4.4 Perturbations of Schwarzschild

The nonlinear 3D PITT code has been calibrated in the regime of small perturbations of a Schwarzschild spacetime [311Jump To The Next Citation Point, 312Jump To The Next Citation Point] by measuring convergence with respect to independent solutions of the Teukolsky equation [290]. By decomposition into spherical harmonics, the Teukolsky equation reduces the problem of a perturbation of a stationary black hole to a 1D problem in the (t,r) subspace perturbations for a component of the Weyl tensor. Historically, the Teukolsky equation was first solved numerically by Cauchy evolution. Campanelli, Gómez, Husa, Winicour, and Zlochower [77Jump To The Next Citation Point, 174Jump To The Next Citation Point] have reformulated the Teukolsky formalism as a double-null characteristic evolution algorithm. The evolution proceeds on a family of outgoing null hypersurfaces with an ingoing null hypersurface as inner boundary and with the outer boundary compactified at future null infinity. It applies to either the Weyl component Ψ0 or Ψ4, as classified in the Newman–Penrose formalism. The Ψ0 component comprises constraint-free gravitational data on an outgoing null hypersurface and Ψ4 comprises the corresponding data on an ingoing null hypersurface. In the study of perturbations of a Schwarzschild black hole, Ψ0 is prescribed on an outgoing null hypersurface − 𝒥, representing an early retarded time approximating past null infinity, and Ψ4 is prescribed on the inner white hole horizon ℋ −.

The physical setup is described in Figure 5View Image. The outgoing null hypersurfaces extend to future null infinity ℐ+ on a compactified numerical grid. Consequently, there is no need for either an artificial outer boundary condition or an interior extraction worldtube. The outgoing radiation is computed in the coordinates of an observer in an inertial frame at infinity, thus avoiding any gauge ambiguity in the waveform.

View Image

Figure 5: The physical setup for the scattering problem. A star of mass M has undergone spherically-symmetric collapse to form a black hole. The ingoing null worldtube 𝒩 lies outside the collapsing matter. Inside 𝒩 (but outside the matter) there is a vacuum Schwarzschild metric. Outside of 𝒩, data for an ingoing pulse is specified on the initial outgoing null hypersurface 𝒥 −. As the pulse propagates to the black-hole event horizon + ℋ, part of its energy is scattered to + ℐ.

The first calculations were carried out with nonzero data for Ψ4 on − ℋ and zero data on − 𝒥 [77Jump To The Next Citation Point] (so that no ingoing radiation entered the system). The resulting simulations were highly accurate and tracked the quasi-normal ringdown of a perturbation consisting of a compact pulse through ten orders of magnitude and tracked the final power-law decay through an additional six orders of magnitude. The measured exponent of the power law decay varied from ≈ 5.8, at the beginning of the tail, to ≈ 5.9 near the end, in good agreement with the predicted value of 2ℓ + 2 = 6 for a quadrupole wave [238].

The accuracy of the perturbative solutions provide a virtual exact solution for carrying out convergence tests of the nonlinear PITT null code. In this way, the error in the Bondi news function computed by the PITT code was calibrated for perturbative data consisting of either an outgoing pulse on − ℋ or an ingoing pulse on 𝒥 −. For the outgoing pulse, clean second-order convergence was confirmed until late times in the evolution, when small deviations from second order arise from accumulation of roundoff and truncation error. For the Bondi news produced by the scattering of an ingoing pulse, clean second-order convergence was again confirmed until late times when the pulse approached the r = 2 M black-hole horizon. The late-time error arises from loss of resolution of the pulse (in the radial direction) resulting from the properties of the compactified radial coordinate used in the code. This type of error could be eliminated by using the characteristic AMR techniques under development [237Jump To The Next Citation Point].

4.4.1 Close approximation white-hole and black-hole waveforms

The characteristic Teukolsky code has been used to study radiation from axisymmetric white holes and black holes in the close approximation. The radiation from an axisymmetric fissioning white hole [77Jump To The Next Citation Point] was computed using the Weyl data on − ℋ supplied by the conformal horizon model described in Section 4.3, with the fission occurring along the axis of symmetry. The close approximation implies that the fission takes place far in the future, i.e., in the region of ℋ − above the black-hole horizon ℋ+. The data have a free parameter η, which controls the energy yielded by the white-hole fission. The radiation waveform reveals an interesting dependence on the parameter η. In the large η limit, the waveform consists of a single pulse, followed by ringdown and tail decay. The amplitude of the pulse scales quadratically with η and the width decreases with η. As η is reduced, the initial pulse broadens and develops more structure. In the small η limit, the amplitude scales linearly with η and the shape is independent of η.

