Notes on Relativity and Cosmology - Physics Department, UCSB

196 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME
8.1.1 Straight Lines in Curved Space
Eventually Einstein found a useful analogy with something that at first sight
appears quite different – a curved surface. The idea is captured by the question
“What is a straight line on a curved surface?”
To avoid language games, mathematicians made up a new word for this idea:
“geodesic.” A geodesic can be thought of as the “straightest possible line on a
curved surface.” More precisely, we can define a geodesic as a line of minimal
distance – the shortest line between two points 2 . The idea is that we can define
a straight line to be the shortest line between two points.
Actually, there is another definition of geodesic that is even better, but requires
more mathematical machinery to state precisely. Intuitively, it just captures
the idea that the geodesic is ‘straight.’ It tells us that a geodesic is the path on
a curved surface that would be traveled, for example, by an ant (or a person)
walking on the surface who always walks straight ahead and does not turn to
the right or left.
As an example, suppose you stand on the equator of the earth, face north, and
then walk forward. Where do you go? If you walk far enough (over the ocean,
etc.) you will eventually arrive at the north pole. The path that you have
followed is a geodesic on the sphere.
Alice
Bob
Note that this is true no matter where you start on the equator. So, suppose
there are in fact two people walking from the equator to the north pole, Alice
and Bob. As you can see, Alice and Bob end up moving toward each other. So,
if we drew a diagram of this process using Alice’s frame of reference (so that
her own path is straight), it would look like this:
Alice
Bob
2 Technically a geodesic is a line of locally minimal distance, meaning that the line is shorter
than any nearby line.