Extremal black hole weather

We consider weakly nonlinear gravitational perturbations of a near-extremal Kerr black hole governed by the second-order vacuum Einstein equation. Using the GHZ [for , ], these are parametrized by a Hertz potential. We make an ansatz for the Hertz potential as a series of zero-damped quasinormal modes (QNMs) with time-dependent amplitudes, and derive a nonlinear dynamical system for them. We find that our dynamical system has a time-independent solution within the near-horizon scaling limit. This equilibrium solution is supported on axisymmetric modes, with amplitudes scaling as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>c</a:mi></a:mrow><a:mrow><a:mo>ℓ</a:mo></a:mrow></a:msub><a:mo>∼</a:mo><a:msup><a:mrow><a:mi>C</a:mi></a:mrow><a:mrow><a:mi>low</a:mi></a:mrow></a:msup><a:msup><a:mrow><a:mn>2</a:mn></a:mrow><a:mrow><a:mo>−</a:mo><a:mfrac><a:mrow><a:mo>ℓ</a:mo></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:mfrac></a:mrow></a:msup><a:msup><a:mrow><a:mo>ℓ</a:mo></a:mrow><a:mrow><a:mo>−</a:mo><a:mfrac><a:mrow><a:mn>7</a:mn></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:mfrac></a:mrow></a:msup></a:mrow></a:math> for large polar angular momentum mode number <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo>ℓ</c:mo></c:math>, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>C</e:mi><e:mi>low</e:mi></e:msup></e:math> is a cumulative amplitude of the low <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo>ℓ</g:mo></g:math> modes. We interpret our result as evidence that the dynamical evolution will approach, for a parametrically long time as extremality is approached, a distribution of mode amplitudes exponentially suppressed in <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo>ℓ</i:mo></i:math>, hence as the end point of an inverse cascade. It is reminiscent of weatherlike phenomena in certain models of atmospheric dynamics of rotating bodies. During the timescale considered, the decay of the QNMs themselves plays no role given their parametrically long half-life. Hence, our result is due entirely to weakly nonlinear effects.