ADM mass

ADM mass
Puzhok · CC BY-SA 3.0 · source
Name	ADM mass
Field	General relativity
Introduced	1959
Introduced by	Arnowitt, Deser, Misner

ADM mass The ADM mass is a conserved quantity defined for asymptotically flat spacetimes in general relativity, named after Richard Arnowitt, Stanley Deser, and Charles Misner. It provides a global measure of the total energy (including gravitational energy) of an isolated system and plays a central role in analyses by institutions and researchers studying black holes, gravitational waves, and cosmological models. Landmark studies by groups at Princeton, Caltech, and Cambridge connect the ADM mass to problems addressed by physicists such as Stephen Hawking, Roger Penrose, John Wheeler, and Robert Geroch.

Definition

The ADM mass is defined for an asymptotically flat spacetime that approaches Minkowski space at spatial infinity, a setting extensively considered in work at Princeton University, Harvard University, University of Cambridge, California Institute of Technology, and elsewhere. Early formulations were developed in the context of the Hamiltonian approach by Arnowitt, Deser, and Misner, and were influential for later contributions by Richard Feynman, Freeman Dyson, and Abdus Salam. The definition uses a 3+1 split of spacetime employed in numerical relativity codes created at Max Planck Institute for Gravitational Physics, Jet Propulsion Laboratory, and LIGO Laboratory teams. In mathematical relativity, researchers at Institute for Advanced Study, Rutgers University, Columbia University, and University of Oxford refined the rigorous conditions guaranteeing existence and positivity, with proofs connected to work by Edward Witten and Shing-Tung Yau.

Mathematical Formulation

The ADM mass arises from the Hamiltonian constraint and surface integrals at spatial infinity used by ADM in the canonical formulation developed at Princeton University and discussed in seminars at Institute for Advanced Study. The standard expression involves an integral of first derivatives of the spatial metric over a large sphere, a technique employed by researchers at University of Chicago and Yale University in analytic studies. Rigorous treatments appear in texts associated with Cambridge University Press and lecture notes by authors at Courant Institute, University of California, Berkeley, and Imperial College London. Mathematical analysis connects the ADM surface integral to concepts investigated by Michael Atiyah, Isadore Singer, and Karen Uhlenbeck in global analysis. The positivity of the ADM mass was proved in contexts related to the work of Roger Penrose, Stephen Hawking, and Edward Witten, with ties to the Yamabe problem and studies at ETH Zurich and Princeton University geometry groups.

Physical Interpretation and Properties

Physically, the ADM mass equals the total energy measured by an observer at spatial infinity and is invariant under asymptotic symmetries examined by Hermann Bondi, Ray Sachs, and Ezra Newman. Its conservation properties are central to analyses by LIGO Laboratory researchers and to theoretical work by Kip Thorne, Jacob Bekenstein, and Subrahmanyan Chandrasekhar on compact objects. The ADM mass is nonnegative under energy conditions considered by Stephen Hawking and Roger Penrose in singularity theorems; this nonnegativity was proven in approaches by E. Witten and further analyzed by mathematicians at Princeton University and Columbia University. The quantity distinguishes between the rest energy of isolated systems studied by astrophysicists at Harvard-Smithsonian Center for Astrophysics and the radiative losses cataloged by observatories such as LIGO, VIRGO, and KAGRA.

Computation and Examples

Computing the ADM mass in practice is performed in numerical relativity simulations by groups at Caltech, Massachusetts Institute of Technology, Max Planck Institute for Gravitational Physics, and collaborations such as the Einstein Toolkit consortium. Examples include Schwarzschild and Kerr spacetimes studied in classic papers by Roy Kerr and Karl Schwarzschild, where the ADM mass equals the parameter M in those solutions, and binary black hole mergers simulated by teams at Caltech and MIT show ADM mass decreasing by the radiated energy observed by LIGO Laboratory. Applications to neutron star models employ microphysics informed by groups at Argonne National Laboratory, Brookhaven National Laboratory, and Lawrence Berkeley National Laboratory. Analytical computations appear in monographs published by Oxford University Press and Cambridge University Press and in reviews by researchers at CERN and FERMILAB.

Relation to Other Mass Definitions

The ADM mass is related to, but distinct from, other notions such as the Bondi mass introduced by Hermann Bondi and the Komar mass associated with Arthur Komar; comparisons were explored by researchers at University of Chicago and Stanford University. Bondi mass describes energy at null infinity in analyses by Ray Sachs and Ezra Newman, while Komar mass applies in stationary spacetimes and was used in work by Brandon Carter and Subrahmanyan Chandrasekhar. Quasilocal masses proposed by James Brown, James York, and Robert Bartnik provide intermediate concepts examined at Rutgers University and University of Toronto. The interplay among these definitions informed studies by physicists at Princeton University, Yale University, University of Michigan, and University of California, Santa Barbara.

Applications in General Relativity

The ADM mass underpins conservation laws used in gravitational-wave astronomy by collaborations like LIGO, VIRGO, and KAGRA and in theoretical work on black hole thermodynamics by Jacob Bekenstein, Stephen Hawking, and Don Page. It is used in constraint-solving algorithms in numerical relativity developed by groups at Max Planck Institute for Gravitational Physics, Caltech, and Cambridge University and in stability analyses of spacetimes pursued at Institute for Advanced Study and Princeton University. The ADM mass also features in cosmological investigations by researchers at NASA, European Space Agency, and Space Telescope Science Institute regarding isolated systems in asymptotically flat approximations, and in mathematical studies by teams at IHES, École Normale Supérieure, and National University of Singapore.

Category:General relativity