Vaidya metric

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

$${\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)dt^{2}+\left(1-{\frac {2M}{r}}\right)^{-1}dr^{2}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right).}$$

(1)

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

$${\displaystyle t=u+r+2M\ln \left({\frac {r}{2M}}-1\right)\qquad \Rightarrow \quad dt=du+\left(1-{\frac {2M}{r}}\right)^{-1}dr\;,}$$

(2)

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

$${\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)du^{2}-2dudr+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right);}$$

(3)

or, we could instead employ the "advanced(/ingoing)" null coordinate ${\displaystyle v}$ by

$${\displaystyle t=v-r-2M\ln \left({\frac {r}{2M}}-1\right)\qquad \Rightarrow \quad dt=dv-\left(1-{\frac {2M}{r}}\right)^{-1}dr\;,}$$

(4)

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

$${\displaystyle ds^{2}=-\left(1-{\frac {2M}{r}}\right)dv^{2}+2dvdr+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right).}$$

(5)

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

$${\displaystyle ds^{2}=-\left(1-{\frac {2M(u)}{r}}\right)du^{2}-2dudr+r^{2}\left(d\theta ^{2}+\ sin^{2}\theta \,d\phi ^{2}\right),}$$

(6)

$${\displaystyle ds^{2}=-\left(1-{\frac {2M(v)}{r}}\right)dv^{2}+2dvdr+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right).}$$

(7)

The extended metrics Eq(6) and Eq(7) are respectively the "retarded(/outgoing)" and "advanced(/ingoing)" Vaidya metrics.[1][2] It is also sometimes useful to recast the Vaidya metrics Eqs(6)(7) into the form

$${\displaystyle ds^{2}={\frac {2M(u)}{r}}du^{2}+ds^{2}({\text{flat}})={\frac {2M(v)}{r}}dv^{2}+ds^{2}({\text{flat}})\,,}$$

(8)

where

$${\displaystyle {\begin{aligned}ds^{2}({\text{flat}})&=-du^{2}-2dudr+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)\\&=-dv^{2}+2dvdr+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)\\&=-dt^{2}+dr^{2}+r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)\end{aligned}}}$$

represents the metric of flat spacetime.

As for the "retarded(/outgoing)" Vaidya metric Eq(6),[1][2][3][4][5] the Ricci tensor has only one nonzero component

$${\displaystyle R_{uu}=-2{\frac {M(u)_{,\,u}}{r^{2}}}\,,}$$

(9)

while the Ricci curvature scalar vanishes, ${\displaystyle R=g^{ab}R_{ab}=0}$ because ${\displaystyle g^{uu}=0}$. Thus, according to the trace-free Einstein equation ${\displaystyle G_{ab}=R_{ab}=8\pi T_{ab}}$, the stress–energy tensor ${\displaystyle T_{ab}}$ satisfies

$${\displaystyle T_{ab}=-{\frac {M(u)_{,\,u}}{4\pi r^{2}}}l_{a}l_{b}\;,\qquad l_{a}dx^{a}=-du\;,}$$

(10)

where ${\displaystyle l_{a}=-\partial _{a}u}$ and ${\displaystyle l^{a}=g^{ab}l_{b}}$ are null (co)vectors (c.f. Box A below). Thus, ${\displaystyle T_{ab}}$ is a "pure radiation field",[1][2] which has an energy density of ${\textstyle -{\frac {M(u)_{,\,u}}{4\pi r^{2}}}}$. According to the null energy conditions

$${\displaystyle T_{ab}k^{a}k^{b}\geq 0\;,}$$

(11)

we have ${\displaystyle M(u)_{,\,u}<0}$ and thus the central body is emitting radiation.

