holes not being black, and the universe not having any singularities but being completely self-contained and
without a boundary. The trouble is, as explained in Chapter 7, that the uncertainty principle means that even
“empty” space is filled with pairs of virtual particles and antiparticles. These pairs would have an infinite amount
of energy and, therefore, by Einstein’s famous equation E = mc2, they would have an infinite amount of mass.
Their gravitational attraction would thus curve up the universe to infinitely small size.
Rather similar, seemingly absurd infinities occur in the other partial theories, but in all these cases the infinities
can be canceled out by a process called renormalization. This involves canceling the infinities by introducing
other infinities. Although this technique is rather dubious mathematically, it does seem to work in practice, and
has been used with these theories to make predictions that agree with observations to an extraordinary degree
of accuracy. Renormalization, however, does have a serious drawback from the point of view of trying to find a
complete theory, because it means that the actual values of the masses and the strengths of the forces cannot
be predicted from the theory, but have to be chosen to fit the observations.
In attempting to incorporate the uncertainty principle into general relativity, one has only two quantities that can
be adjusted: the strength of gravity and the value of the cosmological constant. But adjusting these is not
sufficient to remove all the infinities. One therefore has a theory that seems to predict that certain quantities,
such as the curvature of space-time, are really infinite, yet these quantities can be observed and measured to
be perfectly finite! This problem in combining general relativity and the uncertainty principle had been
suspected for some time, but was finally confirmed by detailed calculations in 1972. Four years later, a possible
solution, called “supergravity,” was suggested. The idea was to combine the spin-2 particle called the graviton,
which carries the gravitational force, with certain other particles of spin 3/2, 1, ., and 0. In a sense, all these
particles could then be regarded as different aspects of the same “superparticle,” thus unifying the matter
particles with spin . and 3/2 with the force-carrying particles of spin 0, 1, and 2. The virtual particle/antiparticle
pairs of spin . and 3/2 would have negative energy, and so would tend to cancel out the positive energy of the
spin 2, 1, and 0 virtual pairs. This would cause many of the possible infinities to cancel out, but it was
suspected that some infinities might still remain. However, the calculations required to find out whether or not
there were any infinities left uncancelled were so long and difficult that no one was prepared to undertake them.
Even with a computer it was reckoned it would take at least four years, and the chances were very high that
one would make at least one mistake, probably more. So one would know one had the right answer only if
someone else repeated the calculation and got the same answer, and that did not seem very likely!
Despite these problems, and the fact that the particles in the super-gravity theories did not seem to match the
observed particles, most scientists believed that supergravity was probably the right answer to the problem of
the unification of physics. It seemed the best way of unifying gravity with the other forces. However, in 1984
there was a remarkable change of opinion in favor of what are called string theories. In these theories the basic
objects are not particles, which occupy a single point of space, but things that have a length but no other
dimension, like an infinitely thin piece of string. These strings may have ends (the so-called open strings) or
they may be joined up with themselves in closed loops (closed strings) Figure 11:1 and Figure 11:2.
Figures 11:1 & 11:2
A particle occupies one point of space at each instant of time. Thus its history can be represented by a line in
space-time (the “world-line”). A string, on the other hand, occupies a line in space at each moment of time. So
its history in space-time is a two-dimensional surface called the world-sheet. (Any point on such a world-sheet
can be described by two numbers, one specifying the time and the other the position of the point on the string.)
The world-sheet of an open string is a strip: its edges represent the paths through space-time of the ends of the
string Figure 11:1. The world-sheet of a closed string is a cylinder or tube Figure 11:2: a slice through the tube
is a circle, which represents the position of the string at one particular time.
Two pieces of string can join together to form a single string; in the case of open strings they simply join at the
ends Figure 11:3, while in the case of closed strings it is like the two legs joining on a pair of trousers Figure
11:4.
Figure 11:3
.
Figure 11:4
Similarly, a single piece of string can divide into two strings. In string theories, what were previously thought of
as particles are now pictured as waves traveling down the string, like waves on a vibrating kite string. The
emission or absorption of one particle by another corresponds to the dividing or joining together of strings. For
example, the gravitational force of the sun on the earth was pictured in particle theories as being caused by the
emission of a graviton by a particle in the sun and its absorption by a particle in the earth Figure 11:5.
Figures 11:5 & 11:6
In string theory, this process corresponds to an H-shaped tube or pipe Figure 11:6 (string theory is rather like
plumbing, in a way). The two vertical sides of the H correspond to the particles in the sun and the earth, and the
horizontal crossbar corresponds to the graviton that travels between them.
String theory has a curious history. It was originally invented in the late 1960s in an attempt to find a theory to
describe the strong force. The idea was that particles like the proton and the neutron could be regarded as
waves on a string. The strong forces between the particles would correspond to pieces of string that went
between other bits of string, as in a spider’s web. For this theory to give the observed value of the strong force
between particles, the strings had to be like rubber bands with a pull of about ten tons.
In 1974 Joel Scherk from Paris and John Schwarz from the California Institute of Technology published a paper
in which they showed that string theory could describe the gravitational force, but only if the tension in the string
were very much higher, about a thousand million million million million million million tons (1 with thirty-nine
zeros after it). The predictions of the string theory would be just the same as those of general relativity on
normal length scales, but they would differ at very small distances, less than a thousand million million million
million millionth of a centimeter (a centimeter divided by 1 with thirty-three zeros after it). Their work did not
receive much attention, however, because at just about that time most people abandoned the original string
theory of the strong force in favor of the theory based on quarks and gluons, which seemed to fit much better
with observations. Scherk died in tragic circumstances (he suffered from diabetes and went into a coma when
no one was around to give him an injection of insulin). So Schwarz was left alone as almost the only supporter
of string theory, but now with the much higher proposed value of the string tension.
