We have shown that higher dimensional Reissner-Nordstr\"om-de Sitter black holes are gravitationally unstable for large values of the electric charge and cosmological constant in $D \geq 7$ space-time dimensions. We have found the shape of the slightly perturbed black hole at the threshold point of instability. Why only $D=4, 5$ and 6 dimensional worlds are favorable as to the black stability remains unknown.
Deep Dive into Instability of higher dimensional charged black holes in the de-Sitter world.
We have shown that higher dimensional Reissner-Nordstr"om-de Sitter black holes are gravitationally unstable for large values of the electric charge and cosmological constant in $D \geq 7$ space-time dimensions. We have found the shape of the slightly perturbed black hole at the threshold point of instability. Why only $D=4, 5$ and 6 dimensional worlds are favorable as to the black stability remains unknown.
characteristic size of extra dimensions, one can describe the black hole by the Schwarzschild-Tangherlini metric [7]. When charged particles collide, a charged black holes must be formed. At the same time, recent observational data suggests the non-zero values of the cosmological constant in the Universe, so that the non-vanishing vacuum energy of the world must influence the formation of black holes. Thus, more general black hole background would be the Reissner-Nordström-de Sitter (RNdS) generalization of the Schwarzschild-Tangherlini metric. Yet, there is no traditional uniqueness theorem for D > 4-spacetimes, so that the important physical criteria that selects from all higher dimensional "black" objects (such as black holes, string, branes, rings, and saturns) is their stability: unstable objects cannot exist or need some mechanism of stabilization.
Nevertheless, the stability analysis of D ≥ 5 black holes became feasible relatively recently [8], [10], [11]. This reduction was performed for the D-dimensional Reissner-Nordström-de Sitter black holes in [8] in the general form.
Yet, the stability of the Reissner-Nordström black holes was proven analytically only for D = 4, 5 space-time dimensions [8]. The perturbation equations can be treated separately for all three types, called scalar, vector and tensor, according to the rotation group on the (D -2)-sphere. When D = 4, we know the scalar type as polar and the vector type as axial, while the tensor type is usually a pure gauge. The higher dimensional cases were addressed in our earlier paper [12], where the stability of the D-dimensional Schwarzschild-de Sitter black holes was proved. Recently the stability of Reissner-Nordström-anti-de Sitter black holes (without dilaton) was shown in [13]. In addition, in [12] the numerical data for the quasinormal modes for vector and tensor types of gravitational perturbations of Reissner-Nordström-de Sitter (RNdS) black holes was given. Yet there it was claimed erroneously that Reissner-Nordström-de Sitter black holes are stable for all values of charge and Λ-term. In fact, in [12] for one particular, and most cumbersome, type of gravitational perturbations, the scalar type, one considered the effective potential, which corresponds to the perturbations of FIG. 1: The effective potentials V-for ρ = 0.8, q = 0.9, ℓ = 2, 3, 4 (blue, green red respectively). As ℓ grows the peak becomes higher and the negative gap decreases.
the Einstein equations with the frozen Maxwell field (see Eq. 8 in [12]). This approximation is valid when the charge of the black holes Q is considerably less than the black holes mass M , yet it is inappropriate for highly charged black holes. In the present paper we consider the dynamic behavior of the wave equation, which corresponds to the complete perturbations of the Einstein-Maxwell equations, given by Eq. (5.61), (5.63 b) in [9]. Basic formulae. The metric of the D = d + 2dimensional RNdS black holes is given by the line element
where dΩ d is the line element on a unit d-sphere, f (r
(
The equation of motion for gravitational perturbations of scalar type can be reduced to the wave-like equation
where the tortoise coordinate r * is defined as dr * = dr/f (r) and the effective potential is a function of black hole parameters, r and of a multipole number ℓ that comes from separation of angular variables
The explicit form of V ± can be found in [9], formulas (5.61-5.63). The scalar type of gravitational perturbations, corresponding to the V -potential is the only type for which the stability cannot be proved analytically [8], [9]. The potential V -reduces to the potential for pure gravitational perturbations, when Q = 0. On the contrary, V + reduces to pure electromagnetic perturbations propagating on the D-dimensional Schwarzschild background in the limit of vanishing charge. We shall imply that Ψ ∼ e -iωt , ω = ω Re -iω Im , so that ω Im > 0 corresponds to a stable (decayed) mode, while ω Im < 0 corresponds to an unstable (growing) mode. If the effective potential V (r) is positive definite everywhere outside the black hole event horizon, the differential operator d 2 /dr 2 * + ω 2 is a positive self-adjoint operator in the Hilbert space of the square integrable functions of r * , and, any solution of the wave equation with compact support is bounded, what implies stability. An important feature of the gravitational perturbations is that the effective potential V -(see Fig. 1), which governs the scalar type of the perturbations, has negative gap for the lower values of the multi-pole numbers ℓ. Higher ℓ simply increase the top of the potential barrier, and are usually more stable [14]. Thus, we shall check here those values of ℓ, for which the negative gap is present, and therefore the stability is not guaranteed.
Numerical Method. We shall study the evolution of the black hole perturbations of scalar “-” type in time domain using a numerical characteristic integration method [15],
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