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

$${\displaystyle \Psi _{2}=-{\frac {M(u)}{r^{3}}}\qquad \Phi _{22}=-{\frac {M(u)_{\,,\,u}}{r^{2}}}\;.}$$

(12)

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

$${\displaystyle \theta _{(\ell )}=-(\rho +{\bar {\rho }})={\frac {2}{r}}\,,\quad \theta _{(n)}=\mu +{\bar {\mu }}={\frac {-r+2M(u)}{r^{2}}}\;.}$$

(13)

Suppose ${\textstyle F:=1-{\frac {2M(u)}{r}}}$, then the Lagrangian for null radial geodesics ${\displaystyle (L=0,{\dot {\theta }}=0,{\dot {\phi }}=0)}$ of the "retarded(/outgoing)" Vaidya spacetime Eq(6) is

$${\displaystyle L=0=-F{\dot {u}}^{2}+2{\dot {u}}{\dot {r}}\,,}$$

where dot means derivative with respect to some parameter ${\displaystyle \lambda }$. This Lagrangian has two solutions,

$${\displaystyle {\dot {u}}=0\quad {\text{and}}\quad {\dot {r}}={\frac {F}{2}}{\dot {u}}\;.}$$

According to the definition of ${\displaystyle u}$ in Eq(2), one could find that when ${\displaystyle t}$ increases, the areal radius ${\displaystyle r}$ would increase as well for the solution ${\displaystyle {\dot {u}}=0}$, while ${\displaystyle r}$ would decrease for the solution ${\textstyle {\dot {r}}={\frac {F}{2}}{\dot {u}}}$. Thus, ${\displaystyle {\dot {u}}=0}$ should be recognized as an outgoing solution while ${\textstyle {\dot {r}}={\frac {F}{2}}{\dot {u}}}$ 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

$${\displaystyle l^{a}=(0,1,0,0)\,,\quad n^{a}=\left(1,-{\frac {F}{2}},0,0\right)\,,\quad m^{a}={\frac {1}{{\sqrt {2}}\,r}}(0,0,1,i\,\csc \theta )\,,}$$

and the dual basis covectors are therefore

$${\displaystyle l_{a}=(-1,0,0,0)\,,\quad n_{a}=\left(-{\frac {F}{2}},-1,0,0\right)\,,\quad m_{a}={\frac {r}{\sqrt {2}}}(0,0,1,\sin \theta )\,.}$$

In this null tetrad, the spin coefficients are

$${\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \varepsilon =0}$$

$${\displaystyle \rho =-{\frac {1}{r}}\,,\quad \mu ={\frac {-r+2M(u)}{2r^{2}}}\,,\quad \alpha =-\beta ={\frac {-{\sqrt {2}}\cot \theta }{4r}}\,,\quad \gamma ={\frac {M(u)}{2r^{2}}}\,.}$$

The Weyl-NP and Ricci-NP scalars are given by

$${\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0\,,\quad \Psi _{2}=-{\frac {M(u)}{r^{3}}}\,,}$$

$${\displaystyle \Phi _{00}=\Phi _{10}=\Phi _{20}=\Phi _{11}=\Phi _{12}=\Lambda =0\,,\quad \Phi _{22}=-{\frac {M(u)_{\,,\,u}}{r^{2}}}\,,}$$

Since the only nonvanishing Weyl-NP scalar is ${\displaystyle \Psi _{2}}$, the "retarded(/outgoing)" Vaidya spacetime is of Petrov-type D. Also, there exists a radiation field as ${\displaystyle \Phi _{22}\neq 0}$.

For the "retarded(/outgoing)" Schwarzschild metric Eq(3), let ${\textstyle G:=1-{\frac {2M}{r}}}$, and then the Lagrangian for null radial geodesics will have an outgoing solution ${\displaystyle {\dot {u}}=0}$ and an ingoing solution ${\textstyle {\dot {r}}=-{\frac {G}{2}}{\dot {u}}}$. Similar to Box A, now set up the adapted outgoing tetrad by

$${\displaystyle l^{a}=(0,1,0,0)\,,\quad n^{a}=\left(1,-{\frac {G}{2}},0,0\right)\,,\quad m^{a}={\frac {1}{{\sqrt {2}}\,r}}(0,0,1,i\,\csc \theta )\,,}$$

