Bosonic fields on rotating black hole spacetimes are subject to amplification by superradiance , which induces exponentially-growing instabilities ( the ‘ black hole bomb ’ ) in two scenarios : if the black hole is enclosed by a mirror , or if the bosonic field has rest mass .
Here we present a time-domain study of the scalar field on Kerr spacetime which probes ultra-long timescales up to t \lesssim 5 \times 10 ^ { 6 } M , to reveal the growth of the instability .
We describe an highly-efficient method for evolving the field , based on a spectral decomposition into a coupled set of 1+1D equations , and an absorbing boundary condition inspired by the ‘ perfectly-matched layers ’ paradigm .
First , we examine the mirror case to study how the instability timescale and mode structure depend on mirror radius .
Next , we examine the massive-field , whose rich spectrum ( revealed through Fourier analysis ) generates ‘ beating ’ effects which disguise the instability .
We show that the instability is clearly revealed by tracking the stress-energy of the field in the exterior spacetime .
We calculate the growth rate for a range of mass couplings , by applying a frequency-filter to isolate individual modal contributions to the time-domain signal .
Our results are in accord with previous frequency-domain studies which put the maximum growth rate at \tau ^ { -1 } \approx 1.72 \times 10 ^ { -7 } ( GM / c ^ { 3 } ) ^ { -1 } for the massive scalar field on Kerr spacetime .