Black hole in the expanding universe with arbitrary power-law expansion

We present a time-dependent and spatially inhomogeneous solution that interpolates the extremal Reissner-Nordström (RN) black hole and the Friedmann-Lemaître-Robertson-Walker (FLRW) universe with arbitrary power-law expansion. It is an exact solution of the $D$-dimensional Einstein-Maxwell-dilaton system, where two Abelian gauge fields couple to the dilaton with different coupling constants, and the dilaton field has a Liouville-type exponential potential. It is shown that the system satisfies the weak energy condition. The solution involves two harmonic functions on a ($D-1$)-dimensional Ricci-flat base space. In the case where the harmonics have a single-point source on the Euclidean space, we find that the spacetime describes a spherically symmetric charged black hole in the FLRW universe, which is characterized by three parameters: the steepness parameter of the dilaton potential $n_T$, the U(1) charge $Q$, and the nonextremality $\tau$. In contrast with the extremal RN solution, the spacetime admits a nondegenerate Killing horizon unless these parameters are finely tuned. The global spacetime structures are discussed in detail.

Figure 1

Conformal diagrams of a flat FLRW universe $a=(\overline{t}/{\overline{t}}_0{)}^p$ for (1) $0<p<1/2$, (2) $p=1/2$, (3) $1/2<p<1$, (4) $p=1$, and (5) $p>1$. The dotted and dotted-dashed lines denote the trapping horizon, $r_{\mathrm{TH}}(\overline{t})=(da/d\overline{t}{)}^{-1}$, and the big bang singularity at $a=0$, respectively. The cases (2) and (4) correspond, respectively, to the radiation-dominant universe $P=\rho /(D-1)$ and the marginally accelerating universe driven by the curvature term $\rho \propto a^{-2}$. The cosmological horizon, $r_{\mathrm{CH}}(\overline{t})=(p-1{)}^{-1}{\overline{t}}_0(\overline{t}/{\overline{t}}_0{)}^{1-p}$, is abbreviated to CH, which exists only in the strictly accelerating case ($p>1$). The shaded regions corresponding to $r\to \infty$ approximate our original spacetime.

Figure 2

Typical throat geometries for the ${\mathrm{AdS}}_2\times S^{D-2}$ spacetime (left) and the extremal RN spacetime (right). The double line stands for the degenerate horizon. Only the vicinity of the shaded point at $r=0$ with $t$ kept finite approximates our original solution. Whereas, the domains bounded by (blue) shaded lines in each figure are locally isometric [5].

Figure 3

Typical plots of circumference radius $\tilde{R}$ (vertical axis) against $\tilde{r}$ (horizontal axis) for case I-(i) ($n_T=1$ with $\tau =1$), case II-(i) ($n_T=2$ with $\tau =3/2$), case II-(ii) ($n_T=2$ with $\tau =1$), case II-(iii) ($n_T=2$ with $\tau =1/2$), case III-(i) ($n_T=3$ with $\tau =3$), case III-(ii) ($n_T=3$ with $\tau ={\tau }_{\mathrm{cr}}$), and case III-(iii) ($n_T=3$ with $\tau =1$). The curve of case I-(i) is the representative for case I. The thick lines denote trapping horizons, across which the $\tilde{R}=\mathrm{\text{constant}}$ surface changes the signature. The region “spacelike” (and, respectively, “timelike”) stands for the domain where the $\tilde{R}=\mathrm{\text{constant}}$ surfaces are spacelike (timelike). In case III-(i), the bounded region tends to shrink as $\tau$ decreases and eventually degenerate into a point shown in case III-(ii).

Figure 4

Plots of $F_{+}$ and $F_{-}$ for case I (left), case II (middle), and case III (right) (see Table  for classification). The equality $f(\tilde{R})=0$ then possesses two distinct solutions for case I and case II-(i), a single solution for cases II-(ii), II-(iii), and III-(iii), three distinct solutions for case III-(i), and two solutions with the larger one being degenerate for case III-(ii).

Figure 5

Conformal diagrams of the maximally extended near-horizon geometries. The global structures are the same as those for (A) the nonextremal RN-AdS, (B) the zero-mass RN spacetime with a “hyperbolic horizon,” (C) the negative mass SdS (or the overcharged RNdS), (D) the nonextremal RNdS and (E) the extremal RNdS. The thick dotted-dashed lines denote the central singularity at $\tilde{t}=-1/r$. The shaded regions approximate our original solution (4.1).

Figure 6

Behaviors of radial null geodesics for the accelerating universe ($n_T=3$ with $\tau =3$) in the outside region (left) and the inside region (right). The figures show the circumference radius versus the affine parameter. The red, blue, and purple lines stand for ${\tilde{R}}_{-}\sim 0.82$, ${\tilde{R}}_{+}\sim 1.56$, and ${\tilde{R}}_c\sim 4.03$, respectively. The diagrams in the top, upper middle, lower middle, bottom rows correspond to future-directed ingoing null geodesics, future-directed outgoing null geodesics, past-directed ingoing null geodesics, and past-directed outgoing null geodesics. We have shown the null geodesics for the initial values $\tilde{r}(0)=1$ (left) and $\tilde{r}(0)=-1/10$ (right).

Figure 7

Conformal diagrams of the present spacetime (2.12) for cases I–III. The dotted (green) curves stand for the $R=\mathrm{\text{constant}}$ surfaces. The thick (red) curves represent the trapping horizons (TH) and (blue) lines denote the cosmological horizons (CH). The dotted-dashed (gray and black) curves correspond to the singularities of $r=-Q$ and $t=t_s(r)$, respectively. Panel (a), where the universe is decelerating, has been argued in the previous paper [11]. For panel (b), the universe undergoes a marginal acceleration $a\propto \overline{t}$. It admits an internal null infinity ${\mathcal{I}}_{\mathrm{in}}^{+}$ at which $r=0$ and $t\to \infty$ with $R\to \infty$. Panels (c)–(e) correspond to the accelerating universe. Panel (d) shows that there appear three horizons: the black hole horizon $R_{+}$, the “white hole horizon” $R_{-}$, and the cosmological horizon denoted by CH (not $R_c$). In Panel (e), the horizon is degenerate, while in panel (c) the spacetime is overcharged to give a globally naked singularity. Panels (a), (d), and (e) are extensible across the surface $R_{-}$ with a contracting universe in a continuous but nonanalytic manner.

Figure 8

Relation between $\tau$ and the ratio of two energy densities at the horizon in the double logarithmic plot. This illustrates the almost linear relationship between ${\tau }^2$ and ${\rho }^{(\mathrm{em})}/{\rho }^{(\Phi )}{|}_H$.