$${\displaystyle l_{a}=(-1,0,0,0)\,,\quad n_{a}=\left(-{\frac {G}{2}},-1,0,0\right)\,,\quad m_{a}={\frac {r}{\sqrt {2}}}(0,0,1,\sin \theta )\,.}$$

so the spin coefficients are

$${\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \varepsilon =0}$$

$${\displaystyle \rho =-{\frac {1}{r}}\,,\quad \mu ={\frac {-r+2M}{2r^{2}}}\,,\quad \alpha =-\beta ={\frac {-{\sqrt {2}}\cot \theta }{4r}}\,,\quad \gamma ={\frac {M}{2r^{2}}}\,,}$$

and the Weyl-NP and Ricci-NP scalars are given by

$${\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0\,,\quad \Psi _{2}=-{\frac {M}{r^{3}}}\,,}$$

$${\displaystyle \Phi _{00}=\Phi _{10}=\Phi _{20}=\Phi _{11}=\Phi _{12}=\Phi _{22}=\Lambda =0\,.}$$

The "retarded(/outgoing)" Schwarzschild spacetime is of Petrov-type D with ${\displaystyle \Psi _{2}}$ being the only nonvanishing Weyl-NP scalar.

As for the "advanced/ingoing" Vaidya metric Eq(7),[1][2][6] the Ricci tensors again have one nonzero component

$${\displaystyle R_{vv}=2{\frac {M(v)_{,\,v}}{r^{2}}}\,,}$$

(14)

and therefore ${\displaystyle R=0}$ and the stress–energy tensor is

$${\displaystyle T_{ab}={\frac {M(v)_{,\,v}}{4\pi r^{2}}}\,n_{a}n_{b}\;,\qquad n_{a}dx^{a}=-dv\;.}$$

(15)

This is a pure radiation field with energy density ${\textstyle {\frac {M(v)_{,\,v}}{4\pi r^{2}}}}$, and once again it follows from the null energy condition Eq(11) that ${\displaystyle M(v)_{,\,v}>0}$, 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

$${\displaystyle \Psi _{2}=-{\frac {M(v)}{r^{3}}}\qquad \Phi _{00}={\frac {M(v)_{\,,\,v}}{r^{2}}}\;.}$$

(16)

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

$${\displaystyle \theta _{(\ell )}=-(\rho +{\bar {\rho }})={\frac {r-2M(v)}{r^{2}}}\,,\quad \theta _{(n)}=\mu +{\bar {\mu }}=-{\frac {2}{r}}\;.}$$

(17)

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 ${\displaystyle r=2M(v)}$ is always a marginally outer trapped horizon (${\displaystyle \theta _{(\ell )}=0\;,\theta _{(n)}<0}$).

Suppose ${\displaystyle {\tilde {F}}:=1-{\frac {2M(v)}{r}}}$, then the Lagrangian for null radial geodesics of the "advanced(/ingoing)" Vaidya spacetime Eq(7) is

$${\displaystyle L=-{\tilde {F}}{\dot {v}}^{2}+2{\dot {v}}{\dot {r}}\,,}$$

which has an ingoing solution ${\displaystyle {\dot {v}}=0}$ and an outgoing solution ${\textstyle {\dot {r}}={\frac {\tilde {F}}{2}}{\dot {v}}}$ in accordance with the definition of ${\displaystyle v}$ 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

$${\displaystyle l^{a}=\left(1,{\frac {\tilde {F}}{2}},0,0\right)\,,\quad n^{a}=(0,-1,0,0)\,,\quad m^{a}={\frac {1}{{\sqrt {2}}\,r}}(0,0,1,i\,\csc \theta )\,,}$$

and the dual basis covectors are therefore

$${\displaystyle l_{a}=\left(-{\frac {\tilde {F}}{2}},1,0,0\right)\,,\quad n_{a}=(-1,0,0,0)\,,\quad m_{a}={\frac {r}{\sqrt {2}}}(0,0,1,\sin \theta )\,.}$$

