Curved space

I mentioned that this imaginary 2Dworld need not be infinite in
extent and would therefore have an edge, some border defining its
boundary. We will see later on that universes do not have edges
and so 2Dworld must presumably go on forever. It turns out that
this need only be the case (going on forever that is) if 2Dworld is
flat, which is what I have assumed so far. What if the inhabitants
of 2Dworld lived on the surface of a sphere? Their space is now
curved and is no longer infinite in size. After all, a sphere has
a certain finite surface area which clearly does not have an edge
since the 2D’ers can move anywhere in this universe without ever
reaching a point beyond which they cannot go. The important and
rather tricky concept to appreciate here is that although 2Dworld
is the surface of a 3D sphere, the inside of the sphere and all the
space outside the surface need not even exist as far as the 2D’ers
are concerned. So, in a sense, the analogy with humans living on
the surface of the Earth should not be taken too strongly since we
are clearly 3D beings stuck to the surface of a 3Dobject. The 2D’ers
only have access to the 2D surface. The interior of the sphere does
not even exist for them.The interesting question I would like to address next is
whether the 2D’ers would know that their space is curved.
One way for them to find out would be the way we can prove
that the Earth is not flat: by having someone set off in one direction
and eventually get back to the starting point coming from the
opposite direction having been all the way round the globe. Of
course we now regularly send astronauts into orbit who can look
back and see that the Earth is round, but the inhabitants of the 2D
universe are imprisoned in their surface and cannot move up out
of their world to look down on it. There is another way they could
check whether their world was curved.
We learn at school that if we add up the values of the three
interior angles of any triangle we always get 180 degrees. It does
not matter how large or small we draw the triangle or what shape
it is; the answer will always be the same. If it is a right-angled
triangle then the other two angles must also add up to 90 degrees.
If one of the angles is obtuse with a value of, say, 160 degrees,
then the other two angles must together make up the remaining
20 degrees, and so on. But before you become too complacent
having comfortably negotiated this bit of geometry, allow me to
spoil things by stating that this business of angles of a triangle
always adding up to 180 degrees is only true if the triangle is drawn
on a flat surface! A triangle drawn on a sphere has angles which
always add up to more than 180 degrees. Here is a simple example
which demonstrates what I mean. To help you see this you will
need a ball and a felt tip pen.
Imagine an explorer beginning a journey at the North Pole. He
heads off in a straight line due South (when you are at the North
Pole the only direction you can head is south) passing through
the eastern tip of Canada then down the western Atlantic. He is,
of course, careful to steer clear of the Bermuda Triangle since he
believes all that superstitious nonsense. He keeps heading south
until he reaches the equator somewhere in northern Brazil. Once
at the equator, he turns left and heads East across the Atlantic, now
moving in a straight line along the equator. He reaches the coast
of Africa and carries on to Kenya by which time he has had quite
enough of the hot, humid climate and decides to turn left and head North again. He travels up through Ethiopia, Saudi Arabia, the
Middle East, all the way up through Eastern Europe and back to
the North Pole.
If you have made a rough trace of his route you will see that he
has completed a triangle (figure 1.6(b)). Look closely at the three
angles. On reaching the equator and turning left, he had made a
right angle (90 degrees). But when he finally left the equator to
head back north he made another right angle. These two angles,
therefore already add up to 180 degrees. But we have not included
the angle he has made at the North Pole with the two straight lines
of his outward and inward journeys. These should also roughly
make a 90 degree angle, although of course the size of this angle
depends on how far he has travelled along the equator. I have
chosen it so that he has traced a triangle, joining three straight
lines, with three right-angles adding up to 270 degrees.
Such a triangle is a special case but the basic rule is that any
triangle drawn on the surface of a sphere will have angles adding
up to more than 180 degrees. For instance, a triangle joining Paris,
Rome and Moscow will have angles adding up to slightly over 180
degrees. This tiny departure from 180 degrees is because such a
triangle does not cover a significant fraction of the total surface
area of the Earth and is thus almost flat.
Getting back to the 2D’ers, they can use the same technique
to check whether their space is curved. They would head off in a
2D rocket from their home planet travelling in a straight line until
they reach a distant star. There, they will turn through some fixed
angle and head off towards another star. Once at the second star
they would turn back home. Having traced a triangle they would
measure the three angles. If these came to more than 180 degrees3
they could deduce that they lived in curved space.they could deduce that they lived in curved space.
Another property, which you may remember from school, is
that the circumference of a circle is given by pi times its diameter.
