Arnowitt–Deser–Misner

Arnowitt–Deser–Misner
Name	Arnowitt–Deser–Misner
Known for	ADM formalism, canonical gravity, Hamiltonian formulation

Arnowitt–Deser–Misner is the conventional name for a canonical formulation of general relativity developed by Richard Arnowitt, Stanley Deser, and Charles W. Misner that recasts Einstein field equations into Hamiltonian form, enabling analysis using methods from Hamiltonian mechanics, canonical quantization, and differential geometry. The ADM approach provided definitions of conserved quantities such as energy and momentum in asymptotically flat spacetimes and influenced developments in quantum gravity, numerical relativity, and cosmology, linking to work by researchers at institutions such as Princeton University, Institute for Advanced Study, and Massachusetts Institute of Technology.

History and background

The ADM formulation emerged in the late 1950s and early 1960s through collaborations among Richard Arnowitt, Stanley Deser, and Charles W. Misner, building on earlier mathematical structures from Albert Einstein's original papers, the Hamiltonian analyses of Paul Dirac, and geometric methods popularized by Bernhard Riemann and Élie Cartan. Early presentations at venues like Princeton University, Brandeis University, and conferences organized by International Centre for Theoretical Physics helped disseminate the formalism to communities around John Wheeler, Bryce DeWitt, and Roger Penrose, who applied ADM ideas in contexts including the ADM–DeWitt metric and the Wheeler–DeWitt equation associated with canonical quantum gravity. The historical thread links to efforts by Arthur Eddington, Hermann Weyl, Felix Klein, and later to numerical efforts at NASA, Caltech, and Max Planck Institute that exploited ADM for computational simulations.

ADM formalism

The ADM formalism foliates a four-dimensional Lorentzian manifold into spacelike hypersurfaces parameterized by a time coordinate, employing variables such as the induced three-metric, lapse function, and shift vector introduced in the Hamiltonian density, influenced by methods used by Carl Gustav Jacob Jacobi and Joseph-Louis Lagrange in classical mechanics. This foliation connects to mathematical structures studied by Elie Cartan, André Lichnerowicz, and Yvonne Choquet-Bruhat, and to coordinate choices familiar from the Schwarzschild solution, Kruskal–Szekeres coordinates, and Friedmann–Lemaître–Robertson–Walker metric used in cosmology. ADM rewrites the Einstein–Hilbert action in first-order form, paralleling techniques of Ludwig Boltzmann and David Hilbert and making contact with the phase-space analyses of Leonard Euler and William Rowan Hamilton.

ADM energy, mass, and momentum

ADM provided rigorous definitions of total energy, mass, and linear momentum for asymptotically flat spacetimes, commonly called ADM mass and ADM momentum, extending concepts earlier explored in the context of the Schwarzschild metric and the Komar mass used in stationary spacetimes; these constructions influenced positivity proofs by Richard Schoen and Shing-Tung Yau and the positive energy theorem later connected to techniques by Edward Witten. The ADM mass plays a central role in studies of gravitational collapse related to the Oppenheimer–Snyder collapse, constraints in the Penrose singularity theorem, and the characterization of black hole spacetimes including Kerr metric, Reissner–Nordström metric, and perturbative analyses by Teukolsky. The ADM momentum links to conserved quantities used in the study of gravitational radiation first discussed by Hermann Bondi, Rainer K. Sachs, and Felix A. E. Pirani.

ADM Hamiltonian and constraints

The ADM Hamiltonian splits into constraint equations—the Hamiltonian constraint and the momentum (diffeomorphism) constraints—echoing canonical constraints in Dirac's theory of constrained systems and the BRST methods developed later by Becchi, Rouet, and Stora. Solution techniques employ elliptic PDE theory advanced by Sergiu Klainerman, Demetrios Christodoulou, and Yvonne Choquet-Bruhat, and link to boundary terms analyzed by Geroch, Ashtekar, and Andersson. The constraint algebra under spatial diffeomorphisms relates to structure functions studied in the context of the Lie algebra of vector fields and connects to quantization programs pursued by Lee Smolin, Carlo Rovelli, and Abhay Ashtekar in loop quantum gravity and to path integral approaches by Stephen Hawking, James Hartle, and Gerard 't Hooft.

Applications and extensions

ADM underpins many applications: numerical simulations of binary black hole mergers and gravitational wave emission by groups at Caltech, MIT, and Albert Einstein Institute; post-Newtonian expansions used by Luc Blanchet and Thibault Damour for waveform modeling; Hamiltonian approaches to cosmological perturbation theory relevant to Alan Guth, Andrei Linde, and Paul Steinhardt; and semiclassical and canonical quantization attempts linked to the Wheeler–DeWitt equation and work by John Wheeler, Bryce DeWitt, and Stanley Deser. ADM variables have been adapted in formulations like the Arnowitt–Deser–Misner–York boundary term introduced by James W. York Jr. and in connection with conserved charges in asymptotically anti-de Sitter spacetimes studied by Juan Maldacena and Edward Witten in the context of the AdS/CFT correspondence.

Criticisms and alternative formulations

Critiques of ADM emphasize problems with time in canonical gravity famously articulated by Carlo Rovelli and Julian Barbour as the "problem of time", challenges in defining observables highlighted by Bryce DeWitt and Karel Kuchař, and issues with nonrenormalizability encountered in perturbative treatments by Goroff and Sagnotti and in asymptotic analyses by Steven Weinberg. Alternative formulations include the covariant Palatini action used by Élie Cartan-inspired approaches, the Ashtekar variables developed by Abhay Ashtekar and extended by Thomas Thiemann, shape dynamics advocated by Julian Barbour and Flavio Mercati, and effective field theory methods by Donoghue and John F. Donoghue that reframe gravitational interactions in perturbative quantum contexts; these alternatives intersect with lattice approaches pioneered by Tullio Regge, causal dynamical triangulations by Jan Ambjørn, and string-theoretic frameworks developed by Michael Green, John Schwarz, and Edward Witten.

Category:General relativity