In this note, the ideas employed in [1] to treat the problem of an ellipsoid intersected by a plane are applied to the analogous problem of a hyperboloid being intersected by a plane. The curves of intersection result...In this note, the ideas employed in [1] to treat the problem of an ellipsoid intersected by a plane are applied to the analogous problem of a hyperboloid being intersected by a plane. The curves of intersection resulting in this case are not only ellipses but rather all types of conics: ellipses, hyperbolas and parabolas. In text books of mathematics usually only cases are treated, where the planes of intersection are parallel to the coordinate planes. Here the general case is illustrated with intersecting planes which are not necessarily parallel to the coordinate planes.展开更多
We study the quasinormal modes of Boulware-Deser-Wheeler black hole in Einstein-Gauss-Bonnet gravity theory within the hyperboloidal framework.The effective potentials for the test Klein-Gordon field and gravitational...We study the quasinormal modes of Boulware-Deser-Wheeler black hole in Einstein-Gauss-Bonnet gravity theory within the hyperboloidal framework.The effective potentials for the test Klein-Gordon field and gravitational perturbations of scalar,vector,and tensor types are thoroughly investigated and put into several typical classes.The effective potentials for the gravitational perturbations have more diverse behaviors than those in general relativity,such as double peaks,the existence of a negative region adjacent to or far away from the event horizon.These lead to the existence of unstable modes(Imω<0),and the presence of gravitational wave echoes.These rich phenomena are inherent in Einstein-Gauss-Bonnet theory,rather than artificially introduced by hand.What's more,the(in)stability of quasinormal modes is studied in the frequency domain and time domain,respectively.For the frequency domain,the pseudospectrum is used to account for the instability of the spectrum.For the time domain,we add a small bump to the effective potential,and find that the new waveform does not differ significantly from the original one,where the comparison is characterized by the so-called mismatch functions.This means that quasinormal modes are stable in the time domain regardless of the shapes of the original effective potentials.In this way,our study reveals the non-equivalence of the stability of quasinormal modes in the frequency domain and the time domain.Besides,we also numerically investigate Price's law at both finite distances and infinity with the assistance of the hyperboloidal approach.展开更多
文摘In this note, the ideas employed in [1] to treat the problem of an ellipsoid intersected by a plane are applied to the analogous problem of a hyperboloid being intersected by a plane. The curves of intersection resulting in this case are not only ellipses but rather all types of conics: ellipses, hyperbolas and parabolas. In text books of mathematics usually only cases are treated, where the planes of intersection are parallel to the coordinate planes. Here the general case is illustrated with intersecting planes which are not necessarily parallel to the coordinate planes.
基金supported by the National Key R&D Program of China(Grant No.2022YFC2204603)the National Natural Science Foundation of China(Grant Nos.12475063,12075232,12247103,12235019,and11821505)。
文摘We study the quasinormal modes of Boulware-Deser-Wheeler black hole in Einstein-Gauss-Bonnet gravity theory within the hyperboloidal framework.The effective potentials for the test Klein-Gordon field and gravitational perturbations of scalar,vector,and tensor types are thoroughly investigated and put into several typical classes.The effective potentials for the gravitational perturbations have more diverse behaviors than those in general relativity,such as double peaks,the existence of a negative region adjacent to or far away from the event horizon.These lead to the existence of unstable modes(Imω<0),and the presence of gravitational wave echoes.These rich phenomena are inherent in Einstein-Gauss-Bonnet theory,rather than artificially introduced by hand.What's more,the(in)stability of quasinormal modes is studied in the frequency domain and time domain,respectively.For the frequency domain,the pseudospectrum is used to account for the instability of the spectrum.For the time domain,we add a small bump to the effective potential,and find that the new waveform does not differ significantly from the original one,where the comparison is characterized by the so-called mismatch functions.This means that quasinormal modes are stable in the time domain regardless of the shapes of the original effective potentials.In this way,our study reveals the non-equivalence of the stability of quasinormal modes in the frequency domain and the time domain.Besides,we also numerically investigate Price's law at both finite distances and infinity with the assistance of the hyperboloidal approach.