In this null tetrad, the spin coefficients are

$${\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \gamma =0}$$

$${\displaystyle \rho ={\frac {-r+2M(v)}{2r^{2}}}\,,\quad \mu =-{\frac {1}{r}}\,,\quad \alpha =-\beta ={\frac {-{\sqrt {2}}\cot \theta }{4r}}\,,\quad \varepsilon ={\frac {M(v)}{2r^{2}}}\,.}$$

The Weyl-NP and Ricci-NP scalars are given by

$${\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0\,,\quad \Psi _{2}=-{\frac {M(v)}{r^{3}}}\,,}$$

$${\displaystyle \Phi _{10}=\Phi _{20}=\Phi _{11}=\Phi _{12}=\Phi _{22}=\Lambda =0\,,\quad \Phi _{00}={\frac {M(v)_{\,,\,v}}{r^{2}}}\;.}$$

Since the only nonvanishing Weyl-NP scalar is ${\displaystyle \Psi _{2}}$, the "advanced(/ingoing)" Vaidya spacetime is of Petrov-type D, and there exists a radiation field encoded into ${\displaystyle \Phi _{00}}$.

For the "advanced(/ingoing)" Schwarzschild metric Eq(5), still let ${\textstyle G:=1-{\frac {2M}{r}}}$, and then the Lagrangian for the null radial geodesics will have an ingoing solution ${\displaystyle {\dot {v}}=0}$ and an outgoing solution ${\textstyle {\dot {r}}={\frac {G}{2}}{\dot {v}}}$. Similar to Box C, now set up the adapted ingoing tetrad by

$${\displaystyle l^{a}=\left(1,{\frac {G}{2}},0,0\right)\,,\quad n^{a}=(0,-1,0,0)\,,\quad m^{a}={\frac {1}{{\sqrt {2}}\,r}}(0,0,1,i\,\csc \theta )\,,}$$

$${\displaystyle l_{a}=\left(-{\frac {G}{2}},1,0,0\right)\,,\quad n_{a}=(-1,0,0,0)\,,\quad m_{a}={\frac {r}{\sqrt {2}}}(0,0,1,\sin \theta )\,.}$$

so the spin coefficients are

$${\displaystyle \kappa =\sigma =\tau =0\,,\quad \nu =\lambda =\pi =0\,,\quad \gamma =0}$$

$${\displaystyle \rho ={\frac {-r+2M}{2r^{2}}}\,,\quad \mu =-{\frac {1}{r}}\,,\quad \alpha =-\beta ={\frac {-{\sqrt {2}}\cot \theta }{4r}}\,,\quad \varepsilon ={\frac {M}{2r^{2}}}\,,}$$

and the Weyl-NP and Ricci-NP scalars are given by

$${\displaystyle \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0\,,\quad \Psi _{2}=-{\frac {M}{r^{3}}}\,,}$$

$${\displaystyle \Phi _{00}=\Phi _{10}=\Phi _{20}=\Phi _{11}=\Phi _{12}=\Phi _{22}=\Lambda =0\,.}$$

The "advanced(/ingoing)" Schwarzschild spacetime is of Petrov-type D with ${\displaystyle \Psi _{2}}$ being the only nonvanishing Weyl-NP scalar.

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

• Both metrics are of Petrov-type D with ${\displaystyle \Psi _{2}}$ being the only nonvanishing Weyl-NP scalar (as calculated in Boxes A and B).

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

• First of all, the mass parameter ${\displaystyle M}$ for Schwarzschild is a constant, while for Vaidya ${\displaystyle M(u)}$ is a u-dependent function.
• Schwarzschild is a solution to the vacuum Einstein equation ${\displaystyle R_{ab}=0}$, while Vaidya is a solution to the trace-free Einstein equation ${\displaystyle R_{ab}=8\pi T_{ab}}$ with a nontrivial pure radiation energy field. As a result, all Ricci-NP scalars for Schwarzschild are vanishing, while we have ${\displaystyle \Phi _{00}={\frac {M(u)_{\,,\,u}}{r^{2}}}}$ 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.

• Schwarzschild metric
• Null dust solution