It turns out that a
mathematical model involving imaginary time predicts not only effects we have
already observed but also effects we have not been able to measure yet
nevertheless believe in for other reasons. So what is real and what is
imaginary? Is the distinction just in our minds?

Einstein's classical (i.e., nonquantum) general theory of relativity combined
real time and the three dimensions of space into a four-dimensional spacetime.
But the real time direction was distinguished from the three spatial directions;
the world line or history of an observer always increased in the real time
direction (that is, time always moved from past to future), but it could
increase or decrease in any of the three spatial directions. In other words, one
could reverse direction in space, but not in time.

On the other hand, because imaginary time is at right angles to real time, it
behaves like a fourth spatial direction. It can therefore have a much richer
range of possibilities than the railroad track of ordinary real time, which can
only have a beginning or an end or go around in circles. It is in this imaginary
sense that time has a shape.

To see some of the possibilities, consider an imaginary time spacetime that is a
sphere, like the surface of the Earth. Suppose that imaginary time was degrees
of latitude. Then the history of the universe in imaginary time would begin at
the South Pole. It would make no sense to ask, "What happened before the
beginning?" Such times are simply not defined, any more than there are points
south of the South Pole. The South Pole is a perfectly regular point of the
Earth's surface, and the same laws hold there as at other points. This
suggests that the beginning of the universe in imaginary time can be a regular
point of spacetime, and that the same laws can hold at the beginning as in the
rest of the universe. (The quantum origin and evolution of the universe will be
discussed in the next chapter.)

Another possible behavior is illustrated by taking imaginary time to be degrees
of longitude on the Earth. All the lines of longitude meet at the North and
South Poles. Thus time stands still there, in the sense that an increase of
imaginary time, or of degrees of longitude, leaves one in the same spot. This is
very similar to the way that ordinary time appears to stand still on the horizon
of a black hole. We have come to recognize that this standing still of real and
imaginary time (either both stand still or neither does) means that the
spacetime has a temperature, as I discovered for black holes.

Not only does a black hole have a temperature, it also behaves as if it has a
quantity called entropy. The entropy is a measure of the number of internal
states (ways it could be configured on the inside) that the black hole could
have without looking any different to an outside observer, who can only observe
its mass, rotation, and charge. This black hole entropy is given by a very
simple formula I discovered in 1974. It equals the area of the horizon of the
black hole: there is one bit of information about the internal state of the
black hole for each fundamental unit of area of the horizon. This shows that
there is a deep connection between quantum gravity and thermodynamics, the
science of heat (which includes the study of entropy). It also suggests that
quantum gravity may exhibit what is called holography.

Excerpted from The Universe in a Nutshell by Stephen Hawking Copyright 2001 by Stephen Hawking. Excerpted by permission of Bantam, a division of Random House, Inc. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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