The Schwarzschild metric as the static and spherically symmetric solution to Einstein's equation reads
To remove the coordinate singularity of this metric at , one could switch to the Eddington–Finkelstein coordinates. Thus, introduce the "retarded(/outgoing)" null coordinate 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 in Eqs(3) and Eq(5) from a constant to functions of the corresponding null coordinate, and respectively, thus
The extended metrics Eq(6) and Eq(7) are respectively the "retarded(/outgoing)" and "advanced(/ingoing)" Vaidya metrics. It is also interesting and sometimes useful to recast the Vaidya metrics Eqs(6)(7) into the form
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
Box A: Analyses of Vaidya metric in an "outgoing" null tetrad
Suppose , then the Lagrangian for null radial geodesics of the "retarded(/outgoing)" Vaidya spacetime Eq(6) is
where dot means derivative with respect to some parameter . This Lagrangian has two solutions,
According to the definition of in Eq(2), one could find that when increases, the areal radius would increase as well for the solution , while would decrease for the solution . Thus, should be recognized as an outgoing solution while serves as an ingoing solution. 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. Such an outgoing adapted tetrad can be set up as
Since the only nonvanishing Weyl-NP scalar is , the "retarded(/outgoing)" Vaidya spacetime is of Petrov-type D. Also, there exists a radiation field as .
Box B: Analyses of Schwarzschild metric in an "outgoing" null tetrad
For the "retarded(/outgoing)" Schwarzschild metric Eq(3), let , and then the Lagrangian for null radial geodesics will have an outgoing solution and an ingoing solution . Similar to Box A, now set up the adapted outgoing tetrad by
As for the "advanced/ingoing" Vaidya metric Eq(7), 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 , and once again it follows from the null energy condition Eq(11) that , so the central object is absorbing null dusts. 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 is always a marginally outer trapped horizon ().
Box C: Analyses of Vaidya metric in an "ingoing" null tetrad
Suppose , then the Lagrangian for null radial geodesics of the "advanced(/ingoing)" Vaidya spacetime Eq(7) is
which has an ingoing solution and an outgoing solution in accordance with the definition of in Eq(4). 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. Such an ingoing adapted tetrad can be set up as
Since the only nonvanishing Weyl-NP scalar is , the "advanced(/ingoing)" Vaidya spacetime is of Petrov-type D, and there exists an radiation field encoded into .
Box D: Analyses of Schwarzschild metric in an "ingoing" null tetrad
For the "advanced(/ingoing)" Schwarzschild metric Eq(5), still let , and then the Lagrangian for the null radial geodesics will have an ingoing solution and an outgoing solution . Similar to Box C, now set up the adapted ingoing tetrad by
However, there are three clear differences between the Schwarzschild and Vaidia metric:
First of all, the mass parameter for Schwarzschild is a constant, while for Vaidya is a u-dependent function.
Schwarzschild is a solution to the vacuum Einstein equation , while Vaidya is solution is to the trace-free Einstein equation with a nontrivial pure radiation energy field. As a result, all Ricci-NP scalars for Schwarzschild are vanishing, while we have for Vaidya.
Schwarzschild has 4 independent Killing vector fields, including a timelike one, and thus is a static metric, while Vaidya has only 3 independent Killing vector fields regarding the spherical symmetry, and consequently is nonstatic. Consequently, the Schwarzschild metric belongs to Weyl's class of solutions while the Vaidya metric does not.
While the Vaidya metric is an extension of the Schwarzschild metric to include a pure radiation field, the Kinnersley metric constitutes a further extension of the Vaidya metric; it describes a massive object that accelerates in recoil as it emits massless radiation anisotropically. The Kinnersley metric is a special case of the Kerr-Schild metric, and in cartesian spacetime coordinates it takes the following form:
where for the duration of this section all indices shall be raised and lowered using the "flat space" metric , the "mass" is an arbitrary function of the proper-time along the mass's world line as measured using the "flat" metric, and describes the arbitrary world line of the mass, is then the four-velocity of the mass, is a "flat metric" null-vector field implicitly defined by Eqn. (20), and 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 satisfies the identity Grinding out the Einstein Tensor for the metric and integrating the outgoing energy-momentum flux "at infinity," one finds that the metric describes a mass with proper-time dependent four-momentum that emits a net <<link:0>> at a proper rate of as viewed from the mass's instantaneous rest-frame, the radiation flux has an angular distribution where and are complicated scalar functions of and their derivatives, and is the instantaneous rest-frame angle between the 3-acceleration and the outgoing null-vector. The Kinnersley metric may therefore be viewed as describing the gravitational field of an accelerating photon rocket with a very badly collimated exhaust.
In the special case where is independent of proper-time, the Kinnersley metric reduces to the Vaidya metric.
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, as opposed to the corresponce between Vaidya and Schwarzschild metrics.