Scalarized black holes in the presence of the coupling to Gauss-Bonnet gravity

In this paper, we study static and spherically symmetric black hole (BH) solutions in the scalar-tensor theories with the coupling of the scalar field to the Gauss-Bonnet (GB) term, $\xi (\varphi )R_{\mathrm{GB}}$, where $R_{\mathrm{GB}}≔R^2-4R^{\alpha \beta }{R}_{\alpha \beta }+R^{\alpha \beta \mu \nu }{R}_{\alpha \beta \mu \nu }$ is the GB invariant and $\xi (\varphi )$ is a function of the scalar field $\varphi$. Recently, it was shown that in these theories scalarized static and spherically symmetric BH solutions, which are different from the Schwarzschild solution and possess the nontrivial profiles of the scalar field, can be realized for certain choices of the coupling functions and parameters. These scalarized BH solutions are classified in terms of the number of nodes of the scalar field. It was then pointed out that in the case of the pure quadratic order coupling to the GB term, $\xi (\varphi )=\eta {\varphi }^2/8$, scalarized BH solutions with any number of nodes are unstable against the radial perturbation. In order to see how a higher order power of $\varphi$ in the coupling function $\xi (\varphi )$ affects the properties of the scalarized BHs and their stability, we investigate scalarized BH solutions in the presence of the quartic order term in the GB coupling function, $\xi (\varphi )=\eta {\varphi }^2(1+\alpha {\varphi }^2)/8$. We clarify that the existence of the higher order term in the coupling function can realize scalarized BHs with zero nodes of the scalar field which are stable against the radial perturbation.

Figure 1

The solutions $\psi (r)$ to Eq. (16) for the quadratic order coupling to the GB term (10) with the boundary conditions ${\psi }_0=1.0$ and ${\psi }_{\infty }=0$ are shown as the function of $r/M$. The red, blue, and green curves correspond to the scalar field solutions with zero, one, and two nodes of the scalar field, respectively.

Figure 2

The absolute value of the scalar charge $|Q|$ divided by $\sqrt{\eta }$ is shown as the function of the mass $M$ divided by $\sqrt{\eta }$ for ${\psi }_0=1.0$. We note that $Q>0$ for the scalar field solutions with an even number of nodes, and $Q<0$ for those with an odd number of nodes. The red, blue, and green curves correspond to the scalar field solutions with zero, one, and two nodes, respectively. The dashed lines from the bottom correspond to the solutions with zero, one, and two nodes for the pure quartic order coupling $\xi (\varphi )=\lambda {\varphi }^4/8$, respectively. For $\alpha <0$, we have chosen the minimal value of $\alpha$ to $-0.5+{10}^{-7}$.

Figure 3

For the coupling (24), the effective potential $V_{\mathrm{eff}}(r)$ multiplied by $M^2$ for the radial perturbation ($\ell =0$) is shown as the function of $r/M$ for ${\psi }_0=1.0$ for the scalar field solutions with zero, one, and two nodes, respectively. In the top panel, the black, red, blue, magenta, and green curves correspond to the scalar field solutions with zero nodes for $(\eta /M^2,\alpha )=(2.14,0.30)$, (2.90,0), $(3.32,-0.10)$, $(3.402,-0.115)$, $(3.59,-0.15)$, respectively. Similarly, in the bottom-left panel, the red, blue, and green curves correspond to the scalar field solutions with one node for $(\eta /M^2,\alpha )=(19.5,0)$, $(31.6,-0.40)$, $(53.5,-0.49)$, and in the bottom-right panel, the red, blue, and green curves correspond to the scalar field solutions with two nodes for $(\eta /M^2,\alpha )=(50.9,0)$, $(66.4,-0.35)$, $(105.9,-0.49)$, respectively.

Figure 4

The absolute value of the scalar charge $|Q|$ divided by $\sqrt{\eta }$ is shown as the function of the mass $M$ divided by $\sqrt{\eta }$. In the left larger panel, the red, blue, and green curves correspond to the scalarized BH solutions with zero, one, and two nodes of the scalar field, respectively. The right smaller panel corresponds to the enlarged display for the scalarized BH solutions with one and two nodes. We note that for ${\psi }_0>0$, $Q>0$ for the scalarized BH solutions with an even number of nodes, and $Q<0$ for those with an odd number of nodes. For all the cases, as the value of ${\psi }_0$ increases, the value of $|Q|$ also increases.

Figure 5

The effective energy density ${\rho }^{(\mathrm{eff})}(r)$, the effective radial pressure $p_r^{(\mathrm{eff})}(r)$, and the effective tangential pressure $p_t^{(\mathrm{eff})}(r)$ defined in Eq. (43), which are multiplied by $r_h^2$, are shown as the functions of $r/r_h$ for ${\psi }_0=0.01$. The top panel corresponds to the cases of the scalarized BH solutions with zero nodes of the scalar field, while the bottom-left and bottom-right panels correspond to the cases of the scalarized BH solutions with one and two nodes of the scalar field, respectively. In each panel, the red, blue, and green curves correspond to ${\rho }^{(\mathrm{eff})}(r)$, $p_r^{(\mathrm{eff})}(r)$, and $p_t^{(\mathrm{eff})}(r)$, respectively.

Figure 6

For the scalarized BHs with zero nodes of the scalar field, the amplitude $\psi (r)$, the effective energy density ${\rho }^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, the effective radial pressure $p_r^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, and the effective tangential pressure $p_t^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$ are shown as the functions of $r/r_h$ for ${\psi }_0=0.01$. In each panel, the red, blue, and green curves correspond to the scalarized BH solutions for $(\eta /r_h^2,\alpha )=(0.725,0)$, (0.338,10 000), $(7.31,-4990)$, respectively.