The value of pi, we are told, is not open to negotiation. There is
a button on most pocket calculators that gives pi up to 10 decimal
places (3.1415926536), but most of us remember it as 3.14. OK, Iadmit that I remember it to the ten decimal places that a calculator
shows, but that is only because I use it in my work so often,
which is no different to remembering an important phone number.
However, I have a mathematician friend who knows pi to 30
decimal places. Other than that he is quite normal. We are taught
that pi is what is called a mathematical constant. It is defined as the
ratio of two numbers: the circumference and the diameter of any
circle in flat space. If our explorer were to walk round the Arctic
Circle, which has a diameter that he could measure with accuracy
(it being twice the distance fromtheArctic circle to the North Pole),
then he would find upon multiplying this value for the diameter by
pi (which is thewaytoworkout circumferences of circles) hewould
get a value which was slightly bigger than the true circumference
of the Arctic Circle. The Earth’s curvature means that the Arctic
Circle is smaller than it would be if the Earth were flat.
The properties of triangles and circles that we learn at school
are what are known as Euclidean geometry, or ‘flat geometry’.
The 3D geometry of spheres, cubes and pyramids is also part of
Euclidean geometry if they are imbedded in flat 3D space. Their
properties change if the 3D space is curved, in a way similar to the
way the properties of triangles and circles change when they are
drawn on a curved 2D space such as the surface of a sphere. So,
our 3D space may well be curved but we do not need to visualize
a fourth dimension to ‘see’ this curvature. We can measure it
indirectly by studying the geometry of 3D space and solid objects
within it. In practice, we never see any deviation from Euclidean
geometry because we live in a part of the Universe where space is
so nearly flat we can never detect any curvature. This is analogous
to trying to detect the curvature of the Earth by drawing a triangle
on a football field. Of course, a football field is not completely
smooth. Likewise, space contains regions of curvature here and
there as we will see in the next chapter.
What if a fourth dimension of space does exist beyond our
three? What can we say about its properties? The best way is to
begin by acknowledging that the fourth dimension is to us what
the third dimension is to the 2D’ers. Imagine you are standing at
the centre of a large circle marked out on flat ground such as the centre circle of a football pitch. If you now walk in a straight line
in any direction you will be heading towards the perimeter of the
circle. This is called a radial direction because when you reach the
perimeter you will have travelled along the circle’s radius. On the
other hand a bird sitting in the centre of the circle can move along
the third dimension: upwards. If it flies straight up then it will be
moving away from all parts of the circle at all times.
Now add another dimension to this example and imagine the
bird at the centre of a sphere (say a spherical cage). Whichever
direction the bird now flies in, it will be moving towards the bars
of the cage, and all directions for it are now radial. Just as in the 2D
example of the circle where the bird could move along the third
dimension away from the circle, we can now see what it would
mean tomovealong the fourth dimension. Starting fromthe centre
of the cage it is the direction in which the bird would have to fly
in order to be moving away from all points in the cage at the same
time! This is not a direction that we can ever visualize since, as I
have mentioned before, our brains are only three-dimensional. So
what wouldwesee ifwehad a magical bird, capable of utilizing the
fourth dimension, trapped in a cage? We would see it disappear
from view and then rejoin our 3D space somewhere else, possibly
outside the cage. It would look as astonishing to us as our 3D skills
would look to the 2D’ers were we to pluck objects out of their 2D
space.
Another interesting effect of using a higher dimension is what
happens when objects are flipped over. Imagine you were able to
lift a 2D’er out of his world, turn him over so that his left and right
sides are swapped over, then put him back. Things would be quite
confusing for him for a while. He will not feel any different but
everything around him will be on the wrong side. He would have
to adjust to living in a world where the 2D sun no longer rises from
the right as it used to, but from the left. And he now has to walk
in the opposite direction to get to work from his home.
Things are more amusing if you consider what it would be like
for you if a 4D being where to pluck you out of our 3D world and
flip you over. For a start, people would notice something slightly
different about your appearance since your face now looks to them how it used to appear to you in a mirror. When you next look in a
mirror, you will also see the difference. This is because nobody’s
face is symmetrical. The left side of our faces differs from the right.
Maybe one eye is slightly lower than the other or, like me, your
nose is slightly bent to one side, or you have a mole on one cheek,
and so on. But this is only the start of your problems. Everything
around you appears back to front. All writing will be backwards,
clock hands will go round anticlockwise and you will now be lefthanded
if you were right-handed before. One way of testing how
things would be like would be for you to go round viewing the
world through a mirror. It would take a while before you stopped
bumping into things.