Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.

/pdf/quasi-normal-modes-of-stars-and-black-holes-3f5f558e4h.pdf

Quasi-normal modes of stars and Black holes

Kaluza-Klein gravity

Gravitational waves emitted by perturbed black holes or relativistic stars are dominated by `quasinormal ringing', damped oscillations at single frequencies which are characteristic of the underlying system. These quasinormal modes have been studied for a long time, often with the intent of describing the time evolution of a perturbation in terms of these modes in a way very similar to a normal-mode analysis. In this review, we summarize how quasinormal modes are defined and computed. We will see why they have been regarded as closely analogous to normal modes, and discover why they are actually quite different. We also discuss how quasinormal modes can be used in the analysis of a gravitational wave signal, such as will hopefully be detected in the near future.

https://iopscience.iop.org/article/10.1088/0264-9381/16/12/201/pdf

Quasinormal modes: the characteristic `sound' of black holes and neutron stars

Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstr\"om, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.

https://arxiv.org/pdf/gr-qc/9909058v1.pdf

Quasi-Normal Modes of Stars and Black Holes

Dark matter and background light

Space-Time-Matter

Particle masses and the cosmological constant in Kaluza-Klein theory

We unify the gravitational field with its source by considering a new type of 5D manifold in which space and time are augmented by an extra dimension which induces 4D matter. The classical tests of relativity are satisfied, and for solitons we obtain new effects which can be tested astrophysically. The canonical cosmological models are in agreement with observations, and we gain new insight into the nature of the big bang. Our inference is that the world may be pure geometry in 5D.

A theory of space, time and matter

A 5D Machian generalization of Einstein's theory of gravitation is described in which the cosmological constant appears as the contribution of the rest of the universe to the gravitational field of an “isolated” source. This work amounts to a Machian interpretation of the Kaluza-Klein theory. The standard cosmological models are derived in this framework, and it is shown that - in conformity with the interpretation presented here - the cosmological constant does not directly contribute to the geometric description of the universe as a whole.

A Machian interpretation of the cosmological constant

The spectral properties of the resonant oscillations of a Schwarzschild black hole are studied. The analogy with the Coulomb wave equation is developed, and it is shown that the high-overtone quasi-normal modes are intimately connected with the long-range nature of the gravitational interaction. This is followed by an explicit construction of the quasi-normal eigenfunctions in analogy with the bound states of the inverted black-hole potentials. The special gravitational quasi-normal mode is discussed and its relationship with the algebraically special solution of the gravitational perturbations is clarified.

/pdf/on-the-spectrum-of-oscillations-of-a-schwarzschild-black-3ywidai9oy.pdf

Hongya Liu

Papers

Space-Time-Matter

Particle masses and the cosmological constant in Kaluza-Klein theory

A theory of space, time and matter

A Machian interpretation of the cosmological constant

On the spectrum of oscillations of a Schwarzschild black hole