On the Meaning of Various Mass Definitions for Asymptotically Flat Spacetimes

TL;DR

This paper analyzes different mass definitions in asymptotically flat spacetimes, clarifying their physical interpretations and relationships, especially distinguishing between inertial and active gravitational masses, and examining their effects on observable phenomena.

Contribution

It identifies which mass definitions correspond to inertial versus active gravitational mass and explores their implications in various gravitational phenomena.

Findings

ADM mass corresponds to inertial mass

Møller mass corresponds to active gravitational mass

In vacuum or under energy conditions, inertial and active masses are equal

Abstract

The mass contained in an arbitrary spacetime in general relativity is not well defined. However, for asymptotically flat spacetimes various definitions of mass have been proposed. In this paper I consider eight masses and show that some of them correspond to the active gravitational mass while the others correspond to the inertial mass. For example, the ADM mass corresponds to the inertial mass while the M$\o$ller mass corresponds to the active gravitational mass. In general the inertial and active gravitational masses are not equal. If the spacetime is vacuum at large $r$ the Einstein equations force the inertial and active gravitational masses to be the same. The Einstein equations also force the masses to be the same if any matter that extends out to large $r$ satisfies the weak, strong or dominant energy condition. I also examine the contributions of the inertial and active gravitational masses to the gravitational redshift, the deflection of light, the Shapiro time delay, the precession of perihelia and to the motion of test bodies in the spacetime.