5.3 Scalar field

In 2003, Choptuik, Hirschmann, Liebling and Pretorius reported on numerical evolutions at the black hole threshold of an axisymmetric massless scalar field [54Jump To The Next Citation Point]. In axisymmetry with scalar field matter there is no angular momentum and only one polarisation of gravitational waves. The slicing condition is maximal slicing and the spatial gauge, in cylindrical coordinates, is gzz = gρρ, gzρ = 0, similar to the gauge used by Abrahams and Evans. The Hamiltonian constraint is solved at every time step, and the time derivatives of the spatial gauge conditions are substituted into the momentum constraints to obtain second-order elliptic equations for the two shift components. Thus the evolution is partially constrained. Adaptive mesh-refinement was used in the numerical time evolution.

The initial data were either time-symmetric or approximately ingoing, with the scalar field either symmetric or antisymmetric in z. In the symmetric case, even strongly non-spherical data were attracted to the known spherical critical solution for the massless scalar field. Scaling with the known γ was observed in the Ricci scalar. However, with sufficiently good fine-tuning to the black hole threshold, the approximately spherical region that approaches the critical solution suffers an l = 2 (and by ansatz m = 0) instability and splits into two new spherical regions which again approach the critical solution. The spatial separation of the two new centres is related to the smallest length scale that developed prior to the branching. There is evidence that with increasing fine-tuning each of these centres splits again. The antisymmetric initial data cannot approach a single spherical critical solution, but the solution splits initially into two approximately spherical regions where the critical solution is approached (up to an overall sign in the scalar field). The separation of these initial two centres is determined by the initial data, but there is evidence that they in turn split.

All this is consistent with the assumption that the spherical critical solution has, besides the known one spherical unstable mode, precisely one further l = 2 unstable mode. (Without the restriction to axisymmetry, if such a mode exists, it would be 5-fold degenerate with m = =–2, ..., 2.) This contradicts the calculation of the perturbation spectrum in [154Jump To The Next Citation Point]. Choptuik and co-workers do not state with certainty that the mode they see in numerical evolutions is a continuum mode, although they have no indication that it is a numerical artifact. The growth rate of the putative mode is measured to be λ ≃ 0.1–0.4, which should be compared with the growth rate λ ≃ 2.7 of the spherical mode and the relatively small decay rate of λ ≃ − 0.02 claimed in [154] for the least damped mode, which is also an l = 2 mode. We observe that the range of τ in Figures 6 and 7 of [54Jump To The Next Citation Point] is about 10, and over this range the plot of the amplitude of the l = 2 perturbation against the log-scale coordinate τ seems equally consistent with linear growth in τ as with exponential growth. (In the notation of [54Jump To The Next Citation Point], τ denotes proper time and τ∗ the accumulation point of echos, so that the log-scale coordinate τ used in this review corresponds to − ln(τ − τ∗) in the notation of [54].)

An interesting extension was made in [55] by considering a complex scalar field giving rise to an axisymmetric spacetime with angular momentum. The stress-energy tensor of a complex scalar field Ψ is

( ) Tab = 1- Ψ,aΨ¯,b + ¯Ψ,aΨ,b − gab¯Ψ,cΨ,c . (43 ) 2
An axisymmetric spacetime with azimuthal Killing vector ξ admits a conserved vector field Tabξb, so that the total angular momentum
∫ a b J = Tabξ n dV (44 ) Σ
is independent of the Cauchy surface Σ. With the ansatz
Ψ(ρ,z, t,φ ) = Φ(ρ,z, t)eimφ (45 )
in adapted coordinates where ξ = ∂∕∂ φ and with m being an integer, the stress-energy tensor becomes axisymmetric and hence compatible with the Einstein equations for an axisymmetric spacetime.

For any Σ tangent to ξ, in particular a hypersurface t = constant, the angular momentum density measured by a normal observer becomes

j = T ξanb = − im-(Π ¯Φ − ¯ΠΦ ) = mA2na δ , (46 ) ab 2 ,a
where Φ = Aei δ with δ and A being real and Π = naΦ ,a. By comparison the energy density measured by a normal observer is
a b ¯ a¯ ρ = Tabn n = Π Π + Da ΦD Φ, (47 )
where Da is the derivative operator projected into Σ. This means that the ratio of energy density to angular momentum density can be adjusted arbitrarily in the initial data, including zero angular momentum for a Φ that is real (up to a constant phase). On the other hand, even in the absence of angular momentum a purely real Φ obeys a wave equation with an explicit m2 ∕ρ2 centrifugal term. For the same reason, regular solutions must have Φ ∼ ρm on the axis, and there are no spherically symmetric solutions. Intuitively speaking, the centrifugal force resisting collapse appears unrelated to the angular momentum component of the stress-energy tensor in a way that differs from what one would expect in rotating fluid collapse or rotating (non-axisymmetric) vacuum collapse.

For all initial data in numerical evolutions, a critical solution is approached that is discretely self-similar with log-scale period Δ ≃ 0.42. (By ansatz this solution is axisymmetric but spherical symmetry is ruled out and so the critical solution cannot be the Choptuik solution.) The same critical solution is approached in particular for initial data with Π = 0 and hence no angular momentum, and initial data where Π = ¯Φ and hence with large angular momentum. A scaling exponent of γ ≃ 0.11 is observed in the Ricci scalar in subcritical evolutions. The critical solution is purely real (up to an initial data-dependent constant phase) and hence has no angular momentum. Only m = 1 was investigated, but it is plausible that a different critical solution exists for each integer m.

Far from the black hole threshold 2 J ∼ M in the final black hole, but nearer the black hole threshold, 6 J ∼ M, where J and M are measured on the apparent horizon when it first forms. 2 J ∕M → 0 is compatible with a non-rotating critical solution.

In the absence of angular momentum, the wave equations for the real and imaginary part of Φ decouple. Assuming that the background critical solution is purely real with δ = 0 and A ∼ 1, and angular momentum is provided by a perturbation with λτ δ ∼ e, one would expect 2 2τ ρ ∼ |∇A | ∼ e and (1+λ)τ j ∼ A|∇ δ| ∼ e. Integrating over a region of size −3τ e when the black hole forms, we find M ∼ e−τ∗ and J ∼ e(−2+λ)τ∗. Then J ∼ M 6 would imply λ = − 4.

Olabarrieta et al. [169] study a similar system in spherical symmetry, by arranging 2l+1 scalar fields given by ϕlm = ψ(r,t)Ylm (𝜃,φ) for m = − l,...l with the same ψ (r,t) for all values of m so that the total stress-energy tensor and the spacetime are spherically symmetric. Note that not the ϕlm are added but their stress-energies, and each value of l describes a different matter content. ψ then sees a centrifugal barrier in its evolution equation, but there is no angular momentum in the spacetime. DSS critical behaviour is found, and the logarithmic echoing period Δ and mass scaling exponent γ both decrease approximately exponentially with l. It is observed empirically that the radius r0 of maximum compactness during evolution (a measure of the scale of the initial data) and the accumulation time of echos T∗ (measured from the initial data) obey T ∗ ≃ r0∕(4.35 Δ ) for all l and families of initial data. UpdateJump To The Next Update Information

Lai [141] has studied type I critical phenomena for boson (massive complex scalar field) stars in axisymmetry, the first study of type I in axisymmetry. He finds that the subcritical end state is a boson star with a large amplitude fundamental mode oscillation.

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