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| Vaidya metric | |
|---|---|
| Name | Vaidya metric |
| Introduced by | P. C. Vaidya |
| Year | 1951 |
| Coordinates | (u,r,θ,φ) |
| Signature | (-,+,+,+) |
| Symmetries | spherical symmetry, null flux |
Vaidya metric The Vaidya metric is an exact solution of the Einstein field equations describing a spherically symmetric spacetime with a radial null fluid, introduced by P. C. Vaidya in 1951. It generalizes the Schwarzschild metric by allowing the mass parameter to vary along a null coordinate, thereby modelling radiating or accreting compact objects in contexts related to Oppenheimer–Snyder collapse, Eddington–Finkelstein coordinates, and astrophysical scenarios considered by Subrahmanyan Chandrasekhar. The solution has played a central role in studies associated with the Tolman–Bondi solution, Israel junction conditions, and investigations of cosmic censorship hypotheses debated by Roger Penrose and Stephen Hawking.
The original construction by P. C. Vaidya extended earlier work on null dust solutions linked to research by J. L. Synge and analyses of radiative spacetimes by Hermann Bondi, Arno Penzias, and contemporaries exploring gravitational radiation. The metric uses an outgoing or ingoing null coordinate similar to retarded or advanced times used in the Eddington and Finkelstein formulations; it is widely referenced alongside the Reissner–Nordström metric, Kerr metric, and models of dynamical horizons studied by James Bardeen, Wald, and Ashtekar.
Starting from the assumption of spherical symmetry and a stress–energy tensor of null dust, Vaidya imposes that the only nonzero stress component is along a null direction, analogous to models by Hans Stephani and connected to energy conditions examined by Hawking and Ellis. In standard outgoing null coordinates (u,r,θ,φ) the line element reads ds^2 = -(1-2M(u)/r) du^2 - 2 du dr + r^2 dΩ^2, where M(u) is a mass function comparable to parameters in the Vaidya–Bonner extensions and the angular sector dΩ^2 mirrors that in Schwarzschild and Kottler metrics. The stress–energy tensor T_{μν} = (dM/du)/(4π r^2) l_μ l_ν involves a null vector l_μ analogous to radiation fields in studies by H. Bondi and W. Kinnersley.
Physically the metric represents a spherically symmetric body emitting or absorbing null radiation, a model relevant to theoretical treatments by Oppenheimer and Snyder of gravitational collapse and to semiclassical analyses by Hawking and Unruh. The energy flux is concentrated on radial null geodesics like those in analyses by Chandrasekhar and influences horizon dynamics much as in discussions by Bekenstein and Page. The metric satisfies the weak and dominant energy conditions for monotonic mass functions M(u) studied in work by Israel and violates them when exotic fluxes are introduced as in models considered by Visser.
Setting M(u)=M_0 reduces the metric to the Schwarzschild metric in Eddington–Finkelstein coordinates, recovering the static black hole limit central to the No-hair theorem and analyses by Wheeler. For charged null fluids one obtains the Reissner–Nordström–Vaidya family related to results by Bonnor and Vaidya–Bonnor, while taking slow variation yields perturbative connections to linearized studies by Regge and Wheeler and quasinormal mode analyses by Kokkotas and Schutz. The extremal or evaporating limits connect to the Hawking evaporation paradigm studied by Page and semiclassical backreaction approaches by York.
The solution is used to model imploding or exploding null shells and to test the cosmic censorship conjecture proposed by Roger Penrose through explicit collapse scenarios examined by Joshi and Datt. It provides tractable examples for examining apparent horizon formation, event horizon tracking as in work by Hayward and Niemeyer, and matching conditions across radiating surfaces using the Darmois–Israel junction conditions developed by Werner Israel. The Vaidya family underlies toy models of black hole evaporation, pair creation in strong fields studied by Gibbons and Hawking, and numerical relativity setups referenced by Pretorius.
Generalizations include charged versions by Bonnor and rotation-inspired attempts connecting to Kerr–Vaidya constructions explored by Patel and Dadhich, higher-dimensional analogues relevant to Kaluza–Klein and Brane world scenarios investigated by Randall and Sundrum, and inclusion of cosmological constants yielding Vaidya–de Sitter spacetimes linked to the ΛCDM model discussions. Null fluids with pressure, anisotropy, or nontrivial equations of state connect to studies by Herrera and Santos, while quantum-inspired modifications appear in work by Ashtekar, Bojowald, and researchers in loop quantum gravity.
Mathematically the solution admits coordinate systems analogous to Eddington–Finkelstein and double-null coordinates used by Christodoulou and Dafermos in rigorous global analysis. Trapped surfaces, marginally trapped tubes, and dynamical horizons in Vaidya spacetimes have been classified following criteria set by Hayward and Booth, and causal structure diagrams mirror Penrose diagrams employed by Penrose and Hawking. The Newman–Penrose formalism applied to the metric yields scalars used in studies by Newman and Penrose; curvature invariants reduce to expressions comparable to those in Schwarzschild and Reissner–Nordström contexts, facilitating singularity and horizon analyses pursued by Geroch and Hawking.