Notes on Relativity and Cosmology - Physics Department, UCSB

8.1. A RETURN TO GEOMETRY 199
This is much like the usual calculus trick of building up a curved line from little
pieces of straight lines. In the present context with more than one dimension,
this process has the technical name of “differential geometry.”
8.1.3 From curved space to curved spacetime
The point is that this process of building a curved surface from flat ones is just
exactly what we want to do with gravity! We want to build up the gravitational
field out of little pieces of “flat” inertial frames. Thus, we might say that gravity
is the curvature of spacetime. This gives us the new language that Einstein was
looking for:
1) (Global) Inertial Frames ⇔ Minkowskian Geometry ⇔ Flat Spacetime: We
can draw it on our flat paper or chalk board and geodesics behave like straight
lines.
2) Worldlines of Freely Falling Observers ⇔ Straight lines in Spacetime
3) Gravity ⇔ The Curvature of Spacetime
Similarly, we might refer to the relation between a worldline and a line of simultaneity
as the two lines being at a “right angle in spacetime 3 .” It is often nice
to use the more technical term “orthogonal” for this relationship.
By the way, the examples (spheres, funnels, etc.) that we have discussed so
far are all curved spaces. A curved spacetime is much the same concept. However,
we can’t really put a curved spacetime in our 3-D Euclidean space. This
is because the geometry of spacetime is fundamentally Minkowskian, and not
Euclidean. Remember the minus sign in the interval? Anyway, what we can
do is to once again think about a spacetime diagram for 2+1 Minkowski space
– time will run straight up, and the two space directions (x and y) will run to
the sides. Light rays will move at 45 degree angles to the (vertical) t-axis as
usual. With this understanding, we can draw a (1+1) curved spacetime inside
this 2+1 spacetime diagram. An example is shown below:
3 As opposed to a right angle on a spacetime diagram drawn in a given frame.