Aguiar Alves, Níckolas de, and Bruno Arderucio Costa. 2025. “The Measure of a Mass.” Essay written for the Gravity Research Foundation 2025 Awards for Essays on Gravitation. arXiv: 2503.18963 [gr-qc].
Abstract: The concept of mass is central to any theory of gravity. Nevertheless, defining mass in general relativity is a difficult task, and even when it can be accomplished, we still need to investigate whether the typical properties of mass in Newtonian gravity are still true in Einsteinian gravity. In this essay, we discuss "the measure of a mass" in relativity by considering some of the many different definitions (Komar, ADM, and Bondi) and how they are related. Finally, we discuss when and whether the mass is positive, as is usually expected, and which physical properties of matter and gravity can ensure this result.
Tags: General Relativity and Energy Conditions
Remark: This is a simplified version of the work linked above, written to be an extended abstract or less formal account of the results. For further details, please check the official publication. Since this publication is a short essay, the summary below is relatively short as well. This account is written independently by Níck Aguiar Alves only and may not fully reflect the points of view of other coauthors.
Within Newtonian gravity, the term “mass” refers to a measure of the power of gravitational attraction of an object. It can be quantified by computing the monopole term for the gravitational field. Mass is also the sole way of interacting with the Newtonian gravitational field, and thus this concept plays a very prominent role in any description of gravitational phenomena.
Despite this, defining “mass” in general relativity is a much more subtle task. Since the gravitational field and spacetime are now two facets of the same physical entity, it is not as straightforward to compute what we would like to call a monopole. Furthermore, time is no longer absolute, which means evaluating a monopole moment at a fixed time may have an ambiguous meaning.
In this essay, we discuss the difficulties in defining mass and how to address them in a few different scenarios for asymptotically flat spacetimes. These different notions of mass are compared and interpreted to pursue a deeper understanding of what “mass” means in relativistic contexts.
Once a few possible definitions of mass have been agreed upon, we shift our focus to discussing a key physical property of mass: its positivity. Different possible arguments lead to the conclusion that a suitably defined notion of mass should be nonnegative. One of the main routes is to assume microscopic masses are always positive in a suitable sense (i.e., an energy condition is obeyed) and this is used to conclude that macroscopic masses are always positive as well. A second route, recently considered by the authors in collaboration with Landulfo, uses stability arguments to show that negative masses need to be short-lived.