Black-hole solutions with scalar hair in Einstein-scalar-Gauss-Bonnet theories

In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general coupling function between the scalar field and the quadratic Gauss-Bonnet term, we investigate the existence of regular black-hole solutions with scalar hair. Based on a previous theoretical analysis, which studied the evasion of the old and novel no-hair theorems, we consider a variety of forms for the coupling function (exponential, even and odd polynomial, inverse polynomial, and logarithmic) that, in conjunction with the profile of the scalar field, satisfy a basic constraint. Our numerical analysis then always leads to families of regular, asymptotically flat black-hole solutions with nontrivial scalar hair. The solution for the scalar field and the profile of the corresponding energy-momentum tensor, depending on the value of the coupling constant, may exhibit a nonmonotonic behavior, an unusual feature that highlights the limitations of the existing no-hair theorems. We also determine and study in detail the scalar charge, horizon area, and entropy of our solutions.

Figure 1

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha e^{-\varphi }$.

Figure 2

The metric components $g_{tt}$ and $g_{rr}$ (left plot) and the Gauss-Bonnet term $R_{\mathrm{GB}}^2$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha e^{-\varphi }$.

Figure 3

The scalar charge $D$ as a function of the near-horizon value ${\varphi }_h$ (left plot) and of its mass $M$ (right plot), for $f(\varphi )=\alpha e^{-\varphi }$.

Figure 4

The horizon area $A_h$ and entropy $S_h$ of the black hole (left plot) and their ratios to the corresponding Schwarzschild values (right plot) in terms of the mass $M$ for $f(\varphi )=\alpha e^{-\varphi }$.

Figure 5

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha {\varphi }^2$.

Figure 6

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha {\varphi }^4$.

Figure 7

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha {\varphi }^2$ and various values of the coupling constant $\alpha$.

Figure 8

The scalar charge $D$ (left plot) and the ratios $A_h/A_{\mathrm{Sch}}$ and $S_h/S_{\mathrm{Sch}}$ (right plot) in terms of the mass $M$, for $f(\varphi )=\alpha {\varphi }^2$.

Figure 9

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha \varphi$.

Figure 10

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha {\varphi }^3$.

Figure 11

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha {\varphi }^3$ and a variety of values of the coupling constant $\alpha$.

Figure 12

The scalar charge $D$ (left plot) and the ratios $A_h/A_{\mathrm{Sch}}$ and $S_h/S_{\mathrm{Sch}}$ (right plot) in terms of the mass $M$, for $f(\varphi )=\alpha \varphi$.

Figure 13

The scalar charge $D$ (left plot) and the ratios $A_h/A_{\mathrm{Sch}}$ and $S_h/S_{\mathrm{Sch}}$ (right plot) in terms of the mass $M$, for $f(\varphi )=\alpha {\varphi }^3$.

Figure 14

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha /\varphi$.

Figure 15

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha /{\varphi }^2$.

Figure 16

The scalar charge $D$ (left plot) and the ratios $A_h/A_{\mathrm{Sch}}$ and $S_h/S_{\mathrm{Sch}}$ (right plot) in terms of the mass $M$, for $f(\varphi )=\alpha /\varphi$.

Figure 17

The scalar field $\varphi$ (left plot) and the energy-momentum tensor $T_{\mu \nu }$ (right plot) in terms of the radial coordinate $r$, for $f(\varphi )=\alpha \ln \varphi$.

Figure 18

The scalar charge $D$ (left plot) and the ratios $A_h/A_{\mathrm{Sch}}$ and $S_h/S_{\mathrm{Sch}}$ (right plot) in terms of the mass $M$, for $f(\varphi )=\alpha \ln \varphi$.