Since there was no incoming radiation, the above model gave the physically-appropriate boundary conditions for a white-hole fission (in the close approximation). From a time reversed view point, the system corresponds to a black-hole merger with no outgoing radiation at future null infinity, i.e., the analog of an advanced solution with only ingoing but no outgoing radiation. In the axisymmetric case studied, the merger corresponds to a head-on collision between two black holes. The physically-appropriate boundary conditions for a black-hole merger correspond to no ingoing radiation on 𝒥 − and binary black-hole data on ℋ+. Because 𝒥 − and ℋ+ are disjoint, the corresponding data cannot be used directly to formulate a double null characteristic initial value problem. However, the ingoing radiation at − 𝒥 supplied by the advanced solution for the black-hole merger could be used as Stage I of a two stage approach to determine the corresponding retarded solution. In Stage II, this ingoing radiation is used to generate the analogue of an advanced minus retarded solution. A pure retarded solution (with no ingoing radiation but outgoing radiation at + ℐ can then be constructed by superposition. The time reflection symmetry of the Schwarzschild background is key to carrying out this construction.

This two stage strategy has been carried out by Husa, Zlochower, Gómez, and Winicour [174Jump To The Next Citation Point]. The superposition of the Stage I and II solutions removes the ingoing radiation from 𝒥 − while modifying the close approximation perturbation of + ℋ, essentially making it ring. The amplitude of the radiation waveform at + ℐ has a linear dependence on the parameter η, which in this black-hole scenario governs the energy lost in the inelastic merger process. Unlike the fission waveforms, there is very little η-dependence in their shape and the amplitude continues to scale linearly even for large η. It is not surprising that the retarded waveforms from a black-hole merger differ markedly from the retarded waveforms from a white-hole merger. The white-hole process is directly visible at + ℐ whereas the merger waveform results indirectly from the black holes through the preceding collapse of matter or gravitational energy that formed them. This explains why the fission waveform is more sensitive to the parameter η, which controls the shape and timescale of the horizon data. However, the weakness of the dependence of the merger waveform on η is surprising and has potential importance for enabling the design of an efficient template for extracting a gravitational wave signal from noise.

4.4.2 Fissioning white hole

In the purely vacuum approach to the binary black-hole problem, the stars that collapse to form the black holes are replaced either by imploding gravitational waves or some past singularity as in the Kruskal picture. This avoids hydrodynamic difficulties at the expense of a globally-complicated initial-value problem. The imploding waves either emanate from a past singularity, in which case the time-reversed application of cosmic censorship implies the existence of an anti-trapped surface; or they emanate from ℐ −, which complicates the issue of gravitational radiation content in the initial data and its effect on the outgoing waveform. These complications are avoided in the two stage approach adopted in the close-approximation studies described in Section 4.4.1, where advanced and retarded solutions in a Schwarzschild background can be rigorously identified and superimposed. Computational experiments have been carried out to study the applicability of this approach in the nonlinear regime [136].

From a time-reversed viewpoint, the first stage is equivalent to the determination of the outgoing radiation from the fission of a white hole in the absence of ingoing radiation, i.e., the physically-appropriate “retarded” waveform from a white-hole fission. This fission problem can be formulated in terms of data on the white-hole horizon ℋ − and data representing the absence of ingoing radiation on a null hypersurface − J, which emanates from − ℋ at an early time. The data on − ℋ is provided by the conformal horizon model for a fissioning white hole. This allows study of a range of models extending from the perturbative close approximation regime, in which the fission occurs inside a black-hole event horizon, to the nonlinear regime of a “bare” fission visible from ℐ+. The study concentrates on the axisymmetric spinless fission (corresponding in the time reversed view to the head-on collision of non-spinning black holes). In the perturbative regime, the news function agrees with the close approximation waveforms. In the highly nonlinear regime, a bare fission was found to produce a dramatically sharp radiation pulse, which then undergoes a damped oscillation. Because the black-hole fission is visible from ℐ+, it is a more efficient source of gravitational waves than a black-hole merger and can produce a higher fractional mass loss!

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