In the context of relativity, proper distance is the actual spatial separation between two events or points as measured by an observer at a single moment in time—along a hypersurface of constant time.
Unlike coordinate distance, which depends on the observer’s frame of reference, proper distance is defined in a specific frame where both points are considered at the same time. This requires choosing a “slice” of spacetime—a hypersurface—where time is held constant, and then measuring the distance across that surface.
Proper distance is especially important in cosmology, where it helps define the spatial separation between galaxies or other objects in the universe at a given cosmic time. It also plays a role in general relativity, helping distinguish between the geometry of space and the full four-dimensional curvature of spacetime.
While time and space can be distorted by gravity or motion, proper distance offers a way to measure pure spatial separation, assuming simultaneity in a chosen reference frame.