asymptotically flat spacetimes
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| asymptotically flat spacetimes | |
|---|---|
| Name | asymptotically flat spacetimes |
| Field | General relativity |
| Related | Schwarzschild metric, Kerr metric, ADM formalism, Penrose diagram |
asymptotically flat spacetimes Asymptotically flat spacetimes describe isolated gravitational systems whose metric approaches a fixed background at large distances, providing a framework for defining total energy, momentum and radiation. They appear in studies of black holes, gravitational waves and global structure, connecting solutions like the Schwarzschild and Kerr metrics to asymptotic frameworks used by Arnowitt, Deser and Misner and by Roger Penrose. Work by Bondi, Sachs and Newman established key notions of radiative flux and symmetry at null infinity, influential in later developments by Hawking, Geroch and Christodoulou.
Definition and Properties
Precise definitions of asymptotic flatness use conditions on the metric and curvature that relate to the Minkowski background and to boundary constructions employed by Penrose, ADM and Bondi. In the Hamiltonian approach of Arnowitt, Deser and Misner, boundary terms introduced by Regge and Teitelboim ensure well-defined energy and momentum for isolated systems such as those modeled by Schwarzschild, Reissner–Nordström and Kerr–Newman solutions. Global properties studied by Hawking, Ellis and Geroch connect causal structure, geodesic completeness and singularity theorems developed with Penrose and Hawking to asymptotic behavior. Technical regularity conditions often reference Sobolev estimates used by Christodoulou, Klainerman and Friedrich in proving stability and decay.
Coordinate and Conformal Approaches
Two main frameworks are coordinate expansions around spatial infinity used in ADM analysis and Penrose's conformal compactification using a conformal factor to attach null infinity. The ADM method, associated with Arnowitt, Deser and Misner and Regge–Teitelboim boundary conditions, employs asymptotically Cartesian coordinates popularized in numerical relativity by Pretorius, Baumgarte and Shapiro. The conformal approach, originated by Penrose and later extended by Friedrich, uses conformal field equations and null infinity structure explored by Bondi, Metzner, Sachs and Newman–Penrose to analyze radiative degrees of freedom. Work by Walker, Penrose and Geroch on conformal boundaries informs treatments by Ashtekar, Henneaux and Wald of conserved quantities and symplectic structures.
Examples and Families of Solutions
Canonical examples include the Schwarzschild and Kerr families studied by Schwarzschild and Kerr, with charged analogues Reissner–Nordström and Kerr–Newman appearing in work by Nordström and Newman. Stationary vacuum spacetimes explored by Carter and Israel illustrate uniqueness theorems proven by Robinson and Mazur; families with matter include Tolman–Oppenheimer–Volkoff models and Vaidya radiating solutions used by Vaidya and Bonnor. Asymptotically flat initial data constructions originate in Lichnerowicz and York formulations, with gluing techniques developed by Corvino, Schoen and Chruściel producing compactly supported perturbations matching exact solutions like Kerr. Dynamic examples of gravitational collapse studied by Oppenheimer, Snyder, Penrose and Christodoulou connect to critical phenomena researched by Choptuik.
Asymptotic Symmetries and the Bondi–Metzner–Sachs Group
Asymptotic symmetry groups at null infinity generalize Poincaré invariance to the Bondi–Metzner–Sachs group discovered by Bondi, van der Burg, Metzner and Sachs, with structure influenced by work of Newman and Penrose. The BMS group contains supertranslations that complicate angular momentum definitions debated in analyses by Ashtekar, Streubel, Dray and Streubel, and Barnich and Troessaert. Extensions invoking soft theorems were proposed by Strominger and connected to scattering amplitudes studied by Weinberg and 't Hooft. Matching conditions across spatial infinity studied by Geroch and Frauendiener relate BMS charges to ADM quantities defined by Arnowitt, Deser and Misner.
Energy, Momentum, and Conserved Quantities
Total energy and momentum in asymptotically flat contexts are formalized by the ADM mass, Bondi mass and Komar integrals associated with Arnowitt, Deser, Misner, Bondi, Bronstein and Komar. Positive energy theorems proven by Schoen, Yau and Witten establish nonnegativity of ADM mass for suitable initial data, with contributions by Penrose and Gibbons in quasi-local mass proposals. Angular momentum and center-of-mass definitions developed by Regge, Teitelboim and Beig involve parity conditions and supertranslation ambiguities addressed by Christodoulou, Klainerman and Ashtekar. Conservation laws for radiative losses use Bondi mass-loss formulae derived by Bondi, van der Burg and Metzner and refined by Sachs.
Gravitational Radiation and Null Infinity
Radiation in asymptotically flat spacetimes is analyzed at null infinity using the Bondi–Sachs formalism, Newman–Penrose scalars and peeling behavior introduced by Sachs, Newman and Penrose. Observational connections to LIGO, Virgo and KAGRA experiments link waveform models based on post-Newtonian expansions from Blanchet, Damour and Iyer, and numerical relativity simulations by Pretorius, Campanelli and Baker. Memory effects tied to soft theorems were studied by Zel'dovich, Polnarev and Christodoulou and revisited in modern scattering analyses by Strominger and Pasterski. Matching of near-zone dynamics to far-zone radiation employs techniques by Blanchet, Damour, Burke and Thorne.
Mathematical Results and Existence Theorems
Rigorous existence and stability results for asymptotically flat solutions include the global nonlinear stability of Minkowski space proven by Christodoulou and Klainerman and alternative treatments by Lindblad, Rodnianski and Bieri. Local and global existence theorems for Einstein’s equations with asymptotically flat data were established by Choquet-Bruhat, York and Friedrich using conformal methods. Gluing and obstruction results by Corvino, Schoen, Isenberg and Mazzeo construct initial data interpolating between arbitrary and exact solutions; uniqueness and rigidity statements invoked by Israel, Carter and Mazur govern stationary black hole classifications. Ongoing research by Klainerman, Luk, Dafermos and Rodnianski addresses stability of Kerr and nonlinear scattering in asymptotically flat settings.