In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts.
It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the "radiating(shining) Schwarzschild metric".
The Schwarzschild metric as the static and spherically symmetric solution to Einstein's equation reads To remove the coordinate singularity of this metric at
by and Eq(1) could be transformed into the "retarded(/outgoing) Schwarzschild metric" or, we could instead employ the "advanced(/ingoing)" null coordinate
by so Eq(1) becomes the "advanced(/ingoing) Schwarzschild metric" Eq(3) and Eq(5), as static and spherically symmetric solutions, are valid for both ordinary celestial objects with finite radii and singular objects such as black holes.
It turns out that, it is still physically reasonable if one extends the mass parameter
Following the calculations using Newman–Penrose (NP) formalism in Box A, the outgoing Vaidya spacetime Eq(6) is of Petrov-type D, and the nonzero components of the Weyl-NP and Ricci-NP scalars are It is notable that, the Vaidya field is a pure radiation field rather than electromagnetic fields.
The emitted particles or energy-matter flows have zero rest mass and thus are generally called "null dusts", typically such as photons and neutrinos, but cannot be electromagnetic waves because the Maxwell-NP equations are not satisfied.
By the way, the outgoing and ingoing null expansion rates for the line element Eq(6) are respectively Suppose
Now, we can construct a complex null tetrad which is adapted to the outgoing null radial geodesics and employ the Newman–Penrose formalism for perform a full analysis of the outgoing Vaidya spacetime.
, the "retarded(/outgoing)" Vaidya spacetime is of Petrov-type D. Also, there exists a radiation field as
, and then the Lagrangian for null radial geodesics will have an outgoing solution
Similar to Box A, now set up the adapted outgoing tetrad by
As for the "advanced/ingoing" Vaidya metric Eq(7),[1][2][6] the Ricci tensors again have one nonzero component and therefore
and the stress–energy tensor is This is a pure radiation field with energy density
As calculated in Box C, the nonzero Weyl-NP and Ricci-NP components of the "advanced/ingoing" Vaidya metric Eq(7) are Also, the outgoing and ingoing null expansion rates for the line element Eq(7) are respectively The advanced/ingoing Vaidya solution Eq(7) is especially useful in black-hole physics as it is one of the few existing exact dynamical solutions.
For example, it is often employed to investigate the differences between different definitions of the dynamical black-hole boundaries, such as the classical event horizon and the quasilocal trapping horizon; and as shown by Eq(17), the evolutionary hypersurface
, then the Lagrangian for null radial geodesics of the "advanced(/ingoing)" Vaidya spacetime Eq(7) is
Now, we can construct a complex null tetrad which is adapted to the ingoing null radial geodesics and employ the Newman–Penrose formalism for perform a full analysis of the Vaidya spacetime.
, the "advanced(/ingoing)" Vaidya spacetime is of Petrov-type D, and there exists a radiation field encoded into
, and then the Lagrangian for the null radial geodesics will have an ingoing solution
Similar to Box C, now set up the adapted ingoing tetrad by
it takes the following form: where for the duration of this section all indices shall be raised and lowered using the "flat space" metric
along the mass's world line as measured using the "flat" metric,
is a "flat metric" null-vector field implicitly defined by Eqn.
implicitly extends the proper-time parameter to a scalar field throughout spacetime by viewing it as constant on the outgoing light cone of the "flat" metric that emerges from the event
and integrating the outgoing energy–momentum flux "at infinity," one finds that the metric
as viewed from the mass's instantaneous rest-frame, the radiation flux has an angular distribution
The Kinnersley metric may therefore be viewed as describing the gravitational field of an accelerating photon rocket with a very badly collimated exhaust.
Since the radiated or absorbed matter might be electrically non-neutral, the outgoing and ingoing Vaidya metrics Eqs(6)(7) can be naturally extended to include varying electric charges, Eqs(18)(19) are called the Vaidya-Bonner metrics, and apparently, they can also be regarded as extensions of the Reissner–Nordström metric, analogously to the correspondence between Vaidya and Schwarzschild metrics.