Quasi-Normal Modes of Stars and Black Holes

TL;DR

This article surveys quasi-normal modes of black holes and relativistic stars within general relativity, emphasizing their role as gravitational-wave fingerprints and their mathematical underpinnings. It develops a framework in which QNMs are poles of Green functions arising from perturbation equations, and it reviews numerical, semi-analytic, and time-domain techniques for BH and NS oscillations. It discusses how QNM spectra encode black-hole parameters and stellar structure, how modes are excited in astrophysical events, and the prospects for detecting and interpreting QNM signals to constrain fundamental physics. The authors advocate a synergistic approach combining perturbation theory with numerical relativity and outline directions toward second-order perturbations, expanded mode catalogs, and optimized detectors for high-frequency stellar modes and low-frequency black-hole modes.

Abstract

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-Nordström, 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.

Figures (7)

• Figure 1: QNM ringing after the head-on collision of two unequal mas s black holes AB98. The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.
• Figure 2: The spectrum of QNM for a Schwarzschild black-hole, for $\ell=2$ (diamonds) and $\ell=3$ (crosses) AL92. The 9th mode for $\ell=2$ and the 41st for $\ell=3$ are "special", i.e. the real part of the frequency is zero ($s=i\omega$).
• Figure 3: A graph which shows all the $w$-modes: curvature, trapped and interface both for axial and polar perturbations for a very compact uniform density star with $M/R=0.44$. The black hole spectrum is also drawn for comparison. As the star becomes less compact the number of trapped modes decreases and for a typical neutron star ($M/R=0.2$) they disappear. The $Im(\omega)=1/damping$ of the curvature modes increases with decreasing compactness, and for a typical neutron star the first curvature mode nearly coincides with the fundamental black hole mode. The behavior of the interface modes changes slightly with the compactness. The similarity of the axial and polar spectra is apparent.
• Figure 4: The response of a Schwarzschild black hole as a Gaussian wave packet impinges upon it. The QNM signal dominates the signal after $t\approx 70M$ while at later times (after $t\approx 300M$) the signal is dominated by a power-law fall-off with time.
• Figure 5: Time evolution of axial perturbations of a neutron star, here only axial $w$-modes are excited. It is apparent in panel D that the late time behavior is dominated by a time tail. In the left panels (A and C) the star is ultra compact ($M/R=0.44$) and one can see not only the curvature modes but also the trapped modes which damp out much slower. The stellar model for the panels (B and D) is a typical neutron star ($M/R=0.2$) and we can see only the first curvature mode being excited.