The ingoing Vaidya metric describes a spherically symmetric spacetime sourced by radial null matter, using advanced time and areal radius . In geometrized units it is . A constant mass function reduces it to the Schwarzschild metric in ingoing null coordinates; a varying mass function changes the curvature and the radial null geodesics.
The two radial null geodesic families are and . The first has affine radius, while the second has tangent obeying , so advanced time is generally a nonaffine parameter. An affine parameter for the outgoing family satisfies .
An outgoing radial null geodesic of the Vaidya metric prescribed as determines the mass along that curve by . This inverse relation comes from the radial null condition. It determines a function of advanced time wherever the prescribed curve is differentiable and has positive radius.
Writing , the radial Christoffel symbols are , and . The angular couplings are , , with factors for the corresponding terms, , and . The lower indices are symmetric. The plus sign of follows from the ingoing cross term.
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The Vaidya metric is a solution to the Einstein field equations in general relativity that describes the spacetime geometry around a radiating body. It is particularly useful for modeling scenarios where a star or another massive object is losing mass due to radiation, which can occur during supernovae, for example. The Vaidya solution is an extension of the Schwarzschild solution, which describes the gravitational field outside a non-radiating, spherically symmetric massive body.