In 1984 interest in strings suddenly revived, apparently for two reasons. One was that people were not really
making much progress toward showing that supergravity was finite or that it could explain the kinds of particles
that we observe. The other was the publication of a paper by John Schwarz and Mike Green of Queen Mary
College, London, that showed that string theory might be able to explain the existence of particles that have a
built-in left-handedness, like some of the particles that we observe. Whatever the reasons, a large number of
people soon began to work on string theory and a new version was developed, the so-called heterotic string,
which seemed as if it might be able to explain the types of particles that we observe.
String theories also lead to infinities, but it is thought they will all cancel out in versions like the heterotic string
(though this is not yet known for certain). String theories, however, have a bigger problem: they seem to be
consistent only if space-time has either ten or twenty-six dimensions, instead of the usual four! Of course, extra
space-time dimensions are a commonplace of science fiction indeed, they provide an ideal way of overcoming
the normal restriction of general relativity that one cannot travel faster than light or back in time (see Chapter
10). The idea is to take a shortcut through the extra dimensions. One can picture this in the following way.
Imagine that the space we live in has only two dimensions and is curved like the surface of an anchor ring or
torus Figure 11:7.
Figure 11:7
If you were on one side of the inside edge of the ring and you wanted to get to a point on the other side, you
would have to go round the inner edge of the ring. However, if you were able to travel in the third dimension,
you could cut straight across.
Why don’t we notice all these extra dimensions, if they are really there? Why do we see only three space
dimensions and one time dimension? The suggestion is that the other dimensions are curved up into a space of
very small size, something like a million million million million millionth of an inch. This is so small that we just
don’t notice it: we see only one time dimension and three space dimensions, in which space-time is fairly flat. It
is like the surface of a straw. If you look at it closely, you see it is two-dimensional (the position of a point on the
straw is described by two numbers, the length along the straw and the distance round the circular direction).
But if you look at it from a distance, you don’t see the thickness of the straw and it looks one-dimensional (the
position of a point is specified only by the length along the straw). So it is with space-time: on a very small scale
it is ten-dimensional and highly curved, but on bigger scales you don’t see the curvature or the extra
dimensions. If this picture is correct, it spells bad news for would-be space travelers: the extra dimensions
would be far too small to allow a spaceship through. However, it raises another major problem. Why should
some, but not all, of the dimensions be curled up into a small ball? Presumably, in the very early universe all
the dimensions would have been very curved. Why did one time dimension and three space dimensions flatten
out, while the other dimensions remain tightly curled up?
One possible answer is the anthropic principle. Two space dimensions do not seem to be enough to allow for
the development of complicated beings like us. For example, two-dimensional animals living on a
one-dimensional earth would have to climb over each other in order to get past each other. If a two-dimensional
creature ate something it could not digest completely, it would have to bring up the remains the same way it
swallowed them, because if there were a passage right through its body, it would divide the creature into two
separate halves: our two-dimensional being would fall apart Figure 11:8. Similarly, it is difficult to see how there
could be any circulation of the blood in a two-dimensional creature.
Figure 11:8
There would also be problems with more than three space dimensions. The gravitational force between two
bodies would decrease more rapidly with distance than it does in three dimensions. (In three dimensions, the
gravitational force drops to 1/4 if one doubles the distance. In four dimensions it would drop to 1/5, in five
dimensions to 1/6, and so on.) The significance of this is that the orbits of planets, like the earth, around the sun
would be unstable: the least disturbance from a circular orbit (such as would be caused by the gravitational
attraction of other planets) would result in the earth spiraling away from or into the sun. We would either freeze
or be burned up. In fact, the same behavior of gravity with distance in more than three space dimensions
means that the sun would not be able to exist in a stable state with pressure balancing gravity. It would either
fall apart or it would collapse to form a black hole. In either case, it would not be of much use as a source of
heat and light for life on earth. On a smaller scale, the electrical forces that cause the electrons to orbit round
the nucleus in an atom would behave in the same way as gravitational forces. Thus the electrons would either
escape from the atom altogether or would spiral into the nucleus. In either case, one could not have atoms as
we know them.
It seems clear then that life, at least as we know it, can exist only in regions of space-time in which one time
dimension and three space dimensions are not curled up small. This would mean that one could appeal to the
weak anthropic principle, provided one could show that string theory does at least allow there to be such
regions of the universe – and it seems that indeed string theory does. There may well be other regions of the
universe, or other universes (whatever that may mean), in which all the dimensions are curled up small or in
which more than four dimensions are nearly flat, but there would be no intelligent beings in such regions to
observe the different number of effective dimensions.
Another problem is that there are at least four different string theories (open strings and three different closed
string theories) and millions of ways in which the extra dimensions predicted by string theory could be curled
up. Why should just one string theory and one kind of curling up be picked out? For a time there seemed no
answer, and progress got bogged down. Then, from about 1994, people started discovering what are called
dualities: different string theories and different ways of curling up the extra dimensions could lead to the same
results in four dimensions. Moreover, as well as particles, which occupy a single point of space, and strings,
