Prescriptions for measuring and transporting local angular momenta in general relativity

TL;DR

This paper introduces an operational method for measuring and transporting local linear and angular momenta in curved spacetimes, with applications to asymptotically flat spacetimes and potential implications for understanding BMS charges.

Contribution

It presents a new measurement procedure based on spacetime curvature and an improved transport method for local momenta, enhancing previous approaches.

Findings

Measurement procedure yields well-defined momenta near infinity in stationary spacetimes.

Transport method becomes path independent in certain limiting regimes.

Analysis of holonomy reveals curvature effects on momentum transport.

Abstract

For observers in curved spacetimes, elements of the dual space of the set of linearized Poincar\'e transformations from an observer's tangent space to itself can be naturally interpreted as local linear and angular momenta. We present an operational procedure by which observers can measure such quantities using only information about the spacetime curvature at their location. When applied by observers near spacelike or null infinity in stationary, vacuum, asymptotically flat spacetimes, there is a sense in which the procedure yields the well-defined linear and angular momenta of the spacetime. We also describe a general method by which observers can transport local linear and angular momenta from one point to another, which improves previous prescriptions. This transport is not path independent in general, but becomes path independent for the measured momenta in the same limiting regime. The transport prescription is defined in terms of differential equations, but it can also be interpreted as parallel transport in a particular direct-sum vector bundle. Using the curvature of the connection on this bundle, we compute and discuss the holonomy of the transport law. We anticipate that these measurement and transport definitions may ultimately prove useful for clarifying the physical interpretation of the Bondi-Metzner-Sachs charges of asymptotically flat spacetimes.