Thin accretion discs around spherically symmetric configurations with nonlinear scalar fields

We study stable circular orbits (SCO) around static spherically symmetric configuration of general relativity with a nonlinear scalar field (SF). The configurations are described by solutions of the Einstein-SF equations with monomial SF potential $V(\varphi )=|\varphi {|}^{2n}$, $n>2$, under the conditions of the asymptotic flatness and behavior of SF $\varphi \sim 1/r$ at spatial infinity. We proved that under these conditions, the solution exists and is uniquely defined by the configuration mass $M>0$ and scalar “charge” $Q$. The solutions and the space-time geodesics have been investigated numerically in the range $n\le 40$, $|Q|\le 60$, $M\le 60$. We focus on how nonlinearity of the field affects properties of SCO distributions (SCOD), which, in turn, affect topological form of the thin accretion disk around the configuration. Maps are presented showing the location of possible SCOD types for different $M$, $Q$, $n$. We found a lot of differences from the Fisher-Janis-Newman-Winicour metric (FJNW) dealing with the linear SF, though basic qualitative properties of the configurations have much in common with the FJNW case. For some values of $n$, a topologically new SCOD type was discovered that is not available for the FJNW metric. Like FJNW, all images of accretion disks have a dark spot in the center (mimicking the ordinary black hole), either because there is no SCO near the center or due to the strong deflection of photons near the singularity, although fine features other than a black hole can appear with a special choice of $M$, $Q$.

Figure 1

Contours of the equal $\eta$ as function of $M$, $Q$ for $n=3$ (left panel) and $n=14$ (right panel). The solid white line corresponds to critical value $\eta =3$. If $Q\to 0$, then $\eta \to \infty$.

Figure 2

Typical dependencies $\tilde{L}(r)$ and $U_{\mathrm{eff}}(r)$ in the case of $S3$ type; $\eta <3$ and $U_{\mathrm{eff}}(r)$ is infinite for $r\to 0$. The left panel shows that there can exist several circular orbits with the same $\tilde{L}(r)=L$. Roots of this equation correspond to SCO if they belong to the intervals where ${\tilde{L}}^{\prime }(r)>0$. The right panel shows effective potential with three minima ($S3$ type). Three different SCO rings can exist near $L\sim 6.3$ and about fixed $M$, $Q$, $n$ indicated in the figure.

Figure 3

Boundary radii of SCO regions as a function of $Q$ for some fixed $M$, $n$. Vertical dashed lines separate areas of different SCOD types. In the left panel, we have the $S1$ area between two branches of the bifurcation curve; here, arbitrary SCO radii are allowed. The other allowed region of radii are according to definition of $U1$ and $S2$. The right panel shows the example with larger $M$, $Q$, $n$, when another type is present, namely $S3$; corresponding parameters fall into the yellow region on Figs. 8, 9, 10, 11.

Figure 4

Boundary radii $r_b$ of SCO regions as a functions of $Q$ for $M=1$ and several fixed values of $n$. The continuous blue line on the left panel represents the FJNW solution and the orange straight line corresponds to the Schwarzschild BH case ($r_{\mathrm{ISCO}}=6M$). The first reshaping occurs at $n_1^{*}\approx 2.38$; then the right branch moves to the right and then starts backtracking to the left branch when $n>n_2^{*}\approx 7.53$. We found $n_3^{*}\approx 13.12$ and $n_4^{*}\approx 13.13$. For $n\ge n_5^{*}\approx 13.15$, the form of the branches is represented by solid ($n=15$) and dotted ($n=23$) lines on the right panel; here, we also have the $S3$ type (four-valued function $r_b(Q)$ on some interval of $Q$).

Figure 5

Boundary radii $r_b$ of SCO regions as a functions of $M$ for $Q=1$ and several fixed values of $n$. The first reconnection of two branches occurs at $n\approx 2.62$; dash-dotted ($n=2.4$) and dotted ($n=2.5$) lines show the typical form of branches before the reshaping (left panel), and the lines with $n=2.625$ and $n=3$ show the typical forms after the reshaping. The right panel demonstrates the branches around the second reconnection near $n\approx 13.86$.

