Notes on Relativity and Cosmology - Physics Department, UCSB

200 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME
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Note that one can move along the surface in either a timelike manner (going
up the surface) or a spacelike manner (going across the surface), so that this
surface does indeed represent a (1+1) spacetime. The picture above turns out
to represent a particular kind of gravitational field that we will be discussing
more in a few weeks. To see the similarity to the gravitational field around
the earth, think about two freely falling worldlines (a.k.a. “geodesics,” the
straightest possible lines) that begin near the middle of the diagram and start
out moving straight upward. Suppose for simplicity that one geodesic is on one
side of the fold while the second is on the other side. You will see that the two
worldlines separate, just as two freely falling objects do at different heights in
the earth’s gravitational field. Thus, if we drew a two-dimensional map of this
curved spacetime using the reference frame of one of these observers, the results
would be just like the spacetime diagram we drew for freely falling stones at
different heights! This is a concrete picture of what it means to say that gravity
is the curvature of spacetime.
Well, there is one more subtlety that we should mention...... it is important to
realize that the extra dimension we used to draw the picture above was just a
crutch that we needed because we think best in flat spaces. One can in fact talk
about curved spacetimes without thinking about a “bigger space” that contains
points “outside the spacetime.” This minimalist view is generally a good idea,
as we will discuss more in the sections below.
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8.2 More on Curved Space
Let us remember that the spacetime in which we live is fundamentally four
(=3+1) dimensional and ask if this will cause any new wrinkles in our story. It