We analytically determine the quasinormal mode (QNM) frequencies of a black hole with quadrupole moment in the eikonal limit using the lightring method. The generalized black holes that are discussed in this work possess arbitrary quadrupole and higher mass moments in addition to mass and angular momentum. Static collapsed configurations with mass and quadrupole moment are treated in detail and the QNM frequencies associated with two such configurations are evaluated to linear order in the quadrupole moment. Furthermore, we touch upon the treatment of rotating systems. In particular, the generalized black hole that we consider for our extensive QNM calculations is a completely collapsed configuration whose exterior gravitational field can be described by the HartleThorne spacetime [Astrophys. J. 153, 807834 (1968)10.1086/149707]. This collapsed system as well as its QNMs is characterized by mass M, quadrupole moment Q and angular momentum J, where the latter two parameters are treated to first and second orders of approximation, respectively. When the quadrupole moment is set equal to the relativistic quadrupole moment of the corresponding Kerr black hole, J2/(Mc2), the HartleThorne QNMs reduce to those of the Kerr black hole to second order in angular momentum J. Using ringdown frequencies, one cannot observationally distinguish a generalized HartleThorne black hole with arbitrary quadrupole moment from a Kerr black hole provided the dimensionless parameter given by  QMc2J2  c2/(G2M4) is sufficiently small compared to unity
Download TeX format
