Analysis of the Fisher solution

We study the $d$-dimensional Fisher solution which represents a static, spherically symmetric, asymptotically flat spacetime with a massless scalar field. The solution has two parameters, the mass $M$ and the “scalar charge” $\Sigma$. The Fisher solution has a naked curvature singularity which divides the spacetime manifold into two disconnected parts. The part which is asymptotically flat we call the Fisher spacetime, and another part we call the Fisher universe. The $d$-dimensional Schwarzschild-Tangherlini solution and the Fisher solution belong to the same theory and are dual to each other. The duality transformation acting in the parameter space $(M,\Sigma )$ maps the exterior region of the Schwarzschild-Tangherlini black hole into the Fisher spacetime which has a naked timelike singularity, and interior region of the black hole into the Fisher universe, which is an anisotropic expanding-contracting universe and which has two spacelike singularities representing its “big bang” and “big crunch.” The big bang singularity and the singularity of the Fisher spacetime are radially weak in the sense that a 1-dimensional object moving along a timelike radial geodesic can arrive to the singularities intact. At the vicinity of the singularity the Fisher spacetime of nonzero mass has a region where its Misner-Sharp energy is negative. The Fisher universe has a marginally trapped surface corresponding to the state of its maximal expansion in the angular directions. These results and derived relations between geometric quantities of the Fisher spacetime, the Fisher universe, and the Schwarzschild-Tangherlini black hole may suggest that the massless scalar field transforms the black hole event horizon into the naked radially weak disjoint singularities of the Fisher spacetime and the Fisher universe which are “dual to the horizon.”

Figure 1

Duality diagram. Point $O$ represents the Fisher solution defined by the mass $M_o$ and the scalar charge ${\Sigma }_o$. Sector I represents its dual nonnegative mass solutions ($M_o^{\prime }\ge 0$). One such dual Fisher solution is defined by the mass $M_o^{\prime }$ and the scalar charge ${\Sigma }_o^{\prime }$. Sector II represents dual negative mass solutions ($M_o^{\prime }<0$) which we do not consider here.

Figure 2

(a): Maximal proper time ${\lambda }_{1,2}$ as a function of $S$ for the fixed value of the mass $\mu =1$ and $d=4$, 5, 6. The indices 1 and 2 correspond to ${\lambda }_1$ and ${\lambda }_2$, respectively. (b): Area ${\mathcal{A}}_{*}$ as a function of $S$ for $\mu =1$ and $d=4$, 5, 6. In any dimension, the minimal value of ${\mathcal{A}}_{*}$ corresponds to $S\approx 0.834$ and for $S\approx 0.611$ the value of ${\mathcal{A}}_{*}$ equals to the horizon surface area of the Schwarzschild-Tangherlini black hole of $\mu =M=1$ and $\Sigma =0$, ($S=1$).

Figure 3

Misner-Sharp energy $M(R)$ and the “local (Newtonian) gravitational potential energy” $U(R)$ for $M=1$, $S=1/2$, and $d=4$. The maximum $U_m=U(R_m)$ corresponds to $R_m=2·3^{3/4}$ and is equal to $1/8$, [46].

Figure 4

(a): Proper distance $L_e$ as a function of $S$ for the fixed value of the mass $M=1$ and $d=4$, 5, 6. (b): Area ${\mathcal{A}}_e$ as a function of $S$ for $M=1$. In any dimension, the minimal value of ${\mathcal{A}}_e$ corresponds to $S\approx 0.834$ and for $S\approx 0.611$ the value of ${\mathcal{A}}_e$ equals to the horizon surface area of the Schwarzschild-Tangherlini black hole of $M=1$ and $\Sigma =0$, ($S=1$).

Figure 5

Radial null geodesics in the Schwarzschild-Tangherlini spacetime of $M^{\prime }=2$, ${\Sigma }^{\prime }=0$, [see, (31)] and $d=4$. The behavior of the geodesics is generic for other values of $d>4$. The black hole event horizon is located at $R=r_o=4$. It separates the exterior $I$ and interior $II$ regions. The spacelike singularity is located at $R=r=0$. The direction of local time is illustrated by the future null cones.

Figure 6

Radial null geodesics in the Fisher spacetime of $M=1$, $S=1/2$, and $d=4$. The behavior of the geodesics is generic for other values of $d>4$ and $S\in [0,1)$. The areal radius corresponding to zero value of the Misner-Sharp energy is given by $R_e=3\sqrt{3}/2$. The timelike singularity is located at $R(r_o)=0$. The direction of local time is illustrated by the future null cones.

Figure 7

Radial null geodesics in the Fisher universe of $\mu =1$, $S=1/2$, and $d=4$. The behavior of the geodesics is generic for other values of $d>4$ and $S\in [0,1)$. The marginally trapped surface is located at ${\mathcal{R}}_{*}=3^{3/4}$. The spacelike singularities corresponding to $\rho ={\rho }_o$ and $\rho =0$ are located at $\mathcal{R}=0$. The direction of local time is illustrated by the future null cones.

Figure 8

Penrose diagram for the maximally extended Schwarzschild-Tangherlini spacetime. Each interior point in the diagram represents a ($d-2$)-dimensional sphere.

Figure 9

Penrose diagram for the Fisher spacetime. Each interior point in the diagram represents a ($d-2$)-dimensional sphere.

Figure 10

Penrose diagram for the Fisher universe. Each interior point in the diagram represents a ($d-2$)-dimensional sphere. The marginally trapped surface of the Fisher universe is schematically illustrated by the infinite line $\rho ={\rho }_{*}$.

Figure 11

Embedding diagrams for $d=4$. (a): Exterior region of the Schwarzschild-Tangherlini black hole of $M^{\prime }=2$ and ${\Sigma }^{\prime }=0$, [see, (31)]. The dashed circle of the radius $R(r_o)=r_o$ represents its event horizon. (b): Fisher spacetime corresponding to $M=1$ and $S=1/2$. The point $R(r_o)=0$ represents the naked timelike singularity. The dotted circle of the radius $R_e$ represents the region where the Misner-Sharp energy is zero. The dashed circle of the radius $R_m$ represents the region where the “local (Newtonian) gravitational potential energy” $U(R)$ is minimal (see Fig. 3). The diagrams are qualitatively generic for other values of $d>4$.

Figure 12

Embedding diagrams for $d=4$. (a): Interior region of the Schwarzschild-Tangherlini black hole of $M^{\prime }=2$ and ${\Sigma }^{\prime }=0$, [see, (31)]. The dashed circle $\mathcal{R}({\rho }_o)={\rho }_o=r_o$ represents its event horizon and the point $\mathcal{R}(0)=0$ represents its spacelike singularity. (b): Fisher universe corresponding to $\mu =1$ and $S=1/2$. The points $\mathcal{R}({\rho }_o)=0$ and $\mathcal{R}(0)=0$ represent its spacelike singularities corresponding to the universe’s big bang and big crunch, respectively. The dashed circle of the radius ${\mathcal{R}}_{*}$ represents the marginally trapped surface (85). For $S=0$ the diagram is symmetric with respect to the circle. The diagrams are qualitatively generic for other values of $d>4$.