A novel exact magnetic black hole solution in four-dimensional extended scalar-tensor-Gauss-Bonnet theory

In this work the first exact asymptotically flat static and spherically symmetric black hole solution for $(3+1)$-dimensional ESTGB is presented, with a model of nonlinear electrodynamics – that reduces to Maxwell’s theory in the weak field limit and satisfies the weak energy condition – as a source. The solution has a nonzero magnetic charge, and scalar hair, which turns out to be dependent of the magnetic charge. It is characterized by the ADM mass $m$ and the magnetic charge $q$. Depending on the range of these parameters, the solution describes black holes with different structure. In the case $m\ge 0$ and $q\ge 0$, it shares many of the characteristics of the Schwarzschild solution. For $m>0$ and $q<0$, it is akin to the Reissner-Nordstr"om metric. In the case $m=0$, it represents a purely magnetic black hole.