Black hole solutions in $f(R)$ gravity coupled with nonlinear Yang-Mills field

It is shown that in the static, spherically symmetric spacetime, the problem of metric $f(R)$ gravity coupled with nonlinear Yang-Mills (YM) field constructed from the Wu-Yang ansatz as source can be solved in all dimensions. By nonlinearity it is meant that the YM Lagrangian depends arbitrarily on its invariant. A particular form is considered to be in the power-law form with limit of the standard YM theory. The formalism admits black hole solutions with single or double horizons in which $f(R)$ can be obtained, in general numerically. In 6-dimensional case we obtain an exact solution given by $f(R)=\sqrt{R}$ gravity that couples with the YM field in a consistent manner.

Figure 1

The plot of 6-dimensional $f(R)$ [Fig. 1a] and $A(r)$ [Fig. 1b]. We choose $\Lambda =0$, and four different values of $\eta$ (${\eta }_A$, ${\eta }_B$, ${\eta }_c$, and ${\eta }_D$) are depicted as plots A, B, C and D. From Fig. 1b it can be seen that in A, B, and C we have single, while in D double horizons.

Figure 2

The 5-dimensional plots of $A(r)$, $f(r)$ and $R(r)$ from Eq. (68), for a variety of parameters given in Figs. 2a, 2b, 2c, 2d. Since $s=\frac{5}{4}$ in this particular case, the source is the PYM field with Lagrangian $L\sim F^{5/4}$. These are all black hole solutions with inner and outer horizons. A general analytic expression for $f(R)$ seems out of our reach.

Figure 3

The plot of the metric function $A(r)$ corresponding to the conformally invariant case from Eq. (68) in $d=6$ and $s=\frac{3}{2}$, for a set of $\eta$ parameters. Black hole formations with single/double horizons are explicitly seen. Specifically, Fig. 3a for $0\le \eta \le 0.80$ and Fig. 3b for $0.9\le \eta \le 1.0$.