Vaidya metric

From Schwarzschild to Vaidya metrics

The Schwarzschild metric as the static and spherically symmetric solution to Einstein's equation reads

( 1 ) d s 2 = − ( 1 − 2 M r ) d t 2 + ( 1 − 2 M r ) − 1 d r 2 + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) .

To remove the coordinate singularity of this metric at r = 2 M , one could switch to the Eddington–Finkelstein coordinates. Thus, introduce the "retarded(/outgoing)" null coordinate u by

( 2 ) t = u + r + 2 M ln ⁡ ( r 2 M − 1 ) ⇒ d t = d u + ( 1 − 2 M r ) − 1 d r ,

and Eq(1) could be transformed into the "retarded(/outgoing) Schwarzschild metric"

( 3 ) d s 2 = − ( 1 − 2 M r ) d u 2 − 2 d u d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) ;

or, we could instead employ the "advanced(/ingoing)" null coordinate v by

( 4 ) t = v − r − 2 M ln ⁡ ( r 2 M − 1 ) ⇒ d t = d v − ( 1 − 2 M r ) − 1 d r ,

so Eq(1) becomes the "advanced(/ingoing) Schwarzschild metric"

( 5 ) d s 2 = − ( 1 − 2 M r ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) .

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 M in Eqs(3) and Eq(5) from a constant to functions of the corresponding null coordinate, M ( u ) and M ( v ) respectively, thus

( 6 ) d s 2 = − ( 1 − 2 M ( u ) r ) d u 2 − 2 d u d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) ,

( 7 ) d s 2 = − ( 1 − 2 M ( v ) r ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) .

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

( 8 ) d s 2 = 2 M ( u ) r d u 2 + d s 2 ( flat ) = 2 M ( v ) r d v 2 + d s 2 ( flat ) ,

where d s 2 ( flat ) = − d u 2 − 2 d u d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) = − d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) = − d t 2 + d r 2 + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) represents the metric of flat spacetime.

Outgoing Vaidya with pure Emitting field

As for the "retarded(/outgoing)" Vaidya metric Eq(6), the Ricci tensor has only one nonzero component

( 9 ) R u u = − 2 M ( u ) , u r 2 ,

while the Ricci curvature scalar vanishes, R = g a b R a b = 0 . Thus, according to the trace-free Einstein equation G a b = R a b = 8 π T a b , the stress–energy tensor T a b satisfies

( 10 ) T a b = − M ( u ) , u 4 π r 2 l a l b , l a d x a = − d u ,

where l a = − ∂ a u and l a = g a b l b are null (co)vectors (c.f. Box A below). Thus, T a b is a "pure radiation field", which has an energy density of − M ( u ) , u 4 π r 2 . According to the null energy conditions

( 11 ) T a b k a k b ≥ 0 ,

we have M ( u ) , u < 0 and thus the central body is emitting radiations.

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

( 12 ) Ψ 2 = − M ( u ) r 3 Φ 22 = − M ( u ) , u r 2 .

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

( 13 ) θ ( ℓ ) = − ( ρ + ρ ¯ ) = 2 r , θ ( n ) = μ + μ ¯ = − r + 2 M ( u ) r 2 .

Ingoing Vaidya with pure absorbing field

As for the "advanced/ingoing" Vaidya metric Eq(7), the Ricci tensors again have one nonzero component

( 14 ) R v v = 2 M ( v ) , v r 2 ,

and therefore R = 0 and the stress–energy tensor is

( 15 ) T a b = M ( v ) , v 4 π r 2 n a n b , n a d x a = − d v .

This is a pure radiation field with energy density M ( v ) , v 4 π r 2 , and once again it follows from the null energy condition Eq(11) that M ( v ) , v > 0 , 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

( 16 ) Ψ 2 = − M ( v ) r 3 Φ 00 = M ( v ) , v r 2 .

Also, the outgoing and ingoing null expansion rates for the line element Eq(7) are respectively

( 17 ) θ ( ℓ ) = − ( ρ + ρ ¯ ) = r − 2 M ( v ) r 2 , θ ( n ) = μ + μ ¯ = − 2 r .

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 r = 2 M ( v ) is always a marginally outer trapped horizon ( θ ( ℓ ) = 0 , θ ( n ) < 0 ).

Comparison with the Schwarzschild metric

As a natural and simplest extension of the Schwazschild metric, the Vaidya metric still has a lot in common with it:

However, there are three clear differences between the Schwarzschild and Vaidya metric:

Kinnersley metric

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 x μ it takes the following form:

( 18 ) g μ ν = η μ ν − 2 m ( u ( x ) ) r ( x ) − 3 σ μ ( x ) σ ν ( x )

( 19 ) r ( x ) = σ μ ( x ) λ μ ( u ( x ) )

( 20 ) σ μ ( x ) = X μ ( u ( x ) ) − x μ , η μ ν σ μ ( x ) σ ν ( x ) = 0

where for the duration of this section all indices shall be raised and lowered using the "flat space" metric η μ ν , the "mass" m ( u ) is an arbitrary function of the proper-time u along the mass's world line as measured using the "flat" metric, d u 2 = η μ ν d X μ d X μ , and X μ ( u ) describes the arbitrary world line of the mass, λ μ ( u ) = d X μ ( u ) / d u is then the four-velocity of the mass, σ μ ( x ) is a "flat metric" null-vector field implicitly defined by Eqn. (20), and u ( x ) 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 X μ ( u ) , and satisfies the identity λ μ ( u ( x ) ) ∂ μ u ( x ) = 1. Grinding out the Einstein Tensor for the metric g μ ν and integrating the outgoing energy-momentum flux "at infinity," one finds that the metric g μ ν describes a mass with proper-time dependent four-momentum P μ = m ( u ) λ μ ( u ) that emits a net <<link:0>> at a proper rate of − d P μ / d u ; as viewed from the mass's instantaneous rest-frame, the radiation flux has an angular distribution A ( u ) + B ( u ) cos ⁡ ( θ ( u ) ) , where A ( u ) and B ( u ) are complicated scalar functions of m ( u ) , λ μ ( u ) , σ μ ( u ) , and their derivatives, and θ ( u ) 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 λ μ is independent of proper-time, the Kinnersley metric reduces to the Vaidya metric.

Vaidya-Bonner 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,

( 18 ) d s 2 = − ( 1 − 2 M ( u ) r + Q ( u ) r 2 ) d u 2 − 2 d u d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) ,

( 19 ) d s 2 = − ( 1 − 2 M ( v ) r + Q ( v ) r 2 ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 ⁡ θ d ϕ 2 ) .

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.