Abstract
We study the spontaneous scalarization of a standard conducting charged sphere embedded in Maxwell-scalar models in flat spacetime, wherein the scalar field $\phi$ is nonminimally coupled to the Maxwell electrodynamics. This setup serves as a toy model for the spontaneous scalarization of charged (vacuum) black holes in Einstein-Maxwell-scalar (generalized scalar-tensor) models. In the Maxwell-scalar case, unlike the black hole cases, closed-form solutions exist for the scalarized configurations. We compute these configurations for three illustrations of nonminimal couplings: one that \textit{exactly} linearizes the scalar field equation, and the remaining two that produce nonlinear continuations of the first one. We show that the former model leads to a runaway behaviour in regions of the parameter space and neither the Coulomb nor the scalarized solutions are stable in the model; but the latter models can heal this behaviour producing stable scalarized solutions that are dynamically preferred over the Coulomb one. This parallels reports on black hole scalarization in the extended-scalar-Gauss-Bonnet models. Moreover, we analyse the impact of the choice of the boundary conditions on the scalarization phenomenon. Dirichlet and Neumann boundary conditions accommodate both (linearly) stable and unstable parameter space regions, for the scalar-free conducting sphere; but radiative boundary conditions always yield an unstable scalar-free solution and preference for scalarization. Finally, we perform numerical evolution of the full Maxwell-scalar system, following dynamically the scalarization process. They confirm the linear stability analysis and reveal that the scalarization phenomenon can occur in qualitatively distinct ways.
Highlights
Since such an oscillator has only a stable equilibrium point at the minimum of the potential, we anticipate that only the solution with φ0 = 0 everywhere could be stable. One can relate this instability to the linear nature of (28), which allows the amplitude of the scalar field to grow without bound. This is reminiscent of the instability observed for the scalarized black holes in extended-scalar-tensor-Gauss-Bonnet gravity, for a quadratic coupling function, that leads to a scalar field equation which is linear in the scalar field [8, 9, 12, 13, 17, 30, 31]
In this paper we have investigated spontaneous scalarization of a charged, conducting sphere in Maxwell-scalar (Ms) models
As we have explained, Ms models can parallel features observed in the gravitational context, such as the coupling dependence on the existence of stable scalarized solutions
Summary
Compact objects such as neutron stars and black holes can undergo a spontaneous scalarization phenomenon in asymptotically flat spacetimes, in scalar-tensor theories of gravitation, see e.g. [1,2,3,4,5,6,7] for scalarization of relativistic stars and e.g., [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40] for black hole scalarization. Even in the absence of gravity, a nonminimal coupling of the scalar field to Maxwell’s electromagnetism, f (φ)F μνFμν, where f (φ) is a regular function of φ, may allow scalarization of a charged object, such as a conducting sphere [46] (see [60]) This occurs in Maxwell-scalar (Ms) models on Minkowski spacetime, which provide one of the simplest arenas to study the spontaneous scalarization phenomenon. We will show that the choice of the boundary conditions can impact significantly on the instability of the conducting sphere These findings will result from a linear perturbation theory analysis, which is performed for the scalarized solutions, but will be confirmed by fully nonlinear numerical evolution.
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