Semianalytical approach for the Vaidya metric in double-null coordinates

We reexamine here a problem considered in detail before by Waugh and Lake: the solution of spherically symmetric Einstein’s equations with a radial flow of unpolarized radiation (the Vaidya metric) in double-null coordinates. This problem is known to be not analytically solvable; the only known explicit solutions correspond to the constant mass case (Schwarzschild solution in Kruskal-Szekeres form) and the linear and exponential mass functions originally discovered by Waugh and Lake. We present here a semianalytical approach that can be used to discuss some qualitative and quantitative aspects of the Vaidya metric in double-null coordinates for generic mass functions. We present also a new analytical solution corresponding to $(1/v)$-mass function and discuss some physical examples.

Figure 1

Constant $r/\kappa$ curves for the case $A(v)={\kappa }^2/v$. The values of $r/\kappa$ shown are, from bottom to top, 0, 0.5, 0.6, 1, 1.5, 2, 2.5 and 3. Future is to right and up. The corresponding conformal diagram is inserted. Here, $r=0$ is a naked singularity.

Figure 2

In the region below the generic monotonic function $r=2A(v)$ (the apparent horizon), all solutions have $r^{\prime }<0$. Any solution that enters into this region will reach the singularity at $r=0$ with finite $v$. On the other hand, solutions confined to the $r^{\prime }>0$ region always escape from the singularity and reach ${\mathcal{I}}^{+}$. In this case, supposing that the mass function $A(v)$ has an asymptotic value such that $r_{\mathrm{i}}(v)<2A(v)<r_{\mathrm{i}}(v)$, there exists an event horizon (the dashed line on the conformal diagram) somewhere between the solutions $r_{\mathrm{i}}(v)$ and $r_{\mathrm{e}}(v)$. The vector $T={\partial }_v-{\partial }_u$ points to the future for the ingoing radiation case.

Figure 3

For $\lambda >1/16$, all solutions are concave for $v>0$. Hence, all null trajectories along $v$ necessarily reach the singularity at $r=0$. The figure corresponds to the case $\lambda =1/10$. The conformal diagram is inserted.

Figure 4

For $\lambda =1/16$, we have two relevant lines in the $(r,v)$ plane: $r=v/4$ and $r=v/8$. The first one plays the role of an event horizon, separating the two regions of distinct qualitative behavior for the null trajectories along $v$. Beyond the horizon, all solutions escape to infinity. Inside, all solutions reach the singularity $r=0$ with finite $v$. The conformal diagram is inserted. The singularity at $r=0$ is a shell-focusing one.

Figure 5

With $\lambda <1/16$, we have two horizons $r_{-}$ and $r_{+}$. Any solution starting above $r_{+}$ always escapes to ${\mathcal{I}}_{+}$. Solutions starting inside the inner event horizon $r_i$ always reach the singularity at $r=0$. Between the horizons, solutions start in the naked singularity but escape to ${\mathcal{I}}_{+}$. The figure corresponds to the case $\lambda =1/20$. The conformal diagram is inserted.

Figure 6

The case $A(v)=m(1+\tanh \rho v)/2$. The solutions tend asymptotically to $r=0$ when $v\to -\infty$. A zoom of the range $-3<v<-2$ illustrating such asymptotic behavior is inserted. In this case, $r=0$ is always singular for $v>-\infty$. This plot corresponds to $m\rho =1$; the situation is qualitatively identical for other cases. The conformal diagram is inserted.

Figure 7

Conformal diagram for an evaporating black hole with mass $A(u)$ given by (21). The region I corresponds to the black hole, and can be obtained, by using the transformations (19), from the ingoing radiation case. Region II is the empty space left after the vanishing of the black hole at $u=0$. The event horizon is the line EH. The dashed CH line is a Cauchy horizon. Beyond it, we have breakdown of predictability due to the naked singularity left by the black hole.