Notes on Relativity and Cosmology - Physics Department, UCSB

8.2. MORE ON CURVED SPACE 201
turns out to create only a few. The point is that curvature is fundamentally
associated with two-dimensional surfaces. Roughly speaking, the curvature of
a four-dimensional spacetime (labelled by x, y, z, t) can be described in terms
of xt curvature, yt curvature, etc. associated with two-dimensional bits of the
spacetime. However, this is relativity, in which space and time act pretty much
the same. So, if there is xt, yt, and zt curvature, there should also be xy,
yz, and xz curvature! This means that the curvature can show up even if we
consider only straight lines in space (determined, for example, by stretching out
a string) in addition to the effects on the motion of objects that we have already
discussed. For example, if we draw a picture showing spacelike straight lines
(spacelike geodesics), it might look like this:
Y
Two geodesics
X
So, curved space is as much a part of gravity as is curved spacetime. This is nice,
as curved spaces are easier to visualize. Let us now take a moment to explore
these in more depth and build some intuition about curvature in general.
Curved spaces have a number of fun properties. Some of my favorites are:
C ≠ 2πR: The circumference of a circle is typically not 2π times its radius. Let
us take an example: the equator is a circle on a sphere. What is it’s
center? We are only supposed to consider the two-dimensional surface of
the sphere itself as the third dimension was just a crutch to let us visualize
the curved two-dimensional surface. So this question is really ‘what point
on the sphere is equidistant from all points on the equator?’ In fact, there
are two answers: the north pole and the south pole. Either may be called
the center of the sphere.
Now, how does the distance around the equator compare to the distance
(measured along the sphere) from the north pole to the equator? The arc
running from the north pole to the equator goes 1/4 of the way around
the sphere. This is the radius of the equator in the relevant sense. Of
course, the equator goes once around the sphere. Thus, its circumference
is exactly four times its radius.
A ≠ πR 2 : The area of a circle is typically not π times the square of its radius.
Again, the equator on the sphere makes a good example. With the radius