Notes on Relativity and Cosmology - Physics Department, UCSB

8.1. A RETURN TO GEOMETRY 197
By the way, the above picture is not supposed to be a spacetime diagram. It is
simply supposed to be a map of part of the (two dimensional) earth’s surface,
on which both paths have been drawn. This particular map is drawn in such a
way that Alice’s path appears as a straight line. As you probably know from
looking at maps of the earth’s surface, no flat map will be an accurate description
globally, over the whole earth. There will always be some distortion somewhere.
However, a flat map is perfectly fine locally, say in a region the size of the city
of Syracuse (if we ignore the hills).
Now, does this look or sound at all familiar?
What if we think about a similar experiment involving Alice and Bob walking
on a funnel-shaped surface:
Alice
Bob
In this case they begin to drift apart as they walk so that Alice’s map would
look like this:
Alice
Bob
So, we see that straight lines (geodesics) on a curved surface act much like
freely falling worldlines in a gravitational field. It is useful to think through
this analogy at one more level: Consider two people standing on the surface of
the earth. We know that these two people remain the same distance apart as
time passes. Why do they do so? Because the earth itself holds them apart and
prevents gravity from bringing them together. The earth exerts a force on each
person, keeping them from falling freely.
Now, what is the analogy in terms of Alice and Bob’s walk across the sphere
or the funnel? Suppose that Alice and Bob do not simply walk independently,
but that they are actually connected by a stiff bar. This bar will force them to
always remain the same distance apart as they walk toward the north pole. The
point is that, in doing so, Alice and Bob will be unable to follow their natural
(geodesic) paths. As a result, Alice and Bob will each feel some push or pull
from the bar that keeps them a constant distance apart. This is much like our