Bondi MassEdit
Bondi mass is a notion of total energy in the gravitational field of an isolated system, defined in the framework of general relativity for spacetimes that are asymptotically flat. It arises from the Bondi–Sachs approach to the equations of gravity at large distances, where the geometry of null infinity encodes the energy and radiation content of the system. Named after Hermann Bondi and colleagues who developed the formalism in the 1960s, Bondi mass situates itself at future null infinity, where outgoing radiation can be cleanly analyzed and separated from the local dynamics near the sources.
In the Bondi–Sachs formulation, the spacetime metric is written in coordinates adapted to outgoing light rays, often called Bondi coordinates. The key objects include a mass aspect m(u, θ, φ) defined on each cross-section of null infinity and a radiative quantity known as the news tensor NAB that encodes the presence of gravitational waves. The physical Bondi mass M(u) is obtained by averaging the mass aspect over the celestial sphere at a given retarded time u. The formalism makes explicit how A) energy content changes due to radiation, and B) how the gravitational field itself participates in the energy accounting of the system.
A central result in this framework is the Bondi mass loss formula: the rate of change of the Bondi mass with retarded time is negative and is governed by the square of the news tensor. In practical terms, when gravitational waves are emitted by the system, energy is carried away to infinity, and the Bondi mass decreases accordingly. This provides a precise mechanism by which the system loses energy through radiation, while the local dynamics near the sources can be quite complex. The conceptual picture is that the distant observer, looking toward null infinity, measures a diminishing total energy as waves radiate outward.
Bondi mass is closely connected to the broader structure of spacetime symmetries at infinity. The asymptotic symmetry group of asymptotically flat spacetimes at null infinity, the BMS group, generalizes the familiar Poincaré symmetries and plays a role in how energy and angular momentum are defined in this setting. The interplay between these symmetries and the radiative content of the spacetime helps explain subtleties such as gravitational memory, where a passing wave leaves a lasting imprint on the relative positions of test masses even after the wave has passed.
From a physical standpoint, Bondi mass is expected to be nonnegative under reasonable energy conditions, and it reduces to something akin to the total energy of the system in appropriate limits. In particular, for spacetimes that settle down to a stationary configuration at early and late times, the Bondi mass approaches the ADM mass, which is defined at spatial infinity. The positivity properties of Bondi mass have been established in tandem with broader positivity theorems in general relativity, linking the energy content of the gravitational field to global geometric conditions.
The Bondi framework has been extended and refined in various directions. Researchers have explored the precise relations between Bondi mass, ADM mass, and other quasi-local definitions of energy; they have analyzed how the formalism behaves in spacetimes that are not perfectly isolated or not strictly asymptotically flat; and they have studied how the concept adapts when a cosmological constant is present. The mass aspect and the radiative degrees of freedom are also studied in the context of the gravitational radiation community, where observations of waves from astrophysical sources—such as binary black hole mergers detected by gravitational waves—provide empirical anchors for these theoretical constructs.
Despite its successes, Bondi mass is not without controversy or active debate. Some researchers emphasize that energy in general relativity is inherently tied to the chosen asymptotic structure and that definitions like Bondi mass depend on coordinates and gauge choices in subtle ways. Others point to alternative, quasi-local notions of energy that aim to assign energy content to finite regions of spacetime without reference to infinity, arguing that such definitions can capture more intuitive notions of energy for realistic configurations. The distinction between global quantities defined at null infinity and local or quasi-local measures of energy remains a fruitful area of ongoing research. In addition, discussions about how the asymptotic symmetries (the BMS group) reflect physical observables—such as memory effects and soft graviton theorems—continue to inspire both mathematical and physical commentary about what Bondi mass really tells us about the energy carried by gravitational radiation.
In practical terms, Bondi mass provides a clean, radiation-aware accounting of energy for isolated systems in a regime where spacetime is nearly flat far from sources. It links the emission of gravitational waves to a concrete change in energy content, and it anchors the theoretical understanding of how gravity carries energy away from astrophysical events, complementing the more global notions of energy that appear in other parts of general relativity.
Origins and definitions
The development of the Bondi–Sachs formalism, including the use of null infinity and Bondi coordinates, is a cornerstone of modern gravitational theory. See Bondi–Sachs formalism and null infinity for foundational treatments.
The mass aspect m(u, θ, φ) on cross-sections of null infinity and its relation to the Bondi mass M(u) are central to practical calculations in asymptotically flat spacetimes. See mass aspect and ADM mass for related energy concepts.
The radiative content is encoded in the news tensor NAB, whose square enters the Bondi mass loss formula. See news tensor and gravitational waves for context.
The Bondi mass connects to the broader portrait of spacetime symmetries at infinity, notably the BMS group BMS group.
Mass loss and radiation
The Bondi mass loss formula expresses how outgoing gravitational radiation reduces the mass over retarded time. See Bondi mass loss formula for an explicit statement and derivations.
Gravitational memory and soft theorems connect the radiative content to observable imprints at infinity, linking the abstract mass concept to potentially measurable effects gravitational memory effect.
In the limit of stationary configurations or early times, Bondi mass approaches the ADM mass, tying null-infinity results to spatial-infinity quantities. See ADM mass for the ADM perspective.
Positivity and interpretation
Under standard energy conditions, Bondi mass is nonnegative, and positivity results tie into the general positive energy philosophy in general relativity. See positive energy theorem and discussions of energy in GR for broader context.
The interpretation of Bondi mass depends on the asymptotic structure of spacetime, raising questions about its universality in non-idealized cosmologies. Researchers explore how to translate these ideas to more general settings or to quasi-local constructs of energy.
Extensions and debates
Extensions of the Bondi framework consider spacetimes with different asymptotics, cosmological constants, or more general boundary conditions. See discussions around asymptotically flat spacetime and related topics.
Debates persist about the relative merits of global versus quasi-local energy notions, how to best account for gravitational radiation in realistic astrophysical scenarios, and how to reconcile different definitions within a single physical narrative. See quasi-local mass for competing approaches.