6.3 Other dimensions

Much work has been invested recently in considering physics in other dimensions. Motivation comes from various ideas including string theory (more dimensions) such as the AdS/CFT correspondence and holography (one fewer dimensions) [156, 159, 181]. Another source of motivation comes from the fact that higher dimensional black holes can have very different properties than those in three spatial dimensions [75Jump To The Next Citation Point]. Perhaps BSs will similarly display novel properties in other dimensions.

In lower dimensional AdS (2+1) spacetimes, early work in 1998 studied exact solutions of boson stars [191, 67, 192]. Higher dimensional scenarios were apparently first considered qualitatively a few years later in the context of brane world models [205]. This discussion was followed with a detailed analysis of the 3, 4, and 5 dimensional AdS solutions [11].

More recently, Ref. [84] considers oscillatons in higher dimensions and measure the scalar mass loss rate for dimensions 3, 4, and 5. They extend this work considering inflationary spacetimes [83].

The axisymmetric rotating BSs discussed in Section 3.5 satisfy a coupled set of nonlinear, elliptic PDEs in two dimensions. One might, therefore, suspect that adding another dimensions will only make things more difficult. As it turns out, however, moving to four spatial dimensions provides for another angular momentum, independent of the one along the z-direction (for example). Each of these angular momenta are associated with their own orthogonal plane of rotation. And so if one chooses solutions with equal magnitudes for each of these momenta, the solutions depend on only a single radial coordinate. This choice results in the remarkable simplification that one need only solve ODEs to find rotating solutions [138].

In [108Jump To The Next Citation Point], they extend this idea by assuming an ansatz for two complex scalar fields with equal magnitudes of angular momentum in the two independent directions. Letting the complex doublet be denoted by Φ, the ansatz takes the form

( ) iωt sin𝜃eiφ1 Φ = ϕ (r)e cos𝜃eiφ2 (85 )
in terms of the two angular coordinates φ1 and φ2. One observes that the BS (i) retains a profile ϕ (r ), (ii) possesses harmonic time dependence, and (iii) maintains single-valuedness in the two angles (the ansatz assumes a rotational quantum number of one). They find solutions that are both globally regular and asymptotically flat but these solutions appear only stable with weak gravitational coupling [108Jump To The Next Citation Point].

The work of [69Jump To The Next Citation Point] makes ingenious use of this 5D ansatz to construct rotating black holes with only a single Killing vector. They set the potential of [108] to zero so that the scalar fields are massless and they add a (negative) cosmological constant to work in anti-de Sitter (AdS). They find solutions for rotating black holes in 5D AdS that correspond to a bar mode for rotating neutron stars in 3D (see also [202] for a numerical evolution of a black hole in higher dimensions, which demonstrates such bar formation). One might expect such a non-symmetric black hole to settle into a more symmetric state via the emission of gravitational waves. However, AdS provides for an essentially reflecting boundary in which the black hole can be in equilibrium. The distortion of the higher dimensional black hole also has a correspondence with the discrete values of the angular momentum of the corresponding boson star. For higher values of the rotational quantum number, the black hole develops multiple “lobes” about its center.

These solutions are extended to arbitrary odd-dimensional AdS spacetimes in [206]. Finding the solutions perturbatively, they explicitly show that these solutions approach (i) the boson star and (ii) the Myers–Perry black-hole solutions in AdS [170] in different limits. See [75] for a review of black holes in higher dimensions.

The same authors of [69Jump To The Next Citation Point] also report on the existence of geons in 3+1 AdS “which can be viewed as gravitational analogs of boson stars” [70] (recall that boson stars themselves arose from Wheeler’s desire to construct local electrovacuum solutions). These bundles of gravitational energy are stable to first order due to the confining boundary condition adopted with AdS. However, these geons and the black-hole solutions above [69] are unstable at higher order because of the turbulent instability reported in [31].

Ref. [21] also studies black-hole solutions in 5D AdS. They find solutions for black holes with scalar hair that resemble a boson star with a BH in its center. See Ref. [92] for a review of charged scalar solitons in AdS.

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