We construct black hole solutions with spin-induced scalarization in a class of models where a scalar field is quadratically coupled to the topological Gauss-Bonnet term. Starting from the tachyonically unstable Kerr solutions, we obtain families of scalarized black holes such that the scalar field has either even or odd parity, and we investigate their domain of existence. The scalarized black holes can violate the Kerr rotation bound. We identify “critical” families of scalarized black hole solutions such that the expansion of the metric functions and of the scalar field at the horizon no longer allows for real coefficients. For the quadratic coupling considered here, solutions with spin-induced scalarization are entropically favored over Kerr solutions with the same mass and angular momentum.

Recently a new class of scalarized black holes in Einstein-Gauss-Bonnet (EGB) theories was discovered. What is special for these black hole solutions is that the scalarization is not due to the presence of matter, but it is induced by the curvature of spacetime itself. Moreover, more than one branch of scalarized solutions can bifurcate from the Schwarzschild branch, and these scalarized branches are characterized by the number of nodes of the scalar field. The next step is to consider the linear stability of these solutions, which is particularly important due to the fact that the Schwarzschild black holes lose stability at the first point of bifurcation. Therefore we here study in detail the radial perturbations of the scalarized EGB black holes. The results show that all branches with a nontrivial scalar field with one or more nodes are unstable. The stability of the solutions on the fundamental branch, whose scalar field has no radial nodes, depends on the particular choice of the coupling function between the scalar field and the Gauss-Bonnet invariant. We consider two particular cases based on the previous studies of the background solutions. If this coupling has the form used in [D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. 120, 131103 (2018)] the fundamental branch of solutions is stable, except for very small masses. In the case of a coupling function quadratic in the scalar field [H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Phys. Rev. Lett. 120, 131104 (2018)], though, the whole fundamental branch is unstable.

We construct rotating black holes in Einstein-scalar-Gauss–Bonnet theory with a quadratic coupling function. We map the domain of existence of the rotating fundamental solutions, we construct radially excited rotating black holes (including their existence lines), and we show that there are angularly excited rotating black holes. The bifurcation points of the radially and angularly excited solutions branching out of the Schwarzschild solution follow a regular pattern.

Novel wormholes are obtained in Einstein-scalar-Gauss-Bonnet theory for several coupling functions. The wormholes may feature a single- or a double-throat geometry and do not demand any exotic matter. The scalar field may asymptotically vanish or be finite, and it may possess radial excitations. The domain of existence is fully mapped out for various forms of the coupling function.

We study the quasinormal modes for scalar, axial and radial perturbations of massive static spherically symmetric wormholes supported by a phantom scalar field, calculating the modes directly and employing the WKB approximation. The spectrum of the quasinormal modes is compared with the spectrum of Schwarzschild black holes. For fixed multipole number $l$ and large mass $M$ the wormhole modes approach their black hole counterparts as $1/M$.

The classical sphaleron is constructed for the full SU(2)×U(1) electroweak theory. Unlike the sphaleron of the SU(2) Yang-Mills-Higgs theory, it is not spherically symmetric due to the coupling to the U(1) field. It is symmetric only under rotations around the z axis and parity reflections. The mixing angle ${\mathrm{\theta }}_{\mathit{W}}$ is varied over the full range 0≤${\mathrm{\theta }}_{\mathit{W}}$≤π/2. When the mixing angle is increased the energy of the sphaleron decreases, and the energy density changes its shape from a sphere at ${\mathrm{\theta }}_{\mathit{W}}$=0 to a very elongated spheroid at large values of the mixing angle. At the physical value of the mixing angle, however, the electroweak sphaleron differs only a little from the spherical sphaleron.