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| Arnowitt–Deser–Misner formalism | |
|---|---|
| Name | Arnowitt–Deser–Misner formalism |
| Field | General relativity |
| Introduced | 1959–1962 |
| Designers | Richard Arnowitt, Stanley Deser, Charles W. Misner |
Arnowitt–Deser–Misner formalism is a Hamiltonian formulation of General relativity developed by Richard Arnowitt, Stanley Deser, and Charles W. Misner that decomposes spacetime into spatial hypersurfaces and time, enabling canonical analysis and quantization attempts of Albert Einstein's field equations. It provides a canonical set of variables, a Hamiltonian density, and constraint equations that connect to the Initial value problem and feed directly into modern Numerical relativity, Quantum gravity programs, and studies of gravitational energy such as the ADM mass. The formalism underpins conceptual and computational work linking Pierre-Simon Laplace-era Hamiltonian mechanics traditions to relativistic field theories examined by figures like Paul Dirac and institutions such as the Institute for Advanced Study.
The formalism was developed in a collaboration among Arnowitt, Deser, and Misner during the late 1950s and early 1960s at venues including the Princeton University corridor and interactions with researchers at Brandeis University, reflecting influence from John Archibald Wheeler, Roger Penrose, John Wheeler's students, and contemporaneous work by Paul Dirac, Yakov Borisovich Zel'dovich, and Evgeny Lifshitz. Early presentations tied into discussions at meetings like the International Congress of Mathematicians and seminars at Harvard University, and the approach gained traction through citations in work by Kip Thorne, Joel Smoller, and Abhay Ashtekar. Historical development intertwined with advances in the study of gravitational radiation by Hermann Bondi, the formulation of conserved quantities by Richard Schoen, and later computational efforts at institutions such as Caltech and Max Planck Institute for Gravitational Physics.
The formalism casts Albert Einstein's equations into a canonical Hamiltonian system by foliating a four-dimensional Lorentzian manifold into a one-parameter family of three-dimensional spacelike hypersurfaces labeled by a time parameter, a technique resonant with methods used by Cahit Arf and formal developments in differential geometry at University of Cambridge. The metric is decomposed into the spatial metric, lapse function, and shift vector, paralleling methods in canonical treatments influenced by Paul Dirac's constrained dynamics and the Hamilton–Jacobi tradition from William Rowan Hamilton and Joseph-Louis Lagrange. Canonical momenta are defined conjugate to the three-metric, leading to Hamiltonian and momentum densities whose structure reflects the action principle akin to formulations used by Noether and later adapted in analytical mechanics curricula at Massachusetts Institute of Technology.
The ADM Hamiltonian is the generator of time translations on the space of spatial metrics and conjugate momenta and consists of constraint terms rather than a positive-definite energy density, echoing analyses by Emmy Noether and subsequent conserved-current investigations by Yakov Zel'dovich. The formulation yields the Hamiltonian constraint and momentum (diffeomorphism) constraints that must vanish on physical solutions, constraints that parallel conditions studied by Élie Cartan in geometry and by Paul Dirac in constrained quantization. The asymptotic analysis used to define conserved quantities like the ADM mass involves concepts developed by Roger Penrose and tools later refined in proofs by Richard Schoen and Shing-Tung Yau.
Addressing the initial value problem requires specifying initial data on a spatial hypersurface that satisfy the ADM constraints, a program that connects to elliptic PDE theory advanced by mathematicians such as André-Louis Besse and Sergiu Klainerman, and to methods introduced by James York for conformal decomposition. Techniques for solving the constraint equations include the conformal transverse-traceless method, the Bowen–York solutions, and gluing constructions developed in work related to Dennis Sullivan and Gerard Besson, enabling construction of physically relevant initial data sets for spacetimes containing black holes, gravitational waves, or cosmological slices as studied in Robert Wald's texts. Analytical existence and uniqueness results draw on results by Yvonne Choquet-Bruhat, Lars Hörmander, and others in hyperbolic PDE theory.
The ADM formalism provided the conceptual foundation for early numerical relativity efforts at laboratories such as Caltech, NASA, and the Max Planck Institute, influencing numerical schemes used by groups led by Frans Pretorius, Manuela Campanelli, and Saul Teukolsky. Although raw ADM evolution equations can be unstable numerically, reformulations like the BSSN system and generalized harmonic coordinates—developed by researchers affiliated with University of Texas at Brownsville and Cornell University—trace conceptual lineage to the ADM split, enabling successful simulations of binary black hole mergers that underlie gravitational-wave detections by LIGO, Virgo, and the KAGRA collaborations. Implementation details have been standardized in community codes such as the Einstein Toolkit and have been integral to waveform modeling used by the LIGO Scientific Collaboration and Virgo Collaboration.
Extensions and alternatives to the ADM framework include the Arnowitt–Deser–Misner-inspired canonical quantization approaches pursued in Loop Quantum Gravity programs associated with Abhay Ashtekar and institutions like Penn State University, as well as covariant and tetrad-based formalisms used in spin-foam models connected to Carlo Rovelli and Lee Smolin. Other related formalisms include the BSSN formulation, generalized harmonic gauge methods tied to Richard Matzner's work, and path-integral approaches that reference techniques from Richard Feynman and semiclassical analyses at CERN. The ADM perspective continues to inform research at centers like Perimeter Institute and in analytic studies by figures such as Piotr Chruściel and Gundlach, linking classical relativistic dynamics to quantum and computational frontiers.