A relativistic center of mass in general relativity

TL;DR

This paper defines a covariant relativistic center of mass and spin for isolated gravitational sources, providing new insights into their motion and momentum in general relativity, challenging previous non-covariant assumptions.

Contribution

It introduces a covariant definition of the center of mass and spin in general relativity and extends these concepts to asymptotically flat spacetimes, analyzing their equations of motion.

Findings

Center of mass and spin definitions are covariant and physically meaningful.

The velocity and Bondi momentum are not proportional in general.

Extra terms are needed for accurate descriptions of gravitational radiation sources.

Abstract

The center of mass and spin for isolated sources of gravitational radiation that move at relativistic speeds are defined. As a first step, we also present these definitions in flat space. This contradicts some general wisdom given in textbooks claiming that such definitions are not covariant and thus, have no physical meaning. We then generalize the definitions to asymptotically flat spacetimes giving their equations of motion when gravitational radiation is emitted by the isolated sources. The resulting construction has some similarities with the Mathisson-Papapetrou equations which describes the motion of the particle in an external field. We analyze the relationship between the center of mass velocity and the Bondi linear momentum and show they are not proportional to each other. A similar situation happens between the total and intrinsic angular momentum when the Bondi momentum vanishes. We claim that extra terms should be added in other approaches to adequately describe the time evolution of isolated sources of gravitational radiation.