4.5 Nonlinear mode coupling
The PITT code has been used to model the nonlinear generation of waveforms by scattering off a
Schwarzschild black hole [255
, 256
]. The physical setup is similar to the perturbative study in Section 4.4.
A radially compact pulse is prescribed on an early time outgoing null hypersurface
and
Schwarzschild null data is given on the interior white hole horizon
, which is causally unaffected
by the pulse. The input pulse is standardized to (
,
) and (
,
)
quadrupole modes with amplitude
. The outgoing null hypersurfaces extend to future null infinity
on a compactified numerical grid. Consequently, there is no need for an artificial outer
boundary. The evolution code then provides the news function at
, in the coordinates of an
observer in an inertial frame at infinity, thus avoiding any gauge ambiguity in the waveform. This
provides a simple setting for how the nonlinearities generated by high amplitudes affect the
waveform.
The study reveals several features of qualitative importance:
- The mode coupling amplitudes consistently scale as powers
of the input amplitude
corresponding to the nonlinear order of the terms in the evolution equations which produce
the mode. This allows much economy in producing a waveform catalog: Given the order
associated with a given mode generation, the response to any input amplitude
can be
obtained from the response to a single reference amplitude.
- The frequency response has similar behavior but in a less consistent way. The dominant
frequencies produced by mode coupling are in the approximate range of the quasinormal
frequency of the input mode and the expected sums and difference frequencies generated by
the order of nonlinearity.
- Large phase shifts, ranging up 15% in a half cycle relative to the linearized waveform, are
exhibited in the news function obtained by the superposition of all output modes, i.e. in the
waveform of observational significance. These phase shifts, which are important for design of
signal extraction templates, arise in an erratic way from superposing modes with different
oscillation frequencies. This furnishes a strong argument for going beyond the linearized
approximation in designing a waveform catalog for signal extraction.
- Besides the nonlinear generation of harmonic modes absent in the initial data, there is also
a stronger than linear generation of gravitational wave output. This provides a potential
mechanism for enhancing the strength of the gravitational radiation produced during, say, the
merger phase of a binary inspiral above the strength predicted in linearized theory.
- In the non-axisymmetric
case, there is also considerable generation of radiation in
polarization states not present in the linearized approximation. In the simulations, input
amplitudes in the range
to
lead to nonlinear generation of the
polarization mode which is of the same order of magnitude as the
mode (which would be the
sole polarization in the linearized regime). As a result, significant nonlinear amplification and
phase shifting of the waveform would be observed by a gravitational wave detector, depending
on its orientation.
These effects arise from the nonlinear modification of the Schwarzschild geometry identified by
Papadopoulos in his prior work on axisymmetric mode coupling [183], reported in Section 3.3.2. Although
Papadopoulos studied nonlinear mode generation produced by an outgoing pulse, as opposed to the case of
an ingoing pulse studied in [255
, 256
], the same nonlinear factors were in play and gave rise to several
common features. In both cases, the major effects arise in the region near
. Analogs of Features 1,
2, 3, and 4 above are all apparent in Papadopoulos’s work. At the finite difference level, both codes respect
the reflection symmetry inherent in Einstein’s equations and exhibit the corresponding selection rules
arising from parity considerations. In the axisymmetric case considered by Papadopoulos, this
forbids the nonlinear generation of a
mode from a
mode, as described in Feature 5
above.
The evolution along ingoing null hypersurfaces in the axisymmetric work of Papadopoulos has
complementary numerical features with the evolution along outgoing null hypersurfaces in the 3D work. The
grid based upon ingoing null hypersurfaces avoids the difficulty in resolving effects close to
encountered with the grid based upon outgoing null hypersurfaces. The outgoing code would require AMR
in order to resolve the quasinormal ringdown for as many cycles as achieved by Papadopoulos. However, the
outgoing code avoids the late time caustic formation noted in Papadopoulos’ work, as well as the
complications of gauge ambiguity and backscattering introduced by a finite outer boundary. One
attractive option would be to combine the best features of these approaches by matching an interior
evolution based upon ingoing null hypersurfaces to an exterior evolution based upon outgoing
null hypersurfaces, as implemented in [164
] for spherically symmetric Einstein–Klein–Gordon
waves.
The waveform of relevance to gravitational wave astronomy is the superposition of modes with different
frequency compositions and angular dependence. Although this waveform results from a complicated
nonlinear processing of the input signal, which varies with choice of observation angle, the response of the
individual modes to an input signal of arbitrary amplitude can be obtained by scaling the response to an
input of standard reference amplitude. This offers an economical approach to preparing a waveform
catalog.