Figure 7

For the scalarized BHs with one node of the scalar field, the amplitude $\psi (r)$, the effective energy density ${\rho }^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, the effective radial pressure $p_r^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, and the effective tangential pressure $p_t^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, are shown as the functions of $r/r_h$ for ${\psi }_0=0.01$. In each panel, the red, blue, and green curves correspond to the scalarized BH solutions for $(\eta /r_h^2,\alpha )=(4.87,0)$, (3.09,10 000), $(20.2,-4990)$, respectively.

Figure 8

For the scalarized BHs with two nodes of the scalar field, the amplitude $\psi (r)$, the effective energy density ${\rho }^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, the effective radial pressure $p_r^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$, and the effective tangential pressure $p_t^{(\mathrm{eff})}(r)$ multiplied by $r_h^2$ are shown as the functions of $r/r_h$ for ${\psi }_0=0.01$. In each panel, the red, blue, and green curves correspond to the scalarized BH solutions for $(\eta /r_h^2,\alpha )=(12.7,0)$, (9.19,10 000), $(36.2,-4990)$, respectively.

Figure 9

The absolute value of the scalar charge $|Q|$ divided by $r_h$ is shown as the function of $(M/r_h-1/2)$. The Schwarzschild solution without the scalar charge corresponds to the origin (0,0). In the top panel, the red, blue, and green curves correspond to the scalarized BH solutions with zero, one, and two nodes of the scalar field, respectively. The bottom-left and bottom-right small panels show the enlarged displays for the scalarized BH solutions with one and two nodes of the scalar field, respectively. In each panel, each curve represents the scalarized BH solutions for the same value of ${\psi }_0$ and the curves from the bottom correspond to those with the smaller values of ${\psi }_0$. The scalarized BH solutions in the pure quadratic coupling model ($\alpha =0$) correspond to the upper edges of the red, blue, and green regions, respectively. The bottom black solid curves in the red, blue, and green regions correspond to the scalarized BH solutions with the same number of nodes for the pure quartic order coupling to the GB term ($\alpha \to \infty$) (29), and the top black dashed curves correspond to those for the pure quadratic order coupling to the GB term ($\alpha =0$) (10). We note that for ${\psi }_0>0$, $Q>0$ for the scalarized BH solutions with an even number of nodes, and $Q<0$ for those with an odd number of nodes.

Figure 10

The scalar charge $Q$ divided by $\sqrt{\eta }$ for scalarized BH solutions with zero nodes of the scalar field is shown as the function of the mass $M$ divided by $\sqrt{\eta }$ for fixed values of $\alpha$. The red, blue, green, black, cyan, magenta, and brown curves correspond to the cases of $\alpha =20000,2000,100,0,-10,-100,-1000$, respectively. The dashed curves represent scalarized BH solutions for each constant value of ${\psi }_0$ for $\alpha >0$, where the upper curves correspond to the larger values of ${\psi }_0$. For $\alpha <0$, we have chosen the maximal value of ${\psi }_0$ to $\sqrt{-1/(2\alpha )}(1-{10}^{-8})$.

Figure 11

The absolute value of the scalar charge $|Q|$ divided by $\sqrt{\eta }$ is shown as the function of the mass $M$ divided by $\sqrt{\eta }$ for ${\psi }_0=0.005$. The red, blue and green curves correspond to the solutions of the scalar field with zero, one, and two nodes, respectively. In the limit of $\alpha \to \infty$, the red and blue curves for the scalarized BH solutions with zero and one node approach the straight lines corresponding to the scalarized BH solutions with the same number of nodes for the pure quartic coupling (29). For $\alpha <0$, we have chosen the minimal value of $\alpha$ to $(1/{\psi }_0^2)(-0.5+{10}^{-7})$.

Figure 12

For scalarized BHs with zero nodes of the scalar field, $\eta /M^2$ is shown as the function of $\alpha$ for several values of ${\psi }_0=\mathcal{O}(0.01)$. The red, blue, and green curves correspond to the cases of ${\psi }_0=0.005$, 0.01, and 0.03, respectively.

Figure 13

In the top panel, the effective potential $U_{\mathrm{eff}}(r)$ multiplied by $r_h^2$ is shown as the function of $r/r_h$ for the scalarized BH solutions with zero nodes of the scalar field for ${\psi }_0=0.005$. The black, red, blue, magenta, and green curves correspond to the cases of the scalarized BHs for $(\eta /r_h^2,\alpha )=(0.558,10000)$, (0.725,0), $(0.803,-3000)$, $(0.8505,-4619)$, $(1.03,-9000)$, respectively. Similarly, the bottom-left and bottom-right panels describe the effective potential $U_{\mathrm{eff}}(r)$ multiplied by $r_h^2$ as the function of the mass $r/r_h$ for the scalarized BH solutions with one and two nodes of the scalar field for ${\psi }_0=0.005$, respectively. In the bottom-left and bottom-right panels, the red, blue, and green curves correspond to the cases of for the scalarized BH solutions with one node for $(\eta /r_h^2,\alpha )=(5.27,-4000)$, $(6.12,-10000)$, $(11.0,-19000)$, and for the scalarized BH solutions with two nodes for $(\eta /r_h^2,\alpha )=(13.4,-4000)$, $(14.9,-10000)$, $(22.9,-19000)$, respectively.