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| Boyer–Lindquist coordinates | |
|---|---|
| Name | Boyer–Lindquist coordinates |
| Introduced | 1967 |
| Developers | Robert H. Boyer, Richard W. Lindquist |
| Field | General relativity, Differential geometry |
Boyer–Lindquist coordinates are a coordinate system used to express the Kerr metric and related solutions in general relativity. They were introduced by Robert H. Boyer and Richard W. Lindquist in 1967 to provide a more tractable form for studying rotating black hole spacetimes, facilitating analysis of horizons, ergospheres, and geodesic motion. The coordinates generalize the Schwarzschild metric's static spherical coordinates to the axisymmetric, stationary case associated with the Kerr–Newman metric and are widely employed in theoretical and numerical work across astrophysics, mathematical physics, and cosmology.
Boyer–Lindquist coordinates recast the axisymmetric solutions discovered by Roy Kerr and extended by Erwin Schrödinger-era techniques into a form adapted to axial symmetry and time independence, paralleling how Karl Schwarzschild's coordinates describe nonrotating solutions. They are central in analyses by researchers such as Subrahmanyan Chandrasekhar, Brandon Carter, and John Archibald Wheeler, and appear in studies influenced by institutions like Princeton University, Cambridge University, and Institute for Advanced Study. The coordinate system plays a role in investigations related to the Penrose process, Hawking radiation, and tests of general relativity with observatories including Event Horizon Telescope, LIGO, and Virgo.
Starting from the metric form found by Roy Kerr in 1963, Boyer and Lindquist performed a coordinate transformation akin to those used by David Hilbert and Willem de Sitter for static metrics to isolate the cross term between time and azimuthal angle. The transformation introduces coordinates (t, r, θ, φ) where t is the asymptotic time coordinate tied to observers at infinity such as in analyses by Arthur Eddington and Finkelstein, r is a radial-type coordinate adapted from Boyer–Lindquist's original mapping, θ is the polar angle familiar from Isaac Newton-era spherical geometry, and φ is the azimuthal coordinate associated with axial Killing vectors studied by Noether and applied by Brandon Carter. The line element takes a block-diagonal form except for an off-diagonal dt dφ term reflecting frame dragging originally emphasized by Lense and Thirring; the metric functions involve parameters of mass M (as in Albert Einstein's solutions) and specific angular momentum a (as in Kerr's original parameterization).
In Boyer–Lindquist coordinates, r ranges over values outside and inside the outer and inner horizons identified by roots of Δ = r^2 - 2Mr + a^2 + Q^2 (the latter when charge is included as in Newman–Penrose extensions like Kerr–Newman metric), while θ ranges from 0 to π and φ is periodic with period 2π as in classical Pierre-Simon Laplace spherical conventions. The coordinates clearly exhibit the event horizon structure studied by Roger Penrose and Stephen Hawking, highlight the ergoregion boundary where g_tt changes sign (central to Roger Penrose's energy-extraction mechanism), and allow conserved quantities associated with Killing vectors linked to Emmy Noether's theorem. The metric determinant and curvature invariants such as the Kretschmann scalar show singular behavior at r = 0 and θ = π/2 corresponding to the ring singularity first noted in Kerr analyses, while coordinate singularities at the horizons can be removed by transformations pioneered by Martin Kruskal and applied in rotating contexts by later researchers.
Boyer–Lindquist coordinates reduce to Schwarzschild coordinates when the rotation parameter a vanishes, and relate to Kruskal–Szekeres coordinates and Eddington–Finkelstein coordinates through null-coordinate transformations used to extend spacetimes across horizons—the extensions echo methods developed by Martin Kruskal, George Szekeres, Arthur Eddington, and David Finkelstein. They can be mapped to Cartesian-like Boyer–Lindquist–Cartesian representations used in numerical relativity codes influenced by groups at California Institute of Technology and Max Planck Institute for Gravitational Physics, and are convertible to tetrad formalisms exploited by Newman–Penrose techniques and by analyses from Chandrasekhar and Penrose. For charged or cosmological-constant generalizations, coordinate relations tie them to the Reissner–Nordström metric and Kerr–de Sitter metric, connecting work from Hermann Weyl and Willem de Sitter.
Boyer–Lindquist coordinates are used to compute geodesics and photon trajectories relevant to imaging and spectrum modeling for projects like Event Horizon Telescope, Very Large Telescope, and XMM-Newton, and underpin ray-tracing efforts by teams at Harvard–Smithsonian Center for Astrophysics and MIT. They facilitate analytic derivations of innermost stable circular orbits (ISCO) used in Shakura–Sunyaev-style accretion disk models applied by Narayan, Thorne, and Bardeen, and inform analyses of superradiant scattering involving concepts from Roger Penrose and Yakov Zel'dovich. Studies of gravitational-wave templates for detectors such as LIGO and KAGRA often begin with Boyer–Lindquist-based backgrounds before perturbative methods by Vishveshwara and Teukolsky are applied; the coordinates also serve in quantum field theory on curved spacetime work by Stephen Hawking, Gerard 't Hooft, and Leonard Susskind.
Although well-suited for analytic work, Boyer–Lindquist coordinates exhibit coordinate singularities at horizons and are not horizon-penetrating like Eddington–Finkelstein coordinates or Kruskal–Szekeres coordinates, limiting their use in numerical simulations by collaborations such as Einstein Toolkit and groups at Max Planck Institute for Gravitational Physics. Extensions include horizon-penetrating forms, Kerr–Schild coordinates used in computational relativity by teams at NASA and European Space Agency, and generalizations to include charge and cosmological constant in Kerr–Newman and Kerr–de Sitter metrics explored by scholars at Princeton University and Cambridge University. Modern developments connect Boyer–Lindquist-based analyses to studies in string theory, AdS/CFT correspondence, and holography undertaken by researchers at Institute for Advanced Study and Perimeter Institute.
Category:Coordinate systems Category:General relativity Category:Black holes