Clarifying spatial distance measurement

TL;DR

This paper investigates how to measure spatial distances in curved spacetime using a 1+3-splitting approach, clarifying the roles of the radar metric and spatial projector with examples from Schwarzschild spacetime.

Contribution

It clarifies the relationship between the radar metric and spatial projector in measuring distances within curved spacetime, emphasizing their applicability on the observer's 3-space.

Findings

Radar metric coincides with the spatial projector on the observer's 3-space.

Both metrics reduce to the spatial metric in the observer's frame.

Examples demonstrate the application in Schwarzschild spacetime.

Abstract

We examine length measurement in curved spacetime, based on the 1+3-splitting of a local observer frame. This situates extended objects within spacetime, in terms of a given coordinate which serves as an external reference. The radar metric is shown to coincide with the spatial projector, but these only give meaningful results on the observer's 3-space, where they reduce to the metric. Examples from Schwarzschild spacetime are given.