$N$-dimensional Vaidya metric with a cosmological constant in double-null coordinates

A recently proposed approach to the construction of the Vaidya metric in double-null coordinates for generic mass functions is extended to the $n$-dimensional ($n>2$) case and to allow the inclusion of a cosmological constant. The approach is based on a qualitative study of the null geodesics, allowing the description of light cones and revealing many features of the underlying causal structure. Possible applications are illustrated by explicit examples. Some new exact solutions are also presented and discussed. The results presented here can simplify considerably the study of spherically symmetric gravitational collapse and mass accretion in arbitrary dimensions.

Figure 1

The solutions $r(u,v)$ of (15) for the Schwarzschild case [$A(v)=m$; see 24]. The dotted lines correspond to the lines of constant $r$. In region I (the interior region), the values of $r$ decrease monotonically upward, varying from $r=2m$ (the horizon corresponding to the degenerated hyperbole $u=u_0$ and $v=v_0$) to the singularity $r=0$. In the exterior region (II), on the other hand, the values of $r$ increase monotonically leftward, varying from $r=2m$ to the spatial infinity $r=\infty$. Some slices of constant $u=\overline{u}$ are depicted by the lines with arrows. The values of $r(\overline{u},v)$ read along such lines correspond to the solutions of (15) with a certain initial condition $r(0)=r_{\overline{u}}$. Since the lines of constant $u$ are portions of the light cone, the function $r(\overline{u},v)$ describes how the light cones cross the $(n-2)$-dimensional spheres of radius $r$. In the interior region, for instance, all null geodesics eventually reach the singularity, whereas in the exterior the null geodesics with constant $u$ escape to the null infinity. Between such regions, there is an event horizon at $u=u_0$. We notice that all this information is coded in the qualitative behavior of the solutions $r(\overline{u},v)$ for the first order ordinary differential equation corresponding to (15) considered along constant $u=\overline{u}$. This is the basic idea of our semianalytical approach.

Figure 2

In the region below the curve $r_{\mathrm{EH}}(v)$ (the apparent event horizon), $r_2<0$. Any solution $r(v)$ of (15) that enters into this region will reach the singularity at $r=0$ with finite $v$. Solutions confined to the $r_2>0$ region always escape from the singularity and tend asymptotically to $r_{\mathrm{CH}}(v)$ (the apparent cosmological horizon). The region above $r_{\mathrm{CH}}(v)$ corresponds to the cosmological exterior; any solution there also tends asymptotically to $r_{\mathrm{CH}}(v)$. The dashed line corresponds to the apparent event horizon for the case of a vanishing $\Lambda$ (the unique solution of (34) for $\Lambda =0$). The conformal diagram is inserted. [The case depicted here corresponds to $n=5$, $\Lambda =1$, and $A(v)=(6/5)\tanh v$, $v\ge 0$. In this work, we do not consider maximal extensions. For a discussion of the maximal conformal diagram for de Sitter and anti–de Sitter black holes, see 39, for instance.]

Figure 3

Conformal diagrams corresponding to the mass function (20) and $\Lambda =0$. Cases (b) and (c) exhibit naked and shell-focusing singularities. For all cases, the curves in $(v,r)$ are similar to the ones presented in 26, corresponding to the $n=4$ case.

Figure 4

The $(u,r)$ plane for an evaporating black hole in an $n$-dimensional de Sitter spacetime. The apparent horizons $r_{\mathrm{EH}}(u)$ and $r_{\mathrm{CH}}(u)$ delimit the regions where the solutions of (42) have qualitatively distinct behavior. Since the black hole vanishes for $u=u_0$, for $u>u_0$ the only remaining horizon corresponds to $r_{\mathrm{CH}}(u)$. The conformal diagram is inserted. For $u<u_0$, it represents a usual black hole. For $u>u_0$, we have the empty space left after the vanishing of the black hole. The dashed line HC (extension of the black-hole horizon EH) is a Cauchy horizon. Beyond it, we have a breakdown of predictability due to the naked singularity left by the black hole. Part of the cosmological exterior region (the region above CH and below HC) is completely insensitive to the singularity. (The case depicted here corresponds to $n=6$, $\Lambda =5/2$, and $m_0=u_0=1$, $u\ge 0$.)

Figure 5

The $(v,r)$ plane for a BTZ black hole accreting mass according to (43). In the region below the curve $r_{\mathrm{EH}}(v)$ (the apparent event horizon), $r_2<0$. The constant $u=\overline{u}$ null geodesics associated with the solutions $r(\overline{u},v)$ of (27) entering into this region reach the conical singularity at $r=0$ with finite $v$. The null geodesics confined to the $r_2>0$ region can escape from the singularity, reaching the null and spatial infinity. The conformal diagram is inserted. (The case depicted here corresponds to $\Lambda =-1$, $B=-1/2$, and $m_i=m_f/9=1$.)