Figure 6

Domains of parameters on $(M,Q)$ plane for different values of $n$. Here and on figures below, white color denotes the $S2$ type, yellow-$S3$, light gray-$U1$, dark gray $S1$. For values of $n\sim 7$, the $S1$ area grows.

Figure 7

For larger $n\gtrsim 7$, the black region ($S1$) shrinks to the origin and $S2$ dominates.

Figure 8

Domains of parameters on the $(n,M)$ plane for different values of $Q$.

Figure 9

The same as on the previous figure for larger $Q$.

Figure 10

Domains of parameters on $(n,Q)$ plane for different values of $M$.

Figure 11

The same as on the previous figure for larger $M$. For arbitrary $M$, there is a domain on $(n,Q)$-plane where $S3$ type exists.

Figure 12

The photon orbits radii as a function of scalar charge $Q$ (left) and configuration mass $M$ (right) for different values of $n$. Blue and orange curves correspond to the cases of FJNW and Schwarzschild metrics. Curves $r_{\mathrm{ph}}$ go below FJNW and tend to them with increasing $n$.

Figure 13

In both panels $n=3$, $M=1$, the red line indicates the equatorial plane. Left: trajectories of photons incident from infinity to the attracting singularity; $Q=0.3$, $\eta =4.89$. Right: the same with the repulsing singularity $Q=2.2$, $\eta =2.33$; in this case, the points near the singularity form a dark spot around the center, imitating a black hole.

Figure 14

AD images for the $U1$ type of SCOD ($M=1$, $Q=0.3$, $n=3$): in full face and for inclination $i=30°$. The white contour corresponds to the photon orbit at $r_{\mathrm{ph}}\approx 2.58$. The ISCO placed at $r_{1(U)}\approx 5.56$ and the outer disk edge was fixed at $r=24$.

Figure 15

The same as on Fig. 14 with inclinations 30° and 60°.

Figure 16

AD images for the $S1$ type ($M=1$, $Q=0.8$, $n=3$); in full face and for inclination $i=30°$. The outer disk edge radius $r=24$. There is a dark spot in the center due to the repulsive character of the naked singularity.

Figure 17

The same as on Fig. 16 with inclinations 60° and 80°.

Figure 18

AD images for the $S2$ type ($M=1$, $Q=2.2$, $n=3$) in full face and for inclination $i=30°$. The outer disk edge radius is $r=24$. SCO radii are in intervals (i) $r\in (0,6.5)$ and (ii) $r\in (14.5,\infty )$. However, the inner SCO region (i) cannot be observed in the left panel due to the repulsive nature of the naked ${\text{singularity}}^{\mathrm{a}}$, but the inner ring in the left panel represents the secondary image of the outer part (ii) of the AD. The region (i) appears as the crescentlike feature on the right panel. ${}^{\mathrm{a}}$ This does not mean that the region (i) cannot be observed from the other directions, e.g., if the disk is observed edge on.

Figure 19

The same as on Fig. 18 with inclinations 60° and 80°. The outer disk edge is $r=24$. The white dashed line here shows the outline of the shadow of the naked singularity when it is illuminated by background light.

Figure 20

AD images for the $S3$ type ($M=2$, $Q=0.99$, $n=14$), $\eta \approx 2.9998$, in full face and for inclination $i=30°$. The outer disk edge radius is $r=24$. SCO radii intervals are (i) $r\in (0,0.22)$—inner SCO region, (ii) $r\in (0.65,2.14)$—intermediate SCO ring, and (iii) $r\in (9,\infty )$—outer unbounded SCO ring. The dark blue rings in the centers represents the secondary image of the outer disk; because of strong lensing effect, it has high surface brightness. The intermediate SCO ring (ii) has two images represented by two red circles that almost merge together. The inner SCO region (i) is invisible in both panels.

Figure 21

The same as on Fig. 20 with inclinations 60° and 80°.