GR 3 1
General relativity, III
Curvature
The “curvature” of a function f(x) at any point x is defined as
d2f dx2
at that point; the second
derivative measures how much f deviates from being a straight-line. For functions of more than one
variable, “curvature” has a less obvious meaning. Curvature, in part, involves a point of view. For
example, if we look at a circle drawn on a sheet of paper it appears to be clearly “curved.” In this case,
we say the circle is “embedded” in a 2-dimensional plane. On the other hand, we can equally well embed
the circle in 1-dimension. To do this, every point of the circle is mapped onto a straight-line seg ment
whose ends are “identified” with each other. So is the circle curved or straight? In particular, what if we
were 1-dimensional bugs confined to crawl around in t he 1-dimensional world of the circle; would we say
our world was curved or straight?
As another example, let’s consider again the Mercator
projection (MP) discussed in GR 2. The MP is a planar
representation of the latitude and longitude coordinates that
identify any point on the surface of a sphere. It is an
embedding of the sphere’s 2D surface in a 2-dimensional
plane, as in the figure to the right. In 2D, the sphere’s latitude
and longitude coordinate system certainly looks like a n ice,
flat, rectangular grid. On the other hand, when viewed from
“outside”—that is, when emb edded in 3D—the sphere’s
surface is clearly curved. So if we were 2D bugs confined to
crawl around on the surface o f a sphere would we say o ur
world was curved or flat?
Consider one more example, namely, a torus (or “doug hnut”)—seen to
the right. When viewed from the perspective of 3D, the torus certainly seems
curved. As for the sphere, we can label any point on th e surface of the torus with
two coordinates (suggested by the curved arrows on the figure)—an angle saying
how far around the outside the point is, and a second saying how far around a
circular cross-section it is. If we embed this coordinate system in a plane it
looks like a nice, flat, rectangular grid. So , once more, if we were 2 D bugs
confined to crawl around on the surface of a torus would we say our world was
curved or flat?
As for the circle in 1D, both the MP for a sphere and th e equivalent projection for a torus are
planar representations that have “identified points.” In the MP figure above, for example, the “north
pole” is the set of points with different longitude coordinates but all with latitude = 90˚. Similarly, the
“south pole” consists of all points with latitude = −90˚. The “anti-meridian” is the set of all points with
longitude = 180˚ or −180˚. Every point on the 180˚ anti-meridian is also on the -180˚ anti-meridian. A
similar situation is true for the 2D embedding of a torus. Identification of points guarantees the surface in
question is closed (that is, you warp around to where you started if you crawl far enough) but it doesn’t
say anything about curvature. The important issue for bugs crawling around on a surface is whether when
they start off traveling parallel to one another they eventually collide despite each crawling in the
straightest paths possible. That won’t happen (by defin ition) if the surface is flat but might if the surface
is curved. Such curvature is called “intrinsic” and has nothing to do with the shap e of the surface when
embedded in different dimensional spaces. Using the criterion for flatness of nonintersecting “parallel”
paths, we conclude that a sphere really is curved, but th at a torus is not. The conv ergence (or not) of
