Stationary Scalar Configurations Research Articles

How short can stationary charged scalar hair be?

It is by now well established that charged rotating Kerr-Newman black holes can support bound-state charged-matter configurations which are made of minimally coupled massive-scalar fields. We here prove that the externally supported stationary charged scalar configurations cannot be arbitrarily compact. In particular, for linearized-charged-massive-scalar fields supported by charged rotating near-extremal Kerr-Newman black holes, we derive the remarkably compact lower bound $({r}_{\text{field}}\ensuremath{-}{r}_{+})/({r}_{+}\ensuremath{-}{r}_{\ensuremath{-}})>1/{s}^{2}$ on the effective lengths of the external charged scalar ``clouds'' [here ${r}_{\text{field}}$ is the radial peak location of the stationary scalar configuration, and ${s\ensuremath{\equiv}J/{M}^{2},{r}_{\ifmmode\pm\else\textpm\fi{}}}$ are, respectively, the dimensionless angular momentum and the horizon radii of the central supporting Kerr-Newman black hole]. Remarkably, this lower bound is universal in the sense that it is independent of the physical parameters (proper mass, electric charge, and angular momentum) of the supported charged scalar fields.

Stationary scalar clouds around maximally rotating linear dilaton black holes

We investigate the wave dynamics of a charged massive scalar field propagating in a maximally rotating (extremal) linear dilaton black hole geometry. We prove the existence of a discrete and infinite family of resonances describing non-decaying stationary scalar configurations (clouds) enclosing these rapidly rotating black holes. The results obtained signal the potential stationary scalar field distributions (dark matter) around the extremal linear dilaton black holes. In particular, we analytically compute the effective heights of those clouds above the center of the black hole.

Extremal Kerr–Newman black holes with extremely short charged scalar hair

The recently proved ‘no short hair’ theorem asserts that, if a spherically-symmetric static black hole has hair, then this hair (the external fields) must extend beyond the null circular geodesic (the “photonsphere”) of the corresponding black-hole spacetime: rfield>rnull. In this paper we provide compelling evidence that the bound can be violated by non-spherically symmetric hairy black-hole configurations. To that end, we analytically explore the physical properties of cloudy Kerr–Newman black-hole spacetimes – charged rotating black holes which support linearized stationary charged scalar configurations in their exterior regions. In particular, for given parameters {M,Q,J} of the central black hole, we find the dimensionless ratio q/μ of the field parameters which minimizes the effective lengths (radii) of the exterior stationary charged scalar configurations (here {M,Q,J} are respectively the mass, charge, and angular momentum of the black hole, and {μ,q} are respectively the mass and charge coupling constant of the linearized scalar field). This allows us to prove explicitly that (non-spherically symmetric non-static) composed Kerr–Newman-charged-scalar-field configurations can violate the no-short-hair lower bound. In particular, it is shown that extremely compact stationary charged scalar ‘clouds’, made of linearized charged massive scalar fields with the property rfield→rH, can be supported in the exterior spacetime regions of extremal Kerr–Newman black holes (here rfield is the peak location of the stationary scalar configuration and rH is the black-hole horizon radius). Furthermore, we prove that these remarkably compact stationary field configurations exist in the entire range s≡J/M2∈(0,1) of the dimensionless black-hole angular momentum. In particular, in the large-mass limit they are characterized by the simple dimensionless ratio q/μ=(1−2s2)/(1−s2).

Rotating black holes can have short bristles

The elegant ‘no short hair’ theorem states that, if a spherically-symmetric static black hole has hair, then this hair must extend beyond 3/2 the horizon radius. In the present paper we provide evidence for the failure of this theorem beyond the regime of spherically-symmetric static black holes. In particular, we show that rotating black holes can support extremely short-range stationary scalar configurations (linearized scalar ‘clouds’) in their exterior regions. To that end, we solve analytically the Klein–Gordon–Kerr–Newman wave equation for a linearized massive scalar field in the regime of large scalar masses.

Stationary scalar configurations around extremal charged black holes

We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordström background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies $$\omega $$ for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies $$\mathcal{R }(\omega )$$ tends to the mass of the field and the imaginary part $$\mathcal{I }(\omega )$$ tends to zero, for any angular momentum quantum number $$\ell $$ . The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.