Rubber space

These properties are altered in the
presence of mass. In order to visualize how space can curve near
a massive object we will employ the trick of throwing away one of
the dimensions of space and think again about the curvature of a
2Dworld.
The best way to understand what happens to space when
we introduce a massive object is to imagine the (2D) space to
be like a sheet of rubber. Imagine rolling a small ball across a
trampoline. It should go across in a straight line. Now what if
you stand still in the middle of the trampoline and get someone
to roll the ball again? You will have made a dent causing the
trampoline’s material to bow down a little. If the ball’s path takes
it close enough to this dent, it will follow the curvature and be bent
round to move in a different direction (figure 2.2). Viewed from
above, it would appear as though you had exerted a mysterious
force on the ball causing it to be attracted towards you and away
from its original straight path. This is how we imagine matter
to curve space around it. The curvature causes other objects to
follow a different path to the one they would in the absence of the
curvature. What has happened on the trampoline is that the ball
is following a geodesic path. This is the preferred path for the ball;
the one that it wants to take most naturally given the curvature of
the trampoline’s material that it encounters. Thus, a geodesic path
is the shortest distance between any two points. So if you are ever
asked what the shortest distance is between two points, don’t say
a straight line. A geodesic is only a straight line when the space
is flat. If the ball had been travelling more slowly along the same
path on the trampoline then it would have been caught in the dip
and would have spiralled inwards towards your feet. So now we can understand Einstein’s interpretation of
gravity. All material bodies warp space around them, by
an amount that depends on how massive they are, and this
warped space then guides all bodies that are moving in it,
making them travel along geodesic paths. Such paths can be understood if you think about the flight path that an aircraft
takes.
A few years ago I flew from London to Tokyo to attend a
physics conference. I looked at my world atlas to get a vague
idea of the countries I would be flying over. I forgot that a map
is a flat projection of the Earth’s curved surface. So although the
shortest distance between two points on the map (say London
and Tokyo) may look like a straight line on paper, to find the true
shortest distance we would need to look on a globe. To do this
place one end of a rubber band on London and the other on Tokyo.
The band will always follow a geodesic line since this will be the
shortest distance between the two points. Any other path would
be longer and the band would have to stretch more. Since it has
a natural tendency to minimize its length it will always find the
route which requires least stretching. Now we see that the flight
path—assuming the pilot wants to minimize fuel consumptionand
is not diverted off the geodesic due to bad weather or a country’s
forbidden airspace—will pass over a region far to the north of both
London and Tokyo, a path that looks curved if you plot it on a flat
map.
Now that I have introduced Einstein’s view of gravity we can
go on to look at some of its more fascinating consequences, such as
a hole in space into which anything can fall and be lost forever: a
black hole. You will discover that such fantastic objects are science
fact not fiction because astronomers are now almost certain that
black holes really exist out in space.
To pave the way for a discussion of black holes we must
first learn a little about how they can form. For this to happen,
space needs to be warped by an incredible amount. This requires
something very dense indeed. Even the whole Earth is not
enough—which, by the way, rules out any possibility of the
Bermuda Triangle being some kind of hole in space that swallows
up unsuspecting ships and aircraft, since a hole of that size would
require a mass much more than that of the whole planet, and we
can easily work out the mass of the Earth from the way it orbits
the Sun.
What we need for some serious warping of space is something
big, such as a star.