# How does the idea of "proper distance" make sense in general relativity?

In special relativity we can define the proper distance between two events as the length of a rod that connects the two events in the rest frame of the rod. However, this seems to require the ability to define a global inertial frame unlike the proper time which can be measured by free-falling between two events.

In other words if I want the proper distance in special relativity, I can set up a rod reaching from event $$A$$ to event $$B$$ and measure it while I am in its rest frame. However if we now work in a gravitational field I can only obtain an inertial frame locally, which seems to prohibit me from measuring the length of the rod in the same way, what is the meaning behind the proper distance under these circumstances?

In GR, you follow geodesics, which are the shortest paths in curved spacetime between two points. The proper distance between two points at either end of a geodesic path P is given by $$\int_P\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$$ where $$g_{\mu\nu}$$ is the GR metric tensor (which changes values as you travel over curved space), and $$dx^\mu$$ is the infinitesimal spacetime coordinate separation between neighboring points along the path P.